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6
votes
0answers
60 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
5
votes
0answers
103 views

Representation theorem for continuous process of finite variation

There is a martingale representation theorem If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that $$ M_t = M_0 + \int_0^t ...
4
votes
0answers
272 views

Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?

The following is the definition of infinitesimal generator from Oksendal. Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
3
votes
0answers
93 views

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let ...
3
votes
0answers
17 views

Continuity in $x$ of $E^x \int_0^{\tau} f(X_t)dt$

Suppose I have a stochastic diffusion $X$. I am studying an expression of the form $u(x):=E^x\int_0^\tau f(X_t)dt$ where $\tau$ is the exit time of $X$ from my bounded open domain $D$. I am also ...
3
votes
0answers
120 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 ...
3
votes
0answers
54 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 ...
3
votes
0answers
481 views

Variance of a Wiener process

Problem statement: a continuous wiener process $w(t)$ with unit incremental variance and $w(0)=0$ is given, and then we check the wiener process at every $h$ seconds, $h>0$ is a positive number. If ...
2
votes
0answers
26 views

Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

Suppose we have an sde of the form: \begin{eqnarray} dX_t=b(X_t)dX_t + \sigma (X_t)dB_t \end{eqnarray} where $b$ and $\sigma$ are Lipschitz. Then we have existence and uniqueness of the solution ...
2
votes
0answers
34 views

How to calculate probability of an event in a stochastic setting?

Let $\left(\, B_{t}\,\right)_{t\ \geq\ 0}$ be a Brownian motion. Calculate the probability of the event: $$ E\equiv\left\{\,\exists\ \epsilon > 0 : \forall\ 0 < h < \epsilon, \max_{t\ \in\ ...
2
votes
0answers
64 views

Most probable path of diffusion process

Suppose we have an Ito diffusion $X_{t}$ on $\mathbb{R}$ given by \begin{align*} dX_{t} = A(X_{t})dt + B(X_{t}) dW_{t} \qquad (1) \end{align*} where $W_{t}$ is a standard Brownian motion. If $B = 1$, ...
2
votes
0answers
64 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, ...
2
votes
0answers
215 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 ...
2
votes
0answers
198 views

Good books on “advanced” stochastic analysis

Any good books suggestion for studding advanced features of stochastic analysis ? Thank's in advance
2
votes
0answers
65 views

Initial Conditions for Finite Difference of PDE

I am having trouble with figuring out what my initial conditions should be for a simple finite difference algorithm I wrote in Matlab. Specifically, I'm trying to show that the regular 1-Dimensional ...
2
votes
0answers
215 views

Explaining Ito formula to an analyst

From the point of view of analysis, what is Ito formula? Is it an integral by substitution, or, a radon-nikodym derivative? Define the probability space $$ \left(C\left(\Bbb ...
2
votes
0answers
45 views

Finite p-th variation implies zero-valued q-th variation.

The Question: Let $X$ be a continuous process, and suppose $0 < p < q$. Prove the case $V_t^p(X) < \infty \implies V_t^q(X) = 0$. Definitions: The standard setup. $\Pi := ...
2
votes
0answers
118 views

How to check if a process is a semimartingale?

Consider the process $X_t = \sum_{i=1}^{N_t} Y_i$. This is a Lévy process, hence Markov and so on ($N_t$ is a Poisson counting process). Now add some diffusion $D$ for each jump $Y_i$ that starts at ...
1
vote
0answers
10 views

Is stochastic modelling a subset of Frequentist and Bayesian points of view?

From what I know of stochastic modelling it seems to me that this technique takes a Frequentist approach. For example, and please correct me if I am wrong, but isn't a Monte Carlo Simulation a ...
1
vote
0answers
27 views

Stochastic integral density of simple functions no1

I am trying to understand proposition 2.6 page p.134 from Karatza's book Brownian motion and stochastic calculus. If $M$ is continuous square integrable martingale on $(\Omega, \mathcal{F}, P)$ and ...
1
vote
0answers
38 views

Stochastic process using Markov chain (thief on the run!!)

I'm given an exercise where we are to simulate a thief escaping from an officer. The thief (let's call him/her T for simplicity) and an officer (O) have four cities to be in. Let's call the cities A, ...
1
vote
0answers
46 views

Write down the HJB equation

Suppose that we have to solve the following optimal control problem \begin{align} V(t,x) = \min_{\alpha}\mathbb{E} \left[\int_{0}^{T}L(t,x,\alpha)dt + F(e^{-\beta t}X^{\alpha}_{T})\right] ...
1
vote
0answers
65 views

Applying Ito lemma to multi dimensional semimartingales

Let $X_t=(X^1,\dots,X^d$) be a d-dimensional semimartingale. Then the formula for Ito's lemma I have found in several places, including wikipedia, is: $f(X_T)-f(X_0)=\sum_{i=1}^d\int_0^T ...
1
vote
0answers
96 views

Write the Hamilton Jacobi Bellman equation

Consider the following stochastic optimal control problem. \begin{equation} V(t,x) = \max_{u}\,\, \log \left(\mathbb{E}\left[\int_{0}^{T} u^{2}(t)dt\right]\right) \end{equation} subject to the ...
1
vote
0answers
33 views

Distribution of Levy driven O-U process

Is there a way to find an analytical expression for $E\left[\exp\left(-\int_0^T \gamma_s ds\right)\right]$, where $d\gamma_t=k(\theta-\gamma_t)dt+\sigma dL_t$, and $L_t$ is a symmetric alpha ...
1
vote
0answers
41 views

