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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$\sup B_t$ has the same distribution as $\sup C_t$ for two brownian motions $B_t, C_t$

Let $(B_t)_{t \ge 0}$ and $(C_t)_{t \ge 0}$ be two standardized brownian motions. Now why is $\sup_{t \ge 0} B_t$ distributed same as $\sup_{t \ge 0} C_t$? This is a result we assumed as trivial ...
5
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1answer
33 views

$W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$?

This is a sample question for the actuarial exam MFE. Let $Z(t)$ be a standard Brownian motion. Let $W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$? The only thing I know is Ito's Lemma. So I ...
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0answers
24 views

FFT Hyperbolic Distribution R

This is my first posting so forgive me if it is not 100% in line with this forum's best practices. I am completing an analysis using ICA as the decomposition technique. I am keeping 4 of the 10 ...
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25 views

Simple Markov property

I want to prove the simple Markov property but I come to a point where I do not see how to conclude. I want to prove $\mathbb{E}_\nu[Z\circ\Theta_t\mid ...
2
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25 views

Eigen function of one Stochastic Process from the eigen function of another Stochastic Process

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...
2
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29 views

Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let ...
2
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19 views

Represent stochastic process as conditional expectation

I try to reduce my problem to the following question, which is stated rather sloppy (without possibly necessary additional conditions). Let $Y_t$ be a real stochastic process for $t \in [0, T]$ and ...
5
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1answer
166 views
+50

Conditional expectation $\mathbb E\left(\exp\left(\int_0^tX_sdB_s\right) \mid \mathcal F_t^X\right)$

I have found a theorem (see below) in two papers an I try to figure how it could be proved. The result seems to be intuitive, but I'm not able to prove it in a rigorous way. Assumptions: Consider a ...
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24 views

Prove that an integral is zero (from Gardiner's Handbook of stochastic methods)

I have troubles in one proof of the book Handbook of stochastic methods by Gardiner. In the paragraph 3.7.3 he writes this integral $\sum_i\int d\vec x \frac{\partial}{\partial ...
1
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1answer
27 views

Central limit theorem - generalizations [closed]

I am looking for some generalizations for the Central limit theorem in the following sense: Let $\phi$ be a function on the natural numbers, under what conditions on $\phi$ $ ...
0
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1answer
21 views

construct a martingale process from any process [closed]

If ${Z_n, n \geq 0}$ is any sequence of integrable random variables, then ${\sum_{i=1}^{n}[Z_i-E(Z_i|Z_{i-1},...,Z_1)]}$ is a martingale relative to the sequence of $\sigma$-fields generated by $Z_i$, ...
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1answer
28 views

How to get the basis of $L^2[0,1]$ from the basis of $L^2[0,2]$

Is there any way to derive orthonormal basis of $L^2[0,1]$ from the orthonormal basis of $L^2[0,2]$? Here $L^2[0,2]$: is space of square integrable centered stochastic process on $\Omega\times[0,2]$, ...
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53 views

about martingale

The definition about martingale process is $E(Z_{n+1}\mid \mathcal F(X_n))=Z_n$, where $\mathcal F(X_n)$ is the $\sigma$ field generated by $X_n$. My question is if $E(Z_{n+1}\mid \mathcal F(X_n) ) = ...
2
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1answer
33 views

Why use stopping times rather than a deterministic sequence to localise a martingale?

I am a beginner on stochastic processes I am wondering why , to localise a martingale, require the existence of one non-decreasing sequence of stopping times [$ \tau_1 , \tau_2$,...] such that the ...
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7 views

How to show the symmetry of a special green's function? (without defining the class of green's functions in general)

Given a two-dimensional simple random walk $ (X_i)_{i\in\mathbb{N}}$ on $ \mathbb{Z}^2 $, a square $ S_N :=\{1,2,\dots, N\} \times \{1,2,\dots, N\} $, and the stopping time $ \tau_{\partial ...
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10 views

Ito Isometry on Multivariable indicator function

The background of this question is a paper written by Morten O.Ravn and Harald Uhlig, titled "On Adjusting The HODRICK-PRESCOTT Filter For The Frequency of Observations" I will very much appreciate ...
1
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1answer
26 views

Calculating $ \mathbb E \left[e^{-\mu W_T } 1_\left( {\min W_t \leq a} \right) \right]$ for a Wiener process

Let $W_t$ be a standard Wiener process, $a$ some real number, and $\chi (x)$ the indicator function. I am trying to calculate the following expectation: $$ \mathbb E \left[e^{-\mu W_T } \chi \left( ...
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11 views

Customers and Anti-Customer Queueing Problem: What is the Customer delete probability

Hello may ask for your help? First the setting: I have got a problem with some queueing theory. The whole problem would be a grid of nodes, all nodes have an operation intensity $\mu_{i,j}$. ...
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1answer
26 views

