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:: Itô's Lemma ::

Sometimes a Stochastic process S can be split into the action of  a random walk process (see article) acting on top of a deterministic drift . In the continuous limit the random walk becomes what is known as a Wiener process X(t). During an interval the process moves deterministically by and randomly by  .  This forms our basic example of a Stochastic Differential Equation (SDE).

No matter how small an interval , there will always be a small contribution from the (t) term. In fact by zooming into the function S(t) one gets something that statistically looks the same (see second graph below). This occurs since the Wiener process has a fractal nature. This is very different from the Ordinary Differential Equation (ODE) where the Wiener term X(t) is absent. For the ODE  we find that as long as is a smooth function then our function S in the interval [t,t+) approaches a straight line as (see first graph below).

Top graph: ODE. Bottom graph: SDE.

In Ordinary calculus if a function V depends on t and on x(t),the total derivative of V is found to be

This is can be obtained in the limit of Taylor expansion of V(x(t),t) by neglecting higher orders of . In this limit both and remain constant within the interval [t,t+), explaining why one ends up with a straight line by zooming into V(x,t).

Now consider V(X(t),t), if we Taylor expand this we obtain:

By taking the limit of its Taylor expansion we obtain

This follows from the fact that the expected value of
is dt, which is the statement that in the
continuous limit, the variance of a simple random walk is proportional to
time. Also due to the rapid changing nature of X(t), neither nor remain constant in the interval [t,t+) in this limit.

V may be a function of S and t. In this case our Taylor expansion yields

This is Itô's lemma. It relates the change in a stochastic function to the change in the stochastic variable.

Written by Raffaello Vardavas.

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