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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Reformulate a SPDE parameterized by space and time as an SDE parameterized by time (as it is possible for PDEs)

Let $d\in\mathbb N$ $\mathcal V_t\subseteq\mathbb R^d$ be a bounded domain for $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ be bijective for $t\ge 0$ with $$\Phi(\;\cdot\;,x_0)\in C^1\left(\mathbb ...
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1answer
17 views

Monotone property of transition density of rotational $\alpha$-stable process

For a Brownian motion $B_t$ in $\mathbb R^d$, the transition density of $B_t$ is the normal distribution $$P_x[B_t\in dy]=(2\pi t)^{-d/2}e^{-\frac{|x-y|^2}{2t}}dy$$ and obviously the density is ...
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8 views

Markovian Gaussian stationary process with continuos paths

Could you, please, help me figure out the following problem. We call a stationary Gaussian process $\xi_t$ (with continuos paths) an Ornstein-Uhlenbeck process if its correlation function ...
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1answer
19 views

Probability of exit from compact set

I have a continuous real valued diffusion $\{ X_t \}_{t\ge0 }$ that is contained in a compact set $[a,b] $of $\mathbb{R}$, where $a > 0$ and. Define the stopping times \begin{equation} \tau_c=\inf ...
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+50

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
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0answers
15 views

Gaussian process via RKHS construction: joint measurability comes for free?

Billingsley's probability and measure and others show the joint measurability of the Brownian motion using the continuity of paths. Makes me wonder if we can say it before saying the continuity, if we ...
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4answers
416 views

Figuring out probabilities with Hidden Markov Models

I'm really new to Math so sorry in advance if this question does not make sense. Also I cross posted this on stats.stackexchange.com also. Background: I'm trying to learn about hidden Markov models ...
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1answer
21 views

convergence in distribution of exponential of a brownian motion

If $(B_t)_{t≥0}$ is a standard Brownian motion, show that, as $t \to \infty$, $$ \left(\int_0^t e^{B_s} \, ds\right)^{1/\sqrt{t}} \text{ converges in distribution to} \ e^{M_1}, $$ where $M_1 = ...
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0answers
30 views

Strong Markov property proof

Let $X$ be a Markov chain with state space $\mathcal{S}$ and denote $\mathbb{N} := \{0,1, \cdots\}$. I need to show that for any stopping time $\tau < \infty$ and any bounded measurable function ...
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1answer
55 views

Itô-formula proof, remainder term.

I have a question about the proof a a certain version of the Itô-formula. First the author defines an Itô-process and states the formula: My question is in regarding the proof. The proof uses ...
2
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1answer
19 views

Geometric Brownian motion with exponential of sum of iid's

Glasserman's "Monte Carlo Methods in Financial Engineering" on p. 265 states that the geometric Brownian motion can be modelled with : $$S(t_n)=S(0) \exp(\sum_{i=1}^n X_i)$$ where $X_i$ are iid. I ...
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1answer
238 views

How well can the maximum of a Gaussian process be approximated by a finite-dimensional Gaussian variable?

Consider a compact set $K$ in $\mathbb{R}^p$, and let $W$ be a mean zero continuous Gaussian process on $K$, meaning that $W$ takes its values in the space of continuous functions from $K$ to ...
2
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1answer
275 views

Progressively Measurable for Rigth Continuous Adapted Processes

Any adapted and right continuous process $X_t$ is progressively measurable. For the above statement, I found proof in several books. They all have similar argument as follows. For a given $t > ...
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2answers
46 views

References on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style to Terence Tao

I learned a lot from prof Tao's notes and books because unlike many authors, he seems to prefer writing more words, explanations and intuitions rather than just mathematical formulae. His approach is ...
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57 views

Characterization of point process, given the number of points

For a point process with independent and identically distributed (i.i.d) inter-renewals, with distribution $p(x)$, we observed $N$ points on $[0,T]$. What is the joint probability distribution ...
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25 views

How to derive the distribution of measurement noise in discrete Kalman filter which is transformed from continuous one?

