Sufficient conditions for optimality for stochastic evolution equations

AbdulRahman Al-Hussein
2013 Statistics and Probability Letters  
Let us consider first a stochastic control problem associated with a stochastic evolution equation of the following type: { dX(t) = (AX(t) + b(X(t), ν(t)))dt + σ(X(t), ν(t))dW (t), t ∈ (0, T ], and a cost function, where ν(·) denotes a control process. This equation is driven by a possibly unbounded linear operator A that generates a C 0 -semigroup on a separable Hilbert space H and a cylindrical Wiener process W on H. We shall try in this talk to find sufficient conditions for optimality for
more » ... is control problem. Secondly, we shall generalize these results to a controlled forward-backward stochastic evolution equation of the type: where Q is a symmetric nonnegative nuclear operator on H. We establish an Itô-Krylov formula for BSDEs. This consists on the validity of Itô's formula for functions having only a generalized second derivative in some L p -space. This formula allows us to prove the existence and/or uniqueness for some QBSDEs with mesurable generator and an only square integrable terminale data. This shows that neither the exponetional integrability of the terminale data nor the continuity of the generator are needed for the existence of solution for the QBSDEs. We also give some examples to show that the convexity of the generator is not necessary for the uniqueness of solutions.Application to the existenece of viscosity solution for quadratic PDEs is also given. This talk is based on joint work with Y. Ouknine and M. Eddahbi. We study the potential theory of a large class of infinite dimensional Lévy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e., excessive functions with compact level sets. Then many techniques from classical potential theory carry over to this infinite dimensional setting, a number of potential theoretic properties and principles can be proved, answering to open problems as, e.g., formulated by R. Carmona in 1980. We apply the technique of controlled convergence to solve the Dirichlet problem with general (not necessarily continuous) boundary data. The talk is based on joint works with Aurel Cornea and Michael Röckner. 6 We present the properties of three Green functions for: 1. general complex "clamped beam"
doi:10.1016/j.spl.2013.05.026 fatcat:vjzjjsp7xrglbnh6qvjvgk4cky