8 research outputs found
Infinite horizon optimal control of forward-backward stochastic differential equations with delay
We consider a problem of optimal control of an infinite horizon system
governed by forward-backward stochastic differential equations with delay.
Sufficient and necessary maximum principles for optimal control under partial
information in infinite horizon are derived. We illustrate our results by an
application to a problem of optimal consumption with respect to recursive
utility from a cash flow with delay
Singularly perturbed forward-backward stochastic differential equations: application to the optimal control of bilinear systems
We study linear-quadratic stochastic optimal control problems with bilinear
state dependence for which the underlying stochastic differential equation
(SDE) consists of slow and fast degrees of freedom. We show that, in the same
way in which the underlying dynamics can be well approximated by a reduced
order effective dynamics in the time scale limit (using classical
homogenziation results), the associated optimal expected cost converges in the
time scale limit to an effective optimal cost. This entails that we can well
approximate the stochastic optimal control for the whole system by the reduced
order stochastic optimal control, which is clearly easier to solve because of
lower dimensionality. The approach uses an equivalent formulation of the
Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs
(FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares
Monte Carlo algorithm and show its applicability by a suitable numerical
example
Adaptive importance sampling with forward-backward stochastic differential equations
We describe an adaptive importance sampling algorithm for rare events that is
based on a dual stochastic control formulation of a path sampling problem.
Specifically, we focus on path functionals that have the form of cumulate
generating functions, which appear relevant in the context of, e.g.~molecular
dynamics, and we discuss the construction of an optimal (i.e. minimum variance)
change of measure by solving a stochastic control problem. We show that the
associated semi-linear dynamic programming equations admit an equivalent
formulation as a system of uncoupled forward-backward stochastic differential
equations that can be solved efficiently by a least squares Monte Carlo
algorithm. We illustrate the approach with a suitable numerical example and
discuss the extension of the algorithm to high-dimensional systems
Mean Field Linear-Quadratic-Gaussian (LQG) Games of Forward-Backward Stochastic Differential Equations
This paper studies a new class of dynamic optimization problems of
large-population (LP) system which consists of a large number of negligible and
coupled agents. The most significant feature in our setup is the dynamics of
individual agents follow the forward-backward stochastic differential equations
(FBSDEs) in which the forward and backward states are coupled at the terminal
time. This current paper is hence different to most existing large-population
literature where the individual states are typically modeled by the SDEs
including the forward state only. The associated mean-field
linear-quadratic-Gaussian (LQG) game, in its forward-backward sense, is also
formulated to seek the decentralized strategies. Unlike the forward case, the
consistency conditions of our forward-backward mean-field games involve six
Riccati and force rate equations. Moreover, their initial and terminal
conditions are mixed thus some special decoupling technique is applied here. We
also verify the -Nash equilibrium property of the derived
decentralized strategies. To this end, some estimates to backward stochastic
system are employed. In addition, due to the adaptiveness requirement to
forward-backward system, our arguments here are not parallel to those in its
forward case.Comment: 21 page
The Riesz representation theorem and weak compactness of semimartingales
We show that the sequential closure of a family of probability measures on
the canonical space of c{\`a}dl{\`a}g paths satisfying Stricker's uniform
tightness condition is a weak compact set of semimartingale measures in
the pairing of the Riesz representation theorem under topological assumptions
on the path space. Similar results are obtained for quasi- and supermartingales
under analogous conditions. In particular, we give a full characterization of
the strongest topology on the Skorokhod space for which these results are true.Comment: V1-V6 differ substantially from v7-v8 in exposition v9 adds lemma
3.5, example 3.9, remark 5.6, proposition 5.7 (i) and some other minor
remark
Weak solutions of backward stochastic differential equations with continuous generator
We prove the existence of a weak solution to a backward stochastic
differential equation (BSDE) Y_t=\xi+\int_t^T f(s,X_s,Y_s,Z_s)\,ds-\int_t^T
Z_s\,d\wien_s in a finite-dimensional space, where is affine
with respect to , and satisfies a sublinear growth condition and a
continuity condition This solution takes the form of a triplet of
processes defined on an extended probability space and satisfying
Y_t=\xi+\int_t^T f(s,X_s,Y_s,Z_s)\,ds-\int_t^T Z_s\,d\wien_s-(L_T-L_t) where
is a continuous martingale which is orthogonal to any \wien. The solution
is constructed on an extended probability space, using Young measures on the
space of trajectories. One component of this space is the Skorokhod space D
endowed with the topology S of Jakubowski