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Hi I think i might have spotted an errata in my SDE textbook but I don't feel confident enough to confirm it. The question is compute $$ dX_t = X_tW_tdW_t $$ $W_t$ being the wiener process.

The textbook's answer is $$ X_t = exp( W_t^ 2 /2 - t/2) $$ which is obviously wrong if we differentiate using Ito's formula (unless i am missing something).

In any case, I am wondering if there is a method to solve this SDE systematically. I have tried integration factor method, exact method, inspection and I still can't find the solution. Any input appreciated.

Chen Ee Woon
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  • If you consider $d(\ln X_t)$, you should be able to get a solution for $X_t$ that involves $\int_0^t W_s^2, ds$. – Minus One-Twelfth Feb 18 '19 at 08:39
  • wow ok thanks. How did you come up with that guess? – Chen Ee Woon Feb 18 '19 at 08:44
  • It's a quite common tactic, also used for solving the geometric brownian motion stochastic differential equation. One intuition for why we might want to guess to consider it is that for our given SDE, we have $\dfrac{dX_t}{X_t} = W_t , dW_t$. The left-hand side should remind you of $d(\ln X_t)$ since in classical calculus, $\dfrac{dx}{x} = d(\ln x)$. – Minus One-Twelfth Feb 18 '19 at 09:30

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