We study the probability that a Lévy process at time T has not yet crossed
a given deterministic function f (called the moving boundary). If f is
constant this question is in the realm of classical fluctuation theory. We
determine for what functions f the asymptotic order (as T tends to
infinity) of this probability is the same as for a constant function.
This is joint work with Tanja Kramm.
We introduce option price expansions for regular barrier options focusing on the down and out case. In the framework of time-dependent local volatility models and martingale assets, we derive new formulas using a method mixing stochastic analysis and PDE approach. We choose a Gaussian proxy model and express the difference with the local volatility model using the PDE associated to the proxy process. Then we smartly combine expansion of the local volatility function, Itô calculus, key relations (involving martingales and convolution simplifications) and PDE arguments to obtain the approximation formulas with tight error estimates using the derivatives of the Gaussian proxy kernel. The accuracy of our formulas is illustrated throughout numerical experiments.
The aim of this work is to propose a practical numerical method to approximate the survival function of the exit time for a piecewise-deterministic Markov process. Our method is based on a recursive formulation of the problem and the discretization of a discrete-time Markov chain naturally embedded within the continuous-time process. Our results are applied to the evaluation of the service time of a structure subject to corrosion.
We study the one-sided exit problem, also known as one-sided barrier problem: given a stochastic process X we would like to find the asymptotic rate for it to stay below 1 for long time.
As underlying process we consider the fractionally integrated Lévy process and random walks. We find polynomial tails, where the leading order, the persistence exponent, is very robust under the choice of the underlying Lévy process.
The talk is based on joint work with Frank Aurzada (Darmstadt).
Considering deterministic time grids for approximating stochastic integrals or discretizing stochastic processes is a standard approach. Here in this work, we allow the grids to be random (sequence of stopping times) and we seek the optimal ones in an asymptotic sense. We establish that, for Ito diffusion models, hitting times of ellipsoids appear universally as optimal among a very large class of stopping times. The proofs rely in particular on original pathwise estimates on Brownian martingale increments and time steps of random grids.
Joint work with N. Landon.
In this talk we will address the rst exit problem from a bounded domain, containing the attractor of a deterministic dynamical system, which is the support of a respective invariant measure, perturbed by multiplicative heavy-tailed Levy noise, in the limit of small intensity. The main application is the asymptotic rst exit problem for the Van-der-Pol oscillator perturbed by multiplicative α-stable noise. This is joint work with Ilya Pavlyukevich, Friedrich-Schiller- Universitat Jena.
Interface conditions appear in many domains where diffusion are involved:
geophysics, population ecology, oceanography, brain imaging, ... In the simplest
cases, these interface conditions are defined as linear relations
on the solution of a PDE and their flux at each side of the interface.
Second-order PDEs with discontinuous coefficients may be rewritten
as PDE with interface conditions.
During this talk, we consider some problems related to diffusion processes evolving in one-dimensional media with interface. They could be described as stopped processes that rebirth in some way at the interface. We then focus on the role played by the local time as well as on simulations issues.
In the early 50’s, Feller characterized fully the analytical structure
of one-dimensional diffusion processes. In particular, he constructed
their Green functions in terms of the two positive fundamental solutions of a second order differential equations associated to their infinitesimal generators. Following Darling and Siegert, he also characterized completely the Laplace transform of their first exit times from an interval in terms of these two fundamental solutions. These type of characterization has been also achieved for the class of spectrally negative Lévy processes and some closely related processes. There is a substantial literature devoted to these problems in this framework, we mention the works of Takacs, Suprun,
Bertoin and Doney to name but a few.
During this mini-course, we aim to pursue Feller’s program for the class of one-dimensional completely asymmetric Markov processes, that is strong Markov processes having jumps only in one direction. More specifically, we will present an original methodology based mostly on potential theoretical arguments to characterize the Laplace of their first exit times from an interval which may occur by a jump. We will also describe, in terms of fundamental q-excessive functions, their resolvent densities (Green functions) whose existence of a nice version is provided. We will first illustrate our approach by recovering easily the well-known fluctuations identities for spectrally negative Lévy processes. Finally, we will present the solution of the first exit-times problems for some generalizations of the Ornstein-Uhlenbeck process and continuous time branching processes with immigration.
We study the behaviour of one-dimensional Langevin equations with linear and non-linear friction driven by α-stable Lévy processes. In the limit of large friction, study the convergence of the solutions in the Skorokhod M_1-topology. In particular this allows us to determine the limiting distribution of their first passage times.
The toolbox of Monte Carlo simulation provides - easy to implement - procedures for estimating
functionals of stochastic processes such as boundary crossing probabilities, c.d.f.'s of first exit times
or prices of certain path dependent options. There exist procedures which generate paths on a
discrete time grid from the exact distributions, but most procedures are biased in the sense that even
on the discrete time grid, the distribution from which the samples are drawn is an approximation.
The MSE (mean squared error) is then the sum of the squared bias and a variance term. If N
univariate random variables are used, n discrete paths of lengths m = N/n are generated, the
variance is of order 1/n, but the MSE is of order (1/N)^a with a < 1. Naive applications of MC often
have a MSE of order (1/N)^(1/2) only!
In the talk I will present as variance reduction technique the method of adaptive control vari- ables. The approximating functional is itself approximated by a functional of a discrete time path of smaller complexity. Although the expectation of the control variable has to be estimated, the combination of expectation and approximation allows an improvement of the convergence rate. Iterating the control variables leads even to a MSE which is O(1/N), the approximation rate of finite-dimensional problems.
Examples of applications and results on approximation rates are given. [pdf]
In 1981 J. Pitman and M. Yor showed that the law of the last passage
time of a linear (transient) diffusion to a constant boundary may be
expressed simply thanks to the transition density of the diffusion.
Therefore, unlike first hitting times for which we generally only know
the Laplace transform, last passage times seems to be easier to grasp
The aim of this talk is then to give a few results regarding the last passage time of a linear diffusion to a curved boundary. We shall first give a general (theoretical) expression for the density of such a random variable, and then discuss some martingales' methods and integral equations' methods to obtain explicit formulae. These results will be illustrated by many examples, mainly with Bessel processes (with drift).
We consider a nearest neighbor random walk on Z which is reflecting at 0 and perturbed when it reaches its maximum. We compute the law of the hitting times and derive many corollaries, especially invariance principles with (rather) explicit descriptions of the asymptotic laws. We obtain also some results on the almost sure asymptotic behavior.
The spiking times of a noisy neuron is usually modeled by the first hitting time of a deterministic threshold by a (non homogeneous) 1D diffusion process. The aim of this work is to construct an efficient algorithm to simulate such f.h.t. We present here an extension of the classical walk on spheres algorithm to non-constant (in time) domains. We first prove the convergence of the algorithm and have a control on its mean complexity in the case of particular diffusions like Brownian Motion and Ornstein Uhlenbeck process.
This is a common work with Samuel Herrmann (Univ. de Bourgogne)
The aim of the talk is to present a new proof of a result on McKean-Vlasov diffusions. Herrmann, Imkeller and Peithmann solved the exit problem of self-stabilizing diffusions by adapting the proof by Freidlin and Wentzell. Here, we study the exit problem of the associated system of particles then we use some strong result of propagation of chaos.