L2 convergence of smooth approximations of stochastic differential equations with unbounded coefficients

The aim of this paper is to obtain convergence in mean in the uniform topology of piecewise linear approximations of Stochastic Differential Equations (SDEs) with $C^1$ drift and $C^2$ diffusion coefficients with uniformly bounded derivatives. Convergence analyses for such Wong-Zakai approximations most often assume that the coefficients of the SDE are uniformly bounded. Almost sure convergence in the unbounded case can be obtained using now standard rough path techniques, although $L^q$ convergence appears yet to be established and is of importance for several applications involving Monte-Carlo approximations. We consider $L^2$ convergence in the unbounded case using a combination of traditional stochastic analysis and rough path techniques. We expect our proof technique extend to more general piecewise smooth approximations.

[1]  György Michaletzky,et al.  On Wong–Zakai approximations with δ–martingales , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[2]  Peter K. Friz,et al.  Rough path limits of the Wong–Zakai type with a modified drift term , 2009 .

[3]  Ian Melbourne,et al.  Smooth approximation of stochastic differential equations , 2014, 1403.7281.

[4]  G. Han,et al.  A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale , 2019, Discrete & Continuous Dynamical Systems - B.

[5]  Paul Malliavin,et al.  Stochastic Analysis , 1997, Nature.

[6]  I. Gyongy,et al.  Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations , 2012 .

[7]  L. Coutin.  Rough paths via sewing Lemma , 2012 .

[8]  Martin Hairer,et al.  A Wong-Zakai theorem for stochastic PDEs , 2014, Journal of the Mathematical Society of Japan.

[9]  Philip Protter,et al.  Approximations of Solutions of Stochastic Differential Equations Driven by Semimartingales , 1985 .

[10]  Stochastic differential equations driven by processes generated by divergence form operators I : a Wong-Zakai theorem , 2006 .

[11]  P. Kloeden,et al.  Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients , 2010, 1010.3756.

[12]  Laure Coutin,et al.  Good rough path sequences and applications to anticipating stochastic calculus , 2007, 0707.4546.

[13]  Adam Bowditch.  Stochastic Analysis , 2013 .

[14]  W. Stannat,et al.  Stochastic partial differential equations: a rough path view , 2014, 1412.6557.

[15]  I. Bailleul,et al.  Non-explosion criteria for rough differential equations driven by unbounded vector fields , 2018, 1802.04605.

[16]  Philip Protter,et al.  Stratonovich stochastic differential equations driven by general semimartingales , 1995 .

[17]  Nicolas Victoir,et al.  Differential equations driven by Gaussian signals , 2007, 0707.0313.

[18]  Sebastian Reich,et al.  McKean-Vlasov SDEs in nonlinear filtering , 2020, ArXiv.

[19]  J. Zabczyk,et al.  Wong-Zakai approximations of stochastic evolution equations , 2006 .

[20]  Wilhelm Huisinga,et al.  Extracting macroscopic stochastic dynamics: Model problems , 2003 .

[21]  F. Flandoli,et al.  Almost sure approximation of Wong-Zakai type for stochastic partial differential equations , 1995 .

[22]  N. Ikeda,et al.  A Class of Approximations of Brownian Motion , 1977 .

[23]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[24]  Global Solutions to Rough Differential Equations with Unbounded Vector Fields , 2012 .

[25]  Dan Crisan,et al.  On a robust version of the integral representation formula of nonlinear filtering , 2005 .

[26]  Grigorios A. Pavliotis,et al.  Multiscale Methods: Averaging and Homogenization , 2008 .

[27]  J. Mémin,et al.  Convergence en loi des suites d'intégrales stochastiques sur l'espace $$\mathbb{D}$$ 1 de Skorokhod , 1989 .

[28]  Limits of the Wong-Zakai Type with a Modified Drift Term , 1991 .

[29]  L. Synge.  North - Holland Publishing Company , 1960 .

[30]  A. Davie,et al.  Individual Path Uniqueness of Solutions of Stochastic Differential Equations , 2011 .

[31]  E. Wong,et al.  ON THE RELATION BETWEEN ORDINARY AND STOCHASTIC DIFFERENTIAL EQUATIONS , 1965 .

[32]  S. Varadhan,et al.  On the Support of Diffusion Processes with Applications to the Strong Maximum Principle , 1972 .

[33]  Antoine Lejay,et al.  An Introduction to Rough Paths , 2003 .

[34]  N. U. Prabhu,et al.  Stochastic Processes and Their Applications , 1999 .

[35]  P. S. Dwyer.  Annals of Applied Probability , 2006 .

[36]  Terry Lyons,et al.  System Control and Rough Paths , 2003 .

[37]  Max L. Warshauer,et al.  Lecture Notes in Mathematics , 2001 .

[38]  Terry Lyons.  Di erential equations driven by rough signals , 1998 .

[39]  E. J. McShane.  Stochastic Differential Equations and Models of Random Processes , 1972 .

[40]  G. Kallianpur,et al.  An Approximation for the Zakai Equation , 2002 .

[41]  H. P..  Annales de l'Institut Henri Poincaré , 1931, Nature.

[42]  P. Protter,et al.  Stochastic Analysis: Characterizing the weak convergence of stochastic integrals , 1991 .

[43]  A. M. Davie,et al.  Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation , 2007, 0710.0772.

[44]  Peter K. Friz,et al.  Multidimensional Stochastic Processes as Rough Paths: Theory and Applications , 2010 .

[45]  Massimiliano Gubinelli.  Controlling rough paths , 2003 .

[46]  京都大学数理解析研究所,et al.  Publications of the Research Institute for Mathematical Sciences , 1965 .

[47]  I. G. BONNER CLAPPISON.  Editor , 1960, The Electric Power Engineering Handbook - Five Volume Set.

[48]  I. Bailleul.  Flows driven by rough paths , 2012, 1203.0888.

[49]  Société de mathématiques appliquées et industrielles,et al.  ESAIM. Probability and statistics , 1997 .