Fractional Brownian Motion Increments and Likelihood Estimation
Summary
The document examines parameter estimation for a stochastic differential equation driven by fractional Brownian motion. It describes an Euler approximation in which each process increment combines a drift term with a scaled fractional Brownian increment, then derives a Gaussian conditional density for an individual step and a product-form likelihood for estimating the drift and diffusion parameters.
Its central concern is dependence: fractional Brownian increments are correlated except in the classical Brownian case, so multiplying the individual conditional densities may not give the joint likelihood. The text provides the covariance structure and the proposed formulas but does not resolve whether the density or likelihood is valid. In particular, the product form assumes independence or an appropriate conditional factorization, which the document itself questions. The setup is therefore useful as a prompt about modeling temporal dependence in estimation, but it is not a complete derivation or empirical validation.
Key ideas
- Fractional Brownian motion has increments whose correlation depends on its Hurst parameter.
- The Euler approximation expresses each process increment using drift and a scaled fractional Brownian increment.
- A Gaussian one-step density is proposed using the conditional mean and variance of that increment.
- The proposed product likelihood raises a dependence issue because fractional Brownian increments are generally correlated.
- The document poses the estimation problem but does not establish that its likelihood formula is correct.
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Full text
# Are the increments of a stochastic process driven by fractional Brownian motion independent?
# 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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.