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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Difficulty in writing a mathematical solution

I find it really difficult to write a mathematical solution for a probability problem. Let me take an example: A counting process $(N_t)_{t\ge 0}$ has the independent increment property if, for ...
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32 views

Renewal process large time behaviour

I have the following question: I have two types of battery, $1$ and $2$. Suppose the lifetime of $1$ is uniformly distributed on the interval $(0,3)$, battery $2$ uniformly distributed on the ...
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24 views

Proof arrival time of n-th event in a Poisson process is a random variable

I have been made the following question: Proof arrival time for $n^{\mathrm{th}}$ event is a random variable in a Poisson process. For this, we define the time of arrival or waiting time as: $$T_n ...
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11 views

Binomial representation of stochastic process

It is common knowledge that a random walk can be represented in the form of a binomial process. Is it possible to represent any generic stochastic process (including non-linear) of the form ...
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1answer
24 views

A uniformly bounded local martingale is a martingale

I was trying to prove that A uniformly bounded local martingale is a martingale. Clearly a bounded local martingale is integrable I know how to show that a lower bounded local martingale is a ...
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20 views

Gaussian process with deterministic bracket

Let $(M_t)_{t\geq 0}$ a continuous Gaussian process that is a martingale with $M_0=0$. Show that $\langle M,M \rangle _t=f(t)$ a.s. where $f$ is a continuous increasing function. In the particular ...
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1answer
29 views

Itô's formula yields an Itô process

In our course on stochastic analysis, we proved the following version of the one-dimensional Itô formula: Let $\{W_t\}_{t\ge 0}$ be a one-dimensional Brownian motion w.r.t. some (right-continuous and ...
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1answer
31 views

Expectation of the time difference between starting times in queueing theory

Consider 2 independent, parallel $M/M/1$ queues $Q_1, Q_2$ with identical arrival rate $\lambda$ (corresponding to an exponential random variable $A \sim \text{Exp}(\lambda)$) and service rate $\mu$ ...
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1answer
33 views

Rescaling time for rates in a nonhomogenous Poisson process

If I have a non homogeneous Poisson process, say $N_{1}(t)$, with rate $\lambda_{1}(t)$ and I rescale the time by some function, $s=f(t)$, is the following correct? $\lambda_{2}(t)$ denotes the rate ...
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20 views

Partial differential equation involving a random process (literature advice)

In articles like this one (end of page one and page two), physicists often tend to treat a random process with discrete time and countable space set as a differentiable function (whose domains are ...
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26 views

Construction of Brownian motion - differentiability

I'm working on the the construction of BM given by Lévy-Ciesielski. The author begin to prove another result and for this reason he assume that BM exists and that it is also differentiable. For this ...
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1answer
17 views

Progressive measurability implies adaptedness

Somehow this statement in the title is obvious according to many textbooks but I couldn't produce a rigorous proof of it. Here is what I have so far. $(X_t)_{t\geq 0}$ being a stochastic process and ...
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1answer
25 views

Girsanov's theorem corollary

Trying to understand the proof of the corollary on the page http://en.wikipedia.org/wiki/Girsanov_theorem It remains for me the show the equality of the quadratic variations $[W, X]_t = 2[[W, X], ...
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17 views

Solving nonlinear first-order difference equation $ d_m = p_0 + p_1d_{m-1} + p_2(d_{m-1})^2 $ (extinction problem) [duplicate]

The steady-state equilibrium is $ d^* = \frac{1-p_1-\sqrt{(p_1-1)^2-4p_0p_2}}{2p_2} $. Based on a plot, I guessed the solution $ d_m = d^*(1-e^{-\alpha m}) $, which is pretty close but not correct. ...
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1answer
17 views

$MN-\langle M,N \rangle$ is uniformly integrable when M, N are $H^2$?

This proof below really seems to have little to do with uniform integrability, what does the DCT application give exactly? Any alternative proofs would be good too
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28 views

Expectation of an ito process

I came across this sub-question as a part of a bigger question, the question itself seems very simple but I'm having hard time figuring out a solution. Just to give a little background, this comes in ...
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40 views

Conditional first passage of a BM

Given a Brownian Motion $W_t$ I am interested to know if the following makes sense. Write $\tau=\inf\{t:W_t<x\}$ as the first passage time which has a known probability $\mathbb{P}[\tau\leq t]$. ...
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1answer
27 views

why does $\lim_{s \to t} P(|L(t)-L(s)|>\epsilon) =0 \implies \Delta L(t)=0$ P.as

Given that a a Levy process $L=L(t)_{0 \leq t \leq T}$ is stochastic continuous i.e $\forall t $ and $\forall \epsilon>0$ we have that $$\lim_{s \to t} P(|L(t)-L(s)|>\epsilon) =0$$ then my ...
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28 views

