Pointwise Second-Order Necessary Conditions for Stochastic Optimal Control with Jump Diffusions

Abstract

In this paper, we establish a second-order necessary conditions for stochastic optimal control for jump diffusions. The controlled system is described by a stochastic differential systems driven by Poisson random measure and an independent Brownian motion. The control domain is assumed to be convex. Pointwise second-order maximum principle for controlled jump diffusion in terms of the martingale with respect to the time variable is proved. The proof of the main result is based on variational approach using the stochastic calculus of jump diffusions and some estimates on the state processes.

Appendix

The following result gives special case of the Itô formula for jump diffusions.

Lemma A.1

(Integration by parts formula for jumps processes) Suppose that the processes $x_{i}(t)$ are given by: for $i=1,2,$ $t\in \left[ 0,T\right] :$

$$\begin{aligned} \left\{ \begin{array}{l} dx_{i}(t)=b\left( t,x_{i}(t),u(t)\right) dt+\sigma \left( t,x_{i}(t),u(t)\right) dW(t)\\ +\int _{Z}\eta \left( t,x_{i}(t_{-}),z\right) {\widetilde{N}}\left( \mathrm{d}z,\mathrm{d}t\right) ,\\ x_{i}(0)=0. \end{array} \right. \end{aligned}$$

Then, we get

$$\begin{aligned} {\mathbb {E}}\left( x_{1}(T)x_{2}(T)\right)&={\mathbb {E}}\left[ \int _{0}^{T}x_{1}(t)\mathrm{d}x_{2}(t)+\int _{0}^{T}x_{2}(t)\mathrm{d}x_{1}(t)\right] \\&\quad +{\mathbb {E}}\int _{0}^{T}\sigma ^{\intercal }\left( t,x_{1}(t),u(t)\right) \sigma \left( t,x_{2}(t),u(t)\right) \mathrm{d}t \\&\quad +{\mathbb {E}}\int _{0}^{T}\int _{Z}\eta ^{\intercal }\left( t,x_{1}(t),z\right) \eta \left( t,x_{2}(t),z\right) \mu (\mathrm{d}z)\mathrm{d}t. \end{aligned}$$

The proof can be performed as in Framstad et al. [7, Lemma 2.1] for the detailed proof of the above Lemma.

Proposition A.2

Let ${\mathcal {G}}$ be the predictable $\sigma -$field on $\Omega \times \left[ 0,T\right]$, $\mu (Z)<\infty$, and f be a ${\mathcal {G}}\times {\mathcal {B}}(Z)-$measurable function such that.

$$\begin{aligned} {\mathbb {E}}\int _{0}^{T}\int _{Z}\left| f\left( s,z\right) \right| ^{2}\mu (\mathrm{d}z)\mathrm{d}s<\infty , \end{aligned}$$

then for all $k\ge 2$, there exists a positive constant $C_{(k,\mu (Z))}>0$ such that

$$\begin{aligned} {\mathbb {E}}\left[ \sup _{0\le t\le T}\left| \int _{0}^{t}\int _{Z}f\left( s,z\right) {\widetilde{N}}(\mathrm{d}z,\mathrm{d}s)\right| ^{k}\right] \le C_{(k,\mu (Z))} {\mathbb {E}}\left[ \int _{0}^{T}\int _{Z}\left| f\left( s,z\right) \right| ^{k}\mu (\mathrm{d}z)\mathrm{d}s\right] . \end{aligned}$$

Proof

See Bouchard et al. [2,  Appendix]. $\square$