For an HJB equation like this,
$$ \frac{dV(t,x)}{dt}+max(H1(V,x),H2(V,x),H3(V,x))=0 $$
I try to solve it with Euler scheme until the actions becomes steady. I find when x<=x0, the optimal action is H1, but when x>x0, at each integration time step, the optimal action switches between H2 and H3. I also find that the diffirence between H2 and H3 is generally smaller than 1e-14, which seems to be round off error. How to make the selection of optimal action more stable when x>x0?
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- Cuchulainn
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Re: How to solve this HJB equation more stably?
Hi,
This nonlinear (ODE/PDE?) equation cannot be solved by classical methods in general due to nonlinearity, an infinity of solutions and stuff.
Euler will probably not be a good approach going forward for various well-known reasons. Plan B could be modified Euler (Heun) or evern {Boost C++, Python} ODEINT. Might be worth a shot:
https://en.wikipedia.org/wiki/Heun%27s_method
But we need monotone, stable and consistent scheme that converges to the viscosity solution. And the solution must be unique.
A similar idea is UVM model
https://onlinelibrary.wiley.com/doi/epd ... wilm.10014
This nonlinear (ODE/PDE?) equation cannot be solved by classical methods in general due to nonlinearity, an infinity of solutions and stuff.
Euler will probably not be a good approach going forward for various well-known reasons. Plan B could be modified Euler (Heun) or evern {Boost C++, Python} ODEINT. Might be worth a shot:
https://en.wikipedia.org/wiki/Heun%27s_method
But we need monotone, stable and consistent scheme that converges to the viscosity solution. And the solution must be unique.
A similar idea is UVM model
https://onlinelibrary.wiley.com/doi/epd ... wilm.10014
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- Cuchulainn
- Posts: 22927
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Re: How to solve this HJB equation more stably?
If you provide a full spec of this DE I can see if I can have a shot using ODEINT and stuff.For an HJB equation like this,
$$ \frac{dV(t,x)}{dt}+max(H1(V,x),H2(V,x),H3(V,x))=0 $$
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- EdisonCruise
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Re: How to solve this HJB equation more stably?
Thank you so much for your suggestions Cuchulainn. It seems that I can solve the equation below instead.
$$ \frac{dV(t,x)}{dt}+max(H1(V,x),H2(V,x)-e,H3(V,x))=0 $$
Where e is a very small number like 1e-12.
In practice, action H2 indeed has marginal cost e compared with H3, though e is difficult to quantify. This way helps to suppress numerical oscillations.
$$ \frac{dV(t,x)}{dt}+max(H1(V,x),H2(V,x)-e,H3(V,x))=0 $$
Where e is a very small number like 1e-12.
In practice, action H2 indeed has marginal cost e compared with H3, though e is difficult to quantify. This way helps to suppress numerical oscillations.
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- Cuchulainn
- Posts: 22927
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Re: How to solve this HJB equation more stably?
Super. You're welcome. Looks like a promising route.
I am learning stuff like this (mostly, slow osmotic flow
). A common theme is to regularise/perturb the PDE by a higher order one (like with singular pertirbations). In your case the perturbation seems to be in the zero-order term (or is there an elliptic PDE hiding in there?)
This thesis might be relevant?
https://core.ac.uk/download/pdf/217192471.pdf
I am learning stuff like this (mostly, slow osmotic flow
). A common theme is to regularise/perturb the PDE by a higher order one (like with singular pertirbations). In your case the perturbation seems to be in the zero-order term (or is there an elliptic PDE hiding in there?)This thesis might be relevant?
https://core.ac.uk/download/pdf/217192471.pdf
Last edited by Cuchulainn on August 26th, 2023, 9:26 am, edited 1 time in total.
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- Cuchulainn
- Posts: 22927
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Re: How to solve this HJB equation more stably?
An interesting question is how convergence (monotone) depends on [$]\varepsilon[$].