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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A stochastic volatility model

An example of stochastic volatility model: $$\begin{cases} \frac{dX_t}{X_t} &= g_t dW_t \\ dg_t &= - k g_t dt + \sigma dZ_t \end{cases} $$ where $Z_t$ and $W_t$ are Brownian motions and ...
5
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1answer
35 views

Marginally Gaussian not Bivariate Gaussian - Ito Integral

Let $(W_t)_{0\leq t\leq 1}$ be a Wiener process defined up to time $1$ on some probability space. Consider the random vector $$\left(W_{1},\int_0^1 \operatorname{sgn}(W_s) \, dW_s\right)=:(W_1,X_1)$$ ...
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24 views

Compute the Mean of the Following Process

Given the following process: $\Delta \ln(St+1)= \mu - (\sigma2/2) + \sigma(\varepsilon(t+1))$ (where both $\mu$ and $\sigma$ squared are of $S$) How does one calculate the mean of $S(t+1)/S(t)$? ...
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18 views

Perron-Frobenius Theorem: Markov Chain -> Matrices

I am interested in finding out a way how to transform the stochastic results of perron-frobenius for markov chains to any matrix. I am aware that perron-frobenius was originally proofed with linear ...
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13 views

Spectral Density of an ARMA process.

For an upcoming Stochastic Processes exam, we have had a sudden brief email about Spectral Density as the lecturer had forgotten to mention it in classes. He states, For an ARMA process with ...
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1answer
24 views

Expected location of Brownian motion on the circle

Intuitively it seems likely that the expected whereabouts of Brownian motion on the unit circle would be the origin $\left(0,0\right)$, at least in the limit as $t\to\infty$. Is this right? Are there ...
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23 views

Probability that Brownian motion falls between two piecewise constant functions

I'll first present the problem, and then describe my motivation: Suppose $a_j \in \mathbb{R}$, $b_j \ge 0$, and $0 = t_0 < t_1 < \cdots < t_J$ are time points. Let $W_t$ be a standard ...
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1answer
43 views

show that the process is a martingale [closed]

maybe you will have an idea how to show that : the process $(exp(X_t-\frac{1}{2}Y_t))$ is a martingale? Where $h \in L^2([0,T])$, $T< \infty$, $X_t=\int_0^th(s)dW_s$ and $Y_t=\int_0^th^2(s)ds$ for ...
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0answers
21 views

What is the probabilty of event A given event B probability in next T time duration?

Assume: a system S with three component: A, B and C. At any moment any component may fail. System MAY fail due to failure of any one component. Given: From the history, we know how many times ...
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21 views

Transition functions induced by Markov processes

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and denote by $(X_t,\mathcal{F}_t)_{t\geq 0}$ a time-continuous Markov process with values in $(E,\mathcal{E})$. For $s<t\in ...
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58 views

Can this approximation result for stochastic processes be modified.(p=1 instead of p=2)?

In McKeans stochastic integrals from 1969 he proves this: You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is ...
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117 views

Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
4
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2answers
54 views

process with integral is martingale

How to show that the process $X_t=tW_t - \int_0^t W_s ds $ is a martingale? I guess I have to use the definition of martingale and properties of Wiener process, but I stack with this integral. ...
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1answer
33 views

Basic Question on Definition of Brownian Motion

I am quite new to discrete and continuous stochastic processes. It seems there is something I don`t understand about definition of Brownian motion. Let $\Omega, \mathcal{F}, \mathbb{P}$ be a ...
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0answers
16 views

Understanding of Second Arcsine law for Brownian motion

Ok I'm trying to understand the second arcsine law which states: Let $g_t:=\sup\{s\leq t:W_s=0\}$, then $$\mathbb{P}(g_t\leq s)=\frac{2}{\pi}\arcsin \left(\sqrt{\frac{s}{t}}\right )$$ This won't be ...
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32 views

Stochastically perturbed fluid flow map ${\rm d}Φ_t(x_0)=u_t(Φ_t(x_0)){\rm d}t+ξ_t(Φ_t(x_0)){\rm d}W_t$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace $(W_t)_{t\ge ...
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2answers
62 views

Discounted price process in Black-Scholes model is a martingale with respect to Q.

