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1answer
48 views

Locally Nondeterministic Property of Brownian Bridge

Could anyone please give ideas or point me out references where I can find any result concerning the locally nondeterministic (LND) property (in the sense of Berman: ...
1
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0answers
108 views

Forming a local martingale with continuous increasing process

If $M_t$ is continuous martingale, we know that there exists quadratic variation process which is continuous and increasing. I am interested to know if the converse is also true. To make it precise ...
1
vote
1answer
70 views

limit of sup of a stochastic integral

Let $W$ be a standard, one-dimensional Brownian motion and $0 < T < \infty$. Show that $$\lim_{\beta \to \infty} \sup_{0\leq t \leq T} |e^{-\beta t }\int_0^t e^{\beta s } dW_s| = 0$$ a.s.
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votes
1answer
99 views

2 dimensional Brownian motion but not 3 dimensional Brownian motion

Let $W_t = (W_t^{(1)},W_t^{(2)},W_t^{(3)})$ be 3 dimensional Brownian motion. Let $X=sgn(W_1^{(1)})sgn(W_1^{(2)})sgn(W_1^{(3)})$. Define a 3 dimensional process $M_t$ as follows : $M_t^{(1)} = ...
1
vote
1answer
159 views

Finding the joint distribution of a random process with memory

I'm modeling a digital system as a random process and attempting to solve for the autocorrelation in order to arrive at the power spectral density of the process. The system is as follows: At any ...
0
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1answer
81 views

exit time and indicator function

let $D$ open set of $\mathbb{R}^{n}$ and $T_{D}=\inf\{t\geq 0 : X_{t}\notin D\} $ be the first exit time from the $D$ and $1_{A}$ is Indicator function of $A \subseteq \partial D$ $$ ...
4
votes
1answer
100 views

What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
5
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1answer
221 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
1
vote
1answer
99 views

Variance of a stochastic integral?

Does there exist a variance formula for stochastic integrals? Suppose we have $dX = \sigma (X) dW + \mu(X) dt$ Do we have a formula for $Var(X_t)$ or an intergral of $X$ against $B$ More ...
2
votes
0answers
63 views

an exetension of Doob's inequality

Doob's inequality gives an estimation of $$\mathbb{P}(\sup_{0\leq t\leq 1}|X_t|\geq\varepsilon)$$ where $X$ is a martingale. Now I wonder how to estimate $$\mathbb{P}(\sup_{0\leq t,s\leq 1, ...
1
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0answers
63 views

question about the sequential continuity of the set of probability measures

I have a question about the sequential continuity of the set of probability measures. Let $\Omega$ be the space of continuous functions defined in $[0,1]$ taking values in $\mathbb{R}$. Let ...
1
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0answers
22 views

A question on Stochastic Approximation

I have just started learning stochastic approximation methods, so the question I'm going to ask may be a trivial one in this field, but I need to know this seriousely. I know, that if $g(x,\xi)$ is a ...
0
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0answers
82 views

A stochastic programming with a chance constraint

Let $X$ be a bounded positive variable with an unknown probability density function (PDF) and $f(X)$ be a differentiable positive function. $$\begin{align*} &\min/\max ...
1
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0answers
41 views

Space of stochastic process $\mathcal M (\mathcal C [0, T], E)$

A simple notation question, what is the precise definition of the space $\mathcal M (\mathcal C [0, T], E)$ ($\mathcal M^p (\mathcal C [0, T], E)$) in the context of stochastic processes where $E$ is ...
3
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0answers
107 views

Example of a regular strong solution of an SDE, which doesn't satisfy a Lyapunov condition?

Let $$dX_t = a(t,X_t) \, dt + b(t, X_t) \, dW_t, \quad t \in [0,T]$$ be a stochastic differential equation, where $W$ is an $m$-dimensional Brownian motion, $X_0 = x \in \mathbb{R}^d$, and the ...
2
votes
0answers
211 views

The most fundamental papers in stochastic analysis

I have soft a question. What papers will be good to on start and allow me to make little step into research, without harm for reader. I am interested in an stochastic analysis. I am looking for ...
0
votes
1answer
113 views

First Order Stochastic Dominance

I am reading up on stochastic dominance(http://en.wikipedia.org/wiki/Stochastic_dominance) and have some questions: PDF and CDF of Gamble A and B look like this. Since the CDF of A is always less ...
0
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1answer
47 views

