diff --git a/docs/pages.jl b/docs/pages.jl index ce1a4bf..5a0650e 100644 --- a/docs/pages.jl +++ b/docs/pages.jl @@ -1,6 +1,7 @@ pages = [ "Home" => "index.md", "Getting started" => "getting_started.md", + "Problems" => "problems.md", "Solver Algorithms" => ["MLP.md", "DeepSplitting.md", "DeepBSDE.md", diff --git a/docs/src/index.md b/docs/src/index.md index 49f6a9d..32c70e0 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -1,8 +1,9 @@ # HighDimPDE.jl -**HighDimPDE.jl** is a Julia package to **solve Highly Dimensional non-linear, non-local PDEs** of the form +**HighDimPDE.jl** is a Julia package to **solve Highly Dimensional non-linear, non-local PDEs** of the forms: +1. Partial Integro Differential Equations: ```math \begin{aligned} (\partial_t u)(t,x) &= \int_{\Omega} f\big(t,x,{\bf x}, u(t,x),u(t,{\bf x}), ( \nabla_x u )(t,x ),( \nabla_x u )(t,{\bf x} ) \big) \, d{\bf x} \\ @@ -12,6 +13,18 @@ where $u \colon [0,T] \times \Omega \to \R$, $\Omega \subseteq \R^d$ is subject to initial and boundary conditions, and where $d$ is large. +2. Parabolic Partial Differential Equations: +```math +\begin{aligned} + (\partial_t u)(t,x) &= f\big(t,x, u(t,x), ( \nabla_x u )(t,x )\big) + + \big\langle \mu(t,x), ( \nabla_x u )( t,x ) \big\rangle + \tfrac{1}{2} \text{Trace} \big(\sigma(t,x) [ \sigma(t,x) ]^* ( \text{Hess}_x u)(t, x ) \big). +\end{aligned} +``` + +where $u \colon [0,T] \times \Omega \to \R$, $\Omega \subseteq \R^d$ is subject to initial and boundary conditions, and where $d$ is large. + +!!! note + The difference between the two problems is that in Partial Integro Differential Equations, the integrand is integrated over **x**, while in Parabolic Integro Differential Equations, the function `f` is just evaluated for `x`. **HighDimPDE.jl** implements solver algorithms that break down the curse of dimensionality, including diff --git a/docs/src/problems.md b/docs/src/problems.md new file mode 100644 index 0000000..a10555c --- /dev/null +++ b/docs/src/problems.md @@ -0,0 +1,8 @@ +```@docs +PIDEProblem +ParabolicPDEProblem +``` + +!!! note + While choosing to define a PDE using `PIDEProblem`, not that the function being integrated `f` is a function of `f(x, y, v_x, v_y, ∇v_x, ∇v_y)` out of which `y` is the integrating variable and `x` is constant throughout the integration. + If a PDE has no integral and the non linear term `f` is just evaluated as `f(x, v_x, ∇v_x)` then we suggest using `ParabolicPDEProblem` \ No newline at end of file diff --git a/docs/src/tutorials/nnstopping.md b/docs/src/tutorials/nnstopping.md index 54fb78a..25edad2 100644 --- a/docs/src/tutorials/nnstopping.md +++ b/docs/src/tutorials/nnstopping.md @@ -1,3 +1,5 @@ +# `NNStopping` + ## Solving for optimal strategy and expected payoff of a Bermudan Max-Call option We will calculate optimal strategy for Bermudan Max-Call option with following drift, diffusion and payoff: