Verifying if a Function is a Radon-Nikodym Derivative for changing the numeraire

External question — Quantitative Finance Stack Exchange Author: Gab Dam Original post: https://quant.stackexchange.com/questions/80396 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I am analyzing the following function within a financial mathematics framework: $$ f(t) = \dfrac{B(S; S) \cdot m(t)}{B(t; S) \cdot m(S)} $$ where: $$ B(t; S) := \mathbb{E}_{t}^{\mathbb{P}} \left[\exp\left(-\int_{t}^{S}r_{f}(u)\, du\right)\right] $$ and $$ m(t) := \exp\left(\int_{0}^{t} r_{f}(u)\, du\right) $$ Definitions : $B(t; S)$ represents the price at time $t$ of a zero-coupon bond maturing at time $S>t$ , given the information available at time $t$ . $m(t)$ is a discount factor related to the risk-free interest rate $r_f(t)$ . I want to determine whether this function can be considered as the Radon-Nikodym derivative $$\dfrac{d\mathbb{P}^{S}}{d\mathbb{P}}_{|\mathfrak{F}_{t}}$$ where $\mathbb{P}^{S}$ is the new probability measure associated to the new numeraire $B(t;S)$ and $\mathfrak{F}_{t}$ is a filtration. My Approach: Notice that in order to define $\dfrac{B(S; S) \cdot m(t)}{B(t; S) \cdot m(S)}$ as the Radon-Nikodym derivative $\dfrac{d\mathbb{P}^{S}}{d\mathbb{P}}_{|\mathfrak{F}_{t}}$ , it is sufficient to verify that: The expression is a $\mathbb{P}^{S}$ -martingale. It has a $\mathbb{P}^{S}$ -expectation equal to one. Here’s how these conditions are met: $B(S; S)$ is equal to 1. The expectation of $\frac{m(t)}{m(S)}$ under $\mathbb{P}^{S}$ is equal to $B(t; S)$ . These properties ensure that the expression has a $\mathbb{P}^{S}$ -expectation equal to one. Moreover, since the expression is equal to one for every $t$ , this implies that it is also a $\mathbb{P}^{S}$ -martingale. Questions: Verification : Is my understanding correct that verifying the martingale property and the expectation is sufficient to establish that this function is a Radon-Nikodym derivative? Martingale Proof : How can I rigorously prove that the expression is indeed a \$mathbb{P}^{S}$-martingale? Additional Considerations : Are there other properties or conditions that I should consider in this context? Any feedback or insights would be greatly appreciated. Thank you!

External answer — Quantitative Finance Stack Exchange Author: Wei Original post: https://quant.stackexchange.com/a/80402 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I note that $$f(t) = \frac{D(S)}{\mathbb E^\mathbb P _t[D(S)]}$$ where $$D(S) = \exp\left(-\int_0^Sr_f(u)du\right).$$ What you actually want is to use $f(0)$ as your Radon-Nikodym derivative. All that is needed for this to be a Radon-Nikodym derivative between probability measures is that it is non-negative and $E^\mathbb P _0 [f(0)] = 1$ , which is plainly true. For more info, see: https://en.wikipedia.org/wiki/Forward_measure .