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

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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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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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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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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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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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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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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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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 ...
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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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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 ...
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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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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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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 $X_t = \mathbb{E}[X \mid \mathcal{F}_t], ...
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$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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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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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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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 $(A_N)_{N \geq 0}$ ...
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Distribution of Hitting Times

I am curious about the following problem: Let the diffusion process $\{X_t\}_{t\ge 0}$ be defined as $$dX_t=c(1-X_t)X_td B_t$$ where $X_0\in (a,b)\subseteq (0,1)$, $c>0$, and $B_t$ is the standard ...
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How is the form of the transition functions of a process define as the solution of a stochastic differential equation?

I´m searching for a prove that solutions of SDE are markovian and i'm trying to find (or understand) in this case (SDE) what form have the transition functions associated to this kind of process. I´ve ...
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A functional of a Lévy process

Does anyone know if there are any papers/results on functionals of the type : $$\int_0^tp(X_s)ds$$ where $X$ is a Lévy process and $p$ is a polynomial. For example, is the distribution of such an ...
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Stochastic process on compact spaces

I just heard some strange reasoning that I would like to understand with your help, let me describe the situation (unfortunately, I hesitated to ask the lecturer about it, because I apparently lacked ...
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Existence of compensator process under the assumption of local integrability and finite variation

I am reading a proof regarding existence of compensators under the assumption of local integrability in which I don't quite understand: Definition: The compensator of a cadlag adapted process $X$ ...
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Different definitions of local p integrability for local martingales

When talking about cadlag (but not continuous) martingales and local martingales in the context of stochastic integration one can come across different definitions depending on the author. These are: ...
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$L^p$ integrable local martingale is still $L^p$ integrable when stopped at localizing stopping times.

Assume that $X$ is $L^p$ integrable for $1\leq p\leq \infty$ (i.e., for all $t$, $X_t\in L^p$) and is also a (Cadlag) local martingale. If $T_n$ is a localizing sequence of stopping times for $X$. Is ...
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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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What is the benefit of stochastic models over deterministic models? [duplicate]

I have posted a similar question earlier and I guess this sounds naive to all of you, but nonetheless let me just ask: Consider I have a simple and deterministic model $M$, with a number of input ...
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What kind of decomposition is $X_{t \wedge L}=\tilde{X}_t+\int_0^{t \wedge L} \frac{d \langle X, M^L \rangle_s}{Z^L_{s^-}}$?

In one of the papers I was reading for my masters thesis I came across a theorem with no references. Theorem: If $(X_t)$ is an $(\mathcal{F}_t)$ martingale then there exists a $(\mathcal{F}^L_t)$ ...
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Monte-Carlo simulation with sampling from uniform distribution

I used to work with Monte-Carlo simulations for a while. In my case, I generated random data for a variety of input parameters according to uniform distributions (with non-negative support), say for ...
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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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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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How can we prove that the derivative of a generalized Hilbert space valued Brownian motion is a Gaussian white noise?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\lambda$ be the Lebesgue measure on $[0,\infty)$ $\mathcal D:=C_c^\infty([0,\infty))$ and $\mathcal D'$ be the dual space of ...
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Existence and uniqueness of solution of a non linear SDE

I have the following SDE: $dX_t=(\mu+X_t^2) dt+e^t dB_t$. What can I say about existence and uniqueness of solutions? I would like to verify the usual conditions of sub-linear growth and Lipschitz, ...
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Locally square integrable (local) martingales

I'm reading Protter and sometimes he says "locally square integrable martingale", and sometimes he says "locally square integrable local martingale", and I wonder if these two are the same. Protter's ...
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Formula for contingent claim similar to European call option but with two dates for option to buy

So in a normal European call option with one maturity date, you'd buy a share of a stock if the price of the stock at the maturity date was higher than the exercise price. How would you come up with a ...
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Why is the expression $E[M_L^*] \leq \inf_{\mu >0}\{\mu^{1/{1-r}} \varphi_r(1)+\mu E[\langle M \rangle_L^{r/2}]\}$ an explicit function of $C_r$

Why is the expression $E[M_L^*] \leq$ $ \inf_{\mu >0}\{\mu^{1/{1-r}}\varphi_r(1)+\mu E[\langle M \rangle_L^{r/2}]\}$ an explicit function of $C_r$ where $C_r$ is a function of $\varphi_r(1)$. In ...