The Annals of Probability

Loss of regularity for Kolmogorov equations

Martin Hairer, Martin Hutzenthaler, and Arnulf Jentzen

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Abstract

The celebrated Hörmander condition is a sufficient (and nearly necessary) condition for a second-order linear Kolmogorov partial differential equation (PDE) with smooth coefficients to be hypoelliptic. As a consequence, the solutions of Kolmogorov PDEs are smooth at all positive times if the coefficients of the PDE are smooth and satisfy Hörmander’s condition even if the initial function is only continuous but not differentiable. First-order linear Kolmogorov PDEs with smooth coefficients do not have this smoothing effect but at least preserve regularity in the sense that solutions are smooth if their initial functions are smooth. In this article, we consider the intermediate regime of nonhypoelliptic second-order Kolmogorov PDEs with smooth coefficients. The main observation of this article is that there exist counterexamples to regularity preservation in that case. More precisely, we give an example of a second-order linear Kolmogorov PDE with globally bounded and smooth coefficients and a smooth initial function with compact support such that the unique globally bounded viscosity solution of the PDE is not even locally Hölder continuous. From the perspective of probability theory, the existence of this example PDE has the consequence that there exists a stochastic differential equation (SDE) with globally bounded and smooth coefficients and a smooth function with compact support which is mapped by the corresponding transition semigroup to a function which is not locally Hölder continuous. In other words, degenerate noise can have a roughening effect. A further implication of this loss of regularity phenomenon is that numerical approximations may converge without any arbitrarily small polynomial rate of convergence to the true solution of the SDE. More precisely, we prove for an example SDE with globally bounded and smooth coefficients that the standard Euler approximations converge to the exact solution of the SDE in the strong and numerically weak sense, but at a rate that is slower then any power law.

Article information

Source
Ann. Probab. Volume 43, Number 2 (2015), 468-527.

Dates
First available in Project Euclid: 2 February 2015

Permanent link to this document
http://projecteuclid.org/euclid.aop/1422885568

Digital Object Identifier
doi:10.1214/13-AOP838

Mathematical Reviews number (MathSciNet)
MR3305998

Zentralblatt MATH identifier
1322.35083

Subjects
Primary: 35B65: Smoothness and regularity of solutions

Keywords
Kolmogorov equation loss of regularity roughening effect smoothing effect hypoellipticity Hörmander condition viscosity solution degenerate noise nonglobally Lipschitz continuous

Citation

Hairer, Martin; Hutzenthaler, Martin; Jentzen, Arnulf. Loss of regularity for Kolmogorov equations. Ann. Probab. 43 (2015), no. 2, 468--527. doi:10.1214/13-AOP838. http://projecteuclid.org/euclid.aop/1422885568.


