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On the speed of convergence of Picard iterations
of backward stochastic differential equations

Martin Hutzenthaler Faculty of Mathematics, University of Duisburg-Essen, Essen, Germany;
      e-mail: martin.hutzenthaler\texttt{a}⃝uni-due.de    Thomas Kruse Institute of Mathematics, University of Gießen, Gießen, Germany;
      e-mail: thomas.kruse\texttt{a}⃝math.uni-giessen.de    Tuan Anh Nguyen Faculty of Mathematics, University of Duisburg-Essen, Essen, Germany;
      e-mail: tuan.nguyen\texttt{a}⃝uni-due.de
Abstract

It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearity converge at least exponentially fast to the solution. In this paper we prove that this convergence is in fact at least square-root factorially fast. We show for one example that no higher convergence speed is possible in general. Moreover, if the nonlinearity is zz-independent, then the convergence is even factorially fast. Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations.

††footnotetext: Key words and phrases: backward stochastic differential equation, Picard iteration, a priori estimate, semilinear parabolic partial differential equation††footnotetext: AMS 2020 subject classification: 65C99, 60H99, 60G99

1 Introduction

Since their introduction by Pardoux & Peng in [17] backward stochastic differential equations (BSDEs) have been extensively studied in the scientific literature and have found numerous applications. For example, BSDEs provide a solution approach for stochastic optimal control problems, BSDEs appear in the pricing and hedging of options in mathematical finance, and BSDEs provide stochastic representations of semilinear parabolic partial differential equations (PDEs).

A standard approach for proving existence results for BSDEs is to construct a contraction mapping whose fixed point is the solution (Y,Z)(Y,Z) of the BSDE. The associated fixed point iterations, the so-called Picard iterations, are a key component of several numerical approximation methods for BSDEs. We refer, e.g., to [2, 3] for numerical approximation methods for BSDEs based on Picard iterations and the least squares Monte Carlo method, we refer, e.g., to [10, 15] for numerical approximation methods for BSDEs based on Picard iterations and adaptive control variates, we refer, e.g., to [4, 9] for numerical approximation methods for BSDEs based on Picard iterations and Wiener chaos expansions, and we refer, e.g., to [6, 13, 7, 14, 11, 1, 12] for numerical approximation methods for BSDEs based on Picard iterations and a multilevel technique. Precise estimates on the speed of convergence of the Picard iterations (Yn,Zn)n∈ℕ0(Y_{n},Z_{n})_{n\in{\mathbbm{N}}_{0}} to the solution (Y,Z)(Y,Z) of the BSDE are essential for the error analyses of these numerical approximation methods for BSDEs where ℕ={1,2,…}{\mathbbm{N}}=\{1,2,\ldots\} and ℕ0=ℕ∪{0}{\mathbbm{N}}_{0}={\mathbbm{N}}\cup\{0\}.

Picard iterations, e.g., of ordinary differential equations converge not only exponentially fast but even factorially fast under suitable assumptions. Picard iterations of BSDEs are known to converge at least square-root factorially fast if the nonlinearity is z-independent; see the proof of [17, Theorem 3.1]. In the general case of z-dependent nonlinearities we have only found results proving that Picard iterations converge at least exponentially fast (see, e.g., [8, Theorem 2.1], [21, Theorem 4.3.1], and [19, Theorem 6.2.1]).

In this article we prove for BSDEs with z-independent and globally Lipschitz continuous nonlinearities that the Picard iterations converge in fact factorially fast. Moreover, we show for BSDEs with z-dependent and globally Lipschitz continuous nonlinearities that the Picard iterations converge at least square-root factorially fast. Somewhat surprisingly this speed of convergence cannot be improved in general. More precisely, we establish for a linear example BSDE a corresponding lower bound. We thereby reveal a phase transition in the speed of convergence of Picard iterations between the z-independent and the z-dependent case. Theorem 1.1 below illustrates the main results of this article.

Theorem 1.1.

Let T∈(0,∞)T\in(0,\infty), d,m∈ℕd,m\in{\mathbbm{N}}, L𝔶,L𝔷∈[0,∞)L_{\mathfrak{y}},L_{\mathfrak{z}}\in[0,\infty), b∈ℝmb\in{\mathbbm{R}}^{m}, let ∥⋅∥:∪n∈ℕℝn→[0,∞)\lVert\cdot\rVert\colon\cup_{n\in{\mathbbm{N}}}{\mathbbm{R}}^{n}\to[0,\infty) satisfy for all n∈ℕn\in{\mathbbm{N}} that ∥⋅∥|ℝn\lVert\cdot\rVert|_{{\mathbbm{R}}^{n}} is the standard norm on ℝn{\mathbbm{R}}^{n}, let ∥⋅∥𝖥:ℝd×m→[0,∞)\lVert\cdot\rVert_{\mathsf{F}}\colon{\mathbbm{R}}^{d\times m}\to[0,\infty) denote the Frobenius norm on ℝd×m{\mathbbm{R}}^{d\times m}, let (Ω,ℱ,ℙ,(𝔽t)t∈[0,T])(\Omega,\mathcal{F},{\mathbb{P}},({\mathbbm{F}}_{t})_{t\in[0,T]}) be a filtered probability space which satisfies the usual conditions111Let T∈(0,∞)T\in(0,\infty) and let 𝛀=(Ω,ℱ,ℙ,(𝔽t)t∈[0,T]){\bf\Omega}=(\Omega,\mathcal{F},{\mathbb{P}},({\mathbbm{F}}_{t})_{t\in[0,T]}) be a filtered probability space. Then we say that 𝛀{\bf\Omega} satisfies the usual conditions if and only if it holds for all t∈[0,T)t\in[0,T) that {A∈ℱ:ℙ​(A)=0}⊆𝔽t=∩s∈(t,T]𝔽s\{A\in\mathcal{F}:{\mathbb{P}}(A)=0\}\subseteq{\mathbbm{F}}_{t}=\cap_{s\in(t,T]}{\mathbbm{F}}_{s}., let f:[0,T]×Ω×ℝd×ℝd×m→ℝdf\colon[0,T]\times\Omega\times{\mathbbm{R}}^{d}\times{\mathbbm{R}}^{d\times m}\to{\mathbbm{R}}^{d} be measurable, assume for all t∈[0,T]t\in[0,T], y,y~∈ℝdy,\tilde{y}\in{\mathbbm{R}}^{d}, z,z~∈ℝd×mz,\tilde{z}\in{\mathbbm{R}}^{d\times m} it holds a.s. that

∥f​(t,y,z)−f​(t,y~,z~)∥≤L𝔶​∥y−y~∥+L𝔷​∥z−z~∥𝖥,\begin{split}\lVert f(t,y,z)-f(t,\tilde{y},\tilde{z})\rVert\leq L_{\mathfrak{y}}\lVert y-\tilde{y}\rVert+L_{\mathfrak{z}}\lVert z-\tilde{z}\rVert_{\mathsf{F}},\end{split} (1)

let W:[0,T]×Ω→ℝmW\colon[0,T]\times\Omega\to{\mathbbm{R}}^{m} be a standard (𝔽t)t∈[0,T]({\mathbbm{F}}_{t})_{t\in[0,T]}-Brownian motion with continuous sample paths, let ξ:Ω→ℝd\xi\colon\Omega\to{\mathbbm{R}}^{d} be 𝔽T{\mathbbm{F}}_{T}-measurable, let Yk:[0,T]×Ω→ℝdY^{k}\colon[0,T]\times\Omega\to{\mathbbm{R}}^{d}, k∈ℕ0∪{∞}k\in{\mathbbm{N}}_{0}\cup\{\infty\}, be adapted with continuous sample paths, let Zk:[0,T]×Ω→ℝd×mZ^{k}\colon[0,T]\times\Omega\to{\mathbbm{R}}^{d\times m}, k∈ℕ0∪{∞}k\in{\mathbbm{N}}_{0}\cup\{\infty\}, be progressively measurable, assume that for all s∈[0,T]s\in[0,T], k∈ℕ0∪{∞}k\in{\mathbbm{N}}_{0}\cup\{\infty\} it holds a.s.  that ∫0T𝔼[∥ξ∥2+∥f(t,0,0)∥2+∥Yt∞∥2+∥Ztk∥𝖥2]dt<∞\int_{0}^{T}{\mathbb{E}}[\|\xi\rVert^{2}+\lVert f(t,0,0)\rVert^{2}+\lVert Y_{t}^{\infty}\rVert^{2}+\lVert Z_{t}^{k}\rVert^{2}_{\mathsf{F}}]\,dt<\infty, Ys0=0Y^{0}_{s}=0, Zs0=0Z^{0}_{s}=0, and

Ysk+1=ξ+∫sTf​(t,Ytk,Ztk)​𝑑t−∫sTZtk+1​𝑑Wt,\displaystyle Y^{k+1}_{s}=\xi+\int_{s}^{T}f(t,Y_{t}^{k},Z_{t}^{k})\,dt-\int_{s}^{T}Z^{k+1}_{t}\,dW_{t}, (2)

and let ek∈[0,∞]e_{k}\in[0,\infty], k∈ℕk\in{\mathbbm{N}}, satisfy for all k∈ℕk\in{\mathbbm{N}} that

ek=(𝔼​[supt∈[0,T](∥Ytk−Yt∞∥2)+∫0T∥Ztk−Zt∞∥𝖥2​𝑑t])1/2.\displaystyle e_{k}=\left({\mathbb{E}}\!\left[\sup_{t\in[0,T]}\left(\left\lVert Y^{k}_{t}-Y_{t}^{\infty}\right\rVert^{2}\right)+\int_{0}^{T}\left\lVert Z^{k}_{t}-Z_{t}^{\infty}\right\rVert^{2}_{\mathsf{F}}\,dt\right]\right)^{\nicefrac{{1}}{{2}}}. (3)

Then

  1. (i)

    there exists c∈[0,∞)c\in[0,\infty) such that for all k∈ℕk\in{\mathbbm{N}} it holds that ek≤ckk!,e_{k}\leq\frac{c^{k}}{\sqrt{k!}},

  2. (ii)

    if, in addition to the above assumptions, it holds that L𝔷=0L_{\mathfrak{z}}=0, then there exists c∈[0,∞)c\in[0,\infty) such that for all k∈ℕk\in{\mathbbm{N}} it holds that ek≤ckk!,e_{k}\leq\frac{c^{k}}{k!}, and

  3. (iii)

    if, in addition to the above assumptions, d=T=1d=T=1, ξ=2m/2​e−∥W1∥22\xi=2^{m/2}e^{-\frac{\lVert W_{1}\rVert^{2}}{2}}, and for all t∈[0,T]t\in[0,T], y∈ℝy\in{\mathbbm{R}}, z∈ℝ1×mz\in{\mathbbm{R}}^{1\times m} it holds a.s. that f​(t,y,z)=z⋅bf(t,y,z)=z\cdot b, then there exists c∈[0,∞)c\in[0,\infty) such that for all k∈ℕ∩[∥b∥2−1,∞)k\in{\mathbbm{N}}\cap[\lVert b\rVert^{2}-1,\infty) it holds that 12​(∥b∥24)⌊k+12⌋​1k!≤ek≤ckk!.\frac{1}{2}\left(\frac{\lVert b\rVert^{2}}{4}\right)^{\lfloor\frac{k+1}{2}\rfloor}\frac{1}{\sqrt{k!}}\leq e_{k}\leq\frac{c^{k}}{\sqrt{k!}}.

Item (i) of Theorem 1.1 is a direct consequence of Proposition 4.1 and Remark 4.2. Item (ii) of Theorem 1.1 follows from Proposition 4.1 and Remark 4.3. Item (i) of Theorem 1.1 and Corollary 2.2 prove Item (iii) of Theorem 1.1. The proof of Item (i) of Theorem 1.1 is based on Lemma 3.1 which shows for all t∈[0,T]t\in[0,T], k∈ℕk\in{\mathbbm{N}}, λ∈(0,∞)\lambda\in(0,\infty) that

𝔼​[eλ​t​|Ytk−Yt∞|2+∫tTeλ​s​|Zsk−Zs∞|2​𝑑s]≤1λ​𝔼​[∫tTeλ​s​|fs​(Ysk−1,Zsk−1)−fs​(Ys∞,Zs∞)|2​𝑑s],\displaystyle\begin{split}&{\mathbb{E}}\!\left[e^{\lambda t}\lvert Y^{k}_{t}-Y^{\infty}_{t}\rvert^{2}+\int_{t}^{T}e^{\lambda s}\lvert Z^{k}_{s}-Z^{\infty}_{s}\rvert^{2}\,ds\right]\\ &\leq\frac{1}{\lambda}{\mathbb{E}}\!\left[\int_{t}^{T}e^{\lambda s}\lvert f_{s}(Y_{s}^{k-1},Z_{s}^{k-1})-f_{s}(Y_{s}^{\infty},Z_{s}^{\infty})\rvert^{2}\,ds\right],\end{split} (4)

based on the Lipschitz continuity of ff, iterating (4) k∈ℕk\in{\mathbbm{N}} times and then setting λ=k\lambda=k to get an upper bound of the form ck/kkc^{k}/\sqrt{k^{k}} for eke_{k}.

We finally discuss some possible consequences of Item (iii) of Theorem 1.1 on the performance of numerical approximation methods for BSDEs based on Picard iterations in high-dimensional situations. To this end we consider a sequence of BSDEs indexed by the dimension m∈ℕm\in{\mathbbm{N}} of the driving Brownian motion WW whose associated Lipschitz constants L𝔷,m∈[0,∞)L_{\mathfrak{z},m}\in[0,\infty), m∈ℕm\in{\mathbbm{N}}, grow for some α∈(0,∞)\alpha\in(0,\infty) like mαm^{\alpha} as m→∞m\to\infty. Item (iii) of Theorem 1.1 shows that it is possible in such a situation that the approximation errors ek,me_{k,m}, k,m∈ℕk,m\in{\mathbbm{N}}, grow faster in the dimension m∈ℕm\in{\mathbbm{N}} than any polynomial in the sense that for all p∈[0,∞)p\in[0,\infty) there exists N∈ℕN\in{\mathbbm{N}} such that for all k∈ℕ∩[N,∞)k\in{\mathbbm{N}}\cap[N,\infty) it holds that lim infm→∞ek,mmp=∞\liminf_{m\to\infty}\frac{e_{k,m}}{m^{p}}=\infty.

The remainder of this article is organized as follows. In Section 2 we provide lower bounds for the convergence speed of Picard iterations. In Section 2.2 we establish in Corollary 2.2 lower bounds for the convergence speed of Picard iterations for a linear example BSDE. In our proof of Corollary 2.2 we employ lower bounds for the convergence speed of Picard iterations for a linear example PDE which we prove in Lemma 2.1 in Section 2.1. In Lemma 3.1 in Section 3 we establish explicit a priori estimates for certain backward Itô processes in appropriate L2L^{2}-norms. In Section 4 we provide upper bounds for the convergence speed of Picard iterations of BSDEs. Proposition 4.1 establishes an explicit bound for the L2L^{2}-distance between the Picard iterations and the solution of a BSDE with a globally Lipschitz continuous nonlinearity. In Remark 4.2 we employ the estimate of Proposition 4.1 to obtain the square root-factorial speed of convergence of Picard iterations. In Remark 4.3 we employ the estimate of Proposition 4.1 to obtain the factorial speed of convergence of Picard iterations in the z-independent case.

2 Lower bounds for the convergence speed of Picard iterations

In this section we provide lower bounds for the convergence speed of Picard iterations of BSDEs. In Lemma 2.1 in Section 2.1 we establish lower bounds for the convergence speed of Picard iterations for a linear example PDE. We employ Lemma 2.1 in our proof of Corollary 2.2 in Section 2.2 to provide lower bounds for the convergence speed of Picard iterations for a linear example BSDE. Corollary 2.2 shows that square-root factorial convergence speed cannot be improved up to exponential factors in the case of zz-dependent drivers. Item (ii) of Theorem 1.1 shows in the case of zz-independent drivers that factorial speed of convergence is possible. Lemma 2.3 observes that factorial speed of convergence cannot be improved up to exponential factors in the case of yy-dependent drivers.

2.1 Lower bounds for the convergence speed of Picard iterations for an example PDE

Lemma 2.1.

Let d∈ℕd\in{\mathbbm{N}}, b=(b1,b2,…,bd)∈ℝdb=(b_{1},b_{2},\ldots,b_{d})\in{\mathbbm{R}}^{d}, let ⟨⋅,⋅⟩:ℝd×ℝd→ℝ\langle\cdot,\cdot\rangle\colon{\mathbbm{R}}^{d}\times{\mathbbm{R}}^{d}\to{\mathbbm{R}} denote the standard scalar product on ℝd{\mathbbm{R}}^{d}, let ∥⋅∥:ℝd→[0,∞)\lVert\cdot\rVert\colon{\mathbbm{R}}^{d}\to[0,\infty) denote the standard norm on ℝd{\mathbbm{R}}^{d}, let (Ω,ℱ,ℙ)(\Omega,\mathcal{F},{\mathbb{P}}) be a probability space, let W=(W1,W2,…,Wd):[0,1]×Ω→ℝdW=(W^{1},W^{2},\ldots,W^{d})\colon[0,1]\times\Omega\to{\mathbbm{R}}^{d} be a standard Brownian motion, let vn:[0,1]×ℝd→ℝv^{n}\colon[0,1]\times{\mathbbm{R}}^{d}\to{\mathbbm{R}}, n∈ℕ0∪{∞}n\in{\mathbbm{N}}_{0}\cup\{\infty\}, satisfy for all t∈[0,1]t\in[0,1], x∈ℝdx\in{\mathbbm{R}}^{d}, n∈ℕn\in{\mathbbm{N}} that v0​(t,x)=0v^{0}(t,x)=0,

vn​(t,x)=𝔼​[2d/2​exp⁡(−∥x+W1−Wt∥22)]+∑k=1n−1∑μ1,μ2,…,μk=1d[(1−t)kk!​bμ1​bμ2​⋯​bμk​𝔼​[2d/2​∂k∂xμ1​∂xμ2​…​∂xμk​exp⁡(−∥x+W1−Wt∥22)]],\displaystyle\begin{split}&v^{n}(t,x)={\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\right]\\ &+\sum_{k=1}^{n-1}\sum_{\mu_{1},\mu_{2},\ldots,\mu_{k}=1}^{d}\Biggl{[}\tfrac{(1-t)^{k}}{k!}b_{\mu_{1}}b_{\mu_{2}}\cdots b_{\mu_{k}}{\mathbb{E}}\!\left[2^{d/2}\tfrac{\partial^{k}}{\partial x_{\mu_{1}}\partial x_{\mu_{2}}\ldots\partial x_{\mu_{k}}}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\right]\Biggr{]},\end{split} (5)

and

v∞​(t,x)=𝔼​[2d/2​exp⁡(−∥x+b​(1−t)+W1−Wt∥22)].v^{\infty}(t,x)={\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\tfrac{\lVert x+b(1-t)+W_{1}-W_{t}\rVert^{2}}{2}\right)\right]. (6)

Then

  1. (i)

    it holds for all t∈[0,1]t\in[0,1], x∈ℝdx\in{\mathbbm{R}}^{d} that v∞∈C∞​([0,1]×ℝd,ℝ)v^{\infty}\in C^{\infty}([0,1]\times{\mathbbm{R}}^{d},{\mathbbm{R}}), v∞​(1,x)=2d/2​e−∥x∥22v^{\infty}(1,x)=2^{d/2}e^{-\frac{\lVert x\rVert^{2}}{2}}, and

