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We are mainly concerned with equations of the form -Lu = f(x,u) + μ, where L is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, f satisfies the monotonicity condition and mild integrability conditions, and μ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric...
In this note we prove that the local martingale part of a convex function f of a d-dimensional semimartingale X = M + A can be written in terms of an Itô stochastic integral ∫H(X)dM, where H(x) is some particular measurable choice of subgradient ∇ f ( x ) off at x, and M is the martingale part of X. This result was first proved by Bouleau in [N. Bouleau, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981) 87–90]. Here we present a new treatment of the problem. We first prove the result for x10ff65;...
We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect...
We consider the problem of the existence of solutions of the random set-valued equation:
(I) , t ∈ [0,T] -a.e.; X₀ = U p.1
where F and U are given random set-valued mappings with values in the space , of all nonempty, compact and convex subsets of the separable Banach space E. Under certain restrictions on F we obtain existence of solutions of the problem (I). The connections between solutions of (I) and solutions of random differential inclusions are investigated.
We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form
The purpose of the paper is to introduce a set-valued Stratonovich integral driven by a one-dimensional Brownian motion. We discuss the existence of this integral and investigate its properties.
We consider the approximate Euler scheme for Lévy-driven stochastic differential equations. We study the rate of convergence in law of the paths. We show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them. For example, when the Lévy measure of the driving process behaves like |z|−1−αdz near 0, for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα. For a similar error when neglecting the...
We consider the approximate Euler scheme for Lévy-driven
stochastic differential equations.
We study the rate of convergence in law of the paths.
We show that when approximating the small jumps by Gaussian
variables, the convergence is much faster than when simply
neglecting them.
For example, when the Lévy measure of the driving process
behaves like |z|−1−αdz near 0, for some α∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα. For a similar error when neglecting the...
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