\section{Multifractional Brownian Motion}
\label{sec:multifracBM}
FBM can only be used in modelling phenomena which are monofractal with same irregularity globally 
and with constant memory as characterised by the constant H\"{o}lder exponent $H$. 
However, for real world systems global self-similarity seldom exists. Fixed scaling only holds for a certain finite range of intervals. 
In addition, empirical data indicates that the scaling exponent or order of self-similarity usually has more than one value.
Thus in many complex heterogeneous systems there exist phenomena which exhibit multifractal properties with variable space and time dependent memory.
\cite{KKK03a,KKK03b,Sun09}
One simple way to generalize a mono-scaling FBM to a multi-scaling FBM (or multifractional Brownian motion, MBM) is to replace the constant H\"{o}lder exponent by
$H(t), t \in \mathbb{R}^{+}$,
a $(0,1)$-valued function with H\"{o}lder regularity 
$r, r > \sup{H(t)}$. 
In general $H(t)$ can be a deterministic or random function, 
and it needs not be a continuous function. 
This time-varying H\"{o}lder exponent $H(t)$ describes the local variations of the irregularity of the MBM. 
Such a generalization of FBM $B_H$ to MBM $B_{H(t)}$ was carried out independently by 
Peltier and L\'{e}vy-V\'{e}hel\cite{PeltierVehel95}
based on the moving-average representation and by 
Benassi {\it et al}\cite{BenassiJaffardRoux97} 
using the harmonizable representation. As expected, these two generalizations of MBM are almost certainly equivalent up to a multiplicative deterministic function of time.
\cite{Cohen99,LimMuniandy00}


MBM does not satisfy the self-similar property and its increments are no longer stationary as a result of the time-dependence of the H\"{o}lder exponent. 
However, one expects $B_{H(t)}$ to behave like FBM locally. 
If an additional condition is imposed on $H(t)$ such that 
$H(t) \in C^r\bigl(\mathbb{R},(0,1)\bigr), t \in \mathbb{R}$ for some positive 
$r$ with $r > \sup{H(t)}$, then it can be shown that $H(t_\circ)$ is almost certainly the H\"{o}lder exponent of the MBM at 
the point $t_\circ$; 
and the local Hausdorff and box dimensions of the graph of $B_{H(t)}$ at $t_\circ$ are almost certainly 
$2 - H(t_\circ)$. 
One can also characterize the above local fractal property by using the following notion. 
A process $Z(t)$ is said to satisfy the locally asymptotically self-similarity at a point $t_\circ$ if 
\begin{align}
  \lim_{\rho \to 0^{+}} \biggl[\frac{Z(t+\rho{u}) - Z(t_\circ)}{\rho^{H(t_\circ)}}\biggr]_{u \in \mathbb{R}} 
& = \bigl(B_{H(t_\circ)}(u)\bigr)_{u \in \mathbb{R}} ,
\label{eq:multifracBM_0010}
\end{align}
where the equality is up to a multipltcative deterministic function of time. 
It can be verified that $B_H(t)$ is locally asymptotically self-similar. 
Thus MBM at a time $t_\circ$ behaves locally like a FBM with H\"{o}lder exponent $H(t_\circ)$. 
Note that the time-dependent H\"{o}lder exponent
has no effect on the long range dependence of the process. 
Just like FBM, $B_{H(t)}$ is a long memory process.

Similar to the case of standard FBM, one can also extend FBM of RL type to its corresponding multifractional process. 
By replacing $H$ by $H(t)$ in (\ref{eq:fracBM_0110}),
one gets MBM of RL type with the following covariance:\cite{LimMuniandy00,Lim01}
\begin{align}
  \Bigl<X_{H(s)}X_{H(t)}\Bigr> & = \frac{{_2F_1\Bigl(1,\frac{1}{2} - H(t),H(s) + \frac{3}{2}, \frac{s}{t}\Bigr)}}
                                     { \bigl(2H(s)+1\bigr)
                                       \Gamma\Bigl(H(s)+\frac{1}{2}\Bigr)
                                       \Gamma\Bigl(H(t)+\frac{1}{2}\Bigr)
                                     }                                    
                                     s^{H(s)+\frac{1}{2}}t^{H(s)+\frac{1}{2}} .
\label{eq:multifracBM_0020}
\end{align}

The two type of MBM (standard and RL) have more properties in common as compared with the corresponding 
two types of FBM. 
They have non-stationary increments, and both are locally asymptotically self-similar with local fractal dimension at a point 
$t_\circ$ given by $2 - H(t_\circ)$, and they are both LRD.


MBM has been applied to model many phenomena which have variable irregurities or variable memory. 
For examples, it is used in modelling network traffic and signal processing,
\cite{Krishna03,Sheng12ch6}
in geophysics for terrain modelling,
\cite{Gaci12,Echelard10}
in financial time series for stochastic volatility modelling,
\cite{Corlay14}
and in modelling anomalous diffusion with variable memory.
\cite{LimMuniandy02,MarguezLago12} 


Finally we remark that MBM can be further generalized. 
Various generalizations of to MBM have been proposed 
\cite{Ayache00,Vehel13}
to allow H\"{o}lder function to be very irregular, and enable the prescription of local intensity of jumps in space or time.
{"config": "arxiv", "file": "1406.7500/multifracBM.tex"}