Viscosity solutions to Hamilton-Jacobi-Bellman equations associated with sublinear L\'evy(-type) processes

Abstract

Using probabilistic methods we study the existence of viscosity solutions to non-linear integro-differential equations $$\partial_t u(t,x) - \sup_{\alpha \in I} \bigg( b_{\alpha}(x) \cdot \nabla_x u(t,x) + \frac{1}{2} \text{tr}\left(Q_{\alpha}(x) \cdot \nabla^2_x u(t,x)\right) +\int_{y \neq 0} \big(u(t,x+y)-u(t,x)-\nabla_x u(t,x) \cdot h(y) \big) \, \nu_{\alpha}(x,dy) \bigg) = 0$$ with initial condition $u(0,x)= \varphi(x)$; here $(b_{\alpha}(x),Q_{\alpha}(x),\nu_{\alpha}(x,dy))$, $\alpha \in I$, $x \in \mathbb{R}^d$, is a family of L\'evy triplets and $h$ is some truncation function. The solutions, which we construct, are of the form $u(t,x) = T_t \varphi(x)$ for a sublinear Markov semigroup $(T_t)_{t \geq 0}$ with representation $$T_t \varphi(x) = \mathcal{E}^x \varphi(X_t):= \sup_{\mathbb{P} \in \mathfrak{P}_x} \int_{\Omega} \varphi(X_t) \, d\mathbb{P}$$ where $(X_t)_{t \geq 0}$ is a stochastic process and $\mathfrak{P}_x$, $x \in \mathbb{R}^d$, are families of probability measures. The key idea is to exploit the connection between sublinear Markov semigroups and the associated Kolmogorov backward equation. In particular, we obtain new existence and uniqueness results for viscosity solutions to Kolmogorov backward equations associated with L\'evy(-type) processes for sublinear expectations and Feller processes on classical probability spaces.