yuima (version 1.1.6)

simulate: Simulator function for multi-dimensional stochastic processes

Description

Simulate multi-dimensional stochastic processes.

Usage

simulate(object, nsim=1, seed=NULL, xinit, true.parameter, space.discretized = FALSE,  increment.W = NULL, increment.L = NULL, method = "euler", hurst, methodfGn = "WoodChan", 	sampling=sampling, subsampling=subsampling, ...)

Arguments

object

an yuima-class, yuima.model-class or yuima.carma-class object.

xinit

initial value vector of state variables.

true.parameter

named list of parameters.

space.discretized

flag to switch to space-discretized Euler Maruyama method.

increment.W

to specify Wiener increment for each time tics in advance.

increment.L

to specify Levy increment for each time tics in advance.

method

string Variable for simulation scheme. The default value method=euler uses the euler discretization for the simulation of a sample path.

nsim

Not used yet. Included only to match the standard genenirc in package stats.

seed

Not used yet. Included only to match the standard genenirc in package stats.

hurst

value of Hurst parameter for simulation of the fGn. Overrides the specified hurst slot.

methodfGn

simulation methods for fractional Gaussian noise.

...

passed to setSampling to create a sampling

sampling

a yuima.sampling-class object.

subsampling

a yuima.sampling-class object.

Value

Details

simulate is a function to solve SDE using the Euler-Maruyama method. This function supports usual Euler-Maruyama method for multidimensional SDE, and space discretized Euler-Maruyama method for one dimensional SDE.

It simulates solutions of stochastic differential equations with Gaussian noise, fractional Gaussian noise awith/without jumps.

If a yuima-class object is passed as input, then the sampling information is taken from the slot sampling of the object. If a yuima.carma-class object, a yuima.model-class object or a yuima-class object with missing sampling slot is passed as input the sampling argument is used. If this argument is missing then the sampling structure is constructed from Initial, Terminal, etc. arguments (see setSampling for details on how to use these arguments).

For a COGARCH(p,q) process setting method=mixed implies that the simulation scheme is based on the solution of the state space process. For the case in which the underlying noise is a compound poisson Levy process, the trajectory is build firstly by simulation of the jump time, then the quadratic variation and the increments noise are simulated exactly at jump time. For the others Levy process, the simulation scheme is based on the discretization of the state space process solution.

Examples

set.seed(123)

# Path-simulation for 1-dim diffusion process. 
# dXt = -0.3*Xt*dt + dWt
mod <- setModel(drift="-0.3*y", diffusion=1, solve.variable=c("y"))
str(mod)

# Set the model in an `yuima' object with a sampling scheme. 
T <- 1
n <- 1000
samp <- setSampling(Terminal=T, n=n)
ou <- setYuima(model=mod, sampling=samp)

# Solve SDEs using Euler-Maruyama method. 
par(mfrow=c(3,1))
ou <- simulate(ou, xinit=1)
plot(ou)


set.seed(123)
ouB <- simulate(mod, xinit=1,sampling=samp)
plot(ouB)


set.seed(123)
ouC <- simulate(mod, xinit=1, Terminal=1, n=1000)
plot(ouC)

par(mfrow=c(1,1))


# Path-simulation for 1-dim diffusion process. 
# dXt = theta*Xt*dt + dWt
mod1 <- setModel(drift="theta*y", diffusion=1, solve.variable=c("y"))
str(mod1)
ou1 <- setYuima(model=mod1, sampling=samp)

