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.

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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 ...
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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 ...
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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: ...
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36 views

Random walk in a random environment

Consider the random walk in a random environment $\{X_n\}$, that is $P(X_{n+1}=z+1|X_n=z)= \alpha_{z}$ and $P(X_{n+1}=z-1|X_n=z)=1- \alpha_{z}$, where $\{\alpha_{z}\}$ is i.i.d random variables, with $...
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31 views

birth-death process

This is from notes I don't understand why $e^{-h(\lambda _i+\mu _i)}=1$? and then the note shows that: But isn't $o(h)$ the probility of combinations such as 2 births and 1 death, 3 births and 2 ...
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14 views

Rate of convergence for martingales, “merging of opinions” results

Let $(\Omega, \mathcal{F})$ be a measurable space, and let $P$ and $Q$ be probability measures on this space. Let $(\mathcal{F}_{n})_{n \in \mathbb{N}}$ be a filtration on $\Omega$. Assuming that the ...
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39 views

Does anyone recognise this non-linear diffusion equation?

I'm doing some work on modelling cell migration, I've derived this particular form of a non-linear diffusion equation to describe the mean behaviour of a stochastic model I'm studying. I was wondering ...
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88 views

Probability that a Lévy-process is unbounded, zero-one law?.

For a Lévy-process, I need to prove that the probability that the trajectories are bounded on $[0,\infty)$ is either 0 or 1. Can you please help me? (The author says that this is a consequence of ...
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17 views

Markov process and Doob-Meyer decomposition

$(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq0}),\mathbb{P})$ - a filtered probability space supporting a 1-dimensional Brownian motion $B=(B_t)_{t\geq0}$, where \begin{equation} \mathcal{F}_t=\sigma(...
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1answer
26 views

Brownian noise perturbing a differential equation

The following one-degree-of-freedom oscillator is given; $$\ddot{x}+kx=w(t),$$ where, $k>0$ and $w(.)$ is a Brownian noise perturbing the system. Assume we want to study boundedness of the ...
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88 views

Etemadi's inequality

In another post an inequality referred to as "Etemadi's Inequality" is mentioned twice - in the original post as well as in the answer. However, the contexts of usage are such as to raise the question ...
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28 views

how to check if a point is deleted from a lattice before adding perturbations

Define lattice L to be either $\mathbb{Z}^2$ or $\mathbb{Z}^2\backslash\{(0,0)\}$. That is, all the points with integer coordinates on the 2D plane with the difference that the origin could be deleted ...
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1answer
27 views

Proving an SDE has a unique strong solution

I have the stochastic differential equation $$dX_t = \ln(1+ X_t^2) \, dt + X_t \, dB_t$$ In this equation, $X_0 = x$, and $x \in\mathbb R$. How can we show that this equation has a unique strong ...
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12 views

Bound moments of SDE

I have a rather messy SDE with solution $x(t)$ and I would like to bound its moments. Now using Ito I obtained approximately the following form: $$ \frac{d}{dt} \mathbb E |x(t)|^k \leq -C \mathbb E |...
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17 views

Expected value of the exponential of a Geometric Brownian motion

I am trying to compute the following expectation: $$ E[ \exp (A_T)], $$ where $A_T = - C \int_{0}^{T} \exp( 2 \alpha W_t - \alpha^2 t) dt $, with $C$ and $\alpha$ positive constants, $W_t$ a standard ...
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15 views

Predictable stopping time

Assume an increasing rightcontinuous $(X_t)_{t\geq 0}$ has the compensator $(A_t)_{t\geq 0}$. As saz pointed out, we want to assume that $A_t$ is continuous. Define the stopping time $\tau_s:=\inf\{t\...
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27 views

Black scholes model for down and out European call option using Monte Carlo

I tried to implement Matlab program computing the price of the European down and out call option using Monte Carlo and Euler discretization scheme. I have initial price S0=50, strike K=50, barrier ...
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19 views

What is the distribution of the maximum of scaled Gaussian random walk?

A Gaussian random walk is the sum of standard normal variables $$Z(n) = \sum_{i=1}^n X_i,$$ where $X_i\sim N(0,1)$. What I mean by a scaled Gaussian random walk is the following: $$ U(n) = \frac{Z(n)}{...
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1answer
27 views

Mean and Variance of SDE

How would I compute the mean and variance of the following SDE? $dX_t = \alpha X_t dt + \sigma dB_t$ I know $E[X_t]$ produces the mean and $E[(X_t)^2]$ produces the variance, but I'm not sure how to ...
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23 views

JL Doob / KL Chung paper: Fields Optionality and Measurability

I was reading the following paper: https://www.jstor.org/stable/2373011?seq=1#page_scan_tab_contents I had a question about a proof that is part of the paper. Here are some of the definitions and ...
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20 views

Finding best strategies in a problem about traffic lights

A problem came up in a course of a conversation with my friend. Suppose we have a street with several traffic lights, placed equidistantly one from another. Time of them being green $t_g$ and time of ...
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33 views

Hitting time or normalized Brownian motion (divided by square root of t)

What I mean by a normalized Brownian motion is the following: $$ U(t) = \frac{W(t)}{\sqrt{t}}, $$ where W(t) is a standard Brownian motion (with $\mu = 0$ and $\sigma = 1$). Is there a name for such a ...
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15 views

How can be a stopped martingale martingale?

