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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Transforming semimartingale to local martingale by change of measure

Consider a continuous $\mathbb{P}$ - semimartingale X which can be decomposed as M+A (M is local martingale and A is bounded variation process). Is it possible to change measure to $\mathbb{Q}$ s.t. ...
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25 views

Differential of the integral of a stochastic process

In the HJM model one considers the forward rates to be on the form $$\mathrm df(t,T) = \alpha(t,T)\,\mathrm dt + \sigma(t,T)\,\mathrm dW(t)$$ In the proof of showing the drift condition on $\alpha$ ...
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87 views

Elementary proof of geometric / negative binomial distribution in birth-death processes

The birth-death process concerns a population of $n_0$ individuals, each of which reproduce and die at a constant rate as time $t$ increases from $t=0$. Each individual splits into two individuals ...
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65 views

Variance of a Gaussian process

I want to prove that if $Y_t$, $0\leq t\leq 1$ is a zero mean Gaussian process such that there exist $a,b$ with $$\operatorname{Var}(Y_t-Y_s) \leq a|t-s|^b, \;\; s,t\in[0,1]$$ then there exists a ...
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42 views

First hitting time in the one-dimensional case by solving a boundary value problem

If have a question about section 3.1 in the paper Kramers' law: Validity, derivations and generalisations by Nils Berglund. (See http://arxiv.org/abs/1106.5799 page 7 - 9) On page 8 it says, that ...
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28 views

Is any FV-Process a special Semimartingale?

Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special ...
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47 views

Poisson Processes Expected Value

"A machine needs two types of components in order to function. We have a stockpile of n type-1 and m type-2 components. Type-i components last for an exponential time with rate $$\mu_{i}$$, before ...
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45 views

lower bounds for a stochastic integral

for all $t \in [0,T]$, consider a stochastic integral as follows: $\int_0^{min \{t^*,T \}} f(t,\omega) dt$ where $f \geq 0$ is a nonnegative stochastic process and $t^*$ is a random stopping time. I ...
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48 views

Inventory Control using Dynamic Programming

I am trying to solve a traditional inventory control stochastic dynamic programming problem where \begin{align} x_{k+1} &= x_k + u_k - w_k\\ w_{N-1} &= \begin{cases} 0 &\text{w.p. } ...
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29 views

Proof that the predictable sigma algebra is also generated by continuous and adapted processes

I'm reading George Lowther's blog and have a question about the proof of lemma 2. We want to verify that the predictable sigma algebra is also generated by the continuous and adapted processes. One ...
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What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have ...
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49 views

Strong law of large numbers for Poisson process

My question regards the strong law of large numbers as stated, e.g., in Ethier and Kurtz (1986, p. 456 Eq. (2.5)), as follows: If $Y$ is a unit Poisson process, then for each $u_0>0$, ...
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37 views

A filtration with usual condition if the process is Càdlàg

$\{ \mathcal F_t \}$ is a natural filtration associated to a process $\{X_t\}_{t \ge1}$. Show $\{ \mathcal F_t \}$ is a filtration with usual conditions if $X_t$ is Càdlàg. Here a function is Càdlàg ...
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3answers
40 views

Stochastic in finance

I need of a undergraduate guide level to study stochastic process with finances. Starting from a review of probability theory. Eg books, papers or posts. I'll apreciate some help.
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103 views

Clarification in stochastic integration

In the book "Stochastic Processes" by Bass R.F. when he constructs the Stochastic Integral, at some point he defines for $Y$ predictable $$||Y||_2= \left(\mathbb E \int_0^{\infty}Y_t^2\text{d} \langle ...
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18 views

Finite state Markov chains and expectations

Let $X_t$ be a finite state Markov chain with generator matrix $Q$. For a give function $f(x)>0$ define: $$ u(t, i) = \mathbb{E}\left[\int_t^T f(X_s) \mathrm{d} s| X_t = i\right] $$ Are there any ...
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32 views

Limit of a left continuous process

Suppose we are given a left continuous process $X=(X_t)_{t\ge 0}$ and define $$Y^n_t=n\int^t_{t-\frac{1}{n}}\mathbf1_{\{|X_{s\vee 0}|\le n\}}X_{s\vee 0}ds$$ Why does it hold that $\lim_nY^n\to X$? ...
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Show $L$ is not a stopping time

Let $L = \sup\{ n : n \le 10; A_n \in B \}$, $B \in \mathcal B$, $\sup\{\emptyset \}=0$. $(A_n)_{n \ge1}$ is a process adapted by a natural filtration $\{\mathcal F_n\}.$ Show that $L$ is NOT a ...
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126 views

Expected value of Stock Price, Poisson Process

I would appreciate a hint regarding the following question (taken from Durret, Essentials of Stochastic Processes, questions 2.38 "Let $S_t$ be the price of stock at time t and suppose that at times ...
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47 views

Why $\int _0 ^t \phi_s ^2 ds < \infty \ \mathbb P \text{-a.e.}$ do not implies $\mathbb E [\int _0 ^t \phi_s ^2 ds] < \infty$?

