[Paper Review] Approximation and stability of solutions of SDEs driven by a symmetric \alpha\ stable process with non-Lipschitz coefficients

Hiroya Hashimoto|arXiv (Cornell University)|Oct 11, 2011

Stochastic processes and financial applications5 references5 citations

TL;DR

This paper establishes Euler-Maruyama approximation and pathwise stability for stochastic differential equations (SDEs) driven by symmetric $α$-stable processes (1 < α < 2) with non-Lipschitz coefficients. Under Komatsu's condition, it proves convergence in $L^\beta$-norm for $\beta \in (1,\alpha)$, implying existence of strong solutions. Under Belfadli-Ouknine's condition, it further establishes pathwise stability in $L^\beta$-norm for $\beta < \alpha$, extending classical results to non-Brownian Lévy processes.

ABSTRACT

Firstly, we investigate Euler-Maruyama approximation for solutions of stochastic differential equations (SDEs) driven by a symmetric \alpha\ stable process under Komatsu condition for coefficients. The approximation implies naturally the existence of strong solutions. Secondly, we study the stability of solutions under Komatsu condition, and also discuss it under Belfadli-Ouknine condition.

Motivation & Objective

• To establish the existence of strong solutions for SDEs driven by symmetric $\alpha$-stable processes with non-Lipschitz coefficients.
• To extend the Euler-Maruyama approximation method to SDEs driven by non-Brownian Lévy processes.
• To investigate pathwise stability of solutions under non-Lipschitz conditions in the context of $\alpha$-stable noise.
• To generalize classical stability and approximation results from Brownian motion SDEs to $\alpha$-stable processes.
• To provide a theoretical foundation for numerical methods and stability analysis in SDEs with heavy-tailed Lévy noise.

Proposed method

• Uses the Euler-Maruyama scheme to approximate solutions of SDEs driven by symmetric $\alpha$-stable processes.
• Applies Komatsu's condition as a non-Lipschitz coefficient condition ensuring pathwise uniqueness and stability.
• Employs Emery's inequality, Burkholder-Davis-Gundy inequality, and Doob's maximal inequality to control sample path increments.
• Uses Giné-Marcus inequality to bound the probability of large deviations in the approximation error.
• Applies Lebesgue's dominated convergence theorem to handle convergence of expectations involving the modulus of continuity $\rho$.
• Introduces a mollified function $u_m = |\cdot|^{\alpha-1} * \phi_m$ and uses Itô's formula with a smooth approximation to analyze the $L^{\alpha-1}$-norm of the solution difference.

Experimental results

Research questions

• RQ1 Does the Euler-Maruyama scheme converge to the true solution in $L^\beta$-norm for SDEs driven by symmetric $\alpha$-stable processes with non-Lipschitz coefficients?
• RQ2 Can pathwise uniqueness and existence of strong solutions be established under Komatsu's condition for $\alpha$-stable SDEs?
• RQ3 Is the solution stable in the pathwise sense under Komatsu's condition as the step size tends to zero?
• RQ4 How does the Belfadli-Ouknine condition affect the stability of solutions in $L^\beta$-norm for $\beta < \alpha$?
• RQ5 Can convergence and uniform integrability be established for the approximation error under the Belfadli-Ouknine condition?

Key findings

• Under Komatsu's condition, the Euler-Maruyama approximation converges to the true solution in $L^\beta$-norm for any $\beta \in (1,\alpha)$ as the mesh size $\|\Delta\| \to 0$.
• The convergence result implies the existence of strong solutions for the SDE driven by a symmetric $\alpha$-stable process.
• Pathwise stability of solutions is established under Komatsu's condition, with $\sup_{0 \leq t \leq T} |X_n(t) - X(t)| \to 0$ in probability and uniform integrability of the $\beta$-th moments.
• Under the Belfadli-Ouknine condition, the solution sequence $X_n(t)$ converges to $X(t)$ in $L^\beta$-norm for any $\beta < \alpha$, with $\lim_{n \to \infty} \mathbb{E}[\sup_{0 \leq t \leq T} |X_n(t) - X(t)|^\beta] = 0$.
• The proof relies on a mollified $L^{\alpha-1}$-norm of the solution difference and estimates via the transition density of the $\alpha$-stable process.
• The paper establishes that the initial condition convergence $\mathbb{E}[|X_n(0) - X(0)|^\alpha] \to 0$ is sufficient for $L^\beta$-stability of the solution path.