Kolmogorov distance between the exponential functionals of fractional Brownian motion

TL;DR

This paper studies how the distribution of exponential functionals of fractional Brownian motion changes with the Hurst index, providing explicit bounds on their Kolmogorov distance using Malliavin calculus.

Contribution

It introduces a method to quantify the law continuity of exponential functionals of fractional Brownian motion with respect to the Hurst index, using explicit bounds.

Findings

Derived explicit bounds on Kolmogorov distance between functionals with different Hurst indexes

Established continuity in law of exponential functionals with respect to Hurst parameter

Applied Malliavin calculus techniques for probabilistic bounds

Abstract

In this note, we investigate the continuity in law with respect to the Hurst index of the exponential functional of the fractional Brownian motion. Based on the techniques of Malliavin's calculus, we provide an explicit bound on the Kolmogorov distance between two functionals with different Hurst indexes.

Equations

$$B_t^H={\int }_0^t{K}_H(t,s)d{W}_s,$$

$$K_{H}(t,s)=c_{H}\bigg{[}\frac{t^{H-\frac{1}{2}}}{s^{H-\frac{1}{2}}}(t-s)^{H-\frac{1}{2}}-(H-\frac{1}{2})\int\limits_{s}^{t}\frac{u^{H-\frac{3}{2}}}{s^{H-\frac{1}{2}}}(u-s)^{H-\frac{1}{2}}du\bigg{]}\,\,\text{if}\,\,H<\frac{1}{2}.$$

$$F_H={\int }_0^T{e}^{as+\sigma B_s^H}ds.$$

$$\underset{x\ge 0}{\sup }|P\left(F_{H_1}\le x\right)-P\left(F_{H_2}\le x\right)|\le C|H_1-H_2|,$$

$$W(h)=\underset{0}{\overset{T}{\int }}h(t)d{W}_t.$$

$$F=f(W(h_1),...,W(h_n)),$$

$$D_{t}F=\sum _{k=1}^n\frac{\partial f}{\partial x_k}(W(h_1),...,W(h_n))h_k(t).$$

$$\|F\|^{2}_{1,2}:=E|F|^{2}+E\bigg{[}\int\limits_{0}^{T}|D_{u}F|^{2}du\bigg{]}.$$

$$E|⟨DF,u{⟩}_{L^2[0,T]}|\le C(u)\Vert F{\Vert }_{L^2(\Omega )},$$

$$E⟨DF,u{⟩}_{L^2[0,T]}=E[F\delta (u)]\text{for any}F\in {\mathbb{D}}^{1,2}.$$

$${\left({\int }_0^T|D_r{F}_H{|}^{2}dr\right)}^{-1}\in L^p(\Omega ),\forall p\ge 1.$$

$$D_r{F}_H={\int }_r^T\sigma K_H(s,r)e^{as+\sigma B_s^H}ds,0\le r\le T.$$

$$D_r{F}_H\ge e^{-|a|T+\sigma \underset{0\le s\le T}{\min }{B}_s^H}{\int }_r^T\sigma K_H(s,r)ds,0\le r\le T.$$

$${\int }_0^T|D_r{F}_H{|}^{2}dr$$

$$={\sigma }^2{e}^{-2|a|T+2\sigma \underset{0\le s\le T}{\min }{B}_s^H}{\int }_0^T\left({\int }_r^T{K}_H(s,r)ds\right)\left({\int }_r^T{K}_H(t,r)dt\right)dr$$

$$={\sigma }^2{e}^{-2|a|T+2\sigma \underset{0\le s\le T}{\min }{B}_s^H}{\int }_0^T{\int }_0^T\left({\int }_0^{s\wedge t}{K}_H(s,r)K_H(t,r)dr\right)dsdt$$

$$={\sigma }^2{e}^{-2|a|T+2\sigma \underset{0\le s\le T}{\min }{B}_s^H}{\int }_0^T{\int }_0^{T}E[B_s^H{B}_t^H]dsdt$$

$$=\frac{T^{2H+2}}{2H+2}{\sigma }^2{e}^{-2|a|T+2\sigma \underset{0\le s\le T}{\min }{B}_s^H}.$$

$${\left({\int }_0^T|D_r{F}_H{|}^{2}dr\right)}^{-1}\le \frac{2H+2}{T^{2H+2}{\sigma }^2}{e}^{2|a|T+2\sigma \underset{0\le s\le T}{\max }(-B_s^H)}.$$

