A new method for solving Kolmogorov equations in mathematical finance

Abstract

For multi-dimensional Fokker-Planck-Kolmogorov equations, we propose a numerical method which is based on a novel localization technique. We present extensive numerical experiments that demonstrate its practical interest for finance applications. In particular, this approach allows us to treat calibration and valuation problems, as well as various risk measure computations.

The Transport‐based Mesh‐free Method: A Short Review

“…In this paper based on [7][8][9], we presented a new analysis of Monte-Carlo-type integration formulas, which is relevant in a variety of applications and leads to sharp error estimates of practical interest.…”

“…Our TMM allows us to solve the above two sets of equations, namely the Fokker-Planck and the Kolmogorov equations. Here, we only outline the arguments and explain how quantitative error estimates are be ensured; we refer the reader to [7][8][9] for further details. We emphasize that the proposed framework can be used for more general problems of hyperbolic-parabolic type, such as the Hamilton-Jacobi equations [6], Euler equations, and Navier-Stokes equations.…”

“…Consider the Fokker-Planck equation (4.2) together with the Monte-Carlo-type error estimate (2.1). Once a kernel K is selected, we can apply the numerical scheme presented earlier in [7], which is a stable and consistent approximation of the Fokker-Planck equation (42) and provides an approximation of the solution . This approximation is a discrete probability measure of the form 1…”

“…for instance [4,5,15]) and shares some analogies with Lagrangian mesh-free methods. Importantly, we can handle both transport terms and diffusive terms, and it can be regarded as a generalization of our earlier work [7].…”

“…The proposed method was tested extensively [7][8][9] in several academic tests. A business case in asset and liability management using the so-called Libor market model [2] was then treated in [11].…”