620 research outputs found
An approximation scheme for semilinear parabolic PDEs with convex and coercive Hamiltonians
We propose an approximation scheme for a class of semilinear parabolic
equations that are convex and coercive in their gradients. Such equations arise
often in pricing and portfolio management in incomplete markets and, more
broadly, are directly connected to the representation of solutions to backward
stochastic differential equations. The proposed scheme is based on splitting
the equation in two parts, the first corresponding to a linear parabolic
equation and the second to a Hamilton-Jacobi equation. The solutions of these
two equations are approximated using, respectively, the Feynman-Kac and the
Hopf-Lax formulae. We establish the convergence of the scheme and determine the
convergence rate, combining Krylov's shaking coefficients technique and
Barles-Jakobsen's optimal switching approximation.Comment: 24 page
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
Numerical solution of Dirichlet problems for nonlinear parabolic equations by probability approach
A number of new layer methods solving Dirichlet problems for semilinear parabolic equations is constructed by using probabilistic representations of their solutions. The methods exploit the ideas of weak sense numerical integration of stochastic differential equations in bounded domain. In spite of the probabilistic nature these methods are nevertheless deterministic. Some convergence theorems are proved. Numerical tests are presented
Numerical methods for nonlinear parabolic equations with small parameter based on probability approach
The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. In spite of the probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter
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