Related papers: Explicit moments of decision times for single- and…
The drift diffusion model (DDM) is a model of sequential sampling with diffusion (Brownian) signals, where the decision maker accumulates evidence until the process hits a stopping boundary, and then stops and chooses the alternative that…
We propose a unifying theoretical framework for the analysis of first-passage time distributions in two important classes of stochastic processes in which the diffusivity of a particle evolves randomly in time. In the first class of…
Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a…
Real data are constrained to finite sampling rates, which calls for a suitable mathematical description of the corrections to the finite-time estimations of the dynamic equations. Often in the literature, lower order discrete time…
We consider the distribution of the duration time, the time elapsed since it began, of a diffusion process given its present position, under the assumption that the process began at the origin. For unbiased diffusion, the distribution does…
The paper presents new simple sharp bounds for transition density functions for time-homogeneous diffusions processes. The bounds are obtained under mild conditions on the drift and diffusion coefficients, extending and substantially…
We consider the drift and diffusion properties of periodically driven renewal processes. These processes are defined by a periodically time dependent waiting time distribution, which governs the interval between subsequent events. We show…
Drift diffusion models (DDMs) have found widespread use in computational neuroscience and other fields. They model evidence accumulation in simple decision tasks as a stochastic process drifting towards a decision barrier. In models where…
When the unconditioned process is a diffusion submitted to a space-dependent killing rate $k(\vec x)$, various conditioning constraints can be imposed for a finite time horizon $T$. We first analyze the conditioned process when one imposes…
We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original…
Decision Transformer (DT), a trajectory modelling method, has shown competitive performance compared to traditional offline reinforcement learning (RL) approaches on various classic control tasks. However, it struggles to learn optimal…
New theorems for the moments of the first passage time of one dimensional nonlinear stochastic processes with an entrance boundary are formulated. This important class of one dimensional stochastic processes results among others from…
This study connects two methods for modeling reaction times (RTs) in choice tasks: (1) the first-hitting time of a simple diffusion model with a single barrier, representing the cognitive process leading to a response, and (2) Generalized…
We propose threshold diffusion processes as unique solutions to stochastic differential equations with step-function coefficients, and obtain explicit expressions for the conditional Laplace transform of the hitting times and the potential…
The problem of eliminating fast-relaxing variables to obtain an effective drift-diffusion process in position is solved in a uniform and straightforward way for models with velocity a function jointly of position and fast variables. A more…
Piecewise Diffusion Markov Processes (PDifMPs) are valuable for modelling systems where continuous dynamics are interrupted by sudden shifts and/or changes in drift and diffusion. The first-passage time (FPT) in such models plays a central…
The distribution of the first-passage time (FPT)$T_a$ for a Brownian particle with drift $\mu$ subject to hitting an absorber at a level $a>0$ is well-known and given by its density $\gamma(t) = \frac{a}{\sqrt{2 \pi t^3} } e^{-\frac{(a-\mu…
Research in psychology and neuroscience has successfully modeled decision making as a process of noisy evidence accumulation to a decision bound. While there are several variants and implementations of this idea, the majority of these…
We consider the first-crossing-time problem through a constant boundary for a Wiener process perturbed by random jumps driven by a counting process. On the base of a sample-path analysis of the jump-diffusion process we obtain explicit…
For the last ten years, almost every theoretical result concerning the expected run time of a randomized search heuristic used drift theory, making it the arguably most important tool in this domain. Its success is due to its ease of use…