Stochastic Analysis and Applications: Numerical Approximation of Solutions to Stochastic Differential Equations with Lévy Noise using Malliavin Calculus Techniques

Project Abstract

This study advances the numerical treatment of stochastic differential equations (SDEs) driven by Lévy processes through the synthesis of Malliavin calculus techniques with high-order approximation schemes. Motivated by the ubiquity of jumps in real-world dynamical systems—from finance and insurance to engineering and environmental sciences—our work addresses the gap between theoretical solvability and practical computability in paths with discontinuities. We develop a cohesive framework that integrates probabilistic representations, regularity analysis, and discretization strategies to obtain accurate and stable numerical solutions for SDEs featuring Lévy noise, including both finite-activity and infinite-activity jump components. The core contribution lies in leveraging Malliavin calculus to derive derivative-based error representations for functionals of Lévy-driven processes, enabling the design of weak and strong approximation schemes with provable convergence rates. We formulate Malliavin-weighted quadrature rules and implement an adaptive time-stepping algorithm that judiciously balances jump-induced irregularities against drift and diffusion effects. Our methodology encompasses a systematic treatment of jump structures via Poisson random measures, compensated integrals, and truncation techniques, accompanied by thorough regularity results in appropriate Sobolev spaces to justify interchange of differentiation and expectation. By exploiting integration-by-parts identities in the Malliavin framework, we obtain efficient estimators for sensitivities and distributional properties, which in turn enhance variance reduction and computational efficiency in Monte Carlo simulations. A key aspect of the research is the rigorous error analysis for both strong (pathwise) and weak (distributional) metrics. We establish a priori estimates for solution regularity, quantify the impact of Lévy measures with heavy tails, and derive explicit constants in convergence rates that reflect jump intensity, jump size distribution, and truncation schemes. The numerical experiments cover a spectrum of Lévy models, including stable, tempered stable, and compound Poisson processes, as well as mixed noise configurations. We demonstrate accuracy improvements over classical Euler–Maruyama-type methods and standard jump-adapted schemes, particularly in scenarios with frequent small jumps and irregular sample paths where Malliavin-informed corrections yield substantial gains. Applications are showcased through problems in quantitative finance (option pricing under jump-diffusion dynamics), epidemiological models with abrupt regime shifts, and physical systems subject to impulsive perturbations. The results provide practitioners with robust, implementable algorithms that deliver reliable error controls and scalable performance. The work not only contributes to the numerical analysis of Lévy-driven SDEs but also broadens the practical use of Malliavin calculus in stochastic simulation, offering a versatile toolkit for researchers dealing with jump processes in complex stochastic environments.

Project Overview

What This Project Is About

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A straightforward introduction to how random processes are modeled and simulated, focusing on equations that describe systems influenced by both continuous fluctuations and sudden jumps. The project investigates numerical methods to approximate solutions to these equations and explains how Malliavin Calculus helps analyze and improve these approximations in the presence of Lévy noise.

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The Problem It Addresses

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Many real-world systems experience random shocks that occur abruptly, not just smooth randomness. Traditional methods struggle to accurately capture these jumps, leading to less reliable predictions. This project aims to develop and test numerical ways to approximate such jump-diffusion models and to understand how small changes in inputs affect the outputs (sensitivity analysis) using Malliavin Calculus concepts.

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Objectives of the Project

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1. Understand stochastic differential equations with Lévy noise at a high level.
2. Learn numerical schemes for approximating solutions of these equations.
3. Study Malliavin Calculus ideas relevant to sensitivity and error analysis.
4. Develop and implement a simple numerical method for a chosen model.
5. Evaluate accuracy through tests and compare with known benchmarks.

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What You Will Do Step by Step

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1. Review basic probability, stochastic processes, and Lévy noise concepts.
2. Choose a representative stochastic differential equation with jumps.
3. Implement a basic numerical scheme to approximate its solution.
4. Incorporate ideas from Malliavin Calculus to study the effect of input changes.
5. Run simulations to test accuracy and stability under different scenarios.
6. Analyze errors and interpret results in simple terms.
7. Document methods, findings, and potential improvements.

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Expected Outcome

A clear, implementable numerical approach for SDEs with Lévy noise and a basic sensitivity analysis framework. Students will gain hands-on coding experience, understand when jump effects matter, and appreciate how Malliavin Calculus informs error and stability considerations.