Significance of the term $\int_0^t X_sdY_s+\int_0^t Y_sdX_s$.

Question

Consider a two dimension Brownian motion $(X_t,Y_t)$ and we can consider Levy's area as $\int_0^t X_sdY_s-\int_0^t Y_sdX_s$. Is there any significance of the term $\int_0^t X_sdY_s+\int_0^t Y_sdX_s$. Any reference would be appreciated. Specifically, can it be some form of area (may be overlapped), anyone using it for some application? In rough path theory Levy area is the second term of signature.

Answer 1

Let $Z_t$ be $\int_0^t X_sdY_s+\int_0^t Y_sdX_s$. $Z$ is somehow less interesting than the area because we expect $Z$ to be the product of $X$ and $Y$.

In the case you are asking specifically about, where $(X,Y)$ is a two-dimensional standard Brownian motion and the integration is Ito integration, then $Z_t$ is $X_tY_t$ by the product rule for Ito processes.

If $X$ and $Y$ are two components of a geometric rough path, then because the shuffle product of $1$ and $2$ is $12+21$, the shuffle identity will mean that $Z$ is just be the product of $X$ and $Y$.