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References

  • [1] Alfonsi, A. (2012). Strong convergence of some drift implicit Euler scheme. Application to the CIR process. Available at arXiv:1206.3855.
  • [2] Barles, G. and Perthame, B. (1987). Discontinuous solutions of deterministic optimal stopping time problems. RAIRO Modél. Math. Anal. Numér. 21 557–579.
  • [3] Burrage, K., Burrage, P. M. and Tian, T. (2004). Numerical methods for strong solutions of stochastic differential equations: An overview. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 373–402.
  • [4] Cambanis, S. and Hu, Y. (1996). Exact convergence rate of the Euler–Maruyama scheme, with application to sampling design. Stochastics Stochastics Rep. 59 211–240.
  • [5] Cerrai, S. (2001). Second Order PDE’s in Finite and Infinite Dimension: A Probabilistic Approach. Lecture Notes in Math. 1762. Springer, Berlin.
  • [6] Clark, J. M. C. and Cameron, R. J. (1980). The maximum rate of convergence of discrete approximations for stochastic differential equations. In Stochastic Differential Systems (Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978). Lecture Notes in Control and Information Sci. 25 162–171. Springer, Berlin.
  • [7] Crandall, M. G., Ishii, H. and Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 1–67.
  • [8] Crandall, M. G. and Lions, P.-L. (1981). Condition d’unicité pour les solutions généralisées des équations de Hamilton–Jacobi du premier ordre. C. R. Acad. Sci. Paris Sér. I Math. 292 183–186.
  • [9] Crandall, M. G. and Lions, P.-L. (1983). Viscosity solutions of Hamilton–Jacobi equations. Trans. Amer. Math. Soc. 277 1–42.
  • [10] Davie, A. M. and Gaines, J. G. (2001). Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations. Math. Comp. 70 121–134.
  • [11] Da Prato, G. (2004). Kolmogorov Equations for Stochastic PDEs. Birkhäuser, Basel.
  • [12] Da Prato, G. and Zabczyk, J. (2002). Second Order Partial Differential Equations in Hilbert Spaces. London Mathematical Society Lecture Note Series 293. Cambridge Univ. Press, Cambridge.
  • [13] Dereich, S., Neuenkirch, A. and Szpruch, L. (2012). An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 1105–1115.
  • [14] Dörsek, P. (2012). Semigroup splitting and cubature approximations for the stochastic Navier–Stokes equations. SIAM J. Numer. Anal. 50 729–746.
  • [15] Elworthy, K. D. (1978). Stochastic dynamical systems and their flows. In Stochastic Analysis (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1978) 79–95. Academic Press, New York.
  • [16] Evans, L. C. (1978). A convergence theorem for solutions of nonlinear second-order elliptic equations. Indiana Univ. Math. J. 27 875–887.
  • [17] Evans, L. C. (1980). On solving certain nonlinear partial differential equations by accretive operator methods. Israel J. Math. 36 225–247.
  • [18] Evans, L. C. (2010). Partial Differential Equations, 2nd ed. Graduate Studies in Mathematics 19. Amer. Math. Soc., Providence, RI.
  • [19] Fang, S., Imkeller, P. and Zhang, T. (2007). Global flows for stochastic differential equations without global Lipschitz conditions. Ann. Probab. 35 180–205.
  • [20] Gīhman, Ĭ. Ī. and Skorohod, A. V. (1972). Stochastic Differential Equations. Springer, New York.
  • [21] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering: Stochastic Modelling and Applied Probability. Applications of Mathematics (New York) 53. Springer, New York.
  • [22] Gyöngy, I. (1998). A note on Euler’s approximations. Potential Anal. 8 205–216.
  • [23] Gyöngy, I. (2002). Approximations of stochastic partial differential equations. In Stochastic Partial Differential Equations and Applications (Trento, 2002). Lecture Notes in Pure and Applied Mathematics 227 287–307. Dekker, New York.
  • [24] Gyöngy, I. and Krylov, N. (1996). Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Related Fields 105 143–158.
  • [25] Gyöngy, I. and Rásonyi, M. (2011). A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients. Stochastic Process. Appl. 121 2189–2200.
  • [26] Hairer, M. (2011). On Malliavin’s proof of Hörmander’s theorem. Bull. Sci. Math. 135 650–666.
  • [27] Higham, D. J. and Kloeden, P. E. (2007). Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems. J. Comput. Appl. Math. 205 949–956.
  • [28] Higham, D. J., Mao, X. and Stuart, A. M. (2002). Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40 1041–1063 (electronic).
  • [29] Hofmann, N., Müller-Gronbach, T. and Ritter, K. (2000). Optimal approximation of stochastic differential equations by adaptive step-size control. Math. Comp. 69 1017–1034.
  • [30] Hofmann, N., Müller-Gronbach, T. and Ritter, K. (2000). Step size control for the uniform approximation of systems of stochastic differential equations with additive noise. Ann. Appl. Probab. 10 616–633.
  • [31] Hörmander, L. (1967). Hypoelliptic second order differential equations. Acta Math. 119 147–171.
  • [32] Hörmander, L. (1990). The Analysis of Linear Partial Differential Operators. I: Distribution Theory and Fourier Analysis, 2nd ed. Grundlehren der Mathematischen Wissenschaften 256. Springer, Berlin.
