Sticky Bessel diffusions

Goran Peskir

Stochastic Processes and their Applications, 2022, vol. 150, issue C, 1015-1036

Abstract: We consider a Bessel process X of dimension δ∈(0,2) having 0 as a slowly reflecting (sticky) boundary point with a stickiness parameter 1/μ∈(0,∞). We show that (i) the process X can be characterised through its square Y=X2 as a unique weak solution to the SDE system dYt=δI(Yt>0)dt+2YtdBtI(Yt=0)dt=12μdℓt0(Y) where B is a standard Brownian motion and ℓ0(Y) is a diffusion local time process of Y at 0, and (ii) the transition density function of X can be expressed in the closed form as a convolution integral involving a Mittag-Leffler function and a modified Bessel function of the second kind. Appearance of the Mittag-Leffler function is novel in this context. We determine a (sticky) boundary condition at zero that characterises the transition density function of X as a unique solution to the Kolmogorov backward/forward equation of X. We also show that the convolution integral can be characterised as a unique solution to the generalised Abel equation of the second kind. Letting μ↓0 (absorption) and μ↑∞ (instantaneous reflection) the closed-form expression for the transition density function of X reduces to the ones found by Feller (1951) and Molchanov (1967) respectively.

Keywords: Slowly reflecting (sticky) boundary behaviour; Bessel process; Stochastic differential equation; Transition probability density function; Mittag-Leffler function; Generalised Abel equation (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.spa.2020.12.009

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