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

Abstract

Let σ > 0, δ ≥ 1, b ≥ 0, 0 < p < 1. Let λ be a continuous and positive function in H 1,2 loc (R + ). Using the technique of moving domains (see Russo and Trutnau (2005) [9]), and classical direct stochastic calculus, we construct for positive initial conditions a pair of continuous and positive semimartingales (R,where the symmetric local times 0 (R −λ 2 ), 0 (Well-known special cases are the (squared) Bessel processes (choose σ = 2, b = 0, and λ 2 ≡ 0, or equivalently p = 1 2 ), and the Cox-Ingersoll-Ross process (i.e. R, with λ 2 ≡ 0, or equivalently p = 1 2 ). The case 0 < δ < 1 can also be handled, but is different. If | p| > 1, then there is no solution.

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

“…If now |2p − 1| > 1, then a solution to (1) does not exist. This improves results of [35,Remark 2.7(i)] where this could only be shown for those λ 2 's for which a solution to (1) could be constructed. In fact, by (i) it holds that ℓ Lemma 2.4 Let X, Y be two continuous semimartingales, with X 0 = Y 0 .…”

“…Weak existence of a solution to (1) as well as for its square root has been established in [35] in various cases. It is worth to remark that we were not able to construct in whole generality a solution in the extreme cases p = 0 and p = 1, and for other than increasing (though nonetheless for constant) λ 2 if −1 < 2p − 1 < 0.…”

“…Recently, in [8], the square root of (1) for σ = 2, b = 0, p ∈ (0, 1), dimension δ ≥ 2, and λ ≡ a ∈ R ++ , was independently of us constructed and further studied. In an earlier paper [33] we considered a more general equation where δ > 0 and λ is a monotone function, and later in [35] we generalized our results and considered weakly differentiable λ. In [8] many interesting properties are derived and applied to the valuation of perpetuities and pricing of weighted Asian options.…”

“…In fact, by (i) it holds that ℓ 0 (R − λ 2 ), ℓ 0+ (R − λ 2 ), ℓ 0− (R − λ 2 ) ≡ 0. Now, in order to conclude see [35,Remark 2.7(i)]. There analytic methods were used.…”

“…These processes form a natural generalization of the classical squared Bessel and CIR processes. Local times on curves already appear in [7], [11], [4], [5], in [27] where an Itô formula involving such local times is derived, and in [33], [35]. For another related Itô-type formula but which is less relevant for the topic of our paper we refer to [12].…”