Volume 11, Issue 1

We consider an extended Merton's problem of optimal consumption and investment in continuous-time with stochastic volatility. The wealth process of the investor is approximated by a particular weak Itô-Taylor approximation called Euler scheme. It is shown that the optimal control of the value function generated by the Euler scheme is an ε -optimal control of the original problem of maximizing total expected discounted HARA utility from consumption.

Stochastic models and Monte Carlo algorithms for simulation of flow through porous media beyond the small hydraulic conductivity fluctuation assumptions are developed. The hydraulic conductivity is modelled as an isotropic random field with a lognormal distribution and prescribed correlation or spectral functions. It is sampled by a Monte Carlo method based on a randomized spectral representation. The Darcy and continuity equations with the random hydraulic conductivity are solved numerically, using the successive over relaxation method in order to extract statistical characteristics of the flow. Hybrid averaging is used: we combine spatial and ensemble avergaing to get efficient numerical procedure. We provide some conceptual and numerical comparison of various stochastic simulation techniques, and focus on the prediction of applicability of the randomized spectral models derived under the assumption of small hydraulic conductivity fluctuations.

In this paper, we extend the techniques used in Grid-based Monte Carlo applications to Grid-based quasi-Monte Carlo applications. These techniques include an N-out-of-M strategy for efficiently scheduling subtasks on the Grid, lightweight checkpointing for Grid subtask status recovery, a partial result validation scheme to verify the correctness of each individual partial result, and an intermediate result checking scheme to enforce the faithful execution of each subtask. Our analysis shows that the extremely high uniformity seen in quasirandom sequences prevents us from applying many of our Grid-based Monte Carlo techniques to Grid- based quasi-Monte Carlo applications. However, the use of scrambled quasirandom sequence becomes a key to tackling this problem, and makes many of the techniques we used in Grid- based Monte Carlo applications effective in Grid-based quasi-Monte Carlo applications. All the techniques we will describe here contribute to performance improvement and trustworthiness enhancement for large-scale quasi-Monte Carlo applications on the Grid, which eventually lead to a high-performance Grid-computing infrastructure that is capable of providing trustworthy quasi-Monte Carlo computation services.

We present an approximation method for discrete time nonlinear filtering in view of solving dynamic optimization problems under partial information. The method is based on quantization of the Markov pair process filter-observation (Π, Y ) and is such that, at each time step k and for a given size N k of the quantization grid in period k , this grid is chosen to minimize a suitable quantization error. The algorithm is based on a stochastic gradient descent combined with Monte Carlo simulations of (Π, Y ). Convergence results are given and applications to optimal stopping under partial observation are discussed. Numerical results are presented for a particular stopping problem: American option pricing with unobservable volatility.

Different Quasi-Monte Carlo algorithms corresponding to the same Monte Carlo algorithm are considered. Even in the case when their constructive dimensions are equal and the same quasi-random points are used, the efficiencies of these algorithms may differ. Global sensitivity analysis provides an insight into this situation. As a model problem two well-known approximations of a Wiener integral are considered: the standard one and the Brownian bridge. The advantage of the Brownian bridge is confirmed.