Explicit Mean-Square Error Bounds for Monte-Carlo and Linear Stochastic Approximation

This paper concerns error bounds for recursive equations subject to Markovian disturbances. Motivating examples abound within the fields of Markov chain Monte Carlo (MCMC) and Reinforcement Learning (RL), and many of these algorithms can be interpreted as special cases of stochastic approximation (SA). It is argued that it is not possible in general to obtain a Hoeffding bound on the error sequence, even when the underlying Markov chain is reversible and geometrically ergodic, such as the M/M/1 queue. This is motivation for the focus on mean square error bounds for parameter estimates. It is shown that mean square error achieves the optimal rate of $O(1/n)$, subject to conditions on the step-size sequence. Moreover, the exact constants in the rate are obtained, which is of great value in algorithm design

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

Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients

On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.Comment: Published in at http://dx.doi.org/10.1214/11-AAP803 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Strong convergence rates for Euler approximations to a class of stochastic path-dependent volatility models

We consider a class of stochastic path-dependent volatility models where the stochastic volatility, whose square follows the Cox-Ingersoll-Ross model, is multiplied by a (leverage) function of the spot price, its running maximum, and time. We propose a Monte Carlo simulation scheme which combines a log-Euler scheme for the spot process with the full truncation Euler scheme or the backward Euler-Maruyama scheme for the squared stochastic volatility component. Under some mild regularity assumptions and a condition on the Feller ratio, we establish the strong convergence with order 1/2 (up to a logarithmic factor) of the approximation process up to a critical time. The model studied in this paper contains as special cases Heston-type stochastic-local volatility models, the state-of-the-art in derivative pricing, and a relatively new class of path-dependent volatility models. The present paper is the first to prove the convergence of the popular Euler schemes with a positive rate, which is moreover consistent with that for Lipschitz coefficients and hence optimal.Comment: 34 pages, 5 figure