ae function

Asymptotic Expansion

Asymptotic expansion of uni-dimensional and multi-dimensional diffusion processes.

ae(
  model,
  xinit,
  order = 1L,
  true.parameter = list(),
  sampling = NULL,
  eps.var = "eps",
  solver = "rk4",
  verbose = FALSE
)

Arguments

• model: an object of yuima-class or yuima.model-class.
• xinit: initial value vector of state variables.
• order: integer. The asymptotic expansion order. Higher orders lead to better approximations but longer computational times.
• true.parameter: named list of parameters.
• sampling: a yuima.sampling-class object.
• eps.var: character. The perturbation variable.
• solver: the solver for ordinary differential equations. One of "rk4" (more accurate) or "euler" (faster).
• verbose: logical. Print on progress? Default FALSE.

Returns

An object of yuima.ae-class

Details

If sampling is not provided, then model must be an object of yuima-class with non-empty sampling.

if eps.var does not appear in the model specification, then it is internally added in front of the diffusion matrix to apply the asymptotic expansion scheme.

Examples

## Not run:

# model
gbm <- setModel(drift = 'mu*x', diffusion = 'sigma*x', solve.variable = 'x')

# settings
xinit <- 100
par <- list(mu = 0.01, sigma = 0.2)
sampling <- setSampling(Initial = 0, Terminal = 1, n = 1000)

# asymptotic expansion
approx <- ae(model = gbm, sampling = sampling, order = 4, true.parameter = par, xinit = xinit)

# exact density
x <- seq(50, 200, by = 0.1)
exact <- dlnorm(x = x, meanlog = log(xinit)+(par$mu-0.5*par$sigma^2)*1, sdlog = par$sigma*sqrt(1))

# compare
plot(x, exact, type = 'l', ylab = "Density")
lines(x, aeDensity(x = x, ae = approx, order = 1), col = 2)
lines(x, aeDensity(x = x, ae = approx, order = 2), col = 3)
lines(x, aeDensity(x = x, ae = approx, order = 3), col = 4)
lines(x, aeDensity(x = x, ae = approx, order = 4), col = 5)
## End(Not run)

Author(s)

Emanuele Guidotti emanuele.guidotti@unine.ch