Differentiating an Integral of Conditional Expectations in an Ito Model

Summary

The document asks how to find the stochastic differential of an integral whose integrand is a conditional expectation of a future Ito process. The proposed answer assumes the filtration is generated by the driving Brownian motion and that the drift and diffusion coefficients are deterministic. Under those assumptions, Brownian Markovianity lets the conditional expectation be represented as a function of time, the future time argument, and the current Brownian state.

The answer then applies Leibniz’s rule to the integral with a time-dependent lower limit. The differential includes an integral of the time derivative of that function and a boundary contribution equal to the negative of the process at the current time. This is a concise result under restrictive assumptions: it does not derive the function explicitly or address more general filtrations, random coefficients, or regularity conditions needed to interchange differentiation and integration.

Key ideas

• With deterministic coefficients and a Brownian-generated filtration, the conditional expectation can be represented using the current Brownian state.
• Leibniz’s rule handles the moving lower limit of the integral.
• The differential contains an integrated time derivative and a negative boundary term.
• The result depends on the stated Markovian and determinism assumptions.

Tags

# Stochastic differential of integral with variable inside


# Stochastic differential of integral with variable inside












Consider a Ito process $$d\sigma_t = F_tdt + G_t dW_t$$ and $$X_t = \int_t^T \phi_t(s)ds$$ where $$\phi_t(s) = \mathbb{E}[\sigma_s | \mathcal{F_t}] \quad \text{with } s \leq t.$$ I need to compute $dX_t$. How can I do that? I tried to set $$\Psi(x, t) = \int_t^T xds$$ and observe that $X_t = \Psi(\phi_t)$, than using Ito formula. But I'm pretty sure this is wrong since I'm bringing out of the integral $\phi$. Could anyone help me? Thank you

## Answer by Daneel Olivaw (score 2, accepted)

https://quant.stackexchange.com/a/82266

Assuming (a) $\mathscr{F}$ is generated by your Brownian motion $W$ and that (b) $F$ and $G$ are deterministic processes, then by Brownian’s markovianity we have: $$\phi_t(s)=f(t,s,W_t)$$ for some function $f$. Then by Leibniz rule for differentiation of integrals: $$\begin{align} \text{d}X_t &=\int_t^T\partial_tf(t,s,W_t)\text{d}s -\phi_t(t)\\ &= \int_t^T\partial_t\mathbb{E}(\sigma_s|\mathscr{F}_t)\text{d}s -\sigma_t \end{align}$$

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