Derivative of self-intersection local time for the sub-bifractional Brownian motion

1.   Introduction and main results

Recently, El-Nouty and Journ$\acute{e}$ ^[3] introduced the process $S^{H, K} = \{S^{H, K}_t, t\geq 0\}$ on the probability space $(\Omega, F, P)$ with indices $H\in (0, 1)$ and $K\in (0, 1],$ named the sub-bifractional Brownian motion (sbBm) and defined as follows:

where $\{B^{H, K}_t, t\in {\textbf{R}}\}$ is a bifractional Brownian motion (bBm) with indices $H\in (0, 1)$ and $K\in (0, 1],$ namely,

and $B^{H, K}_t(1)$ and $B^{H, K}_{-t}(2)$ are independent bifractional Brownian motions in $[0, +\infty)$ with indices $H$$\in(0, 1)$ and $K$$\in (0, 1]$, where bifractional Brownian motion $\{B^{H, K}_t(1), t\geq 0\}$ is a centered Gaussian process, starting from zero, with covariance

with $H\in (0, 1)$ and $K\in (0, 1].$

Clearly, the sbBm is a centered Gaussian process such that $S^{H, K}_0 = 0$, with probability 1, and ${\rm{Var}}(S^{H, K}_t) = (2^{K}-2^{2HK-1})t^{2HK}.$ Since $(2H-1)K-1 < K-1\leq 0,$ it follows that $2HK-1 < K$. We can easily verify that $S^{H, K}$ is self-similar with index $HK$. When $K = 1$, $S^{H, 1}$ is the sub-fractional Brownian motion (sfBm). For more on sub-fractional Brownian motion, we can see Kuang and Xie ^[15,16], Kuang and Liu ^[13,14], Xie and Kuang ^[21] and so on. Straightforward computations show that for all $s, t\geq 0,$

and

where

(See El-Nouty and Journ$\acute{e}$ ^[3]). Kuang ^[11] investigated the collision local time of two independent sub-bifractional Brownian motions. Kuang and Li ^[12] obtained Berry-Esséen bounds and proved the almost sure central limit theorem for the quadratic variation of the sub-bifractional Brownian motion.

The self-intersection local time of fractional Brownian motion (fBm) $B^H = \{B_t^H, t\geq 0\}$ was first studied in Rosen ^[17], and formally defined by

where $D = \{(r, s):0 < r < s < t\}$ and $\delta$ is the Dirac delta function. It was further investigated in Hu ^[4] and Hu and Nualart ^[5]. Jiang and Wang ^[8] considered self-intersection local time and collision local time of bifractional Brownian motion. Chen et al. ^[2] studied renormalized self-intersection local time of bifractional Brownian motion.

The context of derivative for the self-intersection local time is organized as follows. Rosen ^[18] first studied the Brownian motion case, and Yan et al. ^[22] extended this to fractional Brownian motion. On this basis, Jung and Markowsky ^[9,10] considered some in-depth results for derivative of the self-intersection local time of fractional Brownian motion. Jaramillo and Nualart ^[6,7] studied asymptotic properties of the derivative of self-intersection local time of fractional Brownian motion and functional limit theorem for the self-intersection local time of the fractional Brownian motion, respectively. Yan and Yu ^[23] considered the multidimensional fractional Brownian motion case. Yu ^[24] investigated higher order derivative of self-intersection local time for fBm. Shi ^[19] investigated fractional smoothness of derivative of self-intersection local time for bi-fractional Brownian motion.

Moreover, many authors have proposed to use more general self-similar Gaussian processes and random fields as stochastic models, and such applications have raised many interesting theoretical questions about self-similar Gaussian processes and fields in general. Therefore, some generalizations of the fBm has been introduced. However, contrast to the extensive studies on fractional Brownian motion, there has been little systematic investigation on other self-similar Gaussian processes. The main reason for this is the complexity of dependence structures for self-similar Gaussian processes which do not have stationary increments. Thus, it seems interesting to study some extensions of fractional Brownian motion such as sbfBm.

In this paper, we will study the existence of the self-intersection local time and its derivative for the sub-bifractional Brownian motion $S^{H, K} = \{S^{H, K}_t, t\geq 0\}$. They are defined by, respectively,

and

where $D = \{(r, s):0 < r < s < t\}$.

