Consider the following SDE:

$$ \mathbf{y}_{t} = \mathbf{F}_{t,s} \mathbf{x}_{s} + \int_{s}^{t} \mathbf{B}_{t,u} d\mathbf{W}_{u} $$

where the subscripts indicate dependence on $t$, $s$, and $u$ respectively and $\mathbf{F}_{t,s}$ and $\mathbf{B}_{t,u}$ are non-deterministic but with $t,s,u$ dependence.

I want to find the mean, variance, and covariance of the process. I am unsure about the variance and the covariance calculations, and I will show where I get stuck.

The Mean

$$ \boxed{E\left[ \mathbf{y}_{t} \right] = \mathbf{F}_{t,s} E \left[ \mathbf{x}_{s} \right] }$$

The Variance:

$$ V[ \mathbf{y}_{t}] = E \left[ \mathbf{y}_{t} \mathbf{y}_{t}^{T} \right] - E\left[ \mathbf{y}_{t} \right] E \left[ \mathbf{y}_{t} \right]^{T}$$ $$ V[ \mathbf{y}_{t}] = E \left[ \mathbf{y}_{t} \mathbf{y}_{t}^{T} \right] - \mathbf{F}_{t,s}E[ \mathbf{x}_{s} ]E[ \mathbf{x}_{s} ]^{T} \mathbf{F}_{t,s}^{T}$$

Now we need to calculate $E \left[ \mathbf{y}_{t} \mathbf{y}_{t}^{T} \right]$. This is where I get stuck. We have:

$$ E \left[ \mathbf{y}_{t} \mathbf{y}_{t}^{T} \right] = E \left[ \left( \mathbf{F}_{t,s} \mathbf{x}_{s} + \int_{s}^{t} \mathbf{B}_{t,u} d\mathbf{W}_{u} \right) \left( \mathbf{F}_{t,s} \mathbf{x}_{s} + \int_{s}^{t} \mathbf{B}_{t,u} d\mathbf{W}_{u} \right)^{T} \right] $$ $$ = E \left[ \left( \mathbf{F}_{t,s} \mathbf{x}_{s} + \int_{s}^{t} \mathbf{B}_{t,u} d\mathbf{W}_{u} \right) \left( \mathbf{x}^{T}_{s} \mathbf{F}_{t,s}^{T} + \int_{s}^{t} d\mathbf{W}^{T}_{u} \mathbf{B}^{T}_{t,u} \right) \right] $$ $$ = E \left[ \mathbf{F}_{t,s} \mathbf{x}_{s} \mathbf{x}^{T}_{s} \mathbf{F}_{t,s}^{T} \right] + E \left[ \int_{s}^{t} \int_{s}^{t} \mathbf{B}_{t,u} d\mathbf{W}_{u} d\mathbf{W}^{T}_{u} \mathbf{B}^{T}_{t,u} \right] $$ $$ = \mathbf{F}_{t,s} E\left[ \mathbf{x}_{s} \mathbf{x}^{T}_{s} \right] \mathbf{F}_{t,s}^{T} + \int_{s}^{t} \mathbf{B}_{t,u} \mathbf{B}^{T}_{t,u} \mathbf{B}_{t,u} \mathbf{B}^{T}_{t,u} dt $$

Thus the variance is:

$$ V[ \mathbf{y}_{t}] = \mathbf{F}_{t,s} E\left[ \mathbf{x}_{s} \mathbf{x}^{T}_{s} \right] \mathbf{F}_{t,s}^{T} + \int_{s}^{t} \mathbf{B}_{t,u} \mathbf{B}^{T}_{t,u} \mathbf{B}_{t,u} \mathbf{B}^{T}_{t,u} dt - \mathbf{F}_{t,s}E[ \mathbf{x}_{s} ]E[ \mathbf{x}_{s} ]^{T} \mathbf{F}_{t,s}^{T}$$

$$ \boxed{V[ \mathbf{y}_{t}] = \mathbf{F}_{t,s} V\left[ \mathbf{x}_{s} \right] \mathbf{F}_{t,s}^{T} + \int_{s}^{t} \mathbf{B}_{t,u} \mathbf{B}^{T}_{t,u} \mathbf{B}_{t,u} \mathbf{B}^{T}_{t,u} dt }$$

Is this correct? Is it true, by the Iso Isometry, that:

$$ E \left[ \int_{s}^{t} \int_{s}^{t} \mathbf{B}_{t,u} d\mathbf{W}_{u} d\mathbf{W}^{T}_{u} \mathbf{B}^{T}_{t,u} \right] = \int_{s}^{t} \mathbf{B}_{t,u} \mathbf{B}^{T}_{t,u} \mathbf{B}_{t,u} \mathbf{B}^{T}_{t,u} dt $$

If I knew this, then I could probably continue with the covariance computation.



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