Tagged Questions

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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1answer
347 views

expected value of product of stochastic processes

Let $X_t=\sigma \int_0^t e^{-a(t-s)} dW_s$, where $\sigma , a $ are constants. How can I find the expected value of the product of $X_t, X_s$ For t>s, $\mathbb{E}[X_t, X_s]$, and $\mathbb{E}[X_t, ...
1
vote
1answer
71 views

Degree of girraphs

A girraph is an infinite, regular, vertex-transitive graph, on which a random walk is recurrent. The random walk on the square grid returns to the origin with probability 1, and for the cubic grid ...
2
votes
1answer
99 views

Restricted random walk in $\mathbb Z^3$

What is the proability to return to the origin, for a uniform random walk on the integer lattice in $\mathbb Z^3$, if we are restricted to $x \geq 0$? I.e. if we try to step into a negative ...
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2answers
629 views

Gambler's ruin (calculating probabilities--hitting time)

Im meant to produce the transition matrix which I've already done (in the picture) and list the communication classes. But Im not sure how to find the probability regarding the hitting times (see ...
3
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0answers
381 views

Quadratic variation of a Brownian motion up to time $T$ converges to $T$ in $L^2$?

In Stochastic Calculus for Finance II: Continuous-time Models by Steve Shreve, Theorem 3.4.3. Let $W$ be a Brownian motion. Then $[W, W](T) = T$ for all $T > 0$ almost surely. where $[W, ...
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0answers
73 views

Multiplicative functionals for Markov process: discrete time

I read a theorem, stating Let $X_t$ be a Markov process w.r.t. to its natural filtration $(\mathcal F_t)$ on the space of cadlag functions on $\mathbb R_{\geq 0}$ and $(Z_t)_{t\geq 0}$ be a ...
2
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0answers
239 views

Show that the stopped sequence is a martingale

This is a series problem and I'm struggling with the last part. I assume that the last part has nothing to do with previous ones, so i won't put up the other parts. Question is : Let $\tau$ be a ...
2
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1answer
81 views

Irreducible MCs

Why is it that theorems for (discrete) Markov chains always require that the MC concerned is irreducible? Can problems with reducible MCs can be simplified to considering the irreducible components? ...
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2answers
220 views

Reversibility of a Markov Chain

Rephrased question: Is it ever possible for a reducible Markov chain to be reversible?
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0answers
86 views

Solving a non-linear inequality related to geometric Brownian motion

Consider the non linear inequality $$\sum_{i=1}^{n}a_{i}u^{\sum\limits_{j=1}^{i}y_j} > c$$ $$y_j \in \{0,1\}, j=1,2,\dots,n$$ $$a_i \in \mathbb{R}, i=1,2,\dots,n$$ $$n \in \mathbb{N}, u>0, c ...
3
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2answers
738 views

Definition: transient random walk

What exactly does a "transient random walk on a graph/binary tree" mean? Does it mean that we never return to the origin (assuming there is one as for the tree) or just any vertex of the graph or ...
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3answers
252 views

Throwing dice a randomly determined number of times

Let $X$ be the number that shows up when rolling a die. Now throw another dice $X$ times $(Y_1, ..., Y_X)$ and calculate the sum $Z = \sum_{k=1}^X Y_k$. What kind of Stochastic Process is this? How ...
4
votes
2answers
228 views

conditional expectation of a solution to the SDE

Suppose we have an SDE, which is the Wiener process with drift $dr_t=c dt+\sigma dB_t$, where $B_t$ is brownian I want to find $\mathbb{E}[e^{-\int_0^t r_s ds} |r_t=r]$ so my approach is this : ...
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4answers
295 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 ...
6
votes
1answer
2k views

Expected value of the stochastic integral $\int_0^t e^{as} dW_s$

I am trying to calculate a stochastic integral $\mathbb{E}[\int_0^t e^{as} dW_s]$. I tried breaking it up into a Riemann sum $\mathbb{E}[\sum e^{as_{t_i}}(W_{t_i}-W_{t_{i-1}})]$, but I get expected ...
4
votes
1answer
508 views

Is Brownian bridge a Markov process

As in the title, question is whether a Brownian bridge: $X_{t} = B_{t} - tB_{1}$ is a Markov process. I could sort of prove it by the markov property, but not sure whether it's sufficient. Does anyone ...
2
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1answer
62 views

subadditive function of brownian motion

Let $W_t$ be Brownian, and let $g$ be integrable , and odd, and subadditive $g(x+y)\le g(x)+g(y)$. How to show that $g(W_t)$ is a supermartingale? I am not sure how to make use of the subadditive of ...
4
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2answers
313 views

joint distribution of the stochastic process

Let $X_t$ be a solution to the SDE, $dX_t=-aX_t \; dt+\sigma \; dW_t$, $a>0$, $\sigma>0$, $X_0=\text{constant}$ where $W_t$ is Brownian. What is the joint distribution of $(X_t, \int_0^t X_s \; ...
6
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1answer
534 views