$$E|F_{H_1}-F_{H_2}{|}^2\le C|H_1-H_2{|}^2,$$

$${\int }_0^{T}E|D_r{F}_{H_1}-D_r{F}_{H_2}{|}^{2}dr\le C|H_1-H_2{|}^2,$$

$$E|F_{H_1}-F_{H_2}{|}^2$$

$$\displaystyle\leq T\int_{0}^{T}E\big{|}e^{as+\sigma B^{H_{1}}_{s}}-e^{as+\sigma B^{H_{2}}_{s}}\big{|}^{2}ds.$$

$$E|F_{H_1}-F_{H_2}{|}^2$$

$$\le \frac{T{\sigma }^2}{4}{\int }_0^T{\left(E|B_s^{H_1}-B_s^{H_2}{|}^4\right)}^{\frac{1}{2}}{\left(E|e^{as+\sigma B_s^{H_1}}+e^{as+\sigma B_s^{H_2}}{|}^4\right)}^{\frac{1}{2}}ds$$

$$\le \frac{T{\sigma }^2}{4}{\int }_0^T{\left(E|B_s^{H_1}-B_s^{H_2}{|}^4\right)}^{\frac{1}{2}}{\left(8E[e^{4as+4\sigma B_s^{H_1}}]+8E[e^{4as+4\sigma B_s^{H_2}}]\right)}^{\frac{1}{2}}ds$$

$$=\frac{T{\sigma }^2}{4}{\int }_0^T{\left(E|B_s^{H_1}-B_s^{H_2}{|}^4\right)}^{\frac{1}{2}}{\left(8e^{4as+8{\sigma }^2{s}^{2H_1}}+8e^{4as+8{\sigma }^2{s}^{2H_2}}\right)}^{\frac{1}{2}}ds$$

$$\le \frac{T{\sigma }^2}{4}{\int }_0^T{\left(E|B_s^{H_1}-B_s^{H_2}{|}^4\right)}^{\frac{1}{2}}{\left(8e^{4as+8{\sigma }^2{s}^{2H_1}}+8e^{4as+8{\sigma }^2{s}^{2H_2}}\right)}^{\frac{1}{2}}ds.$$

$$\underset{0\le s\le T}{\sup }E|B_s^{H_1}-B_s^{H_2}{|}^2\le C|H_1-H_2{|}^2.$$

$$E|F_{H_1}-F_{H_2}{|}^2\le C|H_1-H_2{|}^2.$$

$$\displaystyle E|D_{r}F_{H_{1}}-D_{r}F_{H_{2}}|^{2}\leq\sigma^{2}T\int_{r}^{T}E\big{|}K_{H_{1}}(s,r)e^{as+\sigma B^{H_{1}}_{s}}-K_{H_{2}}(s,r)e^{as+\sigma B^{H_{2}}_{s}}\big{|}^{2}ds$$

$$\displaystyle\leq 2\sigma^{2}T\int_{r}^{T}|K_{H_{1}}(s,r)-K_{H_{2}}(s,r)|^{2}E[e^{2as+2\sigma B^{H_{1}}_{s}}]+K^{2}_{H_{2}}(s,r)E\big{|}e^{2as+2\sigma B^{H_{1}}_{s}}-e^{as+\sigma B^{H_{2}}_{s}}\big{|}^{2}ds,$$

$${\int }_0^{T}E|D_r{F}_{H_1}-D_r{F}_{H_2}{|}^{2}dr$$

$$\displaystyle+2\sigma^{2}T\int_{0}^{T}E\big{|}e^{2as+2\sigma B^{H_{1}}_{s}}-e^{as+\sigma B^{H_{2}}_{s}}\big{|}^{2}\int_{0}^{s}K^{2}_{H_{2}}(s,r)drds$$

$$=2{\sigma }^{2}T{\int }_0^T{e}^{2as+2{\sigma }^2{s}^{2H_1}}E|B_s^{H_1}-B_s^{H_2}{|}^{2}ds$$

$$\displaystyle+2\sigma^{2}T\int_{0}^{T}E\big{|}e^{2as+2\sigma B^{H_{1}}_{s}}-e^{as+\sigma B^{H_{2}}_{s}}\big{|}^{2}s^{2H_{2}}ds.$$

$$⟨D\phi (Z),D{F}_{H_2}⟩=\psi (Z)⟨DZ,D{F}_{H_2}⟩.$$

Full text

Keywords: Fractional Brownian motion, Exponential functional, Malliavin calculus.

