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Abstract. Kawabata and Yamada (1991) proposed an implicit formulation for Newton's method for an univariate stochastic differential equation (SDEs). Amano (2009) used the linearized equation technique and proposed explicit formulation for the Newton scheme. In this note, we extend the Newton method for univariate SDEs to the multivariate cases. The error analysis is given and some examples are proposed. Results show that the method works well.

Abstract. We propose an alternative method to simulation based recursive importance sampling procedure to estimate the optimal change of measure for Monte Carlo simulations. We consider an algorithm which combines (vector and functional) optimal quantization with Newton-Raphson zero search procedure. Our approach can be seen as a robust and automatic deterministic counterpart of recursive importance sampling (by translation of the mean) by means of stochastic approximation algorithm which may require tuning of the step sequence and a good knowledge of the payoff function in practice. Moreover, unlike recursive importance sampling procedures, the proposed methodology does not rely on simulations so it is quite fast, generic and can come along on the top of Monte Carlo simulations.

Abstract. For higher dimensional cases, the essential spectrum radius of the Perron–Frobenius operator is usually greater than (the definition is given below) even if we restrict its domain to a suitable space. In this article, using algebraic method, we construct a transformation whose essential spectrum radius equals . By this transformation, we can construct low discrepancy sequences.

Abstract. We propose an efficient Monte Carlo approach to compute boundary crossing probabilities (BCP) for Brownian motion and a large class of diffusion processes, the method of adaptive control variables. For the Brownian motion the boundary b (or the boundaries in case of two-sided boundary crossing probabilities) is approximated by a piecewise linear boundary , which is linear on m intervals. Monte Carlo estimators of the corresponding BCP are based on an m -dimensional Gaussian distribution. Let N denote the number of (univariate) Gaussian variables used. The mean squared error for the boundary is of order , leading to a mean squared error for the boundary b of order with , if the difference of the (exact) BCP's for b and is . Typically, for infinite-dimensional Monte Carlo methods, the convergence rate is less than the finite-dimensional . Let be a further approximating boundary which is linear on k intervals. If k is small compared to m , the corresponding BCP may be estimated with high accuracy. The BCP for as control variable improves the convergence rate of the Monte Carlo estimator to with . The constant depends on the correlation of the estimators for and . We show that this method of adaptive control variable improves the convergence rate considerably. Iterating control variables leads to a rate of convergence (of the mean squared error) of order , reducing the problem of estimating the BCP to an essentially finite-dimensional problem.