Distribution of Correlated Stochastic Integrals with Ornstein–Uhlenbeck Processes

Summary

The document poses a probability question about whether the sum of stochastic integrals of independent Ornstein–Uhlenbeck processes against their respective Brownian motions has the same distribution as a single integral whose integrand is the Euclidean norm of those processes, driven by another Brownian motion. It specifies a linear stochastic differential equation and gives the marginal normal distribution of each process as a potentially relevant fact.

The question highlights an important distinction: deterministic-integrand Wiener integrals are normally distributed, but random adapted integrands require more care. It considers Itô’s isometry, which can identify second moments, and an attempted use of Itô’s lemma, but supplies no proof or answer to the distributional claim. As a result, the text is useful as a statement of a stochastic-calculus problem and its modeling setup, rather than as a completed derivation or a trading method.

Key ideas

• The problem compares a sum of integrals against independent Brownian motions with a single integral driven by another Brownian motion.
• Each integrand is an Ornstein–Uhlenbeck process with a stated linear stochastic differential equation.
• A random integrand does not inherit the deterministic-integrand normality argument automatically.
• Itô’s isometry can help compare variances but does not by itself establish equality in distribution.
• The document presents an open question and does not provide a proof or conclusion.

Tags

# Prove the given stochastic integral are equally distributed


# Prove the given stochastic integral are equally distributed












Let $W^i_t$ and $W_t$ be pairwise independent Brownian motions for $i \in \{1, \dots , d\}$.

Let $X_t^i$ be $d$ independent Ornstein–Uhlenbeck processes for $i \in \{1, \dots , d\}$, i.e. each $X_t^i$ fulfills the SDE:

$$dX_t = \frac \gamma 2dW^i_t + \frac \alpha 2 X^i_t dt$$

To show: the following stochastic integrals are equally distributed:

$$\sum\int^T_0X^i_tdW^i_t \overset{d}= \int_0^T\sqrt{\sum (X^t_i)^2} dW_t$$

My attempt: if the integrands were deterministic, the Wiener integrals would be normally distributed with mean zero. However, the integrands are random.

One possibly useful observation is that the solution $X$ of the above Ornstein–Uhlenbeck equation is normally distributed with

$$X \sim \mathcal N \left(X_0e^{\,\alpha t / 2}, \frac {\gamma ^2 (e^{\alpha t} - 1)}{4\alpha} \right)$$

I thought I could apply Ito's isometry at some point, though in this situation it might only help me to calculate the variance of the random variable and not it's distribution.

I also tried to apply Ito's lemma to the differential $d\left(W_t\cdot\sqrt{\sum (X_t^i) ^ 2}\right)$ without any success.

Any help appreciated.

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