Page 672 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 672
PROBABILITY
Hitting times for the Brownian motion and Bessel processes: some new
algorithms
Madalina Deaconu, madalina.deaconu@inria.fr
Inria, France
The aim of this talk is to introduce new numerical methods for the hitting time of some bound-
aries for a class of stochastic processes. We will start by constructing the walk on moving
spheres method for Bessel processes. We will apply this result for the simulation of the Brow-
nian motion hitting times. This algorithm exhibits a new procedure which efficiently approxi-
mate both the hitting time and the hitting position of the stochastic process and can be used in
problems arising from finance, geophysics or neuroscience.
We give also the convergence results and present some numerical examples that permit to
emphasise the efficiency and accuracy of this new method.
This is a joint work with Samuel Herrmann (Dijon University, France).
Generation of first passage times for diffusion processes: an overview of
simulation techniques
Samuel Herrmann, samuel.herrmann@u-bourgogne.fr
University of Burgundy, France
Coauthor: Cristina Zucca
Many biological or physical applications require to simulate random variables with a given
probability distribution. The aim of our study is to focus on a particular random variable: the
first passage time (FPT) of a diffusion process. We introduce (Xt) the unique solution of the
following SDE:
dXt = b(Xt) dt + σ(Xt) dBt, X0 = x,
where (Bt) stands for a one-dimensional Brownian motion and define τL the first passage time
through the level L. We propose an overview of several simulation techniques.
• The classical way is to use efficient algorithms for the simulation of sample paths, like dis-
cretization schemes. Such methods permit to obtain approximations of the first-passage
times as a by-product.
• Another approach based on a random walk on spheroids permit in particular cases to
express the first passage time as the limit of a random walk. It suffices therefore to
describe precisely the convergence of this stochastic process and to introduce a stopping
procedure.
• Finally we present a new rejection sampling algorithm which permits to perform an exact
simulation of the first-passage time for general one-dimensional diffusion processes. The
main ideas are based both on a previous algorithm pointed out by A. Beskos et G. O.
Roberts which uses Girsanov’s transformation and on properties of Bessel paths.
References
[1] A. BESKOS AND G.O. ROBERTS, Exact simulation of diffusions, The Annals of Applied
Probability, 15(4), 2005.
670
Hitting times for the Brownian motion and Bessel processes: some new
algorithms
Madalina Deaconu, madalina.deaconu@inria.fr
Inria, France
The aim of this talk is to introduce new numerical methods for the hitting time of some bound-
aries for a class of stochastic processes. We will start by constructing the walk on moving
spheres method for Bessel processes. We will apply this result for the simulation of the Brow-
nian motion hitting times. This algorithm exhibits a new procedure which efficiently approxi-
mate both the hitting time and the hitting position of the stochastic process and can be used in
problems arising from finance, geophysics or neuroscience.
We give also the convergence results and present some numerical examples that permit to
emphasise the efficiency and accuracy of this new method.
This is a joint work with Samuel Herrmann (Dijon University, France).
Generation of first passage times for diffusion processes: an overview of
simulation techniques
Samuel Herrmann, samuel.herrmann@u-bourgogne.fr
University of Burgundy, France
Coauthor: Cristina Zucca
Many biological or physical applications require to simulate random variables with a given
probability distribution. The aim of our study is to focus on a particular random variable: the
first passage time (FPT) of a diffusion process. We introduce (Xt) the unique solution of the
following SDE:
dXt = b(Xt) dt + σ(Xt) dBt, X0 = x,
where (Bt) stands for a one-dimensional Brownian motion and define τL the first passage time
through the level L. We propose an overview of several simulation techniques.
• The classical way is to use efficient algorithms for the simulation of sample paths, like dis-
cretization schemes. Such methods permit to obtain approximations of the first-passage
times as a by-product.
• Another approach based on a random walk on spheroids permit in particular cases to
express the first passage time as the limit of a random walk. It suffices therefore to
describe precisely the convergence of this stochastic process and to introduce a stopping
procedure.
• Finally we present a new rejection sampling algorithm which permits to perform an exact
simulation of the first-passage time for general one-dimensional diffusion processes. The
main ideas are based both on a previous algorithm pointed out by A. Beskos et G. O.
Roberts which uses Girsanov’s transformation and on properties of Bessel paths.
References
[1] A. BESKOS AND G.O. ROBERTS, Exact simulation of diffusions, The Annals of Applied
Probability, 15(4), 2005.
670
