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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30 views

Hitting time vs supremum of a càdlàg process

A question on stopping times that came up while I was trying to prove Doob's inequality. Let $X$ be a càdlàg, nonnegative submartingale. Define $X^*_t = \sup_{0\le s\le t} X_s$ and, for $K \ge 0$, ...
0
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0answers
26 views

Occupation time of a uniform process

Let $\{\mathcal{X}(\theta ):\theta \in \Theta\}$ be a stochastic process with continuous sample paths, where $\Theta$ is a compact set, and $\mathcal{X}(\theta )$ is uniformly distributed on $[0,1]$. ...
3
votes
1answer
56 views

Question on applying Ito's formula in this proof

I am reviewing this paper and I'm on page 3 of the document, and I'm having trouble with the proof of uniqueness. First off, the version of Ito's lemma I've learned is: if $X_{t}$ is an Ito process ...
2
votes
2answers
85 views

How to prove it is a strictly stationary process?

$ξ(t) = z*sin(ωt + θ)$ where $z$ is a random variable and its distribution is unknown and $θ$ is another random variable that is independent of $z$ and $θ$ is uniformly distributed on $(0, 2\pi)$. ...
6
votes
1answer
39 views

Basic question about the stochastic integral $\int \limits_{0}^{t} X(s) \,dM(s)$

Suppose $(X_{t})_{t \geq 0}$ and $(M_{t})_{t \geq 0 }$ are stochastic processes, where the index is continuous and the probability space is $(\Omega, \Sigma, P)$. We say for each fixed $\omega \in ...
0
votes
0answers
20 views

Probability that running maximum $M(t) > 2B(t)$, where $B(t)$ is Brownian Motion starting at 0

Looking for where to start with this one. Any hints will be appreciated. Probability that running maximum $M(t) > 2B(t)$, where $B(t)$ is Brownian Motion starting at 0.
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1answer
10 views

Show that for martingale and predictable process, it is not possible to gain almost surely in some step

Let $X_t, t = 0, 1,\ldots, T$ be a martingale and $V_t, t = 1,2,\ldots, T$ a predictable process, I want to show that for $t = 1,2,\ldots, T$ we have $$ V_t\cdot (X_t - X_{t-1}) \ge 0 \textrm{ ...
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votes
2answers
45 views

Expectation of first passage time of Brownian Motion [closed]

Let $B(t)$ be Brownian Motion beginning at zero. Define $T_{\alpha} = inf\{t>0 ; B(t) = \alpha\}$ to be the first passage time. I need to calculate the expectation of the first passage time for a ...
1
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0answers
64 views

Doubt Concerning Markov Property

Given a Markovian process $(X_t )_{t\geq 0 }$, is the following property accurate? $$\mathbb E \left[ f(X_{t_1}, X_{t_2},X_{t_3}) \mid \mathcal F ^X_{t_2}\right] = \mathbb E \left[ f(X_{t_1}, ...
1
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0answers
12 views

Inverse of Lumped Markov Process

Suppose we have an (ergodic, with dumping factor $\alpha$) Markov chain $P$ on a graph $G$, and we lump the transition matrix (it's a sort of reduction of the matrix using a equivalence relation of ...
0
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0answers
14 views

How do I parametrise a stochastic matrix?

I have a matrix $\mathbb{t}$ whereby $\sum\limits_j t_{ij} = 1$ and $\sum\limits_i t_{ij} x_i = q_j$ where $x_i$ are the elements of a discrete probability distribution, as are $q_j$, i.e. ...
2
votes
1answer
30 views

BMO martingales

Let $(Y_t)_{t\leq 0}$ be a continuous uniformly integrable martingale. It can be shown that for any $p\geq 1$, the following two properties are equivalent : there is a constant $C$ such that for any ...
1
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1answer
21 views

Condition for the existence of a Markov process from the properties of a semigroup $\{T_t\}$

In the article Diffusion processes with continuous coefficients I (1969, Stroock and Varadhan) one finds the following arguments in pages 26-27 "$(\cdots)$ for any $ \epsilon >0, \sup_{x \in ...
3
votes
1answer
99 views

$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t$ with $x_t$ Ornstein Uhlenbeck process - what to do? [UNRESOLVED]

I consider the following equation: $$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t, \tag{1}$$ where $a=$ constant and where $x_t$ follows an Ornstein Uhlenbeck process (see here under Alternative ...
1
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0answers
64 views

Problem including SDE

I have following problem. Let $Y_{t}$ be an exponential Lévy Process. That is: $$Y_{t} = Y_{0}e^{X_{t}}$$ Where $X_{t}$ is Lévy process. I have a function of $Y_{t}$, $f$ :$\mathbb{R}_{+} \times ...
0
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1answer
57 views

a problem from durret's essentials of stochastic processes

A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means $1,1.5$ and $3$ years. What is the average ...
3
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2answers
54 views

Uniform convergence in distribution

Consider a sequence of stochastic processes, $X_n(x)$ and a limiting process $X(x)$. For a fixed $x$, if $\mathbb{P}(X_n(x) \leq y)$ converges to $\mathbb{P}(X(x) \leq y)$ for continuity points of ...
0
votes
0answers
3 views