# Solve SDEs using Euler-Maruyama method. 
ou1 <- simulate(ou1, xinit=1, true.p = list(theta=-0.3))
plot(ou1)

## Not run: 
# 
# # A multi-dimensional (correlated) diffusion process. 
# # To describe the following model: 
# # X=(X1,X2,X3); dXt = U(t,Xt)dt + V(t)dWt
# # For drift coeffcient
# U <- c("-x1","-2*x2","-t*x3")
# # For diffusion coefficient of X1 
# v1 <- function(t) 0.5*sqrt(t)
# # For diffusion coefficient of X2
# v2 <- function(t) sqrt(t)
# # For diffusion coefficient of X3
# v3 <- function(t) 2*sqrt(t)
# # correlation
# rho <- function(t) sqrt(1/2)
# # coefficient matrix for diffusion term
# V <- matrix( c( "v1(t)",
#                 "v2(t) * rho(t)",
#                 "v3(t) * rho(t)",
#                 "",
#                 "v2(t) * sqrt(1-rho(t)^2)",
#                 "",
#                 "",
#                 "",
#                 "v3(t) * sqrt(1-rho(t)^2)" 
#                ), 3, 3)
# # Model sde using "setModel" function
# cor.mod <- setModel(drift = U, diffusion = V,
# state.variable=c("x1","x2","x3"), 
# solve.variable=c("x1","x2","x3") )
# str(cor.mod)
# 
# # Set the `yuima' object. 
# cor.samp <- setSampling(Terminal=T, n=n)
# cor <- setYuima(model=cor.mod, sampling=cor.samp)
# 
# # Solve SDEs using Euler-Maruyama method. 
# set.seed(123)
# cor <- simulate(cor)
# plot(cor)
# 
# # A non-negative process (CIR process)
# # dXt= a*(c-y)*dt + b*sqrt(Xt)*dWt
#  sq <- function(x){y = 0;if(x>0){y = sqrt(x);};return(y);}
#  model<- setModel(drift="0.8*(0.2-x)",
#   diffusion="0.5*sq(x)",solve.variable=c("x"))
#  T<-10
#  n<-1000
#  sampling <- setSampling(Terminal=T,n=n)
#  yuima<-setYuima(model=model, sampling=sampling)
#  cir<-simulate(yuima,xinit=0.1)
#  plot(cir)
# 
# # solve SDEs using Space-discretized Euler-Maruyama method
# v4 <- function(t,x){
#   return(0.5*(1-x)*sqrt(t))
# }
# mod_sd <- setModel(drift = c("0.1*x1", "0.2*x2"),
#                      diffusion = c("v1(t)","v4(t,x2)"),
#                      solve.var=c("x1","x2")
#                      )
# samp_sd <- setSampling(Terminal=T, n=n)
# sd <- setYuima(model=mod_sd, sampling=samp_sd)
# sd <- simulate(sd, xinit=c(1,1), space.discretized=TRUE)
# plot(sd)
# 
# 
# ## example of simulation by specifying increments
# ## Path-simulation for 1-dim diffusion process
# ## dXt = -0.3*Xt*dt + dWt
# 
# mod <- setModel(drift="-0.3*y", diffusion=1,solve.variable=c("y"))
# str(mod)
# 
# ## Set the model in an `yuima' object with a sampling scheme. 
# Terminal <- 1
# n <- 500
# mod.sampling <- setSampling(Terminal=Terminal, n=n)
# yuima.mod <- setYuima(model=mod, sampling=mod.sampling)
# 
# ##use original increment
# delta <- Terminal/n
# my.dW <- rnorm(n * yuima.mod@[e-mail], 0, sqrt(delta))
# my.dW <- t(matrix(my.dW, nrow=n, ncol=yuima.mod@[e-mail]))
# 
# ## Solve SDEs using Euler-Maruyama method.
# yuima.mod <- simulate(yuima.mod,
#                       xinit=1,
#                       space.discretized=FALSE,
#                       increment.W=my.dW)
# if( !is.null(yuima.mod) ){
#  dev.new()
#  # x11()
#   plot(yuima.mod)
# }
# 
# ## A multi-dimensional (correlated) diffusion process. 
# ## To describe the following model: 
# ## X=(X1,X2,X3); dXt = U(t,Xt)dt + V(t)dWt
# ## For drift coeffcient
# U <- c("-x1","-2*x2","-t*x3")
# ## For process 1
# diff.coef.1 <- function(t) 0.5*sqrt(t)
# ## For process 2
# diff.coef.2 <- function(t) sqrt(t)
# ## For process 3
# diff.coef.3 <- function(t) 2*sqrt(t)
# ## correlation
# cor.rho <- function(t) sqrt(1/2)
# ## coefficient matrix for diffusion term
# V <- matrix( c( "diff.coef.1(t)",
#                "diff.coef.2(t) * cor.rho(t)",
#                "diff.coef.3(t) * cor.rho(t)",
#                "",
#                "diff.coef.2(t)",
#                "diff.coef.3(t) * sqrt(1-cor.rho(t)^2)",
#                "diff.coef.1(t) * cor.rho(t)",
#                "",
#                "diff.coef.3(t)" 
#                ), 3, 3)
# ## Model sde using "setModel" function
# cor.mod <- setModel(drift = U, diffusion = V,
#                     solve.variable=c("x1","x2","x3") )
# str(cor.mod)
# ## Set the `yuima' object.
