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Rate of convergence of approximations of some convex functionals of stochastic differential equations

The rate of convergence of numerical methods for integration of some convex functionals of ordinary stochastic differential equations (SDEs) is discussed. In particular, we answer how rates of p-th mean convergence carry over to rates of weak convergence for non-smooth and convex functionals of SDEs. Qualitative behavior of some numerical approximations such as nonnegativity of balanced implicit Milstein methods (BMMs) is investigated as well. Nonstandard integration techniques such as partial- and linear-implicit ones seem to be the most promissing. As a main result we obtain some justification for the choice of approximation schemes of discounted price functionals and their ingredients of random interest rates and volatility processes involved in dynamic asset pricing.