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(Looking at the question and notation used more closely, the formulation seems to be problematic in couple places.) General Fact Let $W$ be standard Brownian motion with respect to filtration $( \mathscr F_t )_{t \in [0,T]}$. Consider $(L_t)_{t \in [0,T]}$ defined by $$\frac{dL_t}{L_t} = \psi_t dL_t, \; L_0 = 1.$$ In general, $L_t = e^{\int_0^t \psi_s ... 2 So we have$S_n \thicksim^{iid} \ ?\bar{s} = \frac{1}{n}\sum{s_n}$Exponential Suppose$S_n$follows the exponential distribution. $$f(s|\beta) = \frac{1}{\beta} e^{-\frac{s}{\beta}} \quad , \quad 0 \leq s < \infty \quad , \quad \beta > 0$$ Take the simple bivariate case. Say$Z = \frac{S_1 + S_2}{2}$and$W = S_1$. So$S_2 = 2Z - W$and$S_1 ...

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The best introduction to measure-theoretic probability for economics is probably: Chapter 7 Measure Theory and Integration, Recursive Methods in Economic Dynamics by Stokey and Lucas. Its presentation is along mathematical lines but with judicious (and numerous) omissions for economics audience. For example, a measure space is defined but no real concrete ...

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If you want a good book with emphasis on rigorous then An Introduction to Probability: Theory and Application, by William Feller is good source. The book starts completely from the first principles and covers also a lot of applications in statistics. Arguably the book is more suited to graduate as it has a steep curve - the content difficulty increases ...

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I cannot really follow your formulas, what is the logic behind them? Seems to me there is no way to divine three state risk-neutral probabilities from 1 financial instrument's prices. The equation $$p_1 288 + p_2 180 + (1 - p_1 - p_2) 120 = 180$$ is underdetermined, there are infinitely many solutions to it. Now if you solved the equation system  \...

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It is safe to assume that if the consumer passes away, his creditors get his endowment. This actually happens in real life. $B$ should obviously be zero. As far as the second period is concerned you can represent it as a von Neuman-Morgenstern utility from the lottery: $\beta u(c_2)$ with probability $p$ and $0$ with probability $1-p$. I.e. $U(c_1, c_2) = ... 1 Didn't manage to get to a definitve answer in one shot, but it seems to me that Jensen inequality is not going to help much. Build up: You are essentially asking that \begin{equation} E_v \left(u(a - v) \right) \leq E_c \left(u(a-c) |a\leq c \leq b\right) \end{equation} for a function$u$increasing and concave and$[a, b] \subset Supp_v = [x,y]$. ... 1 I'm sorry, this is probably better a comment than an answer, but I don't have sufficient points: In the diagram you've included, Type I and Type II errors are more properly conditional probabilities.$\alpha$= Prob( Reject$H_0$|$H_0$) (= probability of saying not not-pregnant, conditional on actually not-pregnant))$\beta$= Prob( Fail to reject$H_0\$...

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