Itô’s Formula for Banach-space-valued Jump Processes Driven by Poisson Random Measures

Abstract

Abstract. We prove Itô's formula for a general class of functions H : R + × F → G of class C 1,2 , where F, G are separable Banach spaces, and jump processes driven by a compensated Poisson random measure.

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

“…Now we will apply the Itô formula for strong solutions of [19] to the test function Ψ(s, P (U t (v))) for s ∈ [0, t], v ∈ H Ψ(t, X(t)) = Ψ(t, P (U t (X t ))) = Ψ(P (U t (X 0 ))) + t 0 (∂ 1 Ψ)(s, P (U t (X s )))ds [Ψ(s, P (U t (X s )) + P U (t−s) f (v, X(s))) − Ψ(s, P (U t (X s ))) − (∂ 2 Ψ)(s, P (U t (X s ))), P U (t−s) f (v, X(s)) ]β(dv)ds + t 0 H\{0}…”

“…The Itô formula for strong solutions of SPDEs can be derived similarly as for the case of SDEs, see e.g. [13], [22] for the Gaussian case and [19], [18]. Here we derive through Yosida approximation an Itô formula for mild solutions {X x (t), t ≥ 0} for SPDEs driven by a Wiener process and general Lévy processes.…”

Weak Convergence of Finite Element Approximations of Linear Stochastic Evolution Equations with Additive Lévy Noise

“…Alternatively, one could use Itô's formula as proved in [17], but the formula in [32] is more convenient in our setting. In this section, we use Lemma 3.1 to relate our setting to the setting in [32]. It is well-known that the jumps of a Lévy process determine a Poisson random measure on the product space of the underlying time interval and the state space.…”

“…We refer to [32], [40] and the references therein for details on stochastic integration w.r.t. Poisson random measures, compare also [38,Section 8.7].…”

“…(3.13), we first note that for fixed t ∈ [0, T ] the H-valued random variables t 0 E(s)B dL(s) and TT −t E(T − s)B dL(s) have the same distribution, so that v(t, x) := E G x + t 0 E(s)B dL(s) = u(T − t, x), (t, x) ∈ [0, T ] × H. (3.14)Next, we fix x ∈ H and apply Itô's formula[32, Theorem 3.6] to the function y → G(x+y) and the martingale M = (M(t)) t∈[0,T ] := ( t 0 E(s)B dL(s)) t∈[0,T ] ∈ M 2 T (H). Note that M fits into the setting of[32] since it has the representation…”