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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Is a Markov process a random dynamic system?

A random dynamic system is defined in Wikipedia. Its definition, which is not included in this post for the sake of clarity, reminds me how similar a Markov process is to a random dynamic system just ...
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461 views

Right continuous version of a martingale

This is an exercise in chapter 2 of the book "Continuous Martingales and Brownian Motion" by Revuz and Yor: Consider the probability space $([0,1], \mathcal{B}([0,1]), dx)$, where $dx$ denotes ...
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When is a stochastic process defined via a SDE Markovian?

I was wondering when a stochastic process defined via a SDE is Markovian? The SDE may involved Ito integral, Lebesgue integral, jump component, and any other things. The reason I ask this question is ...
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205 views

Proving Galmarino's Test

Galmarino's Test gives a condition equivalent to being a stopping time. It says: Let $X$ be a continuous stochastic process with index set $\mathbb{R}_+$ (i.e. each sample path is a continuous ...
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310 views

A probability question

Suppose $X_1, X_2, ...,$ are IID random variables with $P(X_n=1)=p$ and $P(X_n=2)=1-p$. Let $S_n=\sum_{i=1}^n X_i$. I was wondering how to find $P(S_n \neq z, \forall n \in \mathbb{N})$ for some ...
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205 views

Is a probability of 0 or 1 given information up to time t unchanged by information thereafter?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_n\}_{n \in \mathbb{N}}, \mathbb{P})$, let $A \in \mathscr{F}$. Suppose $$\exists t \in \mathbb{N} \ \text{s.t.} \ E[1_A | \...
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0answers
72 views

No drift brownian motion problem

Given two same brownian motion with no drift and different variances: $$dG_1= \sigma_1 G_1 dW $$ $$dG_2= \sigma_2 G_2 dW $$ and two barriers $P_1 > P_2$ assuming that $ \sigma_1 > \sigma_2 $ ...
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182 views

Survival probability up to time $n$ in a branching process.

Let $\{Z_n : n=0,1,2,\ldots\}$ be a Galton-Watson branching process with time-homogeneous offspring distribution $$\mathbb P(Z_{n,j} = 0) = 1-p = 1 - \mathbb P(Z_{n,j}=2), $$ where $0<p<1$. That ...
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1answer
378 views

Proof that the predictable sigma algebra is also generated by continuous and adapted processes

I'm reading George Lowther's blog and have a question about the proof of lemma 2. We want to verify that the predictable sigma algebra is also generated by the continuous and adapted processes. One ...
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2answers
842 views

Generated $\sigma$-algebras with cylinder set doesn't contain the space of continuous functions

Consider $\mathbb R^{[0,1]}$ the space of all functions from $[0,1]$ to $\mathbb R$ and the cylindrical sigma algebra $\mathcal B$ on it. The question is: how to prove that $C[0,1]\notin \mathcal B$...
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Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
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1k views

Verifying Ito isometry for simple stochastic processes

It is known that stochastic integral must satisfy the isometry property which is $$ \mathbb{E}\left[ \left( \int_0^T X_t~dB_t\right)^2 \right] = \mathbb{E} \left[ \int_0^T X^2_t~dt \right] . $$ I am ...
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2k views

Expectation value of a product of an Ito integral and a function of a Brownian motion

this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated. Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
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575 views

Is a predictable process adapted?

Let us consider a measurable space $(\Omega, \mathcal{F})$, with a filtration $(\mathcal{F}_t)_t$ of sub $\sigma$-algebras of $\mathcal{F}$. The predictable $\sigma$-algebra $\mathcal{P}$ is the $\...
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266 views

The continuity of the expectation of a continuous stochastic procees

Let $X_t$ be a continuous stochastic process on a filtered space $(\Omega, \mathcal F, \mathcal F_t, \mathbb P)$. Is $\mathbb E[X_t]$ necessarily a continuous function? My first answer would be no. ...
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2answers
2k views

A question regarding the hitting time formula in brownian motion

Let $\tau_a=\inf\{t: B_t=a\}$, the hitting time of the standard Brownian motion to reach the boundary $a$. This is easily derived $$E(e^{-\lambda \tau_a})=e^{-|a|\sqrt{2\lambda}}$$ But I am having ...
3
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205 views

