Solving linear parabolic rough partial differential equations. (English) Zbl 1442.65303
Summary: We study linear rough partial differential equations in the setting of P. K. Friz and M. Hairer [A course on rough paths. With an introduction to regularity structures. Cham: Springer (2014; Zbl 1327.60013), Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path W of Hölder regularity \(\alpha\) with \(1 / 3 < \alpha \leq 1 / 2\). Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, we present a simulation study showing the applicability of the proposed algorithm.
MSC:
| 65M75 | Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs |
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
rough paths; rough partial differential equations; Monte Carlo; Feynman-Kac formula; nonparametric regression; nonlinear stochastic systemsCitations:
Zbl 1327.60013References:
| [1] | Amann, Herbert, Ordinary Differential Equations: An Introduction to Nonlinear Analysis, vol. 13 (1990), Walter de Gruyter · Zbl 0708.34002 |
| [2] | Anker, Felix; Bayer, Christian; Eigel, Martin; Ladkau, Marcel; Neumann, Johannes; Schoenmakers, John, SDE based regression for linear random PDEs, SIAM J. Sci. Comput., 39, 3, A1168-A1200 (2017) · Zbl 1372.65012 |
| [3] | Baudoin, Fabrice, An Introduction to the Geometry of Stochastic Flows (2004), World Scientific · Zbl 1085.60002 |
| [4] | Baumann, Manuel; Benner, Peter; Heiland, Jan, Space-time Galerkin POD with application in optimal control of semi-linear parabolic partial differential equations (2016), preprint · Zbl 1392.35323 |
| [5] | Bayer, Christian; Friz, Peter K.; Riedel, Sebastian; Schoenmakers, John, From rough path estimates to multilevel Monte Carlo, SIAM J. Numer. Anal., 54, 3, 1449-1483 (2016) · Zbl 1343.60097 |
| [6] | Bayer, Christian; Horst, Ulrich; Qiu, Jinniao, A functional limit theorem for limit order books with state dependent price dynamics, Ann. Appl. Probab., 27, 5, 2753-2806 (2017), 10 · Zbl 1379.60036 |
| [7] | Benner, Peter; Redmann, Martin, Model reduction for stochastic systems, Stoch. Partial Differ. Equ. Anal. Comput., 3, 3, 291-338 (2015) · Zbl 1331.65017 |
| [8] | Caruana, Michael; Friz, Peter K.; Oberhauser, Harald, A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. Henri Poincaré (C), Anal. Non Linéaire, 28, 1, 27-46 (2011) · Zbl 1219.60061 |
| [9] | Cass, Thomas; Litterer, Christian; Lyons, Terry, Integrability and tail estimates for Gaussian rough differential equations, Ann. Probab., 41, 4, 3026-3050 (2013) · Zbl 1278.60091 |
| [10] | Deya, Aurélien; Gubinelli, Massimiliano; Hofmanova, Martina; Tindel, Samy, A priori estimates for rough PDEs with application to rough conservation laws (2016), preprint · Zbl 1411.60094 |
| [11] | Deya, Aurélien; Gubinelli, Massimiliano; Tindel, Samy, Non-linear rough heat equations, Probab. Theory Relat. Fields, 153, 1-2, 97-147 (2012) · Zbl 1255.60106 |
| [12] | Deya, Aurélien; Neuenkirch, Andreas; Tindel, Samy, A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion, Ann. Inst. Henri Poincaré Probab. Stat., 48, 2, 518-550 (2012) · Zbl 1260.60135 |
| [13] | Diehl, Joscha; Friz, Peter; Stannat, Wilhelm, Stochastic partial differential equations: a rough paths view on weak solutions via Feynman-Kac, Ann. Fac. Sci. Toulouse Sér. 6, 26, 4, 911-947 (2017) · Zbl 1392.35335 |
| [14] | Diehl, Joscha; Oberhauser, Harald; Riedel, Sebastian, A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations, Stoch. Process. Appl., 125, 161-181 (2015) · Zbl 1304.60065 |
| [15] | Friz, Peter; Hairer, Martin, A Course on Rough Paths (2014), Springer · Zbl 1327.60013 |
| [16] | Friz, Peter; Oberhauser, Harald, Rough path limits of the Wong-Zakai type with a modified drift term, J. Funct. Anal., 256, 10, 3236-3256 (2009) · Zbl 1169.60011 |
| [17] | Friz, Peter; Riedel, Sebastian, Integrability of (non-)linear rough differential equations and integrals, Stoch. Anal. Appl., 31, 2, 336-358 (2013) · Zbl 1274.60173 |
| [18] | Friz, Peter K.; Gassiat, Paul; Lions, Pierre-Louis; Souganidis, Panagiotis E., Eikonal equations and pathwise solutions to fully non-linear SPDEs, Stoch. Partial Differ. Equ., Anal. Computat., 5, 2, 256-277 (2017) · Zbl 1377.35293 |
| [19] | Friz, Peter K.; Victoir, Nicolas B., Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge Studies in Advanced Mathematics, vol. 120 (2010), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1193.60053 |
| [20] | Friz, Peter K.; Zhang, Huilin, Differential equations driven by rough paths with jumps, J. Differ. Equ., 264, 10, 6226-6301 (2018) · Zbl 1432.60098 |
| [21] | Györfi, László; Kohler, Michael; Krzyżak, Adam; Walk, Harro, A Distribution-Free Theory of Nonparametric Regression, Springer Series in Statistics (2002), Springer-Verlag: Springer-Verlag New York · Zbl 1021.62024 |
| [22] | Hairer, Martin, Solving the KPZ equation, Ann. Math., 178, 2, 559-664 (2013) · Zbl 1281.60060 |
| [23] | Han, Jiequn; Jentzen, Arnulf; Weinan, E., Solving high-dimensional partial differential equations using deep learning, Proc. Natl. Acad. Sci., 115, 34, 8505-8510 (2018) · Zbl 1416.35137 |
| [24] | Kotelenez, Peter, Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations, vol. 58 (2007), Springer Science & Business Media · Zbl 1159.60004 |
| [25] | Kruse, Raphael, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Lecture Notes in Mathematics, vol. 2093 (2014), Springer · Zbl 1285.60002 |
| [26] | Lyons, Terry, Rough paths, signatures and the modelling of functions on streams (2014), preprint · Zbl 1373.93158 |
| [27] | Lyons, Terry J., Differential equations driven by rough signals, Rev. Mat. Iberoam., 14, 2, 215-310 (1998) · Zbl 0923.34056 |
| [28] | Milstein, G. N.; Tretyakov, M. V., Stochastic Numerics for Mathematical Physics, Scientific Computation (2004), Springer-Verlag: Springer-Verlag Berlin · Zbl 1085.60004 |
| [29] | Milstein, G. N.; Tretyakov, M. V., Solving the Dirichlet problem for Navier-Stokes equations by probabilistic approach, BIT Numer. Math., 52, 1, 141-153 (2012) · Zbl 1387.35460 |
| [30] | Pardoux, Étienne, Filtrage non linéaire et équations aux dérivées partielles stochastiques associées, (Ecole d’Eté de Probabilités de Saint-Flour XIX—1989 (1991), Springer), 68-163 |
| [31] | Strichartz, Robert S., The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations, J. Funct. Anal., 72, 2, 320-345 (1987) · Zbl 0623.34058 |
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