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I have a little problem in a proof. I have to calculate the following conditional expectated value: $$ \mathbb{E} \left[ \varphi_{k}^{\Phi} C_{i,k-i}\vert \mathcal{T}_{t}, \Phi \right].\qquad \qquad (*)$$ We know, that $\varphi_t^{\Phi}$ is a martingale w.r.t. $(\mathcal{T}_t,\Phi)$ and we have the relation $$C_{i,k-i}= C_{i, t-i} \prod_{j=t-i+1}^{k-i} \left( \exp\{\xi_{i,j}\}+1 \right)$$, where the $C_{i,t-i}$ ist $\mathcal{T}_t$ measureable and $\xi_{i,j} \sim \mathcal{N}(\Phi_{j-1},\sigma_{j-1}^2)$, where the $\sigma_j$ are deterministic and $\Phi_j \sim \mathcal{N}(\phi_j, s_j^2)$. The $\Phi_j$ are independent. The martingale is a density process given by \begin{align*} \begin{split} &\varphi_{t}^{\Phi} = \mathbb{E} \left[ \varphi_{n} \vert \mathcal{T}_{t}, \Phi \right]\\ & = \prod_{j=1}^{J}\prod_{l=1}^{(t-j) \wedge I} \exp \left\{ \alpha_{1} \xi_{l,j} - \alpha_{1} \Phi_{j-1}- \alpha_{1}^{2} \frac{\sigma_{j}^{2}}{2} \right\}\\ &\; \;\qquad\qquad\qquad\times \prod_{j=0}^{J-1} \exp \left\{ (I \alpha_{1} + \alpha_{2} ) \Phi_{j} - (I \alpha_{1}+ \alpha_{2} ) \phi_{j}- (I \alpha_{1}+ \alpha_{2})^{2} \frac{s_{j}^{2}}{2} \right\}, \\ \end{split} \end{align*} where $\alpha_1,\alpha_2$ are fixed parameters. So my idea for calculating the expected value was to use the martingale property, (*) and the independence of $\xi_{i,j}$, so that we get $$\varphi_t^\Phi C_{i,t-i} \prod_{j=t-i+1}^{k-i} \mathbb{E} [\exp\{\xi_{i,j} \}+1\vert \mathcal{T}_t, \Phi ].$$ Then - since $\xi_{i,j}$ are normally distributed - we get by using the moment generating function $$\mathbb{E} [\exp\{\xi_{i,j} \}+1\vert \mathcal{T}_t, \Phi]= \exp\{\Phi_{j-1}+\sigma_{j-1}^2/2\}+1.$$

But in the original proof these steps were skipped and they got as solution $$\mathbb{E} [\exp\{\xi_{i,j} \}+1\vert \mathcal{T}_t, \Phi]= \exp\{\Phi_{j-1}+\alpha_1\sigma_{j-1}^2+\sigma_{j-1}^2/2\}+1.$$

So i have no idea what my mistake is.

EDIT:

Well, i recognized right now that we required a positive correlation between $\varphi_{t}^\Phi$ and $\xi_{i,j}$. So my next idea was to calculate the mean as follows $$ \mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y] + Cov[X,Y].$$ For the first part i got $$\varphi_t^\Phi \exp\{\Phi_j+\sigma_j^2/2\},$$ by using, that $\varphi_t^\Phi$ is a martingale and the moment generating function of the $\xi_{i,j}$. So i need to calculate the covariance function but actually i have no idea how to do that in this case.

2nd EDIT:

I tried to calculate the Covariance. Here is my "solution"

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