# What’s the Ito’s lemma of compound Poisson process with two-sided jump and mean-reverting jump size?

In the book Cartea and Jaimungal (Algorithmic and High Frequency Trading, page 249.), the midprice dynamics follows $$dS_t=\sigma dW_t+\epsilon^+ dM^+_t-\epsilon^- dM^-_t$$ (10.22)

where $$M^+_t$$ and $$M^-_t$$ are Poisson processes, with intensity $$\lambda^+$$ and $$\lambda^-$$ respectively. The jump size $$\epsilon^\pm$$ are i.i.d. whose distribution functions are $$F^\pm$$ with finite first moment denoted by $$\varepsilon^\pm= \mathbb{E}[\epsilon^\pm]$$. By applying Ito’s lemma, market maker’s value function $$H(t,x,S,q)$$ can be expressed as

$$0= \partial_tH+\frac{1}{2}\sigma^2 \partial_{SS} H-\phi q^2+\lambda^+ \sup_{\delta^+} (f(\delta^+)\mathbb{E}[H(t,x+(S+\delta^+),S+\varepsilon^+,q-1)-H]+(1-f(\delta^+)) \mathbb{E}[H(t,x,S+\varepsilon^+,q)-H]) +\lambda^- \sup_{\delta^-} (f(\delta^-)\mathbb{E}[H(t,x-(S-\delta^-),S-\varepsilon^-,q+1)-H]+(1- f(\delta^-)) \mathbb{E}[H(t,x,S-\varepsilon^-,q)-H])$$ (10.23)

where $$f(\delta^\pm)$$ is the filled probability of limit order; $$x$$ is total wealth and $$q$$ is inventory.

I want to modify this model by assuming the midprice as a compound Poisson process with two-sided jump and mean-reverting jump size as below:

$$dS_t= dN_t$$ (1)

where $$N_t$$ is a compound Poisson process with constant intensity $$\lambda$$. It can jump up or down depending on sign of $$\alpha_t$$. Its jump size $$J$$ follows a random distribution (e.g. Laplace or normal ) with mean $$\alpha_t$$ and a constant second moment $$\sigma_\alpha$$. $$\alpha_t$$ itself is a mean reverting process like this:

$$d\alpha_t=-\xi\alpha_t dt+\sigma_a dW_t+\eta dM^+_t-\eta dM^-_t$$ (2)

In analogy of Eq (10.23), I think market maker’s value function $$H(t,x,S,\alpha_t,q)$$ can be expressed as:

$$0= \partial_tH -\xi\alpha\partial_\alpha H+\frac{1}{2}\sigma_{\alpha}^2 \partial_{\alpha\alpha} H -\phi q^2+\lambda P(J>0;\alpha_t)\sup_{\delta^+} (f(\delta^+)\mathbb{E}[H(t,x+(S+\delta^+),S+J, \alpha+\eta,q-1)-H]+(1-f(\delta^+)) \mathbb{E}[H(t,x,S+J, \alpha+\eta,q)-H])+ \lambda P(J<0;\alpha_t)\sup_{\delta^-} (f(\delta^-)\mathbb{E}[H(t,x-(S-\delta^-),S+J, \alpha_t-\eta,q+1)-H]+(1- f(\delta^-)) \mathbb{E}[H(t,x,S+J,\alpha_t-\eta,q)-H])$$ (3)

Here $$\lambda^+$$ is updated as $$\lambda P(J>0;\alpha_t)$$ which is the probability that $$J$$>0 given $$\alpha_t$$. By substituting $$H(t,x,S,\alpha_t,q)=x+qS+h(t,\alpha_t,q)$$, the expectation term in Eq. (3) can becomes $$\mathbb{E}[H(t,x+(S+\delta^+),S+J, \alpha+\eta,q-1)-H]= \mathbb{E}[\delta^+ +(q-1)J+h(t,\alpha_t+\eta,q-1)-h(t,\alpha_t,q)]= \delta^+ +(q-1) \mathbb{E}[J|J>0;\alpha_t]+h(t,\alpha_t+\eta,q-1)-h(t,\alpha_t,q)$$ where $$\mathbb{E}[J|J>0;\alpha_t]$$ is the conditional expectation of jump size when $$J>0$$ given $$\alpha_t$$.

So it seems that the key differences between Eq (10.23) and Eq (3) are that we need to further calculate $$P(J>0;\alpha_t), \mathbb{E}[J|J>0;\alpha_t], P(J<0;\alpha_t), \mathbb{E}[J|J<0;\alpha_t]$$. These can be obtained by assuming an appropriate random distribution, which in Eq. (10.23) is unnecessary. Do I miss a compensated Poisson process to vanish the stochastic terms?