What is the general expression for the covariance $cov \left[ X_s X_t \right]$ of a stochastic process given by \begin{equation} dX_t = f(X_t,t)dt + g(X_t,t) dW_t \end{equation} for some general (sufficiently well-behaved) functions $f$ and $g$?

  • $\begingroup$ What do you mean by "general expression"? General expression in terms of ... ? $\endgroup$ – saz Mar 9 '15 at 13:00
  • $\begingroup$ Formally, the solution to the above is just \begin{equation} X(t) = X_0 + \int_0^t f(X,r) dr + \int_0^t g(X,r) dW_r. \end{equation} for some initial value $X_0$. So now I can just calculate $cov \left[ X(s) X(t) \right]$ by plugging in this formal solution. Of course, I cannot evaluate the integrals but I assume that several of the integrals that will appear in this calculation will drop out due to standard rules from stochastic calculus such as \begin{equation} \mathbb{E} \left[ \int f(X,r) dW_r \right] = 0 \end{equation} and possibly many more. $\endgroup$ – user56643 Mar 10 '15 at 2:41
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    $\begingroup$ Well, yes... so have you tried Itô's formula? $\endgroup$ – saz Mar 10 '15 at 6:21
  • $\begingroup$ @ user56643 : You can even use the weaker than ITô's rule, stochastic integration by parts formula (for continuous semimartingale) ? Best regards $\endgroup$ – TheBridge Mar 12 '15 at 7:36
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    $\begingroup$ If $X_0=0$, the covariance of $X_s$ and $X_t$ is $$\int_0^{t\wedge s}E(g(X_u,u)^2)du.$$ $\endgroup$ – Did Mar 13 '15 at 23:37

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