The study on fractional processes started from the fractional Brownian motion (FBM) which was first introduced by Kolmogrov in 1940 and popularized by Mandelbrot and Van Ness [1] in 1968. For a constant $\beta\in(-\frac{1}{2}, \frac{1}{2})$, a FBM $W^{\beta}$ is defined by

$ {{W}^{\beta }}=\int_{R}{\left( I_{-}^{\beta }{{1}_{[0, t]}} \right)\left( s \right)}d{{W}_{s}}=\frac{1}{\Gamma \left( \beta +1 \right)}\int_{-\infty }^{t}{\left( {{\left( t-s \right)}^{^{\beta }}}-\left( -s \right)_{+}^{\beta } \right)}d{{W}_{s}}, $

where $W$ is a standard Brownian motion, $I_{-}^{\beta}$ is the Riemann-Liouville fractional integration operator, $H=\beta+\frac{1}{2}$ is called the Hurst parameter of $W^{\beta}$. FBM exhibits self-similarity and long-range dependence when$0 < \beta < \frac{1}{2}$ while remaining Gaussian, therefore, suits to model driving noise in different applications such as mathematical finance. However, the Hurst parameter $H=\beta+\frac{1}{2}$ is a constant, this property make it unsuitable when some one use it to model some phenomena which do not admit a constant Hölder exponent. To this purpose, [2, 3] independently substitute $\beta$ by a Hölder continuous function $\beta:[0, \infty)\mapsto(-\frac{1}{2}, \frac{1}{2})$ and define the multifractional Brownian motion (MFBM) by

$ \begin{array}{*{35}{l}} {{\widetilde{W}}^{\beta }}\left( t \right)={{W}^{\beta (t)}}\left( t \right)=\int_{R}{\left( I_{-}^{\beta (t)}{{1}_{[0, t]}} \right)\left( s \right)}d{{W}_{s}} \\ \ \ \ \ \ \ \ \ \ =\frac{1}{\Gamma \left( \beta \left( t \right)+1 \right)}\int_{-\infty }^{t}{\left( {{\left( t-s \right)}^{\beta \left( t \right)}}-\left( -s \right)_{+}^{\beta \left( t \right)} \right)}d{{W}_{s}}. \\ \end{array} $

On the other hand, the light tails of FBM make it unsuitable to model the high volatility phenomenon.[4] and [5] defined fractional Lévy processes and noises on a Gel'fand triple. The$\beta$-fractional Lévy process on a Gel'fand$\{X_{t}^{\beta}, t\geq0\}$ ($0 < \beta < \frac{1}{2})$ is defined by

$ X_{t}^{\beta }=\int_{R}{\left( I_{-}^{\beta }{{1}_{[0, t]}} \right)\left( s \right)}d{{X}_{s}}=\frac{1}{\Gamma (\beta +1)}\int_{-\infty }^{t}{\left( {{\left( t-s \right)}^{_{\beta }}}-\left( -s \right)_{+}^{\beta } \right)}d{{X}_{s}}, $

where $X$ is a Lévy process on a Gel'fand with zero mean, continuous covariance operator. LÜ et al.[5] showed that fractional Lévy process has stationary increments, long-range dependence. Moreover, LÜ et al.[7]substitute the $\beta$ parameter of fractional Lévy processes by a Hölder continuous function with respect to time to define multifractional Lévy processes on Gel'fand triple.

In this paper, based on the white noises analysis of 0 mean Levy process with finite moments of any orders given by [6], we give the chaos expansion of multifractional Lévy Processes. Moreover, we derive their Lévy-Hermite transforms and Malliavin derivatives. The paper is organized as follows: In section 2, we recall the basic results about white noises analysis of Lévy processes; In section 3, we we derive the chaos expansion of Multifractional Lévy Processes.

