On the excursions of drifted Brownian motion and the successive passage times of Brownian motion

By using the law of the excursions of Brownian motion with drift, we find the distribution of the $n-$th passage time of Brownian motion through a straight line $S(t)= a + bt.$ In the special case when $b = 0,$ we extend the result to a space-time transformation of Brownian motion.Comment: 4 figures, accepted for publication in Physica

The arctangent law for a certain random time related to a one-dimensional diffusion

For a time-homogeneous, one-dimensional diffusion process $X(t),$ we investigate the distribution of the first instant, after a given time $r,$ at which $X(t)$ exceeds its maximum on the interval $[0,r],$ generalizing a result of Papanicolaou, which is valid for Brownian motion

The first-crossing area of a diffusion process with jumps over a constant barrier

For a given barrier $S$ and a one-dimensional jump-diffusion process $X(t),$ starting from $x<S,$ we study the probability distribution of the integral $A_S(x)= \int_0 ^ {\tau_S(x)}X(t) \ dt$ determined by $X(t)$ till its first-crossing time $\tau_S(x)$ over $S.$ In particular, we show that the Laplace transform and the moments of $A_S(x)$ are solutions to certain partial differential-difference equations with outer conditions. The distribution of the minimum of $X(t)$ in $[0, \tau_S(x)]$ is also studied. Thus, we extend the results of a previous paper by the author, concerning the area swept out by $X(t)$ till its first-passage below zero. Some explicit examples are reported, regarding diffusions with and without jumps

One-dimensional reflected diffusions with two boundaries and an inverse first-hitting problem

We study an inverse first-hitting problem for a one-dimensional, time-homogeneous diffusion $X(t)$ reflected between two boundaries $a$ and $b,$ which starts from a random position $\eta.$ Let $a \le S \le b$ be a given threshold, such that $P( \eta \in [a,S])=1,$ and $F$ an assigned distribution function. The problem consists of finding the distribution of $\eta$ such that the first-hitting time of $X$ to $S$ has distribution $F.$ This is a generalization of the analogous problem for ordinary diffusions, i.e. without reflecting, previously considered by the author

On the first-passage time of an integrated Gauss-Markov process

It is considered the integrated process $X(t)= x + \int _0^t Y(s) ds ,$ where $Y(t)$ is a Gauss-Markov process starting from $y.$ The first-passage time (FPT) of $X$ through a constant boundary and the first-exit time of $X$ from an interval $(a,b)$ are investigated, generalizing some results on FPT of integrated Brownian motion. An essential role is played by a useful representation of $X,$ in terms of Brownian motion which allows to reduces the FPT of $X$ to that of a time-changed Brownian motion. Some explicit examples are reported; when theoretical calculation is not available, the quantities of interest are estimated by numerical computation.Comment: 4 figure

Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion process

We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a previous paper. If X(t) is a one-dimensional diffusion with jumps, starting from a random position η ∈ [a, b], let be τa,b the time at which X(t) first exits the interval (a, b), and πa = P(X(τa,b) ≤ a) the probability of exit from the left of (a, b). Given a probability q ∈ (0, 1), the problem consists in finding the density g of η (if it exists) such that πa = q; it can be seen as a problem of optimization

On the estimation of the persistence exponent for a fractionally integrated brownian motion by numerical simulations

For a fractionally integrated Brownian motion (FIBM) of order alpha is an element of (0, 1], X-alpha(t), we investigate the decaying rate of P(tau(alpha)(S) &gt; t) as t -&gt; +infinity, where tau(alpha)(S) = inf{t &gt; 0 : X-alpha(t) &gt;= S} is the first-passage time (FPT) of X-alpha(t) through the barrier S &gt; 0. Precisely, we study the so-called persistent exponent theta = theta(alpha) of the FPT tail, such that P(tau(alpha)(S) &gt; t) = t(-theta+o(1)), as t -&gt; +infinity, and by means of numerical simulation of long enough trajectories of the process X-alpha(t), we are able to estimate theta(alpha) and to show that it is a non-increasing function of alpha is an element of (0, 1], with 1/4 &lt;= theta(alpha) &lt;= 1/2. In particular, we are able to validate numerically a new conjecture about the analytical expression of the function theta = theta(alpha), for alpha is an element of (0, 1]. Such a numerical validation is carried out in two ways: in the first one, we estimate theta(alpha), by using the simulated FPT density, obtained for any alpha is an element of (0, 1]; in the second one, we estimate the persistent exponent by directly calculating P(max(0)&lt;= s &lt;= tX(alpha)(s) &lt; 1). Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of X-alpha(t) and we find the upper bound of its covariance function