Showing a process is a martingale

Let $\newcommand{\F}{\mathcal F} S_n=S_{n-1} +X_n $ where $S_0=0$ , and $X_k$ are iid, and let $\phi(t)=\mathbb{E}e^{itX_1}$ be the characteristic function of $X_k$. Consider a process ...
2
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1answer
234 views

joint distribution of random vector

I want to find the joint distribution of the random vector $(W_t, \int_0^t W_s \; \mathrm ds)$ where $W_t$ is Brownian motion. I know $W_t \sim N(0,t)$, but I don't know how to calculate the ...
5
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1answer
436 views

“Paradox” of a Poisson process on $\mathbb R$

This question concerns the apparently peculiar behavior around zero of the Poisson process defined over the entire real line $\mathbb R$. A Poisson process $N(A)$ with mean measure $\mu$ on a general ...
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1answer
163 views

Primes and probabilities

Imagine a set $S$ of $10^{12}$ consecutive integers $n, n + 1, n + 2, n + 3, \ldots, n + 10^{12}-1$, where the exact identity of $n$ will be partially determined randomly as described below, and ...
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0answers
87 views

Studying the maxima of columns of a random matrix as a point process

Consider a matrix, $S$, of i.i.d. real RVs : $X_{ij}$ for $1 \leq i \leq s$, $1 \leq j \leq n$. Let $F$ denote the distribution of $X_{ij}$. For $1 \leq j \leq n$, consider $Y_{j}^{(1)} = \max_{i} ...
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1answer
702 views

Relations between Order Statistics of Uniform RVs and Exponential RVs

Say we have $U_1 \dots U_n$ i.i.d. random variables uniform on $[0,1]$ and $Y_1 \dots Y_{n+1}$ i.i.d. random variables distributed as $Y_i \sim Exp(1)$. I know that the joint distribution of the order ...
2
votes
2answers
135 views

what's the relationship between a.s. continuous and m.s. continuous?

suppose that X(t) is a s.p. on T with $EX(t)^2<+\infty$. we give two kinds of continuity of X(t). X(t) is continuous a.s. X(t) is m.s. continuous, i.e. $\lim\limits_{\triangle t \rightarrow ...
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votes
1answer
123 views

Expressing the pdf $F_n(t)$ as an n-fold convolution of the failure time distribution

please I want to show that $F_n(t)=P{(S_n\le t})=\int_0^tF_{n-1}(t-x)f(x)dx.$ My problem here is I do not have enough hypothesis to work with. What I only know is that $F(t)=\int_0^tf(s)ds$ and ...
1
vote
1answer
286 views

Is the norm of a martingale a martingale?

Let $M_n$ be a vector-valued $F_n$-martingale ($M_n:\Omega \rightarrow R^p$). Is then $\lVert M_n \rVert $ also a martingale? I have $E(M_{n+1} | F_n)= M_n$ and liked to say something about $E(\lVert ...
2
votes
1answer
88 views

Number of crossings in a two-dimensional random walk

Given the standard two-dimensional random walk (up, down, left, or right 1 unit with equal probability), what is the expected number of crossings of the origin after $x$ steps? It strikes me as ...
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1answer
91 views

Error committed by linear interpolation of Levy process trajectory

Consider an independent increment process $X_t$, such that $X_t$ follows a continuous distribution. Examples would be a Brownian motion, Gamma process or a stable Levy process. When sampling from ...
6
votes
1answer
381 views

Brownian motion and Fourier series

Let $(B_t)_{t \in [0, \infty)}$ be a Brownian motion. Can you prove me why it can be written as $$B_t= Z_0 \cdot t + \sum_{k=1}^{\infty} Z_k \frac{\sqrt{2} \cdot \sin(k \pi t)}{k \pi}$$ for some ...
1
vote
1answer
101 views

If almost all sample paths of a process converge to a constant, is it okay to say the process itself converges to a constant?

Suppose that $(a_n)$ is a sequence of reals and $(e_n)$ is a sequence of iid r.v.s such that $\Pr(e_n=\pm1)=1/2$. It is well known that $\sum a_ne_n$ converges a.s. to some limit r.v. iff $\sum a_n^2 ...
3
votes
2answers
446 views

A question about Convergence of a product in random variables

Let $\{U_k\}$ be a sequence of independent random variables, with each variable being uniformly distributed over the interval $[0,2]$, and let $X_n = U_1 U_2\cdots U_n$ for $n \geq 1$. (a) Determine ...
3
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1answer
372 views

what's the difference between RDE and SDE?

what's the difference between random differential equation and stochastic differential equation? does stochastic differential equations include random differential equation?
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1answer
69 views

How to prove that all the $a_i$ will be the same after inf operations?