Denote by $\mathcal{S}$$(\mathbb{R})$ the Schwartz space of rapidly decreasing $C^{\infty}$-functions on $\mathbb{R}^{d}$ and by $\mathcal{S}'$$(\mathbb{R})$ the space of tempered distributions, let $\mathcal{F}=B $$(\mathcal{S}'$$(\mathbb{R} ))$ be the Borel $\sigma-$ algebra. By Bochner-Minlos theorem, there exists a probability P on $\mathcal{S}'$$(\mathbb{R} )$ such that

$ \int_{\Omega }{{{e}^{iz\langle \omega ,f\rangle }}}P\left( d\omega \right)=\exp \left\{ \int_{-\infty }^{+\infty }{\psi }\left( zf\left( s \right) \right)ds \right\},z\in \mathbb{R},f\in S(\mathbb{R}), $

(2.1)

where

$ \psi \left( z \right)={{\int }_{\mathbb{R}}}\left[{{e}^{izx}}-1-izx \right]d\nu \left( \text{x} \right), u\in \mathbb{R}, $

$\nu$ is the Lévy measure satisfying $\nu(\{0\})=0$ and

$ \int_{\mathbb{R}}{\left( |x{{|}^{2}}\wedge 1 \right)}d\nu \left( x \right) < \infty, $

where $a\wedge b =\min\{a, b\}.$Moreover, we assume that

$ \int_{|x|>1}{|}x{{|}^{2}}dv\left( x \right)\text{ }\mathsf{ < }\infty . $

For $f\in S\left( \mathbb{R} \right)$, let$\dot X(f)(\omega ): = \langle \omega, f\rangle $, then by (2.1), we have

$ \begin{array}{l} \mathbb{E}\left[{\dot X\left( f \right)} \right] = 0, \\ \mathbb{E}{\left[{\dot X\left( f \right)} \right]^2} = \int_\mathbb{R} {{f^2}\left( y \right)} dy\int_\mathbb{R} {{x^2}} d\nu \left( x \right). \end{array} $

we can extend the definition of $\dot{X}(f)(\omega)$ for $f\in \mathcal{S}(\mathbb{R} ) $ to any $f \in L^{2}(\mathbb{R})$ by choosing $f_{n} \in \mathcal{S}(\mathbb{R} ) $ such that $f_{n}\rightarrow f$ in $L^{2}(\mathbb{R})$ and defining $\dot X\left( f \right)\left( \omega \right): = \mathop {\lim }\limits_{n \to \infty } \dot X\left( {{f_n}} \right)\left( \omega \right)$ (in $L^{2}(P))$. Now define $\eta(t) :=\dot{X}(\chi_{[0, t]}(s)) $, where

$ {\chi _{[0,t]}}\left( s \right) = \left\{ \begin{array}{l} 1,\;\;\;\;\;\;0 < s < t,\\ - 1,\;\;\;\;t s < 0,\\ 0,\;\;\;\;\;\;{\rm{else}}. \end{array} \right. $

The stochastic process $\{\eta(t), t\in \mathbb{R} \}$ has a càdlàg version, denoted by $X$. This process $\{X(t), t\in \mathbb{R} \}$ is a pure jump Lévy process with Lévy measure $\nu$. $X$ admits the stochastic integral representation

$ X\left( t \right)=\int_{0}^{t}{\int_{\mathbb{R}}{x}}\widetilde{N}\left( ds, dx \right), t\ge 0. $

where $ \widetilde{N}((0, t]\times A)=N((0, t]\times A)-t\nu(A) $ is its compensation Poisson measure. In this case, $X$ is a martingale and we call it pure-jump Lévy process.

From now on we assume that the Levy measure $\nu$ satisfies for all $\varepsilon>0$, there exists $\lambda>0$ such that

$ {{\int }_{{{\mathbb{R}}_{0}}\backslash \left( -\epsilon ,\mathsf{\epsilon } \right)}}\exp (\lambda |x|)d\nu (x) < \infty . $

(2.2)

where ${{\mathbb{R}}_{0}}=\mathbb{R}\backslash \left\{ 0 \right\}$. This condition means that $X$ has finite moment of any orders.Let $\{p_{j}(z), j\geq1\}$ be the orthonormal basis for $L^{2}(\nu)$ and $\{e_{i}(t), i\geq1\}$ be the Hermite functions, where $p_{1}(z)=m_{2}^{-1}z$, $m_{2}=\int_{\mathbb{R}}x^{2}\nu(dx)$. Define $\kappa(i, j)=j+\frac{(i+j-2)( i+j-1)}{2} $, $\delta_{\kappa(i, j)}(t, z)=e_{i}(t)p_{j}(z)$. For $\alpha\in \mathcal{J}$, (where $\mathcal{J}=\mathbb{N}^{\mathbb{N}}$), $({\rm{index}})\alpha = j$, $|\alpha|=m$, that is $\alpha=(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{j}, 0, 0, \cdots)$, $\sum\limits_{i=1}^{j}{{{\alpha }_{i}}}=m,{{\alpha }_{j}}\in \mathbb{N}$, we define

