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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Poisson point process convergence
Let Π be a Poisson point process on [0,∞) with intensity measure $\mu$. Assume $μ([0,t])<∞$ for all $t<∞$ and $μ([0,∞))=∞$. Also assume $μ({x})=0$ for all x. Prove ...
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1answer
12 views
Analytic tools in the theory of Galton-Watson processes
The questions basically aims at discussing the relative power of using probability generating functions, moment generating functions and characteristic functions as an example for ...
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1answer
39 views
Help me solve the invariant measure of $Q$
My $Q$ matrix is given by:
\begin{bmatrix}
-\lambda &0 &\lambda &0 &0 &... \\
\mu&-(\lambda+\mu) &0 &\lambda &0 &... \\
0&\mu ...
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1answer
37 views
Order of convergence of a sum
Let $(X_t)_{t\geq 0},\;X_0=0$, be a positive stochastic process such that
\begin{align*}
\mathbb{E}\left[\sum_{n=1}^{\infty}X_t^n\right]=\sum_{n=1}^{\infty}\mathbb{E}[X_t^n]<\infty.
\end{align*}
...
2
votes
1answer
21 views
Conditioning a martingale increment by earlier increments
I have a $L^1$ - martingale ($E[|X|]<\infty$) defined on $(\Omega,\mathcal F , \mathbb P)$, with constant expectation $EX_t$, and I have to prove that $$E\{(X_v-X_u)|(X_t-X_s)\}=0$$ for $0\le ...
2
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1answer
56 views
Stopping times and $\sigma$-algebras
We have the usual $(\Omega, \mathcal{F}, P)$ stochastic basis. Let $\rho, \tau: \Omega \to T \cup \{+\infty\}$ be stopping times and $\mathcal{F}_{\rho}, \mathcal{F}_{\tau}$ their respective ...
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0answers
15 views
The impact of jump on the returns of portfolio and asset pricing
There exsits jumps in financial market. What will be the impact of jump on the returns of portfolio and asset pricing?
Please explain it both academically and plainly. If you can give some excellent ...
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1answer
27 views
A theorem about the Poisson Point process.
In the proof of the Levy-Khintchine theorem, I saw a theorem about the Poisson
point process.
The theorem states that if $\Pi$ is a poission point process on $S$ with
intensity measure $\mu.$ Let ...
6
votes
1answer
475 views
Hitting time of Brownian Motion with a drift
Let $X_t =x+bt+\sqrt{2}W_t$, where $W_t$ is a standard Brownian motion.
Let $T=\inf\{t: |X_t|=1\}$. I am trying to find $\mathbb{E}[T]$ for the case $b\neq0$.
Firstly, I am going to apply Girsanov to ...
1
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1answer
49 views
Stochastic process, Gaussian, with zero mean is a Wiener process
Let $(\Omega, \mathcal F , \mathbb P)$ be a probability space and let $\mathcal F = \{\mathcal F_t\}_{t\ge} $ a filtration. Let $W=\{W_t;t ≥ 0\}$ be a stochastic process adapted to $\mathcal F$. ...
3
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1answer
66 views
Optimal probability measure
Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
2
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2answers
43 views
Chaos in finite field
Let's think about some finite field $\mathbb{F}$. Is it possible to construct a map
$x[n+1] = \mathcal{P}(x[n], x[n-1],...,x[n-k]), \ \ \ \forall x\in\mathbb{F} $
where $\mathcal{P}$ - ...
4
votes
1answer
141 views
Brownian Motion Covariance: max instead of min
It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion.
Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
0
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0answers
30 views
Prove $\mathbb{E}[X_t | \mathcal{F}_s] = \mathbb{E}[X_t | \sigma(\mathcal{F}_s \cup \mathcal{G}_s)] $
We want to prove that if $X_t$ is an $\mathcal{F}_t$ - martingale: $\mathbb{E}[X_t | \mathcal{F}_s] = X_s$ for $s<t$, then it's also a $\sigma(\mathcal{F}_s \cup \mathcal{G}_s)$- martingale. ...
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0answers
14 views
Levy process absolute moment
For a Levy process $(X_t)_{t\geq 0}$, we have $\mathbb{E}[X_t]=t\mathbb{E}[X_t^1]$ and $\text{Var}(X_t)=t\text{Var}(X_t^1)$. Does the same hold for the first absolute moment, i.e. does ...
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0answers
33 views
Construction binary tree
First let $\mu$ be the induced distribution of the random variable $X$ on $(\mathbb{R},\mathcal{B})$ and denote $EX=m$.
