Sim.DiffProc (version 4.9)

MEM.sde: Moment Equations Methods for SDE's

Description

Calculate and numerical approximation of moment equations (Symbolic ODE's of means and variances-covariance) at any time for SDE's (1,2 and 3 dim) for the two cases Ito and Stratonovich interpretations.

Usage

MEM.sde(drift, diffusion, ...)
# S3 method for default
MEM.sde(drift, diffusion, corr = NULL, type = c("ito", "str"), solve = FALSE, 
      parms = NULL, init = NULL, time = NULL, ...)
					
# S3 method for MEM.sde
summary(object, at , ...)

Value

Symbolic ODE's of means and variances-covariance. If solve=TRUE approximate the moment of SDE's at any time.

Arguments

drift

drift coefficient: an expression 1-dim(t,x), 2-dim(t,x,y) or 3-dim(t,x,y,z).

diffusion

diffusion coefficient: an expression 1-dim(t,x), 2-dim(t,x,y) or 3-dim(t,x,y,z).

corr

the correlation coefficient '|corr|<=1' of W1(t) and W2(t) (2d) must be an expression length equal 1. And for 3d (W1(t),W2(t),W3(t)) an expressions length equal 3. See examples.

type

type of process "ito" or "Stratonovich"; the default type="ito".

solve

if solve=TRUE solves a system of ordinary differential equations.

parms

parameters passed to drift and diffusion.

init

the initial (state) values for the ODE system. for 1-dim (m=x0,S=0), 2-dim (m1=x0,m2=y0,S1=0,S2=0,C12=0) and for 3-dim (m1=x0,m2=y0,m3=z0,S1=0,S2=0,S3=0,C12=0,C13=0,C23=0), see examples.

time

time sequence (vector) for which output is wanted; the first value of time must be the initial time.

object, at

an object inheriting from class "MEM.sde" and summaries at any time at.

...

potentially arguments to be passed to methods, such as ode for solver for ODE's.

Author

A.C. Guidoum, K. Boukhetala.

Details

The stochastic transition is approximated by the moment equations, and the numerical treatment is required to solve these equations from above with given initial conditions.

An overview of this package, see browseVignettes('Sim.DiffProc') for more informations.

References

Guidoum AC, Boukhetala K (2020). "Performing Parallel Monte Carlo and Moment Equations Methods for Itô and Stratonovich Stochastic Differential Systems: R Package Sim.DiffProc". Journal of Statistical Software, 96(2), 1--82. doi:10.18637/jss.v096.i02

Rodriguez R, Tuckwell H (2000). A dynamical system for the approximate moments of nonlinear stochastic models of spiking neurons and networks. Mathematical and Computer Modelling, 31(4), 175--180.

Alibrandi U, Ricciardi G (2012). Stochastic Methods in Nonlinear Structural Dynamics, 3--60. Springer Vienna, Vienna. ISBN 978-3-7091-1306-6.

See Also

MCM.sde Monte-Carlo methods for SDE's.

Examples

library(deSolve)
## Example 1: 1-dim
## dX(t) = mu * X(t) * dt + sigma * X(t) * dW(t)
## Symbolic ODE's of mean and variance
f <- expression(mu*x)
g <- expression(sigma*x)
res1 <- MEM.sde(drift=f,diffusion=g,type="ito")
res2 <- MEM.sde(drift=f,diffusion=g,type="str")
res1
res2
## numerical approximation of mean and variance
para <- c(mu=2,sigma=0.5)
t    <- seq(0,1,by=0.001)
init <- c(m=1,S=0)
res1 <- MEM.sde(drift=f,diffusion=g,solve=TRUE,init=init,parms=para,time=t)
res1
matplot.0D(res1$sol.ode,main="Mean and Variance of X(t), type Ito")
plot(res1$sol.ode,select=c("m","S"))
## approximation at time = 0.75
summary(res1,at=0.75)

##
res2 <- MEM.sde(drift=f,diffusion=g,solve=TRUE,init=init,parms=para,time=t,type="str")
res2
matplot.0D(res2$sol.ode,main="Mean and Variance of X(t), type Stratonovich")
plot(res2$sol.ode,select=c("m","S"))
## approximation at time = 0.75
summary(res2,at=0.75)

## Comparison:

plot(res1$sol.ode, res2$sol.ode,ylab = c("m(t)"),select="m",xlab = "Time",
     col = c("red", "blue"))
plot(res1$sol.ode, res2$sol.ode,ylab = c("S(t)"),select="S",xlab = "Time",
     col = c("red", "blue"))
	 