Probability Density Function to Cumulative Density Function

I am reading on Stochastic Dominance (http://en.wikipedia.org/wiki/Stochastic_dominance) and few questions on PDF and CDF. The paragraph I am looking at this: Why is that $P[A\ge x] \ge P[B \ge x] ...
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0answers
51 views

Banach space :space of all adapted processes continuous equipped wih specific norm is complete

Let $\mathbb{B}$ be space of all adapted processes continuous equipped with the norm $\lVert Y\rVert_{\mathbb{B}}^2=E\left[\sup_{t\in [0,T]} |Y_{t}|^{2}\right] < \infty $, ...
3
votes
0answers
53 views

variational inequality

Consider the following dynamics \begin{align} dX_{s} &= a(s,X_{s},Y_{s},Z_{s})ds + \sigma(s,X_{s},Y_{s},Z_{s})dW_s \\ X_{t}&=x \, (\in\mathbb{R}^{n}) \end{align} and the associated payoff ...
1
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0answers
142 views

question about conditional probability and $\sigma$-algebra

I have a question: Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and two random variables $X$ and $Y$. For a Borel-measurable set $\Gamma$, if there exists a measurable function $g$ ...
0
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0answers
84 views

Taylor expansion for a distribution function with a stochastic with a stochastic argument

I have seen the following and I am wondering why I can do this! Given two real valued random variables $X$ and $Y$ with finite variance. Than it holds that \begin{align*} P(X+Y \leq u)=P(X\leq ...
2
votes
1answer
109 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
1
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0answers
44 views

question about time change for filtration

I have a question: Let $T$ be a bounded stopping time and let $(\mathcal{F}_t)_{t\geq 0}$ be a filtration satisfying the usual conditions. Define $\mathcal{G}_t:=\mathcal{F}_{T+t}$, $t\geq 0$. Then ...
1
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0answers
49 views

right continuity of martingale constructed by $X_t=E[X|\mathcal{F}_t]$

$X\in L_1$ is a random variable, and $(\mathcal{F}_t)_{t\geq 0}$ is a filtration satisfying the usual conditions, so could we find a version of martingale defined by $X_t=E[X|\mathcal{F}_t]$. I think ...
2
votes
3answers
236 views

Proof of Levy's zero-one law

Let $(\Omega, \mathcal{F},\mathbb P)$ be a probability space and let $X$ be a random variable in $L^1$. Let $(\mathcal{F}_k)_k$ be any filtration, and define $\mathcal{F}_{\infty}$ to be the minimal ...
1
vote
1answer
227 views

Uniform integrability of a backward submartingale

Let $\{\mathcal{F}_n\}_n$ be a decreasing sequence of sub-$\sigma$-fields of $\mathcal{F}$($\mathcal{F}_{n+1}\subset\mathcal{F}_n$) and let $\{X_n\}_n$ be a backward ...
4
votes
1answer
274 views

Continuous Square integrable martingale Quadratic Variation

We know that given a continuous square integrable martingale there exists unique (up to indistinguishability) continuous, natural and increasing process which is quadratic variation process of the ...
1
vote
1answer
90 views

an estimation of the expected value of a Poisson process

Let $N$ be a Poisson process with intensity $\lambda$, I want to prove that for any $c>0$, $$\limsup_{t\rightarrow\infty}P(\sup_{0\leq s\leq t}(N_s-\lambda s)\geq c\sqrt{\lambda t})\leq ...
2
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1answer
339 views

predictable quadratic covariation from Jacod / Shiryaev

In Limit theorems for stochastic processes, by Jacod and Shiryaev, they state the following theorem: $\mathbf{Theorem}$ To each pair $(M,N)$ of locally square integrable martingales one associates ...
2
votes
1answer
81 views

Lebesgue–Stieltjes integral from 0 to $\infty$ on $\mathbb{R}^+$

In the Stochastic analysis course we encountered the following integral $\int_0^\infty H^2_sd[M,M]_s$, where $H_s$ is a predictable process, $M_s$ is a uniformly integrable martingale in $L^2$, ...
2
votes
1answer
965 views

Distribution of Sum of Two Brownian Motions

How do we find the distribution of the sum of two Brownian Motions? The questions was asked here: Distribution of Brownian motion, and was answered with We can write ...
2
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0answers
175 views

Good books on “advanced” stochastic analysis

Any good books suggestion for studding advanced features of stochastic analysis ? Thank's in advance
1
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2answers
113 views

Generalization of Doob Dynkin for Stochastic processes

Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
2
votes
1answer
288 views

How is Brownian motion predictable?

Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has ...
0
votes
1answer
94 views

Continuous time Stochastic Process stopping time measurability

Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ ...
0
votes
1answer
49 views

Integral: Is there a closed form?

