I am studying the following equation in the context of population dynamics

$$ tag1

dX_t = mu X_t dt + sigma X_t dB ^ H_t

$$

or $ B ^ H $ is the fractional Brownian motion (fBm) of the Hurst parameter $ H in (0,1) $, i.e. 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 classic 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

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

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

In this article, the research thus derives the maximum likelihood function.

Let $ f, g $ to 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 $, of the $ (2) $ can be approximated by the 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

$$

or $ Delta B ^ H_n = B ^ H_ {t_ {n + 1}} – B ^ H_ {t_n} $ (in the mentioned article, $ Delta B ^ H_n $ is not explicitly defined, but I guess the definition is the one I wrote here) is the fGn and $ 0 le n le N-1 $.

The probability density function of $ (X_ {n + 1}, t_ {n + 1}) $ from $ (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 parameter estimates $ 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 $, Therefore

$$ 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 process increments pdf $ 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 process increments $ X $, Defined by $ (3) $, correct?

Regarding the second question, my doubt is that since the fBm increments are not independent, perhaps also the process increments $ X $, Defined by $ (3) $, piloted by the fBm are not independent. If that were the case, then we could not write the joint density of the increments of $ X $ as the product of individual densities. How to prove if the increments of $ X $ are independent or not?