# Tagged Questions

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

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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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### Exponential stability in mean SDE

we consider the stochastic differential equations: for $s\leq t$ $$dX_{t}=f(X_{s})dB_{s}$$ 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 $X_t = \mathbb{E}[X \mid \mathcal{F}_t], t\in[0,\infty]$...
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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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### 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 ...
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### What is the min-max argument in mathematics?

In the proof of a theorem the author says that he would prove a special case using the min-max argument. After reading the proof I could not infer what the min-max argument actually does. Could ...
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### linear combination of infinitely divisible random variables

If $X$ and $Y$ are real valued random variables with infinitely divisible distributions, does $aX + bY$ also have an infinitely distribution ($a, b \in \mathbb{R}$). I've seen this stated in several ...
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### Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E[\langle\xi,φ\rangle]$ an element of the dual space of $C_c^∞(G)$?

Let $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$ $\mathcal D'$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the ...
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### What is a generalized stochastic process? I've found two different definitions. Are they equivalent?

Let $\mathcal D:=C_c^\infty(\mathbb R^d)$ and $\mathcal D'$ be the dual space of $\mathcal D$. What is a generalized stochastic process? I've found two different definitions in some textbooks: ...
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### How to arrive the following results?

I am reading the book "stochastic differential equations and diffusion processes" written by Ikeda and Watanabe. In the chapter IV about uniqueness of stochastic differential equation, there is a ...
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### 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 \rho_n^2(t)=2 \int_0^...
Let $d\in\left\{2,3\right\}$ and $\mathcal V_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $x_0\in\mathcal V_0$ be a particle and [0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto X_t(x_0)\...