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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Expectation Involving Two Values of Geometric Brownian Motion

Not sure this is the best place to ask for verification, but I can't seem to find a derivation anywhere else. I want to calculate $\mathbb{E}[e^{\sigma(W_t + W_s)}]$, where $W_t$ and $W_s$ are two ...
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1answer
42 views

Long memory of stochastic differential equation

It is well known that the solution to an ordinary stochastic differential equation has the Markov property so that if one tries to model some kind of long memory process one has to instead consider ...
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3answers
3k views

Similarity between two probability distribution

I am not sure how to put the question. I am not even sure if this question makes sense at all. I know that the similarity of two discrete (or continuous) distributions can be quantified by ...
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1answer
36 views

Finite expectation - Difference between these two statements?

Main Question: Let $\{X_n, n=0,1,2...\}$ be a stochastic process. How are following two statements different? \begin{align} E[|X_n|]<\infty \text{ for all } n \end{align} and \begin{align} ...
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1answer
51 views

continuous RV from discrete RV

So I am reading some notes in stochastic processes and I don't really understand the solution of this problem: Problem: Let $(\Omega,F,\mathbb{P})$ be a probability space where $\Omega$ is the set ...
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1answer
210 views

Power spectral density of convolution of stochastic processes

I was wondering what it is the result of convolving two WSS processes in terms of power spectral densities. I know that, the output $Y(t)$ of a generic linear time invariant system with impulse ...
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0answers
56 views

Finding a pre-visible process

Question: Let $W_t$ be a standard brownian motion under P with filtration $\mathscr F_t$. Let: $$ M_t=\mathbb E[W_T^2|\mathscr F_t] $$ Show that $M$ is a P martingale. This is simple enough using ...
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1answer
232 views

Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
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1answer
43 views

Fubini Question in context of Independence

I am trying to show that if $X_t$ is some process and there is a function $p$ such that $$P[(X_{t_1},...,X_{t_n}) \in A_1 \times...\times A_n] = \int_{A_1 \times...\times A_n} ...
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0answers
30 views

characterising attractors for master equations

I have a master equation for $(x,y,z)$ with the constraint $x+y+z=N$. $x$ can be regarded as the number of animal of a certain species in the whole system. In other words, I have a differential ...
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1answer
62 views

Random process $X(t) = 10 \cos(Wt + A)$.

I am doing some exercises based on random process, but I can't find a way out on this: Let $X(t) = 10 \cos(Wt + A)$, where W is a Gaussian aleatory variable with parameters $N(10,2)$ and $A$ is ...
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1answer
175 views

Probability of getting SUCCESS AND FAILURE at number n-1 and n trial

In a sequence of Bernoulli trials let $u_n$ be the probability that the combination SF occurs for the first time at the trials number n-1 and n. To find the generating function I wrote the following ...
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1answer
145 views

Problem on Solving Stochastic Differential Equation

Let $(Xt)$ be a solution to the equation $dX_t = aX_t dt + \sqrt{(1+X_t^2)} dW_t$ where $W_t$ is a Brownian motion process at time t Let $Y = F(X_t)$ for a certain function $F$. Find $F$ for which ...
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1answer
41 views

American Put question

If the interest rate is zero. Then show that the optimal exercise for an american put option is always the terminal time. That is, it is equivalent to a european put option.
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1answer
30 views

limite presque sure

I just want to know why for a continue process X such $X_{t} \rightarrow Z$ p .s when $t \rightarrow \infty$ then lim inf $X_{s}^{2}$=Z when p .s $t \rightarrow \infty$ inf is on $\frac{t}{2}\leq s ...
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0answers
88 views

Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Just to avoid misunderstanding, ...
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1answer
182 views

Probability of a trajectory in Markov processes

I need help with a simple formula! (My question is taken from here, pag 26 eq 1.112. ) Consider a Markov Process with associated Master Equation: \begin{equation*} ...
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41 views

Poisson process different type of events

Suppose that it arrives people to a store according to a poisson process with rate $\lambda = 6$/hour , females arrive with probability $0.6$ and male with $0.4$. What is the probability that there ...
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1answer
689 views

