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### asymptotic variance of sample autocorrelation of two iid random variables

I am trying to prove that the variance of the sample lag-1 autocorrelation $$\hat{\rho}=\frac{\sum_{t=1}^n(x_t-\bar{x})(x_{t-1}-\bar{x})}{\sum_{t=1}^n(x_{t-1}-\bar{x})^2}$$ for an i.i.d. R.V is ...
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### Sufficient condition for local martingale property of stochastic integral

Is the following correct and/or a (simple) known result? Let $X$ be a local martingale and $H$ an integrand for $X$, such that the stochastic integral $\int H\cdot dX\ge x$ for some random variable. ...
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### Doob's inequality for martingale “convolution”

Let $(X_t, t \in \mathbb{N})$ be a martingale, and let $a \leq b \leq T \in \mathbb{N}$ be constants. Is there something like Doob's inequality for $\mathbb{E} \sup_{a \leq t \leq b} X_t(X_T-X_t)$, i....
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### a dominated convergence theorem for martingale (II)

The question is presented in a dominated convergence theorem for martingale Let $\{(X_1^n, X_2^n)\}_n$ be a sequence of martingales defined some probability space. (which means $E[X_2^n|X_1^n]=X_1^n$...
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### Can $<.>$ of a martingale determine it only?

Let $\Omega$ be the space of continuous functions defined on $[0,1]$. Define the canonical process $B$ by $$B_t(\omega)=\omega_t,~ \forall\omega\in\Omega$$ Let us equip $\Omega$ with the usual ...
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### Some constants in Martingale Stein inequality

Dear all, the following is a special case of Stein inequalities for martingales. $\textbf{Theorem}$ Let $(\Omega, \mathbb{P})$ be a (standard) probability space equipped with a filtration of ...
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### Best introduction to probability spaces, convergence, spectral analysis

I'm not sure if this stuff all falls under what most would just term "probability", but I'm researching applied macroeconomics and need to get a handle on the following concepts ASAP: probability ...
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### Lower-semicomputable supermartingales with bounded increments

I'm interested in whether Levin and Solomonoff's results on "universal semimeasures" can be extended to other settings. One case that especially interests me is finding "universal" strategies in the ...
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### Martingale part of the discontinuous put payoff

I need the martingale part of the put payoff (not $C^2$..). Where $S_t=exp(\sigma W_t -\frac{\sigma^2t}{2})$ $d[(S_t -K)^+ ]$ ?? I guess I need to use local times but how?
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### Supermartingales and convergence

These feel like basic enough questions, but I don't know where to find the answer. Let $X_1,X_2,X_3,\dots$ be a supermartingale such that $|X_{n+1} - X_n| < K$ for all $n$ ($K$ fixed). Does the ...
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### Martingales in both discrete and continuous setting

I am wondering, polynomials like $S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...
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### Is stopped brownian motion not a martingale ?

In page 45 of the book "Financial Derivatives In Theory and Practice by P.J.Hunt and J.E.Kennedy, it seems to me that the author says the stopped Brownian Motion is not a martingale as follows. (...
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One can read about Walsh's construction of martingale integral in the paper (pp.16-23) www.math.utah.edu/~davar/ps-pdf-files/SPDEBookDK.pdf For $U,V\in \mathcal{B}(\mathbb{R}\times \mathbb{R}^+), \... 1answer 346 views ### Stieltjes integrals of predictable processes I am looking for a direct proof of the fact that, roughly speaking, if$S=S_0+A+M$is an$L^2$semimartingale, and$M$(the martingale part) has the martingale representation property, then for any ... 5answers 3k views ### Brownian motion, martingales, Markov Chains - Rosetta Stone What are the most fundamental/useful/interesting ways in which the concepts of Brownian motion, martingales and markov chains are related? I'm a graduate student doing a crash course in ... 0answers 336 views ### Change of Time in Stochastic Integral Hi everyone, Let's be given$I(0,t)$a Stochastic Integral with respect to a local martingale$ M_t$of the form :$I(0,t)=\int_0^t h(s_-)dM_s$with$h\in L(M)$(for example$h$is an adapted ... 1answer 1k views ### Distribution of running maximum of a local martingale Let$(\Omega, \mathcal{F}, \mathbb{P}, \mathcal{F}_t)$be a given probability space with usual conditions, on which$W$is a standard Brownian motion. For$x \ge 0$, consider$$X(t) = x + \int_0^t \... 2answers 466 views ### Path continuity for (closed) martingales? Take a time interval$[0,T]$, and a filtered probability space$(\Omega,P,\mathcal{F},\mathcal{F}_t)$. If$X \in L^1(\mathcal{F}_T)$, then$M_t = E [X \ | \ \mathcal{F}_t]$is a martingale. If I ... 2answers 945 views ### Is the truncated Brownian motion of the class DL? Let$W$be a standard Brownian motion under given probability space. For a given constant$a$,$W^a$is a truncated Brownian motion by stopping time$T^a = \inf(t>0:W(t) = a)$. That is,$W^a(t) = ...
I am trying to prove that a certain sequence of Markov chains $x^N_k$ converges towards a diffusion process. The invariant measure of $x^N$ is $\pi^N$ and the Markov chain $x^N$ is started in ...