1,721,040 research outputs found
Inverse First-Passage Problems of a Diffusion with Resetting
No full textWe address some inverse problems for the first-passage place and the
first-passage time of a one-dimensional diffusion process with stochastic resetting.
This type of diffusion is characterized by the fact that a reset to the position xR
can occur according to a homogeneous Poisson process with rate r > 0. As regards
the inverse first-passage place problem, for random initial position belonging to an
interval (0, b) with finite b > 0 (and fixed r and xR belonging to (0, b)), let τ be
the first time at which the process exits the interval (0, b), and π0 be the probability
of exit from the left end of (0, b). Given a probability q, the inverse first-passage
place problem consists in finding the density g of the initial position, if it exists, such
that π0 = q. Concerning the inverse first-passage time problem, for random positive
starting point (and fixed r and xR > 0), let again τ be the first-passage time of
the process through zero. For a given distribution function F(t) on the positive real
axis, the inverse first-passage time problem consists in finding the density g of the
starting point, if it exists, such that P(τ ≤ t) = F(t), t > 0. In addition to the case
of random initial position, we also study the case when the initial position and the
resetting rate r are fixed, whereas the reset position xR is random. For all types of
inverse problems considered, several explicit examples of solutions are reported
Some Remarks on the Mean of the Running maximum of Integrated Gauss-Markov Processes and Their First-Passage Times.
No full textExplicit formulae for the mean of the running maximum of
conditional and unconditional Brownian motion are found; these formulae are used to obtain the mean, a(t), of the running maximum of an integrated Gauss-Markov process X(t). Moreover, the connection between
the moments of the first-passage-time of X(t) and a(t) is investigated.
Some explicit examples are reported
On the joint distribution of first-passage time and first-passage area of drifted Brownian motion
No full textFor drifted Brownian motion X(t) = x − μt + Bt (μ > 0) starting from x > 0,
we study the joint distribution of the first-passage time below zero, τ (x), and the firstpassage
area, A(x), swept out by X till the time τ (x). In particular, we establish differential
equations with boundary conditions for the joint moments E[τ (x)mA(x)n], and we present
an algorithm to find recursively them, for any m and n. Finally, the expected value of the
time average of X till the time τ (x) is obtaine
On the entropy of fractionally integrated gauss–markov processes
No full textThis paper is devoted to the estimation of the entropy of the dynamical system {Xα (t), t ≥ 0}, where the stochastic process Xα (t) consists of the fractional Riemann–Liouville integral of order α ∈ (0, 1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα (t) is a decreasing function of α ∈ (0, 1)
The mean of the running maximum of an integrated Gauss-Markov process and the connection with its first-passage time.
No full textWe find explicit formulae for the mean of the running maximum of conditional
and unconditional Brownian motion; they are used to obtain the
mean, a(t), of the running maximum of an integrated Gauss–Markov
process. Then, we deal with the connection between the moments of
its first-passage-time and a(t). As explicit examples, we consider integrated
Brownian motion and integrated Ornstein–Uhlenbeck process
An inverse first-passage problem revisited: the case of fractional Brownian motion, and time-changed Brownian motion
No full textWe revisit an inverse first-passage time (IFPT) problem, in the cases
of fractional Brownian motion, and time-changed Brownian motion.
Let be a one dimensional continuous stochastic process starting
from a random position , let be an assigned continuous
boundary, such that with probability one, and F an assigned distribution
function. The IFPT problem here considered consists in finding the
distribution of such that the first-passage time of X(t) below S(t)
has distribution F. We study this IFPT problem for fractional
Brownian motion and a constant boundary ; we also obtain
some extension to other Gaussian processes, for one, or two, time dependent boundaries
On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations
No full textFor a fractionally integrated Brownian motion (FIBM) of order (Formula presented.) (Formula presented.) we investigate the decaying rate of (Formula presented.) as (Formula presented.) where (Formula presented.) is the first-passage time (FPT) of (Formula presented.) through the barrier (Formula presented.) Precisely, we study the so-called persistent exponent (Formula presented.) of the FPT tail, such that (Formula presented.) as (Formula presented.) and by means of numerical simulation of long enough trajectories of the process (Formula presented.) we are able to estimate (Formula presented.) and to show that it is a non-increasing function of (Formula presented.) with (Formula presented.) In particular, we are able to validate numerically a new conjecture about the analytical expression of the function (Formula presented.) for (Formula presented.) Such a numerical validation is carried out in two ways: in the first one, we estimate (Formula presented.) by using the simulated FPT density, obtained for any (Formula presented.) in the second one, we estimate the persistent exponent by directly calculating (Formula presented.) Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of (Formula presented.) and we find the upper bound of its covariance function
Fractionally integrated Gauss-Markov processes and applications
No full textWe investigate the stochastic processes obtained as the fractional Riemann-Liouville inte-gral of order alpha is an element of (0 , 1) of Gauss-Markov processes. The general expressions of the mean, variance and covariance functions are given. Due to the central role, for the fractional inte-gral of standard Brownian motion and of the non-stationary/stationary Ornstein-Uhlenbeck processes, the covariance functions are carried out in closed-form. In order to clarify how the fractional order parameter alpha affects these functions, their numerical evaluations are shown and compared also with those of the corresponding processes obtained by ordi-nary Riemann integral. The results are useful for fractional neuronal models with long range memory dynamics and involving correlated input processes. The simulation of these fractionally integrated processes can be performed starting from the obtained covariance functions. A suitable neuronal model is proposed. Graphical comparisons are provided and discussed
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