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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Differentiability of $E[1_{\tau > T} \mid X_t = x]$ where $X_t$ is a Lévy process

Let $X$ be a finite-variation Lévy process which starts at $X_0>0$, has positive drift, and has only downward jumps. Also define a stopping time $\tau := \inf(0\leq t \leq T: X_t<0)$, the first ...
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1answer
11 views

Solution to sde with specfic mean

Goal: I'm attempting to work backwards to recover an SDE as follows: Example: $e^{\mu t}$ is the mean of the geometric Brownian Motion, which solves the SDE: \begin{equation} dS_t = \mu S_t dt + ...
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2answers
97 views

Stochastic variables independent given Tau

Say we have a filtration $(\mathbb{F}_s)$, and a stopping time $\tau$ w.r.t. to that filtration.Let $X_t$ be a continuous stochastic process (not required to be adapted to the mentioned filtration), ...
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1answer
58 views

Bernoulli Process Question

Suppose on any given day you receive mail in your mailbox with common probability $p$. Assume that whether mail is put in the mailbox or not is independent from day to day. Find the ...
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34 views

What can you tell me about backward Brownian motion?

I'm trying to understand "backward Brownian motion" and how it relates to standard Brownian motion. In this paper, they construct a solution to Burgers Equation (transformed via Cole-Hopf) with ...
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1answer
20 views

Doob Meyer decomposition for Super-martingales

Let $Z$ be a super-martingale with usual Doob-Meyer decomposition: $Z=M-A$. Is it true that : $A\leq M$ and therefore: $\mathbb{E}[A^2]\leq \mathbb{E}[M^2]$ ?
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64 views

Define the random process $U(t) = A$ where $A$ is uniform over $[-1,1]$

Define the random process $U(t) = A$ where $A$ is uniform over $[-1,1]$. How would one sketch a sample realization of this?? Can someone give me a simple idea so I could attempt my own definition? ...
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2answers
60 views

Markov chains and conditioning on impossible events

Consider a Markov chain $(X_0,X_1,\ldots)$ with a state space $S\equiv\{s_1,s_2\}$ and the following matrix of “transition probabilities” (I will explain the use of quotation marks below): ...
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1answer
19 views

Stationarity and Ergodicity vs. Memorylessness

A (discrete) memoryless information source is (usually) defined as a collection of random variables that are independent and identically distributed. My question is, does memorylessness imply ...
2
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1answer
21 views

Master equation to Fokker Planck equation

How can we convert a given master equation to a Fokker Planck equation.Is there any general method for this transformation?
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18 views

Overlap between disks generated through a Poisson process

I have a homogeneous spatial Poisson process on $[0-d; 0-d]$ with an expected number of points equal to $n$. I use the points to generate disks of radius $r$ without any overlap constraints. I am ...
3
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43 views

Doob's inequalities for not necessarily right-continuous martingales

In Revuz and Yor, they denote $\mathbb{H}^2$ the space of $L^2$-bounded martingales, and $H^2$ the space of continuous $L^2$-bounded martingales. They state "... by Doob's inequality ... ...
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1answer
27 views

Is there a way to estimate moments of strong solution to SDE

Suppose the SDE $$\mathsf dX_t =b(t,X_t)\mathsf dt + \sigma(t,X_t)\mathsf dW_t,\; X_0 = x$$ where $t\in[0,T]$ has a strong solution. I know in general we can't find an explicit formula for the ...
2
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2answers
35 views

a problem regarding conditional probability and binomial distribution.

Die A has 4 red and 2 white faces whereas die B has 2 red and 4 white faces . A coin (fair) is tossed once . If it falls head , the game is carried on by throwing die A alone. If it falls tail die B ...
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1answer
51 views

Calculation with Ito processes, what is $ds \, dt$, $dW_t \, ds$ and $dW_s \, dW_t$?

I am working on an exercise and I am not sure how to deal with these 3 cases... For example, is $ds \, dt=0$? I know $(dt)^2=0$, but I am not sure when it is 2 different variables. And what about ...
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1answer
33 views

Martingale CLT: “without loss of generality”?

(Hopefully last in a long series of posts from the "I don't have Rick Durrett's brain" department... apologies.) In Durrett's proof of a a simple martingale CLT (Theorem 8.8.3, p. 341), he loses me ...
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13 views

Autoregressive process and average return time

I am back in my maths courses but I doesn't arrive to assess the average return time of an $\operatorname{AR}(1)$ process. Has someone any idea? And after for an $\operatorname{AR}(2)$? Suppose ...
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2answers
36 views

Show that $X_tY_t=X_0Y_0+\int_0^tX_sdYs+\int_0^tY_sdX_s+[X,Y](t)$, where $X_t,Y_t$ are Ito processes

So I have done this exercise and the proof holds, but I really don't believe it can be correct because the proof is worth twice as much as other exercises. I am also not 100% sure if $d_s ...
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0answers
11 views

