# Questions tagged [stochastic-processes]

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

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### Random Two-Player Asymmetric Game

About half a year ago I asked a question on MSE about a random two player game. At the time, the question received some attention and some progress was made, but was not resolved completely. I have ...

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### Distribution Functions in Poisson Process

We consider definition of Poisson processes that satisfy Condition 0 and 1 according to Billingsley section 23 . How find out the densities of $A_t , B_t , L_t$ defined as in problems section of ...

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### Total offspring of Poisson multitype branching process

A normal branching process $Z_n$ initialized with $Z_0=1$ and offspring generated from $Pois(p),p<1,$ has a total progeny / total off spring distribution
$$X=\sum_{n=0}^\infty Z_n$$
$X\in \mathbb{...

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votes

**1**answer

81 views

### Heavy tail central limit theorem

I am looking for a proof based on characteristic functions for the generalized central limit theorem when the second moment does not exist, in which case one ends up with a power law rather than a ...

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**1**answer

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### A balls into bins problem with combinatorial constraints

We are given $m$ balls and $n$ bins, with $m \ge n$. Each bin can contain at most $c$ balls (we assume that $c$ is an even integer). In a sequential fashion, at each time step, one ball is placed into ...

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### How to judge the solution process of an SDE to lie on the sphere?

Consider the following SDE on $\mathbf R^d$:
\begin{equation}\tag{*}
dX_t^i = -\frac{d-1}{2}X_t^i dt + \sum_{j=1}^d(\delta^{ij}-X_t^iX_t^j)dW_t^j, \quad i=1,2,...,d,
\end{equation}
where $W = (W^1,W^2,...

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### A modification of Kolmogorov's continuity criterion for $C_{tem}$

I am wondering about how to prove a modification of Kolmogrov's continuity criterion in order to also being able to quantify the growth behaviour of the process. In particular, I am interested in the ...

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**1**answer

137 views

### Feynman-Kac formula for lattice heat equation with non-diagonal potential

Suppose that $X$ is the continuous-time simple symmetric random walk on the lattice $\mathbb Z^d$ (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let
$$u(t,x):=\mathbf E\...

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**1**answer

177 views

### Obtaining generator matrix and first-passage time distribution for CTMC?

Setup:
I have a model of a biological process described by two ODEs as follows:
$$\dot{X_1} = (\beta_1-d-1)X_1 + 2X_1^2 - X_1^3 + dX_2$$
$$\dot{X_2} = (\beta_2-d-1)X_2 + 2X_2^2 - X_2^3 + dX_1$$
I ...

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### Convergence rate of the smallest eigenvalue of an integral of a multivariate squared Brownian Motion

I am interested in deriving the convergence rate of the smallest eigenvalue of a sequence of random matrices with diverging dimension. More precisely, let $W_n(r)$ represent an $n$-dimensional ...

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### Comparison of Rademacher processes

Suppose that $T$ is a bounded set in $\mathbb{R}^n$ and $f,g$ are two nonnegative functions such that $0\leq f(x)\leq g(x)$ for all $x\geq 0$.
Let $\epsilon_1,\epsilon_2,\dots,$ be a Rademacher ...

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### Derivative of stochastic process in $L^p$ coincides with sample path derivative

In the article Random ordinary differential equations, by J.L. Strand (1970) (it is available at https://core.ac.uk/download/pdf/82447522.pdf), it is stated the following result, which relates ...

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**1**answer

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### If a strong Markov process reaches a Borel set a.s., can it be restarted from that set?

Let $X$ be a strong Markov process on $E$, and $B\in \mathcal B(E)$. Suppose that, for some $x\in E$,
$$
P_x(\exists t\ge0 \text{ such that } X_t\in B)=1.
$$
My question: Does there exist a stopping ...

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### Sturm-Liouville-like Eigenproblem

Consider the piecewise-deterministic Markov process on $\mathbf{R}$ which
moves according to the vector field $\phi (x) = 1$,
experiences events at rate $\lambda(x) = 1$, and
at events, jumps ...

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**1**answer

81 views

### Large time behavior of Girsanov type Geometric Brownian Motion with time-dependent drift and diffusion

Recall the Geometric Brownian Motion $X={\rm e}^{\mu W+\left(\sigma-\frac{\mu^2}{2}\right)t}$. If $\sigma<\frac{\mu^2}{2}$, then $X$ tends to 0 almost surely. But if we consider the following case,
...