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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Convergence in Distribution of CDF but not PDF

I came across this example that demonstrates the convergence of the CDF of a random variable converging in distribution but not the convergence of the PDF. I just have two questions: Why is the PDF ...
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43 views

Convergence in distribution does not mean PDF of $X_{n}$ converges to PDF of X

I'm trying to come up with a simple and concise example that demonstrates even though $X_{n} \overset{d}\to X$, the PDF of $X_{n}$ does not converge to the PDF of X. The research I've done so far has ...
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9 views

Natural $\sigma$-algebra on the co-domain of a stochastic flow

Say that we have a stochastic local flow $X$. This is a map that sends $\omega$ into a collection of maps $\phi_{s,t}$ on $s,t\in \mathbb{R}$ which form a local flow. I don't understand how $X$ ...
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2answers
38 views

Skorohod Representation Theorem Application

I'm reading through the Skorohod representation theorem and finding this proof a little bit difficult to understand. My understanding of the proof in bullet point form is as follows: Define a new ...
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16 views

Uniform stochastic process

Is it possible to construct a stochastic process $X_t$, $t \in [0, +\infty)$ such that for each $t$, $X_t$ be uniform on, say, $[0,1]$ by using Kolmogorov's compatibility theorem ?
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Stopping Times, the $\inf$ is not a stopping time

I'm having a hard time figuring out why the infimum of a sequence of stopping times is not necessarily a stopping time itself. Indeed, the justification my book gives me is that: Given $(\mathcal ...
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23 views

Show that $W(t)$ is almost surely non-differentiable at $t=0$

Show that $W_t$ is almost surely non-differentiable at $t=0$. Of course, $W(t)$ denotes a standard Wiener process. It is enough to show that $$P(\{\omega : \exists \epsilon>0 \: \forall \delta ...
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24 views

Compare moments of $\int_0^t h(B_s) ds$ and $\int_0^t h(\sqrt{s}Z)ds$ for $(B_t)$ Brownian motion and $Z$ standard normal

If we let $B_t$ be a standard Brownian motion and $\sqrt{t}Z$, where $Z$ is our standard normal random variable, we know that they have the same distribution. However, how can I show that the process ...
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34 views

Left-continuity of a Lévy filtration

The natural filtration $(\mathcal{F}_t^X)_{t\geq 0}$ of a Lévy process $X$ is right-continuous, but what about left-continuity? A Lévy process is quasi left-continuous at time $t$ which says that ...
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10 views

Are smooth functions with compact support a core for a generator of a Feller process?

I was reading an article, when the following was stated: Assume we have partial generator defined on some open subset $U\subset \mathbb{R}^n$ $$ A=\sum_{i=1}^n ...
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Conditional expectation on Function space

This question is from a notation in section 13.4 of the book "Linear and Nonlinear Filtering for Scientists and Engineers" By N U Ahmed In this section, the author is deriving the Zakai ...
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36 views

How to find the variance of $\int_0^t B_s^2 ds$ where $B_s$ is a standard Brownian motion random variable?

I am trying to find the variance of $\int_0^t B_s^2 ds$ where $B_s$ is a standard Brownian motion random variable. My approach is to represent the integral as a sum. However, I am not sure how this ...
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1answer
23 views

Proof for convergence in distribution implying convergence in probability for constants

I'm trying to understand this proof (also in the image below) that proves if $X_{n}$ converges to some constant $c$ in distribution that this implies it converges in probability too. Specifically, my ...
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1answer
31 views

Stopping time - Stochastic Processes

Suppose I have a sequence of random variables $X_{1} ... X_{n}$ such that $X_{i}$ takes a random value from the set ${-1,0,1}$ with equal probability. Now take the sum of the sequence $X_{1} ... ...
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40 views

Prove that $P(\xi_{n+1}>t) = e^{-\lambda t} \sum_{k=0}^n \frac{(\lambda t)^k}{k!}$

Let $\xi_n = \eta_1 + \dots + \eta_n$ where $\eta_i$ are iid random variables exponentially distributed with the rate $\lambda>0$. In the book there is a proof that $P(\xi_{n+1}>t) = ...
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25 views

Solve this simple Linear SDE?

