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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damped harmonic oscillator driven by a stochastic momentum (not force)

Could you give references for solutions or solutions to the following problem: Given: damped harmonic oscillator driven by stochastic force of very short duration (= stochastic momentum). Find: ...
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148 views

How to construct the strong solution to the SDE $dX_{t}=\sqrt{X_{t}}dW_{t}$?

Given the SDE: $dX_{t}=\sqrt{X_{t}}dW_{t},$ $\ X_{0}=1$ , where $W_{t}$ is a 1-d Brownian motion. I was told that this SDE has a unique strong solution, but I don't know how to construct it. I know ...
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214 views

rate of convergence of mean ,variance & skewness estimators

We are asked a question which of mean,variance or skewness converges faster. At first I thought it was straight forward answer: mean->variance->skewness. But I am not sure anymore because I read ...
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176 views

how to show that the price process is a martingale

Suppose I have an $d$-dimensional semimartingale $S=\{S_t\}$ with $t\in[0,T]$ under $P$. $S $ need not to be continuous (RCLL can be assumed). Suppose $Q$ is an equivalent measure w.r.t. $P$ such that ...
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1answer
210 views

Conditional Probability & Stochastic process Example

I'm taking this example from a paper I'm reading. I'm having trouble understanding the logic, and I'm hoping someone can help me. Let $ v_{t}=a \ x_{t}+u_{t} $ where $a$ is some constant and $u$ is ...
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86 views

Formulae to about Moment and Cross-moments of Stratanovitch Iterated Integrals

The title is a bit long but quite explicit, I am looking for a reference where the moments and cross moment Stratanovitch Iterated Integrals defined as : $E[J_n(1).J_p(1)]$ with $p\not=n$ With : ...
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140 views

Definition of Doob martingale

From Wikipedia A Doob martingale is a generic construction that is always a martingale. Specifically, consider any set of random variables $$ \vec{X}=X_1, X_2, ..., X_n $$ taking values in ...
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162 views

Expectation of the following random variable related to random walk and stopping time

Assume that $S_n=\sum_{i=1}^n X_i$ where $X_i$ are iid r.v. with finite mean $E(X_1)=\mu$. Assume that $\tau$ is a stopping time with finite expectation. What's the expectation of $\tau S_\tau$? Is ...
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162 views

calculate a conditional expectation

Doing some exercises from a mathematical finance book, I got stuck at the following point. It is a purely probability question. Let $S_t^1 = \sigma W_t$, where $W_t$ is a brownian motion and ...
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158 views

Programming a set of binary switches where the lifetime of a given state is exponentially distributed

Imagine I have $(s_1, ..., s_N) \in S$ binary switches in a panel that can be switched between states $0$ and $1$. Initially, we flip all of the switches to the $0$ state. Now, for each of $(t_1, ...
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74 views

convergence in law of an exponential brownian motion

I have a queston about the convergence in law of the following stochastic processe: $$\left\{I_t=\left(\int_0^te^{B_s}ds\right)^{1/\sqrt{t}}\right\}_{t\geq 0}$$ with $\{B_t\}_{t\geq 0}$ is a ...
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225 views

Correlated diffusion processes and covariance matrix

I'm really noob in maths topics so I hope you will excuse me if I use terms which aren't correct. I would like to simulate $n$ dimensional diffusion processes with $n$ noises. Each process has its ...
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1answer
110 views

Variable t times a Wiener Process W(1/t)

If $W(t)$ is a Wiener process and $V(t) = t\cdot W(1/t)$ is it possible to say that Since $W(1/t)\space \sim N(0,1/t)$ that $V(t) \sim t\cdot N(0,1/t)$? And if so then is $t\cdot N(0,1/t) = ...
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1answer
670 views

Poisson Process Arrival Probability

Just a quick question regarding two Poisson Processes: Let $X_t$ and $Y_t$ be two independent Poisson Processes with rate parameters $\lambda_1$ and $\lambda_2$, respectively, measuring the number ...
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115 views

Martingale Decomposition

Is it possible to decompose a discrete-time martingale $(M_n)$ uniquely into two processes $$M_n=M_n^I+A_n$$ where $(M^I_n)$ is a martingale with independent increments and $(A_n)$ is a martingale? If ...
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2answers
127 views

Brownian motion: Show $\lim \sum W_{i} (W_{i+1}-W_{i})=\frac12 W^2_t-\frac12 t$ in probability.

Let $\{t_i\}_{i=1}^n$ be a partition of $[0,t]$ and $W$ a standard Brownian motion. Write $W_i$ for $W_{t_i}$. Show $$ \lim \sum W_{i} (W_{i+1}-W_i)=\frac12 W^2_t-\frac12 t $$ where the limit is in ...
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299 views

Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?

