Why Equivalent Measures Cannot Simply Rescale an Itô Diffusion’s Volatility

Summary

The document asks whether an Itô diffusion can retain its path behavior while a probability measure equivalent to the original one changes its volatility. The questioner proposes defining a new measure through a ratio of integrated squared volatilities, using quadratic variation as a bridge between the diffusion and its speed, but is unsure how to construct or validate the Radon–Nikodym derivative.

No solution or supporting derivation is provided; the text is an open mathematical question. Its useful lesson is that changing measure requires a valid, positive density with expectation one, and standard equivalent changes of measure alter drift under suitable conditions rather than quadratic variation. Since quadratic variation is a path property preserved under equivalent measures, the proposed approach may not permit a different diffusion coefficient for the same process. The post does not specify assumptions or resolve whether a time change or a different process is intended.

Key ideas

• The post asks whether an equivalent probability measure can change an Itô diffusion’s volatility.
• It proposes relating measures to integrated squared volatility through quadratic variation.
• A Radon–Nikodym derivative must define a valid probability measure.
• The document offers no derivation or resolution of the proposed construction.

Tags

# Ito Diffusion with Change of Measure


# Ito Diffusion with Change of Measure












Let $(X_t)$ be an Ito diffusion with speed $(V_t)$, under a probability measure P. Could there exist a change of measure to a probability measure Q, with Q ~ P, under which $(X_t)$ is an Ito diffusion with a different speed $(V'_t)$?

I've figured out a way such that QV(X) = $\displaystyle\int_{0}^{T} V_t^2 dt.$ With probability 1. Also by Radon-Nikodym derivative of Q with respect to P such that $T = \frac{dQ}{dP}$. So we can get $\frac{dQ}{dP}$ = $\frac{\displaystyle\int_{0}^{T} (V'_t)^2 dt}{\displaystyle\int_{0}^{T} V_t^2 dt}$ But I'm stuck after that. If anyone can give any ideas of how to continue, or if the way I did was incorrect. Thanks

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