A martingale has constant expectation, but if we stop a martingale, then after the stopping the process becomes a constant. So how can it remain a martingale?
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21 views

Convergence of random variable 5

If $Q < \frac{n}{m^2} X_n$ where $X_n$ is a sequence of random variables, $X_n \xrightarrow{a.s}1$, $0\leq Q \leq1$, $m=\omega(\sqrt{n})$ (The $\omega$ denotes the order, see here). Then, how can ...
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19 views

Exponential martingale, Lévy-process and stopping times, definition quesiton.

I feel there is some ambiguity for the definition of the exponential martingale for a levy process which I do not understand. For a Lévy process it can be shown that $E[e^{iuX_t}]=e^{t\eta(u)}$, ...
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19 views

Identity of Running maximum

let $X_t$ denotes a arithmetic Brownian motion process. I am wondering if the following identity is true ? $$ \mathrm{P}\left[\sup_{0 \le s \le t} \mathrm{e}^{X_t} < x\right] = \mathrm{P}\left[\...
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44 views

Almost sure convergence (correctness of an argument)

Is this statement correct? If $X_n \xrightarrow{a.s} c$, where $X_n$ is a sequence of random variables and $c$ is a constant, then we can conclude that since almost sure convergence implies on ...
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1answer
43 views

Finding the limit $\lim_{t\to ∞} \mathbb{E}[R_t]$ of an SDE

I have the SDE $$dR_t = (1 - \beta R_t)dt + \sigma dB_t$$ In this equation, $R_0 = r$ in which $r > 0$ Can someone please help me find the $\lim_{t\to ∞} \mathbb{E}[R_t]$? Thus far I have ...
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63 views

Differentiability of a simple value function driven by a diffusion

Consider a diffusion given by, $d X_t = \mu(X_t) dt + \sigma(X_t) dB_t$ $X_0 = x$. Suppose the functions $\mu$ and $\sigma$ are as follows - $f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \...
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1answer
42 views

Computing the expectation value of a stochastic process

I have a stochastic differential equation for which I have solved the process X$_t$. The SDE is as follows: $$ dX_t = \left( r\mu X_t + \frac{r(r-1)} 2 \sigma^2 X_t \right) \, dt + r\sigma X_t\,dB_t, ...
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Why is a Brownian Motion completly specified by its increments?

Without any formal base frame I want to know why a Brownian Motion is completly specified by its increments? Even though I think here is no need to give a Reference you can see the following. ...
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1answer
48 views

What does it mean for a pdf to have this property?

What does it mean for a probability density function $f(x)$ to have the following property? $$1+\int_{x=0}^{\infty}x^2 \left(\frac{f'(x)^2}{f(x)}-f''(x)\right)dx>0$$ I have tried a lot to ...
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1answer
14 views

Reference needed for properties of Convergence of Random Variables

Does anybody know a good reference for properties of convergence of random variables? For example, if $X_n$ converges almost surely (a.s) to $X$ and if $Y_n$ converges a.s to $Y$, then $X_n Y_n$ ...
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21 views

Property of renewal point processes

For a renewal process where $f(t)$ is the number of arrivals in time $t$ and $S_k$ is the $k^{th}$ time of arrival, how can we show: $$f(\alpha S_k)/k \xrightarrow{\text{a.s.}}\alpha $$ as $k \...
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1answer
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 ...
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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 ...
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1answer
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 ...
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1answer
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 ...
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3answers
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, ...
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1answer
98 views

quadratic variation of brownian motion doesn't converge almost surely

I just came across the following remark: If $(B_t)_{t\geq0}$ is a one dimensional Brownian motion and if we have a subdivison $0=t_0^n<...<t_{k_n}^n=t$ such that $\sup_{1\leq i\leq k_n}(t_i^n-t_{...
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37 views

The hitting time $T-\tau^{l}$ has the same distribution as $\min\{\tau^{f},T\}$ regarding an Poisson Process.

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq T}$ is a Filtration, with $T < \infty$. On that prbability space we want to ...
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26 views

Comparison of stopped sub- and supermartingales when the future is discounted

Suppose we have a submartingale $X=\{X_t\}_t$ and a supermartingale $Y=\{Y_t\}_t$ which are adapted to the same filtration on a bounded set and have a common initial value $X_0=Y_0$. Suppose that $\...
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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 ...
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20 views

Does a linear operator on probability measures determine a Markov kernel?

Let $(\Omega, \mathcal{F})$ be a measurable space. Let $M$ be a linear operator on the space of probability measures on $(\Omega, \mathcal{F})$, i.e. for $\alpha \in [0,1]$ and probability measures $...
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45 views

Why are we allowed to multiply the differential form of an SDE by a function?

Suppose for example that we have the following SDE: $$dX_{t} = a(X_{t})\,dt + b(X_{t}) \,dB_{t}. $$ What rigorous justification is there for then saying, for example, that we can multiply both sides ...
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65 views

Will this integral be progressively measurable?

Assume you have a function: $F(t,x,\omega)$: $[0,T]\times E\times \Omega \rightarrow \mathbb{R}$, which is predictable (predictable is explained below). Each of the three spaces can be viewed as 3 ...
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24 views

Notation in the stochastic derivatives in the mean square sense

The stochastic limit $X$ in the mean square sense is given the definition: For a row (sequence?) of stochastic variables $X_n$ if $\displaystyle\lim_{n\to\infty}E\{(X_n-X)^2\}$ = 0 and we write $\...
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2answers
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 ...
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1answer
49 views

Binary Stochastic Programming with Independent or Positively Correlated Co-efficients

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 ...
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24 views

Mean Square Displacement of Alternating Random Walk

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 ...