A monotone numerical integration method for mean-variance portfolio optimization under jump-diffusion models

In the parametric approach, a parametric stochastic model is postulated, e.g. diffusion dynamics, and then is calibrated to market-observed data.²²2Recently, data-driven (i.e. non-parametric) methods have been proposed for portfolio optimization under different optimality criteria, including mean-variance [39, 10, 51]. Nonetheless, monotonicity of NN-based methods has not been established. A key concern about, and perhaps also a criticism against, MV portfolio optimization in a parametric setting is its potential lack of robustness to model misspecification error. This criticism originated from the fact that, in single-period settings, MV portfolio optimization can provide notoriously unstable asset allocation strategies arising from small changes in the underlying asset parameters ([50, 61, 54, 8]). Nonetheless, in the case of multi-period MV optimization, research findings indicate that, when the risky asset dynamics are allowed to follow pure-diffusion dynamics (e.g. GBM) or any of the standard finite-activity jump-diffusion models commonly encountered in financial settings, such as those considered in this work, the pre-commitment and time-consistent MV outcomes of terminal wealth are generally very robust to model misspecification errors [68].

It is well-documented in the finance literature that jumps are often present in the price processes of risky assets (see, for example, [15, 58]). In addition, findings in previous research work on MV portfolio optimization (pre-commitment and time-consistency strategies) also indicate that (i) jumps in the price processes of risky assets, such as Merton model [49] and the Kou model [36], and (ii) realistic investment constraints, such as no-bankruptcy or leverage, have substantial impact on efficient frontiers and optimal investment strategies of MV portfolio optimization [19, 65]. Furthermore, the results of [46] show that the effects of stochastic volatility, with realistic mean-reverting dynamics, are not important for long-term investors with time horizons greater than 10 years.

Furthermore, for multi-period MV optimization, it is documented in the literature that the composition of the risky asset basket remains relatively stable over time, which suggests that the primary question remains the overall risky asset basket vs. the risk-free asset composition of the portfolio, instead of the exact composition of the risky asset basket. See the available analytical solutions for multi-asset time-consistent MV problems (see, for example, [81]) as well as pre-commitment MV problems (see for example [37]). Therefore, it is reasonable to consider a well-diversified index, instead of a single stock or a basket of stocks, as common in the MV literature [66, 69, 68, 21, 67, 65]. This is the modeling approach adopted in this work, resulting in a low dimensional multi-period MV optimization problem.

In general, many industry practitioners find implementing monotone finite difference methods for jump-diffusion models to be complex and time-consuming, particularly when striving to utilize central differencing as much as possible, as proposed in [73]. As well-noted in the literature (e.g. [56, 59]), many seemingly reasonable finite difference discretization schemes can yield incorrect solutions. In addition, while the concept of (strict) monotonicity in numerical schemes is directly tied to the discrete no-arbitrage principle, making it easy to comprehend, weak monotonicity is less clear, which further hinders its application in practice. Moreover, the convergence analysis of weakly monotone schemes is often complex, potentially introducing additional obstacles to their practical application.

This paper aims to fill the aforementioned research gap through the development of an efficient, easy-to-implement and monotone numerical integration method for MV portfolio optimization under a realistic setting. This setting involves the simultaneous application of different types of investment constraints and jump-diffusion dynamics for the price processes of risky assets. In our method, the guarantee of the key monotonicity property comes with a trade-off: a modest level of analytical tractability is required for the jump sizes. Essentially, this stipulates that the associated jump-diffusion model should provide a closed-form expression for European vanilla options. This condition, however, conveniently accommodates a number of widely used models in finance, marking it as a relatively mild requirement in practice. To demonstrate this, we presents results for two popular distributions for jump sizes in finance: the normal distribution as put forth by [49], and the asymmetric double-exponential distribution as proposed in [36]. Although we focus on the pre-commitment strategy case, the proposed method can be extended to time-consistent MV optimization in a straightforward manner by enforcing a time-consistency constraint as in [65].

The main contributions of the paper are as follows.

