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Sim.DiffProc (version 2.5)

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].

Examples

Run this code
BMIto1(N=1000,T=1)
## comparison with BMIto2
 system.time(BMIto1(N=10^4,T=1))
 system.time(BMIto2(N=10^4,T=1))

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