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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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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3answers
471 views

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 ...
4
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2answers
309 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 ...
4
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2answers
203 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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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 $ ...
3
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1answer
203 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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372 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
823 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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180 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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871 views

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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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 ...
4
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0answers
562 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 $\...
4
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1answer
665 views

What is the difference between stochastic calculus and stochastic analysis?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
4
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1answer
258 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. ...
3
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1answer
196 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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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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1answer
3k views

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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1answer
187 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 | \...
2
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1answer
74 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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2answers
36 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 ...
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0answers
47 views

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
61 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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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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1answer
132 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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1answer
82 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
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 ...
0
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1answer
104 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 ...
8
votes
1answer
448 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
359 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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2answers
709 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
2k views

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. ...
5
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1answer
516 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 $...
4
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1answer
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 ...
4
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1answer
86 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
638 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
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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 ...
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: $\...
2
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1answer
314 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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1answer
39 views

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 ...
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1answer
1k views

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 ...
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
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 ...
2
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1answer
460 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 $\...
2
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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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2answers
136 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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0answers
47 views

Independence of increments of some processes

I am stuck on this question: Let $(B_t)$ be a standard Brownian motion. Define $$ (\tau_1)_t := \inf \{s \geq 0 : B_s = t \} ; \quad (\tau_2)_t := \inf \{s \geq 0 : B_s > t \}. $$ Any ideas how ...
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1answer
798 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.
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1answer
47 views

The Continuity of Correlation Coefficient of a Continuous Stochastic Process

Given a continuous stochastic process with respect to time with finite variance at a given $t$. Does it necessarily imply, as $d\to 0$, 1) the covariance between $t$ and $t+d$ approaches the variance ...
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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
211 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,...