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

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.

Value

Note

There may be missing information in the simulate description. Please contribute with suggestions and fixings.

Author(s)

The YUIMA Project Team

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@model@noise.number, 0, sqrt(delta))
my.dW <- t(matrix(my.dW, nrow=n, ncol=yuima.mod@model@noise.number))

## 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@model@noise.number, 0, sqrt(delta))
my.dW <- t(matrix(my.dW, nrow=n, ncol=yuima.obj@model@noise.number))

## 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)

Results