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Covariance of increments of fractional Gaussian noise

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$begingroup$

Let $B^H(t)$ be a fractional Brownian motion with Hurst parameter $Hin (0,1)$. We define fractional Gaussian noise as $X(t)=B^H(t+1)-B^H(t)$. We know the fBm has covariance $R(s,t)=E(B^H(t)B^H(s))=frac12(t^2H+s^2H-|t-s|^2H)$. We can build up the covariance of increments of fBm from this and then even further we can ask about:

$$E((X(t)-X(s))(X(v)-X(u))$$

Is there any non horrible way of writing down what this covariance should be? I can write it in terms of a bunch of $R$s but nothing better.

probability probability-theory stochastic-processes stochastic-calculus

share|cite|improve this question

edited Apr 2 at 18:10

user658409

asked Apr 2 at 17:27

user658409user658409

486

$endgroup$

add a comment | 

0

1

$begingroup$

Let $B^H(t)$ be a fractional Brownian motion with Hurst parameter $Hin (0,1)$. We define fractional Gaussian noise as $X(t)=B^H(t+1)-B^H(t)$. We know the fBm has covariance $R(s,t)=E(B^H(t)B^H(s))=frac12(t^2H+s^2H-|t-s|^2H)$. We can build up the covariance of increments of fBm from this and then even further we can ask about:

$$E((X(t)-X(s))(X(v)-X(u))$$

Is there any non horrible way of writing down what this covariance should be? I can write it in terms of a bunch of $R$s but nothing better.

probability probability-theory stochastic-processes stochastic-calculus

share|cite|improve this question

edited Apr 2 at 18:10

user658409

asked Apr 2 at 17:27

user658409user658409

486

$endgroup$

add a comment | 

$begingroup$

Let $B^H(t)$ be a fractional Brownian motion with Hurst parameter $Hin (0,1)$. We define fractional Gaussian noise as $X(t)=B^H(t+1)-B^H(t)$. We know the fBm has covariance $R(s,t)=E(B^H(t)B^H(s))=frac12(t^2H+s^2H-|t-s|^2H)$. We can build up the covariance of increments of fBm from this and then even further we can ask about:

$$E((X(t)-X(s))(X(v)-X(u))$$

Is there any non horrible way of writing down what this covariance should be? I can write it in terms of a bunch of $R$s but nothing better.

probability probability-theory stochastic-processes stochastic-calculus

share|cite|improve this question

edited Apr 2 at 18:10

user658409

asked Apr 2 at 17:27

user658409user658409

486

$endgroup$

share|cite|improve this question

edited Apr 2 at 18:10

user658409

asked Apr 2 at 17:27

user658409user658409

486

share|cite|improve this question

edited Apr 2 at 18:10

user658409

asked Apr 2 at 17:27

user658409user658409

486

asked Apr 2 at 17:27

user658409user658409

486

asked Apr 2 at 17:27

user658409user658409

486