Questions about stochastic analysis or stochastic calculus, for example the Ito integral. See https://en.wikipedia.org/wiki/Stochastic_calculus

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22 views

Ito Formula for increments of Ito Processes

Let $X_{t}=X_{0}+\int_{0}^{t}a_{s}ds+\int_{0}^{t}\sigma_{s}dW_{s}$, $W_{t}$ is a standard BM. How can I apply Ito formula to $(X_{t}-X_{s})^{2}$? Should I use a multidimensional version?
2
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1answer
46 views

Definition/Construction of Wiener Measure

I want to make sure I understand this rigorously: Assume we already know that Brownian motion $B_t$ on $[0,\infty)$ exists/how to construct it. Every $\sigma$-field considered is implicitly assumed ...
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0answers
23 views

Compound Process and its compensator

I have always implicitly thought that for a counting process $N_t$, defining the compound process $$\sum_{i=1}^{N_t} X_i,$$ where $X_i$ are i.i.d, was pretty much equivalent to constructing a ...
2
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1answer
139 views

How to prove convergence of process and stopping time

We consider the random function $X^n=(X^n_t)_{t\geq 0}$ with values in the Skorokhod space $\mathcal{D}$ of càdlàg paths, and suppose that it weakly converges (i.e in distribution) to ...
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0answers
15 views

Correlation and covariance matrices

I have a sample cross correlation defined as, $\hat R^N_{eu}(\tau)=\frac{1}{N}\sum_{t=1}^{N-\tau}e(t+\tau)u(t)$ where $e(t), u(t)$ are independent, then I can use the central limit theorem to show ...
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0answers
16 views

Two ways of understanding $\sigma\{Y_t, 0\leq t \leq T\}$

Given a probability space $(\Omega, \mathcal{F}, P)$ and a stochastic process $Y_t$ with continuous path, are there two ways of understanding $$ \sigma\{Y_s, 0\leq s \leq t\}?$$ First is looking at ...
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0answers
27 views

Conditional expectation on Function space

This question is from a notation in section 13.4 of the book "Linear and Nonlinear Filtering for Scientists and Engineers" By N U Ahmed In this section, the author is deriving the Zakai ...
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18 views

What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
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1answer
48 views

Show the existence of a right-continuous modification

Suppose ($X_t$)$_{t \geq 0}$ is a stochastic process with independent increments and the function $t \rightarrow \mathbb{E}X_t$ is continuous. Prove that $(X_t)$ has a right-continuous modification. ...
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2answers
93 views

If one stochastic process is a modification of another, then they have the same finite probability distribution.

On page 2 in Karatzas and Shreve: Brownian Motion and Stochastic Calculus it is said that a stochastic process $Y$ is a modification of $X$ if for all $t$: $P(X_t=Y_t)=1$. If both are stochastic ...
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1answer
32 views

Indicator function minus probability on an event is a martingale

Define \begin{align} \epsilon_j = \mathbb{1}_{A_j} - \mathbb{P}(A_j) = \begin{cases} 1- \mathbb{P}(A_j) \qquad &\text{if } \omega \in A_j\\ - \mathbb{P}(A_j) \qquad &\text{otherwise }, ...
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1answer
23 views

definition of stochastic process on countable space

I found the following definition of stochastic process: A stochastic process on a countable state space V is a sequence of random variables $(S_n)_{n \in \mathbb{N}}$ taking values in V and defined on ...
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1answer
27 views

Measurability From countable to Uncountable sequences family of functions

I am studying Protter Stochastic Integration and Differential Equations. In a theorem on hitting times Protter has the following theorem. Let $X$ be an adapted cadlag stochastic process, and ...
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1answer
31 views

Geometric Brownian motion with random drift and diffusion

One of my finance professors claims that the following is a meaningful SDE. $$dX_t = \delta_t\mu X_tdt + \delta_t\sigma X_tdW_t$$ Here $W$ is BM and $\mu$ and $\sigma$ are positive real constants. ...
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1answer
21 views

Is a compact set on a Polish space closed?

Proposition: If an exponentially tight familiy $P_\epsilon$ satisfies the LDP (large deviation principle) lower bound for open sets for a rate function I, then I is a good rate function. The proof ...
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1answer
32 views

Why is the Covariance operator continuous?

