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Stochastic optimization/approximation algorithms are widely used to recursively estimate the optimum of a suitable function or its root under noisy observations when this optimum or root is a constant or evolves randomly according to slowly time-varying continuous sample paths. In comparison, this paper analyzes the asymptotic properties of stochastic optimization/approximation algorithms for recursively estimating the optimum or root when it evolves rapidly with nonsmooth (jump-changing) sample paths. The resulting problem falls into the category of regime-switching stochastic approximation algorithms with two-time scales. Motivated by emerging applications in wireless communications, and system identification, we analyze asymptotic behavior of such algorithms. Our analysis assumes that the noisy observations contain a (nonsmooth) jump process modeled by a discrete-time Markov chain whose transition frequency varies much faster than the adaptation rate of the stochastic optimization algorithm. Using stochastic averaging, we prove convergence of the algorithm. Rate of convergence of the algorithm is obtained via bounds on the estimation errors and diffusion approximations. Remarks on improving the convergence rates through iterate averaging, and limit mean dynamics represented by differential inclusions are also presented. [ABSTRACT FROM AUTHOR]

This article is concerned with a numerical method using stochastic approximation approach for an optimal trading (buy and sell) strategy. The underlying asset price is governed by a mean-reverting stochastic process. The objective is to buy and sell the asset so as to maximize an overall expected return. One of the advantages of our approach is that the underlying asset is model free. Only mean reversion is required. Slippage cost is imposed on each transaction. Convergence of the algorithms is provided. Numerical examples are reported to demonstrate the results. [ABSTRACT FROM AUTHOR]

This work is concerned with option pricing. Stochastic approximation/optimization algorithms are proposed and analyzed. The underlying stock price evolves according to two geometric Brownian motions coupled by a continuous-time finite state Markov chain. Recursive stochastic approximation algorithms are developed to estimate the implied volatility. Convergence of the algorithm is proved. Rate of convergence is also ascertained. Then real market data are used to compare our algorithms with other schemes. [ABSTRACT FROM AUTHOR]

Aiming at reduction of complexity, this work is concerned with two-time-scale Markov chains and applications to quasi-birth-death queues. Asymptotic expansions of probability vectors are constructed and justified. Lumping all states of the Markov chain in each subspace into a single state, an aggregated process is shown to converge to a continuous-time Markov chain whose generator is an average with respect to the stationary measures. Then a suitably scaled sequence is shown to converge to a switching diffusion process. Extensions of the results are presented together with examples of quasi-birth-death queues. [ABSTRACT FROM AUTHOR]