236,633 research outputs found
A stochastic maximum principle via Malliavin calculus
This paper considers a controlled It\^o-L\'evy process where the information
available to the controller is possibly less than the overall information. All
the system coefficients and the objective performance functional are allowed to
be random, possibly non-Markovian. Malliavin calculus is employed to derive a
maximum principle for the optimal control of such a system where the adjoint
process is explicitly expressed
Maximum Principle for General Controlled Systems Driven by Fractional Brownian Motions
We obtain a maximum principle for stochastic control problem of general
controlled stochastic differential systems driven by fractional Brownian
motions (of Hurst parameter ). This maximum principle specifies a system
of equations that the optimal control must satisfy (necessary condition for the
optimal control). This system of equations consists of a backward stochastic
differential equation driven by both fractional Brownian motion and the
corresponding underlying standard Brownian motion. In addition to this backward
equation, the maximum principle also involves the Malliavin derivatives. Our
approach is to use conditioning and Malliavin calculus. To arrive at our
maximum principle we need to develop some new results of stochastic analysis of
the controlled systems driven by fractional Brownian motions via fractional
calculus. Our approach of conditioning and Malliavin calculus is also applied
to classical system driven by standard Brownian motion while the controller has
only partial information. As a straightforward consequence, the classical
maximum principle is also deduced in this more natural and simpler way.Comment: 44 page
Maximum Principle for Quasi-linear Backward Stochastic Partial Differential Equations
In this paper we are concerned with the maximum principle for quasi-linear
backward stochastic partial differential equations (BSPDEs for short) of
parabolic type. We first prove the existence and uniqueness of the weak
solution to quasi-linear BSPDE with the null Dirichlet condition on the lateral
boundary. Then using the De Giorgi iteration scheme, we establish the maximum
estimates and the global maximum principle for quasi-linear BSPDEs. To study
the local regularity of weak solutions, we also prove a local maximum principle
for the backward stochastic parabolic De Giorgi class
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