    ∂v∞∂t​(t,x)+12​(△xv∞)​(t,x)+⟨b,(∇xv∞)​(t,x)⟩=0,\frac{\partial v^{\infty}}{\partial t}(t,x)+\frac{1}{2}(\mathop{}\!\mathbin{\bigtriangleup}_{x}v^{\infty})(t,x)+\left\langle b,(\nabla_{x}v^{\infty})(t,x)\right\rangle=0, (7)
  2. (ii)

    it holds for all n∈ℕ0n\in{\mathbbm{N}}_{0}, t∈[0,1)t\in[0,1), x∈ℝdx\in{\mathbbm{R}}^{d} that vn∈C∞​([0,1]×ℝd,ℝ)v^{n}\in C^{\infty}([0,1]\times{\mathbbm{R}}^{d},{\mathbbm{R}}), vn​(1,x)=2d/2​e−∥x∥22v^{n}(1,x)=2^{d/2}e^{-\frac{\lVert x\rVert^{2}}{2}},

    vn+1​(t,x)=𝔼​[2d/2​exp⁡(−∥x+W1−Wt∥22)]+∫t1𝔼​[⟨b,(∇xvn)​(s,x+Ws−Wt)⟩]​𝑑s,\displaystyle\begin{split}v^{n+1}(t,x)&={\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\frac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\right]\\ &\qquad+\int_{t}^{1}{\mathbb{E}}\!\left[\Bigl{\langle}b,(\nabla_{x}v^{n})(s,x+W_{s}-W_{t})\Bigr{\rangle}\right]\!ds,\end{split} (8)

    and

    ∇xvn+1​(t,x)=𝔼​[2d/2​exp⁡(−∥x+W1−Wt∥22)​W1−Wt1−t]+∫t1𝔼​[⟨b,(∇xvn)​(s,x+Ws−Wt)⟩​Ws−Wts−t]​𝑑s,\displaystyle\begin{split}\nabla_{x}v^{n+1}(t,x)&={\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\frac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\frac{W_{1}-W_{t}}{1-t}\right]\\ &\qquad+\int_{t}^{1}{\mathbb{E}}\!\left[\Bigl{\langle}b,(\nabla_{x}v^{n})(s,x+W_{s}-W_{t})\Bigr{\rangle}\frac{W_{s}-W_{t}}{s-t}\right]\!ds,\end{split} (9)
  3. (iii)

    it holds for all n∈ℕn\in{\mathbbm{N}} that

    vn​(0,0)=1+∑i=1⌊n−12⌋(−1)i​∥b∥2​i4i​i!,\displaystyle v^{n}(0,0)=1+\sum_{i=1}^{\lfloor\frac{n-1}{2}\rfloor}\frac{(-1)^{i}\lVert b\rVert^{2i}}{4^{i}i!}, (10)
  4. (iv)

    it holds that

    v∞​(0,0)=exp⁡(−∥b∥24),\displaystyle v^{\infty}(0,0)=\exp\!\left(-\frac{\lVert b\rVert^{2}}{4}\right), (11)

    and

  5. (v)

    it holds for all ϵ∈(0,1)\epsilon\in(0,1), n∈ℕ∩[12​ϵ​∥b∥2−1,∞)n\in{\mathbbm{N}}\cap\bigl{[}\frac{1}{2\epsilon}\lVert b\rVert^{2}-1,\infty\bigr{)} that

    (∥b∥24)⌊n+12⌋​1−ϵ⌊n+12⌋!≤|v∞​(0,0)−vn​(0,0)|≤(∥b∥24)⌊n+12⌋​1⌊n+12⌋!​11−ϵ.\displaystyle\left(\frac{\lVert b\rVert^{2}}{4}\right)^{\lfloor\frac{n+1}{2}\rfloor}\frac{1-\epsilon}{\lfloor\frac{n+1}{2}\rfloor!}\leq\lvert v^{\infty}(0,0)-v^{n}(0,0)\rvert\leq\left(\frac{\lVert b\rVert^{2}}{4}\right)^{\lfloor\frac{n+1}{2}\rfloor}\frac{1}{\lfloor\frac{n+1}{2}\rfloor!}\frac{1}{1-\epsilon}. (12)
Proof of Lemma 2.1.

First note that the Feyman-Kac formula (cf., e.g., [16, Theorem 8.2.1]) proves (i).

Next observe that (5) proves for all ℓ∈{1,2,…,d}\ell\in\{1,2,\ldots,d\}, s∈[0,1)s\in[0,1), x∈ℝdx\in{\mathbbm{R}}^{d}, n∈ℕn\in{\mathbbm{N}} that vn∈C∞​([0,1]×ℝd,ℝ)v^{n}\in C^{\infty}([0,1]\times{\mathbbm{R}}^{d},{\mathbbm{R}}), vn​(1,x)=2d/2​e−∥x∥22v^{n}(1,x)=2^{d/2}e^{-\frac{\lVert x\rVert^{2}}{2}}, and

∂vn∂xℓ​(s,x)−𝔼​[2d/2​∂∂xℓ​exp⁡(−∥x+W1−Ws∥22)]=∂vn∂xℓ​(s,x)−∂∂xℓ​𝔼​[2d/2​exp⁡(−∥x+W1−Ws∥22)]=∑k=1n−1∑μ1,μ2,…,μk=1d[(1−s)kk!​bμ1​bμ2​⋯​bμk​∂∂xℓ​𝔼​[2d/2​∂k∂xμ1​∂xμ2​…​∂xμk​exp⁡(−∥x+W1−Ws∥22)]]=∑k=1n−1∑μ1,μ2,…,μk=1d[(1−s)kk!​bμ1​bμ2​⋯​bμk​𝔼​[2d/2​∂k+1∂xℓ​∂xμ1​∂xμ2​…​∂xμk​exp⁡(−∥x+W1−Ws∥22)]].\displaystyle\begin{split}&\tfrac{\partial v^{n}}{\partial x_{\ell}}(s,x)-{\mathbb{E}}\!\left[2^{d/2}\tfrac{\partial}{\partial x_{\ell}}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{s}\rVert^{2}}{2}\right)\right]=\tfrac{\partial v^{n}}{\partial x_{\ell}}(s,x)-\tfrac{\partial}{\partial x_{\ell}}{\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{s}\rVert^{2}}{2}\right)\right]\\ &=\sum_{k=1}^{n-1}\sum_{\mu_{1},\mu_{2},\ldots,\mu_{k}=1}^{d}\Biggl{[}\tfrac{(1-s)^{k}}{k!}b_{\mu_{1}}b_{\mu_{2}}\cdots b_{\mu_{k}}\tfrac{\partial}{\partial x_{\ell}}{\mathbb{E}}\!\left[2^{d/2}\tfrac{\partial^{k}}{\partial x_{\mu_{1}}\partial x_{\mu_{2}}\ldots\partial x_{\mu_{k}}}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{s}\rVert^{2}}{2}\right)\right]\Biggr{]}\\ &=\sum_{k=1}^{n-1}\sum_{\mu_{1},\mu_{2},\ldots,\mu_{k}=1}^{d}\Biggl{[}\tfrac{(1-s)^{k}}{k!}b_{\mu_{1}}b_{\mu_{2}}\cdots b_{\mu_{k}}{\mathbb{E}}\!\left[2^{d/2}\tfrac{\partial^{k+1}}{\partial x_{\ell}\partial x_{\mu_{1}}\partial x_{\mu_{2}}\ldots\partial x_{\mu_{k}}}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{s}\rVert^{2}}{2}\right)\right]\Biggr{]}.\end{split} (13)

This, the disintegration theorem, and independence of Brownian increments show for all t∈[0,1)t\in[0,1), s∈(t,1)s\in(t,1), x∈ℝdx\in{\mathbbm{R}}^{d}, ℓ∈{1,2,…,d}\ell\in\{1,2,\ldots,d\}, n∈ℕn\in{\mathbbm{N}} that

𝔼​[∂vn∂xℓ​(s,x+Ws−Wt)]−𝔼​[2d/2​∂∂xℓ​exp⁡(−∥x+W1−Wt∥22)]\displaystyle{\mathbb{E}}\!\left[\tfrac{\partial v^{n}}{\partial x_{\ell}}(s,x+W_{s}-W_{t})\right]-{\mathbb{E}}\!\left[2^{d/2}\tfrac{\partial}{\partial x_{\ell}}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\right]
=𝔼​[∂vn∂xℓ​(s,x+Ws−Wt)]−𝔼​[𝔼​[2d/2​∂∂xℓ​exp⁡(−∥z+W1−Ws∥22)]|z=x+Ws−Wt]\displaystyle={\mathbb{E}}\!\left[\tfrac{\partial v^{n}}{\partial x_{\ell}}(s,x+W_{s}-W_{t})\right]-{\mathbb{E}}\!\left[{\mathbb{E}}\!\left[2^{d/2}\tfrac{\partial}{\partial x_{\ell}}\exp\!\left(-\tfrac{\lVert z+W_{1}-W_{s}\rVert^{2}}{2}\right)\right]\bigr{|}_{z=x+W_{s}-W_{t}}\right]
=∑k=1n−1∑μ1,μ2,…,μk=1d[(1−s)kk!bμ1bμ2⋯bμk\displaystyle=\sum_{k=1}^{n-1}\sum_{\mu_{1},\mu_{2},\ldots,\mu_{k}=1}^{d}\Biggl{[}\tfrac{(1-s)^{k}}{k!}b_{\mu_{1}}b_{\mu_{2}}\cdots b_{\mu_{k}}
⋅𝔼[𝔼[2d/2∂k+1∂zℓ​∂zμ1​∂zμ2​…​∂zμkexp(−∥z+W1−Ws∥22)]|z=x+Ws−Wt]]\displaystyle\qquad\qquad\qquad\qquad\qquad\cdot{\mathbb{E}}\biggl{[}{\mathbb{E}}\!\left[2^{d/2}\tfrac{\partial^{k+1}}{\partial z_{\ell}\partial z_{\mu_{1}}\partial z_{\mu_{2}}\ldots\partial z_{\mu_{k}}}\exp\!\left(-\tfrac{\lVert z+W_{1}-W_{s}\rVert^{2}}{2}\right)\right]\bigr{|}_{z=x+W_{s}-W_{t}}\biggr{]}\Biggr{]}
=∑k=1n−1∑μ1,μ2,…,μk=1d[(1−s)kk!​bμ1​bμ2​⋯​bμk​𝔼​[2d/2​∂k+1∂xμ1​∂xμ2​…​∂xμk​∂xℓ​exp⁡(−∥x+W1−Wt∥22)]].\displaystyle=\sum_{k=1}^{n-1}\sum_{\mu_{1},\mu_{2},\ldots,\mu_{k}=1}^{d}\Biggl{[}\tfrac{(1-s)^{k}}{k!}b_{\mu_{1}}b_{\mu_{2}}\cdots b_{\mu_{k}}{\mathbb{E}}\!\left[2^{d/2}\tfrac{\partial^{k+1}}{\partial x_{\mu_{1}}\partial x_{\mu_{2}}\ldots\partial x_{\mu_{k}}\partial x_{\ell}}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\right]\Biggr{]}. (14)

This, the fact that ∀k∈ℕ0,t∈[0,1]:∫t1(1−s)kk!​𝑑s=(1−t)k+1(k+1)!\forall\,k\in{\mathbbm{N}}_{0},t\in[0,1]\colon\int_{t}^{1}\frac{(1-s)^{k}}{k!}\,ds=\frac{(1-t)^{k+1}}{(k+1)!}, and (5) show for all t∈[0,1)t\in[0,1), x∈ℝdx\in{\mathbbm{R}}^{d}, n∈ℕn\in{\mathbbm{N}} that

∫t1𝔼​[⟨b,(∇xvn)​(s,x+Ws−Wt)⟩]​𝑑s=∑μk+1=1d∫t1𝔼​[bμk+1​∂vn∂xμk+1​(s,x+Ws−Wt)]​𝑑s\displaystyle\int_{t}^{1}{\mathbb{E}}\!\left[\Bigl{\langle}b,(\nabla_{x}v^{n})(s,x+W_{s}-W_{t})\Bigr{\rangle}\right]\!ds=\sum_{\mu_{k+1}=1}^{d}\int_{t}^{1}{\mathbb{E}}\!\left[b_{\mu_{k+1}}\tfrac{\partial v^{n}}{\partial x_{\mu_{k+1}}}(s,x+W_{s}-W_{t})\right]\!ds
=∑k=0n−1∑μ1,μ2,…,μk+1=1d∫t1(1−s)kk!​bμ1​bμ2​⋯​bμk+1​𝔼​[2d/2​∂k+1∂xμ1​∂xμ2​…​∂xμk+1​exp⁡(−∥x+W1−Wt∥22)]​𝑑s\displaystyle=\sum_{k=0}^{n-1}\sum_{\mu_{1},\mu_{2},\ldots,\mu_{k+1}=1}^{d}\int_{t}^{1}\tfrac{(1-s)^{k}}{k!}b_{\mu_{1}}b_{\mu_{2}}\cdots b_{\mu_{k+1}}{\mathbb{E}}\!\left[2^{d/2}\tfrac{\partial^{k+1}}{\partial x_{\mu_{1}}\partial x_{\mu_{2}}\ldots\partial x_{\mu_{k+1}}}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\right]\!ds
=∑k=0n−1∑μ1,μ2,…,μk+1=1d[(1−t)k+1(k+1)!​bμ1​bμ2​⋯​bμk+1​𝔼​[2d/2​∂k+1∂xμ1​∂xμ2​…​∂xμk+1​exp⁡(−∥x+W1−Wt∥22)]]\displaystyle=\sum_{k=0}^{n-1}\sum_{\mu_{1},\mu_{2},\ldots,\mu_{k+1}=1}^{d}\Biggl{[}\tfrac{(1-t)^{k+1}}{(k+1)!}b_{\mu_{1}}b_{\mu_{2}}\cdots b_{\mu_{k+1}}{\mathbb{E}}\!\left[2^{d/2}\tfrac{\partial^{k+1}}{\partial x_{\mu_{1}}\partial x_{\mu_{2}}\ldots\partial x_{\mu_{k+1}}}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\right]\Biggr{]}
=∑k=1n∑μ1,μ2,…,μk=1d[(1−t)kk!​bμ1​bμ2​⋯​bμk​𝔼​[2d/2​∂k∂xμ1​∂xμ2​…​∂xμk​exp⁡(−∥x+W1−Wt∥22)]]\displaystyle=\sum_{k=1}^{n}\sum_{\mu_{1},\mu_{2},\ldots,\mu_{k}=1}^{d}\Biggl{[}\tfrac{(1-t)^{k}}{k!}b_{\mu_{1}}b_{\mu_{2}}\cdots b_{\mu_{k}}{\mathbb{E}}\!\left[2^{d/2}\tfrac{\partial^{k}}{\partial x_{\mu_{1}}\partial x_{\mu_{2}}\ldots\partial x_{\mu_{k}}}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\right]\Biggr{]}
=vn+1​(t,x)−𝔼​[2d/2​exp⁡(−∥x+W1−Wt∥22)].\displaystyle=v^{n+1}(t,x)-{\mathbb{E}}\Bigl{[}2^{d/2}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\Bigr{]}. (15)

This, the fact that ∀t∈[0,1],x∈ℝd:v0​(t,x)=0\forall\,t\in[0,1],x\in{\mathbbm{R}}^{d}\colon v^{0}(t,x)=0, and (5) show for all t∈[0,1)t\in[0,1), x∈ℝdx\in{\mathbbm{R}}^{d}, n∈ℕ0n\in{\mathbbm{N}}_{0} that

vn+1​(t,x)=𝔼​[2d/2​exp⁡(−∥x+W1−Wt∥22)]+∫t1𝔼​[⟨b,(∇xvn)​(s,x+Ws−Wt)⟩]​𝑑s.\displaystyle v^{n+1}(t,x)={\mathbb{E}}\Bigl{[}2^{d/2}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\Bigr{]}+\int_{t}^{1}{\mathbb{E}}\!\left[\Bigl{\langle}b,(\nabla_{x}v^{n})(s,x+W_{s}-W_{t})\Bigr{\rangle}\right]\!ds. (16)

Next note that Stein’s lemma proves for all ℓ∈{1,2,…,d}\ell\in\{1,2,\ldots,d\}, x∈ℝdx\in{\mathbbm{R}}^{d}, s∈(0,1]s\in(0,1], t∈[0,s)t\in[0,s), h∈C1​(ℝd,ℝ)h\in C^{1}({\mathbbm{R}}^{d},{\mathbbm{R}}) with supy∈ℝd(|h​(y)|+|∂h∂yℓ​(y)|)<∞\sup_{y\in{\mathbbm{R}}^{d}}\bigl{(}\lvert h(y)\rvert+\lvert\tfrac{\partial h}{\partial y_{\ell}}(y)\rvert\bigr{)}<\infty that

∂∂xℓ​𝔼​[h​(x+Ws−Wt)]=𝔼​[∂h∂xℓ​(x+Ws−Wt)]=𝔼​[h​(x+Ws−Wt)​Wsℓ−Wtℓs−t].\displaystyle\frac{\partial}{\partial x_{\ell}}{\mathbb{E}}\!\left[h(x+W_{s}-W_{t})\right]={\mathbb{E}}\!\left[\frac{\partial h}{\partial x_{\ell}}(x+W_{s}-W_{t})\right]={\mathbb{E}}\!\left[h(x+W_{s}-W_{t})\frac{W^{\ell}_{s}-W^{\ell}_{t}}{s-t}\right]. (17)

This, (16), differentiation under integrals, the fact that ∀ℓ∈{1,2,…,d},n∈ℕ0,s∈[0,1):supx∈ℝd[exp⁡(−∥x∥22)+|∂∂xℓ​exp⁡(−∥x∥22)|+|⟨b,(∇xvn)​(s,x)⟩|+|∂∂xℓ​⟨b,(∇xvn)​(s,x)⟩|]<∞\forall\,\ell\in\{1,2,\ldots,d\},n\in{\mathbbm{N}}_{0},s\in[0,1)\colon\sup_{x\in{\mathbbm{R}}^{d}}\bigl{[}\exp(-\frac{\lVert x\rVert^{2}}{2})+\lvert\frac{\partial}{\partial x_{\ell}}\exp(-\frac{\lVert x\rVert^{2}}{2})\rvert+\lvert\langle b,(\nabla_{x}v^{n})(s,x)\rangle\rvert+\lvert\frac{\partial}{\partial x_{\ell}}\langle b,(\nabla_{x}v^{n})(s,x)\rangle\rvert\bigr{]}<\infty, and the fact that ∀t∈[0,1],x∈ℝd:v0​(t,x)=0\forall\,t\in[0,1],x\in{\mathbbm{R}}^{d}\colon v^{0}(t,x)=0 show for all ℓ∈{1,2,…,d}\ell\in\{1,2,\ldots,d\}, n∈ℕ0n\in{\mathbbm{N}}_{0}, t∈[0,1)t\in[0,1), x∈ℝdx\in{\mathbbm{R}}^{d} that

∂vn+1∂xℓ​(t,x)=∂∂xℓ​𝔼​[2d/2​exp⁡(−∥x+W1−Wt∥22)]+∫t1∂∂xℓ​𝔼​[⟨b,(∇xvn)​(s,x+Ws−Wt)⟩]​𝑑s\displaystyle\frac{\partial v^{n+1}}{\partial x_{\ell}}(t,x)=\frac{\partial}{\partial x_{\ell}}{\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\right]+\int_{t}^{1}\frac{\partial}{\partial x_{\ell}}{\mathbb{E}}\!\left[\Bigl{\langle}b,(\nabla_{x}v^{n})(s,x+W_{s}-W_{t})\Bigr{\rangle}\right]\!ds
=𝔼​[2d/2​exp⁡(−∥x+W1−Wt∥22)​W1ℓ−Wtℓ1−t]\displaystyle={\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{t}\rVert^{2}}{2}\right)\frac{W_{1}^{\ell}-W_{t}^{\ell}}{1-t}\right]
+∫t1𝔼​[⟨b,(∇xvn)​(s,x+Ws−Wt)⟩​Wsℓ−Wtℓs−t]​𝑑s.\displaystyle\qquad\qquad\qquad+\int_{t}^{1}{\mathbb{E}}\!\left[\Bigl{\langle}b,(\nabla_{x}v^{n})(s,x+W_{s}-W_{t})\Bigr{\rangle}\frac{W^{\ell}_{s}-W^{\ell}_{t}}{s-t}\right]\!ds. (18)

This and (16) show (ii).