Why $\phi =(\phi_t)_{t \in [0,T]}$ is a progressive mesurable stochastic process do not implies $\mathbb E [\int _0 ^t \phi_s ^2 ds] < \infty$? I know that if $X$ is a positive random variable ...
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Intuition for random variable being $\sigma$-algebra measurable?

Is there some sort of intuition or a good ilustrative example for random variables being $\sigma$-algebra measurable? I understand the definition, but when looking at martingales, the meaning of ...
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28 views

Discrete random variable with probability generating function problem. Help!

Suppose $X$ is a discrete random variable with probability generating function: $G_X(\theta)$ = $2(3-\theta)^{-1}$ 1) If $Y$ = $X^{2}$ write down $P(Y=k)$ for $0\leq k \leq 10$, and find $E(Y)$ ...
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19 views

Finding Simple Branching Process Recursive Generating Function

Say we have the following PMF for a simple branching process and want to find the eventual extinction probability, $$ P(Z_{1,1} = 0) = 0.25$$ $$ P(Z_{1,1} = 1) = 0.25$$ $$ P(Z_{1,1} = 2) = 0.50$$ ...
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69 views

Exercise from Rogers and Williams's Diffusions, Markov processes and martingales

I'm stuck trying to do an exercise (see below) in the first volume of the book by Rogers and Williams and any help would be great (my actual question is right at the end). Let $E$ be a locally ...
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36 views

Book suggestions alongside Adventures in Stochastic Processes by Resnick

I am currently taking a SP course following Resnick's book. Are there any other books with exercises (and possibly solutions) I could also look at?
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38 views

Counterintuitive Markov chain problem

In class, my professor said that given a Markov chain $\{X_k\}$ it intuitively should be true that $P(X_{k+1} = x_{k+1} \, \mid \, X_0 = a_0, \dots, X_{k-1}= a_{k-1}) = P(X_{k+1} = x_{k+1}\, \mid ...
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51 views

Question about Markov chain

We know that if $\{X_n\}$ is a Markov chain, then $X_{n+1}$ is independent with the past states $X_0,\ldots,X_{n-1}$ given current state $X_n$, that is ...
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27 views

Markov Chain Conditional Probability

A Markov chain has the transition probability matrix as follows. $$To$$ $$ From \begin{matrix} STATES& 0 & 1 & 2 \\ 0 & 0.6 & 0.3 & 0.1 \\ 1 ...
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Strictly stationary exponential Ornstein-Uhlenbeck process?

Can one define the initial value of the exponential Ornstein-Uhlenbeck process $r$, defined by $$r(t) = e^{y(t)}\quad\text{with}\quad dy(t) = k(θ −y(t)) \mathrm dt+\sigma \mathrm dW(t),$$ such that ...
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Evaluation of $\mathbb E[\int _{t_1} ^{t_2} f(s, X_s^{t,x} )ds \mid \mathcal F _{t_1} ]$ for a markovian SDE solution.

Given a probability space $(\Omega, \mathcal F , \mathbb P)$, a filtration $\mathbb F = (\mathcal F _t )_{t\geq 0}$ and $\mathbb F$-adapted brownian motion $W=(W_t)_{t \geq 0}$, consider $X^{t,x}= ...
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33 views

Squared Poisson Martingale

I know that $M_t=N_t-\lambda t$ is a martingale for $N_t$ a rate $\lambda$ poisson process and that for a brownian motion, $B_t^2-t$ is a martingale. I'm wondering, is there something similar for ...
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How to find restrictions of Autoregressive Lag Model (ADL)

All of my textbooks mention restrictions for AD models but don't explicitly say what they mean by "restrictions" and I'm having a hard time grasping what they mean. $y_t = \beta_0 +\phi y_{t-1} + ...
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Question regarding Markov chains

I have this problem: Random variables $U_1,U_2,...$ are i.i.d with the distribution, $P(0)=0.1, P(1)=0.3,P(2)=0.2,P(3)=0.4.$ Consider a new sequence $X=(X_n=X(n))$ defined as $X(0)=0, ...
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70 views

Brownian motion conditional probability

If $B$ is the standard brownian motion and $a,b >0$ I want to show, using the reflection principle $$\mathbb{P}\left(B_t\geq a-b | \inf_{s\leq t} B_s \geq -b\right) = \frac{\mathbb P(|B_t+x|\leq ...
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83 views

Computing cross variation of independent brownian motions

I am familiar with computing the quadratic variation of Brownian motion, but was confused when the text I'm working through introduced cross variation of independent Brownian motions. the notation is ...
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How could we know the Dirichlet allocation is describing the topic rather than something else?

