Fractional Hida Malliavin Derivatives and Series Representations of Fractional Conditional Expectations

# Fractional Hida Malliavin Derivatives and Series Representations of Fractional Conditional Expectations

## Abstract

We represent fractional conditional expectations of a functional of fractional Brownian motion as a convergent series in space. When the target random variable is some function of a discrete trajectory of fractional Brownian motion, we obtain a backward Taylor series representation; when the target functional is generated by a continuous fractional filtration, the series representation is obtained by applying a ”frozen path” operator and an exponential operator to the functional. Three examples are provided to show that our representation gives useful series expansions of ordinary expectations of target random variables.

Sixian Jin1, Qidi Peng2, Henry Schellhorn3

Keywords: Fractional Brownian motion ; Malliavin calculus ; fractional Hida Malliavin derivative ; fractional Clark-Hausmann-Ocone formula
MSC: 60G15 ; 60G22 ; 60H07

## 1 Introduction

Fractional Brownian motion (fBm), nowadays considered as one of the most natural extensions of the classical Brownian motion (Bm), was first introduced by Kolmogorov in 1940 [13]. It is popularized by Mandelbrot and Van Ness [16] in 1968, by being introduced to mathematical finance, to take into account the effect of exogenous arrivals. Today, the mathematical analysis and applications of fBms have advanced enormously in a variety of fields and at all levels, such as stock returns modeling, fractal geometry, signal processing, geology, geography and statistical biology, etc. FBm is almost applicable in every domain where one can apply Bm. However, unlike Bm, some new mathematical issues arise when replacing Bm by such an extension, which is mainly due to the complex covariance structure of its increments. For example, fBm has correlated increments for . This results that evaluating conditional expectations of functions or functionals of fBm is a notoriously difficult problem. Grippenberg and Norros [8] provided a technical and difficult approach to calculate the conditional mean of fBm. Fink et al. [5] also addressed this problem when studying the price of a zero-coupon bond in a fractional bond market. Since fBm is generally not a Markov process, both authors restricted themselves to calculate conditional expectations given the current value of , and not given the whole path of preceding .

This paper presents a different and original way to evaluate expectations of functionals of fBm. It is based on the Malliavin calculus with respect to fBm as presented in Biagini et al. [4], Chapter 2 and Chapter 3. Since the 70’s, Malliavin calculus plays an important role in analysis on the Wiener space as well as in the study on stochastic differential equations. The main advantage of Malliavin calculus is that, it allows to give sufficient conditions for the distribution of a random variable to have a smooth (differentiable) density with respect to Lebesgue measure and to give bounds for this density and its derivatives. With this technique, one of the main fruits from Malliavin calculus is the fractional conditional expectation and its representation by Clark-Hausmann-Ocone formula (see e.g. [4]). By using the so-called fractional Clark-Hausmann-Ocone formula, our first main result represents the fractional conditional expectations of functions of fBm’s discrete trajectory as a convergent series in . Our second main result is more general under a different sufficient condition for the convergence. It is obtained from the fact that, the fBm has the ”martingale” property under fractional conditional expectation. This property leads to an exponential expansion of the fractional conditional expectations. The latter result partially extends the work of Schellhorn and Jin [12], who proved this representation for conditional expectations of a functional of Bm. It is worth noting that, the fBm can be divided into three very different classes according to the values of , and . When , the corresponding fBm is persistent, which means that its increments are positively correlated (the increase of the increments is likely to be followed by another increase). A huge number of phenomena can be modeled in terms of this class of processes, such as the level of the optimum dam sizing, the logarithm of the stock return and financial turbulence [16]. As in [4], Chapter 3, we only consider the case where in this work. We remark that a similar study can be done for classes of fBm with for the future.

The structure of this paper is as follows. In Section 2, we introduce some definitions, notations and known results on Malliavin calculus related to fBm, which are needed for the next sections. In Section 3, we present our two main results, which are i) a generalization to fBm of the backward Taylor expansion obtained in [12], and ii) the exponential formula itself. We note that both series representations can be used both for numerical applications and for solving some algebraic problems. In section 4, we show 3 applications of the exponential formula, respectively to the fractional Merton model of interest rates, to a special case of the fractional Cox-Ingersoll-Ross model of interest rates, and to the characteristic function of geometric fBm.

## 2 Preliminaries

### 2.1 Fractional Brownian Motion

A real-valued standard fBm can be defined independently and equivalently using a moving average representation [16] and a harmonizable representation [20]. In fact, up to a multiplicative scaling factor, fBm is the unique Gaussian, self-similar, with stationary increments process. Therefore, a standard fBm can be defined from the uniqueness of its covariance structure:

###### Definition 2.1

A standard fBm with Hurst index is the unique centered Gaussian process with almost surely continuous non-differentiable sample path and with covariance function: for any ,

 E[BHsBHt]=12(|s|2H+|t|2H−|t−s|2H).

