Ito integral of an arbitrary f(Wiener process)

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The discussion focuses on the computation of the Itô integral of an arbitrary function f(W_T), where W_T represents a standard Wiener process. The integral is expressed as X_T = ∫_0^t f(W_s)dW_s. Participants explore the application of Itô's lemma, noting challenges in identifying a suitable function f(x) that aligns with the lemma's requirements. The conversation highlights the necessity of understanding the stochastic calculus framework to effectively utilize Itô's lemma in this context.

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saminator910
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How would you take the Ito integral of an arbitrary [itex]f(W_T)[/itex] where [itex]W_T[/itex] is a standard Wiener process

[itex]X_T=\int^t_0 f(W_s)dW_s[/itex]

would you somehow use Ito's lemma? I have attempted, but it doesn't seem to make sense...
[itex]dX_T=f(W_T)dW_T[/itex], There doesn't seem to be a [itex]f(x)[/itex] that makes sense for this in Ito's lemma.

[itex]df = \frac{\partial f}{\partial x}dX_T + \frac{\partial^2 f}{\partial x^2}dX_T^2[/itex]
 
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