Related papers: Brownian excursions an Parisian barrier options: a…
For normally reflected Brownian motion and for simple random walk on independently growing in time d-dimensional domains, d>=3, we establish a sharp criterion for recurrence versus transience in terms of the growth rate.
We develop a numerical method for the martingale analogue of the Benamou--Brenier optimal transport problem, which seeks a martingale interpolating two prescribed marginals which is closest to the Brownian motion. Recent contributions have…
We present an exact solution for the probability density function $P(\tau=t_{\min}-t_{\max}|T)$ of the time-difference between the minimum and the maximum of a one-dimensional Brownian motion of duration $T$. We then generalise our results…
We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in…
We calculate crossing probabilities and one-sided last exit time densities for a class of moving barriers on an interval $[0,T]$ via Schwartz distributions. We derive crossing probabilities and first hitting time densities for another class…
The present paper is concerned with the integral of the absolute value of a Brownian motion with drift. By establishing an asymptotic expansion of the space Laplace transform, we obtain series representations for the probability density…
Basic properties of Brownian motion are used to derive two results concerning birth-death chains. First, the probability of extinction is calculated. Second, sufficient conditions on the transition probabilities of a birth-death chain are…
Brownian escape is key to a wealth of physico-chemical processes, including polymer folding, and information storage. The frequency of thermally activated energy barrier crossings is assumed to generally decrease exponentially with…
We consider a class of stochastic processes containing the classical and well-studied class of Squared Bessel processes. Our model, however, allows the dimension be a function of the time. We first give some classical results in a larger…
We investigate the unique stationary measure of a positive recurrent reflecting Brownian motion in the upper half-plane, where the direction of reflection is constant on each half-axis. The Laplace transform of the stationary distribution…
We consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $\tau=t_{\min}-t_{\max}$ between the time $t_{\min}$ of the global…
The construction of the paths of all possible Brownian motions (in the sense of Knight) on a half line or a finite interval is reviewed.
The prime number theorem provided the chief impulse for complex Tauberian theory, in which the boundary behavior of a transform in the complex plane plays a crucial role. We consider Laplace transforms of bounded functions. Our Tauberian…
This note illustrates how a simple random walk with possibly long jumps is related to fractional powers of the Laplace operator. The exposition is elementary and self-contained.
Questions of flux regulation in biological cells raise renewed interest in the narrow escape problem. The often inadequate expansions of the narrow escape time are due to a not so well known fact that the boundary singularity of Green's…
The mean first-passage time (MFPT) for a Brownian particle to surmount a potential barrier of height $\Delta U$ is a fundamental quantity governing a wide array of physical and chemical processes. According to the Arrhenius Law, the MFPT…
In this note we consider the maximization of the expected terminal wealth for the setup of quadratic transaction costs. First, we provide a very simple probabilistic solution to the problem. Although the problem was largely studied, as far…
We consider the use of random walks as an approach to obtain connection coefficients for higher-order Bernoulli and Euler polynomials. In particular, we consider the cases of a $1$-dimensional linear reflected Brownian motion and of a…
Fractional Brownian motion has become a standard tool to address long-range dependence in financial time series. However, a constant memory parameter is too restrictive to address different market conditions. Here we model the price…
Donsker's theorem shows that random walks behave like Brownian motion in an asymptotic sense. This result can be used to approximate expectations associated with the time and location of a random walk when it first crosses a nonlinear…