With sampling time $T$, and a continuous measuring model: $$ \begin{align} y(t) &= Cx(t)+v(t) \\ v(t) & \sim \text{N}(0,R_c) \end{align} $$ we can change it into a practical discrete one, ...
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15 views

Kolmogorov forward and Backwards equation interpretation

Let $\lambda_i$ be the sojourn rate of state i, $q_{ij}$ be the transition rate form i to j, and $p_{ij}$ be the transition probability from i to j. The Kolmogorov Forward and backwards equation are ...
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1answer
14 views

Why does Euler-Maruyama method use a square root of the time step

Euler-Maruyama method is supposed to be an extension of the Euler method for ODE, but applied to SDE. This means that if we have an equation: $$ dY_t = Y_t dW_t $$ where $W_t$ is the Wiener process, ...
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22 views

Uniqueness of Predictable Quadratic Covariation

In order to prove that $\langle M,N \rangle$ is the only process which is continuous and has bounded variation such that \begin{align} M_tN_t - \langle M,N \rangle_t \end{align} is a continuous ...
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1answer
36 views

A question about Bernoulli process ( maybe conceptual)?

Question: Al performs an experiment comprising a series of independent trials. On each trial, he simultaneously flips a set of three fair coins. Whenever all three coins land on the same side in any ...
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23 views

probability of splitted exponentially distributed random variables

Let $X$ be a exponentially distributed random variable(time interval) with mean of $u$ And $Y$ be a exponentially distributed random variable (time interval)with mean of $\lambda$ And $s$ be a ...
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2answers
22 views

A relation between first passage time and occupation time

Let's think about a discrete time Markov chain $X_t$ with only one recurrent state. Let $T$ be the random variable that is the number of steps taken from a given state $i$ to the recurrent state (ie. ...
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21 views

Hypothesis Testing on Renewal Processes

We have time $[0,T]$ to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of ...
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36 views

Quaternion Kalman Filter Process Noise

I'm implementing a extended Kalman filter using quaternions. I've extended this paper to deal with my custom observations. My state space is analogous to the one in the previous paper : $ ...
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1answer
26 views

Simple Random Walk - Why are these two events the same?

Let $S = (S_n)_{n \geq 1}$ be a simple random walk. We denote the hitting time of a point $b$ by $\tau_b = \min \{i \geq 1 : S_i \geq b\}$. My text says that the events $\displaystyle\{\max_{k \leq ...
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3answers
800 views

Sensor fusioning in Kalman filter

I'm interested, how is the dual input in a sensor fusioning setup in a Kalman filter modeled? Say for instance that you have an accelerometer and a gyro and want to present the "horizon level", like ...
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1answer
83 views

Hidden Markov Models and Viterbi Algorithm: Fair and Biased Die

So following is the problem that I am trying to solve using Viterbi algorithm and HMM: Before attempting to write a program, I want to do this problem by hand for the first 3 observations($651$). ...
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1answer
42 views

How long does it take two identical hidden Markov models run on same observations to forget their initial distributions (if ever)?

Let $H_1$ and $H_2$ be two instances of a finite Hidden Markov Model (HMM) $H$. That is, $H_1$ and $H_2$ have identical state spaces $Q$ as well as identical transition $A$ and emission probabilities ...
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2answers
66 views

Is there a proof that the observations of a hidden Markov chain is not itself a Markov chain?

Suppose $\{X_n\}$ is the hidden Markov chain, and $\{Y_n\}$ is the series of observations, where $\mathbb{P}\{Y_n = j| X_n = i\}$ is the same for all $n$ (please correct me if I have not stated the ...
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9 views

Identification of Infinite Dimensional State in Hidden Markov Model

Consider a hidden markov model (HMM) where the state, $X_t(\alpha)$, is a stochastic distribution over $\alpha \in \mathbb{R}_+$ and one observes a signal $Y_t$, which is simply a moment of this ...
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72 views

Transition matrix in left-right hidden semi-Markov model

I'm developing a hidden semi-Markov model left-right . In a left-right model a sequence of $M$ states starts in state $1$ and ends in state $M$, with no repetition of states. Since the model is ...
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25 views

How to Compute Distributions in Hidden Markov Models?

Let $Z_1, Z_2, ..., Z_n$ be the latent variables, and $X_1, X_2, ... X_n$ be the observed ones in a hidden Markov models. Let's assume that the parameters of the hidden Markov models are known: the ...
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1answer
37 views

How to solve for the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$?

