# Product rule of stochastic exponents

we know that for standard exponents, $(e^x)(e^y)=e^{(x+y)}$. What is the product rule for stochastic exponents?

$E_n(U)E_n(V)=E_n(U+V+[U,V])$ where $U$ and $V$ are stocchastic sequences, $E_n$ is the stochastic exponent and $[U,V]_n$=$\sum_{k=1}^n(\Delta_Uk*\Delta_Vk)$.

That is the rule, but whats the proof!? thx :)

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What's the definition of $E$ you use? – Ilya Jan 28 at 12:40

## 1 Answer

I assume that by "stochastic exponents", you mean stochastic exponentials, as in Doleans-Dade exponential semimartingales (at least, this definition is consistent with the product rule you mention). The proof of the property

$$\mathcal{E}(X)\mathcal{E}(Y) = \mathcal{E}(X+Y+[X,Y])$$

for semimartingales $X$ and $Y$ can be found as the proof of Theorem II.38 of P. Protter's book "Stochastic integration and Differential equations", 2nd edition.

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