BMIto1: Properties of the stochastic integral and Ito Process [1]
Description
Simulation of the Ito integral(W(s)dW(s),0,t).
Usage
BMIto1(N, T, output = FALSE)
Arguments
N
size of process.
T
final time.
output
if output = TRUE write a output to an Excel (.csv).
Value
data frame(time,Ito,sum.Ito) and plot of the Ito integral.
Details
However the Ito integral also has the peculiar property, amongst others, that : $$integral(W(s)dW(s),0,t) = 0.5 * (W(t)^2 - t )$$
from classical calculus for Ito integral with w(0) = 0.
The follows from the algebraic rearrangement : $$integral(W(s)dW(s),0,t) = sum ( W(t)*(W(t+1)-W(t)),0,t)$$
See Also
BMIto2 simulation of the Ito integral[2], BMItoC properties of the stochastic integral and Ito processes[3], BMItoP properties of the stochastic integral and Ito processes[4], BMItoT properties of the stochastic integral and Ito processes[5].