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Observations on the Theory and Empirics of Mathematical Finance

[PDS] Chain Rule in Ito Calculus

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Chain Rule in Ito Calculus

Given two stochastic processes w_1(t) and w_2(t) driven by different Brownian Motions X_1(t) and X_2(t) as

\begin{aligned} w_1(t) &= \int{g_1(\tau) dX_1(\tau)}\\ w_2(t) &= \int{g_2(\tau) dX_2(\tau)}\end{aligned}

or alternatively, writing them in ‘short-hand’ (their SDE form) as:

\begin{aligned} dw_1(t) &= g_1(t) dX_1(t) \\ dw_2(t) &= g_2(t) dX_2(t)\end{aligned}

Then Ito’s lemma in 2-D tells us that function f(w_1(t), w_2(t)) = w_1(t)w_2(t) will satisfy:

\displaystyle \begin{aligned} d f(w_1(t),w_2(t)) &= \frac{\partial f}{\partial w_1}dw_1(t)+\frac{\partial f}{\partial w_2}dw_2(t)+\frac{\partial^2 f}{\partial w_1^2} dw_1^2(t) \\& \hspace{6pc} +\frac{\partial^2 f}{\partial w_2^2}dw_2^2(t)+ \frac{\partial^2 f}{\partial w_1\partial w_2}dw_1(t)dw_2(t) \end{aligned}

Given f(w_1(t), w_2(t)) = w_1(t)w_2(t), the following will hold:

\begin{aligned} \frac{\partial^2 f}{\partial w_1^2} &=0 \\ \frac{\partial^2 f}{\partial w_2^2}&= 0\\ \frac{\partial^2 f}{\partial w_1\partial w_2}&= 1\end{aligned}

With this we can now simplify the expression for df(w_1(t), w_2(t)) as:

\boxed{d f(w_1(t),w_2(t)) = w_2(t)dw_1(t)+ w_1(t)dw_2(t)+ dw_1(t)dw_2(t)}

This describes the Chain Rule in Ito calculus.

We can, of course, further simplify the above and write:

\begin{aligned} df(w_1(t),w_2(t)) &= w_2(t)dw_1(t)+ w_1(t)dw_2(t)+ dw_1(t)dw_2(t) \\&= w_2(t)g_1(t)dX_1(t) + w_1(t)g_2(t)dX_2(t) + g_1(t)g_2(t) dX_1(t)dX_2(t) \\&= w_2(t)g_1(t)dX_1(t) + w_1(t)g_2(t)dX_2(t) + \rho g_1(t)g_2(t)dt\end{aligned}

where \rho is the correlation between the two Brownian Motions X_1(t) and X_2(t).

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Written by Vineet

February 11, 2013 at 11:48 pm

Posted in Teaching: PDS

Tagged with , ,

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