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It is the former. Martingales are defined by filtration and probability space. The probability space for the filtration need not be the same as the probability space for the martingale. I think? That's what my prof said (iirc).


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Were we doing physics and we said there was an arithmetic Brownian motion we could indeed have a drift rate other than $\mu=r$ and it would make sense. Suppose, for example, that a fluid is moving at velocity $v$ and we have a random walk of particle in it. This would be reality. The whole point of using SDEs in finance is to identify what ought to be ...


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I thought this was an interesting example to add. It concerns a "ratio model" of habit (as opposed to a "difference" model of habit). See, for example, Abel (1990, American Economic Review). Let $$ x_t = \lambda \int_{-\infty}^t e^{-\lambda(t-s)} c_s ds. $$ (For context, $x_t$ is a log habit index that is given by a geometric average of past consumption, ...


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You might want to give us the exact statement of the author. Let the Wiener process $W_{s}$ be a r.v. from $\left(\mathcal{F}_{s},\Omega\right)\to\left(\mathcal{B}\left(\mathbb{R}\right),\mathbb{R}\right)$. The Borel-$\sigma$-algebra $\mathcal{B}\left(\mathbb{R}\right)$ contains all intervals of the form $\left[x,y\right]$ for $x\neq y\in\mathbb{R}$, ...


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The above question was a typo due to the author -- the expression should be evaluated as \begin{equation} E(t|\mathcal{F}_{s}^{W}) = t \end{equation} due to the reasoning in the question. Sorry for the noise.


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I am not sure if I understood your question correctly but I will try to answer it anyway. If you have a standard normal random vector $z \sim N(\mathbb{0},I_n)$ (where $z,0 \in \mathbb{R}^{n\times1}$ and $I_n \in \mathbb{R}^{n\times n}$ is the identity matrix) and you want to transform it into a multivariate normal $x \sim N(\mu,\Sigma)$ you do it the ...


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Let me first start with your very last equation. There, you just stumbled into an application of the Wong-Zakai decomposition: http://projecteuclid.org/euclid.aoms/1177699916 It essentially says that if a sequence of processes $( M_n )$ converges weakly to a martingale $M$, then the sequence of processes, $ t \mapsto \int_0^t \Gamma(s)dM_n(s) $ converges ...



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