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https://github.com/QuantEcon/SimpleDifferentialOperators.jl

Library for simple upwind finite differences
https://github.com/QuantEcon/SimpleDifferentialOperators.jl

differential-equations julia mathematics

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Library for simple upwind finite differences

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README 

        
# SimpleDifferentialOperators



[![Build Status](https://travis-ci.com/QuantEcon/SimpleDifferentialOperators.jl.svg?branch=master)](https://travis-ci.com/QuantEcon/SimpleDifferentialOperators.jl)

[![Codecov](https://codecov.io/gh/QuantEcon/SimpleDifferentialOperators.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/QuantEcon/SimpleDifferentialOperators.jl)



[![](https://img.shields.io/badge/docs-stable-blue.svg)](https://QuantEcon.github.io/SimpleDifferentialOperators.jl/stable)



## Overview

This is a package to return discretized differential operators subject to various boundary conditions.  It is intended to be a "simple" stopgap as more advanced implementations (e.g. [DiffEqOperators.jl](https://github.com/JuliaDiffEq/DiffEqOperators.jl/) ) mature.  This package is also not intended to provide a "higher-level" interface for constructing the equations.  See [EconPDEs.jl](https://github.com/matthieugomez/EconPDEs.jl/) for a package intended to make translation of the sorts of equations used in economics more direct.



### Example

#### Bellman equation



Consider constructing the corresponding infinitesimal generator for the following stochastic differential equation:



![SDE](https://render.githubusercontent.com/render/math?math=%24dx_t%20%3D%20%5Cmu%20dt%20%2B%20%5Csigma%20dWt%24)



with some constant `μ` and `σ >= 0`, and `W_t` Brownian Motion subject, with reflecting barriers at `x=0` and `x=1`, i.e., `v'(0) = v'(1) = 0`.



If the payoff is in state `x` is `f(x)` and `ρ` is the discount rate, then the bellman equation for the expected present discounted value of payoffs fulfills



![Bellman](https://render.githubusercontent.com/render/math?math=%24%5Crho%20v(x)%20%3D%20f(x)%20%2B%20%5Cpartial_x%20v(x)%20%2B%20%5Cfrac%7B%5Csigma%5E2%7D%7B2%7D%5Cpartial_%7Bxx%7Dv(x)%24)



subject to ![BC](https://render.githubusercontent.com/render/math?math=%24%5Cpartial_x%20v(0)%20%3D%200%2C%5C%2C%20%5Cpartial_x%20v(1)%20%3D%200%20%24)



Written in operator form, define the differential operators



![Operator](https://render.githubusercontent.com/render/math?math=%5Cmathcal%7BL%7D%20%5Cequiv%20%5Crho%20-%20%5Cmu%20%5Cpartial_x%20-%20%5Cfrac%7B%5Csigma%5E2%7D%7B2%7D%5Cpartial_%7Bxx%7D)



then the Bellman equation can be written as



![Bellman with Operator](https://render.githubusercontent.com/render/math?math=%24%5Cmathcal%7BL%7D%20v(x)%20%3D%20f(x)%24)



This package provides components to discretize differential operators.  To implement directly,



```julia

using SimpleDifferentialOperators, LinearAlgebra

f(x) = x^2

μ = -0.1 # constant negative drift

σ = 0.1

ρ = 0.05

M = 100 # size of interior nodes

x̄ = range(0.0, 1.0, length = (M+2)) # extended grid

x = interiornodes(x̄) # interior grid



# discretize L = ρ - μ D_x - σ^2 / 2 D_xx on the interior

# subject to reflecting barriers at 0 and 1

bc = (Reflecting(), Reflecting())

L_x = μ*L₁₋bc(x̄, bc) - σ^2 / 2 * L₂bc(x̄, bc)

L = I * ρ - L_x

## solve the value function on the interior

v = L \ f.(x)

```



To extrapolate the interior solution to the boundary, one can call `extrapolatetoboundary` as follows:



```julia

v̄ = extrapolatetoboundary(x̄, v, bc) 

```



## Documentation



To install, run `] add SimpleDifferentialOperators` on Julia 1.3+.



For more usage information, see the docs badge above.



Detailed derivations and more applications can be found [here](https://github.com/QuantEcon/SimpleDifferentialOperators.jl/releases/download/dev/discretized-differential-operator-derivation.pdf).



## Troubleshooting



* As a reminder, the package requires **Julia 1.1 or later.**



* If you discover a bug in the code or math, please file an issue in this repo with the label "bug."



* The same holds for feature requests, with the appropriate label.