# Tagged Questions

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

18 views

### Feller boundary conditions

The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...
23 views

### Integrated Brownian Bridge is a Gaussian Process

Let $W(t),t \in [0,1]$ be a (Standard) Wiener Process. The Brownian Bridge $B(t), t \in [0,1]$ can be constructed via $B(t):=W(t) - t \cdot W(1)$ and is a Gaussian process with zero mean and ...
28 views

### How to show that $S_k = \inf \{t \geq 0 | \|X(t)\| \geq k \} \to \infty$ as $k \to \infty$ a.s.

stack.exchangers! I am currently working my way through the proof given by Karatzas and Shreve (1988) of the Feynman-Kac Theorem (Theorem 5.7.6). However, I am missing out on the following problem: ...
36 views

35 views

### How to deduce the expectation of a stochastic equation [closed]

I am having a difficult time deducing the expectation, $\mathbb{E}[R_t]$, of the following stochastic equation: $$dR_t = (1 - \beta R_t)dt + \sigma dB_t$$ $R_0 = r$, with $r > 0$. Please help me ...
51 views

### Finding a unique strong solution

I am brushing up on my stochastic approximation. I am having a hard time with the following problem. I have the equation dX$_t$ = ln(1+ X$_t^2$)dt + X$_t$dB$_t$ X$_0$ = x, with x ∈ ℝ I know that ...
31 views

### Finding Stochastic processes

I have the following differential equation dX$_t$ = (r$\mu$X$_t$ + $\frac{r(r-1)}{2}σ^2X_t$)dt + rσX$_t$dB$_t$, X$_0$ = x, with x > 0. Here, r>0. I am having trouble figuring out how to find the ...
52 views

### Almost sure convergence equivalence

Are the following statements equivalent? $$a) X(t)/t\xrightarrow{a.s} c$$ $$b) X(t)\xrightarrow{a.s} t c$$ where $c$ is a constant and $X(t)$ is a sequence of random variable. By ...
266 views

### What does it mean to integrate a Brownian motion with respect to time?

I am reading about stochastic process, but could not make sense if one equation I encountered. Can anyone help me understand it? The equation states that suppose R(s) is an interest rate process, ...
98 views

30 views

### Difference between stationarity and independence properties for Brownian motion

What is the difference between the stationarity and independence properties of proving that a stochastic process $W(t)$ is Brownian motion? I only understand that for stationarity, we're trying to ...
20 views

61 views

### Is there a boundary in probability for Brownian motion?

For a standard Brownian motion $W_t$ and a given crossing probability $\alpha < 1$, I want to have a boundary function $f(t) > 0$, such that the probability that $W_t$ ever crosses the boundary ...
A manufacturer can select a maximum of $N$ stores to fulfill orders from a total of $M$ stores who are looking for inventory, $N\le M$. The case when $N\geq M$ is trivially solved when all stores ...
Consider a 1D random walk with varying steps: the length of the steps is $A$ a fraction $\gamma$ of the time, and $B$ the rest of the time. If $\gamma = 0$, the mean squared displacement approaches ...