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2019 BSDE representation and randomized dynamic programming principle for stochastic control problems of infinite-dimensional jump-diffusions
Elena Bandini, Fulvia Confortola, Andrea Cosso
Electron. J. Probab. 24(none): 1-37 (2019). DOI: 10.1214/19-EJP333

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.

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Elena Bandini. Fulvia Confortola. Andrea Cosso. "BSDE representation and randomized dynamic programming principle for stochastic control problems of infinite-dimensional jump-diffusions." Electron. J. Probab. 24 1 - 37, 2019. https://doi.org/10.1214/19-EJP333

Information

Received: 3 October 2018; Accepted: 9 June 2019; Published: 2019
First available in Project Euclid: 10 September 2019

zbMATH: 07107388
MathSciNet: MR4003134
Digital Object Identifier: 10.1214/19-EJP333

Subjects:
Primary: 49L25, 60H10, 60H15, 93E20

JOURNAL ARTICLE
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