Set

and

Now we state our main results as follows.

Theorem 1.1. Existence of self-intersection local time.

Define:

for $H\in (0, 1)$ and $K\in (0, 1].$ Then $\alpha_{t, \epsilon}(y)$ converges in $L^2(\Omega, P)$ as $\epsilon\rightarrow 0,$ we denote the limit by $\alpha_{t}(y)$.

Theorem 1.2. Existence of the derivative of self-intersection local time.

Define:

If $H\in (0, 1), K\in (0, 1],$ and $0 < 2HK < 1,$ then $\alpha_{t, \epsilon}^{'}(y)$ converges in $L^2(\Omega, P)$ as $\epsilon\rightarrow 0,$ we denote the limit by $\alpha_{t}^{'}(y)$.

Theorem 1.3. Let $n\geq 1$ be an arbitrary but fixed integer, $t, \tilde{t}\in [0, T]$ and as $0 < 2HK < 1.$

(1) For any $\tau\in \left(0, \mathit{{min}}\left\{1, \frac{1}{HK}-2\right\} \right)$, there exists a positive constant $C$, such that

where $y_1, y_2\in {\boldsymbol{R}}.$

(2) For any $\gamma < 1-2HK,$ there exists a positive constant $C$, such that

where $y\in {\boldsymbol{R}}.$

In what follows, we will use $m$ to denote unspecified positive and finite constants whose value may be different in each occurrence.

2.   Some useful lemmas

In this section, we give some useful lemmas in order to prove the Theorems 1.1–1.3.

Lemma 2.1. For all constants $0 < a < b$, $S^{H, K}$ is strongly locally $\varphi$-nondeterministic on$I = [a, b]$ with $\varphi(r) = r^{2HK}$. That is, there exist positive constants $c_1$ and $r_0$ such that for all $t\in I$ and all $0 < r\leq \min\{t, r_0\}$,

Proof. See Kuang ^[11].

From the local nondeterminism (see Berman ^[1], Xiao ^[20]), we have the following property: if $0\leq t_1 < t_2 < \cdots < t_n < T$, then there is a constant $m > 0$ such that

for any $u_i\in {\textbf{R}}, i = 2, 3, \dots, n.$

Lemma 2.2. Let

and

Case 2.1. If $(r, s, r^{'}, s^{'})\in D_1: = \{(r, s, r^{'}, s^{'})|0 < r < r^{'} < s < s^{'} < t\},$ denoting $a = r^{'}-r, b = s-r^{'}, c = s^{'}-s,$ then we have

(1)

where $C_1$ and $C_2$ are given by (1.5).

(2) There exists a positive constant $m$, such that

where $\mu = \mu_1.$

(3) When $0 < 2HK < 1,$ there exists a positive constant $m$, such that

Case 2.2. If $(r, s, r^{'}, s^{'})\in D_2: = \{(r, s, r^{'}, s^{'})|0 < r < r^{'} < s^{'} < s < t\},$ denoting $a = r^{'}-r, b = s^{'}-r^{'}, c = s-s^{'},$ then we have

(1)

where $C_1$ and $C_2$ are given by (1.5).

(2) There exists a positive constant $m$, such that

where $\mu = \mu_2.$

(3) When $0 < 2HK < 1,$ there exists a positive constant $m$, such that

Case 2.3. If $(r, s, r^{'}, s^{'})\in D_3: = \{(r, s, r^{'}, s^{'})|0 < r < s < r^{'} < s^{'} < t\},$ denoting $a = s-r, b = r^{'}-s, c = s^{'}-r^{'},$ then we have

(1)

where $C_1$ and $C_2$ are given by (1.5).

(2) There exists a positive constant $m$, such that

where $\mu = \mu_3.$

(3) When $0 < 2HK < 1,$ for $\alpha, \beta > 0$ with $\alpha+\beta = 1$, there exists a positive constant $m$, such that

Proof. The proof of this lemma is given in the Appendix since its proof is long.

