Volume 21, Issue 3

We establish and examine the deep connection between highly stratified random cubature formulas and quasi-Monte Carlo methods. A class of such formulas, designed to exactly integrate the introduced generalized s -dimensional Haar system, is shown to have additional variance reduction compared to the known theoretical upper bound. We propose several equivalent expressions for the variance within the standard quasi-Monte Carlo setting. The theory of random cubatures is supplemented with both refined versions of known results and completely new facts.

An inverse problem of reconstructing the two-dimensional coefficient of the wave equation is solved by a stochastic projection method. We apply the Gel'fand–Levitan approach to reduce the nonlinear inverse problem to a family of linear integral equations. The stochastic projection method is applied to solve the relevant linear system. We analyze the structure of the problem to increase the efficiency of the method by constructing an improved initial approximation. A smoothing spline is used to treat the random errors of the method. The method has low cost and memory requirements. Results of numerical calculations are presented.

The Heston model is appealing as it possesses a stochastic volatility term as well as semi-closed formulas for pricing European options. Unfortunately, few simulation schemes for this model can handle the violation of the Feller Condition. An algorithm based on the exact scheme of Broadie and Kaya to simulate price paths under the Heston model is introduced. In order to increase the speed of their exact method, we use a gamma approximation. According to Stewart, Strijbosch, Moors and Batenburg, it is possible to approximate a complex gamma convolution (similar to the representation given by Glasserman and Kim) by a simple moment-matched gamma distribution. We also perform a review of popular simulation schemes for the Heston model and validate our approach through a simulation study. The gamma approximation scheme appears to yield small biases on European and Asian option prices when compared to the most popular schemes.

This paper is devoted to the estimation of the accuracy and reliability (in L p ( T ) metrics) for the calculation of improper integrals depending on a parameter t using the Monte Carlo method. For this estimates we use the theory of 𝐅 ψ (Ω) spaces.

In this paper we generalize an explicit numerical scheme for the CIR process that we have proposed before. The advantage of the new proposed scheme is that preserves positivity and is well posed for a (little bit) broader set of parameters among the positivity preserving schemes. The order of convergence is at least logarithmic in general and for a smaller set of parameters is at least 1/4.