• (i)

  We present a recursive and localized formulation of the pre-commitment MV portfolio optimization under a realistic context that involves the simultaneous application of different types of investment constraints and jump-diffusion dynamics [36, 49]. Over each rebalancing interval, the key component of the formulation of MV portfolio optimization is a convolution integral involving a conditional density of the logarithm of amount invested in the risky asset.

• (ii)

  Through a known closed-form expression of the Fourier transform of the underlying transition density, we derive an infinite series representation for this density in which all the terms of the series are non-negative and readily computable explicitly. Therefore, the convolution integral can be approximated in a straightforward manner using a monotone integration scheme via a composite quadrature rule. A significant benefit of the proposed method compared to existing monotone finite difference methods is that it offers a computational complexity that is an order of magnitude lower. Utilizing the Toeplitz matrix structure, we propose an efficient implementation of the proposed monotone integration scheme via FFTs.

• (iii)

  We mathematically demonstrate that the proposed monotone scheme is also $\ell_{\infty}$-stable and pointwise consistent with the convolution integral formulation. We rigourously prove the pointwise convergence of the scheme as the discretization parameter approach zero. As the rebalancing time interval approaches zero, we intuitively demonstrate that the proposed scheme converges to a continuously observed impulse control formulation for MV optimization in the form of a Hamilton-Jacobi-Bellman Quasi-Variational Inequality.

• (iv)

  All numerical experiments are conducted using model parameters calibrated to inflation-adjusted, long-term US market data (89 years), enabling realistic conclusions to be drawn from the results. Numerical experiments demonstrate an agreement with benchmark results obtained by FD method and Monte Carlo simulation of [19].

Although we focus specifically on monotone integration methods for multi-period MV portfolio optimization, our comprehensive and systematic approach could serve as numerical and convergence analysis framework for the development of similar monotone integration methods for other multi-period or continuously observed control problems in finance.

In Section 2, we describe the underlying dynamics and a multi-period rebalancing framework for MV portfolio optimization. A localization of the pre-commitment MV portfolio optimization in the form of an convolution integral together with appropriate boundary conditions are presented in Section 3. Also therein, we present an infinite series representation of the transition density. A simple and easy-to-implement monotone numerical integration method via a composite quadrature rule is described in Section 4. In Section 5, we mathematically establish pointwise convergence the proposed integration method. Section 6 explore possible convergence between the proposed scheme and a Hamilton-Jacobi-Bellman Quasi-Variational Inequality arising from continuously observed impulse control formulation for MV optimization. Numerical results are given in Section 4. Section 8 concludes the paper and outlines possible future work.

Define $\mathcal{T}_{M}$ as the set of $M$ predetermined, equally spaced rebalancing times in $\left[0,T\right]$,

We adopt the convention that $t_{M}=T$ and the portfolio is not rebalanced at the end of the investment horizon $t_{M}=T$. The evolution of the portfolio over a rebalancing interval $[t_{m-1},t_{m}]$, $t_{m-1}\in\mathcal{T}_{M}$, can be viewed as consisting of three steps as follows. Over $[t_{m-1},t_{m-1}^{+}]$, $(S_{t},B_{t})$, change according to some rebalancing strategy (i.e. an impulse control). Over the time period $[t_{m-1}^{+},t_{m}^{-}]$, there is no intervention by the investor according to some control (investment strategy), and therefore $(S_{t},B_{t})$ are uncontrolled, and are assumed to follow some dynamics for all $t\in[t_{m-1}^{+},t_{m}^{-}]$. Over $[t_{m}^{-},t_{m}]$, the settlement (payment or receipt) of interest due for the time period $[t_{m-1},t_{m}]$. In the following, we first discuss stochastic modeling for $(S_{t},B_{t})$ over $[t_{m-1}^{+},t_{m}^{-}]$, then describe settlement of interest and modelling of rebalancing strategies using impulse controls.

