Probabilistic methods for semilinear  partial differential equations.  Applications to finance

Dan Crisan; Konstantinos Manolarakis

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Journal of Applied Mathematics and Stochastic Analysis

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A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient.

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A hull and white formula for a general stochastic volatility jump-diffusion model with applications to the study of the short-time behavior of the implied volatility.

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Numerical schemes for multivalued backward stochastic differential systems

Lucian Maticiuc, Eduard Rotenstein (2012)

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We define approximation schemes for generalized backward stochastic differential systems, considered in the Markovian framework. More precisely, we propose a mixed approximation scheme for the following backward stochastic variational inequality: $d Y_{t} + F (t, X_{t}, Y_{t}, Z_{t}) d t \in \partial φ (Y_{t}) d t + Z_{t} d W_{t},$ where ∂φ is the subdifferential operator of a convex lower semicontinuous function φ and (X t)t∈[0;T] is the unique solution of a forward stochastic differential equation. We use an Euler type scheme for the system of decoupled forward-backward...

Stochastic differential equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem

Antoine Lejay (2006)

ESAIM: Probability and Statistics

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We show in this article how the theory of “rough paths” allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.

Dynamical properties and characterization of gradient drift diffusions.

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Reflected forward-backward SDEs and obstacle problems with boundary conditions.

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A converse comparison theorem for BSDEs and related properties of $g$ -expectation.

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Euler schemes and half-space approximation for the simulation of diffusion in a domain

Emmanuel Gobet (2001)

ESAIM: Probability and Statistics

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This paper is concerned with the problem of simulation of ${(X_{t})}_{0 \leq t \leq T}$ , the solution of a stochastic differential equation constrained by some boundary conditions in a smooth domain $D$ : namely, we consider the case where the boundary $\partial D$ is killing, or where it is instantaneously reflecting in an oblique direction. Given $N$ discretization times equally spaced on the interval $[0, T]$ , we propose new discretization schemes: they are fully implementable and provide a weak error of order $N^{- 1}$ under some conditions....