When a Brownian motion is no longer a Brownian motion after change in the filtration

Let $(B_t,\mathcal{F}_t)$ be a Brownian motion with respect to the filtration $(\mathcal{F}_t)_{t\geq 0}$ and $(B_t,\mathcal{H}_t)$ be a Brownian motion with respect to the filtration ...
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0answers
12 views

Criterions for strong mixing

I just read about $\alpha$ (or strong) mixing (as defined here http://arxiv.org/pdf/math/0511078.pdf on pages 2 and 5). Assume, I've some random variables $(X_i)_{i \geq 1}$ which are not independent ...
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9 views

representing random variables on a standard space

Let $(\Omega, \mathcal{F}, \Bbb{P})$ be a probability space and $X:\Omega \to {C}^d([0,\infty), \Bbb{R}^d)$ and $B:\Omega \to {C}^d([0,\infty), \Bbb{R}^r)$ random variables. I would like to ...
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2answers
39 views

Show that $\mathbb{E}\left(\int_0^1X_n(t)dW(t)-\int_0^1X(t)dW(t)\right)^2\to 0$

I tried to shove this question in another one of my posts as a follow-up question, but I deleted my comment and will post it instead. I would really appreciate if someone could help me spot out and ...
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1answer
49 views

Principal component analysis (PCA) results in sinusoids, what is the underlying cause?

Background I'm analysing a data set of $M$ flow measurements (volume per time). The flows go from zero mL/s gradually to higher values and back to zero again, thus: their shapes ideally look like a ...
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2answers
31 views

Show that $\mathbb{E}[X_t]=X_0e^{-ct}$ if $X_t=X_0e^{-ct}+\sigma e^{-ct}\int_0^te^{cs}dW_s$, $X_0\in\mathbb{R}$

so I know the result is trivially correct, but I am being asked to prove it. I tried using a theorem, but it seems rather contradictory. Thanks in advance! Question: Show that ...
2
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1answer
33 views

Calculate the distance $d_{\mathcal{H}}(X_n,X):=\mathbb{E}\left(\int_0^{\infty}(X_n(t)-X(t))^2dt\right)$ for all $n\ge 1$

I have done this exercise but I have done something wrong because I don't get the correct result for the next part of this exercise (this is part B). I posted something earlier that is related to ...
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2answers
47 views

Quadratic variation of the Ornstein-Uhlenbeck process

Let $(X_t)_{t\geq 0}$ be the zero-mean Ornstein-Uhlenbeck process such that $X_0 = 0$ almost surely, i.e. $$X_t = \sigma e^{-\alpha t}\int_0^t e^{\alpha s}\,dB_s \quad \qquad (\triangle)$$ On the ...
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1answer
27 views

Joint distribution of Brownian motion and its running maximum

$B$ being standard Brownian motion, its running maximum is defined as $M_t = \sup_{0\leq s\leq t} B_s$. I am trying to follow the proof of the following result but I don't understand some of the steps ...
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12 views

Boundary measure and integral relations

On the article (On the existence and uniqueness of diffusion processes with Wentzell’s boundary conditions - Watanabe & Takanobu) there is a definition of $(A,L)$ process on a domain $D$ The ...
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1answer
33 views

Stationary distribution in continuous-time Markov chain

Consider a barbershop with one barber who can cut hair at rate 4 (people per hour), and three waiting chairs. Customers arrive at rate 5 per hour. Customers who arrive to a fully occupied shop ...
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1answer
16 views

Show that $X_n\in\mathcal{H}$, where $\mathcal{H}:=\{h(t):h(t)\text{ is an adapted process, }\mathbb{E}[\int_0^{\infty}h^2(t)dt]<\infty\}$

I am not sure if I got this exercise right... I have 2 questions: Have I obtained the final result correctly? If so, I used Wolfram Alpha to obtain the value of the series, but how else can I obtain ...
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1answer
11 views

Existence of a localizing sequence of stopping times for a continuous local martingale

I have a a question about continuous local martingales: the definition of continuous local martingale says that a continuous process $X_s$ is continuous local martingale if there is non decreasing ...
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28 views

Differentiating Stochastic Integral

I was wondering how to write the following integral in differential form: $$\int^t_0 f(s,t)dW_s$$ where $W_s$ is a standard Brownian Motion. In my understanding, if $f(s,t)$ can be written as ...
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1answer
23 views

Lévy-Khintchine formula and Taylor expansion

I have trouble finding out why this condition $\int_{\mathbb{R}\backslash\{0\}} \min(1, x^2 ) \nu(dx) < \infty$ in the Lévy-Khintchine formula is necessary. The Lévy-Khintchine formula is ...
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1answer
23 views

Show that $\mathbb{E}\left[c_{\tau\wedge n}X_{\tau\wedge n}-\sum_{i=1}^{\tau\wedge n}c_i\mathbb{E}(X_i-X_{i-1}\mid\mathcal{F}_{i-1})\right]\le 0$