Lemma 2.3. Let $H\in(0, 1), K\in(0, 1],$ as $0 < 2HK < 1,$ we have

where $D^2: = \{(r, s, r^{'}, s^{'})|0 < r < s < t, 0 < r^{'} < s^{'} < t\}$, and $\lambda, \rho$ and $\mu$ are given in Lemma 2.2.

Proof. Since $D^2\cap \{r < r^{'}\} = D_1\cup D_2\cup D_3,$ where $D_1, D_2$ and $D_3$ are given in Lemma 2.2, it is suffice to show that

For $i = 1,$ by (2.4) and (2.5), we obtain

since $0 < 2HK < 1.$

For $i = 2,$ by (2.7) and (2.8), we obtain

since $0 < 2HK < 1.$

For $i = 3,$ by (2.10) and (2.11), we obtain

Since $0 < 2HK < 1,$ then $2(1-HK) = 2-2HK > 1.$ We first choose $\alpha > 0,$ such that $2\alpha (1-HK) < 1$, and we have

which imply

The proof of Lemma 2.3 is now complete.

3.   Proofs of theorems

In this section, we will prove Theorems 1.1–1.3.

3.1.   Proof of Theorem 1.1.

Proof. We prove the theorem in two steps.

Step 1. Show that for each $\epsilon > 0, \alpha_{t, \epsilon}(y)\in L^2(\Omega, P).$ In fact, by (1.8) and (1.10), we have

Let $D^2: = \{(r, s, r^{'}, s^{'})|0 < r < s < t, 0 < r^{'} < s^{'} < t\}$. Then,

where

Denote $M = {\rm{Var}}\left(\xi(S^{H, K}_s-S^{H, K}_r)+\eta(S^{H, K}_{s^{'}}-S^{H, K}_{r^{'}})\right)$, by (2.2), we obtain

(1) if $(r, s, r^{'}, s^{'})\in D_1,$ then

Thus,

since $0 < HK < 1.$

(2) if $(r, s, r^{'}, s^{'})\in D_2,$ then

Thus,

since $0 < HK < 1.$

(3) if $(r, s, r^{'}, s^{'})\in D_3,$ then

Thus,

since $0 < HK < 1.$ Hence, for any $H\in (0, 1)$ and $K\in (0, 1]$, we have

Step 2. Show that $\{\alpha_{t, \epsilon}(y), \epsilon > 0\}$ is a Cauchy sequence in $L^2(\Omega, P)$. Since the proof of Step 2 is similar to that of Theorem 3.1 in Jiang and Wang ^[8], we omitted the details.

Therefore we obtain $\lim_{\epsilon\rightarrow 0}\alpha_{t, \epsilon}(y)$ exists in $L^2$ and the theorem follows.

3.2.   Proof of Theorem 1.2.

Proof. By (1.9) and (1.11), we get

where $\lambda, \rho$ and $\mu$ are given in Lemma 2.2,

By Lemma 2.3, as $0 < 2HK < 1,$ we have

which implies $\alpha_{t, \epsilon}^{'}(y)\in L^2(\Omega, P)$ for each $\epsilon > 0.$

Similar to Step 2 of the proof of Theorem 1.1, we can prove that $\{\alpha_{t, \epsilon}^{'}(y), \epsilon > 0\}$ is a Cauchy sequence in $L^2(\Omega, P)$. Here we omitted the details. Therefore the proof of Theorem 1.2 is completed.

3.3.   Proof of Theorem 1.3.

Proof. (1) It is enough to prove that

holds for every $t\in [0, T]$ and $n\geq 1.$

Since

and

Hence

for any $\tau\in(0, 1)$.

