Differential of a Time Integral Driven by a Stochastic Process

Summary

The document asks how to derive the stochastic process for an accumulated quantity defined as the time integral of a function of a price process. The price is modeled with a stochastic differential equation whose drift is proportional to price and whose volatility may depend on time and the current price. The accumulated process is the ordinary time integral of a deterministic function evaluated along that random price path.

The questioner expands the price process inside the integral and proposes applying Itô’s lemma to the resulting expression, but suspects the proposed differential is incorrect. The document contains no accepted answer or derivation, so it does not establish a final formula. Its useful lesson is the modeling distinction: an integral with respect to time accumulates its integrand, while an Itô integral arises from integration with respect to Brownian motion. A valid derivation must handle the time integral’s dependence on the evolving random state carefully; the source itself leaves that step unresolved.

Key ideas

• The accumulated process is defined as a time integral of a function evaluated along a stochastic price path.
• The price process has a drift and a Brownian component, with state-dependent volatility permitted.
• The question challenges a naive application of Itô’s lemma to the entire accumulated expression.
• No solution is supplied, so the document does not confirm a differential formula.

Tags

# Stochastic differential of a time integral


# Stochastic differential of a time integral












Suppose that $S$ follows a geometric brownian motion: $$ dS(u) = r S(u)du + S(u)\sigma(u,S(u))dW(u) , $$ with $r$ a deterministic constant, and let the process $Z$ be defined by: $$ Z(t) = \int_0^t g(u, S(u))du, $$ where $g$ is a deterministic function. I would like to find the stochastic process followed by $Z$.

Following the idea shown here, I compute: $$ S(t) = S(0) + \int_0^t \alpha(u,S(u))S(u)du + \int_0^t \sigma(u,S(u))S(u)dW(u), $$ and for $Z$: $$ Z(t) = \int_0^t g(u, S(0)+ \int_0^u \alpha(v,S(v))S(v)dv + \int_0^u\sigma(v,S(v))S(v)dW(v))du, $$ where all the integrals are deterministic, except $\int_0^u \sigma(v,S(v))S(v)dW(v)$ which is an Ito integral. At this point, I am not sure on how to differentiate appropriately the equation above in order to get the stochastic process for $Z$.

Naively, I would just take advantage of Ito's lemma to write: $$ dZ(t) = \left(g(t, S(t)) + \int_0^t g_S(u,S(u))du\cdot \alpha(t,S(t))S(t) + \frac12 \int_0^t g_{SS}(u,S(u))du \cdot\sigma^2(t,S(t))S(t)^2 \right)dt + \int_0^t g_S(u,S(u))du \cdot\sigma(t,S(t))S(t)dW(t), $$ but this does not seem to lead to the correct answer.

What is the satisfactory way of formalising the stochastic process for $Z$?

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