# set.seed(123)
# obj.sampling <- setSampling(Terminal=Terminal, n=n)
# yuima.obj <- setYuima(model=cor.mod, sampling=obj.sampling)
# 
# ##use original dW
# my.dW <- rnorm(n * yuima.obj@[e-mail], 0, sqrt(delta))
# my.dW <- t(matrix(my.dW, nrow=n, ncol=yuima.obj@[e-mail]))
# 
# ## Solve SDEs using Euler-Maruyama method.
# yuima.obj.path <- simulate(yuima.obj, space.discretized=FALSE, 
#  increment.W=my.dW)
# if( !is.null(yuima.obj.path) ){
#   dev.new()
# #  x11()
#   plot(yuima.obj.path)
# }
# 
# 
# ##:: sample for Levy process ("CP" type)
# ## specify the jump term as c(x,t)dz
# obj.model <- setModel(drift=c("-theta*x"), diffusion="sigma",
# jump.coeff="1", measure=list(intensity="1", df=list("dnorm(z, 0, 1)")),
# measure.type="CP", solve.variable="x")
# 
# ##:: Parameters
# lambda <- 3
# theta <- 6
# sigma <- 1
# xinit <- runif(1)
# N <- 500
# h <- N^(-0.7)
# eps <- h/50
# n <- 50*N
# T <- N*h
# 
# set.seed(123)
# obj.sampling <- setSampling(Terminal=T, n=n)
# obj.yuima <- setYuima(model=obj.model, sampling=obj.sampling)
# X <- simulate(obj.yuima, xinit=xinit, true.parameter=list(theta=theta, sigma=sigma))
# dev.new()
# plot(X)
# 
# 
# ##:: sample for Levy process ("CP" type)
# ## specify the jump term as c(x,t,z)
# ## same plot as above example
# obj.model <- setModel(drift=c("-theta*x"), diffusion="sigma",
# jump.coeff="z", measure=list(intensity="1", df=list("dnorm(z, 0, 1)")),
# measure.type="CP", solve.variable="x")
# 
# set.seed(123)
# obj.sampling <- setSampling(Terminal=T, n=n)
# obj.yuima <- setYuima(model=obj.model, sampling=obj.sampling)
# X <- simulate(obj.yuima, xinit=xinit, true.parameter=list(theta=theta, sigma=sigma))
# dev.new()
# plot(X)
# 
# 
# 
# 
# ##:: sample for Levy process ("code" type)
# ## dX_{t} = -x dt + dZ_t
# obj.model <- setModel(drift="-x", xinit=1, jump.coeff="1", measure.type="code", 
# measure=list(df="rIG(z, 1, 0.1)"))
# obj.sampling <- setSampling(Terminal=10, n=10000)
# obj.yuima <- setYuima(model=obj.model, sampling=obj.sampling)
# result <- simulate(obj.yuima)
# dev.new()
# plot(result)
# 
# ##:: sample for multidimensional Levy process ("code" type)
# ## dX = (theta - A X)dt + dZ,
# ##    theta=(theta_1, theta_2) = c(1,.5)
# ##    A=[a_ij], a_11 = 2, a_12 = 1, a_21 = 1, a_22=2
# require(yuima)
# x0 <- c(1,1)
# beta <- c(.1,.1)
# mu <- c(0,0)
# delta0 <- 1
# alpha <- 1
# Lambda <- matrix(c(1,0,0,1),2,2)
# cc <- matrix(c(1,0,0,1),2,2)
# obj.model <- setModel(drift=c("1 - 2*x1-x2",".5-x1-2*x2"), xinit=x0,
# solve.variable=c("x1","x2"), jump.coeff=cc, measure.type="code",
#  measure=list(df="rNIG(z, alpha, beta, delta0, mu, Lambda)"))
# obj.sampling <- setSampling(Terminal=10, n=10000)
# obj.yuima <- setYuima(model=obj.model, sampling=obj.sampling)
# result <- simulate(obj.yuima,true.par=list( alpha=alpha, 
#  beta=beta, delta0=delta0, mu=mu, Lambda=Lambda))
# plot(result)
# 
# 
# # Path-simulation for a Carma(p=2,q=1) model driven by a Brownian motion:
# carma1<-setCarma(p=2,q=1)
# str(carma1)
# 
# # Set the sampling scheme
# samp<-setSampling(Terminal=100,n=10000)
# 
# # Set the values of the model parameters
# par.carma1<-list(b0=1,b1=2.8,a1=2.66,a2=0.3)
# 
# set.seed(123)
# sim.carma1<-simulate(carma1,
#                      true.parameter=par.carma1,
#                      sampling=samp)
# 
# plot(sim.carma1)
# 
# 
# 
# # Path-simulation for a Carma(p=2,q=1) model driven by a Compound Poisson process.
# carma1<-setCarma(p=2,
#                  q=1,
#                  measure=list(intensity="1",df=list("dnorm(z, 0, 1)")),
#                  measure.type="CP")
# 
# # Set Sampling scheme
# samp<-setSampling(Terminal=100,n=10000)
# 
# # Fix carma parameters
# par.carma1<-list(b0=1,
#                  b1=2.8,
#                  a1=2.66,
#                  a2=0.3)
# 
# set.seed(123)
# sim.carma1<-simulate(carma1,
#                      true.parameter=par.carma1,
#                      sampling=samp)
# 
# plot(sim.carma1)
# ## End(Not run)