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

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**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.

$$ {dX_t=\mu(X_t)dt+\sigma(X_t)dW_t},with probability 1-q(X_t)dt$$

$$ {dX_t=Y_t-X_t},with probability q(X_t)dtp(Y_t)$$

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.