Simulator function for multi-dimensional stochastic processes
Simulate multi-dimensional stochastic processes.
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, ...)
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 samplingsampling: a yuima.sampling-class object.subsampling: a yuima.sampling-class object.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.
yuima-class object.The YUIMA Project Team
In the simulation of multi-variate Levy processes, the values of parameters have to be defined outside of simulate function in advance (see examples below).
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)