Construct an arbitrage opportunity in a multi-period model

I am currently revising for my exam in Financial Mathematics, and I could not solve this question: For $T > 1$, consider a $T$-period model with a single risky asset and a bank account which pays ...
1
vote
0answers
92 views

Tail of hitting times for Brownian motion on the circle

For $y\in \mathbb R/\mathbb Z$ and $\varphi\in C([0,\infty);\mathbb R/\mathbb Z)$ let $T_{y}(\varphi) \ := \ \inf\{t>0: \varphi_t = y \} \ \ \ $ (first time the path $\varphi$ hits $y$) ...
1
vote
0answers
113 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
0answers
67 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
vote
0answers
23 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 ...
1
vote
0answers
42 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 ...
1
vote
0answers
53 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 $, ...
1
vote
0answers
143 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 $g$ ...
1
vote
0answers
46 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
vote
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 ...
1
vote
0answers
40 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
0answers
65 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 ...
1
vote
0answers
42 views

How do you convert an infintesimal generator of a Markov process to a transition function?

Suppose a continuous-time continuous-step Markov stochastic process $X_t$ has infinitesimal generator $\mu(x, t)$, $\sigma(x, t)$ ($\mu$, $\sigma$, and $X_0$ are known). How can we use this ...
1
vote
0answers
871 views

Stochastic integral: Interchanging the order of expectation and integration

Let $B$ be a standard Brownian motion and $$ X_t=\int_0^t f_s ds+\int_0^t g_s dB_s, $$ where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$. Is it true that $$ ...
1
vote
0answers
254 views

What kind of process is a locally bounded process?

Definition of locally bounded process is on http://planetmath.org/encyclopedia/LocallyBoundedProcess.html On that website, it says any discrete-time predictable process is locally bounded. How can I ...
1
vote
0answers
106 views

Martingale Decomposition

Is it possible to decompose a discrete-time martingale $(M_n)$ uniquely into two processes $$M_n=M_n^I+A_n$$ where $(M^I_n)$ is a martingale with independent increments and $(A_n)$ is a martingale? If ...
1
vote
0answers
73 views

Applicability of Itô's Lemma for $g\in \mathcal{C}^2((0,1)^2)\cap \mathcal{C}_0([0,1]^2)$

Let the domain be $[0,1]^2$. And let $W^x_t$ be the standard Brownian Motion started in $x\in [0,1]^2$ with absorbption on $\partial [0,1]^2$ and choose some $g\in \mathcal{C}^2((0,1)^2)\cap ...
1
vote
0answers
227 views

existence of a strong solution for a sde

Suppose we want to study a SDE of the form $$ dX_t = a(t,X_t)dt + b(t,X_t)dW_t$$ and $X_0=Y$, on a filtered probability space $(\Omega,\mathcal{F}, \mathbb{F},P)$ and where $W$ is a $(P,\mathbb{F})$ ...
1
vote
0answers
205 views

stochastic exponential

I was able to show the following: $X$ a semimartingale, $X_0=0$ then the SDE $$ dZ_t=Z_tdX_t$$ with $ Z_0=1$ has the unique solution $Z_t:=\exp{(X_t-\frac{1}{2}\langle X\rangle_t)}$. I was able ...
1
vote
0answers
75 views

Langevin equation with uniform noise

Given the Langevin equation written in the form: $$\ddot{x}(t)+\lambda \dot{x}(t)=\mu(t)$$ if $\mu(t)$ is noise with gaussian $pdf$, the solution is well known in therms of the spectrum of the ...
0
votes
0answers
16 views

markov property in Durrett's textbook

Assume $B_t(\omega)=\omega(t),\omega\in (C,\mathcal{C},\mathbb{P}^x)$ is a B.M.(C is the continuous function space ,$\mathcal{C}$ is generated by the coordinate maps) In Durrett's textbook,he proved ...
0
votes
0answers
25 views

Equality of two spaces of stochastic processes

Let $(\Omega, \mathcal F, P, \mathcal F_t)$ be a filtered probability space. Consider two spaces $M$ and $S$ defines as follows: $M$ is a collection of all continuous $\mathcal F_t$-adapted processes ...
0
votes
0answers
5 views

Derive bound for a function satisfying stochastic differential equations/inequalities

I have a function $U(t)=\frac{1}{2}\|{x}(t)\|^2$ satisfying $ d{U} \le -\frac{\lambda_0}{2}dt+\frac{\sqrt{2T}}{N} \sum_i {dB_i(t)} \cdot {x}, \ \ if\ \ \|{x}\|\ge d_0,$ $d{U} \le \lambda_1 ...
0
votes
0answers
29 views

Processes adapted to time changes

I have a question regarding a passage in Chapter X of "Calcul Stochastique et Problèmes de Martingales"J.Jacod(1979). In (10.13) they define an adapted process $X$ to the time change $\tau(t)$ as a ...
0
votes
0answers
36 views

Ito formula for integral function

Let $$dS_t = \mu S_t dt + \sigma S_t dW_t$$ where $W_t$ is a Wiener process. Let $$Z_t = e^{-r(T-t)} \int_{t}^{T}{h(u,S(u))du} = g(t,S)$$ where $h$ is a known function of $t$ and $S$. How can we ...