For the next step let 00=10^{0}=1, let Hk:ℝ→ℝH_{k}\colon{\mathbbm{R}}\to{\mathbbm{R}}, k∈ℕ0k\in{\mathbbm{N}}_{0}, satisfy for all k∈ℕ0k\in{\mathbbm{N}}_{0}, x∈ℝx\in{\mathbbm{R}} that

Hk​(x)=∑ℓ=0⌊k2⌋[k!​(−1)ℓℓ!​(k−2​ℓ)!​xk−2​ℓ2ℓ]H_{k}(x)=\sum_{\ell=0}^{\lfloor\frac{k}{2}\rfloor}\left[\frac{k!(-1)^{\ell}}{\ell!(k-2\ell)!}\frac{x^{k-2\ell}}{2^{\ell}}\right] (19)

and for every n∈ℕ0∪{−1}n\in{\mathbbm{N}}_{0}\cup\{-1\} let n!!∈ℕn!!\in{\mathbbm{N}} satisfy that n!!=∏k=0⌈n/2⌉−1(n−2​k)n!!=\prod_{k=0}^{\left\lceil{{n}/{2}}\right\rceil-1}(n-2k). A well-known fact on Hermite polynomials shows for all x∈ℝx\in{\mathbbm{R}}, k∈ℕ0k\in{\mathbbm{N}}_{0} that dkd​xk​(e−x22)=e−x22​Hk​(x).\tfrac{d^{k}}{dx^{k}}(e^{-\frac{x^{2}}{2}})=e^{-\frac{x^{2}}{2}}H_{k}(x). Furthermore, a well-known fact on moments of normally distributed random variables shows for all k∈ℕ0k\in{\mathbbm{N}}_{0}, ℓ∈[0,k]∩ℕ0\ell\in[0,k]\cap{\mathbbm{N}}_{0} that 1π​∫ℝz2​k+1−2​ℓ​e−z2​𝑑z=0\tfrac{1}{\sqrt{\pi}}\int_{{\mathbbm{R}}}z^{2k+1-2\ell}e^{-z^{2}}\,dz=0 and

1π​∫ℝz2​k−2​ℓ​e−z2​𝑑z=[12​π​σ2​∫ℝz2​k−2​ℓ​e−z22​σ2​𝑑z]|σ2=12=(2​k−2​ℓ−1)!!2k−ℓ.\displaystyle\frac{1}{\sqrt{\pi}}\int_{{\mathbbm{R}}}z^{2k-2\ell}e^{-z^{2}}\,dz=\left[\frac{1}{\sqrt{2\pi\sigma^{2}}}\int_{{\mathbbm{R}}}z^{2k-2\ell}e^{-\frac{z^{2}}{2\sigma^{2}}}\,dz\right]\Bigr{|}_{\sigma^{2}=\frac{1}{2}}=\frac{(2k-2\ell-1)!!}{2^{k-\ell}}. (20)

This, (19), and the binomial theorem imply for all k∈ℕ0k\in{\mathbbm{N}}_{0} that

𝔼​[2​exp⁡(−|W11|22)​H2​k​(W11)]=12​π​∫ℝ2​e−z22​[∑ℓ=0k((2​k)!​(−1)ℓℓ!​(2​k−2​ℓ)!​z2​k−2​ℓ2ℓ)]​e−z22​𝑑z\displaystyle{\mathbb{E}}\!\left[\sqrt{2}\exp\!\left(-\frac{\lvert W_{1}^{1}\rvert^{2}}{2}\right)H_{2k}(W_{1}^{1})\right]=\frac{1}{\sqrt{2\pi}}\int_{{\mathbbm{R}}}\sqrt{2}e^{-\frac{z^{2}}{2}}\left[\sum_{\ell=0}^{k}\left(\frac{(2k)!(-1)^{\ell}}{\ell!(2k-2\ell)!}\frac{z^{2k-2\ell}}{2^{\ell}}\right)\right]e^{-\frac{z^{2}}{2}}\,dz
=∑ℓ=0k[(2​k)!​(−1)ℓℓ!​(2​k−2​ℓ)!​2ℓ​(1π​∫ℝz2​k−2​ℓ​e−z2​𝑑z)]=∑ℓ=0k[(2​k)!​(−1)ℓℓ!​(2​k−2​ℓ)!​2ℓ​(2​k−2​ℓ−1)!!2k−ℓ]\displaystyle=\sum_{\ell=0}^{k}\left[\frac{(2k)!(-1)^{\ell}}{\ell!(2k-2\ell)!2^{\ell}}\left(\frac{1}{\sqrt{\pi}}\int_{{\mathbbm{R}}}z^{2k-2\ell}e^{-z^{2}}\,dz\right)\right]=\sum_{\ell=0}^{k}\left[\frac{(2k)!(-1)^{\ell}}{\ell!(2k-2\ell)!2^{\ell}}\frac{(2k-2\ell-1)!!}{2^{k-\ell}}\right]
=∑ℓ=0k(2​k)!​(−1)ℓ​(2​k−2​ℓ−1)!!ℓ!​(2​k−2​ℓ)!​2k=∑ℓ=0k(2​k)!​(−1)ℓℓ!​(2​k−2​ℓ)!!​2k\displaystyle=\sum_{\ell=0}^{k}\frac{(2k)!(-1)^{\ell}(2k-2\ell-1)!!}{\ell!(2k-2\ell)!2^{k}}=\sum_{\ell=0}^{k}\frac{(2k)!(-1)^{\ell}}{\ell!(2k-2\ell)!!2^{k}}
=∑ℓ=0k(2​k)!​(−1)ℓℓ!​(k−ℓ)!​2k−ℓ​2k=(2​k)!4k​k!​∑ℓ=0kk!​(−2)ℓ(k−ℓ)!​ℓ!=(2​k)!4k​k!​(−1)k\displaystyle=\sum_{\ell=0}^{k}\frac{(2k)!(-1)^{\ell}}{\ell!(k-\ell)!2^{k-\ell}2^{k}}=\frac{(2k)!}{4^{k}k!}\sum_{\ell=0}^{k}\frac{k!(-2)^{\ell}}{(k-\ell)!\ell!}=\frac{(2k)!}{4^{k}k!}(-1)^{k} (21)

and

𝔼​[2​exp⁡(−|W11|22)​H2​k+1​(W11)]=12​π​∫ℝ2​e−z22​[∑ℓ=0⌊(2​k+1)/2⌋((2​k+1)!​(−1)ℓℓ!​(2​k+1−2​ℓ)!​z2​k+1−2​ℓ2ℓ)]​e−z22​𝑑z=0.\displaystyle\begin{split}&{\mathbb{E}}\!\left[\sqrt{2}\exp\!\left(-\frac{\lvert W_{1}^{1}\rvert^{2}}{2}\right)H_{2k+1}(W_{1}^{1})\right]\\ &=\frac{1}{\sqrt{2\pi}}\int_{{\mathbbm{R}}}\sqrt{2}e^{-\frac{z^{2}}{2}}\left[\sum_{\ell=0}^{\lfloor(2k+1)/{2}\rfloor}\left(\frac{(2k+1)!(-1)^{\ell}}{\ell!(2k+1-2\ell)!}\frac{z^{2k+1-2\ell}}{2^{\ell}}\right)\right]e^{-\frac{z^{2}}{2}}\,dz=0.\end{split} (22)

Thus, it holds for all k∈ℕ0k\in{\mathbbm{N}}_{0} that

𝔼​[2​exp⁡(−|W11|22)​Hk​(W11)]=𝟙2​ℕ0​(k)​k!​(−1)⌊k/2⌋4k/2​⌊k/2⌋!.{\mathbb{E}}\!\left[\sqrt{2}\exp\!\left(-\frac{\lvert W_{1}^{1}\rvert^{2}}{2}\right)H_{k}(W_{1}^{1})\right]=\mathbbm{1}_{2{\mathbbm{N}}_{0}}(k)\frac{k!(-1)^{\lfloor k/2\rfloor}}{4^{k/2}\lfloor k/2\rfloor!}. (23)

Furthermore, the combinatorial interpretation of multinomial coefficients yields for all k∈ℕk\in{\mathbbm{N}}, α∈(ℕ0)k\alpha\in({\mathbbm{N}}_{0})^{k} with |α|=k\lvert\alpha\rvert=k that

#​[⋂i=1d{(μ1,μ2,…,μk)∈{1,2,…,d}k:#​{ℓ∈{1,2,…,k}:μℓ=i}=αi}]=k!α1!​α2!​⋯​αd!.\displaystyle\begin{split}&\#\left[\bigcap_{i=1}^{d}\Bigl{\{}(\mu_{1},\mu_{2},\ldots,\mu_{k})\in\{1,2,\ldots,d\}^{k}\colon\#\{\ell\in\{1,2,\ldots,k\}\colon\mu_{\ell}=i\}=\alpha_{i}\Bigr{\}}\right]\\ &=\frac{k!}{\alpha_{1}!\alpha_{2}!\cdots\alpha_{d}!}.\end{split} (24)

This, the fact that W1,W2,…,WdW^{1},W^{2},\ldots,W^{d} are independent, (5), and the fact that ∀x∈ℝ,k∈ℕ0:dkd​xk​(e−x22)=e−x22​Hk​(x)\forall\,x\in{\mathbbm{R}},k\in{\mathbbm{N}}_{0}\colon\frac{d^{k}}{dx^{k}}(e^{-\frac{x^{2}}{2}})=e^{-\frac{x^{2}}{2}}H_{k}(x) show for all n∈ℕn\in{\mathbbm{N}} that

vn​(0,0)−[∏ℓ=1d𝔼​[2​exp⁡(−(W1ℓ)22)]]=vn​(0,0)−𝔼​[2d/2​exp⁡(−∥W1∥22)]\displaystyle v^{n}(0,0)-\left[\prod_{\ell=1}^{d}{\mathbb{E}}\!\left[\sqrt{2}\exp\!\left(-\frac{(W_{1}^{\ell})^{2}}{2}\right)\right]\right]=v^{n}(0,0)-{\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\frac{\lVert W_{1}\rVert^{2}}{2}\right)\right]
=∑k=1n−1∑μ1,μ2,…,μk=1d[1k!​bμ1​bμ2​⋯​bμk​𝔼​[2d/2​∂k∂xμ1​∂xμ2​…​∂xμk​exp⁡(−∥x+W1∥22)]|x=0]\displaystyle=\sum_{k=1}^{n-1}\sum_{\mu_{1},\mu_{2},\ldots,\mu_{k}=1}^{d}\Biggl{[}\frac{1}{k!}b_{\mu_{1}}b_{\mu_{2}}\cdots b_{\mu_{k}}{\mathbb{E}}\!\left[2^{d/2}\frac{\partial^{k}}{\partial x_{\mu_{1}}\partial x_{\mu_{2}}\ldots\partial x_{\mu_{k}}}\exp\!\left(-\frac{\lVert x+W_{1}\rVert^{2}}{2}\right)\right]\Bigr{|}_{x=0}\Biggr{]}
=∑k=1n−1∑α∈(ℕ0)d:|α|=k[∏ℓ=1d𝔼​[bℓαℓ​2αℓ!​∂αℓ∂xℓαℓ​exp⁡(−(xℓ+W1ℓ)22)]|xℓ=0]\displaystyle=\sum_{k=1}^{n-1}\sum_{\alpha\in({\mathbbm{N}}_{0})^{d}\colon\lvert\alpha\rvert=k}\left[\prod_{\ell=1}^{d}{\mathbb{E}}\!\left[\frac{b_{\ell}^{\alpha_{\ell}}\sqrt{2}}{\alpha_{\ell}!}\frac{\partial^{\alpha_{\ell}}}{\partial x_{\ell}^{\alpha_{\ell}}}\exp\!\left(-\frac{(x_{\ell}+W_{1}^{\ell})^{2}}{2}\right)\right]\Bigr{|}_{x_{\ell}=0}\right]
=∑k=1n−1∑α∈(ℕ0)d:|α|=k[∏ℓ=1d𝔼​[bℓαℓ​2αℓ!​exp⁡(−(W1ℓ)22)​Hαℓ​(W1ℓ)]].\displaystyle=\sum_{k=1}^{n-1}\sum_{\alpha\in({\mathbbm{N}}_{0})^{d}\colon\lvert\alpha\rvert=k}\left[\prod_{\ell=1}^{d}{\mathbb{E}}\!\left[\frac{b_{\ell}^{\alpha_{\ell}}\sqrt{2}}{\alpha_{\ell}!}\exp\!\left(-\frac{(W_{1}^{\ell})^{2}}{2}\right)H_{\alpha_{\ell}}(W_{1}^{\ell})\right]\right]. (25)

This, (23), and the multinomial theorem show for all n∈ℕn\in{\mathbbm{N}} that

vn​(0,0)=∑k=0n−1∑α∈(ℕ0)d:|α|=k∏ℓ=1d[bℓαℓαℓ!​𝟙2​ℕ0​(αℓ)​αℓ!​(−1)⌊αℓ/2⌋4αℓ/2​⌊αℓ/2⌋!]\displaystyle v^{n}(0,0)=\sum_{k=0}^{n-1}\sum_{\alpha\in({\mathbbm{N}}_{0})^{d}\colon\lvert\alpha\rvert=k}\prod_{\ell=1}^{d}\left[\frac{b_{\ell}^{\alpha_{\ell}}}{\alpha_{\ell}!}\mathbbm{1}_{2{\mathbbm{N}}_{0}}(\alpha_{\ell})\frac{\alpha_{\ell}!(-1)^{\lfloor\alpha_{\ell}/2\rfloor}}{4^{\alpha_{\ell}/2}\lfloor\alpha_{\ell}/2\rfloor!}\right]
=∑i=0⌊n−12⌋∑β∈(ℕ0)d:|β|=i∏ℓ=1dbℓ2​βℓ​(−1)βℓ4βℓ​βℓ!=∑i=0⌊n−12⌋[(−1)i4i​∑β∈(ℕ0)d:|β|=i∏ℓ=1dbℓ2​βℓβℓ!]\displaystyle=\sum_{i=0}^{\lfloor\frac{n-1}{2}\rfloor}\sum_{\beta\in({\mathbbm{N}}_{0})^{d}\colon\lvert\beta\rvert=i}\prod_{\ell=1}^{d}\frac{b_{\ell}^{2\beta_{\ell}}(-1)^{\beta_{\ell}}}{4^{\beta_{\ell}}\beta_{\ell}!}=\sum_{i=0}^{\lfloor\frac{n-1}{2}\rfloor}\left[\frac{(-1)^{i}}{4^{i}}\sum_{\beta\in({\mathbbm{N}}_{0})^{d}\colon\lvert\beta\rvert=i}\prod_{\ell=1}^{d}\frac{b_{\ell}^{2\beta_{\ell}}}{\beta_{\ell}!}\right]
=∑i=0⌊n−12⌋[(−1)i4i​i!​∑β∈(ℕ0)d:|β|=i[i!​b12​β1​b22​β2​…​bd2​βdβ1!​β2!​⋯​βd!]]=∑i=0⌊n−12⌋(−1)i​∥b∥2​i4i​i!.\displaystyle=\sum_{i=0}^{\lfloor\frac{n-1}{2}\rfloor}\left[\frac{(-1)^{i}}{4^{i}i!}\sum_{\beta\in({\mathbbm{N}}_{0})^{d}\colon\lvert\beta\rvert=i}\left[\frac{i!b_{1}^{2\beta_{1}}b_{2}^{2\beta_{2}}\ldots b_{d}^{2\beta_{d}}}{\beta_{1}!\beta_{2}!\cdots\beta_{d}!}\right]\right]=\sum_{i=0}^{\lfloor\frac{n-1}{2}\rfloor}\frac{(-1)^{i}\lVert b\rVert^{2i}}{4^{i}i!}. (26)

This establishes (iii).

Next observe that for all a∈ℝa\in{\mathbbm{R}}, ℓ∈{1,2,…,d}\ell\in\{1,2,\ldots,d\} it holds that

𝔼​[exp⁡(−(a+W1ℓ)22)]=12​π​∫ℝe−(a+z)22​e−z22​𝑑z=12​π​∫ℝe−a22​e−a​z​e−z2​𝑑z=12​π​∫ℝe−a22+14​a2​e−(z+12​a)2​𝑑z=12​π​∫ℝe−a24​e−y22​12​𝑑y=e−a242.\begin{split}&{\mathbb{E}}\!\left[\exp\!\left(-\frac{(a+W^{\ell}_{1})^{2}}{2}\right)\right]=\frac{1}{\sqrt{2\pi}}\int_{{\mathbbm{R}}}e^{-\frac{(a+z)^{2}}{2}}e^{-\frac{z^{2}}{2}}\,dz=\frac{1}{\sqrt{2\pi}}\int_{{\mathbbm{R}}}e^{-\frac{a^{2}}{2}}e^{-az}e^{-z^{2}}\,dz\\ &=\frac{1}{\sqrt{2\pi}}\int_{{\mathbbm{R}}}e^{-\frac{a^{2}}{2}+\frac{1}{4}a^{2}}e^{-(z+\frac{1}{2}a)^{2}}\,dz=\frac{1}{\sqrt{2\pi}}\int_{{\mathbbm{R}}}e^{-\frac{a^{2}}{4}}e^{-\frac{y^{2}}{2}}\frac{1}{\sqrt{2}}\,dy=\frac{e^{-\frac{a^{2}}{4}}}{\sqrt{2}}.\end{split} (27)

This, (6), and the fact that W1,W2,…,WdW^{1},W^{2},\ldots,W^{d} are independent prove that

v∞​(0,0)=𝔼​[2d/2​exp⁡(−∥b+W1∥22)]=2d/2​∏ℓ=1d𝔼​[exp⁡(−(bℓ+W1ℓ)22)]=e−∥b∥24.v^{\infty}(0,0)={\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\frac{\lVert b+W_{1}\rVert^{2}}{2}\right)\right]=2^{d/2}\prod_{\ell=1}^{d}{\mathbb{E}}\!\left[\exp\!\left(-\frac{(b_{\ell}+W^{\ell}_{1})^{2}}{2}\right)\right]=e^{-\frac{\lVert b\rVert^{2}}{4}}. (28)

This shows (iv).

