Abstract

We propose an approximation scheme for a class of semilinear parabolic equations that are convex and coercive in their gradients. Such equations arise often in pricing and portfolio management in incomplete markets and, more broadly, are directly connected to the representation of solutions to backward stochastic differential equations. The proposed scheme is based on splitting the equation into two parts, the first corresponding to a linear parabolic equation and the second to a Hamilton--Jacobi equation. The solutions of these two equations are approximated using, respectively, the Feynman--Kac and the Hopf--Lax formula. We establish the convergence of the scheme and determine the convergence rate, combining Krylov's shaking coefficients technique and the Barles--Jakobsen optimal switching approximation.

Keywords

  1. splitting
  2. Feynman--Kac formula
  3. Hopf--Lax formula
  4. viscosity solutions
  5. shaking coefficients technique
  6. optimal switching approximation

MSC codes

  1. 35K65
  2. 65M12
  3. 93E20

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Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 165 - 191
ISSN (online): 1095-7138

History

Submitted: 5 July 2018
Accepted: 4 November 2019
Published online: 9 January 2020

Keywords

  1. splitting
  2. Feynman--Kac formula
  3. Hopf--Lax formula
  4. viscosity solutions
  5. shaking coefficients technique
  6. optimal switching approximation

MSC codes

  1. 35K65
  2. 65M12
  3. 93E20

Authors

Affiliations

Funding Information

National Natural Science Foundation of China-Yunnan Joint Fund https://doi.org/10.13039/501100011002 : 11771158
Royal Society https://doi.org/10.13039/501100000288 : 170137
Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universität Freiburg https://doi.org/10.13039/501100003190

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