Probability Density of Convolution of Two Random Processes or Variables

Suppose that we have two stationary random processes $x(t)$ and $y(t)$ with probability density functions $f_{x}(x)$ and $f_{y}(y)$ respectively. Now suppose we form: $z(t) = x(t) \ast y(t)$ What is ...
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1answer
76 views

Brownian Motion with drift (stupid question)

How do you prove that $$ \lim_{t\to +\infty} (B_t+ct)=+\infty $$ almost surely? $(B_t)_t$ is the standard Brownian Motion starting from $0$.
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0answers
75 views

Kolmogorov theorem

I just want to know why for a gaussian process X this inequality lead to apply Kolmogorov Centov Theorem. Thanks. $\mathbb{E}(X_{t}-X_{s})^{2}\leq c \vert t-s\vert^{2}$
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1answer
95 views

The Vacisek Model and the short rate process

I am trying to do some calculations related to the Vacisek model, but I think I am mixing up concepts and I'm not getting to any solution. Let me explain what the problem is. The Vacisek model ...
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2answers
48 views

Markov property for a Stochastic Process

My question: Every Stochastic Process $X(t), t\geq 0$ with space states $\mathcal{S}$ and independent increments has the Markov property, i.e, for each $\in \mathcal{S}$ and $0\leq t_0\leq< ...
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1answer
52 views

Property of Brownian Motion's paths

We are considering a Brownian Motion $(B_t)_t$ with values in $\mathbb{R} $ starting from $x$ defined on the stochastic basis: $$(\Omega,\mathcal{E},(\mathcal{F}_t)_t,\mathbb{P}^x)$$ Then, let's ...
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1answer
265 views

Derive the Black– Scholes formula for the European call option.

Consider the standard Black–Scholes model. Derive the Black– Scholes formula for the European call option. thanks for help.
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57 views

Integrated gaussian process

I just want to know what kind of phenomenon a integrated gaussian process ($Y_{t}=\int_{0}^{t}X_{s}ds$ where X is a gaussian process) can modelize. Thanks.
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98 views

marginal distribution of Ornstein Uhlenbeck process

I am learning the OU process. For now, what I can understand is that the OU process is the strong solution of a SDE $d\sigma²(t)=-\lambda \sigma²(t)dt+dz(\lambda t)$ where z is the compound possion ...
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1answer
48 views

Justification of Poisson postulates

This may be a dumb question. The Poisson postulates are: $P(n=1,h) = \lambda h + o(h)$ $\sum\limits_{i=2}^{\infty}P(n=i,h) = o(h)$ Events in nonoverlapping intervals are independent What ensures ...
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1answer
25 views

Random node observation

The problem is as follows: In a two dimensional plane, nodes are randomly distributed with intensity $\rho$. Each node in the network swings between two states: available, non-avaialable for ...
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0answers
31 views

A question on recurrent events

In a sequence of Bernoulli trials let E occur when the accumulated number of successes equal to $c$ times the number of failures where $c$ is a positive integer. I need to show that E is persistent if ...
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1answer
39 views

I want to show $\operatorname{Cov}(X(t),X(s))=\min(s,t)- \frac{st}{T}.$

i have this Equation with Condition $X\left(0\right)=a $ and $ 0\le t \lt T$ $$dX\left(t\right)=\frac{b-X\left(t\right)}{t-T}dt+dB\left(t\right)$$ I solved and got $$X\left(t\right)= ...
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59 views

Changes in the transition matrix of a Markov chain

In most or all Markov chain theories that I know of assumes that the transition matrix does not change over time. But what if certain changes are expected to occur at certain times in the transition ...
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1answer
49 views

Integral of Ito integral

I'm trying to calculate the distribution of $\int_{t_{n}}^{t_{n+1}}\sigma_{1}(\tau)\int_{t_{n}}^{\tau}\sigma_{2}(t)dW_{t}d\tau$ where $W_{t}$ is a brownian motion, that is ...
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1answer
45 views

Optimization of a Sum of Variables

Let there be variables $A$, $B$, $C$, $D$, and $E$ such that a total of $N$ points is allocated among the variables: $A$+$B$+$C$+$D$+$E$=$N$, $N$∈$ℝ$. Let the corresponding point values returned by ...
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1answer
484 views

Regular matrix and regular stochastic matrix

We know that : A matrix is regular if its determinant is non zero. A stochastic matrix is regular if at a certain power all elements are positive. Question is how can I make the link between the ...
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1answer
132 views

How find stochastic logarithm of $B^2(t)+1$.