We also define for all $A\in G_{n+1}$ and $\omega\in X^{-1}(A)$
...
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1answer
51 views
Backward martingale property of quadratic variation
Let $\pi_n$ denotes a refining sequence of partitions of a finite closed interval (refining means $\pi_n\subset\pi_{n+1}).$ And we denote $\pi_n B = \sum_{t_i\in \pi_n}(B_{t_{i+1}}-B_{t_i})^2$, where ...
0
votes
1answer
22 views
Proving weak existence of CIR process
Consider the following SDE
$$ X_t = x + \int_0^t \theta (\mu -X_s) ds + \int_0^t\kappa \sqrt{|X_s|} dW_s $$
where W is a brownian motion. I'm trying to show a weak solution exists, does anyone have ...
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1answer
176 views
Linear birth death process, probability of extinction by time t
I have a linear birth death process with birth rates $\lambda n$ and death rates $\mu n$ .
Let r(t) be the probability of extinction by time t.
If there is 1 individual alive at time 0 explain why
...
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2answers
29 views
Generalization of Doob Dynkin for Stochastic processes
Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
0
votes
1answer
26 views
Examples of convergence of random variables
First, let's recall the definitions of 4 different types of convergence:almost surely, in $r$th mean, in probability and in distribution:
$X_n\xrightarrow{a.s.}X$ if $\{\omega \in ...
1
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2answers
291 views
Sum of two stopping times is a stopping time?
Let $\sigma$ and $\tau$ be two stopping times in $\mathscr{F}_t$ and let this filtration satisfy all the usual conditions.
Question: Is $\sigma + \tau$ a stopping time?
Attempt at a solution:
I ...
3
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2answers
468 views
First time passage decomposition for continuous time Markov chain
For discrete time finite Markov chain, the first passage time $T_j$ to visit state $j$, is determined from the recurrence equation:
$$
p^{(n)}_{ij} = \sum_{k=0}^n f_{ij}^{(k)} p^{(n-k)}_{jj} =
...
1
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0answers
41 views
Exponential Levy process
We assume that the stochastic process L is a Levy process with the predictable characteristics triplet $(b,c,\nu)$. Which integrability conditions we should assume for the new stochastic process
...
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1answer
107 views
About exponential martingales
Consider the stochastic process defined by
$$ Z_t = \frac{1}{\sqrt{1-t}} \exp \left( \frac{-B^2_t}{2\left( 1-t\right)} \right ) , t \geq 0$$
where $ \left(B_t\right)_{t\geq 0}$ is a real standard ...
0
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1answer
36 views
Continuous Non negative martingale converging to 0
Is there any (non trivial) continuous non negative martingale which converges to 0?
0
votes
1answer
36 views
Continuous time Stochastic Process stopping time measurability
Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ ...
3
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0answers
30 views
lower bound of expectation of stochastic differential equation
I'm looking for a lower bound on the expected value of a smooth, non-negative, increasing function $\mathbb{E}f(X_t)$, $f(0)=0$ of the solution to a stochastic differential equation $X_t = x + ...
1
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1answer
69 views
Reverse Hölder Continuity and Hausdorff dimension
Let $f$ be a function on $[0,1]$. Say that $f$ is reverse Hölder continuous of exponent $\beta > 0$ if there is a $C >0$ such that for any $s<t\in [0,1]$, there exists $s',t'\in [s,t]$ such ...
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0answers
17 views
Show $B_{t}^{2}$ is a weak solution of a stochastic differential equation. [closed]
Let $B_{T}$ be a Brownian motion in $\mathbb{R}$.
Show that $X_{t} = B_{t}^{2}$ is a weak solution of the stochastic differential equation
$dX_{t} = dt + 2\sqrt{|X_{t}|}d\tilde{B_{t}}$
where ...
1
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2answers
55 views
Moment generating function of a stochastic integral
Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then:
$$
\mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
1
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1answer
39 views
How is Brownian motion predictable?
Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has ...
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1answer
112 views
Solving Stochastic Differential Equations
Can anyone help me with the following SDE?
Solve the following stochastic differential equation:
$$dY_t=aY_tdt+(b(t)+cY_t)dB_t$$
with $Y_0=0$.
Hint: Try a solution of the form $Z_tH_t$ where $Z_t = ...
