A Comparison of Two Settings for Stochastic Integration with Respect to Lévy Processes in Infinite Dimensions

The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise

“…Applebaum and Wu [17], Truman and Wu [47], Rockner and Zhang [38]. The readers are also referred to the monograph by Peszat and Zabczyk [37] and the review paper [10] for more details. However, to the best of our knowledge, no one has yet addressed the use of positivity for the solution of an SPDE perturbed by a Lévy noise.…”

Stochastic one layer shallow water equations with Lévy noise

“…It is clear that for every f ∈ F 2 ν,T (X) the process I π t (f ) t∈[0,T ] is still a purely discontinuous X-valued L 2 -martingale and that (20) continues to hold. By the BDG inequality (see, e.g., [10]) for every p ∈ [1, ∞) there exists a constant C p > 0 such that for every F t -stopping time τ and for every f ∈ F 2 ν,T (X) we have…”

“…Let E 0 := {y ∈ U : 0 < |y| U < 1} and f ∈ F 2 ν,T (X). The stochastic integral (0,t] E0 f (s, z)d π(s, z) represents stochastic integration with respect to the process n=1 P n in the Lévy-Khinchin decomposition (13); see [10] for details. The noise in equations (3a) and (3b) will be driven by two independent Lévy processes L 1 and L 2 defined on a probability space (Ω, F, P).…”

“…For notational convenience will assume that the cylindrical Wiener processes W 1 and W 2 have the same reproducing kernel Hilbert space U and take values in the same space U . We will also assume that π 1 and π 2 have the same intensity measure, which has the form dt ⊗ dν for some σ-finite measure ν on E. For more details on Lévy processes the reader is referred to [24], [32] and the review article [10]; and the references therein.…”

“…With E := U \ {0} and E 0 := {ξ ∈ U : 0 < |ξ| U < 1}, the stochastic integrals in (24) represent stochastic integration with respect to the jumps of a Lévy process outside of the unit ball. See [10] for an explanation of how the stochastic integrals (24) generalize stochastic integration with respect to the compound Poisson process P 0 in the Lévy-Khinchin decomposition (13) of a Lévy process. We now introduce the notions of solution that will be used in this article.…”