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### Is Ito’s lemma applicable to a diffusion process with transition probability?

Posted: February 14th, 2020, 8:15 am
I want to model a continuous variable $$X_t$$ by a stochastic process. With probability $$1-q(X_t)dt$$ at an infinitesimal period $$dt$$, it is a diffusion process. However, with probability $$q(X_t)dt$$, $$X_t$$ may jump to $$Y_t$$. The probability density function of $$Y_t$$ is $$p(Y_t)$$. If I am not wrong, $$X_t$$ can be written as below:
$${dX_t=\mu(X_t)dt+\sigma(X_t)dW_t},with probability 1-q(X_t)dt$$
$${dX_t=Y_t-X_t},with probability q(X_t)dtp(Y_t)$$
Actually, I want to model the ask queue length of level I limit order book with the above process (by transforming a discrete queue length to a continuous variable for space reduction) and try to write an HJB equation for large-tick asset execution/market making. Is that possible? Any suggestion is appreciated.

### Re: Is Ito’s lemma applicable to a diffusion process with transition probability?

Posted: February 14th, 2020, 4:14 pm
Yes, what you are describing is a jump-diffusion. There is an Ito's lemma, a generator, an evolution PIDE, etc, etc. There are many books discussing such processes. Cont and Tankov is good. So is "Option Valuation under Stochastic Volatility II"

### Re: Is Ito’s lemma applicable to a diffusion process with transition probability?

Posted: February 18th, 2020, 10:02 am
Thank you, Alan.