Let us first deal with the product insider the expectation. Using the method of sample configuration as in Jung and Markowsky ^[10]. Fix such an ordering and let $l_1\leq l_2\leq \cdots\leq l_{2n}$ be a relabeling of the set $\{r_1, s_1, r_2, s_2, \cdots, r_n, s_n\}$. We obtain

where the $u_j$$^{, }$s are properly chosen linearly combinations of the $\xi_j$$^{, }$s to make (3.4) an equality. By local non-deterministic property for $S_t^{H, K}$, we have

Fixing $\xi_j,$ we choose $j_1$ to be the smallest value such that $u_{j_1}$ contains $\xi_j$ as a term and then choose $j_2$ to be the smallest value strictly larger than $j_1$ such that $u_{j_2}$ does not contain $\xi_j$ as a term. Thus,

Let $a_j = l_{j+1}-l_{j}$ (with $l_0 = 0),$by (3.3), (3.5) and (3.6), we deduce

where $d\xi = d\xi_1d\xi_2\cdots d\xi_n, \ \ dl = dl_1dl_2\cdots dl_{2n}, \ \ u_0 = u_{2n} = 0,$ and we use the inequality

Expanding the product in the numerator of (3.7) gives us the sum of a number of terms of the form

where $m_j = 0, 1$ or $2$ (terms containing $|u_0|$ or $|u_{2n}|$ are equal to 0). Thus, it is suffice to show that

is finite. We make a linear transformation changing this into an integral with respect to variables $u_{k_1}, u_{k_2}, \cdots, u_{k_n}\in \{u_1, \cdots, u_{2n-1}\}$ which span $\{\xi_1, \xi_2, \cdots, \xi_n\}$ in order to bound (3.9) by

This is finite if we choose $\tau$ such that $\frac{1}{HK}-(1+\tau) > 1$, namely $\tau < \frac{1}{HK}-2.$

(2) It is enough to prove that

holds for $t, \tilde{t}\in [0, T]$, every $y\in {\textbf{R}}$ and $n\geq 1.$

Since

Hence,

where $\tilde{D} = \{(r, s):0 < r < s < \tilde{t}\}$ and we use the H$\ddot{o}$lder's inequality in the last inequality with $\gamma < 1-2HK.$

Using the similar method as in the proof of (1) in Theorem 1.3, $\Lambda$ is bounded by

where $E^n = \{ 0 < l_1 < \cdots < l_{2n} < t\}.$

Since $1-\gamma > 2HK,$ there exists a constant $m > 0,$ such that

Thus, we finished the proof of Theorem 1.3.

4.   Conclusions

We mainly study the existence of the self-intersection local time and its derivative for the sub-bifractional Brownian motion. Moreover, we prove its derivative is H$\ddot{o}$lder continuous in space variable and time variable, respectively. In the future, we will consider its higher order derivative.

Acknowledgments

Research supported by the Natural Science Foundation of Hunan Province under Grant 2021JJ30233.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix

In this section we prove Lemma 2.2.

Proof. By (1.4), we obtain (2.3), (2.6) and (2.9) easily. Since the proofs of (2.4), (2.7) and (2.10) are similar to those of Hu ^[4]and Hu and Nualart ^[5], we omitted the details. Thus we only need to prove (2.5), (2.8) and (2.11).

In order to prove (2.5), let $e = r$, by (1.3), we have

Hence,

For $\Delta_{1, 1},$ we obtain

since $0 < K\leq 1$.

For $\Delta_{1, 2},$ we get

since $0 < 2HK < 1,$ where we use the inequality $x^{\alpha}-y^{\alpha}\leq |x-y|^{\alpha}$ for any $x > 0, y > 0, 0 < \alpha < 1.$

For $\Delta_{1, 3},$ we deduce,

Therefore (2.5) holds from (A.2)–(A.4).

In order to prove (2.8), let $e = r$, by (1.3) and $(A.1)$, we have

For $\Delta_{2, 1},$ we obtain

since $0 < K\leq 1$.

For $\Delta_{2, 2},$ we get

since $0 < 2HK < 1.$

For $\Delta_{2, 3},$ we deduce,

since $0 < 2HK < 1.$ Therefore (2.8) holds from (A.5)–(A.7).

In order to prove (2.11), let $e = r$, by (1.3) and $(A.1)$, we have

For $\Delta_{3, 1},$ we obtain

since $0 < K\leq 1$.

For $\Delta_{3, 2},$ and for $\alpha, \beta > 0$ with $\alpha+\beta = 1,$ we get

since $0 < 2HK < 1.$

For $\Delta_{3, 3},$ we deduce,

since $0 < 2HK < 1.$ Therefore (2.11) holds from (A.8)–(A.10). Thus we have finished the proof of Lemma 2.2.