35 research outputs found
Well Posedness of Operator Valued Backward Stochastic Riccati Equations in Infinite Dimensional Spaces
We prove existence and uniqueness of the mild solution of an infinite
dimensional, operator valued, backward stochastic Riccati equation. We exploit
the regularizing properties of the semigroup generated by the unbounded
operator involved in the equation. Then the results will be applied to
characterize the value function and optimal feedback law for a infinite
dimensional, linear quadratic control problem with stochastic coefficients
Stochastic Maximum Principle for Optimal Control ofPartial Differential Equations Driven by White Noise
We prove a stochastic maximum principle ofPontryagin's type for the optimal
control of a stochastic partial differential equationdriven by white noise in
the case when the set of control actions is convex. Particular attention is
paid to well-posedness of the adjoint backward stochastic differential equation
and the regularity properties of its solution with values in
infinite-dimensional spaces
Stochastic maximum principle for optimal control of SPDEs
In this note, we give the stochastic maximum principle for optimal control of
stochastic PDEs in the general case (when the control domain need not be convex
and the diffusion coefficient can contain a control variable)
Ergodic BSDEs under weak dissipative assumptions
In this paper we study ergodic backward stochastic differential equations
(EBSDEs) dropping the strong dissipativity assumption needed in the previous
work. In other words we do not need to require the uniform exponential decay of
the difference of two solutions of the underlying forward equation, which, on
the contrary, is assumed to be non degenerate. We show existence of solutions
by use of coupling estimates for a non-degenerate forward stochastic
differential equations with bounded measurable non-linearity. Moreover we prove
uniqueness of "Markovian" solutions exploiting the recurrence of the same class
of forward equations. Applications are then given to the optimal ergodic
control of stochastic partial differential equations and to the associated
ergodic Hamilton-Jacobi-Bellman equations
Optimal control of two scale stochastic systems in infinite dimensions: the BSDE approach
In this paper we study, by probabilistic techniques, the convergence of the value function for a two-scale, infinite-dimensional, stochastic controlled system as the ratio between the two evolution speeds diverges.
The value function is represented as the solution of a backward stochastic differential equation (BSDE) that it is shown to converge towards a reduced BSDE. The noise is assumed to be additive both in the slow and the fast equations for the state. Some non degeneracy condition on the slow equation are required. The limit BSDE involves the solution of an ergodic BSDE and is itself interpreted as the value function of an auxiliary stochastic control problem on a reduced state space
Ergodic Control of Infinite Dimensional stochastic differential equations with Degenerate Noise
The present paper is devoted to the study of the asymptotic behavior of the value functions of both finite and infinite horizon stochastic control problems and to the investigation of their relation with suitable stochastic ergodic control problems. Our methodology is based only on probabilistic techniques, as for instance the so-called randomization of the control method, thus avoiding completely analytical tools from the theory of viscosity solutions. We are then able to treat with the case where the state process takes values in a general (possibly infinite dimensional) real separable Hilbert space and the diffusion coefficient is allowed to be degenerate