A sharp c\`adl\`ag property for jump diffusions and dynamic programming principle

Abstract

Given a standard Brownian motion $W$ and a stationary Poisson point process $p$ with characteristic measure $\nu(dz)$, both with values in ${\mathbb R}^d$, we consider the following SDE of It\^o's type \begin{align*} &dX_{t}=b\left(r, X_{r}\right) dr + \alpha\left(r, X_{r} \right) dW_r+ \int_{ |z| \le 1}g\left(X_{r-},r,z\right)\tilde{N}_p\left(dr,dz\right) + \int_{ |z| >1 } f\left(X_{r-},r,z\right){N}_p\left(dr,dz\right), \end{align*} where $X_s=x\in\mathbb{R}^d,\,0\le s \le t \le T.$ Here $N_p$ [resp., $\tilde{N}_p$] is the Poisson [resp., compensated Poisson] random measure associated with $p$. We require the coefficients $b$, $\alpha$, and $g$ to satisfy Lipschitz--type conditions in the $x-$variable, ensuring the existence of a pathwise unique strong solution $X = (X^{s,x}_t)_{t\ge s}$. We prove, in particular, that there exists a sharp version of $X$, i.e., there exists an almost sure event $\Omega'$ such that, for every $\omega \in \Omega'$, the map $(s,x,t)\mapsto X^{s,x}_t(\omega)$ is c\`adl\`ag in $s$ (for $t$ and $x$ fixed), c\`adl\`ag in $t$ (for $s$ and $x$ fixed) and continuous in $x$ (for $s$ and $t$ fixed). In the case of SDEs with only small jumps, i.e., with $f\equiv0$, this result solves an open problem which also appears in Kunita's book on stochastic flows. In our proof, we deal with non-separable spaces of c\`adl\`ag functions involving supremum norms and, when $f\equiv 0$, we employ an extension of the c\`adl\`ag criterion by Bezandry and Fernique, which is proved in the appendix. We then extend our approach to encompass controlled SDEs having also a large-jumps component. Using our sharp stochastic flow we obtain a new dynamic programming principle, whose proof is of independent interest