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I found this equation from Eq (10) in the paper Hedge and Speculate: Replicating Option Payoffs with Limit and Market Orders by Alvaro Cartea, Luhui Gan, and Sebastian Jaimungal.

I think I can solve the first equation which is for the use of limit order, but I have difficulty is solving the last two equations, due to the Expectation operator, the utility function and an interpolation operator may also require.

This may be like the American option pricing problem, but for American option pricing, a direct comparison of the option price can be implemented quite straightforward. I am not sure if there is any general numerical scheme can solve this problem. Any reference is appreciated.