I have been presented a proof that the discounted price process in the Black and Scholes formula is a martingale, but there is something important omitted, and I am not able to fill in the gap. I will ...
0
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0answers
12 views

what does a time-reversible Markov matrix look like

I think an unitary symmetric matrix P satisfies the condition for time-reversible Markov chain. Since $P^2=I$ we have $P=P^{-1}$. Let v be the eigenvector with eigrnvalue $\lambda=1$, then we have ...
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0answers
12 views

Replicant portfolio with commissions (Jarrow rudd)

I have created a Jarrow Rudd three for a call option that I know how to replicate with a portfolio. A replicating portfolio of a option works this way: At time 0 we form a replicating portfolio ...
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40 views

Feller property of Ito diffusion

Consider the following Ito diffusion $X_t$ satisfying $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x,$$ with Lipschitz coefficients $b,\sigma$. It can be shown that if $g$ is bounded and continuous, ...
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9 views

What does fixed regressor say about our linearity condition?

The linearity condition states that $y_i=(\vec{x}_i)^{T}\vec{\beta}$ for all $i$. Now, if we have fixed regressors, $\{\vec{x}_1,\vec{x}_2,\cdots\}$, our linearity condition only says for those ...
2
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1answer
15 views

Definition of Cylindrical Brownian Motion and Spatial Correlation

From Gawarecki and Mandrekar, Stochastic Differential Equations in Infinite Dimensions: We call a family $\{ W_t \}_{t\geq 0}$ defined on a filtered probability space $(\Omega, \mathcal{F}, ...
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1answer
18 views

A random variable is mapping from sample space to real numbers. How about random process?

A random variable is mapping from sample space to real numbers. How about random process? Can we think of the simplest random process as again a mapping from sample space to real numbers, with the ...
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0answers
17 views

What are the differences in linearity in Non-stochastic and Stochastic Regression?

I have been confused with the differences between stochastic and non-stochastic explanatory variables for a while. I was able to write down some of my understanding and seek approval or comments about ...
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3answers
43 views

Archer Poisson Process problem

An archer wishes to shoot an arrow at a target. The prospective flight path of the arrow is subject to birds flying past at random times, according to a Poisson process with rate $\mu$ per second. To ...
2
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1answer
22 views

Are these facts about the Poisson process correct?

Before studying theorems one by one, I want to check whether it is right what I know about Poisson process. Let $\left\{N(t)\right\}$ be a Poisson process. Then the number of the event occur during ...
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1answer
19 views

Clarification regarding heat kernel for Brownian motion on a manifold

Let $X$ be Brownian motion on a Riemannian manifold $M$ starting at $x\in M$, D a domain and $f$ a bounded continuous function on $D$. Define $\tau_D$ to be the first exit time of $X$ from $D$. ...
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41 views

Mixing convergence

Given a process $X_n \xrightarrow{d} X$ on some probability space $(\Omega,\mathcal{A},P)$. If for every $B \in \mathcal{A}$ it holds, that $$ \lim_{n\rightarrow \infty} P(X_n\in A,B)=P(X\in A)P(B) $$ ...
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31 views

Can a process be both chaotic and stochastic?

I have been reading about processes like evolution in biology. As the environmental variables that effect an individual's fitness and its survival, are stochastic, they effect the individuals in an ...
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22 views

Calculation of renewal function $R(t) = \sum{F^n(t)}$?

My textbook defines the renewal function $R(t) = E[N_t] = \sum_{n=0}^\infty F^n(t)$, where $F^n(t)$ appears to be the n-fold convolution of $F$ with itself. $F$ is the distribution of the interrenewal ...
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2answers
84 views

Splitting Poisson process formal proof

Let $\{X_t\}_{t\ge 0}$ be a Poisson Process with parameter $\lambda$. Suppose that each event is type 1 with probability $\alpha$ and type 2 with probability $1-\alpha$. Let $\{X^{(1)}_t\}_{t\ge 0}$ ...
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0answers
26 views

Why is this a beta distribution?

I'm given a circle with point $A$ defined by $(x,y)$. Then $T=1-d[O,A]$, so $T=1-\sqrt{(x^2+y^2)}$. Asked to find: $P[T<=u]$ $E[T]$ $Var(T)$ Alright, so $d[O,A]$ has the CDF $u^2$. So, for ...
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1answer
24 views

Equal distribution only for finite dimensional distributions

Two processes $(X_t)_{t \in T}$, $(Y_t)_{t \in T}$ are known to be equal in distribution if and only if they agree on all finite-dimensional distributions, i.e., for all $t_1$, $t_2$, $\ldots$, $t_n$, ...
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0answers
7 views

discrete delayed renewal process

I would like to prove that any delayed renewal process $N^∗$ satisfies the following property: Define $A_k$ = {Renewal at time k}. Then conditionally on $A_k$, the process $N(n) = N^∗(k + n) − ...
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1answer
17 views

What is the probability between two merged Poisson process? [closed]

Question: A store opens at $t =0$ and potential customers arrive in a Poisson manner at an average arrival rate of $λ$ potential customers per hour. As long as the store is open, and independently ...
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0answers
20 views

How to Prove that a (Centered) Gaussian Process is Markov if and only if this Equation Holds?

