Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve

Abstract

Let σ, δ > 0, b ≥ 0. Let λ 2 : R + → R + , be continuous, and locally of bounded variation. We develop a general analytic criterion for the pathwise uniqueness ofwhere p ∈ (0, 1), and ℓ 0 t (R−λ 2 ) is the symmetric semimartingale local time of R−λ 2 . The criterion is related to the existence of nice (Kummer) functions for the time dependent infinitesimal generator of R. As a corollary we obtain explicit sufficient conditions for pathwise uniqueness. These are expressed in terms of λ 2 , its derivative, and the parameters σ, δ, b, p.2000 Mathematics Subject Classification: Primary: 60H10, 60J60, 60J55; Secondary: 35K20.

Weak existence of the squared Bessel and CIR processes with skew reflection on a deterministic time-dependent curve

“…Further, as shown in [12], a solution to (1) always stays positive when started with positive initial condition. So one can discard the absolute value under the square root in (1) exactly as one can do in the case of classical squared Bessel and CIR processes.…”

“…Note that a stronger version of (32) holding for more general λ 2 is obtained in [12] by purely probabilistic methods. One observes immediately the discontinuity of the local times in the space variable, thus we provide another example of diffusion with discontinuous local time (see e.g.…”

“…From [12] we know that {s ≥ 0|X s − κ(s) = 0} is P y -a.s. of Lebesgue measure zero for E-q.e. y ∈ E.…”

“…below (6), and Remarks 2.4 and 2.8(ii) for more general λ). It is remarkable that pathwise uniqueness of (1) could in particular be deduced in [12] for 2 p − 1 < 0 if (λ 2 ) ≥ σ 2 4 (δ − bλ 2 ). Thus for p < 1 2 and any strictly decreasing λ which is bounded below by the mean-reverting level δ b .…”

“…(1) and (2) are truly worth to be studied. The parallel work [12] which deals with pathwise uniqueness of (1) shows that stochastic calculus is possible, even with such highly singular equations.…”