The following is the definition of infinitesimal generator from Oksendal. Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
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1answer
99 views

How do I find the variance of a linear combination of terms in a time series?

I'm working through an econometrics textbook and came upon this supposedly simple problem early on. Suppose you win $\$1$ if a fair coin shows heads and lose $\$1$ if it shows tails. Denote the ...
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65 views

When is this reversible diffusion on the integer lattice non-exploding?

Let $U\in C^{\infty}(\mathbb R^n;\mathbb R)$ and consider a continuos time Markov chain on the scaled integer lattice $\delta\mathbb Z^n$ with jump rates given by $r_{\delta}(x,y) := \begin{cases} ...
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150 views

Behavior of explosive random process

Inspired somewhat by this problem, I've been investigating the behavior under iteration of the following discrete random process: Given $n\in\mathbb{N}$, choose an integer from $\{0,1,\ldots,n\}$ ...
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1answer
180 views

Sum of random subsequence generated by coin tossing

Let $(\pi_1, \pi_2, \cdots)$ be an infinite sequence of real numbers such that $\forall i\; \pi_i > 0$ and $\sum_i \pi_i = 1$. This can be thought of as a probability over natural numbers. Let ...
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691 views

Probability of Gambler's Ruin with Unequal Gain/Loss

I've spent some time reading other questions about Gambler's Ruin, but couldn't find the answer I was looking for. In most questions, it is assumed the Gambler wins \$1 or loses \$1. I'm curious how ...
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88 views

Existence and Unicity Weak, Strong, Pathwise, In Law, etc… for SDE's

I feel always confused with weak, strong, pathwise unicity and or existence for Stochastic Differential Equations. It is mainly my own fault and I should do something about it. But it would be much ...
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1answer
345 views

Prove Yor's formula not using Itô?

In a book is a exercise to prove Yor's formula for stochastic exponential, i.e. $$\mathcal{E}(X+Y)\exp{(\langle X,Y\rangle)}=\mathcal{E}(X)\mathcal{E}(Y)$$ where ...
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174 views

Show that $O_t$ is a Gaussian Process

Let $B_t$ be a Brownian motion process. Let $$O_t = e^{-\alpha t} \int^t_0 e^{\alpha s} dB_s$$ Find $\mathsf{E}[O_t]$ and show that $O_t$ is a Gaussian process. I think ...
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1answer
82 views

Why $\lambda\equiv\nu(\mathbb{R})$ for compound poisson process?

I've seen this notation $\lambda\equiv\nu(\mathbb{R})$ in the book of Tankov and Cont for compound poisson process. I thought before that $\lambda$ (jump intensity) can be choosen independently of ...
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1answer
195 views

calculate the expectation of a modified compound poisson process

Let $t\in [0,a]$, $X_t:=\sum_{i=1}^{N_t}Y_i$ be a compounded Poisson process, i.e. $N_t$ a Poisson process with parameter $\lambda>0$ and $Y_i$ are iiid with distribution $\mu$. Now let ...
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56 views

an inequality about self-similar process

I have a question in my homework: A continuous process $X$ is said to be self-similar if for every $\lambda>0$, $(X_{\lambda t})_{t\geq 0}$ has the same law as $(\lambda X_t)_{t\geq 0}$. Let $X$ ...
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272 views

Null recurrence in Discrete Time Markov Chains

Is it possible to have null recurrent states if the number of states is finite? If so, I would appreciate a small example (a 2x2 or 3x3 transition probability matrix would be nice).
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285 views

Books of Random Numbers. How were the numbers generated?

In the mid 1940's I believe, the RAND corporation published a book with a million random numbers (from a normal distribution). This was before Marsaglia, so considering the primitive state of their ...
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2answers
411 views

discrete stochastic integral is a local martingale

I want to prove the following statement in discrete time: Let $(X_k)$ be a local martingale and $(h_k)$ a predictable process, then the discrete stochastic integral is also a local martingale. ...
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172 views

Question about a proof of Ansel-Stricker in a paper

I was working through a paper by M. De Donno, which proves the Ansel-Stricker lemma in a different way. The paper can be found here. I've chosen this paper instead of the original one by ...
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1answer
209 views

Good substitutes for Ross's book on Probability Models

I was wondering if there are any FREE good alternatives to Sheldon Ross's Probability Models which are more succinct? Are there any free online resources (websites/PDFs/course notes) which cover more ...
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1answer
116 views

Continuous Time Stochastic Process

I am trying to build a stochastic model where two processes happen randomly with different rates that depend on the status of the system. Imagine you have a grid NxN made of 0 or 1. The 1 elements ...
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2answers
56 views

Finding the probability of a client getting the same token in two consecutive interactions.