Over the time period $[t_{m-1}^{+},t_{m}^{-}]$, in the absence of control (investor’s intervention according to some control strategy), the amounts in the risk-free and risky assets are assumed to have the following dynamics:

Here, as noted earlier, $r_{b}$ and $r_{\iota}$ denote the positive, continuously compounded rates at which the investor can respectively borrow funds or earn on cash deposits (with $r_{b}>r_{\iota}$), while $\mathbb{I}_{\left[A\right]}$ denotes the indicator function of the event $A$; $\{W_{t}\}_{t\in[0,T]}$ is a standard Wiener process, and $\mu$ and $\sigma$ are the real world drift rate and the instantaneous volatility, respectively. The jump term $\sum_{\ell=1}^{\pi(t)}\xi_{\ell}$ is a compound Poisson process. Specifically, $\{\pi(t)\}_{0\leq t\leq T}$ is a Poisson process with a constant finite jump intensity $\lambda\geq 0$; and, with $\xi$ being a random variable representing the jump size, $\{\xi_{\ell}\}_{\ell=1}^{\infty}$ are independent and identically distributed (i.i.d.) random variables having the same same distribution as the random variable $\xi$. In the dynamics (2.2), $\kappa=\mathbb{E}\left[e^{\xi}-1\right]$. Here, $\mathbb{E}[\cdot]$ is the expectation operator taken under a suitable measure. The Poisson process $\{\pi(t)\}_{0\leq t\leq T}$, the sequence of random variables $\{\xi_{\ell}\}_{\ell=1}^{\infty}$, and the Wiener process and $\{W_{t}\}_{0\leq t\leq T}$ are mutually independent.

We consider two distributions for the random variable $\xi$, namely the normal distribution [49] and the asymmetric double-exponential distribution [36]. To this end, let $p(y)$ be the probability density function (pdf) of $\xi$. In the former case, $\xi\sim\text{Normal}\left(\widetilde{\mu},\widetilde{\sigma}^{2}\right)$, so that its pdf is given by

Also, in this case, $\mathbb{E}[e^{\xi}]=\exp(\widetilde{\mu}+\widetilde{\sigma}^{2}/2)$, and hence $\kappa=\mathbb{E}\left[e^{\xi}-1\right]$ can be computed accordingly. In the latter case, we consider an asymmetric double-exponential distribution for $\xi$. Specifically, we consider $\xi\sim\text{Asym-Double-Exponential}(q_{1},\eta_{1},\eta_{2})$, $\left(q_{1}\in(0,1),\,\eta_{1}>1,\,{\color[rgb]{0,0,0}{\eta_{2}>0}}\right)$ so that its pdf is given by

Here $q_{1}$ and $q_{2}=1-q_{1}$ respectively are the probabilities of upward and downward jump sizes. In this case, $\mathbb{E}[e^{\xi}]=\frac{q_{1}\eta_{1}}{\eta_{1}-1}+\frac{q_{2}\eta_{2}}{\eta_{2}+1}$, so $\kappa=\mathbb{E}\left[e^{\xi}-1\right]$ can be computed accordingly.

Discrete portfolio rebalancing is modelled using the discrete impulse control formulation as discussed in for example [19, 65, 66], which we now briefly summarize below. Let $c_{m}$ denote the impulse applied at rebalancing time $t_{m}\in\mathcal{T}_{M}$, which corresponds to the amount invested in the risk-free asset according to the investor’s intervention at time $t_{m}$, and let $\mathcal{Z}$ denote the set of admissible impulse values, i.e. $c_{m}\in\mathcal{Z}$ for all $t_{m}\in\mathcal{T}_{M}$.

Let $X_{t}=\left(S_{t},B_{t}\right),\,t\in\left[0,T\right]$ be the multi-dimensional underlying process, and $x=(s,b)$ denote the state of the system. Suppose that at time-$t_{m}$, the state of the system is $x=\left(s,b\right)=\left(S\left(t_{m}\right),B\left(t_{m}\right)\right)$ for some $t_{m}\in\mathcal{T}_{M}$. We denote by $(S_{t_{m}^{+}},B_{t_{m}^{+}})\equiv\left(s^{+}(s,b,c_{m}),b^{+}(s,b,c_{m})\right)$ the state of the system immediately after the application of the impulse $c_{m}$ at time $t_{m}$, where