Claim: Covariance operator $C_\mu : B^*\to B^{**}$ is continuous. $B$ is a Banach space and $B^*$ its dual space. For clarification the original def. of the covariance operator is: $C_\mu: ...
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13 views

Geometrical Brownian motion Passage time

Recently I have been self-studying stochastic analysis. One of the exercises was to calculate the probabilty of Brownian motion reaching certain level before time T given that W(t)=x. This wasn't that ...
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9 views

what's the relation between characteristics of $(X_1,X_2)$ and characteristics of $X_1$ and $X_2$

I am not clear how to write down of the characteristics of two-dimensional Levy process $(X_1,X_2)$ when the characteristics of $X_1$ and $X_2$ are known. More precisely, let's say $$ X_k(t) = ...
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1answer
35 views

Clarification on Stochastic Exponential

Consider a $d$-dimensional Brownian motion $B=\left(B_1,...,B_d\right)$ whose components are independent and let $A$ be a $d\times d$ squared matrix such that $\sum_{i=1}^dA_{ii}^2=1$. Define the ...
4
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1answer
121 views

Solution for SDE: $dF_t= \beta_t\left(F_t - \alpha\right)dW_t$

I am trying to derive the solution for the following stochastic differential equation, but I must be doing something wrong in my calculations because I can't arrive to the correct solution. The SDE ...
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1answer
34 views

Change of variable in $\varphi(s) = t$, effect in $\mathbb{d} W_t$

I'm a little confused here. If I have the stochastic integral $$ \int_0^T f(t)\,\mathbb{d} W_t $$ and perform the change of variables $t = \varphi(s)$, how will $\mathbb{d} W_t$ transform (where the ...
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0answers
21 views

Intuition behind a market uncertainty represented by a filtered complete probability space?

What is the intuition behind a market uncertainty represented by a filtered complete probability space $(\Omega, F, P, {F_t})$, on which an m-dimensional standard Brownian Motion $W(t) = (W_1 (t), W_2 ...
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44 views

Is knowledge of PDE useful for SDE?

I am a stochastic analysis student and am particularly interested in stochastic differential equations. What always struck me as odd is how little PDE (or even ODE for that matter) seems to have ...
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53 views

Variance of Brownian Integral when the end point is specified

Consider the Brownian $W_u$. Suppose you are only considering realizations of this brownian that verify both $W_0=0$ and, for a specific (given) $t$, $W_t=a$. Under these specific conditions, what is ...
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26 views

Showing the Haar wavelet is a complete and orthonormal sequence within $L_2[0,1]$

I define the mother Haar wavelet to be: \begin{align} \phi(t) = \begin{cases} 1 &\mbox{if } 0 \leq t < 1/2 \\ -1 & \mbox{if } 1/2 \leq t \leq 1 \\ 0 &\mbox{otherwise}. ...
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1answer
20 views

Conserved quantity for system of Stochastic Differential Equations

I'm considering the set of SDEs (in the sense of Ito) $\begin{align*} \mathrm d x &= -yx \mathrm d t+ x^2 \mathrm d B_t \\ \mathrm d y &= -y^2 \mathrm d t + xy \mathrm d B_t\end{align*}$ ...
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9 views

Decouple system of Ito SDEs

Consider $\begin{split} \newcommand{\d}{\mathrm d} \d x &= -yx \d t + x^2 \d B\\ \d y &= -2 y^2 \d t + 2xy \d B. \end{split}$ This is a system of two SDEs driven by the same standard ...
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1answer
34 views

Proof that $\mathbb{d}\,\mathbb{E}(f(X_t)) =\mathbb{E}(\mathbb{d}f(X_t))$

I've read in a paper that, if $f$ is continuous, then $$ \mathbb{d}\,\mathbb{E}(f(X_t)) =\mathbb{E}(\mathbb{d}f(X_t)) $$ where $X_t$ is a stochastic process and $\mathbb{d}$ is a differentiation, ...
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43 views

The limit of the ratio of two stochastic integrals

I am just wondering how to calculate the limit of stochastic integrals. Here is one example: $$ \lim\limits_{N \rightarrow \infty}\dfrac{\int_{0}^{N}B(s)dB(s)}{\int_{0}^{N}B^2(s)ds}$$ where $B(s)$ is ...
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15 views

Do general additive-noise SDEs on $\mathbb{R}^d$ have finite, strictly positive, transition probability densities?

Let $b \colon \mathbb{R}^d \to \mathbb{R}^d$ be a bounded measurable function. Let $\{P_t(x,A): t \geq 0, x \in \mathbb{R}^d, A \in \mathcal{B}(\mathbb{R}^d)\}$ denote the family of transition ...
3
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1answer
69 views

Stochastic Integration with respect to Cauchy Process?

