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Applied stochatic processess

  1. Dec 7, 2013 #1
    Prove the following result for the ito stochastic integral (n>1)
    ∫_(t_0)^t▒〖W^n dW = 1/(n+1) (W^(n+1) (t)- 〗 W^(n+1) (t_0 ))- n/2 ∫_(t_0)^t▒〖W^(n-1 ) dt〗 Hint: apply ito differentiation rule f(W) = W^(n+1) to express W^n dW via dW^(n+1) and W^(n-1) dt (analogue of integration by parts for stochastic calculus)

    => stochastic differential equation: dW_t= A(t,W_t)dt +B(t,W_t)dW

    ∂f(t,W_t)= (∂f/∂t)* dt + ∂f/∂W *(dW_t) + 1/2 ∂^2/∂W^2 * (dW_t ^2)

    We have f(W) = W^(n+1)
    A=0 and B=1

    Can some one help me after this to prove the equation
     
  2. jcsd
  3. Dec 8, 2013 #2

    Ray Vickson

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    Use Ito's Lemma:
    [tex] dX_t = a(X_t,t) dt + b(X_t,t) dW_t \text{ and } Y_t = f(X_t,t)\\
    \text{implies}\\
    dY_t = f_x(X_t,t) dX_t + f_t(X_t,t) dt + \frac{1}{2} f_{xx}(X_t,t) (dX_t)^2 \\
    = f_x(a \, dt + b\, dW) + \frac{1}{2} f_{xx} b^2 dt = \left(a\, f_x + f_t + \frac{1}{2}b^2 \,f_{xx}\right)\, dt + f_x b \,dW[/tex]
    Apply this to ##a = 0, b = 1, f(x) = x^{n+1}##.
     
  4. Dec 8, 2013 #3
    dY_t= af_x *dt + f_t * dt +1/2 b^2 f_xx *dt + f_x *b *dW
    Applying a =0 and b=1
    =0+ f_t * dt+ 1/2 b^2 f_xx *dt + f_x *b *dW
    with f(x) = x^(n+1)
    Now differentiating in terms of x
    = 0+ n(n-1)/2 * x^(n-1) * dt + (n+1)* x^n *dW (f_t * dt =0)
    = n(n-1)/2 * x^(n-1) * dt + (n+1)* x^n *dW
    what happens to left side term dY_t ? what to do to prove the qs?
     
    Last edited: Dec 8, 2013
  5. Dec 8, 2013 #4

    Ray Vickson

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    You should have written
    [tex] dY_t \equiv d W^{n+1} = \frac{n(n+1)}{2} W^{n-1} \, dt + (n+1) W^n \, dW[/tex]
    When you realize that a stochastic DE is basically a shorthand notation for an integral result, you will see that you have everything you need.
     
    Last edited: Dec 8, 2013
  6. Dec 8, 2013 #5
    sorry I did my differentiation wrong
    dW^(n+1) = n(n+1)/2 W^(n+1) dt + (n+1) W^n dW
    dW^(n+1) = (n+1) ( n/2 *W^(n+1) dt + W^n dW)
    dW^(n+1) / (n+1) = ( n/2 *W^(n+1) dt + W^n dW)
    did I miss any terms? After that please?
     
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