[Paper Review] Feynman-Kac formula for Lévy processes with discontinuous killing rate

Kathrin Glau|arXiv (Cornell University)|Feb 26, 2015

Stochastic processes and financial applications23 references3 citations

TL;DR

This paper establishes a rigorous Feynman-Kac representation for variational solutions to parabolic evolution equations driven by time-inhomogeneous Lévy processes with discontinuous killing rates. By leveraging weak solution theory in Sobolev-Slobodeckii spaces and proving continuity and growth conditions on the generator's symbol, the authors provide a mathematically sound foundation for fast deterministic option pricing in Lévy models, including generalized hyperbolic and normal inverse Gaussian processes.

ABSTRACT

The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to compute option prices in Lévy models by solving partial integro differential equations have been developed. In order to provide a solid mathematical foundation for these methods, we derive a Feynman-Kac representation of variational solutions to partial integro differential equations that characterize conditional expectations of functionals of killed time-inhomogeneous Lévy processes. We allow for a wide range of underlying stochastic processes, comprising processes with Brownian part, and a broad class of pure jump processes such as generalized hyperbolic, multivariate normal inverse Gaussian, tempered stable, and $α$-semi stable Lévy processes. By virtue of our mild regularity assumptions as to the killing rate and the initial condition of the partial differential equation, our results provide a rigorous basis for numerous applications, not only in financial mathematics but also in probability theory and relativistic quantum mechanics.

Motivation & Objective

• To establish a mathematically rigorous Feynman-Kac representation for conditional expectations in time-inhomogeneous Lévy processes with discontinuous killing rates.
• To bridge the gap between stochastic processes and deterministic partial integro-differential equations (PIDEs) in financial mathematics by identifying minimal regularity conditions on the killing rate and initial data.
• To provide a theoretical basis for fast deterministic option pricing methods in Lévy models, especially those with jumps and non-smooth payoff structures.
• To extend the classical Feynman-Kac framework to include non-local operators and pseudo-differential operators arising in Lévy processes with general characteristics.
• To validate the use of Galerkin schemes for numerical solution of the resulting PIDEs by proving existence and uniqueness of variational solutions.

Proposed method

• Derives a Feynman-Kac-type representation for weak solutions of time-inhomogeneous PIDEs with discontinuous killing rates using variational methods in Sobolev-Slobodeckii spaces.
• Establishes continuity and growth conditions on the symbol of the Lévy generator to ensure the coercivity and boundedness of the associated bilinear form.
• Applies the theory of parabolic evolution equations in Hilbert spaces to prove existence and uniqueness of variational solutions to the PIDE.
• Uses Fourier analysis and complex extension of the symbol to characterize the generator’s analyticity and continuity in the complex domain.
• Proves that piecewise continuity of the bilinear form implies piecewise continuity of the symbol, ensuring regularity of the solution operator.
• Implements a Galerkin method for numerical solution of the PIDE, demonstrating the effect of the killing rate on option prices.

Experimental results

Research questions

• RQ1 Under what conditions does a Feynman-Kac representation hold for conditional expectations of functionals of time-inhomogeneous Lévy processes with discontinuous killing rates?
• RQ2 How can the connection between stochastic processes and PIDEs be rigorously established when the killing rate is discontinuous and the generator is non-local?
• RQ3 What minimal regularity assumptions on the killing rate and initial condition ensure existence and uniqueness of variational solutions to the associated PIDE?
• RQ4 In what sense can the solution to the PIDE be interpreted as a conditional expectation in the context of financial derivatives pricing?
• RQ5 How do the properties of the Lévy generator’s symbol (e.g., continuity, growth) affect the well-posedness of the PIDE and the validity of the Feynman-Kac formula?

Key findings

• A rigorous Feynman-Kac representation is established for variational solutions of PIDEs with discontinuous killing rates, extending classical results to non-smooth and non-Markovian settings.
• The existence and uniqueness of solutions in Sobolev-Slobodeckii spaces are proven under mild regularity assumptions on the killing rate and initial condition.
• The paper shows that the continuity and growth conditions on the symbol of the generator (e.g., for normal inverse Gaussian or tempered stable processes) are sufficient to ensure the coercivity and boundedness of the bilinear form.
• It is demonstrated that piecewise continuity of the bilinear form implies piecewise continuity of the symbol, which is essential for the well-posedness of the evolution equation.
• The Galerkin method is successfully applied to numerically solve the PIDE, and the impact of the killing rate on option prices is illustrated through numerical experiments.
• The framework supports a wide class of Lévy processes, including generalized hyperbolic, multivariate normal inverse Gaussian, tempered stable, and α-semi-stable processes.