Questions tagged [stochastic-processes]

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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6 views

Stochastic process, stochastic differential equation

A stochastic process $\{X_t , t ≥ 0\}$ satisfies stochastic differential equation $$\frac{dX_t}{X_t} = 3\mu\ dt + 2\sigma dB_t.$$ where $-\infty<\mu<\infty$ and $\sigma>0$ are given ...
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11 views

Association between two random variables with one of them being stochastically larger than the other

I was wondering if that is always true that when given two random variables $A$ and $B$ with a condition that $A$ is stochastically bigger than $B$ (which means that $F_A(a) \leq F_B(b)$), then $\...
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Question on the use of the Markov Kernel for conditional probability

We define a Markov kernel Let $(\Omega_{1},\mathcal{A}_{1})$ and $(\Omega_{2},\mathcal{A}_{2})$ be some measurable spaces. A map $K$ where $K : \Omega_{1}\times \mathcal{A}_{2}\to [0,\...
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Extracting a smaller Markov chain from a larger Markov chain

I am not very familiar with Markov chains, hence the probably ill titled questions. If we have 5 random variables $X, Y, Z, W$ and they form a Markov chain such that $$X \leftrightarrow Y \...
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Infinitesimal generator of the standard Brownian motion

As explained in this Wikipedia page, the infinitesimal generator of the standard Brownian motion is $\frac{1}{2}\Delta$ and for the Brownian motion it has an extra $\partial_t f$ term. Can anybody ...
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Literature request — uniqueness and existence of a specific type of ODE

I am looking for a proof of the existence and uniquenes of ODE's of the type: \begin{equation} \dot{f}(t,x,y) = F(h(t,x), f(t,x,y)), \end{equation} where $f : T \times X ...
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8 views

When does a Markov semigroup preserve differentiability?

Let $E$ be a $\mathbb R$-Banach space (for simplicity, assume $E=\mathbb R$, if you like) and $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal B(E))$. I would like to know under which ...
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Does the jump-process of a Markov-process have a transition function?

Let $X$ be a canonical, right-continuous Markov-process with values in a Polish state space $E$, equipped with Borel $\sigma$-algebra $\mathcal{E}$. Assume $t\mapsto \mathbb{E}_{X_t} f(X_s)$ right-...
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30 views

If $A_t = cos(X_t)$ and $B_t = sin(X_t)$ find the infinitesimal increment for $Y_t = A_t^2 + B_t^2$

If $X_t$ is Brownian motion, I'm not sure how to apply Ito's lemma to get $d Y_t$ for $ Y_t = A_t^2 + B_t^2$ where $A_t = cos(X_t)$ and $B_t = sin(X_t)$ in particular, I get confused because $sin^2(x) ...
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Functions within Expectation of Brownian Motion

I am tasked with the following: $$E[e^{3B_3}|\mathcal{F}_1]$$ For standard Brownian Motion, i.e. the drift ($\mu$) is $0$, and $\sigma^2=1$. What I have done so far is the following: $$E[e^{3B_3}|...
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Decomposition of Gaussian spaces with respect to covariance function

Let $K(t,s):T^2 \to \mathbb{R}$ be a kernel symmetric and type positive (for every $n$ $\sum^n_{i,j}u_iu_jK(t_i,t_j) \geq 0$ and $(u_1,\dots,u_n) \in \mathbb{R}^n$) where $T$ is any set. Thus, it is ...
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question about martingales [closed]

Let $\xi_{i}$ - i.i.d., $\mathbb{P}(\xi_{i} = 1) = \mathbb{P}(\xi_{i} = -2) = \frac{1}{2}$; $X_{i} = \sum_{j=1}^{i} \xi_{j}$. How to find the $\mathbb{P}(\exists n \geq 0: X_{i} = 1)$? Give me some ...
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Convergence of a stochastic process

Suppose that $\{X_n(t),t\in\mathbb{R}\}_{n=0,1,...}$ is a collection of stochastic processes, i.e., for any fixed $n$, we have and stochastic process $\{X_n(t),t\in\mathbb{R}\}$. Assume that for any ...
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The quadratic variation of the Brownian motion almost certainly tends to T

On the segment $[0, T]$, choose $n$ independent points $t_{n,k}$ (each distributed evenly). Prove that the quadratic variation of the Brownian motion on the sequence of random partitions of the ...
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Help with Poisson Stochastic Process

Cars pass along the road in accordance with the Poisson process of intensity $\lambda$ . A pedestrian crosses the road at time $W$ as soon as he sees that there will be no cars during time $T$ (...
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1answer
25 views

The probability of getting $y$ new coupons from a batch of $k$

In the coupon collector process, the goal is to assemble a collection of $n$ distinct coupons, while we get a random coupon at each time. I am looking at a generalization of this problem, where at ...
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Can one change the dimension of a Bessel process by a Girsanov change of measure?

