| 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*}
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.
\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 any {\it Borel set} $B \subset \R^{l}$. The coefficients {$b$, $\alpha$, and $g$} satisfy linear growth and Lipschitz--type conditions in the $x-$variable, and are continuous in the control variable. Moreover, we consider 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]
$,
where $h$ and $j$ are suitable given maps, globally bounded and continuous in the $x-$variable and in the control variable. The DPP is a fundamental tool in stochastic control with applications in physics and mathematical finance.
To prove it, we show the existence of a regular stochastic flow for the previous stochastic equation when the coefficients are independent of the control $a$. Notably, this regularity result is new for jump diffusions even when there is no large--jumps component, i.e., $f\equiv0$ (cf. Kunita`s recent book on stochastic flows and jump diffusions). The proof of the DPP is completed by introducing an approach that relies on a new subclass of controls in $\mathcal{P}_T$. These controls allow us to apply a basic measurable selection theorem by L. D. Brown and R. Purves. |
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