References
- Bain , A. and Crisan , D. 2009 . Fundamentals of Stochastic Filtering , Berlin, Heidelberg, New York : Springer .
- A. Barth, A. Lang, and C. Schwab, Multi-level Monte Carlo finite element method for parabolic stochastic partial differential equations, SAM-Report 2011–30, ETH, Zürich, 2011.
- Buckwar , E. and Sickenberger , T. 2011 . A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods . Math. Comput. Simulation , 81 : 1110 – 1127 . (doi:10.1016/j.matcom.2010.09.015)
- Bujok , K. and Reisinger , C. 2012 . Numerical valuation of basket credit derivatives in structural jump–diffusion models . J. Comput. Finance , 15 ( 4 ) : 115 – 158 .
- Bush , N. , Hambly , B. , Haworth , H. , Jin , L. and Reisinger , C. 2011 . Stochastic evolution equations in portfolio credit modelling . SIAM Financ. Math. , 2 ( 1 ) : 627 – 664 . (doi:10.1137/100796777)
- Carter , R. and Giles , M. B. 2007 . Sharp error estimates for discretisations of the 1D convection/diffusion equation with Dirac initial data . IMA J. Numer. Anal. , 27 ( 2 ) : 406 – 425 . (doi:10.1093/imanum/drl043)
- Giles , M. B. 2008 . Multi-level Monte Carlo path simulation . Oper. Res. , 56 ( 3 ) : 981 – 986 . (doi:10.1287/opre.1070.0496)
- Giles , M. B. and Reisinger , C. 2012 . Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance . SIAM Financ. Math. , 3 ( 1 ) : 572 – 592 . (doi:10.1137/110841916)
- Gyöngy , I. 1999 . Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space–time white noise II . Potential Anal. , 11 : 1 – 37 . (doi:10.1023/A:1008699504438)
- Gyöngy , I. and Nualart , D. 1997 . Implicit schemes for stochastic quasi-linear parabolic partial differential equations driven by space–time white noise . Potential Anal. , 7 : 725 – 757 . (doi:10.1023/A:1017998901460)
- Heath , D. , Jarrow , R. and Morton , A. 1992 . Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation . Econometrica , 60 ( 1 ) : 77 – 105 . (doi:10.2307/2951677)
- Higham , D. J. 2000 . Mean-square and asymptotic stability of the stochastic theta method . SIAM J. Numer. Anal. , 38 ( 3 ) : 753 – 769 . (doi:10.1137/S003614299834736X)
- Higham , D. J. 2000 . A-stability and stochastic mean-square stability . BIT , 40 ( 2 ) : 404 – 409 . (doi:10.1023/A:1022355410570)
- Jentzen , A. and Kloeden , P. E. 2009 . The numerical approximation of stochastic partial differential equations . Milan J. Math. , 77 : 205 – 244 . (doi:10.1007/s00032-009-0100-0)
- Jentzen , A. and Kloeden , P. E. 2010 . Taylor expansions of solutions of stochastic partial differential equations with additive noise . Ann. Probab. , 38 ( 2 ) : 532 – 569 . (doi:10.1214/09-AOP500)
- Jentzen , A. , Kloeden , P. E. and Winkel , G. 2011 . Efficient simulation of nonlinear parabolic SPDEs with additive noise . Ann. Appl. Probab. , 21 : 908 – 950 . (doi:10.1214/10-AAP711)
- Kloeden , P. E. and Platen , E. 1992 . Numerical Solution of Stochastic Differential Equations , Berlin, Heidelberg, New York : Springer .
- Krylov , N. V. 1994 . A -theory of the Dirichlet problem for SPDEs in general smooth domains . Probab. Theory Related Fields , 98 : 389 – 421 . (doi:10.1007/BF01192260)
- Krylov , N. V. and Lototsky , S. V. 1998 . A Sobolev space theory of SPDEs with constant coefficients on a half line . SIAM J. Math. Anal. , 30 ( 2 ) : 289 – 325 .
- Krylov , N. V. and Rozovskii , B. L. 1981 . Stochastic evolution equations . J. Soviet Math. , 14 : 1233 – 1277 . (doi:10.1007/BF01084893)
- Kurtz , T. G. and Xiong , J. 1999 . Particle representations for a class of nonlinear SPDEs . Stoch. Process. Appl. , 83 : 103 – 126 . (doi:10.1016/S0304-4149(99)00024-1)
- Lang , A. 2010 . A Lax equivalence theorem for stochastic differential equations . J. Comput. Appl. Math. , 234 ( 12 ) : 3387 – 3396 . (doi:10.1016/j.cam.2010.05.001)
- Morton , K. W. and Mayers , D. F. 2005 . Numerical Solution of Partial Differential Equations , 2 , Cambridge : Cambridge University Press .
- Müller-Gronbach , T. and Ritter , K. 2007 . An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise . BIT Numer. Math. , 47 : 339 – 418 . (doi:10.1007/s10543-007-0129-9)
- Müller-Gronbach , T. , Ritter , K. and Wagner , T. 2007 . “ Optimal pointwise approximation of a linear stochastic heat equation with additive space–time white noise ” . In Monte Carlo and Quasi-Monte Carlo Methods 2006 , Edited by: Keller , A. , Heinrich , S. and Niederreiter , H. 577 – 589 . Berlin : Springer-Verlag .
- Musiela , M. and Zariphopoulou , T. 2010 . “ Stochastic partial differential equations and portfolio choice ” . In Contemporary Quantitative Finance , Edited by: Chiarella , C. and Novikov , A. 195 – 215 . Berlin, Heidelberg, New York : Springer .
- Pooley , D. M. , Vetzal , K. R. and Forsyth , P. A. 2003 . Remedies for non-smooth payoffs in option pricing . J. Comput. Finance , 6 : 25 – 40 .
- Richtmyer , R. D. and Morton , K. W. 1967 . Difference Methods for Initial-Value Problems , New York : Wiley .
- Roth , C. 2002 . Difference methods for stochastic partial differential equations . Z. Angew. Math. Mech. , 82 ( 11–12 ) : 821 – 830 . (doi:10.1002/1521-4001(200211)82:11/12<821::AID-ZAMM821>3.0.CO;2-L)
- Saito , Y. and Mitsui , T. 1996 . Stability analysis of numerical schemes for stochastic differential equations . SIAM J. Numer. Anal. , 33 ( 6 ) : 2254 – 2267 . (doi:10.1137/S0036142992228409)
- Schönbucher , P. J. 2003 . Credit Derivatives Pricing Models , Chichester : Wiley .
- Szpruch , L. 2010 . Numerical approximations of nonlinear stochastic systems PhD Thesis, University of Strathclyde
- Walsh , J. B. 2005 . Finite element methods for parabolic stochastic PDEs . Potential Anal. , 23 : 1 – 43 . (doi:10.1007/s11118-004-2950-y)
- Winter , C. L. and Tartakovsky , D. M. 2002 . Groundwater flow in heterogeneous composite aquifers . Water Resource Res. , 38 ( 8 ) : 23/1 – 23/11 . (doi:10.1029/2001WR000450)