Stratanovich integral?

[Bridgeport]

I need to compute the following limit in $L^2$


$M_{\triangle_n}=\sum_{i=1}^{n}B(\frac{t_{i-1}+t_i}{2})(B(t_i)-B(t_{i-1}))$

where $B(t)$ is standard Brownian Motion

I tried all kind conditioning, but could not get anything good


Ogromnoe spasibo

[Henrylee]

Bulinskij, Shiryaev. Random theory
стр. 285

$$ \frac{B(t)^2}2-\frac{t}2 $$

And your sum is not right. Instead of

$$ B \left(\frac{t_{i-1}+t_i}{2}\right) $$

it must be

$$ B \left(t_{i-1}\right) $$

(I guess you wanna compute stochastic integral

$$ \int\limits_0^t B(s)\,dB(s) $$

[Bridgeport]

I think, everything is correct in my question. The evaluation is exactly at mid-point. Evaluation point is important since we might loose (gain) martingale property of the transform. I thought that mid-point evaluation is Stratanovich integral (not Ito)

Thank you for the reference I will try to read it through.

Spasibo!

[Henrylee]

Ok, sorry. My mistake. I have understood you wrong.

[Хорхе]

I understand that $t_0=0,t_n = T$. Then your sum indeed converges to the Stratonovich integral

$$ \int_0^T B(t)\circ dB(t) = \frac{B^2(T)}2 $$

To prove, you will need the following:

$$Q_{\triangle_n}=\sum_{i=1}^{n}\big(B(t_i)-B(t_{i-1})\big)^2 \to T,\ |\triangle_n|\to 0.$$