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Differentiating Wiener process

Let $\text dX_t=\mu_t\text dt+\sigma_t\text dW_t$ be an Itô process. Itô's Lemma tells us $$\text df(t,X_t)=\left(f_t+\mu_tf_x+\frac{1}{2}\sigma_t^2f_{xx}\right)\text dt+\sigma_tf_x\text dW_t.$$ You'...
• 16k

Integrated Brownian motion

Your derivation is correct. Even if we fix your obvious typo the formula $$\textstyle\int_t^TW_s\,ds=\int_t^T(T-s)\,dW_s$$ is wrong. There is no doubt that \begin{align} TW_T&=\textstyle\int_0^...
• 2,033
1 vote

Is the Black Scholes PDE actually immediate from Ito's lemma?

The derivations of the Black & Scholes PDE being accurate or not has absolutely nothing to do with whether we write the Ito formula for the call price, in differential form, \tag1 dC=C_t\,dt+C_s\...
• 2,033
1 vote

Solving the SDE for GBM

$f$ as you've described it is differentiable in $t$ -- the derivative is just equal to zero. The larger point is that Ito's lemma applies to a broader class of functions $f(t, X_t)$ -- the one we've ...
• 625

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