Climatique
(1)
\begin{align} d\left(\begin{array}{c} S \\ T \end{array} \right)_t= \left(\begin{array}{c} \mu^S \\ \mu^T \end{array} \right) dt + \left(\begin{array}{cc} \sigma^S & \sigma ^{ST} \\ \sigma^{ST} & \sigma T \end{array}\right) dW_t \end{align}
(2)
\begin{align} I\!\!E \left[ S \varphi(T) \right] \end{align}

On applique la formule d'Itô à $(x,y)\rightarrow x\varphi(y)$

(3)
\begin{align} \nabla = \left(\begin{array}{c} \varphi(y) \\ x \varphi'(y) \end{array} \right) \end{align}
(4)
\begin{align} D^2 = \left(\begin{array}{cc} 0 & \varphi'(y) \\ \varphi'(y) & x \varphi''(y) \end{array} \right) \end{align}
(5)
\begin{align} d \left(S\varphi(T)\right)_t = \left ( \mu . D + \frac{1}{2}\mbox{tr}\left[\sigma\sigma^* D^2 \right]\right) dt + .... dW_t \end{align}
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