Ito-Doeblin Formula – Homepage of William Wu

The Ito-Doeblin Formula is an important formula in stochastic calculus, analogous to the chain rule in ordinary calculus. It has many applications in finance, including the derivation of the Black-Scholes-Merton equation.

Let $f(x)$ be a differentiable function and $W(t)$ be a Brownian motion.

The Ito-Doeblin formula in differential form is:

$\displaystyle\boxed{df(W(t))=f'(W(t))\, dW(t)+\frac{1}{2}f''(W(t))\, dt}$

The main difference (compared to ordinary calculus) is that there is an extra term due to the fact that $W$ has nonzero quadratic variation.

Integrating this expression, we get the Ito-Doeblin formula in integral form:

$\displaystyle\boxed{f(W(t))-f(W(0))=\int_0^t f'(W(u))\, dW(u)+\frac{1}{2}\int_0^t f''(W(u))\, du}$

There is also a slightly more generalized version that allows $f$ to be a function of both $t$ and $x$ .

Ito-Doeblin Formula for Brownian Motion

Let $f(t,x)$ be a function where the partial derivatives $f_t(t,x)$ , $f_x(t,x)$ , and $f_{xx}(t,x)$ are continuous, and let $W(t)$ be a Brownian motion. Then, for all $T\geq 0$ ,

$\displaystyle\boxed{\begin{aligned}f(T,W(T))=&f(0,W(0))+\int_0^T f_t(t,W(t))\, dt\\&+\int_0^T f_x(t,W(t))\, dW(t)+\frac{1}{2}\int_0^T f_{xx}(t,W(t))\, dt\end{aligned}}$

The Ito-Doeblin formula in differential form is:

$\displaystyle\boxed{df(t,W(t))=f_t(t,W(t))\, dt+f_x(t,W(t))\, dW(t)+\frac{1}{2}f_{xx}(t,W(t))\, dt}$

Reference

Shreve, Steven E. Stochastic calculus for finance II: Continuous-time models. Vol. 11. New York: Springer, 2004.

Ito-Doeblin Formula for Brownian Motion

Reference

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