PageRank metaphor as suspended unlimited capacity pools

What do you think about my metaphor of the centrality measure PageRank? PageRank is an algorithm for evaluating node centrality: it's a function $f:G \to R^n$ where $n$ is the number of nodes in the ...
0
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0answers
41 views

Calculating the generator of a weighted transition function

Let $(P_t)_t$ be the transition function of a Feller-Dynkin process $X$. The usual Banach space of functions that the semigroup $(P_t)_t$ is working on is $C_0(E)$, i.e. continuous functions that ...
2
votes
0answers
45 views

Derivation of Backward Kolmogorov Equation

I'm following Kallianpur-Gopinath's textbook "Stochastic analysis and diffusion processes" to study Kolmogorov equations and I got stuck in a step of the derivation of the backward equation. In ...
0
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0answers
35 views

Stopping times and expectation for the symmetric random walk

Let $X_n : \Omega \to \{ -1, 1 \}$ be a random variable with $P(X = -1) = P(X = 1) = 1/2$, like tossing a coin, and $M_n = \sum_{i=1}^n X_n$. Also let $\tau_m : \Omega \to \mathbb N \cup \{ \infty\}$ ...
0
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63 views

Ornstein Uhlenbeck - Noise intensity

The Ornstein Uhlenbeck process is sometimes written as $\tau\frac{dy}{dt} = -y + \sqrt{2\tau}\sigma\xi(t) $ where $\xi$ is a zero-mean Gaussian white noise with autocorrelation function $\langle ...
0
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0answers
16 views

Ergodicity property for continuous-time Harris positive Markov process

The following theorem is Theorem 13.3.3 of Meyn and Tweedie's Markov Chains and Stochastic Stability on page 328 Theorem 13.3.3. If $\Phi$ is positive Harris and aperiodic, then for every initial ...
5
votes
0answers
37 views

Equivalent definitions of Poisson process

Define a Poisson process with parameter $\lambda$ is a counting process $(N(t))_{t\ge 0}$ such that: (i) $N(0)=0$; (ii) It has independent increment property; (iii) $N(t+h)-N(t)$ has Poisson ...
2
votes
0answers
38 views

Cross Variation of two stochastic processes

I am currently working on a stochastic calculus exercise at the moment and I am slightly confused when it comes to finding cross variation. We are given that the process $X_t = W_t^3$ ($W_t$ is ...
0
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0answers
4 views

Which property is sufficient for a Z-Matrix such as it's inverse is pointwise row-diagonally dominant?

I have a Z-Matrix which is pointwise row-diagonally dominant. (that is: $a_{ii} > a_{ik} \forall k \neq i$). This matrix is $(I-\alpha A)$, where A is an adjacence matrix of a graph. Can I say ...
0
votes
1answer
37 views

Is every continuous local martingale a uniform limit of step-processes?

The following question pertains to Wengenroth's textbook "Wahrscheinlichkeitstheorie", de Gruyter 2008 (in German). The covariance (aka compensator) of the continuous local martingales $X, Y \in ...
0
votes
1answer
28 views

Probability of losing everything in N games

Consider a gambler who starts with an initial amount of money of $£i$, obtains $£R$ with probability $p$ and loses $£J$ with probability $q=1-p$. What is the probability that it loses everything if he ...
1
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1answer
21 views

How to show that consecutive hitting times are a sequence of stopping times?

Let $\{X_n:n=0,1,\ldots\}$ be a martingale with respect to a filtration $\{\mathcal F_n\}$. Let $A,B$ be nonempty, disjoint Borel sets and define $T_0=0$, \begin{align} S_n &= \inf\{m\geqslant ...
2
votes
1answer
45 views

Induction proof on independent increment property

Define Poisson process is a renewal process in which the interarrival intervals have exponential distribution. $S_n$ is the arrival epoch of the $n$th arrival, $N(t)$ is the number of arrivals in ...
0
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50 views

Determining the Expected value of a random variable

Suppose we have a Poisson process of parameter $\lambda$. Each event of this Poisson process represents a start date of a period which duration is a random variable that follows an exponential ...
1
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0answers
14 views

Stochastic process reference: limit of compound Poisson with infinite second moment

For $S_{N}=\sum_{n=1}^{N}X_{n}$ where $X_{n}$ are $iid$ with infinite second moment (or infinite second moment in limit $N\rightarrow\infty$), the theory of stable distributions substitutes for the ...
1
vote
1answer
41 views

Lemma 3.1 of Stroock and Varadhan 1969

In page 355 of the article Diffusion processes with continuous coefficients I (Stroock Varadhan - 1969), one finds in lemma 3.1: $$(3.2) \quad\mathbb{P}(\sup_{\gamma \leq u \leq t} \vert \xi(u) - ...
2
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2answers
50 views

“The first time a continuous local martingale grows in absolute value beyond $n$” is a localizing sequence

How can it be shown that, for a continuous local martingale $X$ defined w.r.t. the filtered probability space $(\Omega, \mathcal{A}, P; \mathcal{F})$, the stopping times $\tau_n := \inf \{t \geq 0 ...
0
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0answers
10 views

