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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Brownian motion starts fresh variant

It is a standard result that if $W_t$ is a Brownian Motion and $S$ is a stopping time of the standard filtration $F_t$ then we have that $B_t = W_{S+t} - W_S$ is a Brownian Motion. I quote the ...
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26 views

Chances of winning if only a certain amount of people can participate

So we got 2 million (2_000_000) cornflakes boxes. Hidden in 6 of those cornflakes boxes is a gold nugget. Due to a wrong printing, 500_000 cornflakes boxes don't get into the market. So there are ...
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48 views

What is a.e. a.s

I am reading a paper which uses almost everywhere almost surely (a.e.,a.s.) simultaneously, I am not quite sure what it means then. To be specific, they consider a stochastic process $\{X_t\}$ such ...
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14 views

A question in a textbook about Blumenthal 0-1 law for a general Markov process

This question came up as a result of reading this question . Here is the Blumenthal 0-1 law in the book Stochastic Processes by Richard F. Bass. Proposition 20.8 Let $(X_t , \Bbb{P}^x)$ be a ...
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30 views

Independence Lemma, is it non-trivial?

I'm reading Steven E. Shreve's "Stochastic Calculus for Finance II, Continuous-Time models", and a bit confused on the Independence Lemma (Lemma 2.3.4). The lemma says: Lemma 2.3.4 (Independence). ...
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39 views

Why does a Gaussian process have a gradient whose determinant is Gaussian?

I'm trying to understand something in Adler and Taylor's book, Random Fields and Geometry. Let $T \subset \mathbb{R}^N$ be a compact parameter set (for simplicity, suppose it is a closed hypercube) ...
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29 views

Integral of different types of noise

I have been learning about the Wiener process and read this on Wikipedia; "the Wiener process is used to represent the integral of a Gaussian white noise process" It got me wondering what the ...
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40 views

An exponential martingale

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
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18 views

Writing down the transition matrix of a discrete Markov chain

Please consider the following scenario: One person is walking along a discrete circle induced by $\mathbb{Z}/n\mathbb{Z}$ In each round we roll a dice with $w\in\left\{2,\ldots n\right\}$ sides If ...
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45 views

Geometric BM tends to zero but is strictly positive a.s.?

The process $\{S_t\}_{t\ge0}$ following $dS_t = \sigma S_tdW_t$ with $S_0>0$ has the solution $$S_t=S_0 e^{-\frac12\sigma^2t+\sigma W_t}$$ Now for any $\epsilon>0$ we have $$\mathbb ...
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67 views

Multipe Ito Integrals

Im working on a Lemma 10.8 in the Book "Numerical Solution of Stochastic Differential Equations by Kloeden And Platen" I have been stuck on one point. Can somebody help me to understand how he moved ...
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41 views

Fractional Brownian motion---construction via Hilbert space?

The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms: Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, ...
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74 views

Write the Hamilton Jacobi Bellman equation

Consider the following stochastic optimal control problem. \begin{equation} V(t,x) = \max_{u}\,\, \log \left(\mathbb{E}\left[\int_{0}^{T} u^{2}(t)dt\right]\right) \end{equation} subject to the ...
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1answer
32 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
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49 views

On the confinement of the solution of a SDE

One considers the stochastic differential equation $$dX_{t}=(1-X_{t})X_{t}dB_{t},$$ with $B$ Brownian motion, and one assumes that $0\leq X_{0}\leq1 $. One wants to show that ...
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34 views

Second (centered) moment for martingales

Take the process ${x}_t$ following geometric Brownian motion (GBM) $$x_t=\mu x_t \,dt+\sigma x_t \,dW_t$$ with $x_0>0$ known. It has first moment equal to $$\text{E}[x_t]=x_0 e^{\mu t}$$ and second ...
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31 views

partial derivative of stochastic variable inside an integral

Very simple question, is it correct to take a partial derivative of stochastic variable inside an integral. If not, why? is$ \frac {\partial}{\partial R} \int_q^Q R(v) dv = \int_q^Q dv$ ? where R is ...
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23 views

good primer or intro on quenched stochastic processes

I was hoping that someone could recommend a good intro to "quenched" stochastic processes. I am using quenching in the sense of condensed matter physics where systems can present quenched disorder.
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28 views

A discrete time Markov chain with such a transient state that $\mathbb P(T_i<\infty \ | \ X_0=i) \neq 0$

All examples of discrete time Markov chains my text provides are where $S$ is finite, and as far as I can tell, it makes all transient states have $$\mathbb P(T_i<\infty \ | \ X_0=i) = 0.$$ Are ...
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1answer
28 views

Is there a conditional random variable?

