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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?

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Don't bother. I implemented the strategy and took it live and only lost 4000 dollars after turning over 10 million dollars worth of stock. The Cox process assumption is the flaw in this paper. It might work with a more realistic model like Hawkes processes but the hft traders have this arena locked down IMHO. Good luck. And yes, it was extremely challenging to implement . You have to work backwards from the end of the day, and basically do a brute force iteration for the sup and inf to calculate the value of each grid element. In simulations I reproduced the results of the paper. Live was a different story.

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    $\begingroup$ 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... $\endgroup$
    – JMNQC
    Aug 10, 2022 at 2:40

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