Dirichlet distribution is widely used in document modelling and document clustering. I tried to understand its rational. I read from this article that: Different Dirichlet distributions can be ...
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28 views

Stochastic processes with independent increments

If $\{X_{t}:t\geq 0\}$ is a real-valued stochastic process with independent increments then $\{X_{t}:t\geq0\}$ is a Markov process? Let $\{ \mathcal{F}_{t} \}_{t\geq0} $ be a natural filtration of ...
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Expectation of stochastic differential equation

I have solved the nonlinear stochastic equation $dX_t=\frac{1}{2}a(a-1)X_t^{1-2/a}dt+aX_t^{1-1/a}dW_{t}$, by reducing it to a linear one (change of variables $Y_{t}=X_{t}^{1/a}$ and applying Ito ...
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1answer
78 views

counting process, independent increments, stationarity

I am wondering if this definition of a counting process, implies some properties of the probability distribution associated with the counting process $\{N(t): t \ge 0\}$. definition: N(0)=0 The ...
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29 views

Log normal approximation

I often read in the litterature that for small volatilities, log normal distribution can be approxiamted quite well by a normal distribution. What do you think about that ? Is there any way to ...
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Poisson Process problem

Adventures in Stochastic Processes, Resnick (1992). $$ E(e^{-\lambda Z(t)}) = E(e^{-\lambda\sum_{k=1}^{N(t)} \theta_k (t)}) \\ =E(e^{-\lambda\sum_{k=1}^{N(t)} \xi_k e^{-\beta (t-\Gamma_k)}}) \\ ...
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Compute expectation (Ito integral/calculus)

I am having trouble computing this expectation. Does anyone know how to proceed? $$E\left[e^{2B(t)} \int_0^t s dB(s) \right].$$ Is it 0? I tried expressing $e^{2B(t)}=1+ 2\int_0^t ...
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Die question probability

We are rolling a fair die and let say $S(k)$ is sum of the $k$ rolls. and $P(n)$ is the probability that the sequence $(S(1),S(2),...)$ includes $n$. Find the limit of $P(n)$ How I approached the ...
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Autocorrelation of Radial Stochastic Process with Planar Derivatives

I have a random field $h(\vec{r})$ that depends on $\vec{r}=(x,y)$, such that \begin{equation} \langle h(\vec{r})h(\vec{r}+\vec{r}') \rangle \sim \exp(-||\vec{r}-\vec{r}'||/a^2) \end{equation} where ...
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Finding dynamics of a dividend paying stock under arbitrary numeraire

Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding ...
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48 views

Invariant measures for stochastic processes

I have some doubts about the concept of invariant measure for a stochastic process. Let me introduce a definition. Given $(\Omega, \mathcal{E}, \mathbb{P})$ a measure space, $H$ Hilbert space, let be ...
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Writing $A(t)=1+S_1S_2^{-1}$ as an Ito diffusion process.

Let $W$ be a Wiener process/Brownian motian and let $$ \begin{align} \mathrm{d}S_1 &= 2S_1(t)dt +3S_1(t) dW\\ \mathrm{d}S_2 &= 4S_2(t)dt +5S_2(t) dW \end{align} $$ Now I'd like to write ...
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67 views

A Brownian motion $B$ that is discontinuous at an independent, uniformly distributed random variable $U(0,1)$

Suppose that $\left\{B\left(t\right): t \geq 0\right\}$ is a Brownian motion and $U$ is an independent random variable, which is uniformly distributed on $\left[0,1\right]$. Then the process ...
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27 views

optimization problem in mathmetical finance using convex duality

I'm interested in the application of stochastic processes and stochastic calculus in mathematical finance. In my lecture I often see a certain optimization problem usually of a convex function. ...
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45 views

The “maximum” of a simple random walk

Suppose $S_n$ is a simple random walk started from $S_0=0$. Denote $M_n$ to be the maximum of the walk in the first $n$ steps, i.e. $M_n=\max_{k\leq n}S_k$. Show that $M_n$ is not a Markov chain, but ...