$ \begin{array}{*{35}{l}} \ \ \ {{\delta }^{\otimes \alpha }}({{t}_{1}}, {{z}_{1}}, \cdots, {{t}_{m}}, {{z}_{m}}) \\ =\delta _{1}^{\otimes {{\alpha }_{1}}}\otimes \cdots \otimes \delta _{j}^{\otimes {{\alpha }_{j}}}({{t}_{1}}, {{z}_{1}}, \ldots, {{t}_{m}}, {{z}_{m}}) \\ ={{\delta }_{1}}({{t}_{1}}, {{z}_{1}})\cdots {{\delta }_{1}}({{t}_{{{\alpha }_{1}}}}{{z}_{{{\alpha }_{1}}}})\cdots {{\delta }_{j}}({{t}_{m-{{\alpha }_{j}}+1}}, {{z}_{m-{{\alpha }_{j}}+1}})\cdots {{\delta }_{j}}({{t}_{m}}, {{z}_{m}}). \\ \end{array} $

We set $\delta_{j}^{\otimes0}=1$ and $\delta^{\widehat{\otimes }\alpha}(t_{1}, z_{1}, \cdots, t_{m}, z_{m}):=\delta_{1}^{\widehat{\otimes} \alpha_{1}}\otimes\cdots\otimes\delta_{j}^{\widehat{\otimes} \alpha_{j}}(t_{1}, z_{1}, \ldots, t_{m}, z_{m})$, define

$ {{K}_{\alpha }}={{I}_{|\alpha |}}({{\delta }^{\hat{\otimes }\alpha }}), $

where $I_{n}(f)$ is the $n$-fold iterated integral of $f$ with respect to $X$, $ \otimes $ means tensor product, $\widehat \otimes $means the symmetric tensor product (for more details, see [6]).

Theorem 2.1  (see [6]) Any $F\in L^{2}(P)$ has a unique expansion of the form

$ F=\sum\limits_{\alpha \in \mathcal{J}}{{{c}_{\alpha }}}{{K}_{\alpha }} $

(2.3)

with $c_{\alpha}\in\mathbb{R}$. Moreover,

$ \parallel F\parallel _{{{L}^{2}}(\Omega )}^{2}=\sum\limits_{\alpha \in \mathcal{J}}{\alpha }!c_{\alpha }^{2}. $

If F has the chaos expansion (2.3), its Lévy-Hermite transform of F is defined by

$ \mathcal{H}F(u):=\sum\limits_{\alpha \in \mathcal{J}}{{{c}_{\alpha }}}{{u}^{\alpha }}=\sum\limits_{\alpha \in \mathcal{J}}{{{c}_{\alpha }}}\prod\limits_{k}{u_{k}^{{{\alpha }_{k}}}}, $

(2.4)

where $u=(u_{1}, u_{2}, \ldots)\in \mathbb{C}^{N}$.

In this section, we give the chaos expansion of multifractional Lévy processes. Moreover, we derive their Lévy-Hermite transforms and Malliavin derivatives.

By the definition of multi-fractional Lévy processes on Gel'fand triple given by [7], we can easily define the real-valued multi-fractional Lévy process.

