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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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### 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 ...
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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### 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?
175 views

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 $... 0answers 102 views ### Reference: hitting time of Gaussian process Let$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 ... 0answers 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: $$T_x = \mu(\{t \in [0;1] | W(t) > ... 0answers 66 views ### Conditioned Brownian motion? Let U\subseteq (C_0[0,1];\mathbb{R}) be an open subset of the Wiener space satisfying 0<\gamma(U)<1; where \gamma is the Wiener measure and let W_t be the standard (Wiener) coordinate ... 0answers 73 views ### Sufficient conditions for taking limits in stochastic partial differential problems Let's say we have a Cauchy problem:$$ (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), $$where x \in ... 1answer 97 views ### Convergence in distribution of products Suppose that a sequence of random variables Y_n convergence in L^2 to Y, i.e.$$ 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 &... 1answer 43 views ### 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 $$0< ... 1answer 66 views ### About another potential characterization of normal numbers Normal numbers, in a nutshell, are real numbers that have a "uniform" distribution of digits in standard numeration systems (binary, decimal, and so on.) You can find a formal definition and ... 2answers 54 views ### Samples paths are convex Are there stochastic processes with convex sample paths? Suppose C is a given convex set. Is there a real valued stochastic process X_t, t \in C such that the sample path f:C \rightarrow R ... 2answers 82 views ### Backward Stochastic Differential Equation Let W_t be a standard Brownian motion. Let T be the terminal date, X_T=x, and$$ 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 ... 1answer 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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