2010 Mathematics Subject Classification: 60G22, 60H07.

1 Introduction

Let $B^{H}=(B^{H}_{t})_{t\in[0,T]}$ be a fractional Brownian motion (fBm) with Hurst index $H\in(0,1).$ We recall that fBm admits the Volterra represention

[TABLE]

where $(W_{t})_{t\in[0,T]}$ is a standard Brownian motion and for some normalizing constants $c_{H}$ and $c^{\prime}_{H},$ the kernel $K_{H}$ is given by $K_{H}(t,s)=c_{H}s^{1/2-H}\int_{s}^{t}(u-s)^{H-\frac{3}{2}}u^{H-\frac{1}{2}}du$ if $H>\frac{1}{2}$ and

[TABLE]

Given real numbers $a$ and $\sigma,$ we consider the exponential functional of the form

[TABLE]

It is known that this functional plays an important role in several domains. For example, it can be used to investigate the finite-time blowup of positive solutions to semi-linear stochastic partial differential equations [1]. In the special case $H=\frac{1}{2},$ fBm reduces to a standard Brownian motion and a lot of fruitful properties of $F_{\frac{1}{2}}$ can be founded in the literature, see e.g. [4, 5, 8, 11]. In particular, the distribution of $F_{\frac{1}{2}}$ can be computed explicitly. However, to the best our knowledge, it remains a challenge to obtain the deep properties of $F_{H}$ for $H\neq\frac{1}{2}.$

On the other hand, because of its applications in statistical estimators, the problem of proving the continuity in law with respect to $H$ of certain functionals has been studied by several authors. Among others, we refer the reader to [2, 3, 9, 10] and the references therein for the detailed discussions and the related results. Motivated by this observation, the aim of the present paper is to investigate the continuity in law of the exponential functional $F_{H}.$ Intuitively, the continuity of $F_{H}$ with respect to $H$ is not surprising. However, the interesting point of Theorem 1.1 below is that we are able to give an explicit bound on Komogorov distance between two functionals with different Hurst indexes.

Theorem 1.1**.**

For any $H_{1},H_{2}\in(0,1),$ we have

[TABLE]

where $C$ is a positive constant depending on $a,\sigma,T$ and $H_{1},H_{2}.$

2 Proofs

Our main tools are the techniques of Malliavin calculus. Hence, for the reader’s convenience, let us recall some elements of Malliavin calculus with respect to Brownian motion $W,$ where $W$ is used to present $B^{H}$ as in (1.1). We suppose that $(W_{t})_{t\in[0,T]}$ is defined on a complete probability space $(\Omega,\mathcal{F},\mathbb{F},P)$, where $\mathbb{F}=(\mathcal{F}_{t})_{t\in[0,T]}$ is a natural filtration generated by the Brownian motion $W.$ For $h\in L^{2}[0,T],$ we denote by $W(h)$ the Wiener integral

[TABLE]

Let $\mathcal{S}$ denote the dense subset of $L^{2}(\Omega,\mathcal{F},P)$ consisting of smooth random variables of the form

[TABLE]

where $n\in\mathbb{N},f\in C_{b}^{\infty}(\mathbb{R}^{n}),h_{1},...,h_{n}\in L^{2}[0,T].$ If $F$ has the form (2.1), we define its Malliavin derivative as the process $DF:=\{D_{t}F,t\in[0,T]\}$ given by

[TABLE]

We shall denote by $\mathbb{D}^{1,2}$ the closure of $\mathcal{S}$ with respect to the norm

[TABLE]

An important operator in the Malliavin calculus theory is the divergence operator $\delta,$ it is the adjoint of the derivative operator $D.$ The domain of $\delta$ is the set of all functions $u\in L^{2}(\Omega,L^{2}[0,T])$ such that

[TABLE]

where $C(u)$ is some positive constant depending on $u.$ In particular, if $u\in Dom\,\delta,$ then $\delta(u)$ is characterized by the following duality relationship

[TABLE]

In order to be able to prove Theorem 1.1, we need two technical lemmas.

Lemma 2.1**.**

For any $H\in(0,1),$ we have $F_{H}\in\mathbb{D}^{1,2}$ and

[TABLE]

Proof.

