# Are the increments of a stochastic process driven by fractional Brownian motion independent?

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?

• 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. Mar 6, 2020 at 0:58
• @markleeds Thank you very much, I sent you an email Mar 6, 2020 at 7:12