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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Finding a stochastic process that satisfies a few constraints

In a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, let $ \{ \mathcal{F}_t \} $ be a filtration generated by the Brownian motion $W$. Let $\mu$ and $\sigma$ be predictable processes such that ...
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27 views

Proof of equality in Expectation with the Help of a Brownian Motion (Put-Call-Symmetry)

Hey I want to reproduce a proof of Damien Lamberton; proof begins at page 14. Under some assumptions i want to show that \begin{align} \sup_{t\in \mathcal T_{0,T}}\mathbb ...
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1answer
26 views

Poisson process basic problem

I've just started to learn stochastic and I'm stuck with these problems. Don't know how to start solving them. Suppose that cars cross a certain point in the highway in accordance with a Poisson ...
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12 views

Finite expectation of bank account with CIR interest rate model

The CIR interest rate model is $$dr_t=(\theta-ar_t)\,dt+\sigma\sqrt{r_t}\,dW_t\;.$$ The money account with this interest rate is $$e^{\int_0^tr_s\,ds}\;.$$ It is known that ...
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33 views

Pricing/Valuation of American Options

Hi i'm a litte bit confused by the pricing valuation of American options. For simple Assumtions on the Blacksholes Model and no dividends, and constant rates else one can show, that for a given ...
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1answer
30 views

Definition of Itô process

Let $\lambda_t$ and $r_t$ be predictable processes and suppose that $\int_{0}^t | \lambda_s |^2 \,ds < +\infty$, for all $t>0$. We define \begin{equation} Y_t = Y_0 \,\text{exp} \bigg\{ ...
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2answers
39 views

Stationary probability in an M/M/$1$ queue with a lazy server

Customers arrive to a single server queue according to a Poisson process with rate $\lambda$. Each customer requires Exponential($\mu$) service time. In the beginning when there are $0$ ...
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1answer
100 views

Conditions for birth and death process having only finitely many deaths.

Consider a birth and death process on $\mathbb{N}=\left\{0,1,2,\ldots\right\}$, given by the transition probabilities $p(n,n+1)=\lambda_n$ and $p(n,n-1)=\mu_n$ (those are the birth and death rates, ...
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417 views

Gaussian processes versus Bayes rule misinterpretation

I would like to use Gaussian processes (GP) for Bayesian classification of medical data. I think I already understand the basic stuff but I have some uncertainties that are perhaps partly related to ...
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16 views

Question on perpetual American put

Define $u(x):=\sup_{\tau \in T_{0,\infty}}E[e^{-r\tau}(K-S_{\tau})_{+}1_{\tau<\infty}$]. $T_{0,\infty}$ the set of stopping times taking values in $[0,\infty)$ and ...
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13 views

Checking the closeness of probability distributions

Suppose I have a Markov chain that satisfies all the conditions of ergodicity and has a stationary distribution pi. I want to find the time when the probability distribution of the markov chain is ...
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18 views

Unable to verify solution to difference equation $m_x - 2pqm_{x-2} = p^2 + q^2$

I want to verify that the solution to the difference equation $m_x - 2pqm_{x-2} = p^2 + q^2$ with boundary conditions $m_0 = 0$ $m_1 = 0$ is $$m_x = -\frac{1}{2}(\frac{1}{\sqrt{2pq}} ...
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12 views

How can I find the nonlinear and linear MS estimates of y in terms of x and the resulting MS errors?

If $y=x^3$, find the nonlinear and linear MS estimates of $y$ in terms of $x$ and the resulting MS errors? This is what I got for the nonlinear MS estimation: Since $e=E\{[y-C(x)]^2\}$, $C(x)=x^3$ and ...
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9 views

how can I find the resulting MS error with linear estimation?

The problem says: If $\eta_x=\eta_y=0, \sigma_x=\sigma_y=4$ and $\hat{y}=0.2x$ (linear estimate), find $E\{(y-\hat{y})^2\}$. I am doing this, based on Papulis formulas for homogeneous linear estimate: ...
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1answer
14 views

question about brownian hit time and reflexion principle

I have a Brownian motion W(t) I consider 2 events, where T is fixed : A : W(T) is above a, a > 0 B : W(t) hit the level b, b < 0, at least once between 0 and T I am trying to compute P(A and B) ...
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13 views

Numerical integration scheme for stochastic system driven by colored noise (filtered white noise)

I have given quite a few hours to this problem, but I seem to be getting nowhere. Can anyone just give a hint or point towards a text on where to go looking for the concept and solution.
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1answer
41 views

Where does this product of random variables converge to?

Consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ wich are independently normal distributed $N(0,\sigma^2)$. Set $M_0$=1 and $$ M_n =\exp \left( \sum_{i=1}^n X_i - ...
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1answer
42 views

Requirements for square integrable in the Doob-Meyer-Decomposition

Hey i have given a non negative supermartingale $(J_{t})_{t\in[0,T]}$ of Class D. So there exists a Doob meyer decomposition $J_{t}=M_{t}-A_{t}$ where $M_{t}$ is uniformly integrable since $(J_{t})$ ...
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1answer
82 views

Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0, $$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times ...
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1answer
40 views

Conditional expectation of second moment given sum of iid variables.

We have $\xi_i \geq 0$, $\forall i = \overline{1,n}$ (i.i.d. variables). Assume that $S_n = \xi_1 +...+ \xi_n$. It is easy to show that $\mathrm{E} (\xi_1\vert S_n = 1) = \frac{1}{n}$. Now we want ...
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51 views

Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; ...
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10 views

Is stochastic modelling a subset of Frequentist and Bayesian points of view?

From what I know of stochastic modelling it seems to me that this technique takes a Frequentist approach. For example, and please correct me if I am wrong, but isn't a Monte Carlo Simulation a ...
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44 views

Dynamic programming for optimal maximum and optimal minimum

We have a sequence of $a_i$ and a choosing rule that is take the first number $x_t\ge a_t$. The definition is = $$ min\{ t|t \in \{ 1,2,\cdots,n\}\,\,,\,\, x_t\ge a_t\}$$ The sequence $a_i$ is ...
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1answer
14 views

A question on the expectation of Counting Process

Let $N(t)$ denote a counting process, $X_1$, $X_2$, ... denote the inter-arrival time, and $S_1$, $S_2$, ... denote the arrival timestamp. So $S_1=X_1$, $S_2=X_1+X_2$, ... Let $T$ be a constant, so ...
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2answers
23 views

Support of the conditional distribution of a poisson process

I am working on Problem 5.1.8 of this book. It states: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the conditional distribution of ...
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1answer
15 views

What is “subordination” with respect to stochastic processes?

I'm building a model for a panel of counts, $\{n_{kt}\}_{k,t}$. As I read about regression methods for count models and the stochastic processes behind them, the concept of one random variable being ...
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82 views

Probability question involving stochastic process

A stochastic process $\{x_{k}\mid k=1,2,3,...\}$ of zeroes and ones is given with the property that $x_1 = 1, x_2 = 0$ and for every $k>2$ it is true that the probability of the event $x_k = 1$ is ...
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33 views

Modes of convergence for a *continuous-time* stochastic process

I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies $$\mathbb{E}(X_n) \rightarrow 0 $$ as $n \rightarrow \infty$ implies that a subsequence converges ...
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1answer
42 views

Inequality of an expectation (here: perpetual put of an american option)

for a given function $u(x):=\sup_{\tau \in T_{0,\infty}}E[(Ke^{-r\tau}-xe^{\sigma B_{\tau}-(\sigma^{2}\tau)/2})_{+}1_{\tau <\infty}]$ and $x \in [0,\infty)$, K a positive real number, $(B_{t})$ a ...
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12 views

How Can I show that a=A in this linear MS estimation problem?

How can I show that if the constants A,B and a are such that $E\{[y-(Ax+B)]^2\}$ and $E {\{[(y-\eta_y)-a(x-\eta_x)]^2 \}}$ are minimum, then $a=A$. I am trying to use this: $e=e_m$ is minimum if ...
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Autocorrelation of a Markov Chain?