Without any loss of generality, we restrict the fBm to nonnegative-time process and denote the corresponding probability space by , where is the natural filtration generated by fractional Wiener chaos (see e.g. [4], Page 49) and is the corresponding probability measure. Again, as an assumption, we let in the remaining of the paper.

### 2.2 Fractional Hida Malliavin Derivative

Let denote a Hilbert space of random variables equipped with the norm

 ∥F∥L2(PH):=√E|F|2<+∞.

We indicate by the Schwartz space of rapidly decreasing smooth functions on . More precisely,

 S(R+):={f∈C∞(R+): % for any α,β∈N, supx∈R+∣∣xαdβf(x)dxβ∣∣<+∞},

where denotes a space of continuously infinitely differentiable real-valued functions. We equip with the following inner product: for any ,

 Extra open brace or missing close brace

where for any .

Denote by the completion of under the norm . This is a separable Hilbert space of deterministic functions. Note that fBm is an isonormal process and, if two functions , the stochastic integrals with respect to fBm and are well defined, zero mean, Gaussian random variables with covariance

 Cov(∫R+f(s)dBHs,∫R+g(s)dBHs)=⟨f,g⟩H.

Let denote the dual space of . is also called the space of tempered distributions on (see e.g. [19] or [4]). Note that by the Bochner-Milos theorem (see e.g. [4, 9, 14]), the probability measure is the one which allows

 BHt:=:=∫t01dBHu

to be an element in .

For the purpose of presenting an element using convergent Taylor series, we first introduce the notion of fractional Hida Malliavin derivative (see Definition 3.3.1 in [4]).

###### Definition 2.2 (Fractional Hida Malliavin Derivative)

Given an operator and some . is said to have a directional derivative in the direction if, for all , there exists an element in the fractional Hida distribution space (see Definition 3.1.10 in [4]), such that

 limϵ→0G(ν+ϵγ)−G(ν)ϵ=Xν,γ, in (S)∗H.

We say is fractional Hida Malliavin differentiable if there exists a map such that for all , is -integrable and

 Xν,γ=∫R+Ψ(t,ν)γ(t)dt

for all . Then we set, for all ,

 DHtG(ν):=Ψ(t,ν)

and we call the fractional Hida Malliavin derivative with order of on at .

Remark that the fractional Hida Malliavin derivative with respect to fBm extends the classical one (see [18]) with respect to Bm. Also remark that, since (see Definition 3.1.10 in [4]), we will mainly focus on the fractional Hida Malliavin derivative defined on in the sequel.

It is useful to note that the fractional Hida Malliavin derivative possesses some nice properties similar to the classical derivatives. For example, the chain rule is still valid: if , with being some deterministic differentiable function, then for ,

 DHtG(ν)=n∑i=1∂H∂xi(BHt1,...,BHtn)χ[0,ti](t). (2.1)

Now we introduce some other important properties of fractional Hida Malliavin derivative that we will need to construct the convergent series of fractional conditional expectations. The most interesting one is fractional Clark-Hausmann-Ocone formula, which generalizes the classical Clark-Hausmann-Ocone formula. The following statements as well as notations are helpful to the introduction of this formula.

###### Theorem 2.3 (Itô decomposition, see [4], Page 82)

Let , then has the following representation via fractional Wick Itô Skorohod integral (FWISI): there exists a sequence of deterministic functions such that

 Extra open brace or missing close brace

where

the sequence of multiple FWISI is defined in [4], Page 81;

denotes the subspace of symmetric functions in ;

is the norm defined by

 ∥fn∥2H,Rn+:=n!∫R2n+fn(u1,…,un)fn(v1,…,vn)n∏i=1φH(ui,vi)(du)⊗n(dv)⊗n,

with

 (du)⊗n:=dundun−1…du1. (2.2)

###### Definition 2.4

We define , the Hermite polynomial of degree , by and for ,

 hn(x)=(−1)nexp(x22)dndxnexp(−x22), for all x∈R. (2.3)

Remark that the sequence is equivalently defined as solutions of the following equation, valid for all :

 exp(tx−t22)=+∞∑n=0tnn!hn(x). (2.4)
###### Proposition 2.5

Let , then the multiple FWISI exists and is given as

 In(f)=n!∫0≤s1≤…≤sn<+∞f(s1,…,sn)d(BHs)⊗n,

where we denote by

 d(BHs)⊗n:=dBHsndBHsn−1…dBHs1. (2.5)

In particular, if there exists such that for all , then

 In(f)=∥g∥nH,R+hn⎛⎝∫+∞0g(s)dBHs∥g∥H,R+⎞⎠,

where the norm equips the space .

