# Question on Gÿongy' lemma proof

I have some questions regarding a proof of Gÿongy's lemma given in 1

I would like to understand the following passage: $$\int_{s=t_0}^{s=t}\mathbb{E}\left[\delta(X_s-K)\langle dX_s\rangle^2 \right]= \int_{s=t_0}^{s=t}\mathbb{E}\left[\delta(X_s-K) \right] \mathbb{E}\left[\langle dX_s\rangle^2|X_s=K \right]$$

Thanks

If $$X_s \neq K$$ then the delta function gives zero, and the product is zero. So the term only contributes when $$X_s=K$$.

Re-comment, the key to understanding this is the conditional expectation:

$$E \left[ dX_s^2 \mid X_s=K \right] =\frac{E\left[ dX_s^2 \delta(X_s-K)\right]}{E\left [\delta(X_s-K)\right] }$$

Where it might be helpful if you interpret the delta as indicator of $$X_s=K$$

• What I don't understand if why the expectation of a product equals the product of expectations. Is this true because the quadratic variation $\langle dX_s\rangle^2$ is independant from $X_s$ ? – Aguelmame Apr 22 at 17:04
• Added further explanation in the answer. – Magic is in the chain Apr 22 at 17:29
• Ok, that makes sense. Thanks – Aguelmame Apr 22 at 18:09