Processos estocàstics: un curs bàsic

Text docent per l'assignatura Processos Estocàstics del Grau de Matemàtiques. Es presenten algunes de les famílies de processos més utilitzades: els processos de ramificació, el procés de Poisson, el moviment brownià, els processos de Markov a temps continu i els processos de naixement i mort

Weak convergence to a class of two-parameter Gaussian processes from a Lévy sheet

In this paper, we show an approximation in law, in the space of the continuous functions on $[0,1]^2$, of two-parameter Gaussian processes that can be represented as a Wiener type integral by processes constructed from processes that converge to the Brownian sheet. As an application, we obtain a sequence of processes constructed from a Lévy sheet that converges in law towards the fractional Brownian sheet

On the strong convergence of multiple ordinary integrals to multiple Stratonovich integrals

Given $\left\{W^{(m)}(t), t \in[0, T]\right\}_{m \geq 1}$, a sequence of approximations to a standard Brownian motion $W$ in $[0, T]$ such that $W^{(m)}(t)$ converges almost surely to $W(t)$, we show that, under regular conditions on the approximations, the multiple ordinary integrals with respect to $d W^{(m)}$ converge to the multiple Stratonovich integral. We are integrating functions of the type $$f\left(t_1, \ldots, t_n\right)=f_1\left(t_1\right) \cdots f_n\left(t_n\right) I_{\left\{t_1 \leq \cdots \leq t_n\right\}},$$ where for each $i \in\{1, \ldots, n\}, f_i$ has continuous derivatives in $[0, T]$. We apply this result to approximations obtained from uniform transport processes

Convergence of delay equations driven by a Hölder continuous function of order 1/3<β<1/2.

In this article we show that, when the delay approaches zero, the solution of multidimensional delay differential equations driven by a Hölder continuous function of order 1/3 < \beta < 1/2 converges with the supremum norm to the solution for the equation without delay. Finally we discuss the applications to stochastic differential equations

Weak convergence to a class of two-parameter Gaussian processes from a Lévy sheet

In this paper, we show an approximation in law, in the space of the continuous functions on $[0,1]^2$, of two-parameter Gaussian processes that can be represented as a Wiener type integral by processes constructed from processes that converge to the Brownian sheet. As an application, we obtain a sequence of processes constructed from a Lévy sheet that converges in law towards the fractional Brownian sheet

Strong approximations of Brownian sheet by uniform transport processes.

Many years ago, Griego, Heath and Ruiz-Moncayo proved that it is possible to define realizations of a sequence of uniform transport processes that converges almost surely to the standard Brownian motion, uniformly on the unit time interval. In this paper we extend their results to the multi parameter case. We begin constructing a family of processes, starting from a set of independent standard Poisson processes, that has realizations that converge almost surely to the Brownian sheet, uniformly on the unit square. At the end the extension to the d-parameter Wiener processes is presented

Coinfection in a stochastic model for bacteriophage systems

A system modeling bacteriophage treatments with coinfections in a noisy context is analysed. We prove that in a small noise regime, the system converges in the long term to a bacteria-free equilibrium. Moreover, we compare the treatment with coinfection with the treatment without coinfection, showing how coinfection affects the convergence to the bacteria-free equilibrium

Cristalls de spin

L'estudi dels cristalls de spin iniciat a partir dels anys setanta ha tingut darrerament grans avenços en termes matemàtics. En aquest article presentem, de manera divulgativa, els conceptes bàsics d'aquesta àrea, detallem el seu marc matemàtic, descrivim els models clàssics que considerem més importants i expliquem algunes de les tècniques que s'utilitzen per a estudiar-los

A model of continuous time polymer on the lattice

In this article, we try to give a rather complete picture of the behavior of the free energy for a model of directed polymer in a random environment, in which the polymer is a simple symmetric random walk on the lattice Z d, and the environment is a collection {W(t, x);t ≥ 0, x ∈ Z d} of i.i.d. Brownian motions

Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H 1/2

We consider the Cauchy problem for a stochastic delay differential equation driven by a fractional Brownian motion with Hurst parameter H>¿. We prove an existence and uniqueness result for this problem, when the coefficients are sufficiently regular. Furthermore, if the diffusion coefficient is bounded away from zero and the coefficients are smooth functions with bounded derivatives of all orders, we prove that the law of the solution admits a smooth density with respect to Lebesgue measure on R