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EdisonCruise
Topic Author
Posts: 109
Joined: September 15th, 2012, 4:22 am

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

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.

Alan
Posts: 10026
Joined: December 19th, 2001, 4:01 am
Location: California
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### Re: Is Ito’s lemma applicable to a diffusion process with transition probability?

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"

EdisonCruise
Topic Author
Posts: 109
Joined: September 15th, 2012, 4:22 am

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

Thank you, Alan.
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