# Moments of Multivariate SDEs

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.