Definition 3.1  Let $\beta: \mathbb{R^{+}}\rightarrow(0, \frac{1}{2})$ be a measurable function, $X$ be a two-side Lévy process satisfying all the assumptions in Section 2. Define

$ \begin{array}{*{35}{l}} X_{t}^{\beta (t)}:=\int_{R}{\left( I_{-}^{\beta \left( t \right)}{{1}_{[0, t]}} \right)\left( s \right)}d{{X}_{s}} \\ \ \ \ \ \ \ \ \ \ =\frac{1}{\Gamma \left( \beta \left( t \right)+1 \right)}\int_{-\infty }^{t}{\left( {{\left( t-s \right)}^{\beta \left( t \right)}}-\left( -s \right)_{+}^{\beta \left( t \right)} \right)}d{{X}_{s}}, \\ \end{array} $

(3.1)

where $x_{+}=\max\{x, 0\}$, $1_{B}$ is the indicator function on the set $B$, and $I_{-}^{\beta}$ is the Riemann-Liouville fractional integral operator defined by

$ \left( I_{-}^{\beta }f \right)\left( t \right)=\frac{1}{\Gamma \left( \beta \right)}\int_{t}^{\infty }{f\left( s \right)}{{\left( s-t \right)}^{\beta -1}}ds, $

if the integral exist for almost all $x\in \mathbb{R}$ (see [8]), and $\Gamma(\cdot)$ is the Gamma function.

By Theorem 3.2 of [7], we can obtain the one-dimensional distribution of the multi-fractionalLévy process.

Theorem 3.2   Let $X^{\beta}=\{X_{t}^{\beta(t)}, t\geq0\}$ be a $\beta$-multi-fractional Lévy process, then for any $t\geq0$, $X_{t}^{\beta(t)}$ is a 0 mean infinitely divisible R.V. with characteristic function

$ \mathbb{E}\left[\exp \left( izX_{t}^{\beta \left( t \right)} \right) \right]=\exp \left\{ \int{_{\mathbb{R}}}\left[{{e}^{izx}}-1-izx \right]d{{\nu }_{\beta \left( t \right)}}\left( x \right) \right\}, $

(3.2)

where

$ {{\nu }_{\beta \left( t \right)}}\left( B \right)=\int_{\mathbb{R}}{\int_{{{\mathbb{R}}_{0}}}{{{1}_{B}}}}\left\{ \left( I_{-}^{\beta \left( t \right)}{{1}_{[0, t]}} \right)\left( s \right)x \right\}\nu \left( dx \right)ds, \forall B\in \mathcal{B}(\mathbb{R}). $

(3.3)

$\forall s, t\geq0 $,

$ \begin{array}{l} \;\;\mathbb{E}\left[{X_t^{\beta \left( t \right)}X_s^{\beta \left( s \right)}} \right]\\ = {m_2}C\left( {\beta \left( t \right), \beta \left( s \right)} \right)\left( {|t{|^{\beta (t) + \beta (s) + 1}} + |s{|^{\beta (t) + \beta (s) + 1}} - |t - s{|^{\beta (t) + \beta (s) + 1}}} \right) \end{array} $

(3.4)

and

$ C\left( {\beta \left( t \right), \beta \left( s \right)} \right) = \frac{1}{{2\Gamma \left( {\beta \left( t \right) + \beta \left( s \right) + 2} \right)\sin \frac{{\left( {\beta \left( t \right) + \beta \left( s \right) + 1} \right)\pi }}{2}}}. $

Theorem 3.3  Let $X^{\beta}=\{X_{t}^{\beta(t)}, t\geq0\}$ be $\beta$-multi-fractional Lévy process, then for any $t\geq0$, the chaos expansion of $X_{t}^{\beta(t)}$ is

$ X_{t}^{\beta \left( t \right)}=\sum\limits_{i\ge 1}{{{m}_{2}}}\int_{\mathbb{R}}{\left( I_{-}^{\beta \left( t \right)}{{1}_{\left[0, t \right]}} \right)\left( s \right){{e}_{i}}\left( s \right)}ds{{K}_{\varepsilon \left( i, 1 \right)}}, $

(3.5)

where

$ {{K}_{\varepsilon \left( i, 1 \right)}}={{I}_{1}}\left( {{e}_{i}}\left( t \right){{p}_{1}}\left( z \right) \right), $

$\varepsilon(i, 1)=(0, 0, \cdots, 1, 0, \cdots)$ with the 1 on the $\kappa(i, 1)$th place.