Here, $\delta\geq 0$ is a fixed cost⁴⁴4It is straightforward to include a proportional cost into (2.5) as in [19]. However, to focus on the main advantages of the proposed method, we do not consider a proportional cost in this work.; since $\log(\cdot)$ is undefined if $e^{s}+b-c_{m}-\delta\leq 0$, the amount $S_{t_{m}^{+}}$ becomes $\ln\left(\max(e^{s}+b-c_{m}-\delta,\,e^{s_{\scalebox{0.6}{-$\infty$}}})\right)$ for a finite $s_{\scalebox{0.6}{-$\infty$}}\ll 0$.

Associated with the fixed set of rebalancing times $\mathcal{T}_{M}$, defined in (2.1), an impulse control $\mathcal{C}$ will be written as the set of impulse values

and we define $\mathcal{C}_{m}$ to be the subset of the control $\mathcal{C}$ applicable to the set of times $\left\{t_{m},\ldots,t_{M-1}\right\}$,

In a discrete setting, the amount invested in the risk-free asset changes only at rebalancing date. Specifically, over each time interval $\left[t_{m-1},t_{m}\right],\,m=1,\ldots,M$, we suppose the amount invested in the risk-free asset at time $t_{m-1}^{+}$ after rebalancing being $B_{t_{m}^{+}}=b$. For test function $f(S_{t},B_{t},t)$ with both $S_{t}$ and $B_{t}$ varying, we model the change in $f(S_{t},B_{t},t)$ with $(S_{t},B_{t}=b)$ for $t\in[t_{m-1}^{+},t_{m}^{-}]$. Then, the amount in the risk-free asset would jump to $be^{R(b)\Delta t}$ at time $t_{m}$, reflecting the settlement (payment or receipt) of interest due for the time interval $[t_{m-1},t_{m}]$, $m=1,\ldots,M$. Here, we note that, although there is no rebalancing at time $t_{M}=T$, there is still settlement of interest for the interval $[t_{M-1},t_{M}]$.

With the time-$t$ state of the system being $(s,b)$, to include transaction cost, we define the liquidation value $W_{\text{liq}}(t)\equiv W_{\text{liq}}(s,b)$ to be

We strictly enforce two realistic investment constraints on the joint values of $S$ and $B$, namely a solvency condition and a maximum leverage condition. The solvency condition takes the following form: when $W_{\text{liq}}(s,b)\leq 0$, we require that the position in the risky asset be liquidated, the total remaining wealth be placed in the risk-free asset, and the ceasing of all subsequent trading activities. Specifically, assume that the system is in the state $x=\left(s,b\right)\in\Omega^{\infty}$ at time $t_{m}$, where $t_{m}\in\mathcal{T}_{M}$ and

We define a solvency region $\mathcal{N}$ and an insolvency or bankruptcy region $\mathcal{B}$ as follows

The solvency constraint can then be stated as

where, as noted above, $s_{\scalebox{0.6}{-$\infty$}}\ll 0$ and is finite. This effectively means that the investment in the risky asset has to be liquidated, the total wealth is to be placed in the risk-free asset, and all subsequent trading activities much cease.

The maximum leverage constraint specifies that the leverage ratio after rebalancing at $t_{m}$, where $t_{m}\in\mathcal{T}_{M}$, is stipulated by (2.5) must satisfy

for some positive constant $q_{\max}$ typically in the range $\left[1.0,2.0\right]$. Given above the solvency constraint and the maximum leverage constraint, the set of admissible impulse values, namely the set $\mathcal{Z}$ is therefore defined as follows

Based on the definition (2.7), the set of admissible impulse controls is given by