  • [33] Hu, Y. (1996). Semi-implicit Euler–Maruyama scheme for stiff stochastic equations. In Stochastic Analysis and Related Topics, V (Silivri, 1994). Progress in Probability 38 183–202. Birkhäuser, Boston, MA.
  • [34] Hutzenthaler, M. and Jentzen, A. (2014). Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Mem. Amer. Math. Soc. To appear. Available at arXiv:1203.5809.
  • [35] Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2011). Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 1563–1576.
  • [36] Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2012). Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22 1611–1641.
  • [37] Ishii, H. (1989). A boundary value problem of the Dirichlet type for Hamilton–Jacobi equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 16 105–135.
  • [38] Jentzen, A. and Kloeden, P. E. (2009). The numerical approximation of stochastic partial differential equations. Milan J. Math. 77 205–244.
  • [39] Jentzen, A., Kloeden, P. E. and Neuenkirch, A. (2009). Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients. Numer. Math. 112 41–64.
  • [40] Klenke, A. (2008). Probability Theory: A Comprehensive Course. Springer, London.
  • [41] Kloeden, P. E. and Neuenkirch, A. (2013). Convergence of numerical methods for stochastic differential equations in mathematical finance. In Recent Developments in Computational Finance (T. Gerstner and P. Kloeden, eds.) 49–80. World Scientific, Singapore.
  • [42] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
  • [43] Kloeden, P. E., Platen, E. and Schurz, H. (1994). Numerical Solution of SDE Through Computer Experiments. Springer, Berlin.
  • [44] Kolmogoroff, A. (1931). Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104 415–458.
  • [45] Kruse, R. (2012). Characterization of bistability for stochastic multistep methods. BIT 52 109–140.
  • [46] Krylov, N. V. (1991). A simple proof of the existence of a solution to the Itô equation with monotone coefficients. Theory Probab. Appl. 35 583–587.
  • [47] Krylov, N. V. (1999). On Kolmogorov’s equations for finite-dimensional diffusions. In Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998). Lecture Notes in Math. 1715 1–63. Springer, Berlin.
  • [48] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
  • [49] Li, X.-M. and Scheutzow, M. (2011). Lack of strong completeness for stochastic flows. Ann. Probab. 39 1407–1421.
  • [50] Mao, X. and Szpruch, L. (2013). Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients. Stochastics 85 144–171.
  • [51] Maruyama, G. (1955). Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo (2) 4 48–90.
  • [52] Milstein, G. N. (1995). Numerical Integration of Stochastic Differential Equations. Mathematics and Its Applications 313. Kluwer Academic, Dordrecht.
  • [53] Milstein, G. N. and Tretyakov, M. V. (2004). Stochastic Numerics for Mathematical Physics. Springer, Berlin.
  • [54] Milstein, G. N. and Tretyakov, M. V. (2005). Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients. SIAM J. Numer. Anal. 43 1139–1154 (electronic).
  • [55] Müller-Gronbach, T. (2002). The optimal uniform approximation of systems of stochastic differential equations. Ann. Appl. Probab. 12 664–690.
  • [56] Müller-Gronbach, T. and Ritter, K. (2007). Lower bounds and nonuniform time discretization for approximation of stochastic heat equations. Found. Comput. Math. 7 135–181.
  • [57] Müller-Gronbach, T. and Ritter, K. (2008). Minimal errors for strong and weak approximation of stochastic differential equations. In Monte Carlo and Quasi-Monte Carlo Methods 2006 53–82. Springer, Berlin.
  • [58] Neuenkirch, A. and Szpruch, L. (2014). First order strong approximations of scalar SDEs defined in a domain. Numer. Math. Available online.
  • [59] Øksendal, B. (2000). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin.
  • [60] Pardoux, É. and Peng, S. (1992). Backward stochastic differential equations and quasilinear parabolic partial differential equations. In Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991). Lecture Notes in Control and Inform. Sci. 176 200–217. Springer, Berlin.
  • [61] Peng, S. (2010). Nonlinear expectations and stochastic calculus under uncertainty. Available at arXiv:1002.4546v1.
  • [62] Peng, S. (1993). Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim. 27 125–144.
  • [63] Prévôt, C. and Röckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Math. 1905. Springer, Berlin.
  • [64] Röckner, M. (1999). $L^{p}$-analysis of finite and infinite-dimensional diffusion operators. In Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998). Lecture Notes in Math. 1715 65–116. Springer, Berlin.
  • [65] Röckner, M. and Sobol, Z. (2006). Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations. Ann. Probab. 34 663–727.
  • [66] Rümelin, W. (1982). Numerical treatment of stochastic differential equations. SIAM J. Numer. Anal. 19 604–613.
  • [67] Schurz, H. (2006). An axiomatic approach to numerical approximations of stochastic processes. Int. J. Numer. Anal. Model. 3 459–480.
  • [68] Wang, X. and Gan, S. (2013). The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients. J. Difference Equ. Appl. 19 466–490.