25,531 research outputs found
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 optimal control of stochastic evolution equations with recursive utilities
We consider the optimal control problem of stochastic evolution equations in
a Hilbert space under a recursive utility, which is described as the solution
of a backward stochastic differential equation (BSDE). A very general maximum
principle is given for the optimal control, allowing the control domain not to
be convex and the generator of the BSDE to vary with the second unknown
variable . The associated second-order adjoint process is characterized as a
unique solution of a conditionally expected operator-valued backward stochastic
integral equation
Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics
We analyze a stochastic optimal control problem, where the state process
follows a McKean-Vlasov dynamics and the diffusion coefficient can be
degenerate. We prove that its value function V admits a nonlinear Feynman-Kac
representation in terms of a class of forward-backward stochastic differential
equations, with an autonomous forward process. We exploit this probabilistic
representation to rigorously prove the dynamic programming principle (DPP) for
V. The Feynman-Kac representation we obtain has an important role beyond its
intermediary role in obtaining our main result: in fact it would be useful in
developing probabilistic numerical schemes for V. The DPP is important in
obtaining a characterization of the value function as a solution of a
non-linear partial differential equation (the so-called Hamilton-Jacobi-Belman
equation), in this case on the Wasserstein space of measures. We should note
that the usual way of solving these equations is through the Pontryagin maximum
principle, which requires some convexity assumptions. There were attempts in
using the dynamic programming approach before, but these works assumed a priori
that the controls were of Markovian feedback type, which helps write the
problem only in terms of the distribution of the state process (and the control
problem becomes a deterministic problem). In this paper, we will consider
open-loop controls and derive the dynamic programming principle in this most
general case. In order to obtain the Feynman-Kac representation and the
randomized dynamic programming principle, we implement the so-called
randomization method, which consists in formulating a new McKean-Vlasov control
problem, expressed in weak form taking the supremum over a family of equivalent
probability measures. One of the main results of the paper is the proof that
this latter control problem has the same value function V of the original
control problem.Comment: 41 pages, to appear in Transactions of the American Mathematical
Societ
Maximum Principle for Forward-Backward Doubly Stochastic Control Systems and Applications
The maximum principle for optimal control problems of fully coupled
forward-backward doubly stochastic differential equations (FBDSDEs in short) in
the global form is obtained, under the assumptions that the diffusion
coefficients do not contain the control variable, but the control domain need
not to be convex. We apply our stochastic maximum principle (SMP in short) to
investigate the optimal control problems of a class of stochastic partial
differential equations (SPDEs in short). And as an example of the SMP, we solve
a kind of forward-backward doubly stochastic linear quadratic optimal control
problems as well. In the last section, we use the solution of FBDSDEs to get
the explicit form of the optimal control for linear quadratic stochastic
optimal control problem and open-loop Nash equilibrium point for nonzero sum
differential games problem
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