First-passage and escape problems in the Feller process

The Feller process is an one-dimensional diffusion process with linear drift and state-dependent diffusion coefficient vanishing at the origin. The process is positive definite and it is this property along with its linear character that have made Feller process a convenient candidate for the modeling of a number of phenomena ranging from single neuron firing to volatility of financial assets. While general properties of the process are well known since long, less known are properties related to level crossing such as the first-passage and the escape problems. In this work we thoroughly address these questions.

💡 Research Summary

The paper provides a comprehensive treatment of first‑passage and escape problems for the one‑dimensional Feller diffusion, a process defined by the stochastic differential equation
(dX_t = \theta(\mu - X_t),dt + \sigma\sqrt{X_t},dW_t) with (\theta,\mu,\sigma>0). Because the diffusion coefficient vanishes at the origin, the process remains strictly positive, a property that makes it attractive for modeling neuronal inter‑spike intervals, stochastic volatility, and other phenomena where a non‑negative state is required.

The authors begin by formulating the Kolmogorov forward (Fokker‑Planck) equation associated with the SDE and then impose two types of boundary conditions: an absorbing boundary at a prescribed level (a) (first‑passage) and a pair of absorbing/reflecting boundaries at (a) and (b) (escape). By introducing the dimensionless variable (y = 2\theta X/\sigma^2), the forward equation is transformed into a modified Bessel equation. Its general solution is expressed in terms of the modified Bessel functions (I_{\nu}(y)) and (K_{\nu}(y)), where the order (\nu = 2\theta\mu/\sigma^2 - 1) encapsulates the balance between drift strength and diffusion intensity. The sign of (\nu) determines the nature of the origin: for (\nu>0) the origin behaves as a reflecting boundary, whereas for (\nu\le 0) it becomes absorbing, a fact that the paper proves rigorously.

For the first‑passage problem, the Laplace transform of the survival probability is derived, leading to an explicit expression for the probability density function (PDF) of the first‑passage time to level (a):
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