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Assume that we have a process $F(t,T)$ that fulfills the following SDE. $$ dF(t,T) = \sigma(t,T)F(t,T)dW(t) $$ where $t$ is the running time and $T>t$ is called the delivery-time. $\sigma(t,T)$ is a (nice) function and $W(t)$ is a Brownian-Motion.

Is the following statement true? How can I prove it? $$ E_0\left[ \left(\int_0^{t_1} \sigma(s,T_1)F(s,T_1)dW(s) \right)\left(\int_{t_1}^{t_2} \sigma(s,T_2)F(s,T_2)dW(s) \right) \right]= $$ $$ E_0\left[\int_0^{t_1} \sigma(s,T_1)F(s,T_1)dW(s)\right]E_0\left[\int_{t_1}^{t_2} \sigma(s,T_2)F(s,T_2)dW(s) \right]=0 $$

where $t_1 < t_2$.

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    $\begingroup$ You are using a couple of undefined variables here -- what are $F$, $T$, $t$, $W$, $T_1$, $T_2$, etc.? $\endgroup$
    – Stefan Kohl ♦
    Commented Dec 6, 2015 at 21:36
  • $\begingroup$ I have added some explanation of the variables. $\endgroup$
    – Samuel
    Commented Dec 6, 2015 at 22:02

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