Recall that a (squared) Bessel process $X_t$ with the dimension $\delta_0\geq0$ is a solution of the SDE $$d X_t = 2\,\sqrt{X_t}\,d W_t + \delta_0\,d t.$$ A naive application of the Girsanov Theorem ...
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25 views

Finding the expectation of a Wiener Process

My question is on how to find $\mathbb E[W_t^n]$ where $n= 0,1,2,...$ and $W_t$ is a standard normal Wiener process. Would we need to use a moment generating function. Thanks.
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Applyin Itô's formula in function of quadratic variation

I am learning some basic stochastic calculus and came across the following exercise: Consider a local martingale $M$ with continuous trajectories. Let $Z_t = \exp(M_t −0.5[M]_t)$. Show that Z ...
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61 views

Conditional expectation of poisson procces problem

Let $N_{t}^{i}$ - be three independent Poisson processes of intensity $1$. $\tau$ = $\inf\{t: \,N_{t}^{3} = 1\}$, $X^{i}$ = $N_{\tau}^{i}$ (means that $X^{i}$ - the values of the first two processes ...
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Need a help with Markov Chain

We have Markov Chain with continuous time, three conditiions and generator: \begin{equation*} Q = \begin{bmatrix} -3 & 1 & 2\\ 1 & -1 & 0\\ 1 & 0 & -1 \end{bmatrix} \end{...
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38 views

Question about Ito Process. Stochastic Processes

How to prove, that $W_{t/(1-t)}$ at $[0,1)$ is Ito Process ? (Have stohastic differential)
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how to deal with multiple-level conditional probability

Consider a Markov Chain, when I want to prove that $P(X_{n+1} = i_{n+1}|X_{n-1} = i_{n-1}, X_{n-2} = i_{n-2}, \cdots, X_0 = i_0) = P(X_{n+1} = i_{n+1}|X_{n-1} = i_{n-1})$, I tried to use \begin{...
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23 views

Is stopped integral of predictable process predictable?

Assume that $H$ is a predictable process that is locally bounded with localizing sequence $(\tau_n)_n$. And assume that $\langle M, M \rangle$ is a increasing, right-continuous, predictable process. ...
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21 views

Relation between Stopping times

I am having a hard time trying to understand the following relation: Consider that a stopping time is defined by $\{\tau \leq n \} \in \mathcal{F}_{n}$. Now take two stopping times $\phi$ and $\tau$ ...
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Find the Probability density function and prove the solution of the stochastic differential equation

A stochastic process $\{X_t, t \geq 0\}$ satisfies stochastic differential equation $\frac{dX_t}{X_t} = 3μ dt+2σ dB_t$, where $−∞ < μ < ∞$and $σ > 0$ are given constants, and $\{B_t, t \geq 0\...
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36 views

3-dimensional Brownian motion, probability distribution of first hitting time to a sphere

What is the probability density function or probability distribution of the time when 3-dimensional Brownian motion (no drift) starting from origin hits a sphere (ball) centered at the origin for the ...
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19 views

Martingales and OST examples (ABRACADABRA like)

The ABRACADABRA Problem is well known, now I need to understand examples that are like that approach. I think I know how to "calculate" the similar examples, but I would like to understand which ...
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17 views

Let $\mathrm{T}=\inf \left\{n \in \mathbb{N},\left|\mathrm{S}_{n}\right|=N\right\}$. Compute $\mathbb E[T]$

I'm solving this exercise Let $\left(Z_{n}\right)_{n \in \mathrm{N}}$ be a sequence of i.i.d. random variables with common distribution $$\mathbb{P}\left(Z_{1}=1\right)=\frac{1}{2}=\mathbb{P}\left(...
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41 views

How to prove that Markov chain with specific transition probabilities has independent increments?

I have Markov chain $N=\{N(t) \mid t\geq 0 \}$ with the state space $\{0,1,2,\dots\}$. I know that it is homogeneous and transition probabilities are: $$ p_{ij}(s,t)=P(N(t)=j\mid N(s)=i) = p_{i,j}(t-...
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Limit of the hitting time of a Gausian process

Let $(Y_t)$ be a Gaussian process and $\tau_n:=\inf\lbrace t>0:Y_t=n\rbrace$. Does anyone have an idea how I can show that $\lim\limits_{n\to\infty}\tau_n=\infty$ almost surely?
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1answer
14 views

Variance of the difference of Brownian Motions

I have a question about the variance of the following formula: $Var(W(t) - \frac{t}{T}W(T-t))$. Where $W(t)$ is a Brownian motion. I tried the following: $Var(W(t) - \frac{t}{T}W(T-t)) = E[W(t) - \...
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32 views

Verifying stochastic process containing summation of Martingales is a Martingale

Let $M=(M_t)_{t \geq 0}$ be a Martingale with respect to the filtration $\mathcal{F}=(\mathcal{F}_t)_{t \geq 0}$. Assume that $\mathbb{E}(M_t^2)<\infty$ for all $t \geq 0$. Let $0=t_0<t_1<...&...
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1answer
26 views

Unbounded quadratic variation process for bounded continuous martingale

Consider a bounded, continuous martingale $(X_t)_{t\ge 0}$. I was able to show that $(X^2-[X])_{t\ge 0}$ is uniformly integrable, where $[X]$ denotes the quadratic variation. Is there an example of a ...
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Applying Itô's formula on trigonometric identity

I have following issue on my hands: Use Ito's formula to compute the semimartingale decomposition and the quadratic variation $[X_i]_t$ for $X_t= sin^2(B_t) + cos^2(B_t)$ This is quite strange, ...
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Is this solution of martingale problem correct?

