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

Posted: February 14th, 2020, 8:15 am
by EdisonCruise
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
by Alan
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
by EdisonCruise
Thank you, Alan.