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How do one solve $ \int_t^T \exp[\int_0^u-\left( r-\delta_s\right)ds] dW_u $ ?

$\delta(t)$ is a general deterministic function. $r$ is constant.

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    $\begingroup$ Are $r$ and $\delta_u$ deterministic or stochastic? $\endgroup$
    – Olaf
    Dec 10, 2015 at 9:49
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    $\begingroup$ @Olaf $\delta_u$ and $r$ are deterministic, but only $\delta_u$ is allowed to vary. $\endgroup$ Dec 13, 2015 at 10:09

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There is no analytical solutions to this integral. The conclusions we can draw about this integral are that, if $r$ and $\delta$ are deterministic, it is normal and is independent of the information set $\mathscr{F}_t$. These are probably the most needed properties, for example, in the computation of a zero-coupon bond price under the Hull-White interest rate model, as demonstrated in question. What else are you looking for?

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  • $\begingroup$ How do you conclude that there is no analytical solution? I am studying the detailed answer in the link and find it related, but I need more time. $\endgroup$ Dec 13, 2015 at 10:07
  • $\begingroup$ To say there is no analytical solution is based on my feeling. See also the discussion in this question:quant.stackexchange.com/questions/21497/… $\endgroup$
    – Gordon
    Dec 13, 2015 at 15:41

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