Monte Carlo Methods and Applications 

About: Monte Carlo Methods and Applications is an academic journal published by De Gruyter. The journal publishes majorly in the area(s): Monte Carlo method & Hybrid Monte Carlo. It has an ISSN identifier of 0929-9629. Over the lifetime, 601 publications have been published receiving 6035 citations. The journal is also known as: Monte Carlo methods & applications (Internet).

On the discretization schemes for the CIR (and Bessel squared) processes

Abstract: In this paper, we focus on the simulation of the Cox-Ingersoll-Ross processes and present several discretization schemes of both the implicit and explicit types. We study their strong and weak convergence. We also examine numerically their behaviour and compare them to the schemes already proposed by Deelstra and Delbaen and Diop. Finally, we gather all the results obtained and recommend, in the standard case, the use of one of our explicit schemes.

The Law of the Euler Scheme for Stochastic Differential Equations : II. Convergence Rate of the Density

Abstract: In the first part of this work~\cite{Bally-Talay-94-1} we have studied the approximation problem of $\ee f(X_T)$ by $\ee f(X_T^n)$, where $(X_t)$ is the solution of a stochastic differential equation, $(X^n_t)$ is defined by the Euler discretization scheme with step $\fracTn$, and $f(\cdot)$ is a given function, only supposed measurable and bounded. We have proven that the discretization error can be expanded in terms of powers of $\frac1n$ under a nondegeneracy condition of Hormander type for the infinitesimal generator of $(X_t)$. In this second part, we consider the density of the law of a small perturbation of $X_T^n$ and we compare it to the density of the law of $X_T$: we prove that the difference between the densities can also be expanded in terms of $\frac1n$. oindent{\bf AMS(MOS) classification}: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05.

Optimal quadratic quantization for numerics: the Gaussian case

Abstract: Optimal quantization has been recently revisited in numerical integration in high dimension (see [18]), in Control Theory (see [19]) and in Nonlinear Filtering Theory (see [20]). In this paper, we enlight numerical procedures in order to get optimal quadratic quantization of the Gaussian law. We study in particular Newton method in the deterministic case (dimension d = 1) and stochastic gradient in higher dimensional case (d ‚ 2). Some heuristics are provided which concern the step in the stochastic gradient method. Finally numerical examples borrowed to mathematical finance are provided.

Functional quantization for numerics with an application to option pricing

Abstract: We investigate in this paper the numerical performances of quadratic functional quantization with some applications to Finance We emphasize the role played by the so-called product quantizers and the Karhunen-Loeve expansion of Gaussian processes, in particular the Brownian motion We show how to build some efficient functional quantizers for Brownian diffusions We propose a quadrature formula based on a Romberg log-extrapolation of "crude" functional quantization which speeds up significantly the method Numerical experiments are carried out on two European option pricing problems: vanilla and Asian Call options in a Heston stochastic volatility model It suggests that functional quantization is a very efficient integration method for various path-dependent functionals of a diffusion processes: it produces deterministic results which outperforms Monte Carlo simulation for usual accuracy levels

Adaptative Monte Carlo Method, A Variance Reduction Technique

Abstract: In this article we propose an adaptative variance reduction method for Monte Carlo simulations. The method uses importance sampling scheme based on a change of drift. The change of drift is selected adaptatively through the Monte Carlo computation by using a suitable sequence of approximation. We state and prove theoretical results supporting the use of the method. We develop two applications of the procedure for variance reduction in a Monte Carlo computation in finance and in reliability.