# Multifractal Random Walk Model

As discussed above, continuous Multifractal Random Walk (MRW) (Bacry et al. 2000,2001) process is the only continuous stochastic stationary causal process with exact multifractal properties and Gaussian infinitesimal increments. It is defined as a continuous limit

X (t) = lim XAt [t] (8)

At!0

of the discrete random process of following type

t/At t/At

XAt[t] = lAtXAt[kAt] = |a t[k]e! A‘[k],

k = 1

where XAt [0] = 0; |At [k] is iid Gaussian noise with zero mean and variance a2At and! At [k] is another Gaussian process uncorrelated with the first one (Cor [|At [i] ,!At [j]] = 0, 8 i, j). In financial applications process (8) can be interpreted as the process for logarithm of price and process SAtXAt [kAt] can be viewed as the process for log-returns. Process! At [k] can also be considered as a log-volatility.

According to the definition, !At [k] has zero mean and logarithmically decaying covariance function

Cov [!At[kl],!At[k2]] = A2 ln PAt [|ki – k21],

where

L__

PAt [k] = (lkl+1)At;

One can see that here the support of the autocorrelation function of process! At [k] is bounded from above by the value of integral scale L.

Process (8) for scales l < L has strict multifractal properties and parabolic multifractal spectrum

= ( 2C A2)q- yq2- (11)

For scales l > L process (8) has monofractal properties and spectrum = q/2, which is identical to the spectrum of regular Brownian motion with Hurst exponent 1/2.

As discussed, exact multifractal properties comes along with two significant shortcomings. First, the variance of the process! Дг [k]

is infinite in the limit

In order to obtain convergence of variance, the mean value of! Дг [k] was allowed to decrease logarithmically

L

E [!дг [k]] = – Var [!дг [k]] = – A2 ln д. (14)

Such modification provided the “physical meaning” for the variance, but made it difficult to interpret the meaning of producing noise! Дг [k] in applications. The second issue is that multifractal spectrum does not exist for high orders (namely, for orders q > 1/A2). Nevertheless, MRW model is flexible enough and has relatively straightforward way of calibration that is discussed below.

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