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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Properties of Stochastic Differential Equations

Suppose I have an SDE of the form: $$dx_i = x_if(x_1,\cdots,x_n) + \sigma_ix_idW_t $$ By defining $y_i = \log x_i$, I can change the SDE to: $$dy_i = y_i g(y_1,\cdots,y_n) + \sigma_idW_t $$ Both ...
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23 views

Empirical Quantilfunction as Integral Bound

This is my first post, so please be nice ;) I'll try to outline my problem correctly and whilst keep it as short as possible! I have to deal (for my master thesis) with the integral ...
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1answer
18 views

Interpreting Ventzell Boundary conditions

I am trying to understand the article "On boundary conditions for multidimensional Diffusion processes" of A. D. Ventzell (or Wentzell). I copy the images for greater convenience: In the ...
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1answer
32 views

If $P^r$ has all positive entries, then so does $P^n$

Let $P$ be the transition probability matrix of a Markov Chain. Argue that it for some positive integer r, $P^r$ has all positive entries, then so does $P^n$, for all integers $n\geq r$ I ...
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29 views

limit of gaussian process

If I have a sequence of gaussian random process $X_{t}^{n}$ which converge in $L^2$ norm to a process $X_t$ for every $t$. can I say that $X_t$ is also gaussian process? thank you
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Proof of Itō's lemma for the Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
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1answer
34 views

Calculating a stochastic matrix with multiple states

I am struggling with how to calculate the values of a Markov matrix which has multiple states. For example, Imagine an unfair 6 sided dice. The chance of rolling a 1,2,3,4,5 or 6 is 0.3, 0.25, 0.2, ...
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25 views

Stochastic order of the minimum of a stopped sequence of random times

Let $S_k$ and $S'_k$ be two sequences of independent random times such that for each $k \in \mathbb{N}$ it holds that $S'_k \leq_{st} S_k$, that is $S'_k$ is stochastically smaller than $S_k$ which ...
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65 views

Understanding the formula

Let $P$ the transition probability matrix and $\mu$ the row vector of initial distribution. $$P_\mu(X_n=j)=\sum_j\mu(i)p^n(i,j)=\mu p^n(j)$$ I don't want to make a proof of that, I want to ...
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38 views

Show that $\{X_n\}_{n\ge 1}$ is a submartingale with respect to $\{F_n\}_{n\ge 1}$, where $X_n=\left(Z_1+Z_2+…+Z_n\right)^2$

I am trying to do the following exercise from a past exam paper and I am really stuck in it. I know the theory and can prove other cases, but I am not too sure about this one. Any help would be ...
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28 views

Expectation in markov chain

A Markov Chain {$X_n,n\geq0$} with states $0,1,2$, has the transition probability matrix ...
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20 views

Poisson- process inter arrival times

I don't understand the proof of inter arrival timesof a poisson process. Consider the poisson process with rate $\lambda $ and let $\{N (t); t \geq 0\}$ and let $T_1$ be the time up to the first ...
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1answer
36 views

Is {$X_n,n\geq 0$} a markov chain?

Consider a process {$X_n,n=0,1,\dots$}, which takes on the values $0,1,2$. Suppose $$P(X_{n+1}=j|X_n=i,X_{n-1}=i_{n-1},\dots,X_0=i_0)$$ $$=P_{ij}^I,\text{when n is even}$$ ...
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1answer
35 views

$E[|X(t)|]\leq K\implies E[|X(\tau)|]\leq K $?

Let $X(t)$ be a stochastic process. Assume that, for every $t\leq M$, it holds $$E[|X(t)|]\leq K, $$ for some constant $K$. Let now $\tau\in[0,M]$ be random (stopping time). Is it true that also ...
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22 views

Considering the Black and Scholes model, check that $\ln(S_T)=2W_T$ in a particular case

I have the following problem with its solution, but I keep on getting it wrong. I would be really grateful if someone could please explain to me what I am doing wrong. Thanks! It is part C that I ...
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1answer
33 views

Are these implications true for a nonnegative stochastic process $X_t$?

