# Questions tagged [stochastic-analysis]

For questions about stochastic analysis or stochastic calculus, for example the Itô integral.

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### Proof that excessive function are also regular (super-harmonic)

In page 117 of Shiryaev's book optimal stopping rules, he claims that the excessive functions are also regular under some condition and state the proof is analogous to another proof, but I fail to ...
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### Skorokhod topology vs. weak topology [closed]

Let $\mathbb{D}([0,T],V)$ be the collection of the cadlag(right-continuous and has left limit) functions on $[0,T]$ taking values in a Hilbert space $V$. The space $\mathbb{D}$ is equipped with the ...
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### What is the transition semigroup of toroidally wrapped Brownian motion?

More generally, let $(B_t)_{t\ge0}$ be an $\mathbb R^d$-valued Lévy process and $W:=\iota(B)$, where $$\iota:\mathbb R^d\to[0,1)^d\;,\;\;\;x\mapsto x-\lfloor x\rfloor$$ (the floor function is applied ...
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### If $X$ is a time-homogeneous Markov process, is $\tilde X:=X-\lfloor X\rfloor$ Markov as well?

Assume $(X_t)_{t\ge0}$ is an $\mathbb R^d$-valued time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$ on a probability space $(\Omega,\mathcal A,\operatorname P_x)$ ...
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1 vote
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### Expression of a stochastic process at a stopping time: $X_{\tau}$

Let $(X_t)_{t \in [0,\infty)}$ be a real valued stochastic process, let $\mathcal{F}$ be a filtration on $[0,\infty)$. Let $\tau$ be a stopping time of $\mathcal{F}$. Is it possible to give an ...
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### Prove $(g_i ◦ X_i) , i ∈ \{1, . . . , n\}$, are independet random variables (Check my proof please)

Let $(Ω, \mathcal A, \Bbb P)$ a probability space, $n ∈ \Bbb N$ and let $(R_i, \mathcal R_i), (S_i, \mathcal S_i), i ∈ \{1, . . . , n\}$,measuring spaces. Let $X_i: Ω → R_i, i ∈ \{1, . . . , n\}$, ...
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### Bernstein von Mises approximation of posterior density

Is there a version of the Bernstein von Mises theorem that allows us to perform a Gaussian approximation of the posterior density (assuming such a density with respect to the Lebesgue measure exists)? ...
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### Proving martingale properties

I'm new to stochastic processes and have problems understanding martingales, conditional probability, $\sigma$-algebras etc. I have two proofs that I'm now sure how to handle. Problem 1. Prove that an ...
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### Showing Variant of Exponential Martingale is a super-martingale

Let $B_t$ be a standard Brownian motion, and let $a_t$ be progressively measurable and so that $a_t \in [0,1]$ almost surely. It's well known that $e^{\theta B_t - \frac{1}{2} \theta^2 t}$ is a ...
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Let Y and Z be two random variables on $(\Omega,\mathcal{F},P)$, it is well-known that $\sigma(Z)\subset \sigma(Y)$ is equivalent to that there exists some Borel measurable function such that $Z=f(Y)$....
I'm trying to understand an argument my prof made: Given $h:\mathbb R \rightarrow \mathbb R$ convex, we look at $E[h(X)]$. The expected value exists because $h(X)$ is lower-bounded by [some] $l(X)$ ...