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docs: update docs
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ashutosh-b-b committed Feb 15, 2024
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8 changes: 7 additions & 1 deletion docs/src/NNKolmogorov.md
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Expand Up @@ -8,11 +8,17 @@ Modules = [HighDimPDE]
Pages = ["NNKolmogorov.jl"]
```

`NNKolmogorov` obtains a terminal solution for Backward Kolmogorov Equations of the form:
`NNKolmogorov` obtains a
- terminal solution for Forward Kolmogorov Equations of the form:
```math
\partial_t u(t,x) = \mu(t, x) \nabla_x u(t,x) + \frac{1}{2} \sigma^2(t, x) \Delta_x u(t,x)
```
with initial condition given by `g(x)`
- or an initial condition for Backward Kolmogorov Equations of the form:
```math
\partial_t u(t,x) = - \mu(t, x) \nabla_x u(t,x) - \frac{1}{2} \sigma^2(t, x) \Delta_x u(t,x)
```
with terminal condition given by `g(x)`

We can use the Feynman-Kac formula :
```math
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8 changes: 7 additions & 1 deletion docs/src/NNParamKolmogorov.md
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Expand Up @@ -8,11 +8,17 @@ Modules = [HighDimPDE]
Pages = ["NNParamKolmogorov.jl"]
```

`NNParamKolmogorov` obtains a terminal solution for parametric families of Backward Kolmogorov Equations of the form:
`NNParamKolmogorov` obtains a
- terminal solution for parametric families of Forward Kolmogorov Equations of the form:
```math
\partial_t u(t,x) = \mu(t, x, γ_mu) \nabla_x u(t,x) + \frac{1}{2} \sigma^2(t, x, γ_sigma) \Delta_x u(t,x)
```
with initial condition given by `g(x, γ_phi)`
- or an initial condition for parametric families of Backward Kolmogorov Equations of the form:
```math
\partial_t u(t,x) = - \mu(t, x) \nabla_x u(t,x) - \frac{1}{2} \sigma^2(t, x) \Delta_x u(t,x)
```
with terminal condition given by `g(x, γ_phi)`

We can use the Feynman-Kac formula :
```math
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2 changes: 1 addition & 1 deletion src/NNKolmogorov.jl
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@@ -1,5 +1,5 @@
"""
Algorithm for solving Backward Kolmogorov Equations.
Algorithm for solving Kolmogorov Equations.
```julia
HighDimPDE.NNKolmogorov(chain, opt)
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2 changes: 1 addition & 1 deletion src/NNParamKolmogorov.jl
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@@ -1,5 +1,5 @@
"""
Algorithm for solving Backward Kolmogorov Equations.
Algorithm for solving paramateric families of Kolmogorov Equations.
```julia
HighDimPDE.NNKolmogorov(chain, opt)
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