General Transfer Formula for Stochastic Integral with Respect to Multifractional Brownian Motion

Abstract

In this work we show how results from stochastic integration with respect to multifractional Brownian motion (mBm) can be simply deduced from results of stochastic integration with respect to fractional Brownian motion (fBm), by using a “Transfer Principle”. To illustrate this fact, we prove an Itô formula for integral with respect to mBm by deriving it from Itô formula for integral with respect to fBm, of any Hurst index H in (0, 1).

Appendix: Background on the Bochner integral

In order not to weigh down this statement we will only give the necessary tools to proceed. On can refer to [11, p. 247] as well as to [7] for more details about Bochner integral.

Definition 2

(Bochner integral [11], p. 247) Let I be a Borel subset of [0, 1] and $\Phi :={(\Phi _t)}_{t\in I}$ be an ${({\mathcal {S}})}^*$-valued process verifying:

• (i) the process $\Phi$ is weakly measurable on I i.e. the map $t \mapsto<-0.2\,cm<-0.1\,cm \Phi _t,\varphi \ -0.1\,cm>-0.2\,cm>$ is measurable on I, for every $\varphi$ in $({\mathcal {S}})$.

• (ii) there exists $p \in {\textbf{N}}$ such that $\Phi _t \in ({{\mathcal {S}}}_{-p})$ for almost every $t \in I$ and $t \mapsto {\Vert \Phi _t\Vert }_{-p}$ belongs to $L^1(I)$.

Then there exists an unique element in ${({\mathcal {S}})}^{*}$, noted $\int _{I} \Phi _{u} \ du$, called the Bochner integral of $\Phi$ on I such that, for all $\varphi$ in $({\mathcal {S}})$,

$$\begin{aligned} {<< \ \int _{I} \Phi _u \ du,\varphi \ >>} = \int _{I}<< \ \Phi _{u},\varphi \ >> \ du. \end{aligned}$$

(A.1)

In this latter case one says that $\Phi$ is Bochner-integrable on I with index p.

Proposition A.1

If $\Phi -0.1\,cm:I -0.15\,cm \rightarrow -0.1\,cm {({\mathcal {S}})}^{*}$ is Bochner-integrable on I with index p then ${\Vert \int _{I} \Phi _t \ dt \Vert }_{-p} \le \int _{I} {\Vert \Phi _t\Vert }_{-p} \ dt.$

Theorem A.2

([11], Theorem 13.5) Let $\Phi :={(\Phi _t)}_{t\in [0,1]}$ be an ${({\mathcal {S}})}^{*}$-valued process such that:

• (i) $t\mapsto S(\Phi _t)(\eta )$ is measurable for every $\eta$ in ${\mathscr {S}}({\textbf{R}})$.

• (ii) There exist p in ${\textbf{N}}$, b in ${\textbf{R}}^+$ and a function L in $L^1([0,1], dt)$ such that, for a.e. t in [0, 1], $|S(\Phi _t)(\eta )| \le L(t) \ e^{b {| \eta |}^2_{p} }$, for every $\eta$ in ${\mathscr {S}}({\textbf{R}})$.

Then $\Phi$ is Bochner integrable on [0, 1] and $\int ^1_0 \Phi (s) \ ds \in ({{\mathcal {S}}}_{-q})$ for every $q > p$ such that $2 \cdot b\cdot e^2\cdot D(q-p) < 1$, where e denotes the base of the natural logarithm and where $D(r):= \frac{1}{2^{2r}} \ \sum ^{+\infty }_{n=1} \frac{1}{n^{2r}}$, for any r in $(1/2,+\infty )$.