A type of continuous time Markov process

I am looking for a stochastic process model with the following features. It is a continuous time Markov process---modelling, if you like, the evolution of a population. New arrivals are added to ...
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32 views

Modified Stochastic Integral

Suppose one has a stochastic differential equation: $$dX_t = f(X_t) dt + g(X_t)d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda \eta(t) dt + \sigma dW(t)$$ Suppose ...
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1answer
43 views

Deriving a closed form expression for stochastic integral (to show its a martingale)

I have $B_s = $ brownian motion at time $s$. $$ \int_0 ^t B_s \, dB_s$$ $$0 \leq t \leq T$$ And want to check if it is a martingale, first from its closed form expression, and then via conditions on ...
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Expected number of occurences in Poisson process that can stop under certain conditions

We have a fisherman who catches fish according to a Poisson distribution with $\lambda = 0.6$ fish per hour. The fisherman always fishes for at least $2$ hours. If during these $2$ hours he catches ...
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41 views

Superposition of renewal processes: Variance of lifetimes

I've a question concerning the superposition of renewal processes. Assume we have $n$ independent renewal processes with the same lifetime distribution (especially mean $\mu$ and variance $\sigma^2$). ...
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16 views

Will a process with mean step 0 cross every bound infinitely often?

Take a discrete time process with mean step size 0. The steps will take values > ε with positive probability. It seems intuitive that, for any bound, the process will cross the bound infinitely many ...
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1answer
15 views

Expectation of Product Poisson Process

Let $N(t)$ be a Poisson Process with intensity $\lambda$ letting $Y_i$'s be iid random variables is it true that $$E\left(\prod_{i=1}^{N(t)}Y_i\right)=\left(E(Y)\right)^{\lambda t}$$ I know there is ...
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1answer
28 views

Stochastic calculus rules $d(B_t^2) = 2B_t\,dB_t + dt$ - why?

Let $B_t$ = Brownian motion at time $t$ I know that $(dB_t)^2 = dt$ and $d(f(x)) = f'(x)\,dx$ for some differentiable function. Now, I have that $$M_t = B_t^2 - t$$ $$dM_t = d(B_t^2) - d(t)$$ ...
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13 views

Log-likelihood approaches +Inf for a Gaussian process

I am trying to do a standard likelihood maximization for the hyperparameters of a Gaussian Process (details in Chapter 5, Rasmussen & Williams: ...
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28 views

Does Wiener Process to the power of $n$ have independent increments?

I looked at $cov(W_s^n-W_t^n, W_t^n) = \mathbb{E}(W_s^n-W_t^n)(W_t^n)-\mathbb{E}(W_s^n-W_t^n)\mathbb{E}(W_t^n)$, used Binomial theorem and Moments of Normal Distribution to simplify this, but still ...
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1answer
33 views

How to check if integral wrt Brownian motion is a martingale

As in title, I have a process $$X_{t}=\int_{0}^{t}s^{2}dB_{s}$$ I found here a sufficient condition for such integral to be a martingale on the interval. But I am asked if it is a martingale, not ...
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Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk

Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk. My question here mainly has to do with the $F_{t}$ measurability of $M(t)^2 - t$, where $F_{t} = \sigma (X_1 , X_2, ... , ...
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46 views

Prove this expectation of Brownian motion?

Prove $E[(\Delta B_j)^4]=3(\Delta t_j)^2$ where the Delta stands for the change of something i.e $B_j-B_{j-1}=\Delta B_j$ and the $B_j$ stand for the standard Brownian motion I won't show my step ...
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34 views

Differentiate a gaussian process

Let $(X_t)_{t\in [0,1]}$ a centered Gaussian process. Assume the covariance function $K$ to be $\mathcal{C}^2$. Show that for all $t\in [0,1]$, $$\tilde{X}_t=\lim _{h\to 0}\frac{X_{t+h}-X_t}{h}$$ ...
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18 views

Maximum of a Brownian Motion with drift

Consider the following: A Brownian Motion $B_t$, a constant $a\in\mathbb{R}$ and the process $C_t:=B_t+at$. There are wellknown formulas for the distribution of the maximum of a Brownian motion, but ...
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26 views

Proof of the Ballot Theorem Random Walks

Ballot Theorem: For $b>0$ the number of paths (0,0) to (n,b) that do not revisit the x axis is $\frac{b}{n}\mathbb{P}_{0}(S_n=b)$. MY ATTEMPT Now the first step of this path is to the point ...
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1answer
72 views

