[Paper Review] On $L_p$-Solvability of Stochastic Integro-Differential Equations

István Gyöngy, Sizhou Wu|arXiv (Cornell University)|Jul 10, 2019

Stochastic processes and financial applications28 references4 citations

TL;DR

This paper establishes the $L_p$-solvability of a general class of degenerate stochastic integro-differential equations of parabolic type, including the Zakai equation for jump diffusions, by proving existence, uniqueness, and regularity of solutions in Bessel potential spaces. The key contribution is an $L_p$-estimate for the solution in terms of initial and forcing data, valid for all $p \geq 2$ via interpolation, with temporal regularity captured through essential supremum bounds.

ABSTRACT

A class of (possibly) degenerate stochastic integro-differential equations of parabolic type is considered, which includes the Zakai equation in nonlinear filtering for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces.

Motivation & Objective

• To establish the existence and uniqueness of generalized solutions to a broad class of degenerate stochastic integro-differential equations of parabolic type.
• To derive $L_p$-estimates for the solution in Bessel potential spaces $H^s_p$ for all $p \geq 2$ and $s \in [0,m]$.
• To extend the solvability theory to equations driven by jump diffusions and Poisson random measures, including the Zakai equation in nonlinear filtering.
• To characterize the temporal and spatial regularity of the solution through essential supremum estimates in time.
• To develop a solution operator that is bounded and continuous in $L_p$-based function spaces, enabling interpolation to non-integer smoothness indices.

Proposed method

• Use of Itô's formula for jump diffusions, generalizing Krylov's result to include Lévy noise.
• Construction of a priori estimates in Sobolev spaces $W^n_p$ for $p = 2^k$, $k \in \mathbb{N}$, via energy estimates and martingale inequalities.
• Approximation of the original equation by non-degenerate, smooth, compactly supported problems and passage to weak limits in $L_p$-spaces.
• Application of interpolation theory (real interpolation) to extend a priori estimates from $p = 2^k$ to all $p \geq 2$.
• Use of the fact that the essential supremum of a function on $[0,T]$ is the limit of its $L_r$-norms as $r \to \infty$ to derive time-regularity estimates.
• Introduction of cut-off functions $\chi_n(x) = \chi(x/n)$ to localize data and prove convergence of approximating solutions in $L_p$-norms.

Experimental results

Research questions

• RQ1 Under what conditions does a degenerate stochastic integro-differential equation of parabolic type admit a unique generalized solution in $L_p$-spaces?
• RQ2 How can $L_p$-estimates for the solution be extended from $p = 2^k$ to all $p \geq 2$ using interpolation?
• RQ3 What is the precise temporal and spatial regularity of the solution in terms of Bessel potential spaces $H^s_p$?
• RQ4 How does the solution operator mapping initial and forcing data to the solution behave in terms of boundedness and continuity?
• RQ5 Can the essential supremum in time of the solution norm be estimated in terms of the initial and data norms?

Key findings

• The Cauchy problem (1.1)–(1.2) has a unique generalized solution $u$ in the space $\mathbb{U}^m_{r,p}$ for all $p \geq 2$, $r > 1$, and $m \in \mathbb{N}_0$.
• The solution satisfies the a priori estimate $\mathbb{E} \sup_{t \in [0,T]} |u_t|_{H^s_p}^p \leq N \left( \mathbb{E} |\psi|_{H^s_p}^p + \mathbb{E} \mathcal{K}^p_{s,p}(T) \right)$ for all $s \in [0,m]$.
• The solution operator $\mathbb{S}$ is bounded and continuous from $\Psi^m_p \times \mathbb{H}^m_p \times \mathbb{H}^{m+1}_p(l_2) \times \mathbb{H}^{m+i}_p,\mathcal{L}_{p,2}$ to $\mathbb{U}^m_{r,p}$ with norm depending on $d, p, m, T, K, K_\xi, K_\eta$.
• For non-integer $s \in (0,m)$, the solution is strongly cadlag as an $H^s_p$-valued process, established via real interpolation and the interpolation inequality in Theorem 4.1(v).
• The essential supremum estimate in time is derived by taking the limit of $L_r$-norms as $r \to \infty$, yielding $\mathbb{E} \sup_{t \in [0,T]} |u_t|_{H^s_p}^p \leq N \left( \mathbb{E} |\psi|_{H^s_p}^p + \mathbb{E} \mathcal{K}^p_{s,p}(T) \right)$.
• The result holds for $p \in (0,p)$ via Lemma 3.10, extending the estimate to all $p > 0$.