Solution of Stochastic Differential Equations in Finance

Summary

Numerical methods for the solution of stochastic differential equations are essential for the analysis of random phenomena. Strong solvers are neces- sary when exploring characteristics of systems that depend on trajectory-level properties. Several approaches exist for strong solvers, in particular Taylor and Runge-Kutta type methods, although both increase greatly in complication for orders greater than one.

In many financial applications, major emphasis is placed on the proba- bility distribution of solutions, and in particular mean and variance of the distribution. In such cases, weak solvers may suffice, and have the advantage of comparatively less computational overhead, which may be crucial in the context of Monte-Carlo simulation.
Independent of the choice of stochastic differential equation solver, meth- ods of variance reduction exist that may increase computational efficiency. The replacement of pseudorandom numbers with quasirandom analogues from low-discrepancy sequences is applicable as long as statistical independence along trajectories is maintained. In addition, control variates offer an alter- nate means of variance reduction and increases in efficiency in Monte-Carlo simulation of SDE trajectories.

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