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ra_forever8
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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
∫_(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