Volume 20, Issue 2

Abstract. We consider the problem of estimating expected values of functionals of real-valued diffusions over regions in path space that have very small probability. We propose a two-stage importance sampling procedure that first converts the problem into one involving standard Brownian motion and then addresses the rare event problem in this simpler setting. In order to identify an effective yet practical importance measure we propose using a time-dependent deterministic drift that minimizes the relative entropy between the corresponding importance measure and the conditional law of the standard Brownian motion, given that its trajectory lies in the region of interest. We provide numerical evidence that (i) our entropy-based criteria performs favourably with an alternative, but less general and less practical, criteria based on large deviations and (ii) our two-stage procedure performs admirably in cases where the region of interest is so rare that crude estimators fail completely.

Abstract. The great demand for high computational capabilities is omnipresent in every facet of modern financial activities, ranging from financial product pricing, trading and hedging at the front desk on the one end to risk management activities for in house monitoring and legislative compliance on the other. While this demand is met by scalable high performance computing, along with it come new challenges. As a notable proportion of financial computations involve the use of pseudo-random numbers, the engagement of a large number of parallel threads leads to consumption of large amount of pseudo-random numbers, uncovering potential intra-thread and inter-thread correlation that will lead to bias and loss of efficiency in the computation. This paper reviews, in the setting of derivative instrument pricing, the performance of some commonly used scalable pseudo-random number generators constructed based on different parallelization strategies: (1) parameterization ( SPRNG ), (2) sequence-splitting ( TRNG and RngStream ), and (3) cryptography ( Random123 ). In addition, the potential impact of intra-thread and inter-thread correlation in pricing and sensitivity analysis of some common contingent claims via Monte Carlo simulation is examined.

Abstract. Random matrices, that is, matrices whose entries are measurable functions of a random vector Z , are encountered in finite element/difference formulations of a broad range of stochastic mechanics problems. Monte Carlo simulation, the only general method for solving this class of problems, is usual impractical when dealing with realistic problems. A new method is proposed for solving this class of problems. The method can be viewed as a smart Monte Carlo simulation. Like Monte Carlo, it calculates statistics for quantities of interest from deterministic matrices corresponding to samples of Z . In contract to Monte Carlo that uses a large number of samples of Z selected at random, the proposed method uses a small number of samples of this vector selected in an optimal manner. The method is based on stochastic reduced models (SROMs) for Z , i.e., random vectors with finite numbers of samples, and surrogate models expressing quantities of interest as known functions of Z . Theoretical arguments are followed by numerical examples providing statistics for inverses of random matrices, solutions of stochastic algebraic equations, and eigenvalues/eigenvectors of random matrices.

Abstract. We consider a square Gaussian stochastic process. Estimates of the distribution of some functional of this process are obtained. Tests for a hypothesis concerning the form of the covariance function of a Gaussian stochastic process are constructed.

Abstract. We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [`Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE', Ann. Probab., to appear] for representing fully nonlinear HJB equations. This includes in particular numerical resolution for stochastic control problems with controlled volatility, possibly degenerate. Our backward scheme, based on least-squares regressions, takes advantage of high-dimensional properties of Monte Carlo methods, and also provides a parametric estimate in feedback form for the optimal control. A partial analysis of the algorithm error is presented, as well as numerical tests on the problem of option superreplication with uncertain volatilities and/or correlations, including a detailed comparison with the numerical results from the alternative scheme proposed in [J. Comput. Finance 14 (2011), 37–71].