AbstractIn this paper, we consider fuzzy stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm). These equations can be applied in hybrid real-world systems, including randomness, fuzziness and long-range dependence. Under some assumptions on the coefficients, we follow an approximation method to the fractional stochastic integral to study the existence and uniqueness of the solutions. As an example, in financial models, we obtain the solution for an equation with linear coefficients.
Abstract
Backward stochastic differential equations (BSDEs) appear in many problems in stochastic optimal control theory, mathematical finance, insurance and economics.
This work deals with the numerical approximation of the class of Markovian BSDEs where the terminal condition is a functional of a Brownian motion.
Using Hermite martingales, we show that the problem of solving a BSDE is identical to solving a countable infinite-dimensional system of ordinary differential equations (ODEs).
The family of ODEs belongs to the class of stiff ODEs, where the associated functional is one-sided Lipschitz.
On this basis, we derive a numerical scheme and provide numerical applications.
Abstract
This paper shows a general weak approximation method for time-inhomogeneous stochastic differential equations (SDEs) using Malliavin weights. A unified approach is introduced to construct a higher order discretization scheme for expectations of non-smooth functionals of solutions of time-inhomogeneous SDEs. Numerical experiments show the validity of the method.
Due to that the explicit methods in solving stochastic differential equations give instability and inaccurate results, the aim of this paper is to derive an effective implicit method gives higher-order approximate solutions for a stiff stochastic differential equations by using Runge-Kutta
method. It relies on the Stratonovich-Taylor expansion and uses the notion of perturbation and coupling to carry out the method. The validity of this new approximation method is shown by implementing in MATLAB and, showing the convergence of the method graphically.
Fuzzy stochastic differential equations driven by fractional Brownian motion
