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      Questions tagged [stochastic-processes]

      A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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      Distribution of markov chain with a stopping time

      I have a Markov chain $X_0, X_1, ..., X_\tau$, where $X_0$ is sampled from the stationary distribution and $\tau$ is a stopping time. Is it true that for any fixed $k$, $X_k$, given that $k \leq \tau$,...
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      40 views

      Comparing noisy truncated RV with noisy regular RV

      For some reason, I'm having difficulties proving something that is intuitively simple. Assuming I have two a random variable, $x$ and $x^{truncated}$, where $x^{truncated}$ is the truncated version of ...
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      67 views

      Ito's Formula for functions that are $C^2$ a.e

      In the conventional Ito's formula, it is required that $F$ is $C^2$ everywhere. However I've seen mentioning of a slightly weaker condition, where $F$ is $C^1$ everywhere but $C^2$ a.e.. Is there any ...
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      78 views

      Wald Identity for continuous stochastic process

      https://en.wikipedia.org/wiki/Wald%27s_equation Holds for discrete Stochastic process. Is there a version of such equation for continuous stochastic process?
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      175 views

      On Riemann integration of stochastic processes of order $p$

      Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $...
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      Reference: hitting time of Gaussian process

      Let $X_t$ be an OU process and $Y_t$ be the Gaussian process defined by $$ Y_t = y+\int_0^t X_s ds + W_t, $$ for some Brownian motion independent of $X_t$. Let $y,a>0$; is there a large deviation ...
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      60 views

      Beta distribution and Wiener process

      Suppose $W(t)$ is the standard Wiener process on $[0; 1]$ and $\{T_x\}_{x \in \mathbb{R}}$ is a collection of random variables defined by the following relation: $$T_x = \mu(\{t \in [0;1] | W(t) > ...
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      66 views

      Conditioned Brownian motion?

      Let $U\subseteq (C_0[0,1];\mathbb{R})$ be an open subset of the Wiener space satisfying $0<\gamma(U)<1$; where $\gamma$ is the Wiener measure and let $W_t$ be the standard (Wiener) coordinate ...
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      73 views

      Sufficient conditions for taking limits in stochastic partial differential problems

      Let's say we have a Cauchy problem: $$ (1) \hspace{0.5cm} u_t (x,t)+A(u) \cdot u_x (x,t)=\nu \cdot u_{xx} (x,t) + \epsilon \cdot f(u) \cdot W, $$ $$(2) \hspace{0.5cm} u(x,0)=u_0(x), $$ where $x \in ...
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      Convergence in distribution of products

      Suppose that a sequence of random variables $Y_n$ convergence in $L^2$ to $Y$, i.e. $$ E|Y_n-Y|^2\to0\quad \text{as}\quad n\to\infty. $$ Moreover, there exist constants $c_0$ and $c_1$ such that $$ 0 &...
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      Convergence in mean and convergence in distribution

      Suppose a sequence of random variables $X_n$ convergence in distribution to $X$, and $Y_n$ convergence in pth-mean (any $p\geq 1$) to $Y$. Moreover, there exist constants $c_0,c_1$ such that $$ 0< ...
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      66 views

      About another potential characterization of normal numbers

      Normal numbers, in a nutshell, are real numbers that have a "uniform" distribution of digits in standard numeration systems (binary, decimal, and so on.) You can find a formal definition and ...
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      Samples paths are convex

      Are there stochastic processes with convex sample paths? Suppose $C$ is a given convex set. Is there a real valued stochastic process $X_t, t \in C$ such that the sample path $f:C \rightarrow R $ ...
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      Backward Stochastic Differential Equation

      Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and $$ dX_t=f_tdt+B_tdW_t $$ where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated ...
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      102 views

      Stochastic processes and continuity of expectation

      Let $X$ be a stochastic process with a.s. continuous sample paths on $[0, 1]$ such that $\mathbb E [X_t]$ is finite for all $t \in [0, 1]$. Given any non null subset $Y$ of the probability space, ...

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