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The kolmogorov backward equation equation states that the probability density of a random variable $x$ which follows $dx= \mu dt + \sigma dw$

is given by

$-p_t = \mu p_x + 0.5\sigma^2 p_{xx} $

This can be derived by applying Ito's lemma to $p(x,t)$ and setting the $dt$ term =0.

However, it is not clear to me why you can just set the coefficient of $dt =0 $

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  • $\begingroup$ You don’t derive the backwards equation by « setting the dt term to 0 ». Also the density is found by solving the forward equation, not the backwards equation so it is unclear what you are asking actually $\endgroup$ – Ezy Feb 12 at 11:16
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    $\begingroup$ I think dayum is referring to Wikipedia on Itô's lemma ("Informal Derivation"), where the equation is found if u set the coefficient of $dt$ to zero $\endgroup$ – ZRH Feb 12 at 12:15
  • $\begingroup$ Thanks @ZRH. At some other places also, it is derived informally in a discrete case using transition equation $P(x,t) = p * P(x-dx, t-dt) + q * P(x+dx, t-dt)$ where $p$ and $q$ are the probabilities of an up and down move of size $dx$ in next interval $dt$. However I am unclear how this translates to a continuous case. $\endgroup$ – dayum Feb 12 at 19:12
  • $\begingroup$ I have seen a derivation where a trinomial transition equation is written down (up,sideways,down), and the continuous case is derived as limiting case where $dx \rightarrow 0$ and $dt \rightarrow 0$ $\endgroup$ – ZRH Feb 14 at 8:40
  • $\begingroup$ $p(x,t)$ should be a martingale, hence its drift term should be zero and therefore you can set all terms that are multiplied by $dt$ to equal zero. $\endgroup$ – Freelunch Feb 14 at 12:25

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