I wonder whether there is a closed form or way to compute explicitly: $$\int_0^t e^{\alpha s} dB_s$$ where $\alpha$ is just a real number and the integral is in the Itô sense. Thank you very much!
0
votes
1answer
93 views

Solve a special non-linear Backward SDE

It is straigtforward to solve a linear Backward SDE. i.e. $dY_t=Z_tdW_t+ aY_tdt+bZ_tdt$ with $Y_T=\xi$ (where a and b are constants, $\xi$ is bounded Randon Variable.) How can I solve $dY_t=Z_tdW_t+ ...
2
votes
1answer
112 views

Example Martingale not UI

I'm looking for an example of two stopping times $\sigma\leq\tau$ and a martingale $M$ that is bounded in $L^{1}$ but not uniformly integrablem for which the equality ...
1
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0answers
37 views

Variance & Expectation

$X$ is a random variable with values in the set of natural numbers and the Generating function G. In Addition: $t(n) = P(X>n)$. Let $F$ be the generating function of the sequence $\{t(n): n \ge ...
1
vote
1answer
336 views

Haar basis on $L^2(0,1)$ - proof?

I have the following problem. We defined $\mathbb{H}=\{f_0,\quad f_{j,n} \quad j=1,...,2^{n-1} \quad n=1,2,...\}$ where for all $t\in[0,1]$ we put $f_0(t)=1$ and setting $K=2j-1$, $$f_{j,n}(t)=\left\{ ...
1
vote
1answer
38 views

Generating function of a random variable

I've got the following problem: Give the generating function of the random variable $X$ whose mass function is defined by: $$f(m) = P(X=m) = (m+1) p^2 (1-p)^m,$$ where $m$ is a positive integer ...
0
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0answers
41 views

The identity of two parameters derived via conditioning arguments

Suppose I have a random variable $X_1\in\mathbb{R}$ and a random vector $X_2\in\mathbb{R}^d$. Furthermore, there are two measurable functions $f_1$ and $f_2$, and two deterministic vectors $\theta_1, ...
1
vote
1answer
107 views

Why is the following function not càdlàg?

I have constructed the following function but I can't see why it is not càdlàg on $[0,1]$: $$f(x)=\begin{cases} 1, & ...
1
vote
0answers
64 views

Stochastic Exponential: $dZ=-\lambda Z dM + dL$ to $dZ=-\lambda Z dM + Zd\tilde{L}$ while $\tilde{L}$ is still orthogonal to $M$

I have a question concerning the paper http://www.researchgate.net/publication/228648002_No_arbitrage_and_the_growth_optimal_portfolio, Lemma 6.3, which is based on ...
0
votes
1answer
44 views

Stochastic Processes Question

Give an example of a stochastic process $X_{n}$ that is not a Markov chain, such that $P_{y}(N(y)=\infty)=0$ but $E_{y}N(y)=\infty$
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votes
2answers
971 views

Ito Isometry and quadratic variation

Here is a confusion regarding stochastic integrals. Let $Y_t=\int_0^tW_sds$ where $W_t$ is a Brownian Motion. Now $dY_t=W_tdt$. So from this expression one can conclude that $dY_t \cdot ...
-1
votes
1answer
221 views

How can I prove it is a martingale when there is a jump process

Let $N_t$ be a Possion process, $M_t=N_t - \lambda t$ we can easily show that $M_t$ is a martingale. Now $\int_0^t\Phi_udM_u=.....=\sum_{i=1}^{N(t)}\Phi_{\tau_i}-\lambda\int_0^t \Phi(u)du $ $\tau_i$ ...
0
votes
2answers
97 views

One stochastic integrability problem

On a lecture notes, there is a following arguement: To make $\int_0^T \pi_t dW_t$ well-defined, (maybe it means to make $\int_0^T \pi_t dW_t<\infty \ \ a.s.$) we only need $\int_0^T \pi_t^2 ...
3
votes
1answer
76 views

Given the SDE: $dX_t=dB_t+b(X_t) dt$ with $(x,b(x)) \leq 0, \forall x \in \mathbb{R}^n$, prove that $E[|X_t|^2] \leq nt+E[|X_0|^2]$

I'm working on this problem: Given a solution $X_t$ to the SDE $$dX_t=dB_t+b(X_t) dt$$ where $B_t$ is an $n$-dimensional Brownian motion, and $b:\mathbb{R}^n \to \mathbb{R}^n$ a Lipschitz ...