Find the stochastic logarithm of $B^2(t)+1$. I know that for find stochastic logarithm According to Theorem we must use the The following formula $$X(t)=\mathcal L(U)(t)= ...
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2answers
77 views

Is the following Itô-Integral not zero?

is the following statement true: $$\int_0^T t \, dW(t) \neq 0$$ I need it for a counter-example, that one can not change the order of integration between $dW$ and $dP(\omega)$. I thought of taking ...
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1answer
124 views

Partial fraction expansion of generating functions (clarification of a proof)

I give below a part of Feller's. I am struggling to understand how equation 4.8 was derived. Any help will be much appreciated! Thanks
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1answer
77 views

Expectation of $e^{-4B_\tau}$, where $\tau$ is an extended stopping time

This is an specific example so with a bit of luck I can get some general methodology from your answers. I have this stopping time: $$ \tau = \inf\{t \geq 0; B_t < t-2 \} $$ This is a clear ...
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1answer
165 views

Is $t^{-\frac{1}{2}}B_{t^2}$ a Brownian Motion?

I think the title says it all. Let $X_t = t^{-\frac{1}{2}}B_{t^2}$, with $B_t$ being a brownian motion started at $0$. I think I have proved continuity at $0$ by doing the following: $$ X_t = ...
2
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1answer
108 views

Hitting time for Brownian Motion Surplus Process

I'm trying to solve this question for a continuous surplus process. The surplus process is $$U_s=U_0+s-B_s$$ where $B_t$ is a Brownian motion representing payouts, $U_0$ is starting capital, $s$ is ...
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102 views

Problem with stochastic processes book - should I switch.

I've been reading "Essentials of Stochastic Processes" (second edition) by "Richard Durrett" and I quite liked it, it's a nice size book and it's very easy to read. However, and this is quite a big ...
2
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0answers
168 views

Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
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2answers
140 views

Memoryless processes and independence

this is a mere question of definition, that one surely can figure out by conventional means, but maybe someone can just quickly give me the definition. What is a memoryless process? Following the ...
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1answer
67 views

Combination of Wiener Processes

If $W_s$ and $W_t$ are wiener processes, we have that the probability that $W_s$ and $W_t$ attain maximum is (I am concluding this from "running maximum", but I am not sure) ...
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1answer
115 views

Solutions of SDE do not explode when drift term is zero.

Suppose we have $dX_t = \sigma(X_t) dW_t$ where $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is Borel and $W_t$ is a standard one-dimensional Brownian motion. I am trying to show that $X_t$ cannot ...
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2answers
166 views

Joint density of order statistics

I need some help to understand the following proposition (mainly to understand how it is proven): Let $Y_1,Y_2...,Y_n$ be $n$ random variables which are independent, identically distributed random ...
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1answer
39 views

limiting distribution of $Y_t$ in the mean-reverting Ornstein-Uhlenbeck process

The mean reverting Ornstein-Uhlenbeck process is of the equation: $$dX_t=(a-cX_t) \, dt+\sigma \, dW_t$$ If we are told that both $a$ and $c$ are larger than $0$, what then is the limiting ...
2
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0answers
76 views

Is there an information theory for continuous time signals?

Information theory books talk about entropy and mutual information of discrete time processes, such as a sequence of symbols sent with a symbol period $T_s$ and there received sequence. Can we talk ...
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2answers
890 views

Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$ dV_t = - \beta V_t dt + \sigma dW_t $$ with $V_0 = v$, where $W_t$ is a Wiener ...