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September  2021, 26(9): 4927-4962. doi: 10.3934/dcdsb.2020320

## 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 Published  September 2021 Early access  November 2020

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.

Citation: Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4927-4962. doi: 10.3934/dcdsb.2020320
##### 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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [12] P. Henry-Labordère, Counterparty risk valuation: A marked branching diffusion approach, preprint, 17 pages, arXiv: 1203.2369. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [16] A. Kalinin, Markovian integral equations, Ann. Inst. Henri Poincaré Probab. Stat. 56, 1 (2020), 155–174. doi: 10.1214/19-AIHP958.  Google Scholar [17] O. Kallenberg, Foundations of Modern Probability, 2$^nd$ edition, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar [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.  Google Scholar [19] S. Lang, Fundamentals of Differential Geometry, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.  Google Scholar [20] W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar [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.  Google Scholar [22] W. Rudin, Real and Complex Analysis, 3$^rd$ edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar [23] I. Segal, Non-linear semi-groups, Ann. of Math., 78 (1963), 339-364.  doi: 10.2307/1970347.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [12] P. Henry-Labordère, Counterparty risk valuation: A marked branching diffusion approach, preprint, 17 pages, arXiv: 1203.2369. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [16] A. Kalinin, Markovian integral equations, Ann. Inst. Henri Poincaré Probab. Stat. 56, 1 (2020), 155–174. doi: 10.1214/19-AIHP958.  Google Scholar [17] O. Kallenberg, Foundations of Modern Probability, 2$^nd$ edition, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar [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.  Google Scholar [19] S. Lang, Fundamentals of Differential Geometry, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.  Google Scholar [20] W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar [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.  Google Scholar [22] W. Rudin, Real and Complex Analysis, 3$^rd$ edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar [23] I. Segal, Non-linear semi-groups, Ann. of Math., 78 (1963), 339-364.  doi: 10.2307/1970347.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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