Let $(\Omega,\Sigma,\mu)$ be a sample space. Let $F $ be a $\sigma$-subalgebra of $\Sigma$ and let $X$ be a real-valued random variable. So what does it mean $X/F$? How is this defined ...
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23 views

Why does this hold for the mean hitting time?

Let $X$ be a Markov chain and $T_A$ the hitting time. My text uses this in a proof: $$\mathbb E[T_A \ | \ X_0=k ] = \sum_{l\in S} \mathbb P(X_1=l \ | X_0=k\ )(1+\mathbb E[T_A \ | \ X_0=l ])$$ and I ...
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1answer
19 views

How to prove that a jump diffusion has infinite total variation?

We have a jump diffusion: $X_t=bt + \sigma W_t + Y_t$ where b is the drift parameter, $\sigma$ the diffusion parameter, $W_t$ a Wiener process and $Y_t$ a CPP (compound Poisson process). We know ...
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21 views

Simple Random Walk probability of first visit

Consider a particle that moves according to a simple random walk. Denote by $X_n$ the position of the particle immediately after step $n$. Assume that $X_0 = 0$ and that, at each step, the ...
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55 views

Eigenvalue markov chain

I have a questions: We said that if we have a positive recurrent Markov chain, then there is a unique stationary distribution. 1.) Does this mean that if I have several positive recurrent classes, ...
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29 views

Inferring transition rates from continuous markov chain question

A house has 2 rooms of similar sizes with identical air conditioners equipped with thermostats which turn on and off as needed to maintain the temperature in each room to a desired level of 22 ...
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1answer
56 views

random walk with sticky barriers

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are ...
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18 views

Filtrations and sigma algebras [duplicate]

I have a doubt concerning the basilar aspects of the filtrations in the stochastic theory. A filtration is an increasing sequence of $\sigma$-algebras on a measurable space. That is, given a ...
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19 views

Transition matrix in left-right hidden semi-Markov model

I'm developing a hidden semi-Markov model left-right . In a left-right model a sequence of $M$ states starts in state $1$ and ends in state $M$, with no repetition of states. Since the model is ...
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1answer
19 views

Is there an example that shows that the optional stopping theorem fails for finite (unbounded) stopping times?

Is there a martingale $M=(M_t)_{t\geq 0}$ and finite stopping times $S,T$ with $S \leq T$ a.s. such that $\mathrm{E}(|M_T|)<\infty$, but $M_S \neq \mathrm{E}(M_T|\mathcal{F}_S)$ a.s.? I found a ...
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40 views

Transient/Recurrent Markov chain

I am currently studying the concept of recurrent and transient states and was wondering about the following: Is this concept dependent on the initial distribution? Let me take this example: You can ...
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28 views

number of ones with neighbours in a random binary string

Consider a sequence of i.i.d. random variables $(\xi_i)_{1 \leq i \leq L}$ such that $\xi_1 \in \{0,1\}$ and $P(\xi=1)=p$. Introduce the function $N : \{0,1\}^{L} \rightarrow \mathbb{N}$ which counts ...
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29 views

independence at equal and different times

this is a question about stochastic processes. Let's call $A(t)$ and $B(t)$ two stationary processes and denote by $E[*]$ the expectation value. Suppose we know that $E[A(t)B(t)]=0$ for every $t$. The ...
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1answer
61 views

Why this function is continuous?

Let $(\Omega,\Sigma,\mu)$ be a sample space and let $L^2= \lbrace f:\Omega \rightarrow R / \int f^2d\mu <\infty \rbrace$ be a Hilbert space. Let $L_n=L^2\times L^2 \times .... \times L^2$ ($n$ ...
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32 views

Scaled integrated Brownian motion has limit

Let $B$ be a standard Brownian motion and put $$X(t)=\frac{1}{\sqrt{t}}\int_{0}^{t}f(B(s))ds,$$ where $f \in L_1(\mathbb{R}^{1})$ and $\int f(x)dx=1$. Show that $$ \lim_{t \rightarrow \infty} EX(t) ...
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22 views

Can the ergodic theorem for Markov chains be proved with linear algebra?

