# “Сross сubic variation” of two Brownian motions and interpretation of the simulation result

Consider two independent 1-dimensional Brownian motions $$W_{t},B_{t}$$, with an equidistant partition of the interval $$[0,T]$$, and $$n\Delta≡T$$.

How to calculate the expression below? Can we rewrite the limit of the summation when $$\Delta$$ goes to zero as an Ito integral? $$S=\lim_{\Delta\to 0} \sum_{i=0}^{n-1}(\frac{1}{\Delta})⋅(W_{(i+1)\Delta}-W_{i\Delta})^{2}⋅(B_{(i+1)\Delta}-B_{i\Delta})$$

NOTE:

I have done many numerical experiments by Monte-Carlo simulation, I find on interval $$[0,1]$$ the summation converge to a random variable (I guess it is normal) with mean 0 and standard deviation 1.73 (approximately) by the function "normfit()" in MATLAB.

However, I can not guess the final expression of $$S$$ because of the "weird" 1.73.

## 2 Answers

As Bjørn already mentioned, the CLT shows that the limit is indeed normal with variance $$3 T$$. A more interesting case is that of $$I = \lim_{\Delta \to 0} \sum_{i=0}^n f(W_{i\Delta},B_{i\Delta})\Delta^{-1} \delta W_i^2 \delta B_i\;.$$ (Here $$\delta W_i = W_{(i+1)\Delta} - W_{i\Delta}$$.) Here, one would expect the limit to be given by $$I = \int_0^T f(W_s,B_s)\,dB_s + \sqrt 2\int_0^T f(W_s,B_s)\,d\tilde B_s\;,$$ where $$\tilde B$$ is a Brownian motion independent of both $$W$$ and $$B$$, at least if $$f$$ is sufficiently regular (say uniformly Hölder continuous). The reason is that you can again use the CLT to show that, with $$f = 1$$, the triple $$(I, B_T, W_T)$$ converges to $$(B_T + \sqrt 2 \tilde B_T, B_T, W_T)$$. From there it's a simple approximation argument.

• Thanks! And If we assume the function $f$ is smooth and sufficiently regular, could we find something more about $\tilde B_{t}$ ? I'm confused why a Brownian motion independent of both $W$ and $B$ would appear in this question. – Stephen Paul Aug 1 at 11:07

The standard deviation you mention looks like $$\sqrt 3$$.

If you calculate the variance of $$S$$ it is hopefully 3 using known results about Var($$W^2B$$) where $$W$$ and $$B$$ are independent normals.

• Sorry for the typo in my previous version of my question, in my numerical experiment I used interval $T=1$ – Stephen Paul Aug 1 at 7:30
• Morever, what about $$\lim_{\Delta\to 0} \sum_{i=0}^{n-1}f(W_{i\Delta}, B_{i\Delta})(\frac{1}{\Delta})⋅(W_{(i+1)\Delta}-W_{i\Delta})^{2}⋅(B_{(i+1)\Delta}-B_{i\Delta})$$ with a deterministic function $f$ – Stephen Paul Aug 1 at 7:37
• Yes it's more complicated, I edited my (partial) answer. – Bjørn Kjos-Hanssen Aug 1 at 7:46