A centered Gaussian process is Markov if and only if its covariance function $\Gamma: \mathbb{R}\times\mathbb{R} \to \mathbb{R}$ satisfies the equality: ...
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1answer
26 views

A question about merged Poisson process

Question here: Iwana Passe is taking a multiple-choice exam. You may assume that the number of questions is infinite. Simultaneously, but independently, her conscious and subconscious faculties are ...
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0answers
13 views

Conditional expectation of a geometric Brownian motion and stopping time

Let $X$ be a geometric Brownian motion, solution of $dX_t = \mu X_t dt + \sigma X_t dW_t, X_0 > 0$ and ${\cal F}$ its natural filtration. Let's consider $t \geq 0$ and $\tau_a$ the first hitting ...
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22 views

Geometric Brownian motion hitting time

Let $X$ be a geometric Brownian motion $dX_t = \mu X_t dt + \sigma X_t dW_t, X_0 > 0$ and ${\cal F}$ its natural filtration. Let $\tau_a$ be the first hitting time of $a$ by $X$. How can we relate ...
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1answer
28 views

Density of Running Maximum of Drifted Brownian Motion Computation

$\textbf{Proposition}$ The $pdf$ of the Maximum of a Brownian Motion with Drift is given by $$ f_{M_t}(m)={\sqrt{\frac{2}{\pi t}}} ...
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31 views

Brownian motion on a manifold

If I have a manifold $M$ and a chart $\left(x,U\right)$, is it possible to simulate Brownian motion on that manifold by solving an SDE in the chart representation $x\left(U\right)$ and then use the ...
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1answer
26 views

SDE for Brownian motion on a circle [closed]

Brownian motion on a circle can be generated by $\left(\cos\left(B_t\right),\sin\left(B_t\right)\right)$ where $B$ is Brownian motion on the real line. My question is what SDE was solved to get this ...
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81 views

Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.

Let$^1$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace ...
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0answers
23 views

Problem calculating the conditional expectation of an Ito process

let $X_t$ be an Ito process where $X_t = \int_{0}^t v_t dB_t$ where $v_t$ is a stochastic process, $B_t$ is a Wiener Process, $\mathcal{F}_t$ be a filtration: $\sigma\{B_t, 0 \leq t \leq T\}$, and ...
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0answers
20 views

Mean value function of a discrete-time Markov-modulated Poisson process

By "discrete-time Markov-modulated Poisson process" I mean a semi-Markov process $\{(X_n,T_n):n=0,1,\ldots\} $ which satisfies $$T_{n+1}-T_n\mid X_n\sim\operatorname{Exp}(\lambda_{X_n}), $$ with ...
2
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0answers
64 views

Relation between the expected number of visits to a state and reachability in a Markov chain

Let's consider a discrete time Markov chain $X_n$. Let $R_{ij} = \sum_{n=0}^\infty \mathbb{1}_{\{X_n= j | X_0 = i\}}$ be the number of visits to $j$ starting from $i$, and let $f_{ij}$ be ...
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1answer
33 views

Doob's maximal inequality

Let $X$ be a cadlag $L^{p}$ martingale ($p>1$). Let $q$ be the Hölder conjugate of $p$. Let $F$ be a finite subset of $[0,t]$. The following claim appears in a proof of Doob's maximal inequality ...
0
votes
1answer
76 views

Distribution of a process dependent on a Markov chain's states

Consider a Markov chain $X_t$ with state space $\{0,1\}$, initial distribution $$ \begin{array}{l} \mathbf{P}(X_0=1)=\lambda \\ \mathbf{P}(X_0=0)=1-\lambda \end{array} $$ and transition ...
2
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0answers
20 views

Changing domain of the CGF of a stochastic process

Given a stochastic cadlag process $(X_t)_{t\geq 0}$. Define $A_{t}$ as the domain of the cumulant generating function by $f_t(s):=\log E(e^{sX_t})$, for which this expression is welldefined. I search ...
4
votes
1answer
46 views

Ito's formula and Taylor expansions for jumps processes.

Consider some model $$ dX_t = \mu d t + \sigma dW_t $$ where $\mu, \sigma$ are some constants. Now let $f \in C^{1,2}$ and consider $$ Y_t = f(t,X_t). $$ Say we (informally) consider a second order ...