Suppose I have a nonnegative stochastic process $X_t$. Furthermore, suppose the following is true: $$\limsup_t \frac{1}{t}E\left[ \log X(t)\right] \leq a < 0$$ for some constant $a \in ...
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Moment Generating Function for Brownian motion's exit of interval.

Let $B(t)$ be a standard BM. Consider the stopping time $T = \inf\{ t > 0: |B(t)| = a\},$ the usual first exit time of the interval $(-a, a).$ We can see that $\mathbb{E} e^{tT} < \infty$ for ...
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1answer
21 views

A question involving Markov processes

Let $(S, \mathcal{B}, m)$ be a measurable space and $X_p := L^p(S, \mathcal{B}, m)$. Let $T_t \in \mathcal{L}(X_p, X_p)$ be a bounded linear operator defined by $$(T_t f)(x) = \int\limits_S P(t, x, ...
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1answer
32 views

Harmonic functions and Brownian motion

How can I prove that harmonic functions have the mean-value property using Brownian motion ${B_t}$? I know that I need to use the fact that $B_{t\wedge\tau}$ is a martingale where $\tau$ is a ...
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1answer
25 views

A question about Chapman-Kolmogorov equation

I'm reading ''Functional Analysis'' - K. Yosida and at page 379 there is the following claim "The hypothesis that the particle has no memory of the past implies that the transition probability P ...
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28 views

Should a stochastic process satisfy this condition?

Let $(X_t)_{t\ge 0}$ be a stochastic process. Let $$M_n(a_1,\dotsc, a_n; t_1, \dotsc, t_n) = \mathbb{E}e^{\sum_{i=1}^n a_i X_{t_i}},\,$$ where $t_i,\, 1\le i \le n$ are distinct. Is it essential ...
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54 views

Transition matrix proof

Let $P=\begin{bmatrix}1-a&a\\b&1-b\end{bmatrix}$, with $0<a,b<1$. Show that ...
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The Itō integral $\sum_{i=1}^nH_{t_{i-1}}\left(B_{t_i}-B_{t_{i-1}}\right)$ of an simple process $H$ is independent of the choice of $(t_0,\ldots,t_n)$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
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24 views

The jumps of a Poisson process

Let $(X_t)_{t\ge 0}$ be a Poisson process with unit rate. Hence, $X_{t+h}-X_t$ is Poisson distributed with mean $h$. Then, \begin{align*} \sum_{n=1}^\infty P(|X_{t+1/n}-X_t| > .5) &= ...
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30 views

Proof that the Ornstein-Uhlenbeck process is continuous

I am trying to prove the continuity of Ornstein-Uhlenbeck process which is a stationary Gaussian process with covariance kernel $k(x,y) = \exp(-|x-y])$. Let $(X_t)_{t\ge 0}$ be an Ornstein-Uhlenbeck ...
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43 views

Proof the statement

Given a finite aperiodic irreducible Markov Chain, prove that for some $n$ all terms of $P^n$ are positive. I'm little lost in how to prove that, but I know that: $i)$ If a Markov Chain is ...
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17 views

Physical meaning about $\limsup$ of a Stochastic Process

I'm trying to show that a positive $n$-dimensional stochastic process $X_t = (x_1(t), \cdots, x_n(t))$ is nice in that it's well-behaved and controlled (in the sense that the process doesn't grow too ...
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Stochastic matrix proof

Every stochastic $n\times n$ matrix corresponds to a Markov chain for which it is the one-step transition matrix. However, not every stochastic matrix $n\times n$ is the two-step transition ...
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Periodicity of states in Markov Chain

Determine the classes and the periodicity of the various states for a Markov Chain with transition probability matrix ...
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53 views

Transition probability between urn

$N$ black balls and $N$ white balls are placed in two urns so that each urn contains $N$ balls. At each step one ball is selected at random from each urn and the two balls interchange. The ...
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Brownian motion and associated martingales

Under the Wiener measure $\Bbb{W}$ the process $x(t)$ is a brownian motion. This means that $\Bbb{E}[{x(t)-x(s)\mid \mathcal{F}_s}]=0$. Let $P$ be a measure in $C([0,\infty),\Bbb{R}^d)$ such that ...
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Does a counting process allow a state change by more than 1 in a single transition?

