Regular stochastic flow and Dynamic Programming Principle for jump diffusions

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.