How do I solve the following BSDE? $$ \left\{ \begin{aligned} dX_t &=(rX_t+\theta Z_t ) \, dt + Z_t \, dW_t \\ X_T &=\xi \end{aligned} \right. $$ There appears to be nothing online about ...
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27 views

Show that in a Galton-Walton Branching Process, $\phi_n'(s)\to0$ for every $s\in(0,1)$ if $p_0>0$

Let $Z_n$ be the Galton Watson Branching Process. Let $Z_n=\sum_{k=1}^{Z_{n-1}}X_{k,n}$ where $X_{k,n}\sim X$ are iid progeny distribution. If $p_0=P(X=0)>0$ then show that $\forall s\in(0,1)$ we ...
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1answer
14 views

What does it mean to say a RV is fully defined by its distribution

I get that a CDF is used to observe the behaviour of a distribution and therefore the corresponding RV ? But what does it mean to fully defined by its distribution and what does this have to do with ...
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25 views

Proving if $W(t)$ is a weiner process, then $W^2(t)$ is also a Weiner process [duplicate]

I'm trying to solve this question: For a stochastic process to be a Weiner Process it must have these properties: $W(0) = 0$ so $W^2(0) = 0$ $E(W(t)) = 0$ but $E(W^2(t)) = t$ I think this is enough ...
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2answers
88 views

Convergence of mean of an irreducible Markov chain / ergodic theorem

Let $\{X_n\}$ be an irreducible Markov chain on a discrete state space $\mathbb{N}$, that has a stationary distribution $\pi$. Prove or disprove : with probability $1$: $$\lim_{n\rightarrow ...
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1answer
52 views

Markov Chain Questions

I've been stuck on these problems for a while. I keep banging my head against the wall, but my calculations are incorrect each time. I sum the probabilities together for each possibility (it's a ...
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16 views

how different lead values affect linear prediction error?

I really don't expect a full answer. Even a hint or a good reference would be appreciated. Assume a Process is sampled at time instances that are NOT equally distanced in time. Assuming an AR$(p)$ ...
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What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
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42 views

Construction of the Itô-integral in Øksendals book, why is this sequence a Cauchy sequence.

A quick summary of the things regarding my question is this: You have a probability space $(\Omega, \mathcal{F},P)$ and a filtration $\{\mathcal{F}_t\}$. You have a $f(t,\omega): ...
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33 views

Number of $1's$ in a string in terms of successive pairs

Problem. Let $X_n=0$ or $1$ and set $Y_n=(X_n,X_{n+1})$. Set also $\displaystyle \sum_{k=1}^{n}\mathbb{I}_{\{X_k=1\}}$ be the number of times $X_k's$ become $1$, from $X_1$ till $X_n$ ...
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33 views

Showing that the square of Brownian motion, minus time, is a martingale

What exactly are we supposed to do to show what they have given is a martingale. If I try to follow through I'm getting a bit confused. In the third to last line I don't understand how they have ...
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1answer
18 views

Jump diffusion model to work out probabilities

I'm not being able to understand firstly how they determined what the mean and variance of Xt=... is in the last sentence of the solution. Secondly I'm not able to understand where they got the 1 ...
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1answer
11 views

Using Jump diffusion model to find expectation

For this question I'm not able to understand how they got from the second line to the third. So not able to understand how they squared the term in the expectation and then simplified to get the ...
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34 views

Using reflection principle to find probabilities

I'm not able to answer these questions because firstly I don't understand the reflection principle properly. Secondly if someone could provide a visual explanation as to how this process works ...
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15 views

Unconditional variance of a GARCH process and the unconditional mean of an ARMA process

Are the unconditional variance of a GARCH process and the unconditional mean of an ARMA process equal to empirical in theory? From the formulas defining them I don't see why they should be but ...
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41 views

Kummer equation, solution to find optimal value

Suppose V follows the mean reverting process $$dV=η( ̅V-V)Vdt+σVdz$$ I want to find the optimal investment rule, and using Itos's lemma I get that the differential equation that F(V) must satisfy $$ ...
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35 views

Bernoulli and Poisson random variables

I'm reading the following argument which is related to a previous question: I think when $p<1/2$, the statement is not true. Could anybody explain the underlined sentence?
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43 views

question about a changement of probability

Let $(\Omega, F, F_n, \mathbb{P})$ a filtered probability space. Let $v_n=\mu_n+\sigma_n\varepsilon_n$ with $\mu_n ,\sigma_n$ are $F_{n-1}$ measurable and $\varepsilon_n$ ~ $N(0,1)$ are i.i.d Let ...
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1answer
39 views

calculating limit of a markov chain

I want to calculate the following limit $lim_{n \to \infty}\ A={\begin{bmatrix}1 & 0 &0 & 0&0\\1-p & 0 & p & 0&0\\0 & 1-p & 0 & ...
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28 views

Expected value over joint probability measure

I would like to maximise an expected value of a function $g(x_1,\xi_1,...,x_N,\xi_N)$, that depends on several random variables $\xi_1, \xi_2,...,\xi_N$ and corresponding decision variables ...
2
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1answer
40 views

Why is Backward SDE more difficult than forward SDE?