I am trying to find the probability in the following real-world inspired scenario. If I have a finite set of whole numbers from 0 to 4 billion which I call tokens and $n$ clients. Each time a client ...
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1answer
143 views

Conditional expectation of a finite variation process

A simple question: Let $H$ be a cadlag, adapted process and $A$ a process of finite variation. Then also $\int_t^T HdA_t$ is a finite variation process (see "Limit Theorems... "Jacod&Shiryaev ...
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1answer
584 views

Covariance of Gaussian stochastic process

Could someone help me to figure out solutions of following problems?: Let $X = (X_t)_{t \geq 0}$ be a Gaussian, zero-mean stochastic process starting from $0$, i.e. $X_0 = 0$. Moreover, assume that ...
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2answers
784 views

Branching Process Extinction Probability

I'm doing a branching process problem and am not sure I did it correctly. Suppose $X_0 = 1$ and $p_0 = .5, p_1 = .1,$ and $p_3 = .4$ represent the probabilities that zero, one, and three individuals ...
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364 views

Right continuous stochastic process

Can anyone suggest how to prove a right continuous stochastic process is measurable? Thanks Indrajit
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124 views

Certain transformation of Brownian motion is a submartingale

I have a question about a proof in Protter. Let $B$ Brownian motion and $u$ a harmonic (subharmonic) function. Then $u(B)$ is a local martingale (submartingale). I was able to show the case of local ...
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1answer
471 views

Cylindrical sigma algebra and continuous functions.

Consider the space $\mathbb R^{[0,1]}$ of all functions from $[0,1]$ to $\mathbb R$ and the cylindrical sigma algebra $\mathcal B$ on it. I know how to prove that $C[0,1]\not \in \mathcal B$. My ...
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420 views

Is every right continuous local martingale of finite variation constant?

I was reading a chapter in Dellacherie and Meyer. Suppose we have right continuous adapted processes $A$, $A'$ of finite variation. Both are null at zero and the difference is a local martingale. I ...
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176 views

Confusion regarding Stochastic integral

I've a stupid doubt in the construction of stochastic integral of real scalar valued maps. Many times I've seen in books after the stochastic integral is defined in [$0,T$] for the integrand in $L^2$ ...
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207 views

stochastic exponential

For a semimartingale $X$, we want to solve the SDE $dZ_t=Z_{t-}dX_t$. I was able to prove that $$ Z_t:=\exp{(X_t-\frac{1}{2}\langle X\rangle_t^c)}\prod_{0<s\le t}(1+\Delta X_s)\exp{(-\Delta X_s)} ...
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1answer
126 views

Consequence of Doob-Meyer decomposition

Let $\mathcal{H}^2_0$ be the space of all $L^2$ bounded RCLL martingales null at zero. As a consequence of Doob-Meyer we know: For every $M\in \mathcal{H}^2_0$ there exist a unique adapted, ...
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1answer
567 views

Countable state Markov chain: detailed balance consequences

Let $S$ be a countable set and $\pi$ a probability distribution on $S$. A discrete-time Markov chain $(X_n)$ with state space $S$ is said to be in detailed balance with respect to $\pi$ (or simply in ...
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84 views

“To every Q-matrix corresponds a unique Markov process.” Proving uniqueness

"To every Q-matrix corresponds a unique Markov process." I'm trying to understand Klenke's proof of the uniqueness part of this proposition. Klenke's proof Following is an adapted version of ...
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338 views

Computing the mean of a stationary distribution for Wright-Fisher model

The Question Considering a population of $N$ organisms, there are two possible gene types, $A$ or $a$. An $A$ that is drawn ends up becoming an $a$ in the next generation with probability $u$ and ...
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88 views

For $X_{t}=\exp\left\{\left(\mu-r-\frac{\sigma^{2}}{2}\right)t+\sigma W_{t}\right\}$, do we have $\mathbb{E}[\int_{0}^{\tau_{b}}X_{s}dW_{s}]=0$?

Let $X_{t}$ denote the solution to the SDE: $$dX_{t}=(\mu -r)X_t dt+\sigma X_t d W_{t}, \ X_{0}=1$$ i.e. $X_{t}$ is the process: $$X_{t}:=\exp\left\{\left(\mu-r-\frac{\sigma^{2}}{2}\right)t+\sigma ...
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408 views

In a continuous-time Markov process, is the waiting time between jumps a function of the current state?

Two books construct Markov processes from Q-matrices using waiting times and jump chains but differ in whether the waiting times depend on the current state. Can the two be reconciled? Klenke claims ...