Next note that (iii), (iv), and the fact that ∀x∈ℝ:e−x24=1+∑i=1∞(−1)i​x2​i4i​i!\forall\,x\in{\mathbbm{R}}\colon e^{-\frac{x^{2}}{4}}=1+\sum_{i=1}^{\infty}\frac{(-1)^{i}x^{2i}}{4^{i}i!} show for all n∈ℕn\in{\mathbbm{N}} that

v∞​(0,0)−vn​(0,0)=∑i=⌊n−12⌋+1∞(−1)i​∥b∥2​ii!​4i=∑i=⌊n+12⌋∞(−1)i​∥b∥2​ii!​4i.\displaystyle v^{\infty}(0,0)-v^{n}(0,0)=\sum^{\infty}_{i=\lfloor\frac{n-1}{2}\rfloor+1}\frac{(-1)^{i}\lVert b\rVert^{2i}}{i!4^{i}}=\sum^{\infty}_{i=\lfloor\frac{n+1}{2}\rfloor}\frac{(-1)^{i}\lVert b\rVert^{2i}}{i!4^{i}}. (29)

Therefore, for all ϵ∈(0,1)\epsilon\in(0,1), n∈ℕ∩[12​ϵ​∥b∥2−1,∞)n\in{\mathbbm{N}}\cap\bigl{[}\frac{1}{2\epsilon}\lVert b\rVert^{2}-1,\infty\bigr{)}, j∈ℕj\in{\mathbbm{N}} with j=⌊n+12⌋j=\lfloor\tfrac{n+1}{2}\rfloor it holds that 12​ϵ​∥b∥2≤n+1=2​n+12≤2​(⌊n+12⌋+1)=2​(j+1)\frac{1}{2\epsilon}\lVert b\rVert^{2}\leq n+1=2\frac{n+1}{2}\leq 2(\lfloor\frac{n+1}{2}\rfloor+1)=2(j+1), ∥b∥24​(j+1)≤ϵ\frac{\lVert b\rVert^{2}}{4(j+1)}\leq\epsilon,

(v∞​(0,0)−vn​(0,0))​(−1)j=(∥b∥2​jj!​4j−∥b∥2​(j+1)(j+1)!​4j+1)+(∥b∥2​(j+2)(j+2)!​4j+2−∥b∥2​(j+3)(j+3)!​4j+3)+…\displaystyle(v^{\infty}(0,0)-v^{n}(0,0))(-1)^{j}=\left(\frac{\lVert b\rVert^{2j}}{j!4^{j}}-\frac{\lVert b\rVert^{2(j+1)}}{(j+1)!4^{j+1}}\right)+\left(\frac{\lVert b\rVert^{2(j+2)}}{(j+2)!4^{j+2}}-\frac{\lVert b\rVert^{2(j+3)}}{(j+3)!4^{j+3}}\right)+\ldots
=∥b∥2​jj!​4j​(1−∥b∥24​(j+1))+∥b∥2​(j+2)(j+2)!​4j+2​(1−∥b∥24​(j+3))+…\displaystyle=\frac{\lVert b\rVert^{2j}}{j!4^{j}}\left(1-\frac{\lVert b\rVert^{2}}{4(j+1)}\right)+\frac{\lVert b\rVert^{2(j+2)}}{(j+2)!4^{j+2}}\left(1-\frac{\lVert b\rVert^{2}}{4(j+3)}\right)+\ldots
≥∥b∥2​jj!​4j​(1−∥b∥24​(j+1))≥(∥b∥24)j​1j!​(1−ϵ),\displaystyle\geq\frac{\lVert b\rVert^{2j}}{j!4^{j}}\left(1-\frac{\lVert b\rVert^{2}}{4(j+1)}\right)\geq\left(\frac{\lVert b\rVert^{2}}{4}\right)^{j}\frac{1}{j!}(1-\epsilon), (30)

and

|v∞​(0,0)−vn​(0,0)|≤∑i=j∞∥b∥2​ii!​4i≤∥b∥2​jj!​4j​(1+∑i=j+1∞[∥b∥24​(j+1)]i−j)≤∥b∥2​jj!​4j​∑ℓ=0∞ϵℓ=(∥b∥24)j​1j!​11−ϵ.\displaystyle\begin{split}&\lvert v^{\infty}(0,0)-v^{n}(0,0)\rvert\\ &\leq\sum^{\infty}_{i=j}\frac{\lVert b\rVert^{2i}}{i!4^{i}}\leq\frac{\lVert b\rVert^{2j}}{j!4^{j}}\left(1+\sum_{i=j+1}^{\infty}\left[\frac{\lVert b\rVert^{2}}{4(j+1)}\right]^{{}^{i-j}}\right)\leq\frac{\lVert b\rVert^{2j}}{j!4^{j}}\sum_{\ell=0}^{\infty}\epsilon^{\ell}=\left(\frac{\lVert b\rVert^{2}}{4}\right)^{j}\frac{1}{j!}\frac{1}{1-\epsilon}.\end{split} (31)

This establishes (v). The proof of Lemma 2.1 is thus completed. ∎

2.2 Lower bounds for the convergence speed of Picard iterations for an example BSDE

Corollary 2.2.

Let d∈ℕd\in{\mathbbm{N}}, b∈ℝdb\in{\mathbbm{R}}^{d}, let ⟨⋅,⋅⟩:ℝd×ℝd→ℝ\langle\cdot,\cdot\rangle\colon{\mathbbm{R}}^{d}\times{\mathbbm{R}}^{d}\to{\mathbbm{R}} denote the standard scalar product on ℝd{\mathbbm{R}}^{d}, let ∥⋅∥:ℝd→[0,∞)\lVert\cdot\rVert\colon{\mathbbm{R}}^{d}\to[0,\infty) denote the standard norm on ℝd{\mathbbm{R}}^{d}, let (Ω,ℱ,ℙ,(𝔽t)t∈[0,1])(\Omega,\mathcal{F},{\mathbb{P}},({\mathbbm{F}}_{t})_{t\in[0,1]}) be a filtered probability space which satisfies the usual conditions, let W:[0,1]×Ω→ℝdW\colon[0,1]\times\Omega\to{\mathbbm{R}}^{d} be a standard (𝔽t)t∈[0,1]({\mathbbm{F}}_{t})_{t\in[0,1]}-Brownian motion with continuous sample paths, let Yn:[0,1]×Ω→ℝY^{n}\colon[0,1]\times\Omega\to{\mathbbm{R}}, n∈ℕ0∪{∞}n\in{\mathbbm{N}}_{0}\cup\{\infty\}, be adapted with continuous sample paths, let Zn:[0,1]×Ω→ℝdZ^{n}\colon[0,1]\times\Omega\to{\mathbbm{R}}^{d}, n∈ℕ0∪{∞}n\in{\mathbbm{N}}_{0}\cup\{\infty\}, be progressively measurable, and assume for all s∈[0,1]s\in[0,1], n∈ℕ∪{∞}n\in{\mathbbm{N}}\cup\{\infty\} that a.s. it holds that Ys0=0Y^{0}_{s}=0, Zs0=0Z^{0}_{s}=0, ∫0T𝔼​[∥Ztn∥2]​𝑑t<∞\int_{0}^{T}{\mathbb{E}}\bigl{[}\lVert Z^{n}_{t}\rVert^{2}\bigr{]}\,dt<\infty and

Ysn+1=2d/2​e−∥W1∥22+∫s1⟨b,Ztn⟩​𝑑t−∫s1⟨Ztn+1,d​Wt⟩.\displaystyle\begin{split}Y_{s}^{n+1}=2^{d/2}e^{-\frac{\lVert W_{1}\rVert^{2}}{2}}+\int_{s}^{1}\langle b,Z^{n}_{t}\rangle\,dt-\int_{s}^{1}\langle Z^{n+1}_{t},dW_{t}\rangle.\end{split} (32)

Then for all n∈ℕ∩[∥b∥2−1,∞)n\in{\mathbbm{N}}\cap\bigl{[}\lVert b\rVert^{2}-1,\infty\bigr{)} it holds a.s. that |Y0∞−Y0n|≥12​(∥b∥24)⌊n+12⌋​1n!\left\lvert Y_{0}^{\infty}-Y^{n}_{0}\right\rvert\geq\frac{1}{2}\left(\frac{\lVert b\rVert^{2}}{4}\right)^{\lfloor\frac{n+1}{2}\rfloor}\frac{1}{\sqrt{n!}}.

Proof of Corollary 2.2.

Throughout this proof let vn:[0,1]×ℝd→ℝv^{n}\colon[0,1]\times{\mathbbm{R}}^{d}\to{\mathbbm{R}}, n∈ℕ0∪{∞}n\in{\mathbbm{N}}_{0}\cup\{\infty\}, satisfy for all t∈[0,1]t\in[0,1], x∈ℝdx\in{\mathbbm{R}}^{d}, n∈ℕn\in{\mathbbm{N}} that v0​(t,x)=0v^{0}(t,x)=0,

v∞​(s,x)=𝔼​[2d/2​exp⁡(−∥x+b​(1−s)+W1−Ws∥22)],\displaystyle v^{\infty}(s,x)={\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\frac{\lVert x+b(1-s)+W_{1}-W_{s}\rVert^{2}}{2}\right)\right], (33)

and

vn​(s,x)=𝔼​[2d/2​exp⁡(−∥x+W1−Ws∥22)]+∑k=1n−1∑μ1,μ2,…,μk=1d[(1−s)kk!​bμ1​bμ2​⋯​bμk​𝔼​[2d/2​∂k∂xμ1​∂xμ2​…​∂xμk​exp⁡(−∥x+W1−Ws∥22)]].\displaystyle\begin{split}&v^{n}(s,x)={\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\frac{\lVert x+W_{1}-W_{s}\rVert^{2}}{2}\right)\right]\\ &+\sum_{k=1}^{n-1}\sum_{\mu_{1},\mu_{2},\ldots,\mu_{k}=1}^{d}\Biggl{[}\frac{(1-s)^{k}}{k!}b_{\mu_{1}}b_{\mu_{2}}\cdots b_{\mu_{k}}{\mathbb{E}}\!\left[2^{d/2}\frac{\partial^{k}}{\partial x_{\mu_{1}}\partial x_{\mu_{2}}\ldots\partial x_{\mu_{k}}}\exp\!\left(-\frac{\lVert x+W_{1}-W_{s}\rVert^{2}}{2}\right)\right]\Biggr{]}.\end{split} (34)

Then Lemma 2.1 proves

  1. a)

    for all t∈[0,1]t\in[0,1], x∈ℝdx\in{\mathbbm{R}}^{d} that v∞∈C∞​([0,1]×ℝd,ℝ)v^{\infty}\in C^{\infty}([0,1]\times{\mathbbm{R}}^{d},{\mathbbm{R}}) and

    ∂v∞∂t​(t,x)+12​(△xv∞)​(t,x)+⟨b,(∇xv∞)​(t,x)⟩=0,\frac{\partial v^{\infty}}{\partial t}(t,x)+\frac{1}{2}(\mathop{}\!\mathbin{\bigtriangleup}_{x}v^{\infty})(t,x)+\left\langle b,(\nabla_{x}v^{\infty})(t,x)\right\rangle=0, (35)
  2. b)

    for all s∈[0,1]s\in[0,1], x∈ℝdx\in{\mathbbm{R}}^{d}, n∈ℕ0n\in{\mathbbm{N}}_{0} that vn∈C∞​([0,1]×ℝd,ℝ)v^{n}\in C^{\infty}([0,1]\times{\mathbbm{R}}^{d},{\mathbbm{R}}) and

    vn+1​(s,x)=𝔼​[2d/2​exp⁡(−∥x+W1−Ws∥22)]+∫s1𝔼​[⟨b,(∇xvn)​(t,x+Wt−Ws)⟩]​𝑑t,v^{n+1}(s,x)={\mathbb{E}}\!\left[2^{d/2}\exp\!\left(-\tfrac{\lVert x+W_{1}-W_{s}\rVert^{2}}{2}\right)\right]+\int_{s}^{1}{\mathbb{E}}\!\left[\Bigl{\langle}b,(\nabla_{x}v^{n})(t,x+W_{t}-W_{s})\Bigr{\rangle}\right]\!dt, (36)

    and

  3. c)

    for all n∈ℕ∩[∥b∥2−1,∞)n\in{\mathbbm{N}}\cap\bigl{[}\lVert b\rVert^{2}-1,\infty\bigr{)} that

    |vn​(0,0)−v∞​(0,0)|≥12​(∥b∥24)⌊n+12⌋​1⌊n+12⌋!.\lvert v^{n}(0,0)-v^{\infty}(0,0)\rvert\geq\frac{1}{2}\left(\frac{\lVert b\rVert^{2}}{4}\right)^{\lfloor\frac{n+1}{2}\rfloor}\frac{1}{\lfloor\frac{n+1}{2}\rfloor!}. (37)

This and Itô’s formula prove that for all s∈[0,1]s\in[0,1] it holds a.s. that

2d/2​e−∥W1∥22−v∞​(s,Ws)=v∞​(1,W1)−v∞​(s,Ws)=∫s1(∂v∞∂t+12△xv∞)​(t,Wt)​𝑑t+∫s1⟨(∇xv∞)​(t,Wt),d​Wt⟩=−∫s1⟨b,(∇xv∞)​(t,Wt)⟩​𝑑t+∫s1⟨(∇xv∞)​(t,Wt),d​Wt⟩.\displaystyle\begin{split}&2^{d/2}e^{-\frac{\lVert W_{1}\rVert^{2}}{2}}-v^{\infty}(s,W_{s})=v^{\infty}(1,W_{1})-v^{\infty}(s,W_{s})\\ &=\int_{s}^{1}\left(\frac{\partial v^{\infty}}{\partial t}+\frac{1}{2}\mathop{}\!\mathbin{\bigtriangleup}_{x}v^{\infty}\right)\!(t,W_{t})\,dt+\int_{s}^{1}\left\langle(\nabla_{x}v^{\infty})(t,W_{t}),dW_{t}\right\rangle\\ &=-\int_{s}^{1}\left\langle b,(\nabla_{x}v^{\infty})(t,W_{t})\right\rangle dt+\int_{s}^{1}\left\langle(\nabla_{x}v^{\infty})(t,W_{t}),dW_{t}\right\rangle.\end{split} (38)

This, (32), and a standard result on uniqueness of backward stochastic differential equations (cf., e.g., [21, Theorem 4.3.1]) prove for all s∈[0,1]s\in[0,1] that ℙ​(Ys∞=v∞​(s,Ws))=1{\mathbb{P}}\bigl{(}Y^{\infty}_{s}=v^{\infty}(s,W_{s})\bigr{)}=1 and ℙ​(Zs∞=(∇xv∞)​(s,Ws))=1{\mathbb{P}}\bigl{(}Z^{\infty}_{s}=(\nabla_{x}v^{\infty})(s,W_{s})\bigr{)}=1.

Next, we prove by induction on n∈ℕ0n\in{\mathbbm{N}}_{0} that for all n∈ℕ0n\in{\mathbbm{N}}_{0}, s∈[0,1]s\in[0,1] it holds that ℙ​(Ysn=vn​(s,Ws))=1{\mathbb{P}}\bigl{(}Y^{n}_{s}=v^{n}(s,W_{s})\bigr{)}=1 and ℙ​(Zsn=(∇xvn)​(s,Ws))=1{\mathbb{P}}\bigl{(}Z^{n}_{s}=(\nabla_{x}v^{n})(s,W_{s})\bigr{)}=1. First, the fact that ∀t∈[0,1],x∈ℝd:v0​(t,x)=0\forall\,t\in[0,1],x\in{\mathbbm{R}}^{d}\colon v^{0}(t,x)=0 and the fact that ∀s∈[0,1]:ℙ​((Ys0,Zs0)=(0,0))=1\forall\,s\in[0,1]\colon{\mathbb{P}}\bigl{(}(Y^{0}_{s},Z^{0}_{s})=(0,0)\bigr{)}=1 establish the base case n=0n=0. For the induction step ℕ0∋n↦n+1∈ℕ{\mathbbm{N}}_{0}\ni n\mapsto n+1\in{\mathbbm{N}} let n∈ℕ0n\in{\mathbbm{N}}_{0} satisfy for all s∈[0,1]s\in[0,1] that ℙ​(Ysn=vn​(s,Ws))=1{\mathbb{P}}\bigl{(}Y^{n}_{s}=v^{n}(s,W_{s})\bigr{)}=1 and ℙ​(Zsn=(∇xvn)​(s,Ws))=1{\mathbb{P}}\bigl{(}Z^{n}_{s}=(\nabla_{x}v^{n})(s,W_{s})\bigr{)}=1. This, (36), the Markov property of WW, the fact that for all s∈[0,1]s\in[0,1] it holds a.s. that 𝔼​[∫s1⟨Ztn+1,d​Wt⟩|𝔽s]=0{\mathbb{E}}\bigl{[}\int_{s}^{1}\langle{Z}^{n+1}_{t},dW_{t}\rangle|{\mathbbm{F}}_{s}\bigr{]}=0, (32), and adaptedness of Yn+1Y^{n+1} imply that for all s∈[0,1]s\in[0,1] it holds a.s.  that

vn+1​(s,Ws)=𝔼​[2d/2​e−∥W1∥22|𝔽s]+∫s1𝔼​[⟨b,(∇xvn)​(t,Wt)⟩|𝔽s]​𝑑t=𝔼​[2d/2​e−∥W1∥22|𝔽s]+∫s1𝔼​[⟨b,Ztn⟩|𝔽s]​𝑑t=𝔼[2d/2e−∥W1∥22+∫s1⟨b,Ztn⟩dt+∫s1⟨Ztn+1,dWt⟩|𝔽s]=𝔼[Ysn+1|𝔽s]=Ysn+1.\begin{split}&v^{n+1}(s,W_{s})={\mathbb{E}}\Bigl{[}2^{d/2}e^{-\frac{\lVert W_{1}\rVert^{2}}{2}}\big{|}{\mathbbm{F}}_{s}\Bigr{]}+\int_{s}^{1}{\mathbb{E}}\!\left[\Bigl{\langle}b,(\nabla_{x}v^{n})(t,W_{t})\Bigr{\rangle}\big{|}{\mathbbm{F}}_{s}\right]\!dt\\ &={\mathbb{E}}\!\left[2^{d/2}e^{-\frac{\lVert W_{1}\rVert^{2}}{2}}\big{|}{\mathbbm{F}}_{s}\right]+\int_{s}^{1}{\mathbb{E}}\bigl{[}\langle b,{Z}^{n}_{t}\rangle\big{|}{\mathbbm{F}}_{s}\bigr{]}dt\\ &={\mathbb{E}}\!\left[2^{d/2}e^{-\frac{\lVert W_{1}\rVert^{2}}{2}}+\int_{s}^{1}\langle b,Z^{n}_{t}\rangle\,dt+\int_{s}^{1}\langle{Z}^{n+1}_{t},dW_{t}\rangle\middle|{\mathbbm{F}}_{s}\right]={\mathbb{E}}\!\left[Y_{s}^{n+1}\middle|{\mathbbm{F}}_{s}\right]=Y_{s}^{n+1}.\end{split} (39)