I am trying to go through a past exam paper but I don't know how to deal with stopping times since we only did 2 exercises in class... I got stuck, so I would really appreciate if someone could help ...
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0answers
25 views

Interpolation of random processes

Let $\left(\Omega, \mathcal{F}, \left\{\mathcal{F}_{t}\right\}_{t\geq 0}, P \right)$ be a complete probability space with a nondecreasing family of right continuous sub-$\sigma$-algebras ...
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1answer
26 views

Derive the unique solution of $dX_t=\alpha X_t dt+2dW_t,\quad X_0=0$

I have the following question which makes sense when taking into account that $X_0=0$, but I don't get the same result if we use the variable. QUESTION: Consider the SDE $$dX_t=\alpha X_t ...
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0answers
38 views

Process convergence of sum of i.i.d. random variables

I'm interested in the convergence of a stochastic process. Let $(X_i)_{i \geq 1}$ be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. Furthermore, assume $0 < t < T$ for some ...
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1answer
33 views

How to write down the probability space of this stochastic process

Consider infinitely repeated coin-toss. Then the probability space can be written as $\Omega=\{H,T\}^\infty$ with its product $\sigma$-algebra. Now let's assume that after each round, there is ...
2
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1answer
41 views

Compute $\mathbb{E}[\tilde{X}_t]$, where $\tilde{X}_t=X_t=(1-t)\int_0^t\frac{1}{1-s}dW_s$ for $0\le t<1$ and $\tilde{X}_t=0$ for $t=1$

I have the following exercise and I don't really understand the answer. I am going to write my professor's answer first, then a question about what I don't understand about my professor's answer and ...
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59 views

Strong law of large numbers

Suppose $X_i\in\mathcal{L}^2$ with expectation $0$ such that $\sum_{i=1}^\infty \mathbb{E}[X_i^2]/i^2<\infty$ and suppose they are pairwise non correlated. Does then the SLLN still hold?
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1answer
29 views

Calculate $\mathbb{E}[M_{\alpha}^{p}(t)]$ for all $p>0$ and $t>0$, where $M_{\alpha}(t):=e^{\alpha W_t-\frac{\alpha^2}{2}t}$, $t\ge 0$

I am going through this solved problem but I don't understand some steps. My professor is notorious for making errors very often so don't hold back if you think he's wrong... Or if I am wrong. I am ...
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0answers
16 views

Questions on drifts and Girsanov transforms.

I wish to prove the following the statement: "If $\mu$ and $\gamma$ are probability measures on $C([0,\infty), \mathbb{R}^d)$, with $\gamma$ being the standard Weiner measure, $W_t$ being standard ...
3
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1answer
58 views

Markov and strong Markov properties

In my study of strong Markov property of an RCLL canonical Markov process I encounter the following definition: Suppose $Y_t:\omega\rightarrow \omega(t)$ is canonical Markov process with respect to ...
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30 views

Measurability of a random function

Suppose $(U_t)_{t\in[0,1]}$ is a stochastic process such that for every $\{t_1,t_2,\dots ,t_n\}\subset[0,1]$, $$U_{t_1},U_{t_2},\dots ...
2
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0answers
22 views

Probabilistic interpretation for Fokker-Planck equation

It is well known that if $X_t$ is a stochastic process that solves the SDE $$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$ with $W_t$ a Wiener process, then the associated ...
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1answer
60 views

Is this set of random variables a Hilbert space?

Consider a sequence of i.i.d. random variables $\left\{ {{\varepsilon _t}} \right\}_{t = 1}^\infty $ with $E\left( {{\varepsilon _t}} \right) = 0$ and $E\left( {\varepsilon _t^2} \right) = {\sigma ...
2
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1answer
78 views

A simple implication of an approximation theorem by Komlós, Major and Tusnády

I have been reading through the PhD thesis of Professor Aue on change point analysis based on invariance principles. There's a particular argument I have not been able to follow: Let ...
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1answer
40 views

sufficient conditions for a stochastic process to be wide sense stationary

From the page Stationary process, I have the following definition: WSS random processes only require that 1st moment and autocovariance do not vary with respect to time and from the page ...
4
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53 views

Weak convergence of a sequence of stationary distributions to another stationary distribution

Let $\{X_n(t) \in \mathbb{Z}^+\}$ for each $t \in (0,1)$ denote a discrete time Markov chain (with time index $n$ and parameterized by $t$). For each $t$, the Markov chain $\{X_n(t)\}$ has a unique ...
4
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1answer
36 views

Largest jumps of a spectrally positive $\alpha$-stable process

Let $X(.)$ be a (strictly) $\alpha$-stable process (with $\alpha \in (1,2)$). Assume also that $X(.)$ is spectrally positive (its Lévy measure is concentrated in $[0,+\infty)$). I am looking for a ...