Fourth Moment of a Deterministic Itô Integral

Summary

The document asks for the fourth moment of an Itô integral whose integrand is the square root of time. One response treats the integral as a normally distributed random variable because the integrand is deterministic, using the Itô isometry to identify its variance. The Gaussian fourth-moment formula then provides a direct route to the requested quantity.

A second response attempts to simplify the integral by pulling out a time-dependent factor, but it replaces the square-root-of-time integrand with a different one. That step does not preserve the stated integral, so its resulting moment is inconsistent with the question. The normal-distribution argument is the sound method in the document; the exchange is useful partly because it highlights the need to distinguish a time-varying integrand from a constant factor. No broader financial application or empirical evidence is given.

Key ideas

• An Itô integral with a deterministic integrand is normally distributed.
• The Itô isometry gives the variance needed to apply Gaussian moment formulas.
• The integrand must remain inside the integral unless it is constant with respect to time.
• One answer’s simplification changes the stated integrand and therefore does not solve the original problem.

Tags

# Fourth moment of a itos integral


# Fourth moment of a itos integral












$I(t)=\int_0^t \sqrt sdW_s$

What is $E(I(t)^4)$

## Answer by Bjørn Kjos-Hanssen (score 3)

https://quant.stackexchange.com/a/42121

$I(t)=\int_0^t \sqrt tdW_s=\sqrt t \int_0^t dW_s =\sqrt t W_t $ and then $$E(I(t)^4)=E(t^2 W_t^4)=t^2 \cdot 3t^2=3t^4$$ using the 4th moment of the $N(0,\sigma^2=t)$ distribution.

## Answer by Quantuple (score 2)

https://quant.stackexchange.com/a/42122

Note that because the integrand is deterministic this Itô integral is normally distributed with parameters (cf. Itô isometry) $$ I_t := \int_0^t \sqrt{s} dW_s \sim N(0, t^2/2) $$ Now you can just use the results that apply for the moments of a Gaussian variable.

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