Regular stochastic flow and Dynamic Programming Principle for jump diffusions

TL;DR

This paper establishes a strong form of the Dynamic Programming Principle for jump diffusion control problems, proving regularity of stochastic flows and introducing a novel approach using finitely generated step controls.

Contribution

It provides the first regular stochastic flow result for controlled jump diffusions with control-independent coefficients, and introduces a new method for proving DPPs using finitely generated step controls.

Findings

Proved a strong DPP for jump diffusion control problems.

Established regular stochastic flows for controlled SDEs with jumps.

Introduced a novel approach using finitely generated step controls for DPP proof.

Abstract

Given a Brownian motion $W$ and a stationary Poisson point process $p$ with values in ${\mathbb R}^d$, we prove a Dynamic Programming Principle (DPP) in a strong formulation for a stochastic control problem involving controlled SDEs of the form \begin{align} \label{ci1} \nonumber dX_{t}=&\,b(t, X_{t}, a_t) dt + \alpha \left(t, X_{t}, a_t \right) dW_t+ \! \! \int_{ |z| \le 1} g\left(X_{t-},t,z, a_t \right)\widetilde{N}_p\left(dt,dz\right) \\ & + \int_{ |z| >1 } f\left(X_{t-},t,z, a_t \right){N}_p\left(dt,dz\right), \quad \; X_s=x\in\mathbb{R}^d,\,0\le s \le t \le T. \;\;\;\;\;\;\;\;\;\; (1) \end{align} Here $N_p$ [resp., $\widetilde{N}_p$] is the Poisson [resp., compensated Poisson] random measure associated with $p$. We consider arbitrary predictable controls $a \in {\mathcal P}_T$ with values in a closed convex set $C \subset {\mathbb R}^{l}$. The coefficients $b$, $\alpha$, and $g$ satisfy linear growth and Lipschitz--type conditions in the $x-$variable, and are continuous in the control variable. To prove the DPP for the value function $ v(s,x)=\sup_{a \in {\mathcal P}_T} \, \mathbb{E}\big[\int_{s}^{T}h\left(r,X_r^{s,x,a}, a_r\right)dr + j\left(X_T^{s,x,a}\right)\big] $, assuming that $h$ and $j$ are bounded and continuous, we establish the existence of a regular stochastic flow for (1) when the coefficients are independent of the control $a$. Notably, this regularity result is new even when there is no large--jumps component, i.e., $f\equiv0$ (cf. Kunita's recent book on stochastic flows). The proof of the DPP is completed by introducing an approach that relies on a suitable subclass of finitely generated step controls in $\mathcal{P}_T$. These controls allow us to apply a basic measurable selection theorem by L. D. Brown and R. Purves. We believe that this novel method is of independent interest and could be adapted to prove DPPs arising in other stochastic control problems.