I'm reading this problem and its solution I think there is a typo in the highlighted part. It should be $$\begin{array}{l} =\mathbb{E}\left[V_{n}^{2}\right]+\mathbb{E}\left[H_{n}^{2} \mathbb{E}\...
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1answer
26 views

Radial Brownian Motion

Let $X_i(t)$ be standard Brownian motion in 1D. Define the radial Brownian motion as $\displaystyle R(t) = \sqrt{X_1(t)^2 + \cdots + X_N(t)^2}$. How do we lower bound the probability $\mathbb{P}(R(T)...
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$P(\bigcup_{n\ge 1} \{\tau_n=T\})=1$

Hey I have $\tau_n=\sigma_n\wedge\delta_n$ where $\sigma_n=\inf\{t\in[0,T] : \int_0^t|b(s)|^2ds\ge n\}$ and $\delta_n=\inf\{t\in[0,T] : \int_0^tb(s)dW(s)\ge n\}$, $b(s)\in P^2_{[0,T]}=\{f:\Omega\times ...
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Integration with finite variation process

I have some trouble with the following problem: Let $Y$ a real valued random variable bounded and $(A_t)$ a growing process bounded (we have a fixed positive constant $K$ such as $A_\infty \leq K\; a....
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Markov chain convergence in probability

The Markov chain $X_{n}$ takes values 1, 2, 3. The matrix of probability transitions is: $$ \begin{bmatrix} 0.6 & 0.4 & 0\\ 0.2 & 0.5 & 0.3\\ 0 & 0.1 & 0.9 \end{bmatrix} $$ ...
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Show that filtrations are equal

Hey how to show that natural filtrations generated by processes $W(t)$ and $W_Q(t)=\frac{\mu-r}{\sigma}t+W(t)$ are the same where $\mu\in\mathbb{R},r,\sigma>0$? So that $F_t=\sigma\{W(s):0\le s\le ...
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independent Poisson processes

2) Let be $N_{t}^{'}$ and $N_{t}^{''}$ - independent Poisson processes. Is the process $$ X_{t} = (N_{t}^{'} - N_{t}^{''})^2 $$ Markovian?
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36 views

I'm struggling with finding variance of Ito integral.

Find $D \int_{0}^{t}W^{2}_{s}dW_{s}$ My solution is next: variance = $E(\int_{0}^{t}W^{2}_{s}dW_{s})^{2} - (E\int_{0}^{t}W^{2}_{s}dW_{s})^{2}$ Which is equal to (using Ito's isometry principle) = $\...
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1answer
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Solve an Itô Integral by Itô calculus

I saw an example where the following Itô integral was solved by Itô calculus: $\int^{T}_{0}W(t)dW(t)$. They say: let's take the stochastic process $X(t) = W(t)$, which means that $dX(t) = 0 dt + 1 dW(...
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53 views

Find $P(W_{t} < a | W_{2t} > a)$

Here are my thoughts: $P (W_{t} < a | W_{2t} > a) = \frac{P(W_{2t} > 2a, W_{t} < a)}{P(W_{2t} > 2a)} = \frac{P(W_{2t} - W_{t} > a,\, W_{t} < a)}{P(W_{2t} > 2a)}$ Then, taking ...
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54 views

Mathematical expectation of the number of points for Poisson stohastic process

I need to find mathematical expectation of the number of points of Poisson process with parameter $\lambda > 0$ such that: these points $\in [1,2]$ and there are no points of Poisson process in ...
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36 views

stochastic proccess [closed]

1) Find the spectral density of the Ornstein-Uhlenbeck process: $$ e^{-\beta t} W_{\alpha e^{2\beta t}} $$ Where $W_{t}$ - wiener, and $\alpha, \beta > 0$.
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21 views

Is it a good example for local martingale, but not for martingale?

If $B$ is a Brownian-motion in the $\mathcal{F}$ filtration, then the following process is a good example for being a local martingale, but not a martingale?$$S_{t}=\int_{0}^{t}\frac{1}{1-s}dB_{s},\;\;...
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20 views

Find continuous martingale, such that it's quadratic variation is a deterministic continuous function

Given a non decreasing continuous function $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ with $f(0)=0$. I want to find a continuous martingale $(X_t)_{t\ge 0}$ , such that it's quadratic variation is ...
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14 views

Pricing options on several underlying assets using FDM

When implementing a finite difference method to price a European call option where the underlying stock follows the dynamics $dS_t=r S_tdt+\sigma S_tdW_t$ we get a tridiagonal matrix from the finite ...

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