[ABE-L] Seminário de Probabilidade - IM-UFRJ - 8 de junho -

[Maria Eulalia Vares]

Dear colleagues,

Our next seminar will be held on Monday, June 8, from *3:30 p.m. to 4:30
p.m.* (Rio de Janeiro local time). The meeting will take place at *room
C116 - Bloco C - CT – Instituto de Matemática – UFRJ*. There will be no
transmission online.

*Speaker:* Ricardo Rosa (IM-UFRJ)

*Title:* Strong order of convergence of the Euler method for Random ODEs
driven by semi-martingales

*Abstract:* It is well known that the convergence of the Euler method for
approximating the solutions of ordinary differential equations is of order
1, under reasonable regularity assumptions. For Ito diffusion stochastic
differential equations with multiplicative noise, this convergence, in the
mean-square sense, drops to 1/2. What about random ordinary differential
equations (RODEs) $dX_t/dt = f(t, X_t, Y_t)$ driven by a stochastic process
$\{Y_t\}_t$? Previous works estimated the convergence in mean or in
square-mean to be of an order $0 < \theta < 1$ depending on the
$\theta$-Holder regularity of the sample paths of the noise process. This
order was known to increase to $1$ for some types of noise process such as
the Wiener process, but it was still thought to be smaller than 1 for more
general types of processes. In recent work, however, we proved that, in
many more typical cases, further structures on the noise can be exploited
so that the mean-square convergence is of order 1 even in cases when the
sample paths of the noise are discontinuous. More precisely, we prove the
order 1 convergence for any square-integrable semi-martingale noise. This
includes not only Ito diffusion processes, but also point-process noises,
transport-type processes with sample paths of bounded variation, and, more
generally, time-changed Brownian motions. The result follows from
estimating the global error as an iterated integral over both large and
small mesh scales, and then by switching the order of integration to move
the critical regularity to the large scale. In this talk, I intend to
discuss this recent improvement, briefly sketch the main ideas of the proof
and illustrate the results with a few numerical examples, including
examples with fractional Brownian motion noises for which the order of
convergence genuinely drops below 1. This work is a joint work with Peter
Kloeden (Universidade de Tübingen, Alemanha) and was recently published in
ESAIM: Mathematical Modelling and Numerical Analysis.



More complete information about the seminars can be found at:
https://ppge.im.ufrj.br/seminarios-de-probabilidade/

Sincerely,

Giulio Iacobelli and Maria Eulalia Vares


-- 
Maria Eulalia Vares
Professora Titular - Instituto de Matemática - UFRJ
Coordenadora do Programa de Pós-Graduação em Estatística
https://ppge.im.ufrj.br/
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