Itô formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties

Abstract

Abstract. We use Yosida approximation to find an Itô formula for mild solutions {X x (t), t ≥ 0} of SPDEs with Gaussian and non-Gaussian coloured noise, the non Gaussian noise being defined through compensated Poisson random measure associated to a Lévy process. The functions to which we apply such Itô formula are in C 1,2 ([0, T ] × H), as in the case considered for SDEs in [19]. Using this Itô formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such Itô formula to an Itô formula for mild solutions introduced by Ichikawa in [15], and an Itô formula written in terms of the semigroup of the drift operator [6] which we extend before to the non Gaussian case.

Book Review: Yosida approximations of stochastic differential equations in infinite dimensions and applications

“…A thorough discussion of the Itô formula for mild solutions of infinite-dimensional stochastic evolution equations and current research on it (cf. e.g., Ichikawa [4], Da Prato, et al [2], Albeverio, et al [1]) would have provided a more compelling argument for the introduction of Yosida approximations.…”

Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure

“…Let N(dz, dt) be the H-valued Poisson distributed σ -finite measure on the product σ -algebra B(χ ) and B(R + ) with intensity ν(dz)dt, where dt is the Lebesgue measure on B(R + ). In our model problem (1), N(dz, dt) stands for the compensated Poisson random measure defined by N(dz, dt) := N(dz, dt) − ν(dz)dt.…”

“…Note that this scheme is an explicit stable scheme, where the implementation is based on the computation of matrix exponential functions [21]. As the linear implicit and exponential scheme are stable only when the linear operator A is stronger than the nonlinear function F , 1 we also provide the strong convergence of the stochastic exponential Rosenbrock scheme (SERS) [26] for (H ∈ ( 1 2 , 1]), which is very stable when (5) is driven both by its linear or nonlinear parts. However, the model (5) can be unsatisfactory and less realistic.…”

“…As for standard Brownian motion, we can incorporate a non-Gaussian noise such as Lévy process or Poisson random measure to model such events. The corresponding equation is our model equation given in (1). In contrast to SPDE driven by fBm in (5) where at least few numerical schemes exist, numerical schemes for such SPDE of type (1) driven by fBm and Poisson measure have been lacked in scientific literature, to the best of our knowledge.…”

“…In contrast to SPDE driven by fBm in (5) where at least few numerical schemes exist, numerical schemes for such SPDE of type (1) driven by fBm and Poisson measure have been lacked in scientific literature, to the best of our knowledge. In this work, we will also fill the gap by extending the implicit scheme, the exponential scheme and the stochastic exponential Rosenbrock scheme to SPDE of type (1). For SPDE of type (5) and SPDE of type (1), our strong convergence results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains optimal convergence rates of order O(h 2 + Δt) for the exponential integrator and implicit schemes.…”