Moments of Time Integrals Driven by Brownian Motion

Summary

The document asks whether an ordinary time integral whose integrand depends on Brownian motion is meaningful, contrasting it with Ito integrals against Brownian motion. It proposes that the integral’s expectation can be found by integrating the pointwise expectations, and that its second raw moment can be expressed as a double integral of cross-time expectations. The variance then follows from the first two moments.

These are useful moment identities when the integrand is measurable and the relevant integrals are integrable; they do not require the integral to have a known distribution. The post is a question rather than a worked derivation, so it does not establish sufficient conditions or give examples. In practice, exchanging expectation and integration and using the double-integral formula require suitable integrability assumptions. The document concerns stochastic-process mathematics rather than a trading method.

Key ideas

• A time integral of a function of Brownian motion is a meaningful random variable under suitable integrability conditions.
• Its expectation can be obtained by integrating the expected integrand when exchanging expectation and integration is justified.
• The second raw moment depends on cross-time expectations of the integrand.
• The first and second moments can be combined to calculate variance, even when the full distribution is unknown.

Tags

# Is it meaningful to look at $\int f(W_t, t) \,dt$?


# Is it meaningful to look at $\int f(W_t, t) \,dt$?












CONTEXT (can skip):

My textbook looks at two things -

1) Ito integrals for deterministic functions—i.e. $\int f(t) \,dW_t$. We are able to say that they are normally distributed, with a mean of 0 and a variance of $\int f^2(t) \,dt$.

2) Ito integrals for stochastic functions—i.e. $\int f(W_t, t) \,dW_t$. We aren't able to find their distribution in general; but we can conclude that they have a mean of 0, and ito isometry can be used to get a neat expression for their variance.

QUESTION:

Is it meaningful to look at something of the form $\int f(W_t, t) \,dt$?

MY ANSWER (possibly wrong):

I feel the answer is yes. We won't be able to find the distribution of $\int f(W_t, t) \,dt$ in general, but we can say that it will have a mean of $\int E[f(W_t, t)] \,dt$. I also think that the second order raw moment $E[(\int f(W_t, t) \,dt)^2]$ will be given by $\int \int E[f(W_t, t) f(W_s, s)] \,dtds$, which can then be used to find the variance.

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.