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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Application of the Burkholder Davis Gundy inequality

The proof of the Feynman-Kac formula uses a lemma which I need to proof, but I can not figure it out. The lemma is the folllowing: Let $X$ be a weak solution of ...
3
votes
1answer
28 views

Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
0
votes
1answer
24 views

Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each ...
-1
votes
0answers
8 views

Stochastic process $X_t$, $X_t^2$, $X_{t^2}$

Can anyone explain me the difference between such stochastic processes:$X_t$, $X_t^2$, $X_{t^2}$ $X_t$ is let's say normal. How about two others? It's something to do with time, yes?
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0answers
10 views

Doubt concerning Stochastic continuity

I know that a stochastic process $X$ is said to be stochastically continuous if $\forall s$ $$\lim_{t\rightarrow s}\;P(|X(t)-X(s)|>a) = 0.$$. But then it is also true that stochastic continuity ...
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0answers
14 views

Square of a stationary process

I have to prove that if $X_t$ is a stationary process, $X_t^2$ is also. It is easy to prove the part referred to the means but I do not know how to work with covariances. Thanks!
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1answer
21 views

Is $X_t := W_t^2$ a Wiener process for a Wiener process $(W_t)_{t \geq 0}$?

I'm studying for exam and found this exercise which I don't really understand: Suppose $W_t$ is standard Wiener process. Is process $X_t=W_t^2, t\geq0$ a Wiener process? So I need to show that ...
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0answers
41 views

Clarify a question's answer related to random walk. [closed]

I'm studying Problem5.3 and its solution. However, its solution is not clear for me. Please explanatorily show this answer . I need to learn such type of questions. Please help me. Thank you.
4
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1answer
54 views

A question related to reflection principle

Question: $$P(X_1\gt 0, ..., X_n\gt 0, X_n=a-b)=?$$ Its Answer: $= (1,1) \rightarrow (n,a-b) $ that meet neither touch nor cross paths. $=[(1,1) \rightarrow (n,a-b) \ \ \text{all ...
1
vote
1answer
27 views

What is the probability of arrive either A or B at starting point K?

There are two points which are $A$ and $B$. The distance between $A$ and $B$ is $50$ meter. One person goes to $A$ with probability $\frac{1}{6}$, he goes to $B$ with probability $\frac{3}{6}$. And he ...
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0answers
26 views

Markov Chain Detailed Balance property

I am having a hard time to understand the concept of the detailed balance; mostly because of the intermingled notation most of the resources use; which involves constant usage of random and state ...
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0answers
9 views

Higher dimensional SDEs cannot be compressed into a one dimensional SDE.

This question comes from quantitative finance but I think it's true in general outside that setting. I'm trying to make sense of the idea that if a process depends on at least two noises there ...
2
votes
2answers
24 views

What is this conditional probability?

I have been doing some reading for a project on quantitive finance, and I have been seeing a lot of this kind of conditional probabilities on a "$\mathcal{F}_{t_i}$": $$\mathbb{P} ...
2
votes
1answer
30 views

Independence of linear combinations of Brownian motions

Let $0<s\leq t\leq u\leq v$ and $W_x$ be a Brownian motion. Show that $aW_s+bW_t$ and $\frac{1}{v}W_v-\frac{1}{u}W_u$ are independent for $a,b$ satisfying $as+bt=0$. The question seems easy but ...
4
votes
1answer
51 views

Expectation of an Itô integral

I'm interested in computing the following expectation: $$\mathbb{E}\left[W_T\cdot\int_0^T f(s)\mathrm{d}W_s\right].$$ Here $\{W_t\}_{t\ge 0}$ is a standard $\mathbb{R}$-valued Brownian motion and ...
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4answers
55 views

A Stochastic Limiting Inequality Proof

Let $(X_p)_{p\ge 0}$ be a sequence of non-negative random variables with finite mean for each $p\ge 0$. Then $$\liminf_{p\to\infty} X_p^{\frac{1}{p}}\le \liminf_{p\to\infty}E(X_p)^{\frac{1}{p}}$$ ...
0
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1answer
663 views

How to prove difference between two independent poisson process is not a poisson process?

