The Generalized Ito Formula

Recall that Ito’s lemma expresses a twice differentiable function {f} applied to a continuous semimartingale {X} in terms of stochastic integrals, according to the following formula

\displaystyle  f(X) = f(X_0)+\int f^\prime(X)\,dX + \frac{1}{2}\int f^{\prime\prime}(X)\,d[X]. (1)

In this form, the result only applies to continuous processes but, as I will show in this post, it is possible to generalize to arbitrary noncontinuous semimartingales. The result is also referred to as Ito’s lemma or, to distinguish it from the special case for continuous processes, it is known as the generalized Ito formula or generalized Ito’s lemma.

If equation (1) is to be extended to noncontinuous processes then, there are two immediate points to be considered. The first is that if the process {X} is not continuous then it need not be a predictable process, so {f^\prime(X),f^{\prime\prime}(X)} need not be predictable either. So, the integrands in (1) will not be {X}-integrable. To remedy this, we should instead use the left limits {X_{t-}} in the integrands, which is left-continuous and adapted and therefore is predictable. The second point is that the jumps of the left hand side of (1) are equal to {\Delta f(X)} and, on the right, they are {f^\prime(X_-)\Delta X+\frac{1}{2}f^{\prime\prime}(X_-)\Delta X^2}. There is no reason that these should be equal, and (1) cannot possibly hold in general. To fix this, we can simply add on the correction to the jump terms on the right hand side,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle f(X_t) =&\displaystyle f(X_0)+\int_0^t f^\prime(X_-)\,dX + \frac{1}{2}\int_0^t f^{\prime\prime}(X_-)\,d[X]\smallskip\\ &\displaystyle +\sum_{s\le t}\left(\Delta f(X_s)-f^\prime(X_{s-})\Delta X_s-\frac{1}{2}f^{\prime\prime}(X_{s-})\Delta X_s^2\right). \end{array} (2)

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Ito’s Lemma

Ito’s lemma, otherwise known as the Ito formula, expresses functions of stochastic processes in terms of stochastic integrals. In standard calculus, the differential of the composition of functions {f(x), x(t)} satisfies {df(x(t))=f^\prime(x(t))dx(t)}. This is just the chain rule for differentiation or, in integral form, it becomes the change of variables formula.

In stochastic calculus, Ito’s lemma should be used instead. For a twice differentiable function {f} applied to a continuous semimartingale {X}, it states the following,

\displaystyle  df(X) = f^\prime(X)\,dX + \frac{1}{2}f^{\prime\prime}(X)\,dX^2.

This can be understood as a Taylor expansion up to second order in {dX}, where the quadratic term {dX^2\equiv d[X]} is the quadratic variation of the process {X}.

A d-dimensional process {X=(X^1,\ldots,X^d)} is said to be a semimartingale if each of its components, {X^i}, are semimartingales. The first and second order partial derivatives of a function are denoted by {D_if} and {D_{ij}f}, and I make use of the summation convention where indices {i,j} which occur twice in a single term are summed over. Then, the statement of Ito’s lemma is as follows.

Theorem 1 (Ito’s Lemma) Let {X=(X^1,\ldots,X^d)} be a continuous d-dimensional semimartingale taking values in an open subset {U\subseteq{\mathbb R}^d}. Then, for any twice continuously differentiable function {f\colon U\rightarrow{\mathbb R}}, {f(X)} is a semimartingale and,

\displaystyle  df(X) = D_if(X)\,dX^i + \frac{1}{2}D_{ij}f(X)\,d[X^i,X^j]. (1)

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