Minimal Kullback-Leibler Divergence for Constrained Levy-Ito Processes

33 Pages Posted: 18 Jul 2022 Last revised: 3 Aug 2022

See all articles by Sebastian Jaimungal

Sebastian Jaimungal

University of Toronto - Department of Statistics

Silvana M. Pesenti

University of Toronto

Leandro Sánchez-Betancourt

King's College London

Date Written: August 2, 2022

Abstract

Given an n-dimensional stochastic process X driven by P-Brownian motions and Poisson random measures, we seek the probability measure Q, with minimal relative entropy to P, such that the Q-expectations of some terminal and running costs are constrained. We prove existence and uniqueness of the optimal probability measure, derive the explicit form of the measure change, and characterise the optimal drift and compensator adjustments under the optimal measure.

We provide an analytical solution for Value-at-Risk (quantile) constraints, discuss how to perturb a Brownian motion to have arbitrary variance, and show that pinned measures arise as a limiting case of optimal measures. The results are illustrated in a risk management setting - including an algorithm to simulate under the optimal measure - and explore an example where an agent seeks to answer the question: what dynamics are induced by a perturbation of the Value-at-Risk and the average time spent below a barrier on the reference process?

Keywords: Relative entropy, Kullback-Leibler, Levy-Ito processes, Reverse sensitivity, Risk Management, Model Uncertainty, Cryptocurrency

Suggested Citation

Jaimungal, Sebastian and Pesenti, Silvana M. and Sánchez-Betancourt, Leandro, Minimal Kullback-Leibler Divergence for Constrained Levy-Ito Processes (August 2, 2022). Available at SSRN: https://ssrn.com/abstract=4149871 or http://dx.doi.org/10.2139/ssrn.4149871

Sebastian Jaimungal

University of Toronto - Department of Statistics ( email )

100 St. George St.
Toronto, Ontario M5S 3G3
Canada

HOME PAGE: http://http:/sebastian.statistics.utoronto.ca

Silvana M. Pesenti (Contact Author)

University of Toronto ( email )

100 St. George Street
Toronto, Ontario M5S 3G8
Canada

Leandro Sánchez-Betancourt

King's College London ( email )

Strand
London, England WC2R 2LS
United Kingdom

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