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All the following processes are semimartingale. If given a stochastic process $\{X_t:t\geq 0\}$ with $X_t=Y_t+Z_t$ where $Y$ and $Z$ are two other stochastic processes defined on the same probability space as $X$. By Ito lemma we have $$dX_t=dY_t+dZ_t.$$

Can I derive the following? $$1=\frac{dY_t}{dX_t}+\frac{dZ_t}{dX_t}.$$

To motivate why I want the quotient, let's assume $$\phi_t^1dX_t=l_tdY_t,$$ and $$\phi_t^2dX_t=l_tdZ_t,$$to find $\phi_t=\phi_t^1+\phi_t^2$, can i write $$\phi_t=\frac{l_tdY_t}{dX_t}+\frac{l_tdZ_t}{dX_t}=l_t$$?

I am not sure how $\frac{dY_t}{dX_t}$ could be viewed as a Radon-Nikodym derivative.

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    $\begingroup$ What do you mean by those quotients? $\endgroup$ – user223391 Dec 11 '17 at 2:40
  • $\begingroup$ In order to determine if dividing is "okay", first can you tell me what $dX_t$ means? That should give you your answer. $\endgroup$ – user223391 Dec 11 '17 at 2:50
  • $\begingroup$ dX_t=dY_t+dZ_t is just a symbol for X_t=Y_t+Z_t $\endgroup$ – Amira Dec 11 '17 at 2:53
  • $\begingroup$ your question is too general to say much more... $\endgroup$ – cactus314 Dec 11 '17 at 2:58
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I'm guessing $dX_t = X_{t+dt} - X_t$. Then let's try to write a quotient:

$$ \frac{dY_t}{dX_t} = \frac{Y_{t + \delta t} - Y_t}{X_{t + \delta t} - X_t} \stackrel{?}{\in } \mathbb{R}$$

There's no guarantee this fraction has a limit as $dt \to 0$. In real life, all variables are stochastic and we will force this thing to have a limit. Or at least use a best fit line.

There is also something called a Radon-Nikodym derivative for taking the derivative of one measure with respect to another.


It would really help to have more context in your question for formulate a better answer. How does one interpret the meaning of a stochastic derivative?

Have you considered: https://stats.stackexchange.com/ ?

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  • $\begingroup$ I am thinking of the Radon-Nikodym derivative, but I don't know how to proceed. $\endgroup$ – Amira Dec 11 '17 at 2:57

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