Approximation of stationary solutions of Gaussian driven stochastic differential equations

Numerical Approximations to the Stationary Solutions of Stochastic Differential Equations

This thesis investigates the possibility of approximating stationary solutions of stochastic differential equations using numerical methods. We consider a particular class of stochastic differential equations, which are known to generate random dynamical systems. The existence of stochastic stationary solution is proved using global attractor approach. Euler’s numerical method, applied to the s...

Simulation and Approximation of Lévy-driven Stochastic Differential Equations

We consider the problem of the simulation of Lévy-driven stochastic differential equations. It is generally impossible to simulate the increments of a Lévy-process. Thus in addition to an Euler scheme, we have to simulate approximately these increments. We use a method in which the large jumps are simulated exactly, while the small jumps are approximated by Gaussian variables. Using some recent...

Stationary Solutions of Stochastic Differential Equation with Memory and Stochastic Partial Differential Equations

We explore Itô stochastic differential equations where the drift term has possibly infinite dependence on the past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proved if the dependence on the past decays sufficiently fast. The results of this paper ar...

strong approximation for itô stochastic differential equations

in this paper, a class of semi-implicit two-stage stochastic runge-kutta methods (srks) of strong global order one, with minimum principal error constants are given. these methods are applied to solve itô stochastic differential equations (sdes) with a wiener process. the efficiency of this method with respect to explicit two-stage itô runge-kutta methods (irks), it method, milstien method, sem...

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion