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Jorge A. León

Publications and source records attributed to Jorge A. León.

12 recordsLinked to original sources
arXiv2026

Euler scheme for stochastic functional differential equations driven by fractional Brownian motion

In this paper, we apply rough paths techniques to provide an approximation of the solution of stochastic functional differential equations driven by fractional Brownian motion with Hurst parameter $H>1/2$. Here, the involved stochastic integral is the Young one and the coefficient is evaluated in the set of $λ$-Hölder continuous functions on $[-τ,0]$, for some suitable $τ>0$ and $λ\in(1/2,H)$. The rate of convergence of our scheme is $1/n^γ$, for any $γ<2λ-1$. Also, numerical simulations are provided to illustrate our theoretical results.

math.PR↗
arXiv2023

HJB equation for maximization of wealth under insider trading

In this paper, we combine the techniques of enlargement of filtrations and stochastic control theory to establish an extension of the verification theorem, where the coefficients of the stochastic controlled equation are adapted to the underlying filtration and the controls are adapted to a bigger filtration $\mathbf{G}$ than the one generated by the corresponding Brownian motion $B$. Using the forward integral defined by Russo and Vallois \cite{RV1993}, we show that there is a $\mathbf{G}$-adapted optimal control with respect to a certain cost functional if and only if the Brownian motion $B$ is a $\mathbf{G}$-semimartingale. The extended verification theorem allows us to study a financial market with an insider in order to take advantage of the extra information that the insider has from the beginning. Finally, we consider two examples throughout the extended verification theorem. These problems appear in financial markets with an insider.

math.PR↗
arXiv2023

Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates

The Malliavin differentiability of a SDE plays a crucial role in the study of density smoothness and ergodicity among others. For Gaussian driven SDEs the differentiability property is now well established. In this paper, we consider the Malliavin differentiability for the Euler scheme of such SDEs. We will focus on SDEs driven by fractional Brownian motions (fBm), which is a very natural class of Gaussian processes. We derive a uniform (in the step size $n$) path-wise upper-bound estimate for the Euler scheme for stochastic differential equations driven by fBm with Hurst parameter $H>1/3$ and its Malliavin derivatives.

math.PR↗
arXiv2016

Young Differential Equations with Power Type Nonlinearities

In this note we give several methods to construct nontrivial solutions to the equation $dy_{t}=σ(y_{t}) \, dx_{t}$, where $x$ is a $γ$-Hölder $R^{d}$-valued signal with $γ\in(1/2,1)$ and $σ$ is a function behaving like a power function $|ξ|^κ$, with $κ\in(0,1)$. In this situation, classical Young integration techniques allow to get existence and uniqueness results whenever $γ(κ+1)>1$, while we focus on cases where $γ(κ+1)\le 1$. Our analysis then relies on some extensions of Young's integral allowing to cover the situation at hand.

math.PR↗
arXiv2015

Stability for a class of semilinear fractional stochastic integral equations

In this paper we study some stability criteria for some semilinear integral equations with a function as initial condition and with additive noise, which is a Young integral that could be a functional of fractional Brownian motion. Namely, we consider stability in the mean, asymptotic stability, stability, global stability and Mittag-Leffler stability. To do so, we use comparison results for fractional equations and an equation (in terms of Mittag-Leffler functions) whose family of solutions includes those of the underlying equation.

math.PR↗
arXiv2015

On Uniqueness for some non-Lipschitz SDE

We study the uniqueness in the path-by-path sense (i.e. $ω$-by-$ω$) of solutions to stochastic differential equations with additive noise and non-Lipschitz autonomous drift. The notion of path-by-path solution involves considering a collection of ordinary differential equations and is, in principle, weaker than that of a strong solution, since no adaptability condition is required. We use results and ideas from the classical theory of ode's, together with probabilistic tools like Girsanov's theorem, to establish the uniqueness property for some classes of noises, including Brownian motion, and some drift functions not necessarily bounded nor continuous.

math.PR↗
arXiv2014

An Osgood's criterion for a semilinear stochastic differential equation

The purpose of this paper is to give an Osgood's criterion for solutions of semilinear stochastic differential equations of the form $X_{t}=ξ+\int_{0}^{t}b(s,X_{s})ds+\int_{0}^{t}σ(s)X_{s}dW_{s},\ t\geq 0$. Here, $b$ is a non-negative, non-decreasing by components and continuous random field and $σ$ is a predictable and continuous process. Also we present a generalization of the so-called Feller's test whenever $σ\equiv 1$.

math.PR↗
arXiv2013

On the Distribution of Explosion Time of Stochastic Differential Equations

In this paper we use the Itô's formula and comparison theorems to study the blow-up in finite time of stochastic differential equations driven by a Brownian motion. In particular, we obtain an extension of Osgood criterion, which can be applied to some nonautonomous stochastic differential equations with additive Wiener integral noise. In most cases we are able to provide with a method to figure out the distribution of the explosion time of the involved equation.

math.PR↗
arXiv2012

Local Malliavin Calculus for Lévy Processes and Applications

In this paper a Malliavin calculus for Lévy processes based on a family of true derivative operators is developed. The starting point is an extension to Lévy processes of the pioneering paper by Carlen and Pardoux [8] for the Poisson process, and our approach includes also the classical Malliavin derivative for Gaussian processes. We obtain a sufficient condition for the absolute continuity of functionals of the Lévy process. As an application, we analyze the absolute continuity of the law of the solution of some stochastic differential equations.

math.PR↗
arXiv2012

Anticipating Linear Stochastic Differential Equations Driven by a Lévy Process

In this paper we study the existence of a unique solution for linear stochastic differential equations driven by a Lévy process, where the initial condition and the coefficients are random and not necessarily adapted to the underlying filtration. Towards this end, we extend the method based on Girsanov transformation on Wiener space and developped by Buckdahn to the canonical Lévy space.

math.PR↗
arXiv2008

An anticipating Itô formula for Lévy processes

In this paper, we use the Malliavin calculus techniques to obtain an anticipative version of the change of variable formula for Lévy processes. Here the coefficients are in the domain of the anihilation (gradient) operator in the "future sense", which includes the family of all adapted and square-integrable processes. This domain was introduced on the Wiener space by Alòs and Nualart.

math.PR↗