I'm interested in a one-dimensional stochastic process: $$dX_t = f(X_t)dt + g(X_t) dZ_t$$ where $Z_t$ is a Cauchy process and $f,g$ are nice polynomials (I'm looking at an ODE that gets perturbed by ...
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85 views

Multiplication of two stochastic integrals

I was wondering if someone can help me with the concept of stochastic integral multiplication. Consider multiplication of two stochastic integrals $$(\int^T_0f(u)dW_u)(\int^T_0g(s)dW_s)$$ where $W_u$ ...
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0answers
26 views

Construction of Ito Integral: Doubt from Oksendal, Chapter $3$, Page-$27$

In the book "Stochastic Differential Equations" by Oksendal, at the page $27$, in the last few lines he has written Define $g_{n}(t,\omega) := \int_{0}^{t}\psi_{n}(s-t)h(s,\omega)ds$. Then, ...
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23 views

Construction of Ito Integral: doubts from Kuo

In the book "Introduction to Stochastic Integration" by Kuo, at page $46$, he has written: $\int_{a}^{b} E(|f(t)-g_{n}(t)|^{2})dt \\ \leq \int_{a}^{b} \int_{0}^{\infty} e^{-\tau ...
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0answers
19 views

Exponential stability in mean SDE

we consider the stochastic differential equations: for $s\leq t$ \begin{equation} dX_{t}=f(X_{s})dB_{s} \end{equation} where $f:R\rightarrow R$ and $B$ is a one-dimensional Brownian motion. I want ...
2
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0answers
48 views

Show that for every $p >0$, $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$

I am trying to prove that $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$ where $\rho(n)$ is a solution of the following Stochastic differential equation \begin{equation} \rho_n^2(t)=2 ...
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43 views

Prove of the existence of a cylindrical Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be ...
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0answers
25 views

Representation formula for a Hilbert space valued Brownian motion. Prove independence of the real-valued Brownian motions in the expansion.

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be ...
3
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1answer
81 views

Infinitesimal Generator for Stochastic Processes

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The infinitesimal generator $LV(x)$ is defined by: $$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) ...
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1answer
26 views

Proving linear operators are Markov Generators

I am trying to do the following question from Liggett. Let $A$ be a linear operator defined on the space of continuous functions $C(E)$ for a compact set $E$. Define $A$ by $A=T-I$ where $T$ is a ...
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1answer
42 views

$T$ stopping time, does $\mathcal{F}_T \subseteq \mathcal{F}_\infty$?

I am confused about the definition of $\mathcal{F}_T$, where $T$ is a stopping time. From three different books we find two different definitions: (Karatzas and Shreve; Protter) Events $A \in ...
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0answers
46 views

Finding the Levy measure

I am struggling with the derivation of the Lévy-measure of a Gamma-process $X_t$ with law $p_t(x)= \frac{\lambda^{ct}}{\Gamma(ct)}x^{ct-1}e^{-\lambda x}1_{\lbrace x>0 \rbrace }$. The paper I am ...
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0answers
35 views

For a stopping time $T$, prove that $X^T_t = \mathbb{E}\left[X_T\mid \mathcal{F}_t\right]$

We have a sigma-algebra $\mathcal{F}=\mathcal{F}_{\infty}$, a stopping time $T$ and an integrable random variable $X$ and define a martingale by $X_t = \mathbb{E}[X \mid \mathcal{F}_t], ...
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1answer
30 views

local martingales/ Ito formula

I have a problem with following task. Find $(A_t)_{t\ge0}$ a process of bounded variation on bounded intervals, such that $A_0=0$ and process $M_t=W_tsin(\int^t_0W_s^3dW_s)-A_t$ is a local martingale. ...
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1answer
99 views

What is “white noise” and how is it related to the Brownian motion?

In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering ...
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1answer
35 views

A computation using the Ito integral

I was assigned this exercise by my Stochastic Analysis Professor. Exercise. Let $B$ be a one-dimensional Brownian Motion, and consider the following processes: $X_t=\int_0^tB_sds\quad ...
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0answers
17 views

Definition improper Itô's integral.

Let $\{B_t:t\geq 0\}$ be a standard Brownian Motion and let $\{\mathcal{F}_t\}_{t\geq 0}$ be the natural filtration associated to Brownian Motion (that is, $\mathcal{F}_t=\sigma(B_s:0\leq s\leq t)$). ...
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29 views

How can we evaluate the material derivative of the velocity of an particle by means of an Itō formula?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(B_t)_{t\ge 0}$ be a $\mathbb R^d$-valued Brownian motion with respect to ...
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0answers
21 views

Convergence of stochastic processes via convergence of infinitesimal generators

Given a sequence of sequence processes $(X_N(\cdot))_{N \geq 0}$, I want to show this sequence converges to another process $X(\cdot)$ by considering that the sequence of generators $(A_N)_{N \geq 0}$ ...