Differential Equations Driven by Gaussian Signals I

Differential Equations Driven by Gaussian Signals II

Large classes of multi-dimensional Gaussian processes can be enhanced with stochastic Lévy area(s). In a previous paper, we gave sufficient and essentially necessary conditions, only involving variational properties of the covariance. Following T. Lyons, the resulting lift to a ”Gaussian rough path” gives a robust theory of (stochastic) differential equations driven by Gaussian signals with sam...

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

From Rough Path Estimates to Multilevel Monte Carlo

Discrete approximations to solutions of stochastic differential equations are wellknown to converge with “strong” rate 1/2. Such rates have played a key-role in Giles’ multilevel Monte Carlo method [Giles, Oper. Res. 2008] which gives a substantial reduction of the computational effort necessary for the evaluation of diffusion functionals. In the present article similar results are established ...

Numerical solution of second-order stochastic differential equations with Gaussian random parameters

In this paper, we present the numerical solution of ordinary differential equations (or SDEs), from each order especially second-order with time-varying and Gaussian random coefficients. We indicate a complete analysis for second-order equations in special case of scalar linear second-order equations (damped harmonic oscillators with additive or multiplicative noises). Making stochastic differe...

ar X iv : 0 70 8 . 37 30 v 1 [ m at h . PR ] 2 8 A ug 2 00 7 DENSITIES FOR ROUGH DIFFERENTIAL EQUATIONS UNDER HÖRMANDER ’ S CONDITION

We consider stochastic differential equations dY = V (Y ) dX driven by a multidimensional Gaussian process X in the rough path sense. Using Malliavin Calculus we show that Yt admits a density for t ∈ (0, T ] provided (i) the vector fields V = (V1, ..., Vd) satisfy Hörmander’s condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fract...