From ca88d00ed63d6e64be48b8abf2ad149637964d6d Mon Sep 17 00:00:00 2001
From: Matthieu Gomez <gomez.matthieu@gmail.com>
Date: Wed, 12 Jun 2024 13:18:55 -0400
Subject: [PATCH] Update README.md

---
 README.md | 44 +++++++++++++++++++++++++++++++++++++++++---
 1 file changed, 41 insertions(+), 3 deletions(-)

diff --git a/README.md b/README.md
index 537ea9b..8cf0c9f 100644
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+++ b/README.md
@@ -1,9 +1,31 @@
 [![Build status](https://github.com/matthieugomez/InfinitesimalGenerators.jl/workflows/CI/badge.svg)](https://github.com/matthieugomez/InfinitesimalGenerators.jl/actions)
 
+This package provides a set of tools to work with Markov Processes defined on a 1-dimensional Gri
+
+
+
 # Markov Processes
-- `X = DiffusionProcess(x::AbstractVector, μ::AbstractVector, σ::AbstractVector)` creates the discretized Markov Process with drift `μ` and volatility `σ`, on a grid `x` with reflecting boundaries.
-- `generator(X)` returns its associated generator (i.e. the operator `f -> ∂_tE[f(x_t)|x_0=x]`)
-- `stationary_distribution(X)` returns its stationary distribution (i.e. the positive vector `g` such that `g * generator(X) = 0`)
+The package allows you to compute compute expectations involving Markov processes
+
+```julia
+using InfinitesimalGenerators
+# Create a diffusion process (here,  the Ornstein–Uhlenbeck dx = -0.03 * x * dt + 0.01 * dZ_t)
+# Note that the package assumes reflecting boundaries at the limits
+x = range(-1, 1, length = 100)
+μx = .- 0.03 .* x
+σx = 0.01 .* ones(length(x))
+X = DiffusionProcess(x, μx, σx)
+
+
+# Return its stationary distribution 
+g = stationary_distribution(X)
+
+# Return the associated generator as a matrix (i.e. the operator `f -> ∂_tE[f(x_t)|x_0=x]`)
+MX = generator(X)
+
+# Use the generator to compute E[\int_0^T e^{-\int_0^t f(x_s)ds}f(x_s) +  e^{-\int_0^T v(x_s)ds}ψ(x_T) | x_0 = x]
+feynman_kac(MX; t = range(0, 100, step = 1/12), f = zeros(length(x)),  ψ = ones(length(x)), v = zeros(length(x)))
+```
 
 # Additive Functionals
 - `M = AdditiveFunctional(X, μm, σm)` creates, given a discretized Markov Process, the Additive Functional with drift  `μm` and volatility `σm`
@@ -11,5 +33,21 @@
 - `cgf(m)` returns the long run scaled CGF of `m` 
 - `tail_index(m)` returns the tail index of the stationary distribution of `e^m`
 
+
+
+# Derivative
+The package allows you to compute (lazyly) the first and second derivatives on a grid through finite difference schemes
+
+```
+using InfinitesimalGenerators
+x = range(-1, 1, length = 100)
+f = sin.(x)
+FirstDerivative(x, sin, direction = :upward, bc = (0, 0))
+FirstDerivative(x, sin, direction = :downward, bc = (0, 0))
+SecondDerivative(x, sin, bc = (0, 0))
+```
+The argument `bc` refers to the value of the first-derivative at the limit
+
+
 ## Related Packages
 - [SimpleDifferentialOperators](https://github.com/QuantEcon/SimpleDifferentialOperators.jl) contains more general tools to define operators with different boundary counditions. In contrast, InfinitesimalGenerators always assumes reflecting boundaries.