On existence and uniqueness properties for solutions of stochastic fixed point equations

Christian Beck; Lukas Gonon; Martin Hutzenthaler; Arnulf Jentzen

doi:10.3934/dcdsb.2020320

Article Contents

2021, Volume 26, Issue 9: 4927-4962. Doi: 10.3934/dcdsb.2020320

This issue Previous Article A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation Next Article A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition

On existence and uniqueness properties for solutions of stochastic fixed point equations

1.
ETH Zurich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland
2.
University of St. Gallen, Faculty of Mathematics and Statistics, Dufourstrasse 50, 9000 St. Gallen, Switzerland
3.
LMU Munich, Department of Mathematics, Theresienstraße 39, 80333 München, Germany
4.
University of Duisburg-Essen, Faculty of Mathematics, Thea-Leymann-Straße 9, 45127 Essen, Germany
5.
University of Münster, Faculty of Mathematics and Computer Science, Einsteinstraße 62, 48149 Münster, Germany

Received: April 2020

Early access: November 2020

Published: September 2021

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

The Feynman–Kac formula implies that every suitable classical solution of a semilinear Kolmogorov partial differential equation (PDE) is also a solution of a certain stochastic fixed point equation (SFPE). In this article we study such and related SFPEs. In particular, the main result of this work proves existence of unique solutions of certain SFPEs in a general setting. As an application of this main result we establish the existence of unique solutions of SFPEs associated with semilinear Kolmogorov PDEs with Lipschitz continuous nonlinearities even in the case where the associated semilinear Kolmogorov PDE does not possess a classical solution.

Keywords:

Mathematics Subject Classification: Primary:60H30, 60H10;Secondary:45G10, 47D62.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	C. Beck, F. Hornung, M. Hutzenthaler, A. Jentzen and T. Kruse, Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations, J. Numer. Math., 28 (2020), 197-222. doi: 10.1515/jnma-2019-0074.
[2]	C. Beck, M. Hutzenthaler and A. Jentzen, On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations, preprint, 54 pages, arXiv: 2004.03389.
[3]	C. Bender, N. Schweizer and J. Zhuo, A primal-dual algorithm for BSDEs, Math. Finance, 27 (2017), 866-901. doi: 10.1111/mafi.12100.
[4]	C. Burgard and M. Kjaer, Partial differential equation representations of derivatives with bilateral counterparty risk and funding costs, The Journal of Credit Risk, 7 (2011), 1-19. doi: 10.21314/JCR.2011.131.
[5]	S. Crépey, R. Gerboud, Z. Grbac and N. Ngor, Counterparty risk and funding: the four wings of the TVA, Int. J. Theor. Appl. Finance, 16 (2013), 31 pages. doi: 10.1142/S0219024913500064.
[6]	D. Duffie, M. Schroder and C. Skiadas, Recursive valuation of defaultable securities and the timing of resolution of uncertainty, Ann. Appl. Probab., 6 (1996), 1075-1090. doi: 10.1214/aoap/1035463324.
[7]	W. E, M. Hutzenthaler, A. Jentzen and T. Kruse, Multilevel Picard iterations for solving smooth semilinear parabolic heat equations, preprint, 19 pages, arXiv: 1607.03295.
[8]	W. E, M. Hutzenthaler, A. Jentzen and T. Kruse, On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations, J. Sci. Comput., 79 (2019), 1534-1571. doi: 10.1007/s10915-018-00903-0.
[9]	P. Grohs, F. Hornung, A. Jentzen and P. von Wurstemberger, A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations, preprint, 124 pages, arXiv: 1809.02362.
[10]	I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833.
[11]	M. Hairer, M. Hutzenthaler and A. Jentzen, Loss of regularity for Kolmogorov equations, Ann. Probab., 43 (2015), 468-527. doi: 10.1214/13-AOP838.
[12]	P. Henry-Labordère, Counterparty risk valuation: A marked branching diffusion approach, preprint, 17 pages, arXiv: 1203.2369.
[13]	M. Hutzenthaler, A. Jentzen, T. Kruse, T. A. Nguyen and P. von Wurstemberger, Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations, Accepted by Proc. Roy. Soc. London A, 30 pages, arXiv: 1807.01212.
[14]	M. Hutzenthaler, A. Jentzen and P. von Wurstemberger, Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks, Electronic Journal of Probability, 25 (2020), 73 pages, https://doi.org/10.1214/20-EJP423. doi: 10.1007/s13253-019-00378-y.
[15]	M. Hutzenthaler and T. Kruse, Multilevel Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities, SIAM J. Numer. Anal., 58 (2020), 929-961. doi: 10.1137/17M1157015.
[16]	A. Kalinin, Markovian integral equations, Ann. Inst. Henri Poincaré Probab. Stat. 56, 1 (2020), 155–174. doi: 10.1214/19-AIHP958.
[17]	O. Kallenberg, Foundations of Modern Probability, 2$^nd$ edition, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-4015-8.
[18]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2$^nd$ edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.
[19]	S. Lang, Fundamentals of Differential Geometry, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.
[20]	W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.
[21]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[22]	W. Rudin, Real and Complex Analysis, 3$^rd$ edition, McGraw-Hill Book Co., New York, 1987.
[23]	I. Segal, Non-linear semi-groups, Ann. of Math., 78 (1963), 339-364. doi: 10.2307/1970347.
[24]	D. W. Stroock, Lectures on Topics in Stochastic Differential Equations, vol. 68 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin-New York, 1982, with notes by Satyajit Karmakar.
[25]	F. B. Weissler, Semilinear evolution equations in Banach spaces, J. Functional Analysis, 32 (1979), 277-296. doi: 10.1016/0022-1236(79)90040-5.