Reading list to master Numerical Analysis' research literature

As of lately I have been going through many research papers in my current job, and even though I have a Mathematics background at Masters level in Mathematical Finance, I sometimes struggle to follow ...
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51 views

Ski Lift - Expectation Value

The following is an exercise from my textbook. Let $Y$ be a random variable with values in $\mathbb{N}_0$ and $Y_1, Y_2, \dots$ be independent copies of $Y$. Further let $X$ be a markov chain with ...
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1answer
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Why are $dw_1(t) dw_1(t)$=$dt$ and $dw_1(t)dw_2(t)=0$ in shreve's stochastic finance II? [closed]

Refer to http://i.stack.imgur.com/doQuT.png on example 4.6.6 How come $dw_1(t) dw_1(t)$$=$$dt$ and $dw_1(t)dw_2(t)=0$?
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16 views

Independence in Wiener process

I have an exercise in which I have to determine if $W_2$ and $W_1 - W_2$ are independent. Where $(W_t:t>=0)$ is a standard Wiener process. In my notebook I have a hint that I can check this using ...
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49 views

Can these random variables be seen as products of indicator functions? [duplicate]

Spin-off from here. The solution given is that $$E[X_{n+1}|X_n] = 1/2\times 2X_n + 1/2\times 0 = X_n$$ How about using indicator functions? I was thinking that $X_n = 2^n 1_{A_1}$, but I guess ...
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1answer
37 views

Reasoning in “Prove X is a martingale” [duplicate]

From here. The solution given is that $$E[X_{n+1}|X_n] = 1/2\times 2X_n + 1/2\times 0 = X_n$$ Why exactly? In retrospect, I'm not sure I really got it. I'm trying to think about it in terms of ...
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1answer
52 views

$E[X_{n+1}\mid X_n] = E[X_{n+1}\mid\mathscr{F}_n]$

Under what conditions does it hold that $$E[X_{n+1}\mid X_n] = E[X_{n+1}\mid\mathscr{F}_n]$$ if we are given a stochastic process $X = (X_n)_{n \geq 0}$ on a filtered probability space $(\Omega, ...
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18 views

jump-diffusion hitting time

Suppose I have a stochastic process $dS_t= rS_t dt + \sigma S_t dW_t + dJ_t$ where $W_t$ is a brownian motion and $J_t$ a compound poisson process of parameter $\lambda$ with lognormal jump size, ...
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1answer
90 views

Reversible Ito Diffusions

I have given a diffusion equation $$ dX_t = -\nabla V(X_t) \, dt + \sigma dB_t.$$ I found here(1) a characterization when $X_t$ is reversible, aslong as $\sigma=1$. Is this also true for $\sigma ...
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1answer
92 views

Variance of a special random walk

I am trying to find the variance of the following special random walk: Suppose that $U=(U_1,U_2,...)$ is a sequence of independent random variables, each taking values $u$ (for up) and $d$ (for down) ...
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1answer
25 views

Problem with particular proof regarding infinite total variation of Brownian motion [closed]

I have some problems with a proof from the last page of this pdf: Brownian motion has infinite total variation. Could we say that variance is exactly $\frac{c_{1}}{n}$ for some constant $c_{1}$? ...
2
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32 views

2D Brownian Motion — Does this argument work?

Consider a 2D Brownian Motion $(X_1(t),X_2(t))$ starting at $(x_1,x_2) \in \mathbb{R}^2$. For every $s\geq0$, let $$\tau_s = \inf \left\{t \geq 0 \mid X_1(t) - x_1 > s \right\}\qquad Y_s = ...
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27 views

Why aren't these two sets of stochastic processes equal?

I'm learning about stochastic integrals now, and I don't understand the following: If $S$ and $L$ are two classes of processes where: $S=\{f(s,\omega) |f $ is progressively measurable and ...
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1answer
152 views

Independence of the components of a multidimensional Brownian motion

Let $B = (B^1, \dots, B^n)$ be an $n$-dimensional ($n \in \{1, 2, \dots\}$) Brownian motion (i.e. $B = (B_t)_{t \geq 0} \in \Omega \rightarrow (\mathbb{R}^n)^{[0,\infty)}$ has continuous paths, $B_0 = ...
2
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37 views

Calculate mean and correlation of a stochastic process

I am given the Stochastic process $Y_n$, where $n \in Z$ defined by: $ Y_n = X_n - X_{n-1}$ where $X_n$ is a process with independent and identically distributed geometric variables $X_n \sim G(p)$ ...
3
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1answer
25 views

Density Function of Random Variable Related to Brownian Motion

Above is my question. I've done the first two parts, that's no problem. I'm stuck on finding the density of the rv $R = W_1 / M$. I have got as far as $$g(x,y) = \frac{\partial^2}{\partial x ...