## Example2: 2-dim
## dX(t) = 1/mu*(theta-X(t)) dt + sqrt(sigma) * dW1(t),
## dY(t) = X(t) dt + 0 * dW2(t)	 
if (FALSE) {
para=c(mu=0.75,sigma=0.1,theta=2)
init=c(m1=0,m2=0,S1=0,S2=0,C12=0)
t <- seq(0,10,by=0.001)
f <- expression(1/mu*(theta-x), x)  
g <- expression(sqrt(sigma),0)
res2d <- MEM.sde(drift=f,diffusion=g,solve=TRUE,init=init,parms=para,time=t)
res2d

## Exact moment

mu=0.75;sigma=0.1;theta=2;x0=0;y0=0
E_x <- function(t) theta+(x0-theta)*exp(-t/mu)
V_x <- function(t) 0.5*sigma*mu *(1-exp(-2*(t/mu)))
E_y <- function(t) y0+theta*t+(x0-theta)*mu*(1-exp(-t/mu))
V_y <- function(t) sigma*mu^3*((t/mu)-2*(1-exp(-t/mu))+0.5*(1-exp(-2*(t/mu))))
cov_xy <- function(t) 0.5*sigma*mu^2 *(1-2*exp(-t/mu)+exp(-2*(t/mu)))

## 
summary(res2d,at=5)
E_x(5);E_y(5);V_x(5);V_y(5);cov_xy(5)

matplot.0D(res2d$sol.ode,select=c("m1"))
curve(E_x,add=TRUE,col="red") 

## plot

plot(res2d$sol.ode)
matplot.0D(res2d$sol.ode,select=c("S1","S2","C12"))   
plot(res2d$sol.ode[,"m1"], res2d$sol.ode[,"m2"], xlab = "m1(t)",
  ylab = "m2(t)", type = "l",lwd = 2)
hist(res2d$sol.ode,select=c("m1","m2"), col = c("darkblue", "red", "orange", "black"))

## Example3: 2-dim with correlation
## Heston model
## dX(t) = mu*X(t) dt + sqrt(Y(t))*X(t) * dW1(t),
## dY(t) = lambda*(theta-Y(t)) dt + sigma*sqrt(Y(t)) * dW2(t)	
## with E(dw1dw2)=rho


f <- expression( mu*x, lambda*(theta-y) )
g <- expression( sqrt(y)*x, sigma*sqrt(y) )
RHO  <- expression(rho)
res2d <- MEM.sde(drift=f,diffusion=g,corr=RHO)
res2d

## Numerical approximation
RHO <- expression(0.5)
para=c(mu=1,lambda=3,theta=0.5,sigma=0.1)
ini=c(m1=10,m2=2,S1=0,S2=0,C12=0)
res2d = MEM.sde(drift=f,diffusion=g,solve=TRUE,parms=para,init=ini,time=seq(0,1,by=0.01))
res2d

matplot.0D(res2d$sol.ode,select=c("m1","m2")) 
matplot.0D(res2d$sol.ode,select=c("S1","S2","C12")) 
	 
## Example4: 3-dim
## dX(t) = sigma*(Y(t)-X(t)) dt + 0.1 * dW1(t)
## dY(t) = (rho*X(t)-Y(t)-X(t)*Z(t)) dt + 0.1 * dW2(t)
## dZ(t) = (X(t)*Y(t)-bet*Z(t)) dt + 0.1 * dW3(t)
## with E(dw1dw2)=rho1, E(dw1dw3)=rho2 and E(dw2dw3)=rho3


f <- expression(sigma*(y-x),rho*x-y-x*z,x*y-bet*z)
g <- expression(0.1,0.1,0.1)
RHO <- expression(rho1,rho2,rho3)
## Symbolic moments equations
res3d = MEM.sde(drift=f,diffusion=g,corr=RHO)
res3d

## Numerical approximation
RHO <- expression(0.5,0.2,-0.7)
para=c(sigma=10,rho=28,bet=8/3)
ini=c(m1=1,m2=1,m3=1,S1=0,S2=0,S3=0,C12=0,C13=0,C23=0)
res3d = MEM.sde(drift=f,diffusion=g,solve=T,parms=para,init=ini,time=seq(0,1,by=0.01))
res3d

summary(res3d,at=0.25)
summary(res3d,at=0.50)
summary(res3d,at=0.75)

plot(res3d$sol.ode)
matplot.0D(res3d$sol.ode,select=c("m1","m2","m3")) 
matplot.0D(res3d$sol.ode,select=c("S1","S2","S3")) 
matplot.0D(res3d$sol.ode,select=c("C12","C13","C23")) 

plot3D(res3d$sol.ode[,"m1"], res3d$sol.ode[,"m2"],res3d$sol.ode[,"m3"], xlab = "m1(t)",
  ylab = "m2(t)",zlab="m3(t)", type = "l",lwd = 2,box=F)
}