Why is the following example of a Markov process not strong Markov

$X(t) := 0 \;\; (t \leq \tau),\;\; t - \tau\;\;(t \geq \tau)$ with $\tau$ exponentially distributed. Then X has the Markov property but not Strong Markov Property. But why ???? Can someone kindly ...
3
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1answer
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covariance function for Brownian motion

What would the covariance function be of $V(t) = (1-t) B[t/(1-t)]$ if $B(t)$ is standard Brownian motion. Also $t$ is between $0$ and $1$. Thanks for the help! EDIT: Here is where I am stuck: I ...
2
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2answers
37 views

What does it mean for a Poisson point process $\Phi$'s points in $A$, conditioned on $\Phi(A)=k$ to be uniform?

I've read that if $\Phi$ is a Poisson point process (on $\mathbb{R}^d$, say), then conditional on there being $k$ points in some $A \subseteq \mathbb{R}^d$, the positions $X_1,\ldots,X_k$ of these ...
2
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1answer
75 views

Impossible stochastic process

I am trying to prove that a stochastic process with the following properties cannot exist. Let $\{X_t: 0 \leq t \leq 1 \}$ be a stochastic process such that i) $X_s$ and $X_t$ are independent ...
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1answer
194 views

Martingale formulation of Bellman's Optimality Principle

Related question: Deducing an optimal gambling strategy (using martingales). What I tried: For no 2, if $\ln Z_n - n \alpha$ is a supermartingale, then for $m < n$, $$E[\ln Z_n - n \alpha | \...
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1answer
62 views

positive martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}\,dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2\,ds}$$ is a martingale which is positive and has a mean=1, where $\theta_s$ is ...
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1answer
137 views

Asymmetric Random Walk / Prove that $E[T:= \inf\{n: X_n = b\}] < \infty$

Given random variables $Y_1, Y_2, \ldots \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
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2answers
148 views

Solving the SDE $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$

How to solve $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$ together with the initial condition $X(0) = X_0$.
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0answers
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Why do people all the time exploiting almost sure properties of a stochastic process as if they were sure properties?

All the time, I see people working with a given Brownian motion $(B_t)_{t\ge 0}$ on a fixed probability space $(\Omega,\mathcal A,\operatorname P)$ and suddenly exploiting its almost sure properties ...
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1answer
83 views

Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions. However, I have the following statement ...
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1answer
108 views

Flea on a triangle

"A flea hops randomly on the vertices of a triangle with vertices labeled 1,2 and 3, hopping to each of the other vertices with equal probability. If the flea starts at vertex 1, find the probability ...
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27 views

Characterizing superposition of two renewal processes

This is a follow-up question of "When superposition of two renewal processes is another renewal process?". How can we characterize the superposition of two renewal processes? The superposition ...
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453 views

PDE - Feynman-Kac vs. finite difference methods

I've heard that in greater than three dimensions, it's more efficient to solve a second-order parabolic PDE using a Monte-Carlo method based on the Feynman-Kac formula that it is to use finite ...
6
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1answer
361 views

Feller continuity of the stochastic kernel

Given a metric space $X$ with a Borel sigma-algebra, the stochastic kernel $K(x,B)$ is such that $x\mapsto K(x,B)$ is a measurable function and a $B\mapsto K(x,B)$ is a probability measure on $X$ for ...
5
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1answer
517 views

Doob's inequality in probability

In the book "Optimal Stopping and Free-Boundary Problems" there is given Doob's inequality of the following form. Let $X = (X_t,F_t)$ be a submartingale. Then for any $\varepsilon>0$ and each $...
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725 views

Is the condition “sample paths are continuous” an appropriate part of the “characterization” of the Wiener process?

Wikipedia has separate articles on "Brownian motion" and "Wiener process" (http://en.wikipedia.org/wiki/Brownian_motion and http://en.wikipedia.org/wiki/Wiener_process ). I am not an expert, but that ...
5
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1answer
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Why does a time-homogeneous Markov process possess the Markov property?

