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[WIP] Improve Kolmogorov convergence #248

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Jul 1, 2020
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[WIP] Improve Kolmogorov convergence #248

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21 changes: 10 additions & 11 deletions tutorials/advanced/03-kolmogorov_equations.jmd
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Original file line number Diff line number Diff line change
Expand Up @@ -62,8 +62,8 @@ Now we shall define the problem :
We will define the σ and μ by comparing it to the orignal equation. The xspan is the span of initial stock prices.
```julia
d = 1
r = 0.02
sigma = 0.4
r = 0.04
sigma = 0.2
xspan = (80.00 , 115.0)
tspan = (0.0 , 1.0)
σ(du , u , p , t) = du .= sigma.*u
Expand All @@ -75,34 +75,34 @@ Now once we have defined our problem it is necessary to define the parameters fo
sdealg = EM()
ensemblealg = EnsembleThreads()
dt = 0.01
dx = 0.001
dx = 0.01
trajectories = 100000
```

Now lets define our model m and the optimiser
```julia
m = Chain(Dense(d, 8, leakyrelu),Dense(8, 16, leakyrelu),Dense(16 , 8 , leakyrelu) , Dense(8 , 1))
m = Chain(Dense(d, 64, elu),Dense(64, 128, elu),Dense(128 , 16 , elu) , Dense(16 , 1))
use_gpu = false
if CUDAnative.functional() == true
m = fmap(CuArrays.cu , m)
use_gpu = true
end
opt = Flux.ADAM(0.01)
opt = Flux.ADAM(0.0005)
```
And then finally call the solver
```julia
@time sol = solve(prob, NeuralNetDiffEq.NNKolmogorov(m, opt, sdealg, ensemblealg), verbose = true, dt = dt,
dx = dx , trajectories = trajectories , abstol=1e-6, maxiters = 4200 , use_gpu = use_gpu)
dx = dx , trajectories = trajectories , abstol=1e-6, maxiters = 1000 , use_gpu = use_gpu)
```
## Analyzing the solution
Now let us find a Monte-Carlo Solution and plot the both:
```julia
monte_carlo_sol = []
x_out = collect(85:5.00:110.00)
x_out = collect(85:2.00:110.00)
for x in x_out
u₀= [x]
g_val(du , u , p , t) = du .= 0.4.*u
f_val(du , u , p , t) = du .= 0.02.*u
g_val(du , u , p , t) = du .= 0.2.*u
f_val(du , u , p , t) = du .= 0.04.*u
dt = 0.01
tspan = (0.0,1.0)
prob = SDEProblem(f_val,g_val,u₀,tspan)
Expand All @@ -124,11 +124,10 @@ if use_gpu == true
m = fmap(cpu , m)
end
y_out = m(x_model)
y_out = reshape(y_out , 6 , 1)
y_out = reshape(y_out , 13 , 1)
```
And now finally we can plot the solutions
```julia
plot(x_out , y_out , lw = 3 , xaxis="Initial Stock Price", yaxis="Payoff" , label = "NNKolmogorov")
plot!(x_out , monte_carlo_sol , lw = 3 , xaxis="Initial Stock Price", yaxis="Payoff" ,label = "Monte Carlo Solutions")

```