Electronic Journal of Probability

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

Elena Bandini, Fulvia Confortola, and Andrea Cosso

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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.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 81, 37 pp.

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

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Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H15: Stochastic partial differential equations [See also 35R60] 93E20: Optimal stochastic control 49L25: Viscosity solutions

backward stochastic differential equations infinite-dimensional path-dependent controlled SDEs randomization method viscosity solutions

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Bandini, Elena; Confortola, Fulvia; Cosso, Andrea. BSDE representation and randomized dynamic programming principle for stochastic control problems of infinite-dimensional jump-diffusions. Electron. J. Probab. 24 (2019), paper no. 81, 37 pp. doi:10.1214/19-EJP333. https://projecteuclid.org/euclid.ejp/1568080857

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