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?