Klenke defines (Definition 17.3, p. 346) a time-homogeneous Markov process independently, rather than as a special case of a stochastic process that possesses the Markov property (Definition 17.1, p. ...
4
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1answer
640 views

Using Central Limit Theorem

Can anyone help me with it: Using the central limit theorem for suitable Poisson random variables, prove that $$ \lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}=1/2$$ Thanks!
4
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1answer
87 views

Conditional probability branching process

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
4
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1answer
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covariance of integral of Brownian

What is the covariance of the process $X(t) = \int_0^t B(u)\,du$ where $B$ is a standard Brownian motion? i.e., I wish to find $E[X(t)X(s)]$, for $0<s<t<\infty$. Any ideas? Thanks you very ...
3
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1answer
79 views

Lévy Process existence of the expectation of the supremum of the past process.

Given a Lévy Process $X_{t}$ in $\mathbb{R}^{d}$, with $X_{t}^{*}:=\sup_{s\in[0,t]}|X_{s}|$. I want to show, that for $t>0$ with $E[|X_{t}|]<\infty$ for $t>0$, then $E[X_{t}^{*}]<\...
2
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1answer
323 views

A bound for the probability that a Brownian motion stays in an interval

Suppose I have a Brownian motion $X_t$ with $X_0=0$. Let $T$ be the first exit time of the interval $[-1,1]$. I'm trying to get a "quick" lower bound for the probability that $T$ is very large which ...
2
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2answers
137 views

sub martingales and more

This is a problem on sub-martingales. Given : $X_n = X_0 \mathrm{e}^{\mu S_n}$, $n= 1,2,3,\ldots$, where $X_0 > 0$ and where $S_n$ is a symmetric random walk and $\mu$ is greater than zero. We ...
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1answer
464 views

Probability distribution of sign changes in Brownian motion

Let us consider a 1d Brownian motion. Displacements in space will be positive or negative and this is a random variable $U(t)$ that characterizes a random process and that can take just the values $\...
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1answer
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When is a Markov process independent-increment?

An independent-increment stochastic process must be Markov. I am now wondering about the reverse case. Why do some Markov processes fail to be independent-increment? What are some examples of Markov ...
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1answer
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Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E[\langle\xi,φ\rangle]$ an element of the dual space of $C_c^∞(G)$?

Let $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$ $\mathcal D'$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the ...
2
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2answers
207 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
2
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1answer
108 views

Simple question about the definition of Brownian motion

I have a question concerning the definiton of Brownian motion. Usually (e.g. on Wikipdia) one demands a brownian motion $\lbrace B_t\rbrace_{t\in[0,\infty)}$ to satisfy the following condition: $\...
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1answer
176 views

Is the following a martingale?

Let $X_{n}$ be a martingale with respect to a filtration $\mathbb{P}_{n}$. Define: $Y_{n}$ := $X_{n}^{3}$ Is $Y_{n}$ a martingale? Supermartingale?
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114 views

Is $(B_t^2)$ Markov where $(B_t)$ is Brownian motion?

I am pretty sure $(B_{t}^{2})$ not Markov because the squared random walk is not. Showing the square of a Markov process is or isn't Markov I guess I can repeat the method since to be Markov it ...
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1answer
214 views

Evaluating Stratonovich integral from definition

$\bf 3.9.$ Suppose $f\in\mathcal V(0,T)$ and that $t\to f(t,\omega)$ is continuous for a.a. $\omega$. Then we have shown that $$\int\limits_0^T f(t,\omega)dB_t(\omega)=\lim_{\Delta t_j\to0}\sum_jf(t_j,...
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1answer
37 views

Integral representation $B_T^3$

I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me?
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1answer
85 views

A Stochastic Integral Inequality

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ Is there a pair $(\mu,\sigma)$ such that $$\infty&...
1
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1answer
809 views

$X$ is a Geometric random variable find the expectation of $1/X$

Let $X$ be a geometric random variable with parameter $p$, find the expectation of $E[1/X]$. I need help simplifying the series.