[Paper Review] Itô's formula for jump processes in $L_p$-spaces

István Gyöngy, Sizhou Wu|Edinburgh Research Explorer|Apr 29, 2019

Stochastic processes and financial applicationsEconomics, Econometrics and Finance15 references3 citations

TL;DR

This paper establishes an Itô formula for the $L_p$-norm of stochastic processes with jumps, extending Krylov's $L_2$-theory to $L_p$-spaces for $p \geq 2$. It provides a stochastic differential for $|u_t|_{L_p}^p$ driven by Wiener processes and Poisson random measures, enabling existence and uniqueness proofs for SPDEs with Lévy noise in $L_p$-Sobolev spaces.

ABSTRACT

We present an Itô formula for the $L_p$-norm of jump processes having stochastic differentials in $L_p$-spaces. The main results extend well-known theorems of Krylov to the case of processes with jumps, and which can be used to prove existence and uniqueness theorems in $L_p$-spaces for SPDEs driven by Lévy processes.

Motivation & Objective

• To extend Krylov's Itô formula in $L_2$-spaces to $L_p$-spaces for $p \geq 2$ with jump processes.
• To develop a stochastic calculus framework for $L_p$-valued semimartingales driven by Wiener processes and Poisson martingale measures.
• To provide a tool for proving existence and uniqueness of solutions to SPDEs with Lévy noise in $L_p$-Sobolev spaces.
• To generalize Lemma 5.1 and Theorem 2.1 from Krylov (2010) to processes with jumps.
• To establish a priori estimates in $L_p$-norms for stochastic integro-differential equations via the new Itô formula.

Proposed method

• Derives a stochastic differential for $|u_t|_{L_p}^p$ using Itô's formula for $\mathbb{R}^M$-valued semimartingales and a stochastic Fubini theorem.
• Applies a regularization procedure via mollifiers to handle non-smooth processes and derive a priori estimates.
• Uses Hölder and Young inequalities to control martingale and jump terms in $L_p$-norms.
• Establishes bounds on the $L_p$-norm of the solution by estimating the drift, diffusion, and jump components separately.
• Employs a stochastic Fubini theorem to interchange integrals over time, space, and Poisson random measures.
• Applies a limiting argument as the mollification parameter $\varepsilon \to 0$ to pass to the limit in the regularized equation.

Experimental results

Research questions

• RQ1 How can Itô's formula be extended from continuous semimartingales to $L_p$-valued processes with jumps?
• RQ2 What is the correct form of the Itô differential for $|u_t|_{L_p}^p$ when the process includes jumps driven by a Poisson random measure?
• RQ3 Can the resulting formula be used to derive a priori estimates in $L_p$-norms for SPDEs with Lévy noise?
• RQ4 How do the $L_p$-norm estimates depend on the integrability of the drift, diffusion, and jump coefficients?
• RQ5 What conditions ensure the existence and uniqueness of solutions to SPDEs in $L_p$-Sobolev spaces with jump noise?

Key findings

• The paper establishes a new Itô formula for $|u_t|_{L_p}^p$ for $L_p$-valued semimartingales with jumps, generalizing Krylov's $L_2$-result.
• The stochastic differential of $|u_t|_{L_p}^p$ includes drift, diffusion, and jump terms, with explicit expressions involving $L_p$-norms of the coefficients.
• A priori estimate (2.6) is derived: $\mathbb{E} \sup_{t \leq T} |u_t|_{L_p}^p \leq C \left( \mathbb{E} |\psi|_{L_p}^p + \mathbb{E} \int_0^T |h_t|_{L_p(\mathcal{L}_p)}^p dt + \cdots \right)$ with constant $N = N(p,d)$.
• The estimate depends on the $L_p$-norms of the drift, diffusion, and jump coefficients, with time-dependent factors $T^{(p-2)/2}$ and $T^{p-1}$.
• The limiting argument as $\varepsilon \to 0$ in the mollified equation confirms the existence of a cadlag $L_p$-valued solution satisfying the SPDE in the $L_p$-sense.
• The method enables existence and uniqueness results for SPDEs driven by Lévy processes in $L_p$-Sobolev spaces, as applied in [8].