*1.3. Simulation of Multifractal Processes*

The Brownian motion in multifractal time (BMMT) *BH*(*θ*(*t*)) introduced in Mandelbrot et al. (1997) is employed to simulate a multifractal process. The BMMT model shows appealing features that coincide with some stylised features of financial time series. It displays heavy-tails while not necessarily implying infinite variance. It also implies long-term dependence in absolute values of returns while the price increments themselves can remain uncorrelated. By construction, the BMMT is defined as the subordinate model of a fractional Brownian motion *BH*(*t*) with a multifractal process *<sup>θ</sup>*(*t*).

**Definition 2** (Brownian Motion in Multifractal Time)**.** *Brownian motion in multifractal time is defined as*

$$X(t) = B\_H(\theta(t)), \ t \in [0, T]\_\prime$$

*where θ*(*t*) *is a positive multifractal stochastic process and BH is an independent fractional Brownian motion with Hurst parameter H.*

*θ*(*t*) can also be seen as a trading time where the index *t* denotes clock time. We assume the activity time *θ*(*t*) to be the cumulative density function of a random multifractal measure *μ* defined on [0, *<sup>T</sup>*]. That is, *θ*(*t*) is a multifractal process with continuous, non-decreasing paths, and stationary increments.

The *multiplicative multifractal measure* is implemented for simulation. Without loss of generality, we assume that the time series of interest is defined on a compact interval [0, 1]. A multiplicative measure is constructed as follows:


$$\mathbb{P}(\mu\_2[0, 1/b^2] = m\_0 m\_1) = \mathbb{P}(M = m\_0) P(M = m\_1)\_r$$

since the multipliers at different stages are independent. The measure *μ*2 represents the multiplicative measure at Stage 2.

3. Repetition of this scheme generates a sequence of measures (*μk*)*<sup>k</sup>*∈<sup>N</sup> which converges to our desired multiplicative measure *μ* as *k* → ∞.

**Remark 1.** *To preserve the mass at each stage, some restrictions are required on the values of the Ml*,*β*, 1 ≤ *l* ≤ *k. If we strictly assume that* ∑*<sup>b</sup>*−<sup>1</sup> *β*=0 *Ml*,*<sup>β</sup>* = 1 *at each stage, the resulting measure is called microcanonical or micro-conservative. If we loosen the assumption so that the mass at each stage is only conserved "on average", that is,* <sup>E</sup>(∑*<sup>b</sup>*−<sup>1</sup> *β*=0 *Ml*,*β*) = 1, ∀1 ≤ *l* ≤ *k, the resulting measure is called canonical.*

In this paper, we only consider canonical measures, as they impose less restriction on the distribution of *M*. Let

$$t = 0.\eta\_1...\eta\_k = \sum\_{i=1}^k \eta\_i b^{-i}$$

be a *b*-adic number and set Δ*t* = *b*−*k*. The mass on the *b*-adic cell [*t*, *t* + Δ*t*] for a canonical measure at Stage *k* is

$$\mu(\Delta t) = \mu[t, t + \Delta t] = \Omega(\eta\_1, \dots, \eta\_k)M(\eta\_1)M(\eta\_1, \eta\_2)\dots M(\eta\_1, \dots, \eta\_k)\_\*$$

where the random variable Ω represents the total mass. The high-frequency component <sup>Ω</sup>(*η*1, ..., *ηk*) is assumed to have the same distribution as Ω. It captures changes in total mass of the interval caused by stages beyond *k*. As *μ* here is a canonical measure, we assume the multipliers *Ml*,*<sup>β</sup>* (0 ≤ *β* ≤ *b* − 1) satisfy E(∑ *Ml*,*β*) = 1 or equivalently E*M* = 1/*b*.

Since multipliers at different stages of subdivision are independent, <sup>E</sup>[*μ*(<sup>Δ</sup>*t*)*q*] = <sup>E</sup>(Ω*<sup>q</sup>*)[E(*Mq*)]*<sup>k</sup>*, for all *q*. Defining *τ*(*q*) = − log*b* E(*Mq*) and recalling Δ*t* = *<sup>b</sup>*−*k*, we have

$$\mathbb{E}[\mu(\Delta t)^q] = \mathbb{E}(\Omega^q)(\Delta t)^{\tau(q)}.\tag{5}$$

Consequently, the constructed multiplicative measure *μ* is multifractal according to Definition 1. Because Equation (5) only holds for Δ*t* = *b*−*<sup>k</sup>* and *t* = 0.*η*1...*ηk* = ∑*ki*=<sup>1</sup> *<sup>η</sup>ib*−*<sup>i</sup>* being a *b*-adic number, these multiplicative measures are said to be *grid-bound multifractal measures*.

The relation between the distribution of *M* and the scaling function can be inferred from the following theorem.

**Theorem 2** (Calvet et al. 1997)**.** *Define p*(*α*) *to be the continuous density of V* = − log*b M, and pk*(*α*) *to be the density of the k-th convolution product of p. The scaling function of a multiplicative measure satisfies*

$$\pi(q) = \inf\_{a} [aq - \lim\_{k \to \infty} \frac{1}{k} \log\_b [kp\_k(ka)]. \tag{6}$$

In this paper, we have estimated sample scaling functions using the BMMT with a log-normal multiplicative measure. We have chosen this measure because its multifractal spectrum is a better fit for financial time series3. When *V* follows a normal distribution, N (*<sup>λ</sup>*, *σ*<sup>2</sup>) where *λ* > 1, then *M* = *b*−*<sup>V</sup>* follows a log-normal distribution that is, log*b M d*= N (−*λ*, *<sup>σ</sup>*<sup>2</sup>). We then have

$$\mathbb{E}M = \mathbb{E}(\varepsilon^{-V\log b}) = \exp\left(-\lambda \log b + \frac{(\log b)^2 \sigma^2}{2}\right) \mathbb{I}$$

As *M* ∈ (0, <sup>∞</sup>), we need the constraint E(*M*) = 1/*b* to make *M* a canonical measure. This constraint gives a relationship between *λ* and *σ*2, namely log *b* = <sup>2</sup>(*<sup>λ</sup>*−<sup>1</sup>) *σ*<sup>2</sup> .

Based on Theorem 2, the scaling function for the log-normal multiplicative cascade is derived as follows:

$$\pi(q) = \inf\_{a} \left( aq - \lim\_{k \to \infty} \frac{1}{k} \log\_b[kp\_a(ka)] \right) = \lambda q - \frac{\sigma^2 \log b}{2} q^2. \tag{7}$$

This quadratic form reaches its maximum at *q* = *λ σ*<sup>2</sup> log *b* .

The cumulative distribution of the simulated multiplicative measures *μ* gives a multifractal multiplicative cascade, which is taken to be the trading time *θ*(*t*) = *t*0 *μ*([0,*s*])*ds* in a BMMT model *BH*(*θ*(*t*)).
