Convergence of numerical methods for stochastic differential equations in mathematical finance

Many stochastic differential equations that occur in financial modelling do not satisfy the standard assumptions made in convergence proofs of numerical schemes that are given in textbooks, i.e., their coefficients and the corresponding derivatives appearing in the proofs are not uniformly bounded and hence, in particular, not globally Lipschitz. Specific examples are the Heston and Cox-Ingersoll-Ross models with square root coefficients and the Ait-Sahalia model with rational coefficient functions. Simple examples show that, for example, the Euler-Maruyama scheme may not converge either in the strong or weak sense when the standard assumptions do not hold. Nevertheless, new convergence results have been obtained recently for many such models in financial mathematics. These are reviewed here. Although weak convergence is of traditional importance in financial mathematics with its emphasis on expectations of functionals of the solutions, strong convergence plays a crucial role in Multi Level Monte Carlo methods, so it and also pathwise convergence will be considered along with methods which preserve the positivity of the solutions.Comment: Review Pape

Multilevel Monte Carlo methods for applications in finance

Since Giles introduced the multilevel Monte Carlo path simulation method [18], there has been rapid development of the technique for a variety of applications in computational finance. This paper surveys the progress so far, highlights the key features in achieving a high rate of multilevel variance convergence, and suggests directions for future research.Comment: arXiv admin note: text overlap with arXiv:1202.6283; and with arXiv:1106.4730 by other author

Strong convergence of an adaptive time-stepping Milstein method for SDEs with one-sided Lipschitz drift

We introduce explicit adaptive Milstein methods for stochastic differential equations with one-sided Lipschitz drift and globally Lipschitz diffusion with no commutativity condition. These methods rely on a class of path-bounded timestepping strategies which work by reducing the stepsize as solutions approach the boundary of a sphere, invoking a backstop method in the event that the timestep becomes too small. We prove that such schemes are strongly $L_2$ convergent of order one. This convergence order is inherited by an explicit adaptive Euler-Maruyama scheme in the additive noise case. Moreover we show that the probability of using the backstop method at any step can be made arbitrarily small. We compare our method to other fixed-step Milstein variants on a range of test problems.Comment: 20 pages, 2 figure

Semi-Lagrangian discontinuous Galerkin schemes for some first and second-order partial differential equations

Explicit, unconditionally stable, high-order schemes for the approximation of some first- andsecond-order linear, time-dependent partial differential equations (PDEs) are proposed.The schemes are based on a weak formulation of a semi-Lagrangian scheme using discontinuous Galerkin (DG) elements.It follows the ideas of the recent works of Crouseilles, Mehrenberger and Vecil (2010), Rossmanith and Seal (2011),for first-order equations, based on exact integration, quadrature rules, and splitting techniques for the treatment of two-dimensionalPDEs. For second-order PDEs the idea of the schemeis a blending between weak Taylor approximations and projection on a DG basis.New and sharp error estimates are obtained for the fully discrete schemes and for variable coefficients.In particular we obtain high-order schemes, unconditionally stable and convergent,in the case of linear first-order PDEs, or linear second-order PDEs with constant coefficients.In the case of non-constant coefficients, we construct, in some particular cases,"almost" unconditionally stable second-order schemes and give precise convergence results.The schemes are tested on several academic examples