By the representation (1.1), we have $D_{r}B^{H}_{s}=K_{H}(s,r)$ for $0\leq r<s\leq T.$ Hence, $F_{H}\in\mathbb{D}^{1,2}$ and its derivative is given by

[TABLE]

So we can deduce

[TABLE]

As a consequence,

[TABLE]

In the last equality we used the fact that $E[B^{H}_{t}B^{H}_{s}]=\frac{1}{2}(t^{2H}+s^{2H}-|t-s|^{2H}).$ We therefore obtain

[TABLE]

By Fernique’s theorem, we have $e^{2\sigma\max\limits_{0\leq s\leq T}(-B^{H}_{s})}\in L^{p}(\Omega)$ for any $p\geq 1.$ This completes the proof. ∎

Lemma 2.2**.**

For any $H_{1},H_{2}\in(0,1),$ we have

[TABLE]

[TABLE]

where $C$ is a positive constant depending on $a,\sigma,T$ and $H_{1},H_{2}.$

Proof.

By the Hölder inequality we have

[TABLE]

Using the fundamental inequality $|e^{x}-e^{y}|\leq\frac{1}{2}|x-y|(e^{x}+e^{y})$ for all $x,y$ we deduce

[TABLE]

It is known from the proof of Theorem 4 in [7] that there exists a positive constant $C$ such that

[TABLE]

On the other hand, we have $E|B^{H_{1}}_{s}-B^{H_{2}}_{s}|^{4}=3(E|B^{H_{1}}_{s}-B^{H_{2}}_{s}|^{2})^{2}$ because $B^{H_{1}}_{s}-B^{H_{2}}_{s}$ is a Gaussian random variable for every $s\in[0,T].$ So we can conclude that there exists a positive constant $C$ such that

[TABLE]

To finish the proof, let us verify (2.3). By the Hölder and triangle inequalities we obtain

[TABLE]

and hence,

[TABLE]

Notice that $\int_{0}^{s}K^{2}_{H_{2}}(s,r)dr=E|B^{H_{2}}_{s}|^{2}=s^{2H_{2}}.$ Thus the estimate (2.3) follows from (2.2) and (2.4). ∎

Proof of Theorem 1.1. For the simplicity, we write $\langle.,.\rangle$ instead of $\langle.,.\rangle_{L^{2}[0,T]}.$ Borrowing the arguments used in the proof of Proposition 2.1.1 in [6], we let $\psi$ be a nonnegative smooth function with compact support, and set $\varphi(y)=\int_{-\infty}^{y}\psi(z)dz.$ Given $Z\in\mathbb{D}^{1,2},$ we know that $\varphi(Z)$ belongs to $\mathbb{D}^{1,2}$ and making the scalar product of its derivative with $DF_{H_{2}}$ obtains

[TABLE]

Fixed $x\in\mathbb{R}_{+},$ by an approximation argument, the above equation holds for $\psi(z)={\rm 11}_{[0,x]}(z).$ Choosing $Z=F_{H_{1}}$ and $Z=F_{H_{2}}$ we obtain

[TABLE]

[TABLE]

Hence, we can get

[TABLE]

This, together with the fact that $\|DF_{H_{2}}\|^{2}:=\langle DF_{H_{2}},DF_{H_{2}}\rangle>0\,\,a.s.$ gives us

[TABLE]

Taking the expectation yields

[TABLE]

By the Hölder inequality

[TABLE]

Recalling Lemma 2.2, we obtain

[TABLE]

Thanks to Lemma 2.1 we have

[TABLE]

Thus we can obtain (1.2) by checking the finiteness of $E[\delta(u)^{2}],$ where

[TABLE]

It is known from Proposition 1.3.1 in [6] that

[TABLE]

We have

[TABLE]

Furthermore, by the chain rule for Malliavin derivative, we have

[TABLE]

Hence, by the Hölder inequality,

[TABLE]

We now observe that

[TABLE]

Hence,

[TABLE]

and we obtain

[TABLE]

which implies that

[TABLE]

Finally, we have $E\left[\frac{1}{\|DF_{H_{2}}\|^{8}}\right]<\infty$ due to Lemma 2.1. So we can conclude that $E[\delta(u)^{2}]$ is finite. This finishes the proof of Theorem 1.1.

Remark 2.1*.*

Given a bounded and continuous function $\psi,$ with the exact proof of Theorem 1.1, we also have

[TABLE]

This kind of estimates has been investigated by Richard and Talay for the solution of fractional stochastic differential equations. However, Theorem 1.1 in [9] requires $H_{2}=\frac{1}{2}$ and $\psi$ to be Hölder continuous of order $2+\beta$ with $\beta>0.$

Acknowledgments. This research was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03-2019.08.