Dynamic programming cannot be applied directly to (3.1), since no smoothing property of conditional expectation for variance. The technique of [38, 84] embeds (3.1) in a new optimisation problem, often referred to as the embedding problem, which is amenable to dynamic programming techniques. We follow the example of [13, 22] in defining the PCMV optimization problem as the associated embedding MV problem⁵⁵5For a discussion of the elimination of spurious optimization results when using the embedding formulation, see [23].. Specifically, with $\gamma\in\mathbb{R}$ being the embedding parameter, we define the value function $v\left(s,b,t_{m}\right)$, $m=M-1,\ldots,0$ as follows

where $W_{T}$ is given in (2.8), subject to dynamics (2.2) between rebalancing times. We denote by $\mathcal{C}_{m}^{*}$ the optimal control for the problem $PCMV_{\Delta t}(t_{m};\gamma)$, where

For an impulse value $c\in\mathcal{Z}$, we define the intervention operator $\mathcal{M}(\cdot)$ applied at $t_{m}\in\mathcal{T}_{M}$ as follows

By dynamic programming arguments [55, 57], for a fixed embedding parameter $\gamma\in\mathbb{R}$, and $(s,b)\in\Omega^{\infty}$, the recursive relationship for the value function $v(s,b,t_{m})$ in (3.2) is given by

Here, in (3.5b), the intervention operator $\mathcal{M}(\cdot)$ is given by (3.4), with the $\min\{\cdot,\cdot\}$ operator reflecting the optimal choice between no-rebalancing and rebalancing (which is subject to a fixed cost $\delta$ ); (3.5c) reflects the settlement (payment or receipt) of interest due for the time interval $[t_{m-1},t_{m}]$, $m=1,\ldots,M$. In the integral (3.5d) the functions $g(s,s^{\prime};\Delta t)$ denotes the probability density of $s$, the log of the amount invested in the risky asset at a future time ($t_{m}^{-}$), and the information $s^{\prime}$ at the current time ($t_{m-1}^{+}$), given $\Delta t=t_{m}-t_{m-1}$. Also, we note that the fact that amount invested in the risk-free asset does not change in the in the interval $[t_{m-1}^{+},t_{m}^{-}]$ is reflected in (3.5d) since this amount is kept constant ($=b$) on both sides of (3.5d).

It can be shown that $g(s,s^{\prime};\Delta t)$ has the form $g(s-s^{\prime};\Delta t)$, and therefore, in (3.5d), the integral takes the form of the convolution of $g(\cdot)$ and $v(\cdot,t_{m}^{-})$. That is, (3.5d) becomes

Although a closed-form expression for $g(s;\Delta t)$ is not known to exist, its Fourier transform, denoted by $G(\cdot;\Delta t)$, is known in closed-form. Specifically, we recall the Fourier transform pair

A closed-form expression for $G\left(\eta;\Delta t\right)$ is given by

Here, $\Gamma\left(\eta\right)=\int_{-\infty}^{\infty}p(y)~{}e^{i\eta y}~{}dy$, where $p(y)$ is the probability density function of $\ln\left(\xi\right)$ with $\xi$ being the random variable representing the jump multiplier.

The proposed monotone integration method depends on an infinite series representation of the probability density function $g(\cdot)$, which is presented in Lemma 3.1.

A proof of of Lemma 3.1 is given in Appendix A.

The infinite series representation in (3.9) cannot be applied directly for computation as the $k$-th term of the series involves a multiple integral with $\left(\prod_{\ell=1}^{k}p(y_{\ell})\right)$, where $p(y)$ denotes the probability density of $\xi$. To feasibly evaluate this multiple integral analytically, as noted earlier, our method requires a quite modest level of analytical tractability for the random variable $\xi$. To put it more precisely, we need the associated jump-diffusion model of the form $\frac{\mathrm{d}X_{t}}{X_{t}^{-}}=(\mu-\lambda\kappa)\mathrm{d}t+\sigma\,\mathrm{d}W_{t}\ +\mathrm{d}\left(\sum_{\ell=1}^{\pi_{t}}(\xi_{\ell}-1)\right)$ to yield a closed-form expression for European vanilla options. Arguments in [17][Section 3.4.1] can be invoked to demonstrate this relationship. Thus, the requirement of analytical tractability for the random variable $\xi$ is not overly stringent, thereby broadening the applicability of our approach in practice.

