How to solve numerically the IDE of GUILBAUD & PHAM model?

By the Guilbaud & Pham model (Optimal high frequency trading with limit and market orders, 2011), the authors said that integro-differential-equation (IDE) can be easily solved by numerical method.

$$\begin{gathered} \max \left[-\frac{\partial \varphi_{i}}{\partial t}+\left(b \eta y-\frac{1}{2} \sigma^{2}(\eta y)^{2}\right) \varphi_{i}-\sum_{j=1}^{m} r_{i j}(t)\left[\varphi_{j}(t, y)-\varphi_{i}(t, y)\right]\right. \\ -\inf _{\left(q^{b}, \ell^{b}\right) \in \mathcal{Q}_{i}^{b} \times[0, \bar{\ell}]} \lambda_{i}^{b}\left(q^{b}\right)\left[\exp \left(-\eta\left(\frac{i \delta}{2}-\delta 1_{q^{b}=B b_{+}}\right) \ell^{b}\right) \varphi_{i}\left(t, y+\ell^{b}\right)-\varphi_{i}(t, y)\right] \\ -\inf _{\left(q^{a}, \ell^{a}\right) \in \mathcal{Q}_{i}^{a} \times[0, \bar{\ell}]} \lambda_{i}^{a}\left(q^{a}\right)\left[\exp \left(-\eta\left(\frac{i \delta}{2}-\delta 1_{q^{a}=B a_{-}}\right) \ell^{a}\right) \varphi_{i}\left(t, y-\ell^{a}\right)-\varphi_{i}(t, y)\right] \\ \left.\varphi_{i}(t, y)-\inf _{e \in[-\bar{e}, \bar{e}]}\left[\exp \left(\eta|e| \frac{i \delta}{2}+\eta \varepsilon\right) \varphi_{i}(t, y+e)\right]\right]=0, \end{gathered}$$

I know about the finite differential method, but the problem is the summation term. I don't know about the set of solutions $$\varphi = (\varphi_{i})_{\in I}$$ thus, I can't guess what the $$\varphi_{j}$$ is.

How can I solve the DE? Plus, what is optimal control?

• Thank you for your answer. BTW, $\sum_{j=1}^{m} r_{i j}(t)\left[\varphi_{j}(t, y)-\varphi_{i}(t, y)\right]$ This term is really matter... Aug 10, 2022 at 2:40