Relating Numeraire-Based Measure Changes to Girsanov’s Theorem

Summary

The document compares two expressions for changing between equivalent probability measures associated with different numeraires. One expression gives the Radon–Nikodym derivative directly in terms of the two numeraires; the other is written as an exponential involving the difference between the diffusion’s drifts, its volatility, and Brownian motion under the starting measure.

The question asks how to show these formulas are equivalent. It frames the problem through a scalar diffusion and Girsanov’s theorem, but includes no answer, derivation, or reference. The key conceptual link is that changing measure alters the drift while the numeraire ratio specifies the likelihood weight. Establishing equality would require connecting that ratio to the drift adjustment and checking the needed regularity and integrability conditions; those steps are not supplied, so the document poses a mathematical question rather than presenting a complete result.

Key ideas

• The numeraire ratio gives a Radon–Nikodym derivative between two associated measures.
• Girsanov’s theorem expresses the measure change using the diffusion’s drift difference relative to volatility.
• The document asks to establish equivalence between these descriptions but provides no derivation.
• Regularity and integrability conditions for the measure change are not discussed.

Tags

# Equivalence of formulations of Radon Nikodym derivative


# Equivalence of formulations of Radon Nikodym derivative












Let $N$ be a numeraire associated with the probability measure $Q^N$ and $U$ be a numeraire associated with the probability measure $Q^U$, both of which are equivalent to the physical probability measure $Q^0$. The Radon-Nikodym derivative defining the measure $Q^U$ is given by: $$\frac{dQ^U}{dQ^N} = \frac{U_t N_0}{U_0 N_t}$$ Now consider a scalar diffusion process $X$ , whose dynamics under $Q^N$ and $Q^U$ are given respectively by: $$dX_t = \mu_t^N(X_t) dt + \sigma_t dW_t^N, Q^N$$ $$dX_t = \mu_t^U(X_t) dt + \sigma_t dW_t^U, Q^U$$ We can apply Girsanov's theorem to deduce the Radon-Nikodym derivative between $Q^N$ and $Q^U$ from the dynamics of $X$ under the two measures: \begin{align} &\frac{dQ^U}{dQ^N} \\ &= \exp \left( - \frac{1}{2} \int_0^t \lvert \frac{\mu_s^U(X_s) - \mu_s^N(X_s)}{\sigma_s(X_s)} \rvert^2 ds - \int_0^t \frac{\mu_s^U(X_s) - \mu_s^N(X_s)}{\sigma_s(X_s)} dW_s^N \right) \end{align} Can someone help me show how the two formulations are equivalent? I assume that this is something standard so some simple reference would also be appreciated.

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