Is there a general characterization of the autocorrelation metric of a Markov chain? There are some tangential issues as well: do $n$-state transition probabilities obtained through Chapman-Kolmogorov ...
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1answer
21 views

local martingale bounded below by a DL process

Let a continuous adapted process $Z= (Z_t)_{t \geq 0}$ be of class DL if \begin{equation} \{ Z_{\tau \wedge t} : \, \tau \text{ is a stopping time } \} \end{equation} is uniformly integrable, for each ...
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1answer
42 views

Monotone Class Theorem Application

I am trying to proof the following statement. Let $h$ be a bounded, $\mathbb{F}$-predictable process with $\tau$ a $\mathbb{H}$-stopping time, we then like to prove \begin{equation} ...
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1answer
19 views

Bivariate GBM - crosscovariance

I have troubles concerning a correlated bivariate GBM with identical drift and diffusion rates. Let $dX^i_t = \mu X^i_t dt + \sigma X^i_tdW^i_t$ and $E[dW_t ^idW^j_t] = \rho_{i,j}dt$ If $X_0^i = ...
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35 views

Lévy's upward theorem and $\mathcal{L}^p$ convergence.

Lévy's upward theorem: Let $Y \in \mathcal{L}^1(\Omega, \mathcal{F}, P)$, $(\mathcal{F}_n)_{n=1}^{\infty}$ a filtration of $\mathcal{F}$ and $\mathcal{F}_{\infty} = \sigma( \bigcup_{n=1}^{\infty} ...
3
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47 views

Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
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14 views

How close is a Ornstein-Uhlenbeckprocess to Brownian Motion

The Semi-Variance function of an Ornstein-Uhlenbeck (OU) process can be written as: $\gamma(\tau) = \sigma * (1 - \exp(\frac{-\tau}{a})$. If $a \to \infty$ the OU-Process approaches Brownian motion ...
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19 views

Self similar process

I am learning long memory process and came cross the definition of self similar. By definition, process $X(t)$ is self similar if $X(at)=_d a^H X(t)$,$a>0$ and $H$ is Hurst exponent. By equality of ...
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1answer
14 views

Is it an increasing process?

On a probability space $(\Omega,\mathscr{F},\mathbb{P})$ with filtration generated by Brownian motion, there is a progressivley process $(A_t)_{t\in[0,T]}$. If for any stopping times $0\leq \sigma\leq ...
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29 views

Why Gaussian process is not Ergodic in general?

Can anyone use a simple way to explain this? I heard this in class but I do not know why. By Wiki: a random process is ergodic if its statistical properties can be deduced from a ...
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How can I know by inspection that a process is WSS?

I have some codes to generate three different Random Sequences: I am getting a 4x100 matrixes where 4 is the number of samples and 100 is the length of the process. I am getting these results: ...
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9 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
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1answer
28 views

Marginal Probability of Stochastic Process

I have a wide sense stationary stochastic process x(t)=asin(2πf0t)+bcos(2πf0t) where a & b are independent gaussian random variables. How can I find the Marginal probability of x(t)? I am ...
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1answer
17 views

Asymmetric simple random walk?

It comes from the book Probability: Theory and Example. I don't understand the part marked with red line. Why it cannot converge to an interior point of $(a,b)$? Can anyone help? Thanks so much!
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35 views

Finding Conditional Expectation and variance E(Y|X=x)

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x). Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian ...
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1answer
31 views

Property of submartingale and supermartingle?

Is it true that for a submartingale, $$E(X_n) \le E(X_m)$$ for $n \le m$. And for a supermartingale, $$E(X_n) \ge E(X_m)$$ for $n \le m$. If it is true, then why? I feel confused because the ...
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1answer
25 views

Markov property of Brownian motion

There are two statements about Markov property: $B_t $ is Brownian motion and $\mathcal{F}$ is generated by $B$ If $s>0$ and $Y$ is bounded and measuable, then ...
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37 views

Where is the assumption of right continuity used in the following proof?

Lemma:If $X$ be a right-continuous positive local martingale then , $X$ is a generalized super martingale Proof: $\forall s<t$ $$E[X_t\mid F_s]=E[\lim_{n\to\infty} X_{t \wedge\tau_n}\mid F_s] \leq ...
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43 views

Birth immigration process

I'm having some problem with this question. A model for the distribution of the number of goals scored in soccer matches suggests that if n goals have already been scored by time t, then the ...
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21 views

If a stochastic process follows Geometric Brownian Motion, does it imply that it is Log-normally distributed and vice-versa?

This might be a naive question, but it doesn't stop haunting me. Wiki page for GBM writes the SDE for GBM process and shows it follows log-normal distribution. Is it true every time or are there any ...