This, Itô’s formula, the fact that vn+1∈C2​([0,1]×ℝd,ℝ)v^{n+1}\in C^{2}([0,1]\times{\mathbbm{R}}^{d},{\mathbbm{R}}), (34), and (32) show that for all s∈[0,1]s\in[0,1] it holds a.s. that

0=vn+1​(s,Ws)−Ysn+1=−∫s1[(∂vn+1∂t+12△xvn+1)​(t,Wt)−⟨b,Ztn⟩]​𝑑t−∫s1⟨(∇xvn+1)​(t,Wt)−Ztn+1,d​Wt⟩.\displaystyle\begin{split}&0=v^{n+1}(s,W_{s})-Y_{s}^{n+1}=-\int_{s}^{1}\left[\left(\frac{\partial v^{n+1}}{\partial t}+\frac{1}{2}\mathop{}\!\mathbin{\bigtriangleup}_{x}v^{n+1}\right)\!(t,W_{t})-\left\langle b,Z^{n}_{t}\right\rangle\right]dt\\ &\qquad\qquad\qquad-\int_{s}^{1}\left\langle(\nabla_{x}v^{n+1})(t,W_{t})-Z^{n+1}_{t},dW_{t}\right\rangle.\end{split} (40)

This and the uniqueness of the decomposition of continuous semimartingales show for all s∈[0,1]s\in[0,1] that ℙ​(Ysn+1=vn+1​(s,Ws))=1{\mathbb{P}}\bigl{(}Y^{n+1}_{s}=v^{n+1}(s,W_{s})\bigr{)}=1 and ℙ​(Zsn+1=(∇xvn+1)​(s,Ws))=1{\mathbb{P}}\bigl{(}Z^{n+1}_{s}=(\nabla_{x}v^{n+1})(s,W_{s})\bigr{)}=1. This completes the induction step. Induction thus shows for all n∈ℕ0n\in{\mathbbm{N}}_{0}, s∈[0,1]s\in[0,1] that ℙ​(Ysn=vn​(s,Ws))=1{\mathbb{P}}\bigl{(}Y^{n}_{s}=v^{n}(s,W_{s})\bigr{)}=1 and ℙ​(Zsn=(∇xvn)​(s,Ws))=1{\mathbb{P}}\bigl{(}Z^{n}_{s}=(\nabla_{x}v^{n})(s,W_{s})\bigr{)}=1. This and the fact that ℙ​(Y0∞=v∞​(0,W0))=1{\mathbb{P}}\bigl{(}Y^{\infty}_{0}=v^{\infty}(0,W_{0})\bigr{)}=1 imply that for all n∈ℕn\in{\mathbbm{N}} it holds a.s. that Y0n−Y0∞=vn​(0,0)−v∞​(0,0)Y^{n}_{0}-Y_{0}^{\infty}=v^{n}(0,0)-v^{\infty}(0,0). This, the fact that for all k,n∈ℕ0k,n\in{\mathbbm{N}}_{0} with n=2​kn=2k it holds that

⌊n+12⌋!=⌊2​k+12⌋!=k!=1⋅2​⋯​k≤1⋅2​⋯​k⋅(k+1)​(k+2)​⋯​(2​k)=(2​k)!=n!,\begin{split}&\left\lfloor\frac{n+1}{2}\right\rfloor!=\left\lfloor\frac{2k+1}{2}\right\rfloor!=k!\\ &=1\cdot 2\cdots k\leq\sqrt{1\cdot 2\cdots k\cdot(k+1)(k+2)\cdots(2k)}=\sqrt{(2k)!}=\sqrt{n!},\end{split} (41)

the fact that for all k,n∈ℕ0k,n\in{\mathbbm{N}}_{0} with n=2​k+1n=2k+1 it holds that

⌊n+12⌋!=⌊2​k+1+12⌋!=(k+1)!=2⋅3​⋯​(k+1)≤2⋅3​⋯​(k+1)​(k+2)​(k+3)​⋯​(2​k+1)=n!,\begin{split}&\left\lfloor\frac{n+1}{2}\right\rfloor!=\left\lfloor\frac{2k+1+1}{2}\right\rfloor!=(k+1)!\\ &=2\cdot 3\cdots(k+1)\leq\sqrt{2\cdot 3\cdots(k+1)(k+2)(k+3)\cdots(2k+1)}=\sqrt{n!},\end{split} (42)

and (37) imply that for all n∈ℕ∩[∥b∥2−1,∞)n\in{\mathbbm{N}}\cap\bigl{[}\lVert b\rVert^{2}-1,\infty\bigr{)} it holds a.s. that

|Y0n−Y0∞|=|vn​(0,0)−v∞​(0,0)|≥12​(∥b∥24)⌊n+12⌋​1n!.\left\lvert Y_{0}^{n}-Y_{0}^{\infty}\right\rvert=\left\lvert v^{n}(0,0)-v^{\infty}(0,0)\right\rvert\geq\frac{1}{2}\left(\frac{\lVert b\rVert^{2}}{4}\right)^{\lfloor\frac{n+1}{2}\rfloor}\frac{1}{\sqrt{n!}}. (43)

This completes the proof of Corollary 2.2. ∎

The following Lemma 2.3 gives an example where the BSDE solution is an expoential function (see Item (i)) and the Picard approximations are just partial sums of the exponential series (see Item (ii)). Thereby, Lemma 2.3 shows that factorial speed of convergence cannot be improved up to exponential factors in the case of yy-dependent drivers.

Lemma 2.3.

Let T∈(0,∞)T\in(0,\infty) and let Yk∈C​([0,T],ℝ)Y^{k}\in C([0,T],{\mathbbm{R}}), k∈ℕ0∪{∞}k\in{\mathbbm{N}}_{0}\cup\{\infty\}, satisfy for all k∈ℕ0k\in{\mathbbm{N}}_{0}, s∈[0,T]s\in[0,T] that Ys0=1Y_{s}^{0}=1, Ysk+1=1+∫sTYrk​𝑑rY_{s}^{k+1}=1+\int_{s}^{T}Y_{r}^{k}\,dr, and Ys∞=1+∫sTYr∞​𝑑rY_{s}^{\infty}=1+\int_{s}^{T}Y_{r}^{\infty}dr. Then

  1. i)

    it holds for all s∈[0,T]s\in[0,T] that Ys∞=eT−sY_{s}^{\infty}=e^{T-s},

  2. ii)

    it holds for all s∈[0,T]s\in[0,T], n∈ℕ0n\in{\mathbbm{N}}_{0} that Ysn=1+∑k=1n(T−s)kk!Y_{s}^{n}=1+\sum_{k=1}^{n}\frac{(T-s)^{k}}{k!}, and

  3. iii)

    it holds that sups∈[0,T]|Ys∞−Ysn|=∑k=n+1∞Tkk!≥Tn+1(n+1)!\sup_{s\in[0,T]}\lvert Y_{s}^{\infty}-Y_{s}^{n}\rvert=\sum_{k=n+1}^{\infty}\frac{T^{k}}{k!}\geq\frac{T^{n+1}}{(n+1)!}.

Proof of Lemma 2.3.

The fact that ∀s∈[0,T]:Ys∞=1+∫sTYr∞​𝑑r\forall\,s\in[0,T]\colon Y_{s}^{\infty}=1+\int_{s}^{T}Y_{r}^{\infty}\,dr and the substitution rule show for all s∈[0,T]s\in[0,T] that YT−s∞=1+∫T−sTYr∞​𝑑r=1+∫0sYT−r∞​𝑑rY^{\infty}_{T-s}=1+\int_{T-s}^{T}Y_{r}^{\infty}\,dr=1+\int_{0}^{s}Y_{T-r}^{\infty}\,dr. This, the fact that ∀s∈[0,T]:eT−s=1+∫0seT−r​𝑑r\forall\,s\in[0,T]\colon e^{T-s}=1+\int_{0}^{s}e^{T-r}\,dr, and the Picard–Lindelöf theorem show (i). Next, we prove (ii) by induction on n∈ℕ0n\in{\mathbbm{N}}_{0}. The fact that ∀s∈[0,T]:Ys0=1\forall\,s\in[0,T]\colon Y_{s}^{0}=1 shows the base case n=0n=0. For the induction step ℕ0∋n↦n+1∈ℕ{\mathbbm{N}}_{0}\ni n\mapsto n+1\in{\mathbbm{N}} let n∈ℕ0n\in{\mathbbm{N}}_{0} and assume for all s∈[0,T]s\in[0,T] that Ysn=1+∑k=1n(T−s)kk!Y_{s}^{n}=1+\sum_{k=1}^{n}\frac{(T-s)^{k}}{k!}. The assumptions of Lemma 2.3 then show for all s∈[0,T]s\in[0,T] that

Ysn+1=1+∫sTYrn​𝑑r=1+∫sT(1+∑k=1n(T−r)kk!)​𝑑r=1+∑k=1n+1−(T−r)k+1(k+1)!|r=sT=1+∑k=1n+1(T−s)kk!.\displaystyle\begin{split}&Y_{s}^{n+1}=1+\int_{s}^{T}Y_{r}^{n}\,dr=1+\int_{s}^{T}\left(1+\sum_{k=1}^{n}\frac{(T-r)^{k}}{k!}\right)dr\\ &=1+\sum_{k=1}^{n+1}\frac{-(T-r)^{k+1}}{(k+1)!}\Bigr{|}_{r=s}^{T}=1+\sum_{k=1}^{n+1}\frac{(T-s)^{k}}{k!}.\end{split} (44)

This completes the induction step. Induction hence shows (ii). Combining (i), (ii), and the fact that ∀x∈ℝ:ex=1+∑k=1∞xkk!\forall\,x\in{\mathbbm{R}}\colon e^{x}=1+\sum_{k=1}^{\infty}\frac{x^{k}}{k!} yields (iii). The proof of Lemma 2.3 is thus completed. ∎

3 A priori estimates for backward Itô processes

In this section we establish a priori estimates for certain backward Itô processes. Results of this form are well-known in the scientific literature on BSDEs (see, e.g., [17, Proof of Theorem 3.1], [8, Proposition 2.1], [21, Theorem 4.2.1], [18, Proposition 5.2]). Lemma 3.1 below establishes estimates for an Itô process and its diffusion process in terms of the drift process and in terms of the terminal value of the Itô process. The contribution of Lemma 3.1 is to provide explicit universal constants. Moreover, the Itô process in Lemma 3.1 and its drift process are not assumed to be square-integrable and, in particular, the right-hand sides of (46), (47), and (LABEL:eq:b03) are allowed to be infinite (with positive probability). We note that square-integrability of the diffusion process ZZ in Lemma 3.1, however, is in general required; e.g., choose A≡0A\equiv 0 and ZZ such that the Itô isometry does not hold for the Itô integral YTY_{T}.

Lemma 3.1.

Let T∈(0,∞)T\in(0,\infty), d,m∈ℕd,m\in{\mathbbm{N}}, let ⟨⋅,⋅⟩:ℝd×ℝd→ℝ\langle\cdot,\cdot\rangle\colon{\mathbbm{R}}^{d}\times{\mathbbm{R}}^{d}\to{\mathbbm{R}} denote the standard scalar product on ℝd{\mathbbm{R}}^{d}, let ∥⋅∥:ℝd→[0,∞)\lVert\cdot\rVert\colon{\mathbbm{R}}^{d}\to[0,\infty) denote the standard norm on ℝd{\mathbbm{R}}^{d}, let ∥⋅∥𝖥:ℝd×m→[0,∞)\lVert\cdot\rVert_{\mathsf{F}}\colon{\mathbbm{R}}^{d\times m}\to[0,\infty) denote the Frobenius norm on ℝd×m{\mathbbm{R}}^{d\times m}, let (Ω,ℱ,ℙ,(𝔽t)t∈[0,T])(\Omega,\mathcal{F},{\mathbb{P}},({\mathbbm{F}}_{t})_{t\in[0,T]}) be a filtered probability space which satisfies the usual conditions, let W:[0,T]×Ω→ℝmW\colon[0,T]\times\Omega\to{\mathbbm{R}}^{m} be a standard (𝔽t)t∈[0,T]({\mathbbm{F}}_{t})_{t\in[0,T]}-Brownian motion with continuous sample paths, let Y:[0,T]×Ω→ℝdY\colon[0,T]\times\Omega\to{\mathbbm{R}}^{d} be adapted with continuous sample paths, let A:[0,T]×Ω→ℝdA\colon[0,T]\times\Omega\to{\mathbbm{R}}^{d} be measurable, let Z:[0,T]×Ω→ℝd×mZ\colon[0,T]\times\Omega\to{\mathbbm{R}}^{d\times m} be progressively measurable, and assume that for all s∈[0,T]s\in[0,T] it holds a.s. that

∫0T(∥At∥+𝔼​[∥Zt∥𝖥2])​𝑑t<∞andYs=YT+∫sTAt​𝑑t−∫sTZt​𝑑Wt.\displaystyle\int_{0}^{T}\Bigl{(}\lVert A_{t}\rVert+{\mathbb{E}}[\lVert Z_{t}\rVert_{\mathsf{F}}^{2}]\Bigr{)}\,dt<\infty\quad\text{and}\quad Y_{s}=Y_{T}+\int_{s}^{T}A_{t}\,dt-\int_{s}^{T}Z_{t}\,dW_{t}. (45)

Then

  1. (i)

    for all s∈[0,T]s\in[0,T], λ∈(0,∞)\lambda\in(0,\infty) it holds a.s. that

    𝔼​[eλ​s​∥Ys∥2+∫sTeλ​t​∥Zt∥𝖥2​𝑑t|𝔽s]≤𝔼​[eλ​T​∥YT∥2+∫sTeλ​tλ​∥At∥2​𝑑t|𝔽s],\displaystyle{\mathbb{E}}\!\left[e^{\lambda s}\lVert Y_{s}\rVert^{2}+\int_{s}^{T}e^{\lambda t}\lVert Z_{t}\rVert_{\mathsf{F}}^{2}\,dt\Big{|}{\mathbbm{F}}_{s}\right]\leq{\mathbb{E}}\!\left[e^{\lambda T}\lVert Y_{T}\rVert^{2}+\int_{s}^{T}\frac{e^{\lambda t}}{\lambda}\lVert A_{t}\rVert^{2}\,dt\Big{|}{\mathbbm{F}}_{s}\right], (46)
  2. (ii)

    for all s∈[0,T]s\in[0,T], λ∈(0,∞)\lambda\in(0,\infty) it holds a.s. that

    𝔼[supt∈[s,T](eλ​t∥Yt∥2+∫tTeλ​u∥Zu∥𝖥2du)|𝔽s]\displaystyle{\mathbb{E}}\!\left[\sup_{t\in[s,T]}\left(e^{\lambda t}\lVert Y_{t}\rVert^{2}+\int_{t}^{T}e^{\lambda u}\lVert Z_{u}\rVert_{\mathsf{F}}^{2}\,du\right)\middle|{\mathbbm{F}}_{s}\right] ≤34𝔼[eλ​T∥YT∥2+∫sTeλ​tλ∥At∥2dt|𝔽s],\displaystyle\leq 34{\mathbb{E}}\!\left[e^{\lambda T}\lVert Y_{T}\rVert^{2}+\int_{s}^{T}\frac{e^{\lambda t}}{\lambda}\lVert A_{t}\rVert^{2}\,dt\middle|{\mathbbm{F}}_{s}\right], (47)

    and

  3. (iii)

    it holds for all α,λ∈(0,∞)\alpha,\lambda\in(0,\infty) that

    ∫0T[tα−1​eλ​t​𝔼​[∥Yt∥2]Γ​(α)+tα​eλ​t​𝔼​[∥Zt∥𝖥2]Γ​(α+1)]​𝑑t≤eλ​T​Tα​𝔼​[∥YT∥2]Γ​(α+1)+1λ​∫0Teλ​t​tα​𝔼​[∥At∥2]Γ​(α+1)​𝑑t.\displaystyle\begin{split}&\int_{0}^{T}\left[\frac{t^{\alpha-1}e^{\lambda t}{\mathbb{E}}\!\left[\left\lVert Y_{t}\right\rVert^{2}\right]}{\Gamma(\alpha)}+\frac{t^{\alpha}e^{\lambda t}{\mathbb{E}}\!\left[\left\lVert Z_{t}\right\rVert_{\mathsf{F}}^{2}\right]}{\Gamma(\alpha+1)}\right]dt\leq\frac{e^{\lambda T}T^{\alpha}{\mathbb{E}}\!\left[\lVert Y_{T}\rVert^{2}\right]}{\Gamma(\alpha+1)}+\frac{1}{\lambda}\int_{0}^{T}\frac{e^{\lambda t}t^{\alpha}{\mathbb{E}}\!\left[\lVert A_{t}\rVert^{2}\right]}{\Gamma(\alpha+1)}dt.\end{split} (48)
Proof of Lemma 3.1.

