Differentiating Brownian Integrals with Time-Dependent Kernels

Summary

The document explains how to differentiate a stochastic integral when its integrand depends on both the integration variable and the current time. For an integral of a time-varying kernel against Brownian motion, the stated differential has two components: a time drift from integrating the kernel’s partial derivative with respect to time, and a Brownian increment weighted by the kernel evaluated at the current endpoint.

This extends the familiar rule for an integrand that depends only on the integration variable, where the differential is simply the integrand at the endpoint times the Brownian increment. The response assumes differentiability of the kernel in its time argument. The document supplies the formula but no proof, detailed regularity conditions, or examples, so it serves as a concise stochastic-calculus result rather than a full treatment of when the differentiation is valid.

Key ideas

• A Brownian integral with a time-dependent kernel has both a drift term and a stochastic increment term.
• The drift integrates the kernel’s partial derivative with respect to the current time.
• The stochastic term uses the kernel evaluated at the moving upper endpoint.
• The stated result assumes differentiability in the time argument and does not detail further regularity conditions.

Tags

# Problem with derivating integral


# Problem with derivating integral












I have a doubt : I know that if $x_{t}=\int_{0}^{t}\gamma(s)dW_{s}$ (with $W_{s}$ a brownian motion), we have : $dx_{t}=\gamma(t)dW_{t}$ What about if $x_{t}=\int_{0}^{t}\gamma(s,t)dW_{s}$. Do I have to apply a kind of Lieibniz rule to get $dx_{t}$ ? If so what is the result ? Tx !

## Answer by Gordon (score 3, accepted)

https://quant.stackexchange.com/a/26122

We assume that $\gamma(s, t)$ is differentiable with respect to $t$. Then, \begin{align*} dx_t = \left(\int_0^t \frac{\partial\gamma(s, t)}{\partial t} dW_s \right)dt + \gamma(t, t) dW_t. \end{align*}

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.