“The well-known formulas that gives the relation between the generating functions of a sequence and the sequence of its 'tails'”

I'm reading a paper on Branching Processes and the Theory of Epidemics, and the fourth page (p. 262 of the book) the author refers to "the well-known formulas that gives the relation between the ...
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1answer
36 views

$ P(W_t - W_\tau > 0 \text{ and } \tau <t) = \frac{1}{2}P(\tau < t) $ for a stopping time $\tau$

Let $W_t$ be a standard Wiener process and $\tau = \min \lbrace t \geq 0 :W_t \geq a \rbrace$, the first time the process reaches level $a$. By symmetry of the Gaussian distribution we have $$ P(W_t ...
2
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12 views

Mean and variance regime-switching model

Suppose we have the following model for stock price: $$ X_{t}=X_{0}\exp\left(\int_{0}^{t}(r-\frac{1}{2}\sigma_{\epsilon(s)}^2)ds+\int_{0}^{t} \sigma_{\epsilon(s)}dW_{s}\right) $$ This follows a normal ...
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4answers
84 views

Mathematical philosophical questions about the general theory of stochastic processes.

After 6 months spent on what is termed the "general theory" of stochastic processes and after having worked out many nuances of the field, I realized that: The general theory is beautiful ...
2
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1answer
50 views

Compound Poisson Process function expected value

For the calculus of a financial derivatives, I need to compute the next expectation: $$\mathbb{E}\left((\sum_{i=1}^{N_T} (J_i-k))_+\mid J_1+\cdots + J_{N_t}=x \right)$$ where $$(X_t-k)_+= ...
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0answers
44 views

Conditional expectation of stochastic integral with independent components

Let $T$ denote a maturity and $\mathbb{F}$ a filtration. Besides, consider two processes $A$ and $B$ which are mutually independent and are both dependent on (a subfiltration of) $\mathbb{F}$. Does it ...
2
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26 views

Strong Feller property and uniform continuity ( interpreting Stroock Varadhan 1969 )

In the article Diffusion processes with continuous coefficients II (Stroock Varadhan - 1969), the authors begin a section named Strong Feller property with the following: "Let $P(s, x, t , dy) $ be ...
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0answers
32 views

Markov Chains and transition semigoups

I'm trying to figure out what the following statement refers to. A process $X$ is markov with transitions semigroup $(K_t)_{t\geq0}$ and initial distribution $\mu$ if and only if for all ...
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27 views

covariance of a function of Wiener processes

Consider two independent Wiener processes, $W_1$ and $W_2$. The covariance of certain functions of Wiener processes is simple, for example ...
1
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1answer
13 views

Stochastic ordering functionally invariant

I am studying for an exam in actuarial science, where I have the following exercise: Prove that the stochastic order relation $\leq_{\mathrm{st}}$ is functionally invariant; i.e. show that $$X ...
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1answer
52 views

The quadratic variation of $B \cdot B$, where $B$ is a Brownian motion

Let $B$ be a standard, one-dimensional Brownian motion. Can I show that $[B \cdot B] = B^2 \cdot [B]$, using the "fundamental identity of stochastic integration", namely that $[H \cdot X, Y] = H \cdot ...
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spectral density of continuous time stationary random process [on hold]

A continuous time stationary random process $\{X(t)\}$ has spectral density $$f_{X}(\lambda) = \frac{\sigma^2}{\pi} \frac{\alpha}{\alpha^2 + \lambda^2}$$ The process $\{Y(t)\}$ is defined by $$Y(t) = ...
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38 views

Impossible Events, Probability Zero Events, Change of Sample Space, Invariant, Canonical Sample Space?

I am reading this post about probability theory and its foundations by T. Tao, and also this and this post, and they say in essence that the underlying sample space is not that much important. Often ...
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17 views

Convergence of a stochastic integral [duplicate]

Let $(B_t)$ the standard Brownian Motion and $(H_t)$ be an adapted continuous process. Show that $$\frac{1}{B_t}\int _0^tH_sdB_s $$ converge in probability. I guess that the limit is $H_0$ but I ...
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19 views

On stochastic process

What is a linear stochastic process ? is it different from a stationary process ? Can you give me an example of linear discrete stochastic process ? What are the title of good books so that i can ...
2
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17 views

charaterize the $\mathcal{F}_\tau$ a sigma algebra for the stopping time $\tau$

consider a stochastic process $X: [0, \infty) \times \Omega \to \mathbb{R}^d$ We define $\mathcal{F}_t = \sigma(X(s), 0 \leq s \leq t)$ The sigma algebra generated by the sets $\{\omega: X(s,\omega) ...
3
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41 views