This theorem is in my book, let me just say that it is for discrete-time Markov chains, that are time-homogeneous. Ergodic is defined in the book as being positive recurrent and aperiodic. The ...
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18 views

find a probability measure such that $(W_t + \sqrt{3t+2})_t$ is a Wiener process wrt to $P$

Like the title says: suppose $W_t$ is a Wiener process on the space $(\Omega, F, Q)$. We want to find a different probability measure $P$ on $\Omega$ such that $$ (W_t + \sqrt{3t + 2})_{0 \leq t \leq ...
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27 views

Strong Markov property and its meaning

Given a sequence of random variables $(X_n)_n$ (fulfilling the Markov property) and a stopping time $\tau$ such that $P(\tau < \infty)=1$, we have that ...
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29 views

Probability of not reaching completion in Markov process

This question is supposed to be easy but is very hard for me. The Norwegian Skating Association has mass produced certain "collectors' cards" with all $N$ speedskaters (Norwegian as well as ...
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1answer
35 views

Markov chain exercise

Hello i have this Markov chain exercise: Basically we can always move up 1 step, but there is always a possibility that we will go down to the first state 0, the Markov chain consists of N states. ...
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1answer
42 views

Brownian motion motivation of construction

I have read Stochastic Differential Equations by Bernt Oksendal It constructs Brownian motion by Kolmogorov extension theorem by consider $p(t,x,y)=(2\pi t)^{-n/2} e^{- \frac{|x-y|^{2}}{2t}}$ Why ...
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43 views

Poisson/ jump process distribution for process $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$

For the process: $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$, where $X(t)$ is a poisson process with paramater $\lambda$, and: $J_k$ are i.i.d . random variables (jumps). $B(t)$=brownian motion. I want to ...
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31 views

Girsanov theorem conditions

If we have an adapted function $f(t)$ such that $\int_0^t f(s)ds\,<\infty$, then the Girsanov exponent can be defined: $$ Z(t):=\exp\left( \int_0^t f(s)dW(s) - \frac{1}{2} \int_0^t ...
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1answer
29 views

Ito's process and martingale [duplicate]

Let ${W_t}$ be 1 dim Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. My try is below. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why ...
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1answer
25 views

SDE transformation using a primitive of a function?

Consider the following SDEs : (E) : $dX_t = (\alpha b(X_t) + {1\over2}b(X_t)b'(X_t))dt + b(X_t)dB_t$ (E') : $dY_t = \alpha dt + dB_t $ prove that E can be transformed to E' using : $ ...
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44 views

Understanding of Brownian Motion

My background is functional analysis rather than probability, but I would like to understand what is a Brownian motion. Below I'm giving my current understanding, can anyone verify whether I'm ...
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1answer
17 views

Finding number of points in a bounded set when number of points in the unbounded set are known.

Consider a random distribution of points in a Random 2D plane. I would like to find the number of points in a circle within this plane. Can anybody helps in solving the problem? Regards
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30 views

What is the probability that $k$ events have occurred at time $t$, i.e., $\Pr[N(t)=k]$?

Assume that the starting time $T_0=0$. There are $n$ events that occur sequentially at time $T_1$, $T_2$, …, $T_n$, ($T_k\geqslant T_{k-1}$). Suppose the time intervals $\Delta{T_k}\,(\Delta{T_k} = ...
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23 views

Convergence of cadlag functions to a continuous one

I want to prove a convergence result as simple as possible. Using a straight forward approach I can prove the result, but I'm 100% sure that there must be a much simpler (and shorter) argument using ...
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1answer
39 views

Literature on Sabermetrics in baseball

For my bachelor's thesis, I would like to study the use of Sabermetrics in baseball. I was fascinated by the book 'Moneyball: The Art of Winning an Unfair Game' by Michael Lewis, and to me, it ...
2
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1answer
57 views

Strict local martingale implies $\mathbb E_t[S_u]<S_t\ \forall t<u$

Is it true that if $S$ is a strict local martingale (i.e. it is a local martingale but not a true martingale) such that $S_t\ge 0\ \forall t$, then we have $$\mathbb E_t[S_u]<S_t\quad \forall ...