# Tagged Questions

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### Ito integrals and the Euler scheme

I was wondering how to find the solution of the following stochastic integral: $$dY_{t}=a(W_{t},Y_{t})dW_{t}+b(W_{t},Y_{t})dZ_{t}$$ or in integral notation ...
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### Ultrametric space of stochastic filtration

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that ...
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### Transition kernel that is not Markov

Let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be two measurable space. A transition kernel $K$ is a function $K : X \times \mathcal{G} \to \overline{\mathbb{R}}_+$ suche that $K(\cdot,B)$ is measurable ...
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### Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
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### Extensions of the Ito integral

This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate?
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### A minor clarification on completion of $\sigma$-algebras

This is from Karatzas + Shreve Definition: The stochastic process $X$ is adapted to filtration $\{\mathcal{F}_t\}$ if, for each $t\geq 0$, $X_t$ is an $\mathcal{F}_t$-measurable random variable. ...
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### measurability question with regard to a stochastic process

Here are two related exercise from Karatzas and Shreve Let $X$ be a process, every sample path of which is right continuous with left limits. Let $A$ be the event that $X$ is continuous on $[0,t_0)$. ...
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### Can we define probability of an event involving an infinite number of random variables?

Consider a collection $(X_a)_{a\in[0,1]}$ of i.i.d. random variables following the uniform distribution on [0,1]. That is, for every real number $a \in [0,1]$ we have a random variable $X_a$. Can we ...
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### Ergodic for the mean, but not ergodic stochastic process?

Is there an example of a strictly stationary (zero mean, finite variance) stochastic process $(X_t\mid t\in \mathbb{N})$ that satisfies the conclusion of the ergodic theorem, i.e., the sample mean ...
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### Moving boundaries for Ornstein-Uhlenbeck processes

Let $\tau(X_t)$ be the first-passing time to the moving boundary $a(t)$ for an Ornstein-Uhlenbeck process $X_t$. I wonder how general an $a$ can be allowed in order to guarantee that $\tau$ becomes ...
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### Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?

Let say I have a filtration $\mathcal F_i$ with $\mathcal F_1$ contained in $\mathcal F_2$, $\mathcal F_2$ contained in $\mathcal F_3$ and so on...$\mathcal F_n$. $X_i$ is a stochastic process, $X_i$ ...
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### Finding conditional variance

I know the marginal Variance of $\operatorname{Var}(Y) = E(Y^2)- (E(Y))^2$ and conditional variance of $\operatorname{Var}(Y|X)$ is $E((Y-E(Y|X))^2\mid X=x)$. I am trying to expand out the last ...
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### What is the intuition behind Adapted Process

I am reading up on stochastic process and in particular adapted process. I know that if $X_t$ is $F_t$ measurable for each t, then it is an adapted process. But I do not understand the intuition ...