Why an Exponentiated Accrued Integral Has No Itô Diffusion Term

Summary

The document asks why the exponential of an accumulated integral of a state-dependent function has differential equal to the current exponential multiplied by the integrand and elapsed time, without the Brownian and second-derivative terms that appear in Itô’s lemma. The key distinction is that this accumulated quantity is a time integral along the process, not simply a function of the current state to which one applies the usual state-variable form of Itô’s lemma.

Over an infinitesimal time step, the integral changes by its current integrand times the step in time. The exponential therefore changes through ordinary differentiation of that accumulated quantity, giving a finite-variation increment. Although the integrand depends on a stochastic state, its variation over the same infinitesimal step contributes at higher order to this time integral. The post contains the question but no answer, derivation, or example, so it does not establish further conditions or discuss how to apply Itô’s lemma to an augmented state that includes the accumulated integral.

Key ideas

• An accumulated time integral changes at a rate set by its current integrand.
• The exponential of the accumulated integral has a finite-variation differential.
• The standard state-variable form of Itô’s lemma does not apply directly to the accumulated integral as though it were only a function of the current state.
• The post poses the distinction but supplies no worked derivation or answer.

Tags

# Feynman-Kac formula: Ito's lemma for exponentiated integrals $e^{-\int b dr}$


# Feynman-Kac formula: Ito's lemma for exponentiated integrals $e^{-\int b dr}$












Consider the stochastic process $$ dy = f(y,s)ds + g(y,s)dw $$ where, $w$ is Brownian motion. Now consider the following exponentiated integral $$ z_1(s) = \exp \left[ - \int_t^s b(y(r),r) dr \right] $$ This object appears in the Feynman-Kac formula and its derivation (see e.g. Wikipedia and page 3 of this lecture notes). The stochastic differential $d z_1$ is simply $$ dz_1(s) = -z_1 b(y(s),s) ds $$ i.e. this is effectively ordinary differentiation. I was expecting more terms to appear in the above differential due to Ito's lemma, which would give $$ dz_1(s) = \left( \partial_s z_1 + f\partial_y z_1 + \frac{g^2}{2}\partial_y^2 z_1 \right) ds + (g \partial_y z_1) dw $$ and the partial derivatives wrt $y$ would have acted on the integrand $b(y(r),r)$. However, it seems only the $(\partial_s z_1) ds$ term in the RHS above is being retained. Can anyone explain why?

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