snssde1d: Simulation of 1-D Stochastic Differential Equation

In Sim.DiffProc: Simulation of Diffusion Processes

Simulation of 1-D Stochastic Differential Equation

Description

The (S3) generic function snssde1d of simulation of solution to 1-dim stochastic differential equation of Itô or Stratonovich type, with different methods.

Usage

snssde1d(N, ...)
## Default S3 method:
snssde1d(N = 1000, M = 1, x0 = 0, t0 = 0, T = 1, Dt, 
   drift, diffusion, alpha = 0.5, mu = 0.5, type = c("ito", "str"), 
   method = c("euler", "milstein", "predcorr", "smilstein", "taylor", 
   "heun", "rk1", "rk2", "rk3"), ...)	
   
## S3 method for class 'snssde1d'
summary(object, at ,digits=NULL, ...)
## S3 method for class 'snssde1d'
time(x, ...)
## S3 method for class 'snssde1d'
mean(x, at, ...)
## S3 method for class 'snssde1d'
Median(x, at, ...)
## S3 method for class 'snssde1d'
Mode(x, at, ...)
## S3 method for class 'snssde1d'
quantile(x, at, ...)
## S3 method for class 'snssde1d'
kurtosis(x, at, ...)
## S3 method for class 'snssde1d'
min(x, at, ...)
## S3 method for class 'snssde1d'
max(x, at, ...)
## S3 method for class 'snssde1d'
skewness(x, at, ...)
## S3 method for class 'snssde1d'
moment(x, at,  ...)
## S3 method for class 'snssde1d'
cv(x, at,  ...)
## S3 method for class 'snssde1d'
bconfint(x, at,  ...)

## S3 method for class 'snssde1d'
plot(x, ...)
## S3 method for class 'snssde1d'
lines(x, ...)
## S3 method for class 'snssde1d'
points(x, ...)		

Arguments

N	number of simulation steps.
M	number of trajectories (Monte-Carlo).
x0	initial value of the process at time t0.
t0	initial time.
T	ending time.
Dt	time step of the simulation (discretization). If it is missing a default \Delta t = \frac{T-t_{0}}{N}.
drift	drift coefficient: an expression of two variables t and x.
diffusion	diffusion coefficient: an expression of two variables t and x.
alpha, mu	weight of the predictor-corrector scheme; the default alpha = 0.5 and mu = 0.5.
type	if type="ito" simulation sde of Itô type, else type="str" simulation sde of Stratonovich type; the default type="ito".
method	numerical methods of simulation, the default method = "euler".
x, object	an object inheriting from class "snssde1d".
at	time between t0 and T. Monte-Carlo statistics of the solution X_{t} at time at. The default at = T.
digits	integer, used for number formatting.
...	potentially further arguments for (non-default) methods.

Details

The function snssde1d returns a ts x of length N+1; i.e. solution of the sde of Ito or Stratonovich types; If Dt is not specified, then the best discretization \Delta t = \frac{T-t_{0}}{N}.

The Ito stochastic differential equation is:

dX(t) = a(t,X(t)) dt + b(t,X(t)) dW(t)

Stratonovich sde :

dX(t) = a(t,X(t)) dt + b(t,X(t)) \circ dW(t)

The methods of approximation are classified according to their different properties. Mainly two criteria of optimality are used in the literature: the strong and the weak (orders of) convergence. The method of simulation can be one among: Euler-Maruyama Order 0.5, Milstein Order 1, Milstein Second-Order, Predictor-Corrector method, Itô-Taylor Order 1.5, Heun Order 2 and Runge-Kutta Order 1, 2 and 3.

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

Value

snssde1d returns an object inheriting from class "snssde1d".

X	an invisible ts object.
drift	drift coefficient.
diffusion	diffusion coefficient.
type	type of sde.
method	the numerical method used.

Author(s)

A.C. Guidoum, K. Boukhetala.

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

Friedman, A. (1975). Stochastic differential equations and applications. Volume 1, ACADEMIC PRESS.

Henderson, D. and Plaschko,P. (2006). Stochastic differential equations in science and engineering. World Scientific.

Allen, E. (2007). Modeling with Ito stochastic differential equations. Springer-Verlag.

Jedrzejewski, F. (2009). Modeles aleatoires et physique probabiliste. Springer-Verlag.

Iacus, S.M. (2008). Simulation and inference for stochastic differential equations: with R examples. Springer-Verlag, New York.

Kloeden, P.E, and Platen, E. (1989). A survey of numerical methods for stochastic differential equations. Stochastic Hydrology and Hydraulics, 3, 155–178.

Kloeden, P.E, and Platen, E. (1991a). Relations between multiple ito and stratonovich integrals. Stochastic Analysis and Applications, 9(3), 311–321.

Kloeden, P.E, and Platen, E. (1991b). Stratonovich and ito stochastic taylor expansions. Mathematische Nachrichten, 151, 33–50.

Kloeden, P.E, and Platen, E. (1995). Numerical Solution of Stochastic Differential Equations. Springer-Verlag, New York.

Oksendal, B. (2000). Stochastic Differential Equations: An Introduction with Applications. 5th edn. Springer-Verlag, Berlin.

Platen, E. (1980). Weak convergence of approximations of ito integral equations. Z Angew Math Mech. 60, 609–614.

Platen, E. and Bruti-Liberati, N. (2010). Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer-Verlag, New York

Saito, Y, and Mitsui, T. (1993). Simulation of Stochastic Differential Equations. The Annals of the Institute of Statistical Mathematics, 3, 419–432.

See Also

snssde2d and snssde3d for 2 and 3-dim sde.

sde.sim in package "sde".

simulate in package "yuima".

Examples

## Example 1: Ito sde
## dX(t) = 2*(3-X(t)) dt + 2*X(t) dW(t)
set.seed(1234)

f <- expression(2*(3-x) )
g <- expression(1)
mod1 <- snssde1d(drift=f,diffusion=g,M=4000,x0=10,Dt=0.01)
mod1
summary(mod1)
## Not run: 
plot(mod1)
lines(time(mod1),apply(mod1$X,1,mean),col=2,lwd=2)
lines(time(mod1),apply(mod1$X,1,bconfint,level=0.95)[1,],col=4,lwd=2)
lines(time(mod1),apply(mod1$X,1,bconfint,level=0.95)[2,],col=4,lwd=2)
legend("topright",c("mean path",paste("bound of", 95," percent confidence")),
       inset = .01,col=c(2,4),lwd=2,cex=0.8)

## End(Not run)
## Example 2: Stratonovich sde
## dX(t) = ((2-X(t))/(2-t)) dt + X(t) o dW(t)
set.seed(1234)

f <- expression((2-x)/(2-t))
g <- expression(x)
mod2 <- snssde1d(type="str",drift=f,diffusion=g,M=4000,x0=1, method="milstein")
mod2
summary(mod2,at = 0.25)
summary(mod2,at = 1)
## Not run: 
plot(mod2)
lines(time(mod2),apply(mod2$X,1,mean),col=2,lwd=2)
lines(time(mod2),apply(mod2$X,1,bconfint,level=0.95)[1,],col=4,lwd=2)
lines(time(mod2),apply(mod2$X,1,bconfint,level=0.95)[2,],col=4,lwd=2)
legend("topleft",c("mean path",paste("bound of", 95," percent confidence")),
       inset = .01,col=c(2,4),lwd=2,cex=0.8)

## End(Not run)

Sim.DiffProc documentation built on May 29, 2024, 8:09 a.m.