Milstein Approximation for Advection-Diffusion Equations Driven by Multiplicative Noncontinuous Martingale Noises

Abstract

In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, càdlàg, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived in L 2 and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler-Maruyama approximation. Finally, simulations complete the paper.

A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations

“…With these choices of η and σ, all conditions of the randomized Galerkin finite element method (52) with truncated noise are satisfied. In particular, we expect a temporal order as high as 1 2 + min( r 2 , γ) ≈ 1 − ǫ by Theorem 6.4. For the simulation displayed in Figure 1 we chose the parameter values a = 0.9, b = 7, and J = 5 for η J in (64).…”

“…In this section we collect all essential conditions on the stochastic evolution equation (1). Then the main result is stated.…”

“…Under these assumptions the mild solution X : [0, T ]×Ω W → H to (1) is uniquely determined by the variation-of-constants formula X(t) = S(t)X 0 − t 0 S(t − s)f (s, X(s)) ds + t 0 S(t − s)g(s) dW (s) (2) which holds P W -almost surely for all t ∈ [0, T ]. Our assumptions in Section 3 ensure the existence of a unique mild solution to (1). We refer the reader to [7] for a general introduction of the semigroup approach to stochastic evolution equations.…”

Stability and error analysis of an implicit Milstein finite difference scheme for a two-dimensional Zakai SPDE

“…Under suitable regularity, for equations of type (1.4), mean-square convergence of order 1/2 is shown for an Euler semi-discretisation in [Lan10] for square-integrable (not necessarily continuous), infinitedimensional martingale drivers. In contrast, [BL13] allow only for continuous martingales but prove convergence of higher order in space and up to 1 in time, in L p and almost surely, for a Milstein scheme and spatial Galerkin approximation of sufficiently high order; this is extended to advection-diffusion equations with possibly discontinuous martingales in [BL12].…”

Erratum: Almost sure convergence of a semi-discrete Milstein scheme for SPDEs of Zakai type

“…For more details on the simulation of Milstein terms, the reader is referred to Refs [1,2,4]. However, the order of convergence as stated in Theorems 2.3 and 2.4 remains the same because therein Equation (M) was used which already included the mixed terms missing in Equation (10).…”