As a special case, when taking with in Proposition 2.5, we obtain

 Missing or unrecognized delimiter for \right (2.6)

From now on, the FWISI of a continuous-time stochastic process over any time interval , is denoted by .

###### Definition 2.6 (Fractional Conditional Expectation)

Let be represented as (such an expansion exists, due to Definition 3.10.1 in [4])

 F=+∞∑n=0∫Rn+gn(s1,…,sn)d(BHs)⊗n,

with some sequence of functions . Then for , we define the fractional conditional expectation of with respect to by

 ~E[F|FHt]:=+∞∑n=0∫[0,t]ngn(s1,…,sn)d(BHs)⊗n. (2.7)

Remarks: Though different from conditional expectation, fractional conditional expectation has some properties, which are similar to those of classical conditional expectation:

1. For all , is -measurable. Vice versa, if is -measurable, .

2. For any ,

 ~E[~E[F|FHs]|FHt]=~E[F|FHmin{s,t}]. (2.8)
3. From (2.7), for all , is -measurable. By virtue of Lemma 3.10.5 1) in [4] and (2.14), (2.22) in [1] (by taking ), one has,

 ∥∥~E[F|FHt]∥∥L2(PH)≤∥F∥L2(PH). (2.9)

This inequality yields that, if , then .

4. Under the transform , one recovers a property very similar to that of classical martingale. Although fBm is generally not a martingale, it is shown that, for , -a.s..

The following theorem, given in [4], extends the Clark-Hausmann-Ocone Formula from Bm to fBm.

###### Theorem 2.7 (Fractional Clark-Hausmann-Ocone Formula)

Fix , let the random variable be - measurable and Hida Malliavin differentiable, then and

 F=E[F]+∫T0~E[DHtF|FHt]dBHt.

## 3 Main Results

### 3.1 Series Representation via Backward Taylor Expansion

###### Definition 3.1

Fix . We call the set of -measurable random variables which are infinitely fractional Hida Malliavin differentiable. Moreover, for any integer ,

 E[sups1,…,sn∈[0,T]∣∣DHsn…DHs1F∣∣]2<+∞.

To simplify notation, we denote by (identity function) and by the -th composition of the fractional Hida Malliavin derivative.

###### Definition 3.2

Let . Assume for some integer and . is an infinitely differentiable deterministic function. For and , let be the sequence given as:

 ψ(r,tj)0(F)=F; (3.1)

and for , equals

 2−k∑∑Jl=1ql=kJ∏i=1⎛⎝(|tj−ti−1|2H−|tj−ti|2H+|ti−r|2H−|r−ti−1|2H)qiqi!⎞⎠ ×DH,q1t1…DH,qJtJF. (3.2)

Our first main result is given by the following theorem, where a sufficient condition is provided, such that the fractional backward Taylor expansion of the fractional conditional expectation of converges in :
Assumption : Let be given as in Definition 3.2. For some , assume satisfies the following:

 N∑i=0∥∥ ∥∥sup∑Jj=1wj=2N−i∣∣DH,w1t1…DH,wJtJF∣∣∥∥ ∥∥L2(PH)(Ni)2(i!)1/2(N+J−1)!2N−i(N!)2 ×(T2H−r2H+(T−r)2H)N−i(T−r)iH−−−−−→N→+∞0.

Contrary to appearance, this condition is not difficult to check in practice. For example, any verifying

 ∥∥ ∥∥sup∑Jj=1wj=N∣∣DH,w1t1…DH,wJtJF∣∣∥∥ ∥∥L2(PH)≤cN(1/4−ϵ)N

for some , some and for all will satisfy Assumption . Because if so, by taking

 c1=T2H−r2H+(T−r)2H2 and c2=T−r,

the finite partial sum in Assumption can be upper bounded by the following item:

 cN∑i=0(Ni)2(i!)1/2(N+J−1)!(N!)2(2N−i)(1/4−ϵ)(2N−i)cN−i1ci2 ≤c(N⌈N/2⌉)2(N!)1/2(N+J−1)!(2N)(1/2−2ϵ)N(N!)2N∑i=0cN−i1ci2 =c(N!)1/2(N+J−1)!(2N)(1/2−2ϵ)NcN1(⌈N/2⌉!)2((N−⌈N/2⌉)!)21−(c2c−11)N+11−c2c−11 ∼(2πN)1/4(N/e)N/2√2π(N+J−1)((N+J−1)/e)N+J−1π2N2(N/(2e))2N(2N)(1/2−2ϵ)NcN11−(c2c−11)N1−c2c−11 ∼NJ−9/4cN3N2ϵN−−−−−→N→+∞0,

where denotes the ceiling number (the smallest integer upper bound); is some proper constant; and the last approximation is due to the Stirling’s approximation.

