Integrate one sample path of a stochastic process w.r.t. another, returning the Ito integral, the Stratonovich integral, or the "right hand rule".

stochint(f, g, rule = "l")

Arguments

f: numeric vector containing the integrator
g: numeric vector, same length as f, containing the integrand
rule: character vector indicating rule(s). Valid choices are "l", "r", or "c" for "left", "right", and "center", as well as combinations, e.g. c("l","c"). Optional and defaults to "l".

Value

A numeric vector, same length as f, giving the "running integral", i.e. the integral as a function of the upper limit.

Examples

## Integrating a cosine w.r.t. a sine
times <- seq(0,2*pi,length=21)
I <- stochint(cos(times),sin(times))
Ia <- 0.5*times+0.25*sin(2*times)  # Analytical result
matplot(times,I,type="l")
lines(times,Ia,col="blue",lwd=2)


## Integration of Brownian motion w.r.t. itself
times <- seq(0,10,0.01)
BM <- rBM(times)
I <- stochint(BM,BM,c("l","c","r"))
matplot(times,cbind(I$l,0.5*BM^2-0.5*times),type="l",xlab="Time",ylab="Left integral (Ito)",
         main="Integral of B.M. w.r.t itself")

matplot(times,cbind(I$r,0.5*BM^2+0.5*times),type="l",xlab="Time",ylab="Right integral",
         main="Integral of B.M. w.r.t itself")

matplot(times,cbind(I$c,0.5*BM^2),type="l",xlab="Time",ylab="Central integral (Stratonovich)",
         main="Integral of B.M. w.r.t itself")