0
votes
1answer
325 views
Stochastic Calc
(a) Consider the process
$$
d\sqrt{v} = = (\alpha - \beta\sqrt{v})dt + \delta dW
$$
Here $\alpha, \beta,$ and $\delta$ are constants. Using Ito's Lemma show that
$$
dv = (\delta^2 + 2\alpha\sqrt{v} - ...
0
votes
0answers
11 views
Distribution of partial sums of a $L^2$-transformed Gaussian Process
Our assumptions are: $X_t$ is a stationary sequence of standard normal random variables such that $\gamma _X (k)\sim L_{\gamma}(k)k^{2d-1}$ with $d \in (0,1/2)$, where $L_\gamma (k)$ is a slowly ...
1
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1answer
29 views
Defining an equivalent measure starting from a continuous local Martingale
Suppose we have continuous local martingal $L$ given. We define $Z=\mathcal{E}(L)$, the stochastic exponential of $L$. I am interested in finding some condition such that $Z$ defines a density, i.e. I ...
3
votes
1answer
138 views
Expectation of expression involving Brownian motion
How do I compute
$$E\left(tW_t - \int_0^t W_u du \Big| \mathcal{F}_s \right).$$
Given that $W_t$ is standard Brownian motion under the measure $P$ and $\{\mathcal{F}_t, t\ge 0\}$ denotes its ...
0
votes
1answer
696 views
Poisson Process
Customers arrive at a certain facility according to a Poisson process of rate lambda. Suppose that it is known that five customers arrived in the first hour. Each customer spends a time in the store ...
0
votes
1answer
35 views
Finding stationary Distribution
I need to know how to find the stationary distribution for this matrix:
$$
Q=
\begin{bmatrix}
-2 & 2 & 0 & 0 \\
1 & -2 & 1 & 0 \\
0 & 1 & -2 &1\\
0 & 0 ...
5
votes
1answer
184 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 ...
0
votes
1answer
49 views
Product rule of stochastic exponents
we know that for standard exponents, $(e^x)(e^y)=e^{(x+y)}$. What is the product rule for stochastic exponents?
$E_n(U)E_n(V)=E_n(U+V+[U,V])$ where $U$ and $V$ are stocchastic sequences, $E_n$ is the ...
1
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1answer
45 views
finding the probability density function of $ dY_t = - Y_t X_t dW_t$
Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$:
\begin{align}
dY_t &= - Y_t\ ...
0
votes
1answer
20 views
Can an absorbing CTMC be reversible?
Can a CTMC with an absorbing state be reversible? I guess not, as the product of rates through any loop cannot be equal when the loop involves the absorbing state (Kolmogorov criterion). Is my ...
2
votes
1answer
22 views
What is the norm on the functional space used in defining the generator of a homogeneous Markov process?
From Wikipedia:
Given a strongly continuous semigroup $T : \mathbb{R}_+ \to L(B)$ on a Banach space $B$, its infinitesimal generator $A$ of a strongly continuous semigroup $T$ is defined as a ...
1
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2answers
68 views
Standard Brownian Motion
Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$.
Can anyone give me some hint to start the ...
2
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1answer
22 views
Meyer's Theorem in Williams & Rogers
In Diffusions, Markov Processes and Martingales Volume 2 by Rogers and Williams they state the following theorem due to Meyer:
$\mathbf{Theorem }$ Le $M\in\mathcal{M}^2_0$. Then there exists a ...
3
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1answer
135 views
Multidimensional infinitesimal generator of a jump-diffusion
Let $X=\{X_t\}_{t\geq0}$ be an $n$-dimensional Markov process, defined by the SDE
$$dX_t = \mu(t, X_t) \, dt + \sigma(t,X_t) \, dB_t+\beta(t-,X_{t-}) \, dN_t,$$
where $\mu, \sigma$ and $\beta$ are ...
0
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0answers
56 views
Graduate research project in stochastic programming . [closed]
I don't know is this a good question or is this place is right to post this like question or not , but I need keen help, so I'm posting it.
I'm a graduate student & in this semester I've ...
1
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1answer
35 views
The weighted distribution function for combination of two variables
For example, we have two random variables $a$ and $b$. And they have cumulative distribution function $F(x)$ and $H(x)$. We have number $0 < p < 1$.
Suppose, some machine get this random ...
1
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1answer
80 views
A Boundary crossing result for discrete brownian bridge
Let $S_n$ be a random walk with gaussian increments with $S_0=0$, i.e. $S_n-S_{n-1}\sim N(0,1), n\geq 1$. Fix $a>0,b\in \mathbb{R}$ and $c<a+bn$. Define the new process
$$
...
