20,741 research outputs found
Second order backward stochastic differential equations and fully non-linear parabolic PDEs
We introduce a class of second order backward stochastic differential
equations and show relations to fully non-linear parabolic PDEs. In particular,
we provide a stochastic representation result for solutions of such PDEs and
discuss Monte Carlo methods for their numerical treatment.Comment: 26 page
Stochastic Averaging Principle for Dynamical Systems with Fractional Brownian Motion
Stochastic averaging for a class of stochastic differential equations (SDEs)
with fractional Brownian motion, of the Hurst parameter H in the interval (1/2,
1), is investigated. An averaged SDE for the original SDE is proposed, and
their solutions are quantitatively compared. It is shown that the solution of
the averaged SDE converges to that of the original SDE in the sense of mean
square and also in probability. It is further demonstrated that a similar
averaging principle holds for SDEs under stochastic integral of pathwise
backward and forward types. Two examples are presented and numerical
simulations are carried out to illustrate the averaging principle
Solvability and numerical simulation of BSDEs related to BSPDEs with applications to utility maximization.
In this paper we study BSDEs arising from a special class of backward stochastic partial differential equations (BSPDEs) that is intimately related to utility maximization problems with respect to arbitrary utility functions. After providing existence and uniqueness we discuss the numerical realizability. Then we study utility maximization problems on incomplete financial markets whose dynamics are governed by continuous semimartingales. Adapting standard methods that solve the utility maximization problem using BSDEs, we give solutions for the portfolio optimization problem which involve the delivery of a liability at maturity. We illustrate our study by numerical simulations for selected examples. As a byproduct we prove existence of a solution to a very particular quadratic growth BSDE with unbounded terminal condition. This complements results on this topic obtained in [6, 7, 8].numerical scheme; stochastic optimal control; utility optimization; quadratic growth; distortion transformation; logarithmic transformation; BSPDE; BSDE;
Numerical Computation for Backward Doubly SDEs with random terminal time
In this article, we are interested in solving numerically backward doubly
stochastic differential equations (BDSDEs) with random terminal time tau. The
main motivations are giving a probabilistic representation of the Sobolev's
solution of Dirichlet problem for semilinear SPDEs and providing the numerical
scheme for such SPDEs. Thus, we study the strong approximation of this class of
BDSDEs when tau is the first exit time of a forward SDE from a cylindrical
domain. Euler schemes and bounds for the discrete-time approximation error are
provided.Comment: 38, Monte Carlo Methods and Applications (MCMA) 201
Mean Square Polynomial Stability of Numerical Solutions to a Class of Stochastic Differential Equations
The exponential stability of numerical methods to stochastic differential
equations (SDEs) has been widely studied. In contrast, there are relatively few
works on polynomial stability of numerical methods. In this letter, we address
the question of reproducing the polynomial decay of a class of SDEs using the
Euler--Maruyama method and the backward Euler--Maruyama method. The key
technical contribution is based on various estimates involving the gamma
function
Hybrid PDE solver for data-driven problems and modern branching
The numerical solution of large-scale PDEs, such as those occurring in
data-driven applications, unavoidably require powerful parallel computers and
tailored parallel algorithms to make the best possible use of them. In fact,
considerations about the parallelization and scalability of realistic problems
are often critical enough to warrant acknowledgement in the modelling phase.
The purpose of this paper is to spread awareness of the Probabilistic Domain
Decomposition (PDD) method, a fresh approach to the parallelization of PDEs
with excellent scalability properties. The idea exploits the stochastic
representation of the PDE and its approximation via Monte Carlo in combination
with deterministic high-performance PDE solvers. We describe the ingredients of
PDD and its applicability in the scope of data science. In particular, we
highlight recent advances in stochastic representations for nonlinear PDEs
using branching diffusions, which have significantly broadened the scope of
PDD.
We envision this work as a dictionary giving large-scale PDE practitioners
references on the very latest algorithms and techniques of a non-standard, yet
highly parallelizable, methodology at the interface of deterministic and
probabilistic numerical methods. We close this work with an invitation to the
fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European
Journal of Applied Mathematics (EJAM
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
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