The definition of a counting process A counting process is a stochastic process $\{N(t), t ≥ 0\}$ with values that are positive, integer, and increasing: $N(t) ≥ 0$. $N(t)$ is an ...
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27 views

harmonic functions and ito formula

I am trying to prove the mean-value property for harmonic functions in $R^k$ by ito calculus. given $G$ bounded domain and $u$ harmonic function on $G$ then $u(a)=\int_{\partial B_r} u(y)ds(y)$ ...
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Computing an Ito Integral using the Definition

Let $B_t$ be a brownian motion adapted to $\mathcal F_t$. For general $\mathcal F_t$-adapted processes $X_t$ the Ito-integral could be defined as $$ \int_0^t X_s dB_s = \lim_{n\to \infty} \int_0^t ...
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Sufficient conditions for a stochastic process to be continuous?

Let $(X_t)_{t\ge 0}$ be a stochastic process, such that $X_t-X_s \xrightarrow{w} \delta_0$ as $t\to s$. Is that a sufficient condition for $(X_t)_{t\ge 0}$ to be continuous? If not, can you provide ...
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36 views

Quadratic variation of the Brownian motion and Itō's lemma

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
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1answer
42 views

Distribution of the maximum of absolute value of multivariate Gaussian

I am currently working on some simulations. However, I encounter a statistical problem as following. Suppose $ 0 < t_1 < t_2 < \dots < t_m < 1 $ and $ B(t) $ denotes Brownian bridge. ...
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Survival analysis with a parametric model for recurrent events and time-dependent covariates

My goal is to model waiting times between recurrent events with time-dependent covariats with parametric models (Poisson, Weibull, log-normal etc.). This is not an issue time-dependent covaraits as ...
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1answer
40 views

Stochastic process $\exp(W_t - t/2)$ approaches zero for large $t$, but it is a martingale?

The stochastic process $$ S_t = \exp\left( W_t - \frac{1}{2} t \right) $$ is a martingale (for example this could be seen by noting that it solves the SDE $dS_t = S_t dB_t$, which has no drift). But ...
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Why is the solution of a stochastic differential equation wrt the Brownian motion suitable for a model of a disturbed time continuous process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be a Brownian motion on ...
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22 views

Notation in stochastic integrals

There are some notation I don't understand: Given $W_t$, $n$-dimensional Brownian motion, and a smooth function $u:R^n\to R$ my book asserts: $$E^x\left[u(W_0)\right]=u(x)$$ What is the notation ...
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How to extend formula for residue to functional calculus of operators

Suppose $\{X_t\}$ is a stochastic process with the covariance operator $\Gamma$ and the first $d$ eigen values are $\lambda_1\geq\lambda_2\geq \ldots \geq\lambda_d$ with eigen vectors ...
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Convergence in “the” Skorohod topology for monotone functions

Let $x_n$, $x$ be nondecreasing cadlag functions on $[0,T]$. To get $x_n\to x$ in Skorokhods $M_1$-topology, we only have to prove convergence on a dense subset of $[0,T]$ including 0. (see Whitt: ...
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25 views

How to interpret Realized Volatility and TSRV using R

I am looking at some high frequency data and I would like to know how to interpret and compare Realized volatility (RV) and Two Scale Realized Volatility (TSRV). References below. Given X is the log ...
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1answer
56 views

Itô integral with respect to a diffusion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
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1answer
26 views

Example of a markov chain with transient and recurrent states

As the title says, I can't come up with an example of a markov chain with all possible states (transient, positive recurrent and null recurrent). I know that the state space must be infinite, ...
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How do we obtain this inequality? A question concerning an argument in Stroock and Varadhan 1971

The problem comes from the article of Stroock and Varadhan [diffusion processes with boundary conditions (1971) ]. So far I have followed, but in the next page I got lost: I don't follow the ...
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28 views

Itô symmetry for elementary predictable stochastic processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
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1answer
23 views

Prove that the Itô integral for elementary predictable processes builds a martingale

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
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1answer
36 views

Distribution Stopping time under Brownian motions

Considering $W$ the canonical process on $C([0,1],\mathbb{R})$ and the row filtration generated by the coordinate process of $W$, I want to prove that ...