Throughout this proof for every s∈[0,T]s\in[0,T] let Bs∈𝔽sB_{s}\in\mathbb{F}_{s} satisfy that a.s. on BsB_{s} it holds that 𝔼​[‖YT‖2+∫sT‖At‖2​𝑑t|𝔽s]<∞{\mathbb{E}}\!\left[\|Y_{T}\|^{2}+\int_{s}^{T}\|A_{t}\|^{2}\,dt\Big{|}{\mathbbm{F}}_{s}\right]<\infty and a.s. on Ω∖Bs\Omega\setminus B_{s} it holds that 𝔼​[‖YT‖2+∫sT‖At‖2​𝑑t|𝔽s]=∞{\mathbb{E}}\!\left[\|Y_{T}\|^{2}+\int_{s}^{T}\|A_{t}\|^{2}\,dt\Big{|}{\mathbbm{F}}_{s}\right]=\infty, let {e1,e2,…,em}⊆ℝm\{e_{1},e_{2},\ldots,e_{m}\}\subseteq{\mathbbm{R}}^{m} be an orthonormal basis of ℝm{\mathbbm{R}}^{m}, and let α,λ∈(0,∞)\alpha,\lambda\in(0,\infty). First note that (45), Jensen’s inequality, and the Burkholder-Davis-Gundy inequality (see, e.g., [5, Lemma 7.2]) yield that for all s∈[0,T]s\in[0,T] it holds a.s. on BsB_{s} that

𝔼​[supt∈[s,T]∥Yt∥2|𝔽s]≤3​(∥Ys∥2+𝔼​[(∫sT∥At∥​𝑑t)2+supu∈[s,T]∥∫suZt​𝑑Wt∥2|𝔽s])≤12​(∥Ys∥2+𝔼​[T​∫sT∥At∥2​𝑑t|𝔽s]+𝔼​[∫sT∥Zt∥𝖥2​𝑑t|𝔽s])<∞.\begin{split}{\mathbb{E}}\!\left[\sup_{t\in[s,T]}\lVert Y_{t}\rVert^{2}\Big{|}\mathbb{F}_{s}\right]&\leq 3\left(\lVert Y_{s}\rVert^{2}+{\mathbb{E}}\!\left[\left(\int_{s}^{T}\lVert A_{t}\rVert\,dt\right)^{2}+\sup_{u\in[s,T]}\left\lVert\int_{s}^{u}Z_{t}\,dW_{t}\right\rVert^{2}\Big{|}\mathbb{F}_{s}\right]\right)\\ &\leq 12\left(\lVert Y_{s}\rVert^{2}+{\mathbb{E}}\left[T\int_{s}^{T}\lVert A_{t}\rVert^{2}\,dt\Big{|}\mathbb{F}_{s}\right]+{\mathbb{E}}\left[\int_{s}^{T}\lVert Z_{t}\rVert_{\mathsf{F}}^{2}\,dt\Big{|}\mathbb{F}_{s}\right]\right)<\infty.\end{split} (49)

This, the L1L^{1}-Burkholder-Davis-Gundy inequality (e.g., [20, Theorem 1]), the Cauchy-Schwarz inequality, and Hölder’s inequality imply that for all s∈[0,T]s\in[0,T] it holds a.s. on BsB_{s} that

𝔼​[supu∈[s,T]|∫sueλ​t​⟨Yt,Zt​d​Wt⟩||𝔽s]≤8​𝔼​[(∫sTe2​λ​t​[∑i=1m|⟨Yt,Zt​ei⟩|2]​𝑑t)1/2|𝔽s]≤8​𝔼​[(∫sTeλ​t​∥Zt∥𝖥2​𝑑t)1/2​(supt∈[s,T]eλ​t​∥Yt∥2)1/2|𝔽s]≤8​(𝔼​[∫sTeλ​t​∥Zt∥𝖥2​𝑑t|𝔽s]​𝔼​[supt∈[s,T]eλ​t​∥Yt∥2|𝔽s])12.\begin{split}&{\mathbb{E}}\!\left[\sup_{u\in[s,T]}\left\lvert\int_{s}^{u}e^{\lambda t}\langle Y_{t},Z_{t}\,dW_{t}\rangle\right\rvert\Big{|}\mathbb{F}_{s}\right]\leq\sqrt{8}{\mathbb{E}}\left[\left(\int_{s}^{T}e^{2\lambda t}\left[\sum_{i=1}^{m}\left\lvert\left\langle Y_{t},Z_{t}e_{i}\right\rangle\right\rvert^{2}\right]dt\right)^{\nicefrac{{1}}{{2}}}\Big{|}\mathbb{F}_{s}\right]\\ &\leq\sqrt{8}{\mathbb{E}}\!\left[\left(\int_{s}^{T}e^{\lambda t}\left\lVert Z_{t}\right\rVert_{\mathsf{F}}^{2}\,dt\right)^{\!\nicefrac{{1}}{{2}}}\left(\sup_{t\in[s,T]}e^{\lambda t}\lVert Y_{t}\rVert^{2}\right)^{\!\nicefrac{{1}}{{2}}}\Big{|}\mathbb{F}_{s}\right]\\ &\leq\sqrt{8}\left({\mathbb{E}}\!\left[\int_{s}^{T}e^{\lambda t}\left\lVert Z_{t}\right\rVert_{\mathsf{F}}^{2}\,dt\Big{|}\mathbb{F}_{s}\right]{\mathbb{E}}\!\left[\sup_{t\in[s,T]}e^{\lambda t}\lVert Y_{t}\rVert^{2}\Big{|}\mathbb{F}_{s}\right]\right)^{\frac{1}{2}}.\end{split} (50)

This, (45), and (49) yield that for all s∈[0,T]s\in[0,T] it holds a.s. that (𝟙Bs​∫sueλ​t​⟨Yt,Zt​d​Wt⟩)u∈[s,T]\big{(}\mathbbm{1}_{B_{s}}\int_{s}^{u}e^{\lambda t}\langle Y_{t},Z_{t}\,dW_{t}\rangle\big{)}_{u\in[s,T]} is a martingale with respect to ℙ(⋅|𝔽s){\mathbb{P}}(\cdot|\mathbb{F}_{s}) and

𝔼​[𝟙Bs​∫sTeλ​t​⟨Yt,Zt​d​Wt⟩|𝔽s]=0.\begin{split}{\mathbb{E}}\!\left[\mathbbm{1}_{B_{s}}\int_{s}^{T}e^{\lambda t}\langle Y_{t},Z_{t}\,dW_{t}\rangle\Big{|}{\mathbbm{F}}_{s}\right]=0.\end{split} (51)

Next note that (45) and Itô’s formula show that for all s∈[0,T]s\in[0,T] it holds a.s. that

eλ​T​∥YT∥2−eλ​s​∥Ys∥2=∫sTd​(eλ​t​∥Yt∥2)=∫sTλ​eλ​t​∥Yt∥2​𝑑t+∫sTeλ​t​d​(∥Yt∥2)=∫sTλ​eλ​t​∥Yt∥2​𝑑t+∫sTeλ​t​(∥Zt∥𝖥2−2​⟨Yt,At⟩)​𝑑t+∫sT2​eλ​t​⟨Yt,Zt​d​Wt⟩=∫sTeλ​t​∥Zt∥𝖥2​𝑑t+∫sTeλ​tλ​[∥λ​Yt−At∥2−∥At∥2]​𝑑t+∫sT2​eλ​t​⟨Yt,Zt​d​Wt⟩.\displaystyle\begin{split}&e^{\lambda T}\lVert Y_{T}\rVert^{2}-e^{\lambda s}\lVert Y_{s}\rVert^{2}=\int_{s}^{T}d(e^{\lambda t}\lVert Y_{t}\rVert^{2})=\int_{s}^{T}\lambda e^{\lambda t}\lVert Y_{t}\rVert^{2}\,dt+\int_{s}^{T}e^{\lambda t}d(\lVert Y_{t}\rVert^{2})\\ &=\int_{s}^{T}\lambda e^{\lambda t}\lVert Y_{t}\rVert^{2}\,dt+\int_{s}^{T}e^{\lambda t}\left(\lVert Z_{t}\rVert_{\mathsf{F}}^{2}-2\langle Y_{t},A_{t}\rangle\right)dt+\int_{s}^{T}2e^{\lambda t}\langle Y_{t},Z_{t}dW_{t}\rangle\\ &=\int_{s}^{T}e^{\lambda t}\lVert Z_{t}\rVert_{\mathsf{F}}^{2}\,dt+\int_{s}^{T}\tfrac{e^{\lambda t}}{\lambda}\bigl{[}\lVert\lambda Y_{t}-A_{t}\rVert^{2}-\lVert A_{t}\rVert^{2}\bigr{]}\,dt+\int_{s}^{T}2e^{\lambda t}\langle Y_{t},Z_{t}dW_{t}\rangle.\end{split} (52)

This shows that for all s∈[0,T]s\in[0,T] it holds a.s. that

eλ​s​∥Ys∥2+∫sTeλ​t​∥Zt∥𝖥2​𝑑t+∫sTeλ​tλ​∥λ​Yt−At∥2​𝑑t=eλ​T​∥YT∥2+∫sTeλ​tλ​∥At∥2​𝑑t−∫sT2​eλ​t​⟨Yt,Zt​d​Wt⟩.\displaystyle\begin{split}&e^{\lambda s}\lVert Y_{s}\rVert^{2}+\int_{s}^{T}e^{\lambda t}\lVert Z_{t}\rVert_{\mathsf{F}}^{2}\,dt+\int_{s}^{T}\tfrac{e^{\lambda t}}{\lambda}\lVert\lambda Y_{t}-A_{t}\rVert^{2}\,dt\\ &=e^{\lambda T}\lVert Y_{T}\rVert^{2}+\int_{s}^{T}\tfrac{e^{\lambda t}}{\lambda}\lVert A_{t}\rVert^{2}\,dt-\int_{s}^{T}2e^{\lambda t}\langle Y_{t},Z_{t}dW_{t}\rangle.\end{split} (53)

This and (51) show that for all s∈[0,T]s\in[0,T] it holds a.s. on BsB_{s} that

𝔼​[eλ​s​∥Ys∥2+∫sTeλ​t​∥Zt∥𝖥2​𝑑t|𝔽s]≤𝔼​[eλ​T​∥YT∥2+∫sTeλ​tλ​∥At∥2​𝑑t|𝔽s].\displaystyle{\mathbb{E}}\!\left[e^{\lambda s}\lVert Y_{s}\rVert^{2}+\int_{s}^{T}e^{\lambda t}\lVert Z_{t}\rVert_{\mathsf{F}}^{2}\,dt\Big{|}{\mathbbm{F}}_{s}\right]\leq{\mathbb{E}}\!\left[e^{\lambda T}\lVert Y_{T}\rVert^{2}+\int_{s}^{T}\tfrac{e^{\lambda t}}{\lambda}\lVert A_{t}\rVert^{2}\,dt\Big{|}{\mathbbm{F}}_{s}\right]. (54)

This and the definition of BsB_{s}, s∈[0,T]s\in[0,T], prove (i).

Next observe that (LABEL:eq:WIntegralYZ), (51) and (i) yield that for all s∈[0,T]s\in[0,T] it holds a.s. on BsB_{s} that

𝔼​[supt∈[s,T](−2​∫tTeλ​u​⟨Yu,Zu​d​Wu⟩)|𝔽s]=2​𝔼​[supt∈[s,T](∫steλ​u​⟨Yu,Zu​d​Wu⟩)|𝔽s]−2​𝔼​[∫sTeλ​u​⟨Yu,Zu​d​Wu⟩|𝔽s]≤2​8​(𝔼​[∫sTeλ​t​∥Zt∥𝖥2​𝑑t|𝔽s]​𝔼​[supt∈[s,T]eλ​t​‖Yt‖2|𝔽s])12.≤2​8​(𝔼​[eλ​T​‖YT‖2+∫sTeλ​tλ​‖At‖2​𝑑t|𝔽s]​𝔼​[supt∈[s,T]eλ​t​‖Yt‖2|𝔽s])12.\begin{split}&{\mathbb{E}}\bigg{[}\sup_{t\in[s,T]}\left(-2\int_{t}^{T}e^{\lambda u}\langle Y_{u},Z_{u}dW_{u}\rangle\right)\Big{|}\mathbb{F}_{s}\bigg{]}\\ &=2{\mathbb{E}}\bigg{[}\sup_{t\in[s,T]}\left(\int_{s}^{t}e^{\lambda u}\langle Y_{u},Z_{u}\,dW_{u}\rangle\right)\big{|}\mathbb{F}_{s}\bigg{]}-2{\mathbb{E}}\bigg{[}\int_{s}^{T}e^{\lambda u}\langle Y_{u},Z_{u}\,dW_{u}\rangle\big{|}\mathbb{F}_{s}\bigg{]}\\ &\leq 2\sqrt{8}\left({\mathbb{E}}\!\left[\int_{s}^{T}e^{\lambda t}\left\lVert Z_{t}\right\rVert^{2}_{\mathsf{F}}\,dt\big{|}\mathbb{F}_{s}\right]{\mathbb{E}}\!\left[\sup_{t\in[s,T]}e^{\lambda t}\|Y_{t}\|^{2}\big{|}\mathbb{F}_{s}\right]\right)^{\frac{1}{2}}.\\ &\leq 2\sqrt{8}\left({\mathbb{E}}\!\left[e^{\lambda T}\|Y_{T}\|^{2}+\int_{s}^{T}\tfrac{e^{\lambda t}}{\lambda}\|A_{t}\|^{2}\,dt\big{|}\mathbb{F}_{s}\right]{\mathbb{E}}\!\left[\sup_{t\in[s,T]}e^{\lambda t}\|Y_{t}\|^{2}\big{|}\mathbb{F}_{s}\right]\right)^{\frac{1}{2}}.\end{split} (55)

This, (53), (45), and (49) yield that for all s∈[0,T]s\in[0,T] it holds a.s. on BsB_{s} that

𝔼​[supt∈[s,T](eλ​t​∥Yt∥2+∫tTeλ​u​∥Zu∥𝖥2​𝑑u)|𝔽s]≤𝔼​[eλ​T​∥YT∥2+∫sTeλ​tλ​∥At∥2​𝑑t|𝔽s]+𝔼​[supt∈[s,T](−2​∫tTeλ​u​⟨Yu,Zu​d​Wu⟩)|𝔽s]≤𝔼​[eλ​T​∥YT∥2+∫sTeλ​tλ​∥At∥2​𝑑t|𝔽s]+2​8​(𝔼​[eλ​T​∥YT∥2+∫sTeλ​tλ​∥At∥2​𝑑t|𝔽s])12⋅(𝔼[supt∈[s,T](eλ​t∥Yt∥2+∫tTeλ​u∥Zu∥𝖥2du)|𝔽s])12<∞.\begin{split}&{\mathbb{E}}\!\left[\sup_{t\in[s,T]}\left(e^{\lambda t}\lVert Y_{t}\rVert^{2}+\int_{t}^{T}e^{\lambda u}\lVert Z_{u}\rVert^{2}_{\mathsf{F}}\,du\right)\big{|}\mathbb{F}_{s}\right]\\ &\leq{\mathbb{E}}\!\left[e^{\lambda T}\lVert Y_{T}\rVert^{2}+\int_{s}^{T}\frac{e^{\lambda t}}{\lambda}\lVert A_{t}\rVert^{2}\,dt\big{|}\mathbb{F}_{s}\right]+{\mathbb{E}}\left[\sup_{t\in[s,T]}\left(-2\int_{t}^{T}e^{\lambda u}\langle Y_{u},Z_{u}\,dW_{u}\rangle\right)\big{|}\mathbb{F}_{s}\right]\\ &\leq{\mathbb{E}}\!\left[e^{\lambda T}\lVert Y_{T}\rVert^{2}+\int_{s}^{T}\frac{e^{\lambda t}}{\lambda}\lVert A_{t}\rVert^{2}\,dt\big{|}\mathbb{F}_{s}\right]+2\sqrt{8}\left({\mathbb{E}}\left[e^{\lambda T}\lVert Y_{T}\rVert^{2}+\int_{s}^{T}\frac{e^{\lambda t}}{\lambda}\lVert A_{t}\rVert^{2}\,dt\big{|}\mathbb{F}_{s}\right]\right)^{\frac{1}{2}}\\ &\qquad\cdot\left({\mathbb{E}}\!\left[\sup_{t\in[s,T]}\left(e^{\lambda t}\lVert Y_{t}\rVert^{2}+\int_{t}^{T}e^{\lambda u}\lVert Z_{u}\rVert^{2}_{\mathsf{F}}\,du\right)\big{|}{\mathbbm{F}}_{s}\right]\right)^{\frac{1}{2}}<\infty.\end{split} (56)

This and the fact that ∀x,c∈[0,∞):([x≤c+2​8​c​x]⇒[x≤(8+8+1)2​c≤34​c])\forall\,x,c\in[0,\infty)\colon\bigl{(}\bigl{[}x\leq c+2\sqrt{8}\sqrt{c}\sqrt{x}\bigr{]}\Rightarrow\bigl{[}x\leq\bigl{(}\sqrt{8}+\sqrt{8+1}\bigr{)}^{2}c\leq 34c\bigr{]}\bigr{)} imply for all s∈[0,T]s\in[0,T] that a.s. on BsB_{s} it holds that

𝔼[supt∈[s,T](eλ​t∥Yt∥2+∫tTeλ​u∥Zu∥𝖥2du)|𝔽s]\displaystyle{\mathbb{E}}\!\left[\sup_{t\in[s,T]}\left(e^{\lambda t}\lVert Y_{t}\rVert^{2}+\int_{t}^{T}e^{\lambda u}\lVert Z_{u}\rVert_{\mathsf{F}}^{2}\,du\right)\middle|{\mathbbm{F}}_{s}\right] ≤34𝔼[eλ​T∥YT∥2+∫sTeλ​tλ∥At∥2dt|𝔽s].\displaystyle\leq 34{\mathbb{E}}\!\left[e^{\lambda T}\lVert Y_{T}\rVert^{2}+\int_{s}^{T}\frac{e^{\lambda t}}{\lambda}\lVert A_{t}\rVert^{2}\,dt\middle|{\mathbbm{F}}_{s}\right]. (57)

This and the definition of BsB_{s}, s∈[0,T]s\in[0,T], prove (ii).

Next, the fact that ∀t∈[0,T]:tαΓ​(α+1)=∫0tsα−1​d​sΓ​(α)\forall\,t\in[0,T]\colon\frac{t^{\alpha}}{\Gamma(\alpha+1)}=\int_{0}^{t}\frac{s^{\alpha-1}\,ds}{\Gamma(\alpha)}, Tonelli’s theorem, the tower property, and (i) show that

∫0T[tα−1Γ​(α)​eλ​t​𝔼​[∥Yt∥2]+tα​eλ​tΓ​(α+1)​𝔼​[∥Zt∥𝖥2]]​𝑑t\displaystyle\int_{0}^{T}\left[\frac{t^{\alpha-1}}{\Gamma(\alpha)}e^{\lambda t}{\mathbb{E}}\!\left[\left\lVert Y_{t}\right\rVert^{2}\right]+\frac{t^{\alpha}e^{\lambda t}}{\Gamma(\alpha+1)}{\mathbb{E}}\!\left[\left\lVert Z_{t}\right\rVert_{\mathsf{F}}^{2}\right]\right]\!dt
=∫0Ttα−1Γ​(α)​eλ​t​𝔼​[∥Yt∥2]​𝑑t+∫0T∫0tsα−1​eλ​tΓ​(α)​𝔼​[∥Zt∥𝖥2]​𝑑s​𝑑t\displaystyle=\int_{0}^{T}\frac{t^{\alpha-1}}{\Gamma(\alpha)}e^{\lambda t}{\mathbb{E}}\!\left[\left\lVert Y_{t}\right\rVert^{2}\right]\!dt+\int_{0}^{T}\int_{0}^{t}\frac{s^{\alpha-1}e^{\lambda t}}{\Gamma(\alpha)}{\mathbb{E}}\!\left[\left\lVert Z_{t}\right\rVert_{\mathsf{F}}^{2}\right]\!dsdt
=∫0Tsα−1Γ​(α)​eλ​s​𝔼​[∥Ys∥2]​𝑑s+∫0T∫sTsα−1​eλ​tΓ​(α)​𝔼​[∥Zt∥𝖥2]​𝑑t​𝑑s\displaystyle=\int_{0}^{T}\frac{s^{\alpha-1}}{\Gamma(\alpha)}e^{\lambda s}{\mathbb{E}}\!\left[\left\lVert Y_{s}\right\rVert^{2}\right]\!ds+\int_{0}^{T}\int_{s}^{T}\frac{s^{\alpha-1}e^{\lambda t}}{\Gamma(\alpha)}{\mathbb{E}}\!\left[\left\lVert Z_{t}\right\rVert_{\mathsf{F}}^{2}\right]\!dtds
=∫0Tsα−1Γ​(α)​𝔼​[𝔼​[eλ​s​∥Ys∥2+∫sTeλ​t​∥Zt∥𝖥2​𝑑t|𝔽s]]​𝑑s\displaystyle=\int_{0}^{T}\frac{s^{\alpha-1}}{\Gamma(\alpha)}{\mathbb{E}}\!\left[{\mathbb{E}}\!\left[e^{\lambda s}\left\lVert Y_{s}\right\rVert^{2}+\int_{s}^{T}e^{\lambda t}\left\lVert Z_{t}\right\rVert_{\mathsf{F}}^{2}dt\Bigr{|}{\mathbbm{F}}_{s}\right]\right]\!ds
≤∫0Tsα−1Γ​(α)​𝔼​[𝔼​[eλ​T​∥YT∥2+∫sTeλ​tλ​∥At∥2​𝑑t|𝔽s]]​𝑑s\displaystyle\leq\int_{0}^{T}\frac{s^{\alpha-1}}{\Gamma(\alpha)}{\mathbb{E}}\!\left[{\mathbb{E}}\!\left[e^{\lambda T}\lVert Y_{T}\rVert^{2}+\int_{s}^{T}\frac{e^{\lambda t}}{\lambda}\lVert A_{t}\rVert^{2}\,dt\Big{|}{\mathbbm{F}}_{s}\right]\right]\!ds
=eλ​T​𝔼​[∥YT∥2]​(∫0Tsα−1​d​sΓ​(α))+∫0T∫0tsα−1Γ​(α)​eλ​tλ​𝔼​[∥At∥2]​𝑑s​𝑑t\displaystyle=e^{\lambda T}{\mathbb{E}}\!\left[\lVert Y_{T}\rVert^{2}\right]\left(\int_{0}^{T}\frac{s^{\alpha-1}\,ds}{\Gamma(\alpha)}\right)+\int_{0}^{T}\int_{0}^{t}\frac{s^{\alpha-1}}{\Gamma(\alpha)}\frac{e^{\lambda t}}{\lambda}{\mathbb{E}}\!\left[\lVert A_{t}\rVert^{2}\right]dsdt
=eλ​T​Tα​𝔼​[∥YT∥2]Γ​(α+1)+1λ​∫0Teλ​t​tα​𝔼​[∥At∥2]Γ​(α+1)​𝑑t.\displaystyle=\frac{e^{\lambda T}T^{\alpha}{\mathbb{E}}\!\left[\lVert Y_{T}\rVert^{2}\right]}{\Gamma(\alpha+1)}+\frac{1}{\lambda}\int_{0}^{T}\frac{e^{\lambda t}t^{\alpha}{\mathbb{E}}\!\left[\lVert A_{t}\rVert^{2}\right]}{\Gamma(\alpha+1)}\,dt. (58)

This shows (iii). The proof of Lemma 3.1 is thus completed. ∎

4 Upper bounds for the convergence speed of Picard iterations

In this section we provide upper bounds for the convergence speed of Picard iterations of BSDEs. Proposition 4.1 establishes an explicit bound for the L2L^{2}-distance between the Picard iterations and the solution of a BSDE with a globally Lipschitz continuous nonlinearity. Our proof of Proposition 4.1 relies on the a priori estimates for backward Itô processes provided in Lemma 3.1. In Remark 4.2 we employ the estimate of Proposition 4.1 to obtain the square root-factorial speed of convergence of Picard iterations. In Remark 4.3 we employ the estimate of Proposition 4.1 to obtain the factorial speed of convergence of Picard iterations in the z-independent case.

