https://github.com/matthieugomez/InfinitesimalGenerators.jl

A set of tools to work with Markov Processes
https://github.com/matthieugomez/InfinitesimalGenerators.jl

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A set of tools to work with Markov Processes

• Host: GitHub
• URL: https://github.com/matthieugomez/InfinitesimalGenerators.jl
• Owner: matthieugomez
• License: other
• Created: 2019-04-03T20:06:22.000Z (over 5 years ago)
• Default Branch: main
• Last Pushed: 2024-06-22T14:12:24.000Z (5 months ago)
• Last Synced: 2024-06-23T05:49:30.421Z (5 months ago)
• Language: Julia
• Homepage:
• Size: 290 KB
• Stars: 9
• Watchers: 4
• Forks: 2
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE.md

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• awesome-sciml - matthieugomez/InfinitesimalGenerators.jl: A set of tools to work with Markov Processes

README

        
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This package provides a set of tools to work with Markov Processes defined on a 1-dimensional grid

# Markov Processes

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[∫_0^T e^{-∫_0^t v(x_s)ds}f(x_t)dt +  e^{-∫_0^T v(x_s)ds}ψ(x_T) | x_0 = x]

feynman_kac(MX, 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`

- `generator(M)` returns its associated generator (i.e. the operator `f -> ∂_tE[e^{m}f(x_t)|x_0=x]`)

- `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 also allows you to compute (lazy) first and second derivatives of a function on a grid using a finite difference schemes

```julia

using InfinitesimalGenerators

x = range(-1, 1, length = 100)

f = sin.(x)

FirstDerivative(x, f, direction = :upward, bc = (0, 0))

FirstDerivative(x, f, direction = :downward, bc = (0, 0))

SecondDerivative(x, f, bc = (0, 0))

```

The argument `bc` refers to the value of the first-derivative at each limit of the grid. This argument defaults to zero, which is the right condition when solving problems with reflecting boundaries.

## 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.