# Questions tagged [stochastic-processes]

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

1,415
questions

**0**

votes

**1**answer

99 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\} \...

**3**

votes

**1**answer

121 views

### Quadratic variation of sum of random variables

Let $N = (N_t)_{t\geq 0}$ be a Poisson process and consider random variables $Z_n$, $n\in N$. Compute the quadratic variations $[X]_t$ where $X_t = \sum_{n=1}^{N_t}Z_n$.
What I did was plugging $X_t$ ...

**0**

votes

**2**answers

88 views

### Lower bounds on discrete time finite Markov chains hitting probabilities

I am interested in some general theorems related to lower bounds on discrete time finite Markov chains hitting probabilities (preferably ergodic chains , but not necessarily ), with references . ...

**2**

votes

**1**answer

130 views

### Conformal mappings and its singularity

I have a question about singularities of conformal mappings.
Let $\mathbb{H} \subset \mathbb{C} \cong \mathbb{R}^2$ be the upper half-place and let $D$ be a Jordan domain. Let $\varphi:\mathbb{H} \to ...

**2**

votes

**0**answers

89 views

### Conditioning on future events, strong Markov property, independence

I have a question on an argument appearing in this article P.
Setting
Let $S=(1,\infty) \times (-1,1) \subset \mathbb{R}^2$ and let $X=(\{X_t\},\{P_x\}_{x \in S})$ be a diffusion process on $S$. ...

**1**

vote

**1**answer

144 views

### Inequalities for moments of a certain integral

Let $X(t)$ be a stationary Gaussian process, $EX(t)=0$, the correlation function $R(\tau)$ is given. What bounds from above can be given for the $p$-th moment
($p>0, p \in \mathbb{R}$) of the ...

**3**

votes

**0**answers

50 views

### 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 ...

**2**

votes

**0**answers

18 views

### 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 ...

**3**

votes

**1**answer

91 views

### 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{...

**2**

votes

**1**answer

87 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 ...

**4**

votes

**1**answer

217 views

### 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 ...

**1**

vote

**0**answers

106 views

### 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,...

**0**

votes

**0**answers

31 views

### 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 ...

**1**

vote

**1**answer

144 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\...

**1**

vote

**1**answer

190 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 ...