Page 674 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 674
PROBABILITY
tionals of the process. Applications to queuing systems and to machine learning algorithms are
also presented.
Simulation of the time needed by a diffusion process in order to exit from
a given interval (WOMS algorithm)
Nicolas Massin, nicolas.massin@u-bourgogne.fr
Institut de Mathématiques de Bourgogne, France
Coauthor: Samuel Herrmann
For many applications, it is important to describe or simulate the exit time from an interval
for a stochastic process. In the particular context of diffusion processes which are solutions
of stochastic differential equations, time discretization schemes like Euler scheme are usually
used. They permit the simulation of a paths skeleton and lead to an approximation of the exit
time as a by-product. The aim of the talk is to present a completely different approach based
on the study of the Brownian paths. It is possible to find some domains, called for instance
spheroids, such that both the distribution of Brownian exit time from this spheroid and its exit
position are well-known.
From that point, we define an iterative procedure : a walk on spheres for the Brownian
motion and so called WOMS algorithm. We can show that it is hard to find an algorithm as
precise and efficient than this algorithm.
Starting from this particular case, we generalize the iterative procedure in order to deal with
a family of diffusion written as functions of a changed time Brownian motion. This strong rela-
tion permits to define a new WOMS algorithm by deduction.Theoretical results and numerical
illustrations point out the efficiency of such an algorithm. The particular Ornstein-Uhlenbeck
case will be presented in details.
Solving General Itô-Process Hitting-Time Problems with General Moving
Boundaries
Martin Nilsson, from.ecm@drnil.com
RISE Research Institutes of Sweden, Sweden
A spectral method for solving both first-passage time and first-exit time problems for general
Itô processes is presented. The method handles general moving (i.e., time-variable) boundaries
including discontinuities. Results from the application to neuron modelling are given.
The method is built upon the idea of first expressing the problem as a Fokker–Planck equa-
tion over a non-rectangular strip. The Fokker–Planck equation is then spectrally reduced to a
small set of ordinary differential equations which can be solved easily and quickly by standard
solvers.
The reduction is the key step of the method. It observes that the right hand side of the
Fokker–Planck equation
∂
p = Lp
∂t
can be used as the left-hand side of the Sturm-Liouville system
L p = −λ p,
672
tionals of the process. Applications to queuing systems and to machine learning algorithms are
also presented.
Simulation of the time needed by a diffusion process in order to exit from
a given interval (WOMS algorithm)
Nicolas Massin, nicolas.massin@u-bourgogne.fr
Institut de Mathématiques de Bourgogne, France
Coauthor: Samuel Herrmann
For many applications, it is important to describe or simulate the exit time from an interval
for a stochastic process. In the particular context of diffusion processes which are solutions
of stochastic differential equations, time discretization schemes like Euler scheme are usually
used. They permit the simulation of a paths skeleton and lead to an approximation of the exit
time as a by-product. The aim of the talk is to present a completely different approach based
on the study of the Brownian paths. It is possible to find some domains, called for instance
spheroids, such that both the distribution of Brownian exit time from this spheroid and its exit
position are well-known.
From that point, we define an iterative procedure : a walk on spheres for the Brownian
motion and so called WOMS algorithm. We can show that it is hard to find an algorithm as
precise and efficient than this algorithm.
Starting from this particular case, we generalize the iterative procedure in order to deal with
a family of diffusion written as functions of a changed time Brownian motion. This strong rela-
tion permits to define a new WOMS algorithm by deduction.Theoretical results and numerical
illustrations point out the efficiency of such an algorithm. The particular Ornstein-Uhlenbeck
case will be presented in details.
Solving General Itô-Process Hitting-Time Problems with General Moving
Boundaries
Martin Nilsson, from.ecm@drnil.com
RISE Research Institutes of Sweden, Sweden
A spectral method for solving both first-passage time and first-exit time problems for general
Itô processes is presented. The method handles general moving (i.e., time-variable) boundaries
including discontinuities. Results from the application to neuron modelling are given.
The method is built upon the idea of first expressing the problem as a Fokker–Planck equa-
tion over a non-rectangular strip. The Fokker–Planck equation is then spectrally reduced to a
small set of ordinary differential equations which can be solved easily and quickly by standard
solvers.
The reduction is the key step of the method. It observes that the right hand side of the
Fokker–Planck equation
∂
p = Lp
∂t
can be used as the left-hand side of the Sturm-Liouville system
L p = −λ p,
672
