Differentiating an Itō Integral with a Stochastic Integrand

Summary

The document addresses whether differentiating an accumulated stochastic integral requires adding an Itō correction term. For an integral whose integrand is the smooth function g of a process S, the differential of the integral is simply its current integrand multiplied by dS. The proposed extra term involving the derivative of g and quadratic variation is not part of this operation.

Itō's formula applies when deriving the dynamics of a function of a stochastic process, such as g(S) or another transformed state variable. It does not apply a second time to the differential of an integral already defined with respect to S. In the stated model, substituting the stochastic differential for S yields the integral's local dynamics directly. The exchange is concise and gives no broader derivation or assumptions beyond the displayed setup, but it highlights a useful distinction between applying Itō's formula to a process and differentiating a stochastic integral.

Key ideas

• The differential of the integral of g(S) against S is g(S) multiplied by dS.
• Itō's formula is used to derive the dynamics of a function of a stochastic process.
• A quadratic-variation correction is not added when taking the differential of the stochastic integral itself.
• Substituting the SDE for S gives the integral's local dynamics.

Tags

# Derivative of Stochastic Integral


# Derivative of Stochastic Integral












I am trying to take the derivative of the following stochastic integral, $$d\left(\int g(S_t) dS_t \right),$$ where $dS(t) = \sigma S(t) dW_t$ and $g(.)$ is some (smooth) deterministic function. My understanding is that we can't just apply the fundamental theorem of calculus, but instead need to account for QV. My attempt: $$d\left(\int g(S_t) dS_t \right)=g(S_t)dS_t+\frac{1}{2}g'(S_t)(dS_t)^2=g(S_t)S_t\sigma dW_t+g'(S_t)S_t^2\sigma^2dt$$ Is that right?

## Answer by siou0107 (score 6)

https://quant.stackexchange.com/a/63286

No. Itō’s formula helps you derive the dynamics of $f (S_\cdot )$ given the SDE followed by $S$. Here this is not the case. You simply have: $$ \mathrm{d} \left[\int{g(S_t)\mathrm{d}S_t}\right] = g(S_t) dS_t $$

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