[Paper Review] Higher-order derivative of intersection local time for two independent fractional Brownian motions

Jingjun Guo, Yaozhong Hu+1 more|arXiv (Cornell University)|Jun 21, 2017

Stochastic processes and financial applications9 references3 citations

TL;DR

This paper establishes sharp sufficient and necessary conditions for the existence and exponential integrability of the $k$-th order derivative of intersection local time for two independent $d$-dimensional fractional Brownian motions with Hurst parameters $H_1$ and $H_2$. Using a mollified Dirac delta approximation and moment estimates via Fourier analysis and Gamma function bounds, the authors prove that the $k$-th derivative exists in $L^p$ for all $p \in [1,\infty)$ when $\frac{H_1H_2}{H_1+H_2}(|k|+d) < 1$, and that exponential integrability holds with exponent $\beta = \frac{H_1+H_2}{2dH_1H_2}$. This condition is shown to be necessary in certain cases for the existence of the derivative when $k_i$ is even.

ABSTRACT

In this article, we obtain sharp conditions for the existence of the high order derivatives ($k$-th order) of intersection local time $ \widehatα^{(k)}(0)$ of two independent d-dimensional fractional Brownian motions $B^{H_1}_t$ and $\widetilde{B}^{H_2}_s$ with Hurst parameters $H_1$ and $H_2$, respectively. We also study their exponential integrability.

Motivation & Objective

• To determine the precise conditions under which the $k$-th order derivative of the intersection local time of two independent $d$-dimensional fractional Brownian motions exists in $L^p$ for all $p \in [1,\infty)$.
• To establish the exponential integrability of the $k$-th derivative of intersection local time and identify the sharp exponent $\beta$ for which $\mathbb{E}[\exp(C|\widehat{\alpha}^{(k)}(0)|^\beta)] < \infty$.
• To prove that the condition $\frac{H_1H_2}{H_1+H_2}(|k|+d) < 1$ is not only sufficient but also necessary in certain cases, particularly when $k_i$ is an even integer.
• To extend existing results on self-intersection local time and its derivatives to the case of two independent fractional Brownian motions with different Hurst parameters.

Proposed method

• The $k$-th derivative of intersection local time is defined via a mollified approximation of the Dirac delta function $f_\varepsilon^{(k)}(x)$, which is expressed as a Fourier integral involving $p^k e^{ipx} e^{-\varepsilon|p|^2/2}$.
• The existence of $\widehat{\alpha}^{(k)}(0)$ in $L^2$ is established by proving the $L^2$-convergence of the approximating sequence $\widehat{\alpha}^{(k)}_\varepsilon(0)$ as $\varepsilon \downarrow 0$.
• Moment estimates for $\mathbb{E}[|\widehat{\alpha}^{(k)}_\varepsilon(0)|^n]$ are derived using the characteristic function of the difference $B^{H_1}_t - \widetilde{B}^{H_2}_s$, leading to bounds involving Gamma functions and power-law decay.
• The proof leverages the identity $\mathbb{E}[e^{i\langle \xi, B^{H_1}_t - \widetilde{B}^{H_2}_s \rangle}] = \exp\left(-\frac{1}{2}(t^{2H_1} + s^{2H_2})|\xi|^2\right)$ to compute the characteristic function of the difference process.
• A change of variables and polar coordinates are used in the integral $\int_0^T \int_0^T (t^{2H_1} + s^{2H_2})^{-(k+d)/2} dtds$ to analyze convergence, revealing that the integral diverges unless $\frac{H_1H_2}{H_1+H_2}(|k|+d) < 1$.
• The exponential integrability is established by bounding the moment generating function using Stirling's approximation and showing convergence of the series $\sum_n C^n (n!)^{\beta(2-2\kappa_1)-1}$ with $\beta = \frac{H_1+H_2}{2dH_1H_2}$.

Experimental results

Research questions

• RQ1 Under what conditions does the $k$-th order derivative of the intersection local time of two independent $d$-dimensional fractional Brownian motions exist in $L^p$ for all $p \in [1,\infty)$?
• RQ2 What is the sharp exponent $\beta$ for which the exponential moment $\mathbb{E}[\exp(C|\widehat{\alpha}^{(k)}(0)|^\beta)]$ is finite?
• RQ3 Is the condition $\frac{H_1H_2}{H_1+H_2}(|k|+d) < 1$ necessary for the existence of $\widehat{\alpha}^{(k)}(0)$ when $k_i$ is an even integer?
• RQ4 How does the existence and integrability of the derivative depend on the Hurst parameters $H_1$ and $H_2$ when they differ?

Key findings

• The $k$-th order derivative of intersection local time $\widehat{\alpha}^{(k)}(0)$ exists in $L^p(\Omega)$ for all $p \in [1,\infty)$ if and only if $\frac{H_1H_2}{H_1+H_2}(|k|+d) < 1$, which is both sufficient and necessary under certain conditions.
• Exponential integrability holds with exponent $\beta = \frac{H_1+H_2}{2dH_1H_2}$, meaning $\mathbb{E}[\exp(C_{d,k,T}|\widehat{\alpha}^{(k)}(0)|^\beta)] < \infty$ for some $C_{d,k,T} > 0$, establishing strong tail decay.
• When $H_1 = H_2 = \frac{1}{2}$, the exponential integrability exponent reduces to $\beta = \frac{2}{d}$, recovering a known result from earlier literature.
• For the case $k = 0$, the condition reduces to $\frac{H_1H_2}{H_1+H_2}d < 1$, which generalizes the known condition $Hd < 2$ when $H_1 = H_2 = H$, as in Nualart et al. [8].
• The necessity of the condition $\frac{H_1H_2}{H_1+H_2}(|k|+d) < 1$ is proven for the case where $k = (k_i, 0, \dots, 0)$ with $k_i$ even, by analyzing the divergence of the integral $\int_0^T \int_0^T (t^{2H_1} + s^{2H_2})^{-(k+d)/2} dtds$ when the condition fails.
• The proof uses polar coordinates and a change of variables $t = u^{H_2/H_1}$ to reduce the integral to a radial form, showing that convergence depends critically on the exponent of $r$ being greater than $-1$, which yields the condition $\frac{H_2}{H_1} - (k+d)H_2 > -1$, equivalent to the main condition.