IntegratedWienerProcesses

A package implementing tools for working with integrated Wiener processes. A $\nu$-times integrated wiener process on the interval $[0, T]$ is given by

$$ y(t) = \sum_{k=0}^\nu y^{(k)}(0) \frac{t^k}{k!} + \int_0^t \frac{(t-\tau)^\nu}{\nu!} \mathrm{d} w(t), $$

where $y^{(k)}$ is the $k$:th derivative of $y \in \mathbb{R}^d$ and $w$ is a standard Wiener process on $\mathbb{R}^d$. Integrated Wiener processes admit state-space realizations according to

$$ \begin{align} \mathrm{d} x(t) &= A x(t) \mathrm{d} t + B \mathrm{d} w(t), \\ y(t) &= C x(t). \end{align} $$

The state vector is Gauss-Markov with transition distribution given by

$$ x(t+s) \mid x(t) \sim \mathcal{N}( e^{A s} x(t), Q(s) ), $$

where $e^{As}$ is the transition matrix and $Q(s)$ is the process noise covariance. Since $Q(s)$ is positive definite it admits a Cholesky factorization $Q(s) = L(s) L^*(s)$. This pacakge implements functionality to represent integrated Wiener processes and work with their state-space realization.

Functionality

The package defines the following type for representing $\nu$-times integrated Wiener processes.

IWP{T,N} where {T<:Real,N<:Integer}

An instance may be constructed according to

IWP{T}(ndiff::Integer, dim::Integer) # ndiff times differentiable IWP of dimension dim and element type T
IWP(ndiff::Integer, dim::Integer)    # same as IWP{Float64}(ndiff, dim)

Additionally, the following functions are implemented.

ndiff(M::IWP) # number of times M can be differentiated 
dim(M::IWP)   # dimension of M 
ssparams_reverse(M::IWP{T}) # computes the matrices A, B, C when the state vector is the Taylor coefficients in reverse order
state2diff_matrix_reverse(M::IWP{T}, m::Inter) # computes a matrix such that when mutiplied with the state vector the mth derivative is obtained
transition_parameters_cholf_reverse(M::IWP{T}, dt::T) # computes the transition matrix and the Cholesky factor of the process nosie covariance