Next, we demonstrate the applicability of our approach for two popular distributions for $\xi$ in finance: when $\xi$ follow a normal distribution [49] and an asymmetric double-exponential distribution [36].

Here, NorCDF denotes CDF of standard normal distribution $\mathcal{N}(0,1)$. For brevity, we omit a straightforward proof for the log-normal case (3.10) using Equation (A.3). A proof for the log-double exponential case (3.1) is given in Appendix C. For this case, we note that function $Hh_{\ell}(\cdot)$ can be evaluated very efficiently using the standard normal density function and standard normal distribution function via the three-term recursion [1]

In the subsequent section, we present a definition of the localized problem to be solved numerically.

The MV formulation (3.5) is posed on an infinite domain. For the problem statement and convergence analysis of numerical schemes, we define a localized MV portfolio optimsation formulation. To this end, with $s_{\min}^{\dagger}<s_{\min}<0<s_{\max}<s_{\max}^{\dagger}$, $-b_{\max}<0<b_{\max}$, where $|s_{\min}^{\dagger}|$, $|s_{\min}|$, $s_{\max}$, $s_{\max}^{\dagger}$, and $b_{\max}$ are sufficiently large, we define the following spatial sub-domains:

We emphasize that we do not actually solve the MV optimization problem in $\Omega_{b_{\max}}$. However, we may use an approximate value to the solution in $\Omega_{b_{\max}}$, obtained by means of extrapolation of the computed solution in $\Omega_{\scalebox{0.7}{\text{in}}}$, to provide any information required by the MV optimization problem in $\Omega_{\scalebox{0.7}{\text{in}}}$. We also define the following sub-domains:

The solutions within the sub-domains $\Omega_{s_{\min}^{\dagger}}$ and $\Omega_{s_{\max}^{\dagger}}$ are not required for our purposes. These sub-domains are introduced to ensure the well-defined computation of the conditional probability density function $g(\cdot)$ in (3.6) for the convolution integral (3.6) in the MV optimization problem within $\Omega_{\scalebox{0.7}{\text{in}}}$. To simplify our discussion, we will adopt a zero-padding convention going forward. This convention assumes that the value functions within these sub-domains are zero for all time $t$, and we will exclude these sub-domains from further discussions.

Due to rebalancing, the intervention operator $\mathcal{M}(\cdot)$ for $\Omega_{\scalebox{0.7}{\text{in}}}$, defined in (3.4), may require evaluating a candidate value at a point having $s^{+}=\ln(\max(W_{\text{liq}}(s,b)-c,\,e^{s_{\scalebox{0.6}{-$\infty$}}}))$, and $s^{+}$ could be outside $[s_{\min}^{\dagger},s_{\max}^{\dagger}]$, if $s_{\scalebox{0.6}{-$\infty$}}<s^{\dagger}_{\min}$. Therefore, with $|s^{\dagger}_{\min}|$ selected sufficiently large, we assume $s_{\scalebox{0.6}{-$\infty$}}=s^{\dagger}_{\min}$.

We now present equations for spatial sub-domains defined in (3.3). We note that boundary conditions for $s\to-\infty$ and $s\to\infty$ are obtained by relevant asymptotic forms $e^{s}\to 0$ and $e^{s}\to\infty$, respectively, similar to [19]. This is detailed below.

• •

  For $(s,b,T)\in\Omega\times\{T\}$, we apply the terminal condition (3.5a)

  $$v(s,b,T)={\color[rgb]{0,0,0}{\left(W_{\text{liq}}(s,b)-\frac{\gamma}{2}\right)^{2}}}.$$

  (3.16)

• •

  For $(s,b,t_{m})\in\Omega\times\mathcal{T}_{M}$, $m=M-1,\ldots,0$, the intervention result (3.5b) is given by

  $$v\left(s,b,t_{m}\right)~{}=~{}\min\left\{v\left(s,b,t_{m}^{+}\right),\inf_{c\in\mathcal{Z}}\mathcal{M}(c)~{}v\left(s,b,t_{m}^{+}\right)\right\},$$

  (3.17)

  where the intervention $\mathcal{M}(\cdot)$ is defined in (3.4).

