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      Questions tagged [stochastic-processes]

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

      7
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      1k views

      Weighted Poincaré inequality

      Consider a probability distribution $\pi$ with density $e^{-H(x)}$ on $\mathbb{R}$. Let us say that there is a Poincaré inequality with weight $w$ if for any smooth function $\phi$ satisfying $\int \...
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      Hashed coupon collector

      The story: A sport card store manager has $r$ customers, that together wish to assemble a $n$-cards collection. Every day, a random customer arrives and buys his favorite card (that is, each customer ...
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      vote
      0answers
      35 views

      Distribution of a linear pure-birth process's integral

      I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process: $$ Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k \bigg] $$ where $(...
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      509 views

      Proof of the existence of an optimal MDP with a stochastic reward signal?

      I'm following Sutton's book on Reinforcement Learning, and he casually states that "There is always at least one policy that is better than or equal to all other policies" for a given finite MDP. This ...
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      1answer
      66 views

      How to calculate the probability of 2 events happening in time series under only cdf information?

      In time domain $0\rightarrow T$, there are two independent events $A$ and $B$. $B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...
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      151 views

      Existence of stick breaking representations for random measures

      The Dirichlet process has a roughly size ordered representation in terms of beta random variables, called a stick-breaking representation (Sethuraman, 1994). Similar results hold for the beta process, ...
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      1answer
      101 views

      Continuity of green functions

      I have a technical question on a continuity of green function. Setting Let $E$ be a locally compact separable metric space and $m$ a locally finite measure on $E$. Let $X=(\{X_t\}_{t \ge 0},\{P_x\}...
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      votes
      1answer
      191 views

      Large deviation of random walk

      1) Let $\{X_i\}_{i=1}^n$ be i.i.d. such that $\Pr(X_i=1 )=1-\Pr(X_i=-1)=p$. Define the random walk $$ S_i = \sum_{j=1}^iX_j $$ for $i=1,2,\ldots,n$. I am looking for "good" exponential upper bounds ...
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      1answer
      63 views

      Filtration exercise

      I am struggling with 1.7 exercise from the Karatzas, Shreve "Brownian motion and stoch. calulus". Denote by $\mathcal{F}^X_{t_0}$ the natural filtration corresponding to a process $X:[0,\infty)\times ...
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      votes
      1answer
      80 views

      The distribution of the area of a region cut out by chordal SLE?

      Let $\mathbb{D}$ be the unit disc. Let $a,b \in \partial \mathbb{D}$. Let $\gamma$ be a chordal $SLE_{k}$ from $a$ to $b$. For $k \leq 4$, $\gamma$ is a simple curve, and so $\mathbb{D} \setminus \...
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      votes
      1answer
      178 views

      Can we transform $\int_\rho^1 (W_t - W_{t-\rho}) \,dW_t$ to make its law $\rho$-invariant?

      I just bumped into the stochastic integral $$ \int_\rho^1 (W_t - W_{t-\rho}) \,dW_t $$ where $0 < \rho < 1$ is a constant and $W$ is a standard Wiener process. It would be nice if we have a ...
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      0answers
      50 views

      Defining weak solutions to infinitely many SDEs on the same probability space

      Suppose I have an SDE of the form $$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$ which I can solve weakly if I cut off the last integral to range over the set $\{\mid{y}\mid > ...
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      1answer
      82 views

      Obtaining a lower bound on the expectation using the Sudakov-Fernique inequality

      In my work I wish to obtain a lower bound for the term below. Here the expectation is taken over $h$, a standard random Gaussian vector of length $n$. The minimum is taken over all $\{i_1,\dots,i_L\} \...
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      0answers
      103 views

      An application of Girsanov's Theorem

      Let $(W,H,i)$ be the classical Wiener space where $W=C_0([0,1])$, $H$ is the Cameron-Martin space. Let $A= I_{W}+a$ such that $A:W \rightarrow W$ and $a \in L^{0}(\mu,H)$, $a$ has adapted derivative, ...
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      0answers
      61 views

      How to obtain mathematical expectation with the vector as random variable?

      In my study, I wish to get the mathematical expectation for the term below. The vector $\boldsymbol{z} \in \mathcal{C}^{N\times1}$ and $\boldsymbol z \sim \mathcal{CN}\left(\boldsymbol{0},\boldsymbol{...

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