rTrajMou {sdetorus}

Simulation of trajectories for the multivariate OU diffusion

Description

Simulation of trajectories of the multivariate Ornstein–Uhlenbeck (OU) diffusion

dX_t=A(\mu - X_t)dt+\Sigma^\frac{1}{2}dW_t, X_0=x_0

using the exact transition probability density.

Usage

rTrajMou(x0, A, mu, Sigma, N = 100, delta = 0.001)

Arguments

x0

a vector of length p containing initial point.

A

the drift matrix, of size c(p, p).

mu

unconditional mean of the diffusion, a vector of length p.

Sigma

square of the diffusion matrix, a matrix of size c(p, p).

N

number of discretization steps in the resulting trajectory.

delta

time discretization step.

Details

The law of the discretized trajectory at each time step is a multivariate normal with mean meantMou and covariance matrix covtMou. See rTrajOu for the univariate case (more efficient).

solve(A) %*% Sigma has to be a covariance matrix (symmetric and positive definite) in order to have a proper transition density. For the bivariate case, this can be ensured with the alphaToA function. In the multivariate case, it is ensured if Sigma is isotropic and A is a covariance matrix.

Value

A matrix of size c(N + 1, p) containing x0 in the first row and the exact discretized trajectory on the remaining rows.

Examples

set.seed(987658)
data <- rTrajMou(x0 = c(0, 0), A = alphaToA(alpha = c(1, 2, 0.5),
                 sigma = 1:2), mu = c(1, 1), Sigma = diag((1:2)^2),
                 N = 200, delta = 0.1)
plot(data, pch = 19, col = rainbow(201), cex = 0.25)
arrows(x0 = data[-201, 1], y0 = data[-201, 2], x1 = data[-1, 1],
       y1 = data[-1, 2], col = rainbow(201), angle = 10, length = 0.1)