Why the Kolmogorov Backward Equation Sets the Drift Term to Zero

Summary

The document introduces the Kolmogorov backward equation for a diffusion whose state has drift and volatility. It gives the familiar partial differential equation relating the time derivative of a function to its first and second spatial derivatives, and notes that an Itô expansion can be used in its derivation.

Its central question is why the coefficient of the time increment is set to zero. The document does not answer this; it records a point of confusion rather than a completed explanation. In the standard derivation, the backward equation follows from a conditional expectation or pricing representation and the resulting martingale or generator condition, rather than from arbitrarily discarding a drift term. The brief text gives no boundary conditions, payoff setup, or application, so it does not develop the equation into a pricing method or other quantitative-finance result.

Key ideas

• The document states the backward equation for a diffusion with drift and volatility.
• It points to Itô’s lemma as part of the derivation.
• Its main unresolved issue is why the time-increment coefficient must vanish.
• No derivation, boundary condition, or application is provided.

Tags

# kolmogorov backward equation intuition


# 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 $

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.