Lagrange multiplier and minimum variance

Looking into a control variate technique of Monte Carlo simulation I have run into a cost-optimization problem that I'm not quite sure I understand. It seems it has to do with Lagrangian multipliers, ...
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23 views

How to calculate the generator of a continuous time branching process

cells in a population either split or die after an exponentially distributed time with parameter $\lambda + \mu$. The cell split with probability $\frac{\lambda}{\lambda+\mu}$, die with a probability ...
2
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2answers
48 views

Mean Square Error of Monte Carlo

Trying to develop the expression for the Mean Square Error (MSE) of Monte Carlo, I found myself a bit lost when going through a simple proof in the literature. I am working in the context of ...
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23 views

Kac Lemma for staionary ergodic processes

Simple question. I have a stationary ergodic process $U$ on a finite alphaet and I want to prove the Kac's Lemma (see Cover and Thomas - Elements of Information Theory (second ed.) page 445). In the ...
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1answer
54 views

Prove that the difference of continuous and monotonically increasing functions has continuous variation

Let $G:[0,\infty)\to\mathbb{R}$ be continuous and $$V^1_t(G):=\sup\bigcup_{n\in\mathbb{N}}\left\{\sum_{i=0}^{n-1}\left|G_{t_{i+1}}-G_{t_i}\right|:0=t_0\le\cdots\le t_n=t\right\}$$ be the variation ...
1
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1answer
23 views

Question regrading to a definition in Stochastic Calculus for Finance 2 by Shreve

I am confused with a definition in Shreve's Stochastic Caclulus for Finance 2 book. In page 129, Theorem 4.2.2, the Ito isometry theorem. It states that The Ito integral defined before satisfies ...
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9 views

Stochastic delay differential equation

$$dX(t)=F(t, X(t), X(t − τ ))dt + G(t, X(t), X(t − τ ))dW(t)$$ In the stochastic delay differential equation(SDDE) given above, can we assume that the delayed time $τ$ is stochastic process? If so, is ...
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1answer
22 views

Autocovariance Function

I need some help please Let $Y_t$ be stationary zero-mean process. Consider the model $X_t=(1-0.4B)Y_t$ How I find the autocovariance generating function of $X_t$? I multiply both sides by ...
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12 views

Properties of the inverse stable subordinator

I just need a good reference and/or some more well-known results (which I don't know yet) for the following situation. Let $\alpha\in(0,1)$. If I have an $\alpha$-stable subordinator $\mathbf{X}$, ...
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7 views

Table or diagram that classifies stochastic processes and summarizes their relationship?

I am looking for a diagram, table, graph, or something along those lines that classifies stochastic processes and summarizes how they relate to each other. Just to give an idea, I am interested in a ...
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13 views

Nonparametric changepoint detection for point process

This is a replication of a question I've recently asked on Cross Validated. It hasn't received an answer or much attention, so I've posted it here. I have a family of point processes representing ...
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0answers
10 views

Trend modelling interpretation? stochastic deterministic stationary

0 down vote favorite I have estimated the following two models: $$Δy_t=0.015−0.410Δy_{t−1}−0.220Δy_{t−2}$$ and $$Δy_t=0.400+0.00145_t−0.150y_{t−1}−0.325Δy_{t−1}−0.220Δy_{t−2}$$ (Note that $y_t$ is ...
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17 views

$L^p$ Bound for piecewise Brownian motion in the context that $P$ is a solution of the martingale problem

Consider a piecewise Brownian motion $$x_n(t) = x_0 + \int_s^t \sigma^n(u, x_n ) \, dB(u)$$ That is ($\sigma^n(u, x_n) = \sigma\left(\frac{n[u-s]}{n},\, x_n \left(\frac{n(u-s)}{n}\right) ...
2
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34 views

European Call/Put Option of a jump difussion Process

Lets have the next jump difussion Stochastic Process: $$S_t = S_0 e^{\sigma W_t + (v-\frac{\sigma ^2}{2})t}\prod_{i=1}^{N_t}(1+J_i)$$ where $W_t$ is the Brownian Motion, hence $G_t \equiv e^{\sigma ...
2
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1answer
59 views

$\sup_{t \in [0, \infty)} \left|[(H^{(n)} - H) \cdot X, Y]_t \right| \overset{P}{\rightarrow} 0$

1. Notation We start with establishing some (standard, I think) notation. Let $(\Omega, \mathcal{A}, P)$ be a given probability space. For any filtration $\mathcal{G} = (\mathcal{G}_t)_{t \in ...
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30 views

Hitting time vs supremum of a càdlàg process

A question on stopping times that came up while I was trying to prove Doob's inequality. Let $X$ be a càdlàg, nonnegative submartingale. Define $X^*_t = \sup_{0\le s\le t} X_s$ and, for $K \ge 0$, ...