It will come under properties of poisson process in some books. The sum of two independent poisson process can be proved as a poisson process using its memoryless property but how to prove difference ...
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0answers
26 views

Independence of Poisson processes watched only some of the time

Let $(X_t)$ and $(Y_t)$ be independent homogeneous Poisson processes with rates $\lambda,\mu > 0$, and let $t_1, t_2, \dots$ and $t_1', t_2', \dots$ be two increasing sequences of possibly infinite ...
2
votes
1answer
21 views

Uniform integrability of the stopped compensated Poisson process

Let $N(t)$ be a Poisson process of rate $\lambda$ and consider the compensated Poisson process $$\bar{N}(t):= N(t) - \lambda t.$$ It was already shown in another post (Is a compensated Poisson ...
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1answer
13 views

how to solve for Ut stochastic question [on hold]

The process given by dUt = 􀀀-rUtdt + sigmadXt; U0 = u; where r,sigma are constants how to solve this equation for Ut? Thank you
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1answer
14 views

What are the conditions for $E[\int_0^tf(W_s,s)dW_s]=0$?

Let $W_t$ be the standard Brownian Motion. I am interested on the conditions on $f(\cdot)$ that guarantee that the expectation of the Ito integral below is zero: ...
4
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2answers
2k views

Transformation on state-space that preserves Markov property

I am solving a problem in Mathematical Statistics by Jun Shao Let $\{X_n \}$ be a Markov chain. Show that if $g$ is a one-to-one Borel function, then $\{g(X_n )\}$ is also a Markov chain. ...
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0answers
10 views

What is “starting” static distribution?

I'm not sure if I call everything correctly in English in here, but i have a problem with stochastic processes - Markov chains to be more specific. I'm calculating the "starting" stationary ...
0
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1answer
18 views

Example of an adapted but not progressively measurable process

I'm looking for an example of a stochastic process $X$ that is $\mathbb{F}$-adapted, but not progressively measurable. One example I found is the following: $(\Omega, \mathfrak{A}) = (\mathbb{R^+}, ...
1
vote
1answer
16 views

Show that $M_{t}=\max_{0\leq s\leq t}B_{s}$ is adapted to the natural Brownian filtration.

Let $\left(B_{t}\right)_{t\geq0}$ a Brownian motion. Show that $M_{t}=\max_{0\leq s\leq t}B_{s}$ is adapted to the natural Brownian filtration. Remark: I try the following: It suffices to show ...
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0answers
16 views

Skorokhod vs Meyer zheng topology

I am new to the Skorokhod space and I want to know why Meyer-Zheng topology on the space of càdàg functions is weaker than the standard Skorokhod topology. Thanks in advance!
3
votes
1answer
53 views

Stationary distribution of an increasing stochastic process with a cut-off

I have a discrete time stochastic process $\{X_t : t \in T\}$ with continuous state space. Assume $X_0=0$ and increments $\delta_t$ are exponential with mean $\alpha$ (so its parameter is ...
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0answers
7 views

Class properties Markov chain [on hold]

How can we show that an open class in a Markov chain is transient (both for finite and infinite)?
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1answer
418 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
1
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1answer
92 views

Stationary Markov process properties

Let $X$ be a right-continuous process with values in $(E,\mathcal{E})$, defined on $(\Omega, \mathcal{F}_t,P)$. Suppose that $X$ has stationary, independent increments. I now want to show the ...
0
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1answer
26 views

Find probability that payoff function is in $[10,20]$

In moment $t=0$ we bought option with expiration date $T=2$. The payoff function of this option is given by: $$f=(\max_{t\in[0,T]} S_t -110)^{+}$$ where $S_t$ satisfies $$dS_t=15dW_t$$ $$S_0=95$$ ...
0
votes
0answers
7 views

No drift brownian motion and minimization at a given time [on hold]

Given two same brownian motion with no drift and different variances: $$(dG_1/G_1)= \sigma_1dW_g $$ $$(dG_2/G_2)= \sigma_2dW_g $$ At a specific given time $ T = \tau $, how can I tell if ...
2
votes
1answer
590 views

Stopping theorem for continuous martingale

I've a question about a proof in my lecture notes. We want to prove the following theorem. $M=(M_t)_{t\ge 0}$ be a $(P,F)$-martingale, where $P$ is a probability measure and $F=(\mathcal{F}_t)$ a ...
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votes
1answer
30 views

What is the distribution of $B(t_1)+B(t_2)+…+B(t_n)$ [on hold]

$\{ B(t), t\ge 0\}$ is a standard Browian Motion Process. What is the distribution of $B(t_1)+B(t_2)+...+B(t_n)$ ?
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0answers
10 views

linear second moment of zero mean stochastic process with independent, stationary increments

