# Tutorial on the portfolio liquidation problem with PDE deep learning - NeuralPDEs and MTK

**URL:** <https://discourse.julialang.org/t/tutorial-on-the-portfolio-liquidation-problem-with-pde-deep-learning-neuralpdes-and-mtk/122583>\
**Category:** Modelling & Simulations\
**Created:** [November 13, 2024, 1:25pm UTC](https://discourse.julialang.org/t/tutorial-on-the-portfolio-liquidation-problem-with-pde-deep-learning-neuralpdes-and-mtk/122583 "2024-11-13T13:25:17Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![finmod](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/finmod/32/26597_2.png) [@finmod](https://discourse.julialang.org/u/finmod)\
**Post date:** [November 13, 2024, 1:25pm UTC](https://discourse.julialang.org/t/tutorial-on-the-portfolio-liquidation-problem-with-pde-deep-learning-neuralpdes-and-mtk/122583/1 "2024-11-13T13:25:18Z")

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I seek to bring to NeuralPDEs.jl and MTK as a tutorial combining the PDE system and the integro-differential demos the portfolio liquidation problem of the Mean Field Control literature.

Here we consider the model proposed by [Cardaliaguet and Lehalle] ([Mean field game of controls and an application to trade crowding | Mathematics and Financial Economics](https://link.springer.com/article/10.1007/s11579-017-0206-z)) and widely available in Python. Following Cardaliaguet and Lehalle, the Nash equilibrium control is α\_t^∗(q) = \frac{∂\_q v(t,q)}{2κ} , where (v,m) solve the following PDE system:

\begin{equation} \left\{ \begin{aligned} − γμ̄q = ∂\_t v − φq^{2} + \frac{|∂\_q v(t, q)|^2}{4κ} \\ ∂\_t m + ∂\_q \left( m\, \frac{∂\_q v(t,q)}{2κ} \right) = 0\\ μ̄\_t = \int\_{q}^{}\frac{∂\_q v(t,q)}{2κ} m(t,dq) \\ m(0,.) = m\_0 , \, v(T, q) = -Aq^2 \end{aligned} \right. \end{equation}

Note that the mean field interactions are through \bar \mu\_t , which is a non-local (in space) term involving the derivative of the solution to the HJB equation.The initial and  
terminal conditions are imposed by penalization. For the non-local term \bar \mu\_t, we use Monte Carlo samples to estimate the integral.

We solve this PDE system using the DGM. We use the NN architecture proposed with 3 layers and a width of 40. The domains are t \in (0, 1) and q \in (0,10).  
In this example, we use the parameters: T = 1, A = 1, φ = 1, κ = 1, γ = 1, and a Gaussian initial distribution with mean 4 and variance 0.3. To ensure that the neural network for the distribution always outputs positive values, we used on the last layer the _exponential function_ as an activation function.

For the Gaussian distribution, we consider two cases: mean 3 and variance 0.5, and mean −4 and variance 0.3. In both cases, the traders aim to bring their inventory close to zero (but perhaps not exactly equal due to other terms in the cost and due to the price impact). In the first case, it means selling stocks, while in the second case, it means buying stocks.

My initial coding using the docs from NeuralPDEs is:

```julia
using NeuralPDE, Lux, ModelingToolkit, Optimization, OptimizationOptimJL, LineSearches,
      OptimizationOptimisers
using ModelingToolkit: Interval, infimum, supremum

@parameters t, q 
@variables v(..), m(..)
Dt = Differential(t)
Dq = Differential(q)
Ii = Integral(q in DomainSets.ClosedInterval(0.0, 10.0))

μ̄
A = 1.0
γ = 1.0
Φ = 1.0
κ = 1.0

eqs = [
    Dt(v) ~ Φ*q^2 - abs(Dq(v))^2/4*κ - γ*μ̄ *q,
    Dt(m) ~ - Dq(m*Dq(v(t,q))/2*κ)
]

bcs = [
    m(0, q) ~ m0,
    v(1.0, q) ~ -Aq^2,
    μ̄ ~ Ii(Dq(v(t,q))*m(t,Dq)/2*κ
]

# Space and time domains
domains = [t ∈ Interval(0.0, 1.0), q ∈ Interval(0.0, 10.0)]

```

Where do I go from here on the number of NN, the building of the loss including the bcs and the integral term by sampling?

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**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [November 20, 2024, 11:07am UTC](https://discourse.julialang.org/t/tutorial-on-the-portfolio-liquidation-problem-with-pde-deep-learning-neuralpdes-and-mtk/122583/2 "2024-11-20T11:07:27Z")

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> [@finmod](#):
>
> the building of the loss including the bcs and the integral term by sampling?

The process here is exactly what `discretize(pdesys)` does on the PDESystem, thus you do not need to do this yourself. Did you try using the `discretize` function as the tutorials show?