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(a) Dynkin is still a pretty standard reference for this kind of material. (b) The existence of a Markov process with transition function $P$ is an immediate consequence of the Kolmogorov extension theorem, yes. But Kolmogorov might give you a process that is not strong Markov, not right continuous, or bad in other ways. To get a "nice" process with ...


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Since the conditional transition density of an OU process is known explicitly and is Gaussian, I would suggest to use an ML-estimator. Given the observations $X_0,\dots ,X_n$ at time-points $t_0, \dots, t_n$ the log-MLE is \begin{equation} \operatorname{argmin}_\theta \sum_{i=1}^n \frac{X_{i} - m(X_{i-1}, t_i - t_{i-1})}{2 s^2(t_i-t_{i-1})} + \frac{1}{2}\log ...


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First, I wouldn't modelise the markov chain like that , I would consider four states 1 = "player A play", 2 = "player B play", 3 = "player A had won" and 4 = "player B had won" The transition matrix would be $$M = \begin{pmatrix} 9/52 & 10/13 & 3/52 & 0 \\ 10/13 & 9/52 & 0 & 3/52 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 ...


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In general: no, $|| P_t f- f|| \to 0$ does not hold for all $f \in L^\infty$ and a general semigroup $(P_t)_{t\geq 0}$ on the Banach space $L^\infty$. If e.g. $P_t f(x) = f(x+ct)$ for some $c \neq 0$, then if $\hat{x}$ is a discontinuity point of $f$ problems appear: $P_t f(\hat{x}) - f(\hat{x}) = f(\hat{x}+ct) - f(\hat{x})$. This will not converge to $0$ ...



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