How do one solve $ \int_t^T \exp[\int_0^u-( r-\delta_s)ds] dW_u $? Double integral with general deterministic function $\delta(t)$

Question

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.

@Olaf
$\delta_u$ and $r$ are deterministic, but only $\delta_u$ is allowed to vary. — GuestNo3829297, Dec 13 '15 at 10:09

score 1 · Answer 1 · edited Apr 13 '17 at 12:46

1

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?

edited Apr 13 '17 at 12:46

Community

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answered Dec 09 '15 at 21:18

Gordon

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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. – GuestNo3829297 Dec 13 '15 at 10:07
To say there is no analytical solution is based on my feeling. See also the discussion in this question:https://quant.stackexchange.com/questions/21497/how-to-compute-the-stochastic-integral-of-log-normal-process – Gordon Dec 13 '15 at 15:41

How do one solve $ \int_t^T \exp[\int_0^u-( r-\delta_s)ds] dW_u $? Double integral with general deterministic function $\delta(t)$

1 Answers1