What Could Be Causing the Discrepancy in My Fixed Income Mathematics Solution?

[bdw1386]

This problem comes from a book on fixed income finance. The solution is provided, so I gave it a shot and had a slight discrepancy. Probably just due to my rustiness, so hopefully it's an easy one for you guys.

Homework Statement

Differentiate the following expression with respect to t:

<br /> exp[\int_{0}^{t}d\tau \lambda(\tau)]P(t)+R\int_0^tds(-\frac{dP(s)}{ds})exp[\int_0^sd\tau\lambda(\tau)]<br />

Homework Equations

N/A

The Attempt at a Solution

Using the product rule and the FTC on both terms:
<br /> exp[\int_{0}^{t}d\tau \lambda(\tau)]\frac{dP(t)}{dt}+P(t)exp[\int_{0}^{t}d\tau \lambda(\tau)]\lambda(t)<br /> + <br /> R\int_0^tds(-\frac{dP(s)}{ds})exp[\int_0^sd\tau\lambda(\tau)](0)<br /> +<br /> R(-\frac{dP(t)}{dt})exp[\int_0^sd\tau\lambda(\tau)]<br />

The third term falls out, so we get:
<br /> exp[\int_{0}^{t}d\tau \lambda(\tau)]\frac{dP(t)}{dt}+P(t)exp[\int_{0}^{t}d\tau \lambda(\tau)]\lambda(t)<br /> +<br /> R(-\frac{dP(t)}{dt})exp[\int_0^sd\tau\lambda(\tau)]<br />

The textbook matches my solution exactly EXCEPT that the integral in the last term goes from 0 to t, rather than 0 to s. I couldn't figure out why the s changed to a t. The final exp[.] expression comes from the product rule and is simply copied from the original equation:

<br /> x=R\int_0^tds(-\frac{dP(s)}{ds})<br />
<br /> y=exp[\int_0^sd\tau\lambda(\tau)]<br />
<br /> \frac{d}{dt}[xy] = x\frac{dy}{dt}+y\frac{dx}{dt} = y\frac{dx}{dt}<br />
because \frac{dy}{dt} = 0.

What am I missing?

 

[process91]

Your work here appears to be correct.