First time to exit of a continuous Itô process: General moment estimates and L1 -convergence rate for discrete time approximations

We establish general moment estimates for the discrete and continuous exit times of a general It\^{o} process in terms of the distance to the boundary. These estimates serve as intermediate steps to obtain strong convergence results for the approximation of a continuous exit time by a discrete counterpart, computed on a grid. In particular, we prove that the discrete exit time of the Euler scheme of a diffusion converges in the $L_1$ norm with an order $1/2$ with respect to the mesh size.

[1]  G. Maruyama.  Continuous Markov processes and stochastic equations , 1955 .

[2]  M. E. Muller.  Some Continuous Monte Carlo Methods for the Dirichlet Problem , 1956 .

[3]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[4]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal stopping rules , 1977 .

[5]  Alain Bensoussan,et al.  Impulse Control and Quasi-Variational Inequalities , 1984 .

[6]  M. Freidlin.  Functional Integration And Partial Differential Equations , 1985 .

[7]  K. D. Elworthy,et al.  FUNCTIONAL INTEGRATION AND PARTIAL DIFFERENTIAL EQUATIONS (Annals of Mathematics Studies, 109) , 1986 .

[8]  D. Heath,et al.  Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation , 1990, Journal of Financial and Quantitative Analysis.

[9]  Nigel J. Newton.  An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times , 1990 .

[10]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[11]  O. Faure,et al.  Simulation du mouvement brownien et des diffusions , 1992 .

[12]  H. Soner,et al.  Optimal Investment and Consumption with Transaction Costs , 1994 .

[13]  Paolo Baldi,et al.  Exact asymptotics for the probability of exit from a domain and applications to simulation , 1995 .

[14]  The simulation of phase trajectories of a diffusion process in a bounded domain , 1996 .

[15]  G. Milstein.  Weak approximation of a diffusion process in a bounded domain , 1997 .

[16]  On the mean-square approximation of a diffusion process in a bounded domain , 1998 .

[17]  Michael V. Tretyakov,et al.  SIMULATION OF A SPACE-TIME BOUNDED DIFFUSION , 1999 .

[18]  Evelyn Buckwar,et al.  Introduction to the numerical analysis of stochastic delay differential equations , 2000 .

[19]  E. Gobet.  Weak approximation of killed diffusion using Euler schemes , 2000 .

[20]  E. Gobet.  Euler schemes and half-space approximation for the simulation of diffusion in a domain , 2001 .

[21]  Stefan Heinrich,et al.  Multilevel Monte Carlo Methods , 2001, LSSC.

[22]  K. Bichteler.  Stochastic Integration with Jumps , 2002 .

[23]  Grant D. Lythe,et al.  Multidimensional Exponential Timestepping with Boundary Test , 2005, SIAM J. Sci. Comput..

[24]  Annie Millet,et al.  On Discretization Schemes for Stochastic Evolution Equations , 2005, math/0611069.

[25]  Sylvain Maire,et al.  Sequential Control Variates for Functionals of Markov Processes , 2005, SIAM J. Numer. Anal..

[26]  FABIAN M. BUCHMANN,et al.  An Exit Probability Approach to Solving High Dimensional Dirichlet Problems , 2006, SIAM J. Sci. Comput..

[27]  Antoine Lejay,et al.  A Random Walk on Rectangles Algorithm , 2006 .

[28]  E. Gobet,et al.  Stopped diffusion processes: Boundary corrections and overshoot , 2007, 0706.4042.

[29]  B. Bouchard,et al.  Strong approximations of BSDEs in a domain , 2007, 0710.1519.

[30]  Rainer Avikainen,et al.  Convergence Rates for Approximations of Functionals of SDEs , 2007, 0712.3635.

[31]  Emmanuel Gobet,et al.  Discrete Sampling of Functionals of Ito Processes , 2007 .

[32]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[33]  A. Lejay,et al.  An Efficient Algorithm to Simulate a Brownian Motion Over Irregular Domains , 2010 .

[34]  M. Fukasawa.  Discretization error of Stochastic Integrals , 2010, 1004.2107.

[35]  Albert N. Shiryaev,et al.  Optimal Stopping Rules , 2011, International Encyclopedia of Statistical Science.

[36]  E. Gobet,et al.  Almost sure optimal hedging strategy , 2014, 1405.4170.

[37]  T. Faniran.  Numerical Solution of Stochastic Differential Equations , 2015 .