Abstract
We consider a general class of stochastic optimal control problems, where the state process lives in a real separable Hilbert space and is driven by a cylindrical Brownian motion and a Poisson random measure; no special structure is imposed on the coefficients, which are also allowed to be path-dependent; in addition, the diffusion coefficient can be degenerate. For such a class of stochastic control problems, we prove, by means of purely probabilistic techniques based on the so-called randomization method, that the value of the control problem admits a probabilistic representation formula (known as non-linear Feynman-Kac formula) in terms of a suitable backward stochastic differential equation. This probabilistic representation considerably extends current results in the literature on the infinite-dimensional case, and it is also relevant in finite dimension. Such a representation allows to show, in the non-path-dependent (or Markovian) case, that the value function satisfies the so-called randomized dynamic programming principle. As a consequence, we are able to prove that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation, which turns out to be a second-order fully non-linear integro-differential equation in Hilbert space.
Highlights
In the present paper we study a general class of stochastic optimal control problems, where the infinite-dimensional state process, taking values in a real separable Hilbert space H, has a dynamics driven by a cylindrical Brownian motion W and a Poisson random measure π
The key idea of the randomization method consists in randomizing the control process α, by replacing it with an uncontrolled pure jump process I associated with a Poisson random measure θ, independent of W and π; for the pair of processes (X, I), a new randomized intensity-control problem is introduced in such a way that the corresponding value coincides with the original one
We formulate the stochastic optimal control problem on a new probabilistic setting that we introduce, to which we refer as randomized probabilistic setting
Summary
In the present paper we study a general class of stochastic optimal control problems, where the infinite-dimensional state process, taking values in a real separable Hilbert space H, has a dynamics driven by a cylindrical Brownian motion W and a Poisson random measure π. The key idea of the randomization method consists in randomizing the control process α, by replacing it with an uncontrolled pure jump process I associated with a Poisson random measure θ, independent of W and π; for the pair of processes (X, I), a new randomized intensity-control problem is introduced in such a way that the corresponding value coincides with the original one The idea of this control randomization procedure comes from the well-known methodology implemented in [16] to prove the dynamic programming principle, which is based on the use of piece-wise constant policies.
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