Proposition 4.1.

Let T∈(0,∞)T\in(0,\infty), d,m∈ℕd,m\in{\mathbbm{N}}, L𝔶,L𝔷∈[0,∞)L_{\mathfrak{y}},L_{\mathfrak{z}}\in[0,\infty), let 00=10^{0}=1, let ⟨⋅,⋅⟩:ℝd×ℝd→ℝ\langle\cdot,\cdot\rangle\colon{\mathbbm{R}}^{d}\times{\mathbbm{R}}^{d}\to{\mathbbm{R}} denote the standard scalar product on ℝd{\mathbbm{R}}^{d}, let ∥⋅∥:ℝd→[0,∞)\lVert\cdot\rVert\colon{\mathbbm{R}}^{d}\to[0,\infty) denote the standard norm on ℝd{\mathbbm{R}}^{d}, let ∥⋅∥𝖥:ℝd×m→[0,∞)\lVert\cdot\rVert_{\mathsf{F}}\colon{\mathbbm{R}}^{d\times m}\to[0,\infty) denote the Frobenius norm on ℝd×m{\mathbbm{R}}^{d\times m}, let (Ω,ℱ,ℙ,(𝔽t)t∈[0,T])(\Omega,\mathcal{F},{\mathbb{P}},({\mathbbm{F}}_{t})_{t\in[0,T]}) be a filtered probability space which satisfies the usual conditions, let f:[0,T]×Ω×ℝd×ℝd×m→ℝdf\colon[0,T]\times\Omega\times{\mathbbm{R}}^{d}\times{\mathbbm{R}}^{d\times m}\to{\mathbbm{R}}^{d} be measurable, assume that for all t∈[0,T]t\in[0,T], y,y~∈ℝdy,\tilde{y}\in{\mathbbm{R}}^{d}, z,z~∈ℝd×mz,\tilde{z}\in{\mathbbm{R}}^{d\times m} it holds a.s. that

∥f​(t,y,z)−f​(t,y~,z~)∥≤L𝔶​∥y−y~∥+L𝔷​∥z−z~∥𝖥,\begin{split}\left\lVert f(t,y,z)-f(t,\tilde{y},\tilde{z})\right\rVert\leq L_{\mathfrak{y}}\lVert y-\tilde{y}\rVert+L_{\mathfrak{z}}\lVert z-\tilde{z}\rVert_{\mathsf{F}},\end{split} (59)

let W:[0,T]×Ω→ℝmW\colon[0,T]\times\Omega\to{\mathbbm{R}}^{m} be a standard (𝔽t)t∈[0,T]({\mathbbm{F}}_{t})_{t\in[0,T]}-Brownian motion with continuous sample paths, let ξ:Ω→ℝd\xi\colon\Omega\to{\mathbbm{R}}^{d} be 𝔽T{\mathbbm{F}}_{T}-measurable, let Yk:[0,T]×Ω→ℝdY^{k}\colon[0,T]\times\Omega\to{\mathbbm{R}}^{d}, k∈ℕ0∪{∞}k\in{\mathbbm{N}}_{0}\cup\{\infty\}, be adapted with continuous sample paths, let Zk:[0,T]×Ω→ℝd×mZ^{k}\colon[0,T]\times\Omega\to{\mathbbm{R}}^{d\times m}, k∈ℕ0∪{∞}k\in{\mathbbm{N}}_{0}\cup\{\infty\}, be progressively measurable, and assume that for all s∈[0,T]s\in[0,T], k∈ℕ0∪{∞}k\in{\mathbbm{N}}_{0}\cup\{\infty\} it holds a.s.  that ∫0T𝔼​[∥ξ∥2+∥f​(t,0,0)∥2+∥Yt∞∥2+∥Ztk∥𝖥2]​𝑑t<∞\int_{0}^{T}{\mathbb{E}}[\lVert\xi\rVert^{2}+\lVert f(t,0,0)\rVert^{2}+\lVert Y^{\infty}_{t}\rVert^{2}+\lVert Z_{t}^{k}\rVert^{2}_{\mathsf{F}}]\,dt<\infty, Ys0=0Y^{0}_{s}=0, Zs0=0Z^{0}_{s}=0, and

Ysk+1=ξ+∫sTf​(t,Ytk,Ztk)​𝑑t−∫sTZtk+1​𝑑Wt.\displaystyle Y^{k+1}_{s}=\xi+\int_{s}^{T}f(t,Y_{t}^{k},Z_{t}^{k})\,dt-\int_{s}^{T}Z^{k+1}_{t}\,dW_{t}. (60)

Then it holds for all k∈ℕk\in{\mathbbm{N}} that

𝔼​[supt∈[0,T](∥Ytk−Yt∞∥2)+∫0T∥Ztk−Zt∞∥𝖥2​𝑑t]≤35​(T​ek)k​[∑ℓ=0kk!​L𝔶ℓ​L𝔷k−ℓ​Tℓ/2ℓ!​(k−ℓ)!​ℓ!]2​(𝔼​[∥ξ∥2]+Tk​∫0T𝔼​[∥f​(t,Yt∞,Zt∞)∥2]​𝑑t)<∞.\displaystyle\begin{split}&{\mathbb{E}}\!\left[\sup_{t\in[0,T]}\left(\left\lVert Y^{k}_{t}-Y_{t}^{\infty}\right\rVert^{2}\right)+\int_{0}^{T}\left\lVert Z^{k}_{t}-Z_{t}^{\infty}\right\rVert^{2}_{\mathsf{F}}\,dt\right]\\ &\leq 35\left(\frac{Te}{k}\right)^{k}\left[\sum_{\ell=0}^{k}\frac{k!L_{\mathfrak{y}}^{\ell}L_{\mathfrak{z}}^{k-\ell}T^{\ell/2}}{\ell!(k-\ell)!\sqrt{\ell!}}\right]^{2}\left({\mathbb{E}}\bigl{[}\lVert\xi\rVert^{2}\bigr{]}+\frac{T}{k}\int_{0}^{T}{\mathbb{E}}\!\left[\bigl{\lVert}f(t,Y_{t}^{\infty},Z_{t}^{\infty})\bigr{\rVert}^{2}\right]\!dt\right)<\infty.\end{split} (61)
Proof of Proposition 4.1.

First note that (60) proves that for all s∈[0,T]s\in[0,T], k∈ℕ0k\in{\mathbbm{N}}_{0} it holds a.s. that

Ysk−Ys∞=−𝟙{0}​(k)​ξ+∫sT[𝟙ℕ​(k)​f​(s,Ys|k−1|,Zs|k−1|)−f​(s,Ys∞,Zs∞)]​𝑑t−∫sT[Ztk−Zt∞]​𝑑Wt.\displaystyle\begin{split}&Y^{k}_{s}-Y_{s}^{\infty}\\ &=-\mathbbm{1}_{\{0\}}(k)\xi+\int_{s}^{T}\Bigl{[}\mathbbm{1}_{{\mathbbm{N}}}(k)f(s,Y^{\lvert k-1\rvert}_{s},Z^{\lvert k-1\rvert}_{s})-f(s,Y_{s}^{\infty},Z_{s}^{\infty})\Bigr{]}dt-\int_{s}^{T}\bigl{[}Z^{k}_{t}-Z_{t}^{\infty}\bigr{]}dW_{t}.\end{split} (62)

This, the tower property, Tonelli’s theorem, and Lemma 3.1 (applied for every k∈ℕ0k\in{\mathbbm{N}}_{0} with Y←Yk−Y∞Y\leftarrow Y^{k}-Y^{\infty}, A←𝟙ℕ​(k)​f​(⋅,Y|k−1|,Z|k−1|)−f​(⋅,Y∞,Z∞)A\leftarrow\mathbbm{1}_{{\mathbbm{N}}}(k)f(\cdot,Y^{\lvert k-1\rvert},Z^{\lvert k-1\rvert})-f(\cdot,Y^{\infty},Z^{\infty}), Z←Zk−Z∞Z\leftarrow Z^{k}-Z^{\infty} in the notation of Lemma 3.1) prove

  1. (i)

    that for all k∈ℕ0k\in{\mathbbm{N}}_{0}, λ∈(0,∞)\lambda\in(0,\infty) it holds that

    𝔼​[[supt∈[0,T](eλ​t​∥Ytk−Yt∞∥2)]+∫0Teλ​t​∥Ztk−Zt∞∥𝖥2​𝑑t]≤35λ​𝔼​[λ​eλ​T​∥ξ∥2​𝟙{0}​(k)+∫0Teλ​t​∥𝟙ℕ​(k)​f​(t,Yt|k−1|,Zt|k−1|)−f​(t,Yt∞,Zt∞)∥2​𝑑t],\begin{split}&{\mathbb{E}}\!\left[\left[\sup_{t\in[0,T]}\left(e^{\lambda t}\lVert Y^{k}_{t}-Y_{t}^{\infty}\rVert^{2}\right)\right]+\int_{0}^{T}e^{\lambda t}\left\lVert Z^{k}_{t}-Z_{t}^{\infty}\right\rVert^{2}_{\mathsf{F}}\,dt\right]\\ &\leq\frac{35}{\lambda}{\mathbb{E}}\!\left[\lambda e^{\lambda T}\lVert\xi\rVert^{2}\mathbbm{1}_{\{0\}}(k)+\int_{0}^{T}e^{\lambda t}\left\lVert\mathbbm{1}_{{\mathbbm{N}}}(k)f(t,Y^{\lvert k-1\rvert}_{t},Z^{\lvert k-1\rvert}_{t})-f(t,Y_{t}^{\infty},Z_{t}^{\infty})\right\rVert^{2}dt\right],\end{split} (63)

    and

  2. (ii)

    that for all k∈ℕ0k\in{\mathbbm{N}}_{0}, α,λ∈(0,∞)\alpha,\lambda\in(0,\infty) it holds that

    ∫0Ttα−1​eλ​t​𝔼​[∥Ytk−Yt∞∥2]Γ​(α)+tα​eλ​t​𝔼​[∥Ztk−Zt∞∥𝖥2]Γ​(α+1)​d​t≤eλ​T​Tα​𝔼​[∥ξ∥2]​𝟙{0}​(k)Γ​(α+1)+1λ​∫0Teλ​t​tα​𝔼​[∥𝟙ℕ​(k)​f​(t,Yt|k−1|,Zt|k−1|)−f​(t,Yt∞,Zt∞)∥2]Γ​(α+1)​𝑑t.\displaystyle\begin{split}&\int_{0}^{T}\frac{t^{\alpha-1}e^{\lambda t}{\mathbb{E}}\!\left[\left\lVert Y^{k}_{t}-Y_{t}^{\infty}\right\rVert^{2}\right]}{\Gamma(\alpha)}+\frac{t^{\alpha}e^{\lambda t}{\mathbb{E}}\!\left[\left\lVert Z^{k}_{t}-Z_{t}^{\infty}\right\rVert_{\mathsf{F}}^{2}\right]}{\Gamma(\alpha+1)}dt\\ &\leq\frac{e^{\lambda T}T^{\alpha}{\mathbb{E}}\!\left[\lVert\xi\rVert^{2}\right]\mathbbm{1}_{\{0\}}(k)}{\Gamma(\alpha+1)}+\frac{1}{\lambda}\int_{0}^{T}\frac{e^{\lambda t}t^{\alpha}{\mathbb{E}}\!\left[\left\lVert\mathbbm{1}_{{\mathbbm{N}}}(k)f(t,Y^{\lvert k-1\rvert}_{t},Z^{\lvert k-1\rvert}_{t})-f(t,Y_{t}^{\infty},Z_{t}^{\infty})\right\rVert^{2}\right]}{\Gamma(\alpha+1)}\,dt.\end{split} (64)

This, (59), and the fact that ∀α∈ℕ0:Γ​(α+1)=α!\forall\,\alpha\in{\mathbbm{N}}_{0}\colon\Gamma(\alpha+1)=\alpha! show for all α,k∈ℕ0\alpha,k\in{\mathbbm{N}}_{0}, λ∈(0,∞)\lambda\in(0,\infty) that

(∫0Ttα​eλ​tα!​𝔼​[∥f​(t,Ytk,Ztk)−f​(t,Yt∞,Zt∞)∥2]​𝑑t)1/2\displaystyle\left(\int_{0}^{T}\frac{t^{\alpha}e^{\lambda t}}{\alpha!}{\mathbb{E}}\!\left[\bigl{\lVert}f(t,Y^{k}_{t},Z^{k}_{t})-f(t,Y_{t}^{\infty},Z^{\infty}_{t})\bigr{\rVert}^{2}\right]\!dt\right)^{\!\nicefrac{{1}}{{2}}}
≤L𝔶​(∫0Ttα​eλ​tα!​𝔼​[∥Ytk−Yt∞∥2]​𝑑t)1/2+L𝔷​(∫0Ttα​eλ​tα!​𝔼​[∥Ztk−Zt∞∥𝖥2]​𝑑t)1/2\displaystyle\leq L_{\mathfrak{y}}\left(\int_{0}^{T}\frac{t^{\alpha}e^{\lambda t}}{\alpha!}{\mathbb{E}}\!\left[\bigl{\lVert}Y^{k}_{t}-Y_{t}^{\infty}\bigr{\rVert}^{2}\right]\!dt\right)^{\!\nicefrac{{1}}{{2}}}+L_{\mathfrak{z}}\left(\int_{0}^{T}\frac{t^{\alpha}e^{\lambda t}}{\alpha!}{\mathbb{E}}\!\left[\bigl{\lVert}Z^{k}_{t}-Z_{t}^{\infty}\bigr{\rVert}^{2}_{\mathsf{F}}\right]\!dt\right)^{\!\nicefrac{{1}}{{2}}}
≤∑ν=01[L𝔶ν​L𝔷1−νλ(Tα+ν(α+ν)!λeλ​T𝔼[∥ξ∥2]𝟙{0}(k)\displaystyle\leq\sum_{\nu=0}^{1}\Biggl{[}\frac{L_{\mathfrak{y}}^{\nu}L_{\mathfrak{z}}^{1-\nu}}{\sqrt{\lambda}}\Biggl{(}\frac{T^{\alpha+\nu}}{(\alpha+\nu)!}\lambda e^{\lambda T}{\mathbb{E}}\bigl{[}\lVert\xi\rVert^{2}\bigr{]}\mathbbm{1}_{\{0\}}(k)
+∫0Ttα+ν​eλ​t(α+ν)!𝔼[∥𝟙ℕ(k)f(t,Yt|k−1|,Zt|k−1|)−f(t,Yt∞,Zt∞)∥2]dt)1/2].\displaystyle\qquad\qquad+\int_{0}^{T}\frac{t^{\alpha+\nu}e^{\lambda t}}{(\alpha+\nu)!}{\mathbb{E}}\!\left[\bigl{\lVert}\mathbbm{1}_{{\mathbbm{N}}}(k)f(t,Y^{\lvert k-1\rvert}_{t},Z^{\lvert k-1\rvert}_{t})-f(t,Y_{t}^{\infty},Z_{t}^{\infty})\bigr{\rVert}^{2}\right]\!dt\Biggr{)}^{\!\nicefrac{{1}}{{2}}}\Biggr{]}. (65)

This and induction prove for all k∈ℕ∩[2,∞)k\in{\mathbbm{N}}\cap[2,\infty), λ∈(0,∞)\lambda\in(0,\infty) that

