The Liebniz Rule

In considering the derivative of an integral given by

I(x)=\int_{a(x)}^{b(x)} f(x,t)dt

Since

I(x+dx)-I(x) = \int_{a(x+dx)}^{b(x+dx)} f(x+dx,t) dt – \int_{a(x)}^{b(x)} f(x,t) dt

=\int_{a(x+dx)}^{a(x)} f(x+dx,t)dt + \int_{b(x)}^{b(x+dx)}f(x+dx,t)dt+\int_{a(x)}^{b(x)} d_1 f(x,t) dt

= -f(x,t)da(x) + f(x,t) db(x) + \int_{a(x)}^{b(x)} d_1 f(x,t) dt

and so

\frac{dI}{dx} = \int_{a(x)}^{b(x)} f_x(x,t)dt + f(x,b(x)) b_x – f(x,a(x)) a_x

This rule can also be generalized to apply to stochastic differentials, with little modification.

As an application of the rule, if for fixed maturity $T$ the forward rate satisfies

df(t,T) = \alpha(t,T) dt + \sigma(t,T) dW(t)

Then the short rate $r(t) = f(t,t)$ dynamics can be established as

r(t) = f(t,t) = f(0,t) + \int_0^t df(s,t)

=f(0,t) + \int_0^t \alpha(s,t) ds + \int_0^t \sigma(s,t) dW(s)

Using the Leibniz-Ito rule,

dr(t) = \partial_2 f(0,t) + \alpha(t,t)dt + \int_0^t \partial_2 \alpha(s,t)dt ds + \sigma(t,t) dW(t) + \int_0^t \partial_2 \sigma(s,t) dt dW(s)

= (\alpha(t,t) + \partial_2 f(t,t)) dt + \sigma(t,t) dW(t)

Raedwulf ….