Standard and fractional reflected Ornstein–Uhlenbeck processes as the limits of square roots of Cox–Ingersoll–Ross processes

Low-dimensional Cox-Ingersoll-Ross process

“…where ℓ(t, 0) := lim y→0+ ℓ(t, y) is defined by continuity. Finally, we close the gap of [25] mentioned above and obtain a representation of the Skorokhod reflection function for the ROU process in terms of CIR processes of dimension k = 4a σ 2 < 1. It is worth noting that our approach is simpler than the one in [5,6] and is based on the following machinery: we notice that Itô's formula applied to X(t) + ε followed by moving ε ↓ 0 implies that Y = √ X satisfies the equation of the form (1.9) with L represented as an a.s.-limit…”

“…Among notable exceptions, we mention [4,7,9,10] which discussed the SDEs of the type (1.3) when σ 2 4 < a < σ 2 2 . It is worth to note a more recent paper [25] which establishes a connection between Y = √ X and a reflected Ornstein-Uhlenbeck (ROU) process…”

“…Additionally, [25,Theorem 2.4] provides a new representation of L 0 in terms of a limit of the CIR processes: with probability 1, for any positive sequence {ε n , n ≥ 1} such that ε n ↓ 0, n → ∞, and for all T > 0 sup…”

“…The representation of L 0 from [25] described above essentially concerns convergence of the CIR square roots as a → σ 2 4 + and does not cover what happens when a → σ 2 4 −. The reason is that analytic challenges associated to the process Y = √ X are especially acute when 0 < a < σ 2 4 , i.e.…”

Piecewise-Tunneled Captive Processes and Corridored Random Particle Systems

“…This model was later expanded by [15] which investigated the Laplace transform of the first-passage times of the Ornstein-Uhlenbeck diffusions with two-sided boundaries (see [45] for another application in queuing systems). Recently [46] discovered an interesting connection between Cox-Ingersoll-Ross process and ROU, arguing that, with probability 1, the square root of the former converges uniformly to the latter as the mean-reversion parameter tends to 0 in the fractional case. In finance, while [47] focused on interest rate models under reflecting and absorbing boundaries, [48] studied the pricing of barrier options on zero-coupon bonds.…”