I'm studying the following equation $$\tag1 dX_t = \mu X_t dt + \sigma X_t dB^H_t $$ where $B^H$ is the fractional Brownian motion (fBm) of Hurst parameter $H\in(0,1)$, that is a continuous Gaussian process starting at zero, with $B^H_t \sim \mathcal N(0,t^{2H})$ and with covariance $\mathbb E[B^H_t B^H_s] = \frac12(|t|^{2H}+|s|^{2H}-|t-s|^{2H})$.

According to the value of $H$

  • if $H = 1/2$ then $B^H$ is the classical Brownian motion
  • if $H < 1/2$ then increments of $B^H$ are negatively correlated
  • if $H > 1/2$ then increments of $B^H$ are positively correlated

Moreover, the increment process $B^H_{t+1}-B^H_{t}$ is called fractional Gaussian noise (fGn) and has covariance $\gamma(k) = \frac12(|k-1|^{2H}-2|k|^{2H}+|k+1|^{2H})$.

To run numerical simulations, we first have to find estimators for the parameters $\mu$ and $\sigma$.

In this paper the researches derive the maximum likelihood function in this way.

Let $f,g$ be two functions of $X_t$ and of $\theta$, vector of unknown parameters. Consider $$\tag2 dX_t = f(X_t,\theta) dt + g(X_t,\theta) dB^H_t $$ the first and second moments of the increments of $X$ are given by $$ \mathbb E[dX|X,t] = f(X_t,\theta) dt $$ $$ \mathbb E[(dX)^2|X,t] = g^2(X_t,\theta) (dt)^{2H}. $$

Partitioning $[0,T]$ as $0 = t_0 < t_1 < ... < t_N = T$ s.t. $\Delta t = t_{i+1}-t_i = T/N$, the SDE $(2)$ can be approximated by Euler-Maruyama method as $$\tag3 X_0 = x_0,\quad X_{n+1} = X_n + f(X_n,\theta)\Delta t + g(X_n,\theta)\Delta B^H_n $$ where $\Delta B^H_n = B^H_{t_{n+1}}-B^H_{t_n}$ (in the mentioned paper, $\Delta B^H_n$ is not explicitly defined, but I guess that the definition is the one that I wrote here) is the fGn and $0 \le n \le N-1$.

The probability density function of $(X_{n+1}, t_{n+1})$ starting at $(X_n,t_n)$ is then $$\tag4 \color{red}{p_X} = \frac{1}{\sqrt{2\pi g^2(X_n,\theta)(\Delta t)^{2H}}} \exp\Bigg(-\frac{(X_{n+1}-X_n-f(X_n,\theta)\Delta t)^2}{2g^2(X_n,\theta)(\Delta t)^{2H}}\Bigg) $$ and the joint density gives the likelihood function $\mathcal L$, whose maximizers are the estimates of the parameters $\mu$ and $\sigma$.

For the initial sde $(1)$ we have $f(X_t,\theta) = \mu X_t$ and $g(X_t,\theta) = \sigma X_t$, hence $$\tag5 \color{red}{\mathcal L(\mu,\sigma) = \prod_{n=0}^{N-1}} \frac{1}{\sqrt{2\pi\sigma^2X^2_n(\Delta t)^{2H}}} \exp\Bigg(-\frac{(X_{n+1}-X_n-\mu X_n\Delta t)^2}{2\sigma^2X^2_n(\Delta t)^{2H}}\Bigg) $$

The first question is related to the first $\color{red}{\text{red}}$ term: is the formula $(4)$ for the pdf of the increments of the process $X$, defined by $(3)$, correct?

The second question is related to the second $\color{red}{\text{red}}$ term: is the formula $(5)$ for the joint density (likelihood function) of the increments of the process $X$, defined by $(3)$, correct?

About the second question, my doubt is that since the increments of the fBm are not independent, maybe also the increments of the process $X$, defined by $(3)$, driven by the fBm are not independent. If this was the case, then we could not write the joint density of the increments of $X$ as the product of the individual densities. How to prove if the increments of $X$ are independent or not?

  • 2
    $\begingroup$ I emailed an expert in fractional bm and asked her about your question ( I didn't mention your name ). I don't know her but she responded quite quickly and kindly and provided some good insight. But then I emailed her again because her response made some sense to me but not totally. If she responds to the last question, I can forward you the thread if you send me your email off list. my email is my name with a 2 on the end at gmail.com. actually, maybe you want her first response even if she doesn't respond again. it's useful but I still don't quite get it. $\endgroup$
    – mark leeds
    Mar 6, 2020 at 0:58
  • $\begingroup$ @markleeds Thank you very much, I sent you an email $\endgroup$
    – sound wave
    Mar 6, 2020 at 7:12


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