Girsanov Measure Changes at Bounded Stopping Times
Summary
The document asks whether expectations of a function of a process at a stopping time can be transformed between probability measures using the inverse likelihood process at that stopping time. It outlines a conditional-expectation argument: for a bounded stopping time and a random variable measurable with respect to the stopped filtration, the terminal inverse density is conditioned down to its value at the stopping time, provided the required martingale and integrability assumptions hold.
The example uses a Brownian motion with drift under one measure and a likelihood process defining an equivalent measure. The discussion highlights that the identity is not valid for every function without qualification: the stopped variable must be integrable, the stopping time must be bounded by the horizon, and the density process must satisfy the stated measure-change conditions. It poses the generalization question but does not resolve all technical conditions or unbounded stopping times.
Key ideas
- A likelihood ratio can convert expectations between equivalent probability measures.
- For a bounded stopping time, conditioning the inverse density on the stopped filtration yields its value at that stopping time under suitable assumptions.
- The stopped random variable must be measurable at the stopping time and integrable for the expectation argument.
- The document does not establish the identity for arbitrary functions or unbounded stopping times.
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Full text
# Girsanov theorem and stopping time
# Girsanov theorem and stopping time
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, equipped with a filtration $(\mathcal{F})_{0 \leq t \leq T}$ which is a natural filtration of a standard Brownian motion $(W_{t})_{0 \leq t \leq T}$.
According to Girsanov's theorem, the process $X_t \buildrel\textstyle\over={W_t}+ {\lambda}t $ is a standard brownian motion under the $\mathbb{Q}$-measure defined by the Radon-Nikodym Derivative $\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}=Z_T$ where $$Z_t=\exp\left(\frac{1}{2}\lambda^2t-\lambda X_t\right)$$.
If $\tau$ a stopping time with respect to filtration $\mathcal{F}_t$, I want to know if this equality is true for any function f:
$$\mathbb{E}^\mathbb{P}\left[f(X_\tau)\right]=\mathbb{E}^\mathbb{Q}\left[Z_\tau^{-1}f(X_\tau)\right]$$ I found a little proof in book:"MathematicalMethods for Financial Markets by Marc Chesney, Marc Yor et Monique Jeanblanc(2009) see $Proposition 1.7.1.4$ pp.66" I reorganized the demonstration as follows:
Let $X$ be a $\mathbb{P}$-integrable $\mathcal{F}_\mathcal{\tau}$-measurable random variable and $\mathbb{P}$ and $\mathbb{Q}$ be two equivalent probabilities such that $\mathbb{Q}|\mathcal{F}_t = L_t \mathbb{P}|\mathcal{F}_t$ where $L_t$ is a $(\mathbb{P},\mathcal{F}_t)$-martingale, we have under these assumptions and those quoted in the question the following equalities:
$$\mathbb{E}^\mathbb{P}\left[X\right]=\mathbb{E}^\mathbb{Q}\left[L_T^{-1}X\right]=\mathbb{E}^\mathbb{Q}\left[\mathbb{E}^\mathbb{Q}\left[L_T^{-1}X|\mathcal{F}_\tau\right]\right]=\mathbb{E}^\mathbb{Q}\left[X\mathbb{E}^\mathbb{Q}\left[L_T^{-1}|\mathcal{F}_\tau\right]\right]=\mathbb{E}^\mathbb{Q}\left[XL_\tau^{-1}\right]$$
The first equality is justified by the fact that $\mathcal{F}_\tau\subseteq \mathcal{F}_T$ if we assume that the stopping time $\tau$ is bounded by $T$, the second equality is trivial, the last two ties result from the fact that $X$ is $\mathcal{F}_\mathcal{\tau}$-measurable and $L^{-1}$ is a $(\mathbb{Q},\mathcal{F}_t)$-martingale.
What do you think, and how can we generalize this result for any function $\mathbb{P}$-integrable and $\mathcal{F}_\mathcal{\tau}$-measurable ?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.