# kolmogorov backward equation intuition

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 $$dt =0$$

• 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 – Ezy Feb 12 '19 at 11:16
• 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 – ZRH Feb 12 '19 at 12:15
• 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. – dayum Feb 12 '19 at 19:12
• 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 $dt \rightarrow 0$ – ZRH Feb 14 '19 at 8:40
• $p(x,t)$ 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. – Freelunch Feb 14 '19 at 12:25