(∫0Tt0​eλ​t0!​𝔼​[∥f​(t,Ytk−1,Ztk−1)−f​(t,Yt∞,Zt∞)∥2]​𝑑t)1/2\displaystyle\left(\int_{0}^{T}\frac{t^{0}e^{\lambda t}}{0!}{\mathbb{E}}\!\left[\bigl{\lVert}f(t,Y^{k-1}_{t},Z^{k-1}_{t})-f(t,Y_{t}^{\infty},Z_{t}^{\infty})\bigr{\rVert}^{2}\right]\!dt\right)^{\!\nicefrac{{1}}{{2}}}
≤∑ν1,ν2,…,νk−1=01[L𝔶∑i=1k−1νi​L𝔷k−1−∑i=1k−1νiλ(k−1)/2​(∫0Tt∑i=1k−1νi​eλ​t(∑i=1k−1νi)!​𝔼​[∥f​(t,Yt0,Zt0)−f​(t,Yt∞,Zt∞)∥2]​𝑑t)1/2]\displaystyle\leq\sum_{\nu_{1},\nu_{2},\ldots,\nu_{k-1}=0}^{1}\left[\frac{L_{\mathfrak{y}}^{\sum_{i=1}^{k-1}\nu_{i}}L_{\mathfrak{z}}^{k-1-\sum_{i=1}^{k-1}\nu_{i}}}{\lambda^{(k-1)/2}}\left(\int_{0}^{T}\frac{t^{\sum_{i=1}^{k-1}\nu_{i}}e^{\lambda t}}{(\sum_{i=1}^{k-1}\nu_{i})!}{\mathbb{E}}\!\left[\bigl{\lVert}f(t,Y_{t}^{0},Z_{t}^{0})-f(t,Y_{t}^{\infty},Z_{t}^{\infty})\bigr{\rVert}^{2}\right]\!dt\right)^{\!\nicefrac{{1}}{{2}}}\right]
≤∑ν1,ν2,…,νk−1=01[L𝔶∑i=1k−1νi​L𝔷k−1−∑i=1k−1νiλ(k−1)/2∑νk=01L𝔶νk​L𝔷1−νkλ\displaystyle\leq\sum_{\nu_{1},\nu_{2},\ldots,\nu_{k-1}=0}^{1}\Biggl{[}\frac{L_{\mathfrak{y}}^{\sum_{i=1}^{k-1}\nu_{i}}L_{\mathfrak{z}}^{k-1-\sum_{i=1}^{k-1}\nu_{i}}}{\lambda^{(k-1)/2}}\sum_{\nu_{k}=0}^{1}\frac{L_{\mathfrak{y}}^{\nu_{k}}L_{\mathfrak{z}}^{1-\nu_{k}}}{\sqrt{\lambda}}
⋅(T∑i=1kνi(∑i=1kνi)!λeλ​T𝔼[∥ξ∥2]+∫0Tt∑i=1kνi​eλ​t(∑i=1kνi)!𝔼[∥f(t,Yt∞,Zt∞)∥2]dt)1/2]\displaystyle\qquad\qquad\qquad\qquad\cdot\left(\frac{T^{\sum_{i=1}^{k}\nu_{i}}}{(\sum_{i=1}^{k}\nu_{i})!}\lambda e^{\lambda T}{\mathbb{E}}\bigl{[}\lVert\xi\rVert^{2}\bigr{]}+\int_{0}^{T}\frac{t^{\sum_{i=1}^{k}\nu_{i}}e^{\lambda t}}{(\sum_{i=1}^{k}\nu_{i})!}{\mathbb{E}}\!\left[\bigl{\lVert}f(t,Y_{t}^{\infty},Z_{t}^{\infty})\bigr{\rVert}^{2}\right]\!dt\right)^{\!\nicefrac{{1}}{{2}}}\Biggr{]}
=∑ν1,ν2,…,νk=01L𝔶∑i=1kνi​L𝔷k−∑i=1kνiλk/2\displaystyle=\sum_{\nu_{1},\nu_{2},\ldots,\nu_{k}=0}^{1}\frac{L_{\mathfrak{y}}^{\sum_{i=1}^{k}\nu_{i}}L_{\mathfrak{z}}^{k-\sum_{i=1}^{k}\nu_{i}}}{\lambda^{k/2}}
⋅(T∑i=1kνi(∑i=1kνi)!​λ​eλ​T​𝔼​[∥ξ∥2]+∫0Tt∑i=1kνi​eλ​t(∑i=1kνi)!​𝔼​[∥f​(t,Yt∞,Zt∞)∥2]​𝑑t)1/2\displaystyle\qquad\qquad\qquad\cdot\left(\frac{T^{\sum_{i=1}^{k}\nu_{i}}}{(\sum_{i=1}^{k}\nu_{i})!}\lambda e^{\lambda T}{\mathbb{E}}\bigl{[}\lVert\xi\rVert^{2}\bigr{]}+\int_{0}^{T}\frac{t^{\sum_{i=1}^{k}\nu_{i}}e^{\lambda t}}{(\sum_{i=1}^{k}\nu_{i})!}{\mathbb{E}}\!\left[\bigl{\lVert}f(t,Y_{t}^{\infty},Z_{t}^{\infty})\bigr{\rVert}^{2}\right]\!dt\right)^{\!\nicefrac{{1}}{{2}}} (66)

This and (65) show for all k∈ℕk\in{\mathbbm{N}}, λ∈(0,∞)\lambda\in(0,\infty) that

(∫0Teλ​t​𝔼​[∥f​(t,Ytk−1,Ztk−1)−f​(t,Yt∞,Zt∞)∥2]​𝑑t)1/2≤∑ℓ=0k[k!ℓ!​(k−ℓ)!​L𝔶ℓ​L𝔷k−ℓλk/2​(Tℓℓ!​λ​eλ​T​𝔼​[∥ξ∥2]+∫0Ttℓ​eλ​tℓ!​𝔼​[∥f​(t,Yt∞,Zt∞)∥2]​𝑑t)1/2]≤[∑ℓ=0kk!ℓ!​(k−ℓ)!​L𝔶ℓ​L𝔷k−ℓλk/2​Tℓ/2​eλ​T/2ℓ!]​(λ​𝔼​[∥ξ∥2]+∫0T𝔼​[∥f​(t,Yt∞,Zt∞)∥2]​𝑑t)1/2.\displaystyle\begin{split}&\left(\int_{0}^{T}e^{\lambda t}{\mathbb{E}}\!\left[\bigl{\lVert}f(t,Y^{k-1}_{t},Z^{k-1}_{t})-f(t,Y_{t}^{\infty},Z_{t}^{\infty})\bigr{\rVert}^{2}\right]\!dt\right)^{\!\nicefrac{{1}}{{2}}}\\ &\leq\sum_{\ell=0}^{k}\left[\frac{k!}{\ell!(k-\ell)!}\frac{L_{\mathfrak{y}}^{\ell}L_{\mathfrak{z}}^{k-\ell}}{\lambda^{k/2}}\left(\frac{T^{\ell}}{\ell!}\lambda e^{\lambda T}{\mathbb{E}}\bigl{[}\lVert\xi\rVert^{2}\bigr{]}+\int_{0}^{T}\frac{t^{\ell}e^{\lambda t}}{\ell!}{\mathbb{E}}\!\left[\bigl{\lVert}f(t,Y_{t}^{\infty},Z_{t}^{\infty})\bigr{\rVert}^{2}\right]\!dt\right)^{\!\nicefrac{{1}}{{2}}}\right]\\ &\leq\left[\sum_{\ell=0}^{k}\frac{k!}{\ell!(k-\ell)!}\frac{L_{\mathfrak{y}}^{\ell}L_{\mathfrak{z}}^{k-\ell}}{\lambda^{k/2}}\frac{T^{\ell/2}e^{\lambda T/2}}{\sqrt{\ell!}}\right]\left(\lambda{\mathbb{E}}\bigl{[}\lVert\xi\rVert^{2}\bigr{]}+\int_{0}^{T}{\mathbb{E}}\!\left[\bigl{\lVert}f(t,Y_{t}^{\infty},Z_{t}^{\infty})\bigr{\rVert}^{2}\right]\!dt\right)^{\!\nicefrac{{1}}{{2}}}.\end{split} (67)

This and (63) prove for all k∈ℕk\in{\mathbbm{N}}, λ∈(0,∞)\lambda\in(0,\infty) that

𝔼​[supt∈[0,T](eλ​t​∥Ytk−Yt∞∥2)+∫0Teλ​t​∥Ztk−Zt∞∥𝖥2​𝑑t]≤35λ​𝔼​[∫0Teλ​t​∥f​(t,Ytk−1,Ztk−1)−f​(t,Yt∞,Zt∞)∥2​𝑑t]≤35λ​[∑ℓ=0kk!ℓ!​(k−ℓ)!​L𝔶ℓ​L𝔷k−ℓλk/2​Tℓ/2​eλ​T/2ℓ!]2​(λ​𝔼​[∥ξ∥2]+∫0T𝔼​[∥f​(t,Yt∞,Zt∞)∥2]​𝑑t).\displaystyle\begin{split}&{\mathbb{E}}\!\left[\sup_{t\in[0,T]}\left(e^{\lambda t}\left\lVert Y^{k}_{t}-Y_{t}^{\infty}\right\rVert^{2}\right)+\int_{0}^{T}e^{\lambda t}\left\lVert Z^{k}_{t}-Z_{t}^{\infty}\right\rVert^{2}_{\mathsf{F}}\,dt\right]\\ &\leq\frac{35}{\lambda}{\mathbb{E}}\!\left[\int_{0}^{T}e^{\lambda t}\left\lVert f(t,Y^{k-1}_{t},Z^{k-1}_{t})-f(t,Y_{t}^{\infty},Z_{t}^{\infty})\right\rVert^{2}\,dt\right]\\ &\leq\frac{35}{\lambda}\left[\sum_{\ell=0}^{k}\frac{k!}{\ell!(k-\ell)!}\frac{L_{\mathfrak{y}}^{\ell}L_{\mathfrak{z}}^{k-\ell}}{\lambda^{k/2}}\frac{T^{\ell/2}e^{\lambda T/2}}{\sqrt{\ell!}}\right]^{2}\left(\lambda{\mathbb{E}}\bigl{[}\lVert\xi\rVert^{2}\bigr{]}+\int_{0}^{T}{\mathbb{E}}\!\left[\bigl{\lVert}f(t,Y_{t}^{\infty},Z_{t}^{\infty})\bigr{\rVert}^{2}\right]\!dt\right).\end{split} (68)

Furthermore, observe for all k∈ℕk\in{\mathbbm{N}} that

[∑ℓ=0kk!ℓ!​(k−ℓ)!​L𝔶ℓ​L𝔷k−ℓλk/2​Tℓ/2​eλ​T/2ℓ!]2|λ=kT=(T​ek)k​[∑ℓ=0kk!​L𝔶ℓ​L𝔷k−ℓ​Tℓ/2ℓ!​(k−ℓ)!​ℓ!]2.\displaystyle\left[\sum_{\ell=0}^{k}\frac{k!}{\ell!(k-\ell)!}\frac{L_{\mathfrak{y}}^{\ell}L_{\mathfrak{z}}^{k-\ell}}{\lambda^{k/2}}\frac{T^{\ell/2}e^{\lambda T/2}}{\sqrt{\ell!}}\right]^{2}\Biggr{|}_{\lambda=\frac{k}{T}}=\left(\frac{Te}{k}\right)^{k}\left[\sum_{\ell=0}^{k}\frac{k!L_{\mathfrak{y}}^{\ell}L_{\mathfrak{z}}^{k-\ell}T^{\ell/2}}{\ell!(k-\ell)!\sqrt{\ell!}}\right]^{2}. (69)

This and (68) yield for all k∈ℕk\in{\mathbbm{N}} that

𝔼​[supt∈[0,T](∥Ytk−Yt∞∥2)+∫0T∥Ztk−Zt∞∥𝖥2​𝑑t]≤35​Tk​(T​ek)k​[∑ℓ=0kk!​L𝔶ℓ​L𝔷k−ℓ​Tℓ/2ℓ!​(k−ℓ)!​ℓ!​​]2​(kT​𝔼​[∥ξ∥2]+∫0T𝔼​[∥f​(t,Yt∞,Zt∞)∥2]​𝑑t).\displaystyle\begin{split}&{\mathbb{E}}\!\left[\sup_{t\in[0,T]}\left(\left\lVert Y^{k}_{t}-Y_{t}^{\infty}\right\rVert^{2}\right)+\int_{0}^{T}\left\lVert Z^{k}_{t}-Z_{t}^{\infty}\right\rVert^{2}_{\mathsf{F}}\,dt\right]\\ &\leq\frac{35T}{k}\left(\frac{Te}{k}\right)^{k}\left[\sum_{\ell=0}^{k}\frac{k!L_{\mathfrak{y}}^{\ell}L_{\mathfrak{z}}^{k-\ell}T^{\ell/2}}{\ell!(k-\ell)!\sqrt{\ell!}}\frac{}{}\frac{}{}\right]^{2}\left(\frac{k}{T}{\mathbb{E}}\bigl{[}\lVert\xi\rVert^{2}\bigr{]}+\int_{0}^{T}{\mathbb{E}}\!\left[\bigl{\lVert}f(t,Y_{t}^{\infty},Z_{t}^{\infty})\bigr{\rVert}^{2}\right]\!dt\right).\end{split} (70)

Next note that (59) ensures that

(∫0T𝔼​[∥f​(t,Yt∞,Zt∞)∥2]​𝑑t)1/2\displaystyle\left(\int_{0}^{T}{\mathbb{E}}\bigl{[}\lVert f(t,Y_{t}^{\infty},Z_{t}^{\infty})\rVert^{2}\bigr{]}dt\right)^{\nicefrac{{1}}{{2}}} (71)
≤(∫0T𝔼​[∥f​(t,0,0)∥2]​𝑑t)1/2+L𝔶​(∫0T𝔼​[∥Yt∞∥2]​𝑑t)1/2+L𝔷​(∫0T𝔼​[∥Zt∞∥𝖥2]​𝑑t)1/2<∞.\displaystyle\leq\left(\int_{0}^{T}{\mathbb{E}}\bigl{[}\lVert f(t,0,0)\rVert^{2}\bigr{]}dt\right)^{\nicefrac{{1}}{{2}}}+L_{\mathfrak{y}}\left(\int_{0}^{T}{\mathbb{E}}\bigl{[}\lVert Y_{t}^{\infty}\rVert^{2}\bigr{]}dt\right)^{\nicefrac{{1}}{{2}}}+L_{\mathfrak{z}}\left(\int_{0}^{T}{\mathbb{E}}\bigl{[}\lVert Z_{t}^{\infty}\rVert^{2}_{\mathsf{F}}\bigr{]}dt\right)^{\nicefrac{{1}}{{2}}}<\infty.

This, the fact that 𝔼​[∥ξ∥2]<∞{\mathbb{E}}[\lVert\xi\rVert^{2}]<\infty, and (70) complete the proof of Proposition 4.1. ∎

Remark 4.2.

Assume the setting of Proposition 4.1. Then it holds for all k∈ℕk\in{\mathbbm{N}} that

35​(T​ek)k​[∑ℓ=0kk!​L𝔶ℓ​L𝔷k−ℓ​Tℓ/2ℓ!​(k−ℓ)!​ℓ!]2≤35​(max⁡{T2,1}​e​max⁡{L𝔶2,L𝔷2}k)k​[∑ℓ=0kk!(k−ℓ)!​ℓ!]2=35​(4​max⁡{T2,1}​e​max⁡{L𝔶2,L𝔷2}k)k≤35​(4​max⁡{T2,1}​e​max⁡{L𝔶2,L𝔷2})kk!.\displaystyle\begin{split}&35\left(\frac{Te}{k}\right)^{k}\left[\sum_{\ell=0}^{k}\frac{k!L_{\mathfrak{y}}^{\ell}L_{\mathfrak{z}}^{k-\ell}T^{\ell/2}}{\ell!(k-\ell)!\sqrt{\ell!}}\right]^{2}\leq 35\left(\frac{\max\{T^{2},1\}e\max\{L_{\mathfrak{y}}^{2},L_{\mathfrak{z}}^{2}\}}{k}\right)^{k}\left[\sum_{\ell=0}^{k}\frac{k!}{(k-\ell)!\ell!}\right]^{2}\\ &=35\left(\frac{4\max\{T^{2},1\}e\max\{L_{\mathfrak{y}}^{2},L_{\mathfrak{z}}^{2}\}}{k}\right)^{k}\leq 35\frac{\left(4\max\{T^{2},1\}e\max\{L_{\mathfrak{y}}^{2},L_{\mathfrak{z}}^{2}\}\right)^{k}}{k!}.\end{split} (72)
Remark 4.3.

Assume the setting of Proposition 4.1 and assume that L𝔷=0L_{\mathfrak{z}}=0. Then it holds for all k∈ℕk\in{\mathbbm{N}} that

35​(T​ek)k​[∑ℓ=0kk!​L𝔶ℓ​L𝔷k−ℓ​Tℓ/2ℓ!​(k−ℓ)!​ℓ!]2=35​(T​ek)k​[L𝔶k​Tk/2k!]2≤35​(T2​e​L𝔶2)k(k!)2.\displaystyle\begin{split}&35\left(\frac{Te}{k}\right)^{k}\left[\sum_{\ell=0}^{k}\frac{k!L_{\mathfrak{y}}^{\ell}L_{\mathfrak{z}}^{k-\ell}T^{\ell/2}}{\ell!(k-\ell)!\sqrt{\ell!}}\right]^{2}=35\left(\frac{Te}{k}\right)^{k}\left[\frac{L_{\mathfrak{y}}^{k}T^{k/2}}{\sqrt{k!}}\right]^{2}\leq 35\frac{(T^{2}eL_{\mathfrak{y}}^{2})^{k}}{(k!)^{2}}.\end{split} (73)

Acknowledgements

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the research grant HU1889/7-1.

References

  • [1] Beck, C., Hornung, F., Hutzenthaler, M., Jentzen, A., and Kruse, T. Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations. Journal of Numerical Mathematics 28, 4 (2020), 197–222.
  • [2] Bender, C., and Denk, R. A forward scheme for backward SDEs. Stochastic Process. Appl. 117, 12 (2007), 1793–1812.
  • [3] Bender, C., and Zhang, J. Time discretization and Markovian iteration for coupled FBSDEs. The Annals of Applied Probability 18, 1 (2008), 143–177.
  • [4] Briand, P., and Labart, C. Simulation of BSDEs by Wiener chaos expansion. Ann. Appl. Probab. 24, 3 (2014), 1129–1171.
  • [5] Da Prato, G., and Zabczyk, J. Stochastic equations in infinite dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.
  • [6] E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. arXiv:1607.03295 (2016). Accepted in Partial Differential Equations and Applications.
  • [7] E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. Journal of Scientific Computing 79, 3 (2019), 1534–1571.
  • [8] El Karoui, N., Peng, S., and Quenez, M. C. Backward stochastic differential equations in finance. Math. Finance 7, 1 (1997), 1–71.
  • [9] Geiss, C., and Labart, C. Simulation of BSDEs with jumps by Wiener chaos expansion. Stochastic Process. Appl. 126, 7 (2016), 2123–2162.
  • [10] Gobet, E., and Labart, C. Solving BSDE with adaptive control variate. SIAM J. Numer. Anal. 48, 1 (2010), 257–277.
  • [11] Hutzenthaler, M., Jentzen, A., and Kruse, T. Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities. arXiv:1912.02571 (2019). Accepted in Foundations of Computational Mathematics.
  • [12] Hutzenthaler, M., Jentzen, A., Kruse, T., and Nguyen, T. A. Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities. arXiv:2009.02484 (2020).
  • [13] Hutzenthaler, M., Jentzen, A., Kruse, T., Nguyen, T. A., and von Wurstemberger, P. Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations. Proceedings of the Royal Society A 476, 2244 (2020), 20190630.
  • [14] Hutzenthaler, M., and Kruse, T. Multilevel Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities. SIAM Journal on Numerical Analysis 58, 2 (2020), 929–961.
  • [15] Labart, C., and Lelong, J. A parallel algorithm for solving BSDEs. Monte Carlo Methods Appl. 19, 1 (2013), 11–39.
  • [16] Øksendal, B. Stochastic differential equations. Universitext. Springer-Verlag, Berlin, 1985. An introduction with applications.
  • [17] Pardoux, É., and Peng, S. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14, 1 (1990), 55–61.
  • [18] Pardoux, E., and Răşcanu, A. Stochastic differential equations, Backward SDEs, Partial differential equations. Springer, 2016.
  • [19] Pham, H. Continuous-time stochastic control and optimization with financial applications, vol. 61. Springer Science & Business Media, 2009.
  • [20] Ren, Y.-F. On the Burkholder–Davis–Gundy inequalities for continuous martingales. Statistics & probability letters 78, 17 (2008), 3034–3039.
  • [21] Zhang, J. Backward Stochastic Differential Equations. From Linear to Fully Nonlinear Theory. Springer, 2017.