Sufficient Stochastic Maximum Principle for Discounted Control Problem

Abstract

Abstract. In this article, the sufficient Pontryagin's maximum principle for infinite horizon discounted stochastic control problem is established. The sufficiency is ensured by an additional assumption of concavity of the Hamiltonian function. Throughout the paper, it is assumed that the control domain U is a convex set and the control may enter the diffusion term of the state equation. The general results are applied to the controlled stochastic logistic equation of population dynamics.

Ergodic Maximum Principle for Stochastic Systems

“…where p and q i , i = 1, ..., d take values in R n . Due to Hypothesis 1 and estimate (14) the forcing term in the driver is no better than bounded, so that D x f (X s , u s ) ∈ L ∞ R + ; L 2 (Ω; R n ) . Therefore we cannot expect the solution of (34) to be integrable up to infinity but only that p ∈ L ∞ R + ; L 2 (Ω; R n ) .…”

How Do Future Expectations Affect the Financial Sector? Expectations Modeling, Infinite Horizon (Controlled) Random FBSDEs and Stochastic Viscosity Solutions

“…In view of (12) and (13), we may choose a sufficiently small ǫ > 0 such that the inequalities in (30), (32) and…”

“…In the present section we shall formulate these models to motivate now the investigation of the associated stochastic control problem of such FBSDEs systems. Then, moving along the lines of Maslowski and Veverka [32], we shall establish a sufficient stochastic maximum principle for the exponentially λ-weighted control problem that arises from the multidimensional fully coupled FBSDEs (11) with a general control domain. Making in particular use of the duality methodology, we shall define the generalized Hamiltonian which will allow us to state the adjoint system of FBSDEs.…”

“…The last two terms play the role of the Lyapunov function, used also in Maslowski and Veverka [32], needed for the employment of the duality theory to the above optimal control problem; cf. Theorem 5.2 below.…”

Forward - backward stochastic differential equations with random coefficients and applications to finance

“…The infinite time-horizon maximum principle of the discounted control problem was studied later on by Maslowski and Veverka [74]. In fact, this case is widely used in the field of stochastic finance since it leads to maximizing the agent's average discounted utility.…”