I need to explain Backward Stochastic Differential Equation (BSDE) for some non-mathematicians. The audiences are most likely familiar with ODE/PDE as physicists. One concern is probably that why ...
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3answers
61 views

Where to start Stochastic processes

This semester I have the course stochastic processes in university but as our instructor is awful I can't rely on him and I should study this course on my own.To start, I need a suitable book for ...
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26 views

Introduction to stochastic processes by Lawler

I want to know if the book introduction to stochastic processes by Gregory F. Lawler has solution manual or not. I could find a lot of links claiming that on their website we can find the solution ...
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1answer
49 views

A fair die is thrown repeatedly until we obtain the same number twice in a row.

A fair die is thrown repeatedly until we obtain the same number twice in a row. Compute the expected number of throws. For this, I found $6$ finding the transition matrix and using first step ...
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1answer
37 views

Immortal cell lines: how many cells will be alive after $t$ time steps?

I was talking with a friend about a problem in population sizes in a discrete setting: Suppose you have colony of single-celled organisms. Every hour, on the hour, with probability $p$ each cell ...
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24 views

Probability of hitting a barrier

We have a stochastic process $ Y_t= \alpha t+ W_t$ where W is a standard brownian motion. Is there a way to calculate the conditional probability with respect to $Y_1$ for this process to hit a ...
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2answers
40 views

How can I show that the stochastic process $M_t = W_t^3 – 3t W_t$ is a martingale $\mathbb{E}[M_u|F_t]$? [closed]

How can I show that this stochastic process $M_t$ is a martingale $\mathbb{E}[M_u|F_t]$? $W_t$ is a Brownian Motion. $$M_t = W_{t}^3 – 3t W_t$$
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1answer
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expected value problem for a normalized integral of sign

Say we have a Gausian process $X_s=\int_0^sh(x)\,dW(x)$ where $W(x)$ is a Wiener process. Now define $$Z=\frac{\int_0^1\operatorname{sign}(X_s) \, dW(s)}{\int_0^1|X_s| \, ds}$$ Intuitively we have ...
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specifying the joint distribution as a proof technique

The following is a theorem in stochastic process in Pinsky's An introduction to Stochastic Modeling: The proof starts as the following: Here is my question: Could anybody explain why and how ...
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1answer
45 views

How can we identify $\omega\in\Omega$ with a path of Brownian motion $t\rightarrow B_t(\omega)$?

In the Stochastic Differential Euqations written by Oksendal(see page 12), As we shall soon see, the paths of a Brownian motion are (or, more correctly, can be chosen to be) continuous, a.s. ...
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1answer
41 views

Finding marginal density functions for arrival times? (Poisson Process)

The joint density of the first and second arrivals, denoted $W_1,W_2$ is: $$f(w_1,w_2)=\lambda^2 e^{-\lambda w_2}$$ I am asked to find the marginal density functions for $W_1$ and for $W_2$ and also ...
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1answer
36 views

$X$ is a Poisson process and T is a random variable exponentially distributed?

$X = ${$X(t): t>=0$} is a Poisson process (intensity $\lambda$). Independent of $X$, $T$ is a random variable that is exponentially distributed with intensity $\theta$. I need to find the PMF for ...
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23 views

PDE for Brownian Bridge Expectation?

Let $\displaystyle Y(t)=\int_0^t v(s)ds+B(t)$, where $B(t)$ is the standard Brownian motion and $v(t)$ a deterministic function. Compute $m(t,y):=\mathbf E\Big[\max\limits_{s\in[0,t]} ...
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1answer
47 views

Show the existence of a right-continuous modification

Suppose ($X_t$)$_{t \geq 0}$ is a stochastic process with independent increments and the function $t \rightarrow \mathbb{E}X_t$ is continuous. Prove that $(X_t)$ has a right-continuous modification. ...
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1answer
37 views

Doubling strategy

Let {X_n} be simple symmetric random walk. Consider a game where in case of X_n = +1 the gambler wins and in case of X_n = -􀀀1 the gambler loses his stake on game n. The stake on the first game is 1 ...