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.

BSDE representation and randomized dynamic programming principle for stochastic control problems of infinite-dimensional jump-diffusions / Bandini, Elena; Confortola, Fulvia; Cosso, Andrea. - In: ELECTRONIC JOURNAL OF PROBABILITY. - ISSN 1083-6489. - ELETTRONICO. - 24:0(2019), pp. 81.1-81.37. [10.1214/19-EJP333]

BSDE representation and randomized dynamic programming principle for stochastic control problems of infinite-dimensional jump-diffusions

Bandini, Elena;Cosso, Andrea
2019

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.
2019
BSDE representation and randomized dynamic programming principle for stochastic control problems of infinite-dimensional jump-diffusions / Bandini, Elena; Confortola, Fulvia; Cosso, Andrea. - In: ELECTRONIC JOURNAL OF PROBABILITY. - ISSN 1083-6489. - ELETTRONICO. - 24:0(2019), pp. 81.1-81.37. [10.1214/19-EJP333]
Bandini, Elena; Confortola, Fulvia; Cosso, Andrea
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/702695
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