Interpreting a Dirac Delta in a Conditional Expectation

Summary

The document asks how to interpret a step in a proof involving Gyöngy’s lemma, where an expectation weighted by a Dirac delta at a level is rewritten using an expectation at that level. The response frames the delta as concentrating the contribution on states where the process equals the specified level, then expresses the conditional expectation of the squared increment as a ratio: the delta-weighted expectation divided by the expectation of the delta.

This gives an intuition for how level-conditioned quantities can appear in stochastic-process calculations, but the exchange is brief and does not establish the identity rigorously. A Dirac delta is not an ordinary indicator function; for a continuous-valued process, conditioning on an exact level and interpreting delta-weighted expectations generally require a density or a limiting/distributional argument. The post provides no surrounding lemma proof or assumptions, so readers should treat the answer as informal guidance rather than a complete derivation.

Key ideas

• The question concerns delta weighting at a specified process level in a proof of Gyöngy’s lemma.
• The response interprets delta weighting as selecting contributions near the specified level.
• It represents the conditional expectation as a ratio of delta-weighted expectations.
• A Dirac delta is not an ordinary indicator, so the explanation needs technical assumptions for rigor.

Tags

# Question on Gÿongy' lemma proof


# Question on Gÿongy' lemma proof












I have some questions regarding a proof of Gÿongy's lemma given in 1

I would like to understand the following passage: $$ \int_{s=t_0}^{s=t}\mathbb{E}\left[\delta(X_s-K)\langle dX_s\rangle^2 \right]= \int_{s=t_0}^{s=t}\mathbb{E}\left[\delta(X_s-K) \right] \mathbb{E}\left[\langle dX_s\rangle^2|X_s=K \right] $$

Thanks

## Answer by Magic is in the chain (score 4)

https://quant.stackexchange.com/a/45246

If $X_s \neq K $ then the delta function gives zero, and the product is zero. So the term only contributes when $X_s=K$.

Re-comment, the key to understanding this is the conditional expectation:

$ E \left[ dX_s^2 \mid X_s=K \right] =\frac{E\left[ dX_s^2 \delta(X_s-K)\right]}{E\left [\delta(X_s-K)\right] }$

Where it might be helpful if you interpret the delta as indicator of $X_s=K$

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