• •

  For $(s,b,t_{m}^{-})\in\Omega\times\{t_{m}^{-}\}$, $m=M,\ldots,1$, settlement of interest (3.5c) is enforced by

  $$v\left(s,b,t_{m}^{-}\right)~{}=~{}v\left(s,be^{R(b)\Delta t},t_{m}\right),\quad m=M,\ldots,1,\quad\text{ and $v\left(s,\cdot,t_{m}\right)$ is given in \eqref{eq:recursive_b_comp_dom}. }$$

  (3.18)

• •

  For $(s,b,t_{m}^{+})\in\Omega_{b_{\max}}\times\{t_{m}^{+}\}$, where $m=M,\ldots,1$, we impose the boundary condition

  $${\color[rgb]{0,0,0}{v\left(s,b,t_{m}^{+}\right)=\left(\frac{b}{b_{\max}}\right)^{2}v\left(s,\operatorname{sgn}(b)b_{\max},t_{m}^{+}\right).}}$$

  (3.19)

• •

  For $(s,b,t_{m-1}^{+})\in\Omega_{s_{\min}}\times\{t_{m-1}^{+}\}$, where $t_{m-1}\in\mathcal{T}_{M}$, from (3.16), we assume that $v(s,b,t)\approx A_{0}(t)b^{2}$ for some unknown function $A_{0}(t)$, which mimics asymptotic behaviour of the value function as $s\to-\infty$ (or equivalently, $e^{z}\to 0$). Substituting this asymptotic form into the integral (3.5d) gives the boundary condition

  $\displaystyle v(s,b,t_{m-1}^{+})=A_{0}(t_{m}^{-})b^{2}\int_{-\infty}^{\infty}g(s-s^{\prime};\Delta t)~{}\mathrm{d}s^{\prime}=v(s,b,t_{m}^{-}),$

  (3.20)

  where $v(s,b,t_{m}^{-})$ is given by (3.18).

• •

  For $(s,b,t_{m-1}^{+})\in\Omega_{s_{\max}}\times\{t_{m-1}^{+}\}$, where $t_{m-1}\in\mathcal{T}_{M}$, from (3.16), for fixed $b$, we assume that $v(z,b,t)\approx A_{1}(t)e^{2s}$ for some unknown function $A_{1}(t)$, which mimics asymptotic behaviour of the value function as $s\to\infty$ (or equivalently, $e^{z}\to\infty$). We substitute this asymptotic form into the integral (3.5d), noting the infinite series representation of $g(\cdot;\Delta t)$ given Lemma 3.1, and obtain the correspoding boundary condition:

  $$v\left(s,b,t_{m-1}^{+}\right)=v(s,b,t_{m}^{-})~{}e^{\left(\sigma^{2}+2\mu+\lambda\kappa_{2}\right)\Delta t},\quad\kappa_{2}=\mathbb{E}\left[\left(\mathrm{e}^{\xi}-1\right)^{2}\right],$$

  (3.21)

  where $v(s,b,t_{m}^{-})$ is given by (3.18). For a proof, see Appendix B.

• •

  For $(s,b,t_{m-1}^{+})\in\Omega_{\scalebox{0.7}{\text{in}}}\times\{t_{m-1}^{+}\}$, where $t_{m-1}\in\mathcal{T}_{M}$, from the convolution integral $\eqref{eq:recursive_c_comp_int_1}$, we have

  $$v\left(s,b,t_{m-1}^{+}\right)=\int_{s_{\min}^{\dagger}}^{s_{\max}^{\dagger}}v\left(s^{\prime},b,t_{m}^{-}\right)g(s-s^{\prime};\Delta t)~{}\mathrm{d}s^{\prime}.$$

  (3.22)

  where the terminal condition $v\left(s^{\prime},b,t_{m}^{-}\right)$ is given by (3.18). The conditional density $g(\cdot;\Delta t)$ is given by the infinite series in (3.9) (Lemma (3.1)), and is defined on $[s_{\min}^{\ddagger},s_{\max}^{\ddagger}]$.