Equivalent Probability Measures Preserve Null Events, Not Process Variance

Summary

The document clarifies the meaning of equivalence between probability measures in stochastic-process settings. Equivalent measures assign probability zero to the same events, so they agree on which path sets are impossible. For Brownian motion, the response connects this property to preservation of quadratic variation, while warning that it does not imply equal variances for all processes or random variables.

An Ornstein–Uhlenbeck process and a scaled Brownian motion illustrate the distinction: they can have the same diffusion scale and equivalent path measures, yet their values at a given time can have different variances because their dynamics differ. This is a conceptual clarification relevant to measure changes in mathematical finance. The brief answer does not develop the full conditions for equivalence or explain a risk-neutral pricing application; its example is intended to distinguish equivalence from equality of variance.

Key ideas

• Equivalent measures have the same probability-zero events.
• Equivalence does not require all random variables or processes to have equal variance.
• For Brownian motion, the response relates equivalence to preservation of quadratic variation.
• An Ornstein–Uhlenbeck process and scaled Brownian motion illustrate how dynamics can yield different variances despite equivalent measures.

Tags

# Why define the risk free measure as the one equivalent to the real world measure?


# Why define the risk free measure as the one equivalent to the real world measure?












Equivalent is used as a piece of probability jargon that means that the null set of both measures is the same. In the context of stochastic processes, this is the same as saying that the variance term is the same. Is there a reason for describing q as being equivalent to p, rather than saying that q has the same variance as p?

## Answer by Rylan (score 2)

https://quant.stackexchange.com/a/85391

In the context of Brownian motion, it indeed means the quadratic variation is the same. However, the variances of the process are not necessarily equal. Consider $X_t$ following an OU process $$dX_t = \alpha(\mu - X_t)dt + \sigma dW_t$$ versus a (scaled) Brownian motion $$dYt = \sigma dW_t$$

Note that the measures are equivalent (both give the same set of paths zero probabilty) but the variance of $X_T$ vs $Y_T$, viewed at some $t < T$, are different.

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