I am self-learning some Econ papers. Any suggestion will be appreciated. Even though the questions are from an Econ paper, they are math-related. I provide the economic interpretation as background information. Agent is denoted as $A$. The principal controls $\tau$ and $i$. The agent controls $u_t$. $\tau$ is the stopping time of the project. $i_t$ is the investment at each point in time t. $u_t$ is the effort made by the agent at each point in time t.
The model setup is the following:
The real output process is :
$ dX_t^u=\mu dt+v dB_t^{-u} = (\mu+u_t v)dt+vdB_t $
$dX_t^u = (\mu+u_t v)dt+vdB_t$ is the process happened in the real world. After using change of measure, the paper denotes it as $dX_t^u=\mu dt+v dB_t^{-u}$ .
The agent's remaining utility:
$W_t^{A,u}=E_t^{-u}[\int_t^\tau e^{-\gamma(s-t)}[\lambda u_s v ds+di_s]+e^{-\gamma(\tau-t)}R] $
The paper proved that $dc_t=\lambda u_s v ds+di_s$. $c_t$is the consumption of the agent at each point in time t. The agent's remaining utility $W_t^{A,u}$ is simply adding up all the consumption, plus the lump-sum payment R at time $\tau$.
The agent's discounted remaining utility is： $ \hat{W}_t^{A,u}=e^{-\gamma t} W_t^{A,u} $
Assumption 1: Given ($\tau,i$ ), $\mathcal{A}(\tau, i)$ is the set of $\mathbb{F}^B$- adapted process $u \ge 0$ such that: \begin{equation} E^{-u}[(\int_0^{\tau} e^{-\gamma t} \lvert{u_t}\rvert dt+\int_0^\tau e^{-\gamma t} di_t)^2]+E[{\lvert M_\tau^{-u}\rvert}^3+[M_\tau^{-u}]^{-3}]<\infty \end{equation}
By assumption(1) and Martingale Representation Theorem, there exists $Z^{A,u}$ such that
\begin{equation} \hat{W}_t^{A,u}=e^{-\gamma \tau}R+ \int_t^\tau e^{-\gamma s}[\lambda u_s v ds+d i_s]-\int_t^\tau e^{-\gamma s} v Z_s^{A,u} dB_s^{-u} \end{equation}
My questions are the following:
(a) When calculating the agent's expected utility, why does is use $E^{-u}$ instead of $E$. In other words, why does it calculate the expected utility under the adjusted measure instead of the real world measure?
(b) In assumption (1), it says given ($\tau,i$ ), $\mathcal{A}(\tau, i)$ is the set of $\mathbb{F}^B$- adapted process $u \ge 0$. What is the meaning of $\mathbb{F}^B$- adapted process?
(c) Assumption (1) is a technical assumption. It makes sure something in the model is not explosive. Is it some standard assumption in optimal control or optimal stopping time? It says the sum of two expectations are less than infinity. The first expectation is calculated under the adjusted measure $-u$, whereas the second expectation is calculated under the real world measure. Why is it the case? In the second expectation, the first term use the absolute value of $M_\tau^{-u}$, but the second term does not use the absolute value. Is it a typo or it means something? Besides, the paper does not tell me what $M_\tau^{-u}$ denotes.
(d) According to the Wiki's explanation about Martingale Representation Theorem (click here), the integral is from 0 to t. But in equation (2), the integral is from t to $\tau$. At time t, $\int_0^t$ is something we know. So $\int_0^t$ is deterministic. However, $\int_t^\tau$ is stochastic. Why equation (2) use $\int_t^\tau$ here?
(b) According to Wiki (click here), an adapted process is one that cannot "see into the future". An informal interpretation is that $u_t$ is adapted if and only if, for every realization and every t, u_t is known at time t.