Purely nondeterministic weakly stationary processes

I found a necessary and sufficient condition for a stochastic process being purely nondeterministic in Ihara (1993). As follows: A weakly stationary process $X$ is purely non-deterministic if and ...
3
votes
1answer
35 views

Intuitive explanation why finite activity Lévy processes does not have finite moments

I have a question about levy-processes. Let us denote the Lévy measure $\nu$ defined on $\mathbb{R}^d\setminus\{0\}$ and with $N$ the Poisson random measure such that $E[N(t,S)]=t\,\nu(A)$ for all ...
2
votes
1answer
47 views

Natural and completed natural filtration not right-continuous

I'm looking for an example of a stochastic process, such that the natural filtration and the completed natural filtration aren't right-continuous. I defined the process $(Z_t)_{t \geq 0}$ as $Z_t = t ...
0
votes
1answer
25 views

Independence of arrival time and interarrival time

Denote $S_n$ is the arrival time of the $n$ arrival, and $X_{n+1}$ is the waiting time between the $n$th arrival and the $(n+1)$th arrival in a Poisson process. I want to ask of the independence of ...
2
votes
4answers
81 views

What is the statistical equilibrium for this simulation of happy bubbles?

Happy Bubbles I hope this is not too specific or practical, but I just made a simulation of sorts and seem to have hit quite close to an equilibrium (by accident). Now I am wondering if and how you ...
0
votes
1answer
51 views

$e^{X_t - \frac{t^3}{6}}$ is a martingale - show it [closed]

I am trying to use Ito's integral properties to prove it is a martingale, but am getting stuck in the preliminaries. More so, I wanted to confirm, do I use this formula?
0
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2answers
34 views

What is a valid range of applicability of Ito Lemma?

If I have e.g. such process $$ Z_{t}=t^{5}B_{t}+10\int_{0}^{t}sB_{s}ds $$ can I take $$ f(t,x):=t^{5}x+10\int_{0}^{t}sB_{s}ds $$ as a function to which I apply Ito formula? I'm concerned about ...
1
vote
0answers
20 views

Covariance between random variables in a stochastic differential equation

Suppose I have a SDE of the form: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda ...
2
votes
1answer
25 views

how to derive the stochastic differential equation of this process

How can I derive the SDE for the vasicek model : $$r_t = 0.1 + 0.1 e^{-t} + e^{-t}\int_0 ^t e^s dB_s$$ From observation, the SDE vasicek's model is such that: $$dr_t = b(a-r_t)dt + \sigma dB_t$$ ...
2
votes
0answers
33 views

continuous time super martingale

I am trying to prove that if I have super-martingale $(S_t,F_t)_{t\geq0}$ right continuous, and $\tau <\infty$ stopping time that $(S_{\tau \wedge t},F_{\tau \wedge t})$ also super martingale. I ...
5
votes
1answer
33 views

$x_t := a_t -b_t c_t $ , with $dx_t = \theta (\mu-x_t) dt+ \sigma dW_t$

I would like to solve the following equation explicitly using Ito's lemma: $$ x_t := a_t -b_t c_t , $$ where $x_t$ is an Ornstein-Uhlenbeck process (see here) $$ dx_t = \theta (\mu-x_t) dt+ \sigma ...
2
votes
1answer
40 views

Compute the distribution of $S_{N_t}$

In a Poisson process, we define $S_n$ is the arrival epoch of the $n$th occurence, $N_t$ is the number of occurence in $(0,t]$. Compute the distribution of $S_{N_t}$. I made many trials to this ...
2
votes
1answer
16 views

Probability: $p\{X_t\in A\mid \min_{0\leq u\leq t}B_u>a\}=p\{X_t\in A,\min_{0\leq u\leq h} B_u>a\mid \min_{h\leq u\leq t}B_u>a\}$ always work?

Let $(X_t)$ and $(B_t)$ two stochastic processes and $0\leq h\leq t$. Do we always have $$p\left\{X_t\in A\mid \min_{0\leq u\leq t}B_u>a\right\}=p\left\{X_t\in A,\min_{0\leq u\leq h} B_u>a\mid ...
0
votes
2answers
19 views

Limit of Poisson process probability

$X(t)$ is a Poisson process with intensity $\lambda = 3$. I need to compute a limit like: $$\lim_{\Delta t\to 0}\frac{P(X(\Delta t)=1)}{P(X(\Delta t)\ge1)}$$ Any hints?
0
votes
1answer
10 views

Variance of stochastic process [closed]

I have stochastic process like: $Z(t) = X(t) - 0,5t$ where $X(t)$ is a Poisson process with intensity = 0,5. I need to find a variance of stochastic proces $Z(t)$ when $t=2$. Any ideas?
1
vote
1answer
30 views

Compute a conditional expectation with brownian motion

Let $(B_t)_{t\in [0,1]}$ be the standard Brownian Motion. Define $\mathcal{G_t}$ as $(\mathcal{F}_t\vee \sigma (B_1))_+$. Prove that for all $0s\leq t\leq 1$, $$ ...