# Tagged Questions

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### martingale difference

I am trying to solve the following question. {$ξ_k$} is $F_n$-martingale difference (i.e. for every $n$, $E[ξ_n|F_{n-1}]=0$ a.s. ) Also, for every $n$ , $E[ξ_n^2]<\infty$ Show that ...
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### More preliminaries of the Martingale Convergence Theorem

Really struggling with this lemma. Not sure about the general structure of the proof. Why have we chosen g to be orthogonal to all functions of the form 4.3.1? Why should $G(\lambda)=0$, does it ...
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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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### Q-matrix vs. P-matrix description of a Markov chain

Consider a continuous time Markov chain $(X_t)_{t \geq 0}$ on some state space $S$ with transition matrix (P-matrix) $p_t(x,y)$, the probability density of jumping from $x$ to $y$ in time $t$. The ...
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### recurrent events-Probability of even number of successes

Let E be the event of an even number of successes. $u_n$:Probability of E occurring at the nth trial not necessarily for the first time $f_n$:Probability of E occurring at the nth trial for the first ...
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### A Property of the Ito Integral

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then $E[\int^{T}_{S}f dB_t]=0$ Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ...
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### how to compute $E[F(s)]$ where $F(s)=e^{\int_{0}^{s}h dB(t)-\frac{1}{2}\int_{0}^{s}h^2 d\mu}$

I have $$F(s)=e^{\int_{0}^{s}h dB(t)-\frac{1}{2}\int_{0}^{s}h^2 d\mu}$$ and I want to Compute $E[F(s)]$ . $B(t)$ is brownian motion. thanks for help.