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

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.

• Are $r$ and $\delta_u$ deterministic or stochastic? – Olaf Dec 10 '15 at 9:49
• @Olaf $\delta_u$ and $r$ are deterministic, but only $\delta_u$ is allowed to vary. – GuestNo3829297 Dec 13 '15 at 10:09

## 1 Answer

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?

• 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:quant.stackexchange.com/questions/21497/… – Gordon Dec 13 '15 at 15:41