###### Theorem 3.3 (Fractional Backward Taylor Expansion)

Let satisfy Assumption . Define

 Ir:={1, if r∈[0,t1]i, if r∈(ti−1,ti],~{}for 2≤i≤J (3.3)

and . Then the following series is convergent in :

 ~E[F|FHr]=+∞∑l=0(−1)l∑qIr+…+qJ=lqIr∑iIr=0…qJ∑iJ=0J∏k=Ir(−1)ik(~tk−~tk−1)(qk−ik)H(qk−ik)! ×⎛⎜⎝hqIr−iIr(BH~tIr−BH~tIr−1(~tIr−~tIr−1)H)ψ(~tIr−1,~tIr)iIr⎞⎟⎠∘…∘⎛⎜⎝hqJ−iJ(BH~tJ−BH~tJ−1(~tJ−~tJ−1)H)ψ(~tJ−1,~tJ)iJ⎞⎟⎠ (DH,qIrtIr…DH,qJtJF). (3.4)

In particular, when ,

 ~E[F|FHr]=+∞∑l=0(−1)ll∑k=0(−1)k(T−r)(l−k)H(l−k)!hl−k(BHT−BHr(T−r)H)ψ(r,T)k(DH,lTF). (3.5)

The proof is provided in the appendix. To see in a concrete way how to present this convergent series for , we take the following two examples.

###### Example 3.4

Consider the random variable with . One can easily check that it verifies Assumption . We determine for .

By using (3.5),

 ~E[eσBHT|FHr]=+∞∑l=0l∑k=0(−1)l+k(T−r)(l−k)H(l−k)!hl−k(BHT−BHr(T−r)H)ψ(r,T)k(DH,lTeσBHT). (3.6)

Notice, from (3.2), that for all and all ,

 ψ(r,T)k(DH,lTeσBHT)=(T2H−r2H)kk!σk+leσBHT. (3.7)

It follows by (3.6), (3.7) and (2.4) that

 ~E[eσBHT|FHr]=eσBHT+∞∑l=0(−σ(T−r)H)ll!l∑k=0(lk)(−σ(T2H−r2H)(T−r)H)khl−k(BHT−BHr(T−r)H). (3.8)

By the following property of Hermite polynomials (due to a Taylor expansion): for all , and all ,

 l∑k=0(lk)xkhl−k(y)=hl(x+y),

one obtains

 l∑k=0(lk)(−σ(T2H−r2H)(T−r)H)khl−k(BHT−BHr(T−r)H)=hl(−σ(T2H−r2H)+BHT−BHr(T−r)H). (3.9)

Finally, it results from (3.8), (3.9) and (2.8) that

 ~E[eσBHT|FHr] = eσBHT+∞∑l=0(−σ(T−r)H)ll!hl(−σ(T2H−r2H)+BHT−BHr(T−r)H) (3.10) = eσBHr+σ2(T2H−r2H)2.
###### Example 3.5

Consider for some fixed and . Below we provide a backward Taylor expansion of with .

is polynomial of fBms, therefore Assumption is verified for all . By the chain rule (2.1), one gets

 DHtF=2BHtBHT+(BHt)2; DH,2tF=2BHT+4BHt; DH,3tF=6; DH,ltF=0 for l≥4; DHTF=(BHt)2; DH,lTF=0 for l≥2; DHtDHTF=DHTDHtF=2BHt; DHTDH,2tF=2.

Using Theorem 3.3 and elementary calculus leads to: if ,

 Missing or unrecognized delimiter for \left −(T−t)Hh1(BHT−BHt(T−t)H)DHTF+ψ(t,T)1(DHTF) +(t−r)2H2!h2(BHt−BHr(t−r)H)DH,2tF−(t−r)Hh1(BHt−BHr(t−r)H)ψ(r,t)1(DH,2tF) +(t−r)H(T−t)Hh1(BHt−BHr(t−r)H)h1(BHT−BHt(T−t)H)DHtDHTF −(T−t)Hψ(r,t)1(h1(BHT−BHt(T−t)H)DHTDHtF) −(t−r)Hh1(BHt−BHr(t−r)H)ψ(t,T)1(DHtDHTF)−(t−r)3H3!h3(BHt−BHr(t−r)H)DH,3tF −(t−r)2H(T−t)H2!h2(BHt−BHr(t−r)H)h1(BHT−BHt(T−t)H)DH,2tDHTF +(t−r)H(T−t)Hh1(BHt−BHr(t−r)H)ψ(r,t)1(h1(BHT−BHt(T−t)H)DH,2tDHTF) =(BHr)3+BHr(T2H+2t2H−3r2H−(T−t)2H); (3.11)

if ,

 ~E[F|FHr] = F−(T−r)Hh1(BHT−BHr(T−r)H)