Sim.DiffProc (version 2.5)

BMItoP: Properties of the stochastic integral and Ito Process [4]

Description

Simulation of the Ito integral(W(s)^n*dW(s),0,t).

Usage

BMItoP(N, T, power, output = FALSE)

Arguments

N

size of process.

T

final time.

power

constant.

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 : $$integral(W(s)^n*dW(s),0,t) = W(t)^(n+1) /(n+1) - (n/2) * integral(W(s)^n-1 *ds,0,t)$$ from classical calculus for Ito integral with w(0) = 0. The follows from the algebraic rearrangement : $$integral(W(s)^n*dW(s),0,t) = sum ( W(t)^n*(W(t+1)-W(t)),0,t)$$

See Also

BMIto1 simulation of the Ito integral[1], BMIto2 simulation of the Ito integral[2], BMItoC properties of the stochastic integral and Ito processes[3], BMItoT properties of the stochastic integral and Ito processes[5].

Examples

## if power = 1
## integral(W(s) * dW(s),0,t) = W(t)^2/2 - 1/2 * t
 BMItoP(N=1000,T=1,power =1)
## if power = 2
## integral(W(s)^2 * dW(s),0,t) = W(t)^3/3 - 2/2 * integral(W(s)*ds,0,t)
 BMItoP(N=1000,T=1,power =2)