alphaToA {sdetorus} | R Documentation |
Valid drift matrices for the Ornstein–Uhlenbeck diffusion in 2D
Description
Constructs drift matrices A
such that
solve(A) %*% Sigma
is symmetric.
Usage
alphaToA(alpha, sigma = NULL, rho = 0, Sigma = NULL)
aToAlpha(A, sigma = NULL, rho = 0, Sigma = NULL)
Arguments
alpha |
vector of length |
sigma |
vector of length |
rho |
correlation of |
Sigma |
the diffusion matrix of size |
A |
matrix of size |
Details
The parametrization enforces that solve(A) %*% Sigma
is symmetric. Positive definiteness is guaranteed if alpha[3]^2 <
rho^2 * (alpha[1] - alpha[2])^2 / 4 + alpha[1] * alpha[2]
.
Value
The drift matrix A
or the alpha
vector.
Examples
# Parameters
alpha <- 3:1
Sigma <- rbind(c(1, 0.5), c(0.5, 4))
# Covariance matrix
A <- alphaToA(alpha = alpha, Sigma = Sigma)
S <- 0.5 * solve(A) %*% Sigma
det(S)
# Check
aToAlpha(A = alphaToA(alpha = alpha, Sigma = Sigma), Sigma = Sigma)
alphaToA(alpha = aToAlpha(A = A, Sigma = Sigma), Sigma = Sigma)
[Package sdetorus version 0.1.10 Index]