I would like to find the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$. My strategy is to use Ito's general formula with: $$ f(t, B_t) = f(0,0) + \int_0^t \frac{df}{dx}(s, B_s) dB_s + ...
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1answer
28 views

($N_t$) is Poisson process with $\lambda = 1$. Calculate $E(N_2\mid N_1)$ and $E(N_1\mid N_2)$

($N_t$) is a Poisson Process with constant rate $\lambda = 1$. $1)$ Calculate $E(N_2\mid N_1)$: So this is how far I've gotten: Let $N_2 = N_1 + (N_2 - N_1)$ $E(N_2\mid N_1) = E(N_1\mid N_1) ...
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0answers
17 views

Non-uniform convolution with discrete wavelet transform

I understand that if you have a circular N-dimensional convolution matrix, it can be diagonalized with the Fourier transform of the convolution operator. This makes it easy to calculate the density of ...
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1answer
19 views

Are queues CTMC?

The $M/M/1$ queue have all the properties of the countable state continuous time markov chain. Is any general queue also a countable state CTMC?
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21 views

Deduce stochastic differential equation

Let $X$ be a stochastic process with $dX_t = \alpha X_t dt + \sigma X_t dW_t$ and $Y$ a stochastic process with $dY_t = \gamma Y_t dt + \delta Y_t dV_t$, where $W$ and $V$ are independent ...
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13 views

stochastic experiments like galton board?

do you know some other stochastic/statistic experiments like galton board? I'm looking for something that could be build for learners or people who are interested in mathematics; some sort of ...
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Determining weak stationarity of a stochastic process [on hold]

I'm not able to understand how to determine if these processes are weak stationary. For the first process, I'm trying to work out its mean. The epsilon term would be 0 as it's mean is 0, but for ...
3
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0answers
16 views

How can an Ornstein-Uhlenbeck process be shown to be continuous/diffusion?

Given the Ornstein-Uhlenbeck transition pdf (where $t_2\geq t_1 \geq 0$ and $x_2 \geq x_1 \geq 0$ and $\gamma >0$): $$p(x_2,t_2;x_1,t_1) = \frac{1}{\sqrt{2\pi(1-e^{-2\gamma(t_2-t_1)})}}\exp \left( ...
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2answers
44 views

How can we prove that the generalized stochastic process induced by a real-valued Brownian motion is Gaussian?

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...
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16 views

Covariance functional of a generalized real-valued Brownian motion

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...
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1answer
30 views

How do you find the probability of a brownian motion?

If $B(t)$ is a brownian motion what do these two questions mean? 1. What is the probability of $B(2)$ 2. What is the probability of $B(2) \gt B(1)$ I know this is also called a Wiener Process and ...
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1answer
23 views

Convergence of sequence of random variables 2

If I know $\lim\limits_{n \to \infty} \mathbb{P}(X_n<c-\gamma)=0$ for all $\gamma>0$, how can I prove supremum of all reals $\alpha$ for which $\lim\limits_{n \to \infty} \mathbb{P}(X_n\leq ...
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12 views

Comparing two hitting times of Bessel process

Suppose $X$ is a Bessel process of dimension $1 < d \le 3$ with $X_0 = 0$. Then $X$ satisfies the SDE $ dX_t = \frac{d - 1}{2X_t} dt + d W_t$ for some Brownian motion $W_t$. Let $a > 0$. Let ...
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32 views

i have done partial of this problem, need help on this [on hold]

Problem: Start with initial capital $ x$ and consider a continuous time game of betting $1$ on the outcome of the Brownian motion Wt started at x at time 0. Namely, Wt=x+Bt = fortune at time t, where ...
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1answer
23 views

Is this martingale constant 0?

I have a martingale X where $X_0 = 0$ a.s. And for each $\omega$, the path $f(t)=X_t(\omega)$ is of bounded variation in the classical sense. That ...
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16 views

Coupling Brownian Motions

I want to simulate three freight rate indices which are naturally correlated. The freight rate dynamics ($X$) can be modeled as a geometric Brownian motion: $dX_{t} = \mu X_{t}dt + \sigma ...
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1answer
15 views

Can Wiener process be axiomized without normal increments

A common characaterization of Wiener's process is the following which I took directly from Wikipedia: $W_0 = 0$ a.s. $W$ has independent increments: $W_{t+u} - W_t$ is independent of $σ(W_s : s ≤ ...
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25 views

Finding the stationary distribution of specific homogeneous Markov chain and determining its uniqueness

I am presented with $P =\begin{bmatrix} 0.5 & \alpha & \beta \\ \alpha & \beta & 0.5 \\ \beta & 0.5 & \alpha \end{bmatrix}$ where $\alpha+\beta=0.5$ and $\alpha,\beta \in ...