What's the expectation of the following stochastic process?

Question

I have a spot price process for $T>t$ : $S_T=K*C$

With K scalar and C:

$C=\exp{\left[\int_t^T e^{-\alpha(T-s)} (\int_t^s e^{-\beta(s-r)}dW_r^Uds)+\int_t^T e^{-\alpha(T-s)}dW_s^V\right]}$

With:

$<dW_t^V,dW_t^U>=\rho dt$

$\alpha,\beta$ constants.

I'm trying to express $\Bbb{E}_t(S_T|\mathcal F_t)$ to evaluate the forward price but i'm strugling with this double integral.

I tried to use info in Math Stackexchange Post which is quite similar for the first part of C.

Thanks

Answer 1

So for the first part of C , all I'm doing is following the computation of Gordon in the link and replacing :

$(t-1) \rightarrow{t}, t \rightarrow{T}, 0 \rightarrow{t} , s\rightarrow{s}$

So I end up with:

\begin{align*} \ln{C} &= \int_{t}^{T} \int_{t}^{s}e^{-\alpha(T-s)-\beta(s-r)}dW(r) \,ds +... \\ &= \int_{t}^{t} \int_{t}^{T}e^{-\alpha(T-s)-\beta(s-r)}ds\,dW(r) + \int_{t}^{T} \int_{r}^{T}e^{-\alpha(T-s)-\beta(s-r)}ds\,dW(r)+...\\ &= 0 + \frac{e^{-\alpha T}}{\alpha-\beta}\int_{t}^T e^{\beta r} \left[e^{(\alpha -\beta)T}-e^{(\alpha -\beta)r} \right] dW(r)+...\\ &= 0 + \frac{e^{-\beta T}}{\alpha -\beta}\int_{t}^T e^{\beta r} dW(r) - \frac{e^{-\alpha T}}{\alpha -\beta}\int_{t}^T e^{\alpha r} dW(r)+... \end{align*}

From there its not too complicated to compute variance of the process, but I have no idea if this work like that as I dont fully understand the decomposition in the first place.

EDIT: This is working, tested with some closed form solutions on special cases, limit cases and Monte Carlo simulation.