Questions tagged [stochastic-processes]
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
82 questions from the last 365 days
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Explosion of the moment of double stochastic exponential
We consider the stochastic system
$$\frac{dS_t}{S_t}=-R_t\,dW_t,$$
with
$$dR_t=-R_t\,dt-R_t\,dW_t, \quad R_0>0.$$
We conjecture, and would like to show that
$$\mathbb{E}[S_t^2]
=
S_0^2\,\mathbb{E}\...
- 696
2
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0
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53
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On a skew diffusion process
Let $p \in [0,1]$, and let $X=(X_t)_{t \ge 0}$ be a skew Brownian motion with parameter $p$ defined a probability space $(\Omega,\mathcal{F},P)$. It is known that $X$ can be defined as a solution to ...
- 21
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0
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89
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Transfer of stochastic integrals under identical distributions
Let $(\Omega^i, \mathcal{F}^i, \mathbb{P}^i)$ $(i = 1, 2)$ be two probability spaces. On each space we have a predictable process $X^i$ and a continuous semimartingale $Y^i$. Assume that the pairs $(X^...
- 155
5
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0
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95
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Upper and lower left and right derivatives of brownian motion
Let $B$ be a Brownian motion. For $t>0$, let
$$D^*B_t = \limsup_{h \to 0+} \frac{B_{t+h}-B_t}{h}, \quad D_*B_t = \liminf_{h \to 0+} \frac{B_{t+h}-B_t}{h},$$
$${}^*DB_t = \limsup_{h \to 0-} \frac{B_{...
- 4,018
2
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0
answers
56
views
Which processes admit good conditional growth rates bounds?
Fix $T>0$ and let $X_{\cdot}:=(X_t)_{t\ge 0}$ be a semi-martingal adapted to a filtered probability space $(\Omega,\mathcal{F},\mathbb{P},\mathcal{F}_{\cdot})$ which is the unique strong solution ...
- 4,902
5
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3
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255
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Comprehensive research-level reference on martingale concentration inequalities (discrete time)
I am aware that many authors and papers treat particular aspects, but I would prefer sources that collect and contrast the approaches and results in one place, and indicate the relationships between ...
- 63
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35
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Does visibility proportion of a Gaussian graph hypersurface depend only on gradient distribution?
Let
\begin{equation}
h : \mathbb{R}^n \rightarrow \mathbb{R}
\end{equation}
be a stationary mean-zero Gaussian field with almost surely $C^1$ sample paths, and let
\begin{equation}
S = \{ (x, ...
- 11
5
votes
2
answers
291
views
Expected cost of a certain branching process
Consider the following stochastic process: I initially have $m = \lfloor pn \rfloor$ red points and $n$ blue points uniformly distributed in the unit interval $[0,1]$, with $0 < p < 1$. We ...
- 4,263
1
vote
0
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40
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Existence of raw versions for processes adapted to the augmented filtration
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space with $\mathbb{F}=(\mathcal{F}_t)_{0\le t\le ∞}$.
Denote by $\mathcal{F}^{\mathbb{P}}$ the $\mathbb{P}$-completion of ...
- 155
4
votes
1
answer
155
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Asymptotic success probability in a full-information “discrete uniform” secretary problem
Consider the following variant of the full-information secretary problem.
Let $M \in \mathbb{N}$, and suppose we observe a sequence of $M$ i.i.d.\ random variables
$$
X_1, X_2, \dots, X_M,
$$
each ...
3
votes
1
answer
241
views
Do random process and its modification agree almost surely at stopping time?
The question arises when I am learning a proof of the Doob--Meyer decomposition. This theorem is classical and the proofs are standard, but I am not able to find a satisfactory answer from various ...
- 328
1
vote
0
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73
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A lemma on measurability with respect to the filtration on path space
Consider the path space $W = C([0,\infty); \mathbb{R})$ equipped with the topology of uniform convergence on compact sets. Let $\mu$ be a Borel probability measure on $W$, and let the $\sigma$-...
- 30.8k
3
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80
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Tighter bounds on the Expectation of a Supremum over all Markov Kernels
Consider a finite alphabet $\mathcal A$ and let $m,n\in\mathbb N$. For every $x\in \mathcal A^m$ let $Y_x\sim\mathcal U(\mathcal A^n)$, i.e. $Y_x$ is uniformly distributed on the strings of length $n$....
- 71
11
votes
1
answer
636
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A question about random walks on $ \mathbb{R}^2$
Let $\mu$ be a probability measure on $ \mathbb{R}$ with $\mu(\{ 0\})<1$. Otherwise, no further assumptions (e.g. on the existence of moments) are imposed on $\mu$. Let $X_n$ and $Y_n$ denote two ...
- 6,536
5
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1
answer
308
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More on a renewal process
The setup is part of that in the previous question, but the question is now different.
Let $X_1,X_2,\dots$ be independent random variables (r.v.'s) each uniformly distributed over the interval $[0,1]$....
- 147k