Sim.DiffProc (version 2.8)

st.int: Stochastic Integrals

Description

The (S3) generic function st.int of simulation of stochastic integrals of Ito or Stratonovich type.

Usage

st.int(expr, ...)
## S3 method for class 'default':
st.int(expr, lower = 0, upper = 1, M = 1, subdivisions = 1000L, 
               type = c("ito", "str"), ...)

## S3 method for class 'st.int':
summary(object, \dots)
## S3 method for class 'st.int':
time(x, \dots)
## S3 method for class 'st.int':
mean(x, \dots)
## S3 method for class 'st.int':
median(x, \dots)
## S3 method for class 'st.int':
quantile(x, \dots)
## S3 method for class 'st.int':
kurtosis(x, \dots)
## S3 method for class 'st.int':
skewness(x, \dots)
## S3 method for class 'st.int':
moment(x, order = 2, \dots)
## S3 method for class 'st.int':
bconfint(x, level=0.95, \dots)
## S3 method for class 'st.int':
plot(x, \dots)
## S3 method for class 'st.int':
lines(x, \dots)
## S3 method for class 'st.int':
points(x, \dots)

Arguments

expr

an expression of two variables t (time) and w (w: standard Brownian motion).

lower, upper

the lower and upper end points of the interval to be integrate.

M

number of trajectories.

subdivisions

the maximum number of subintervals.

type

Ito or Stratonovich integration.

x, object

an object inheriting from class "st.int".

order

order of moment.

level

the confidence level required.

...

further arguments for (non-default) methods.

Value

• st.int returns an object inheriting from class "st.int".
• X
  the final simulation of the integral, an invisible ts object.
• fun
  function to be integrated.
• type
  type of stochastic integral.
• subdivisions
  the number of subintervals produced in the subdivision process.

newcommand

\CRANpkg

href

http://CRAN.R-project.org/package=#1

pkg

#1

Details

The function st.int returns a ts x of length N+1; i.e. simulation of stochastic integrals of Ito or Stratonovich type. The Ito interpretation is: $$\int_{t_{0}}^{t} f(s) dW_{s} = \lim_{N \rightarrow \infty} \sum_{i=1}^{N} f(t_{i-1})(W_{t_{i}}-W_{t_{i-1}})$$ The Stratonovich interpretation is: $$\int_{t_{0}}^{t} f(s) \circ dW_{s} = \lim_{N \rightarrow \infty} \sum_{i=1}^{N} f\left(\frac{t_{i}+t_{i-1}}{2}\right)(W_{t_{i}}-W_{t_{i-1}})$$ For more details see vignette("SDEs").

References

Ito, K. (1944). Stochastic integral. Proc. Jap. Acad, Tokyo, 20, 19--529. 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.

See Also

snssde1d, snssde2d and snssde3d for 1,2 and 3-dim sde.

Examples

## Example 1: Ito integral
## f(t,w(t)) = int(exp(w(t) - 0.5*t) * dw(s)) with t in [0,1]

fexpr <- expression( exp(w-0.5*t) )
res <- st.int(fexpr,type="ito",M=10,lower=0,upper=1)
res
## res$X 
summary(res)
## Display
plot(res,plot.type="single")
lines(time(res),mean(res),col=2,lwd=2)
lines(time(res),bconfint(res,level=0.95)[,1],col=4,lwd=2)
lines(time(res),bconfint(res,level=0.95)[,2],col=4,lwd=2)
legend("topleft",c("mean path",paste("bound of", 95,"confidence")),
        inset = .01,col=c(2,4),lwd=2,cex=0.8)

## Example 2: Stratonovich integral
## f(t,w(t)) = int(w(s)  o dw(s)) with t in [0,1]

fexpr <- expression( w )
res1 <- st.int(fexpr,type="str",M=10,lower=0,upper=1)
res1
## res1$X 
summary(res1)
## Display
plot(res1,plot.type="single")
lines(time(res1),mean(res1),col=2,lwd=2)
lines(time(res1),bconfint(res1,level=0.95)[,1],col=4,lwd=2)
lines(time(res1),bconfint(res1,level=0.95)[,2],col=4,lwd=2)
legend("topleft",c("mean path",paste("bound of", 95,"confidence")),
       inset = .01,col=c(2,4),lwd=2,cex=0.8)