Proof  Since $I_{-}^{\beta(t)}1_{[0, t]}\in L^{2}(\mathbb{R})$, for any $t\geq0$, $X_{t}^{\beta(t)}\in L^{2}(P)$, then by Theorem 2.1,

$ \begin{array}{l} X_t^\beta = \int_R {(I_ - ^{\beta (t)}{1_{[0, t]}})(s)} d{X_s}\\ \;\;\;\;\;\; = {I_1}(I_ - ^{\beta (t)}{1_{[0, t]}}(s)z)\\ \;\;\;\;\;\; = {I_1}(\sum\limits_i {\langle I_ - ^{\beta (t)}{1_{[0, t]}}(} \cdot ), {e_i}( \cdot ){\rangle _{{L^2}(\mathbb{R})}}{e_i}(s)z)\\ \;\;\;\;\;\; = \sum\limits_i {\langle (} I_ - ^{\beta (t)}{1_{[0, t]}})( \cdot ), {e_i}( \cdot ){\rangle _{{L^2}(\mathbb{R})}}{I_1}({e_i}(s)z)\\ \;\;\;\;\;\; = \sum\limits_{i \ge 1} {{m_2}} {\langle (I_ - ^{\beta (t)}{\chi _{[0, t]}})( \cdot ), {e_i}( \cdot )\rangle _{{L^2}(\mathbb{R})}}{K_{\varepsilon (i, 1)}}\\ \;\;\;\;\;\; = \sum\limits_{i \ge 1} {{m_2}} \int_\mathbb{R} {(I_ - ^{\beta (t)}{\chi _{[0, t]}})(s){e_i}(s)} ds{K_{\varepsilon (i, 1)}}. \end{array} $

Thus we get the desired.

By the following fractional integral by parts formula of operator$I_{\pm}^{\beta}$:

$ \int_{-\infty}^{+\infty}f(s)(I_{+}^{\beta}g)(s)ds= \int_{-\infty}^{+\infty}g(s)(I_{-}^{\beta}f)(s)ds, \ f, g\in \mathcal{S}(\mathbb{R}), $

which can be extended to $f\in L^{p}(\mathbb{R})$, $g\in L^{r}(\mathbb{R})$ with $p>1$, $r>1$ and $\frac{1}{p}+\frac{1}{r}=1+\beta$, where

$ \left( I_{+}^{\beta }f \right)\left( t \right)=\frac{1}{\Gamma (\beta )}\int_{-\infty }^{t}{{{\left( t-s \right)}^{\beta -1}}}f\left( s \right)ds $

(see [8]), (3.5) can be written as

$ X_{t}^{\beta \left( t \right)}={{m}_{2}}\sum\limits_{i\ge 1}{\int_{0}^{t}{I_{+}^{\beta \left( t \right)}}}{{e}_{i}}\left( s \right)ds{{K}_{\varepsilon \left( i, 1 \right)}}. $

By the chaos expansion of $X^{\beta}$, we get

Corollary 3.4  Let $X^{\beta}=\{X_{t}^{\beta(t)}, t\geq0\}$ be $\beta$-multi-fractional Lévy process, then for any $t\geq0$, the Lévy-Hermite transform of $X_{t}^{\beta(t)}$

$ \mathcal{H}X_{t}^{\beta \left( t \right)}\left( u \right)=\sum\limits_{i\ge 1}{{{m}_{2}}}\int_{\mathbb{R}}{\left( I_{-}^{\beta \left( t \right)}{{1}_{\left[0, t \right]}} \right)\left( s \right){{e}_{i}}\left( s \right)}ds{{u}_{\varepsilon \left( i, 1 \right)}}, $

(3.6)

where $u=(u_{1}, u_{2}, \ldots)\in \mathbb{C}^{N}$.

By equality (12.4) of \cite{Nunno}, we can easily get

Proposition 3.5  Let $X^{\beta}=\{X_{t}^{\beta(t)}, t\geq0\}$ be $\beta$-multi-fractional Lévy process, then for any $t\geq0$, the Malliavin derivative of $X_{t}^{\beta(t)}$

$ {{D}_{s, z}}X_{t}^{\beta \left( t \right)}=\left( I_{-}^{\beta \left( t \right)}{{1}_{\left[0, t \right]}} \right)\left( s \right)z. $

(3.7)