The Local Time of the Fractional Ornstein-Uhlenbeck Process

Theorem 2. Let $$ be the fractional Ornstein-Uhlenbeck process. Then, for every t ∈ ℝ⁺ and any x ∈ ℝ, there exist positive and finite constants C₁ and C₂ such that

Proof. Let t ≥ 0 be a fixed point. Following the Fourier analytic approach of Berman [26], we have

Let $$, s ≥ 0, and denote by R(s₁, s₂, …, s_n) the covariance matrix of $$ for different s₁, …, s_n, then we have

By Yan et al. [11], one can write the fractional Ornstein-Uhlenbeck process starting from zero as

where B is a standard Brownian motion with B₀ = 0, and for 0 < u < t

with 1/2 < H < 1, κ_H = (2HΓ((3/2)−H)/Γ(H+(1/2))Γ(2−2H))^1/2, and

with 0 < H < 1/2.

For any r, s ∈ [t, t + h] such that r < s, we have

where the last equality follows from the fact that $$ is measurable with respect to σ(B_u, u ≤ r). Moreover, we can write

Hence, by using the measurability of $$ with respect to σ(B_u, u ≤ s), we have

where, to obtain the second equality, we have used the fact that $$ is independent of σ(B_u, u ≤ s) (by the independence of the increments of the Brownian motion). Combining (31), inequation (35), and inequation (37), we have

where s₀ = 0. Hence, the change of variable V = R^1/2U, U = (u₁, …, u_n) implies that

Hence,

Following from Stirling’s formula, we have  n!/Γ(1 + n(1 − H)) ≤ Aⁿn!^H, n ≥ 2, for a suitable finite number A. So

Following, we first prove that for any K > 0, there exists a positive and finite constant B > 0, depending on t, such that for sufficiently small u

First consider u of the form u = 1/n. By Chebyshev’s inequality and inequation (41), we have

Choose B > C and n₀ large such that for any n ≥ n₀, to dominate (43) by e^−2Kn. Moreover, for u sufficiently small, there exists n ≥ n₀ such that u_n+1 < u < u_n and since n ≥ 1, n/(m + 1) ≥ 1/2. This proves inequation (42).

On the other hand, if we take u(h) = 1/log  log (1/h) and consider h_n of the form 2⁻ⁿ, then inequation (42) implies

for large n. So, following that Borel-Cantelli lemma and monotonicity arguments, we have

Theorem 3. For H_i ∈ (0,1), i = 1,2. Then ℓ_ɛ,T converges in L²(Ω, ℱ, P), as ɛ ↓ 0. Moreover, the limit is denoted by ℓ_T, then ℓ_T ∈ L²(Ω, ℱ, P).

Proof. First we claim that ℓ_ɛ,T ∈ L²(Ω, ℱ, P) for every ɛ > 0. By (48) we have

for all ɛ ∈ (0,1].

Second, we claim that the sequence {ℓ_ɛ,T,  ɛ > 0} is of Cauchy in L²(Ω, ℱ, P). For any θ, ɛ > 0 we have

Thus, dominated convergence theorem yields

as ɛ → 0 and θ → 0, which leads to ℓ_ɛ,T is a Cauchy sequence in L²(Ω, ℱ, P). Consequently, lim _ɛ→0ℓ_ɛ,T exists in L²(Ω, ℱ, P). This completes the proof.

Theorem 4. Let H₁, H₂ ∈ (0,1) and β = min {H₁, H₂}. Then the collision local time ℓ_T satisfies the following estimate:

for all s, t, s < t.

Proof. For any 0 ≤ r, l ≤ T we denote

for a constant C > 0. It follows from (48) that for 0 ≤ s ≤ t ≤ T

Lemma 5 (An and Yan [28]). For any x ∈ [−1,1) we have

where (2n − 2)! ! = 1 · 3 · 5 ⋯ (2n − 1) and (2n − 1)!! = 2 · 4 · 6 ⋯ (2n − 2).

Theorem 6. Let ℓ_T, T ≥ 0 be the collision local time process of two independent fractional Ornstein-Uhlenbeck $$, with respective indices H_i ∈ (0,1). Then ℓ_T is smooth in the sense of the Meyer-Watanabe if and only if

Proof. By Yan et al. [11], we have

where the notation F≍G means that there are positive constants c₁ and c₂ so that

in the common domain of definition for F and G. Hence, following Theorem 2 in An and Yan [28], we have $$ if and only if min {H₁, H₂} < 1/3. Therefore, in order to prove Theorem 6, it only needs to prove: for T ≥ 0, ℓ_T is smooth in the sense of the Meyer-Watanabe if and only if

In fact, for ɛ > 0, T ≥ 0 we denote

and $$. Thus, by Proposition 1 to prove that (67) holds if and only if Φ_Θ(1) < ∞. Clearly, we have

for all T ≥ 0, where we have used the following fact: For two random variables X, Y with joint Gaussian distribution such that E(X) = E(Y) = 0 and E(X²) = E(Y²) = 1 we have (see, for example, Nualart [3])

for all T ≥ 0. This completes the proof.