We present new algorithms for weak approximation of stochastic differential equations driven by pure jump Lévy processes. The method uses adaptive non-uniform discretization based on the times of large jumps of the driving process. To approximate the solution between these times we replace the...

We obtain upper and lower bounds for the density of a functional of a diffusion whose drift is bounded and measurable. The argument consists of using Girsanov’s theorem together with an Itô–Taylor expansion of the change of measure. One then applies Malliavin calculus techniques in a...

In this article, we consider a modification of the Karatzas--Pikovsky model of insider trading. Specifically, we suppose that the insider agent influences the long/medium-term evolution of Black--Scholes type model through the drift of the stochastic differential equation. We say that the...

In this article, we give a brief informal introduction to Malliavin Calculus for newcomers. We apply these ideas to the simulation of Greeks in Finance. First to European-type options where formulas can be computed explicitly and therefore can serve as testing ground. Later, we study the case of...

The main purpose of this article is to propose computational methods for Greeks and the multidimensional density estimation for an asset price dynamics model defined with time-changed Brownian motions. Our approach is based on an application of the Malliavin integration-by-parts formula on the...

We study the problem of density estimation of a non-degenerate diffusion using kernel functions. Thanks to Malliavin calculus techniques, we obtain an expansion of the discretization error. Then, we introduce a new control variate method in order to reduce the variance in the density estimation....

We treat an extension of Jacod's theorem for initial enlargement of filtrations with respect to random times. In Jacod's theorem the main condition requires the absolute continuity of the conditional distribution of the random time with respect to a nonrandom measure. Examples appearing in the...

We study the Euler approximation scheme for solutions of stochastic differential equations with boundary conditions in two different examples: (a) the one-dimensional case with linear boundary condition, and (b) the multidimensional case with constant diffusion coefficient and general boundary...

There exists a non-closed formula for the American put option price and non-trivial computations are required to solve it. Strong efforts have been made to propose efficient numerical techniques but few have strong mathematical reasoning to ascertain why they work well. We present an extension...