Given some real number :$$a_1,a_2,...,a_n$$Every time I chose two of them $a_i$ and $a_j$ and set both of them to $\frac{a_i+a_j}{2}$. Now I have operated $T$ times,when $T$ is infinite,I guess that ...
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vote
1answer
164 views

Clarification on the definition of a closed communicating class

If I have a transition matrix defined as $p_{ij}=1$ for $j=i+1$ and $p_{ij}=0$ otherwise, where the state-space is countably infinite, what would this be? It doesn't look like a communicating class ...
1
vote
1answer
101 views

Why does this algorithm work?

Given a matrix, $P$, why does finding its eigenvalues, say they are $\{\lambda_1, \lambda_2\}$ then the general form of $p_{ij}^{(n)}=A_{ij}\lambda_1^n+B_{ij}\lambda_2^n$? Thanks. Added: Context: $P$ ...
0
votes
1answer
162 views

Discrete Markov chain

Let $(A_n)$ be a discrete Markov chain with transition matrix $M$ and $B_n = A_{(mn)}$ where $m\in \mathbb N$, I want to show that $(B_n)$ is a Markov chain with transition matrix $M^m$. Please help ...
2
votes
0answers
385 views

expectation of supremum of random process

Suppose $(X_{n}(t))_{n\geq 1}$ is a sequence of real valued stochastic processes, and $T>0$ a fixed number. Suppose further that : \begin{equation} \mathbb{E}\left[\displaystyle{\sup_{n>0}}\ ...
1
vote
1answer
672 views

Martingale property under changes of measure

I've been studying martingale property under change of measure and I came up with a following observation. I took a random example with a state space $\Omega = \{ -2,1,2\}$ and two equivalent ...
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2answers
61 views

Markov property and $A=\bigcup_{k=1}^{\infty} A_k$

I am reading Norris's "Markov Chains" and would appreciate an explanation of the following bit. After stating the Markov property, it is said that (on page 4) In general, any event A determined ...
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2answers
111 views

A problem in Poisson Processes

Let $X_n$ be the interarrival times for a Poisson process $\{N_t; t \geq 0\}$ with rate $\lambda$. Is it possible to calculate the probability $P\{ X_k \leq T \text{ for } k \le n, \sum_{k=1}^{n}{X_k} ...
3
votes
1answer
139 views

Measurability question for martingale (Is $S(X_i, Z_i)- E(S(X_i,Z_i) \mid \mathcal{F}_i)$ $\mathcal{F}_i=\{X_1, \dots, X_i \}$-measurable?)

One condition for a martingale $M_k$ with a general filtration $\mathcal{F}_k$ is that the involved random variables $M_k$ are $\mathcal{F}_k$-measurable. Now I have $M_n=Y_1+\dots +Y_n$ and ...
5
votes
2answers
365 views

Is “a fair coin being tossed $n$ times” the same as “$n$ fair coins being tossed once”?

This is possibly a follow-up question to this one: different probability spaces for $P(X=k)=\binom{n}{k}p^k\big(1-p\big)^{ n-k}$? Consider the two models in the title: a fair coin being tossed $n$ ...
4
votes
1answer
183 views

Lower semicontinuity of the indicator function: stochastic processes

Let $X$ be a Markov process given on a metric space $\mathcal X$ by a transition semigroup $P_t$ acting on $\mathbb B(\mathcal X)$ - the set of all bounded and Borel measurable functions. Such a ...
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votes
2answers
371 views

Examples: invariant events

In a couple of books I'm reading chapters devoted to the Ergodic theory. As a setting: $(\Omega,\mathcal F,\mathsf P)$ is a probability space, $X:(\Omega,\mathcal F)\to (S,\mathcal S)$ is random ...
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0answers
134 views

Two identities in probability

I ma reading the book An introduction to stochastic processes in physics by Don Stephen Lemons. I have a question on two identities. One identity is the identity (B.3) in Appendix b page 102. How ...
3
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2answers
479 views

Markov chain with uncountable state space

I'm self-studying probability theory and struggling with understanding Markov chains on uncountable state spaces, notably I would like to solve the following exercise from this book. ...
5
votes
1answer
1k views

Use stochastic calculus (Ito's lemma) to compute the expectation

Calculate $E[\cos(X)e^X]$, where $X\sim N(0,\sigma^2)$. Use stochastic calculus instead of integrating w.r.t the normal density. During the discussion with friends, we believe that we should use ...
3
votes
1answer
256 views

Are hitting times of Brownian motion independent?

Suppose that $B_t$ is a standard Brownian motion. And $T_a$, $T_b$ are the hitting time whereas $a<0$, $b>0$. Then are these two random variables independent?
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0answers
167 views

Observable and unobservable parameters of stochastic processes

Consider the following diffusion process $$ dX_t = \mu\,dt+\sigma(t,X_t)\,dW_t $$ where $X,W$ are 1-dimensional and. Is it true that given a history $(X_s,s\leq t)$ for each $s< t$ one can find ...