We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time τ and provide a novel asymptotic estimate of the cumulative distribution function of t. As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.
The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite-dimensional Brownian motion is in general not a semi-martingale anymore and does in general not satisfy an Itô formula like the solution of a finite-dimensional stochastic ordinary differential equation (SODE). In particular, it is not possible to derive stochastic Taylor expansions as for the solution of a SODE using an iterated application of the Itô formula. Consequently, until recently, only low order numerical approximation results for such a SPDE have been available. Here, the fact that the …
We consider a stochastic partial differential equation (SPDE) on a lattice \partial_t X=(Δ-m^2)X-λX^p+ηwhere $η$ is a space-time Lévy noise. A perturbative (in the sense of formal power series) strong solution is given by a tree expansion, whereas the correlation functions of the solution are given by a perturbative expansion with coefficients that are represented as sums over a certain class of graphs, called Parisi-Wu graphs. The perturbative expansion of the truncated (connected) correlation functions is obtained via a Linked Cluster Theorem as a sums over connected graphs only. The moments…
The Girsanov transform and Kolmogorov equations are two useful methods for studying SPDEs. It is shown that, under suitable conditions, the series expansion obtained from the Girsanov transform coincides with the one generated by an iteration scheme for Kolmogorov equations. We also apply the iteration approach to extend the well posedness theory for Kolmogorov equations beyond the boundedness condition on the nonlinear term.
We study the dependence of mild solutions to linear stochastic evolution equations on Hilbert space driven by Wiener noise, with drift having linear part of the type $A+\varepsilon G$, on the parameter $\varepsilon$. In particular, we study the limit and the asymptotic expansions in powers of $\varepsilon$ of these solutions, as well as of functionals thereof, as $\varepsilon \to 0$, with good control on the remainder. These convergence and series expansion results are then applied to a parabolic perturbation of the Musiela SPDE of mathematical finance modeling the dynamics of forward rates.
An Ito-Skorokhod bi-linear equation driven by infinitely many independent colored noises is considered in a normal triple of Hilbert spaces. The special feature of the equation is the appearance of the Wick product in the definition of the Ito-Skorokhod integral, requiring innovative approaches to computing the solution. A chaos expansion of the solution is derived and several truncations of this expansion are studied. A recursive approximation of the solution is suggested and the corresponding approximation error bound is computed.
We use a conditional Karhunen-Loève (KL) model to quantify and reduce uncertainty in a stochastic partial differential equation (SPDE) problem with partially-known space-dependent coefficient, $Y(x)$. We assume that a small number of $Y(x)$ measurements are available and model $Y(x)$ with a KL expansion. We achieve reduction in uncertainty by conditioning the KL expansion coefficients on measurements. We consider two approaches for conditioning the KL expansion: In Approach 1, we condition the KL model first and then truncate it. In Approach 2, we first truncate the KL expansion and then condi…
We study a reaction-diffusion evolution equation perturbed by a space-time Lévy noise. The associated Kolmogorov operator is the sum of the infinitesimal generator of a $C_0$-semigroup of strictly negative type acting in a Hilbert space and a nonlinear term which has at most polynomial growth, is non necessarily Lipschitz and is such that the whole system is dissipative. The corresponding Itô stochastic equation describes a process on a Hilbert space with dissi- pative nonlinear, non globally Lipschitz drift and a Lévy noise. Under smoothness assumptions on the non-linearity, asymptotics to al…
We consider a nonlinear stochastic partial differential equation (SPDE) in divergence form where the forcing term is a Gaussian noise, that is white in time and colored in space such that the gradient of the solution is Hölder-continuous, but not differentiable. Then, we prove a generalized Taylor expansion of the difference between the solution to the SPDE and the solution to its linearization around a fixed basepoint. The result is reminiscent of the theory of (controlled) rough paths and agrees with the general observation, that, in settings with a rough driver, subtracting the solution to …
In this paper we study the {\it pathwise stochastic Taylor expansion}, in the sense of our previous work \cite{Buckdahn_Ma_02}, for a class of Itô-type random fields in which the diffusion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an "self-exciting" type particularly contains the fully nonlinear stochastic PDEs of curvature driven diffusion, as well as certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion of "$n$-fold" derivatives of a random field, as a fundamental device to cope with the special self…
Quantitative estimates for the top Lyapunov exponents for systems of stochastic reaction-diffusion equations are proven. The treatment includes reaction potentials with degenerate minima. The proof relies on an asymptotic expansion of the invariant measure, with careful control on the resulting error terms. As a consequence of these estimates, synchronisation by noise is deduced for systems of stochastic reaction-diffusion equations for the first time.
This work contributes to the limited literature on estimating the diffusivity or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that the solution is measured locally in space and over a finite time interval, we show that the augmented maximum likelihood estimator introduced in Altmeyer, Reiss (2020) retains its asymptotic properties when used for semilinear SPDEs that satisfy some abstract, and verifiable, conditions. The proofs of asymptotic results are based on splitting the solution in linear and nonlinear parts and fine regularity properties in $L^p$-spaces. The ob…
We prove a stochastic Taylor expansion for SPDEs and apply this result to obtain cubature methods, i. e. high order weak approximation schemes for SPDEs, in the spirit of T. Lyons and N. Victoir. We can prove a high-order weak convergence for well-defined classes of test functions if the process starts at sufficiently regular initial values. We can also derive analogous results in the presence of Lévy processes of finite type, here the results seem to be new even in finite dimension. Several numerical examples are added.