I'm working on the following problem: Let $X$ be a zero mean stochastic process with independent and stationary increments. I want to prove that the function $t \mapsto \mathbb{E}X_t^2$ is linear. I ...
0
votes
3answers
2k views

Ratio Distribution: Poisson Random Variables

Suppose two Poisson processes. For example, during the time interval, $\Delta t_{1} = t_{1} - t_{o} = 50\mu s$ , $x$ photons are incident on a detector with rate $\lambda_{1} = 10$x$10^4 s^{-1}$. At ...
5
votes
1answer
179 views

Convergence of Ornstein-Uhlenbeck process as a scaled Brownian Motion

Let $W$ be a standard Brownian motion. Let $\alpha,\sigma^2 >0$, and let $X_0$ be a $\mathbb{R}$-valued random variable with distibution $\nu$ that is independent of $\sigma(W_t,t\geq 0)$. Now ...
3
votes
1answer
32 views

On random rotational fluctuations in $\mathbb{R}^n$

Imagine first a disk that is mostly stationary, except for random ("thermal" if you like) "rotational fluctuations" around its axis (which is fixed). Something a bit like what's shown in the figure ...
3
votes
1answer
296 views

Local martingale is locally uniformly integrable martingale?

Is a local martingale locally uniformly integrable martingale ? Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale. ...
4
votes
2answers
46 views

Implementing Ornstein–Uhlenbeck in Matlab

I am reading this article on Wikipedia, where three sample paths of different OU-processes are plotted. I would like to do the same to learn how this works, but I face troubles implementing it in ...
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votes
1answer
23 views

counterexample to conditional expectation

Let F,G be some $\sigma-algebra$ is it true that in general $E\left(E\left(X\mid G\right)\mid F\right)\neq E\left(X\mid F\cap G\right)$? I think it's not, however I can't provide a counter example
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21 views

limit of sum of a brownian motion

Let $W_t$ be a wiener process and let $\pi$ be a partition of the segment $[0,T]:0\leq t_1\leq...\leq t_n=T$ I need to show without using the martingale property that the term below tends to $0$ in ...
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votes
1answer
52 views

Markov chain problem 13 [closed]

I have this problem I don't understand, Can you help me, please?
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0answers
8 views

Density of the Absorbed Process

The curiosity arose while reading the Ch.18 of Arbitrage Theory in Continuous Time 3/ed, dedicated to pricing Barrier Options. Definition 18.1 For any $y\in R$, the hitting time of y, $\tau(X,y)$, ...
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0answers
3 views

Limit distribution of absolute value maximum of stationary non-differentiable Gaussian process

Consider a real-valued stationary Gaussian Process $\{ X(t) \colon t \geq 0 \}$ with zero mean and unit variance and covariance function $r$ satisfying $r(t) = 1 - C|t|^{\alpha} + o(|t|^{\alpha}), ...
4
votes
1answer
546 views

Why is it true that the continuous local martingale with quadratic variation “t” is a square integrable continuous martingale?

I am reading Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $M_t$ be a continuous local martingale. On page 157, it wrote that "because $\langle M\rangle_t = t$, we have $M \in ...
3
votes
3answers
48 views

How to show $M_n = X_n^2-n$ is a martingale?

Let $X_n, n = 0, 1, 2, . . .$ denote an unbiased Normal Random Walk. $X_0 = 10$, and $X_{n+1} = X_n + Y_{n+1}$, with $\{Y_n\}$ are i.i.d. $N(0, 1)$. Then how can I show that: A) $M_n = X_n^2-n$ is a ...
3
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1answer
383 views

Continuous local martingale of finite variation is constant

Is a continuous local martingale $M$ of finite variation constant? We know that there exists a sequence of stopping times $T_n\nearrow \infty$ a.s. as $n\to\infty$ such that the stopped process ...
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1answer
20 views

Examples of predictable processes

I am asked to prove that the following processes are predictable. I am used to looking at stochastic processes as sequences of random variables (by fixing time) or as a collection of paths (by fixing ...
0
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1answer
13 views

Inf is not a stopping time in general

If ${\tau_n}$ , $n=1,2,3...$ are stopping times to a given filtration $F_t$, why in general it's not true to claim that $\inf_n {\tau_n}$ is a stopping time also? Thanks