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### Necessary conditions to ensure that stochastic integral is a normal variable

Let $\left(W_t\right)_{t\geq 0}$ be a Brownian motion with respect to filtration $\mathbb{F}=\left(\mathcal{F}_t\right)_{t\geq 0}$. Let $\left(\alpha_t\right)_{t\geq 0}$ be an $\mathbb{F}$-adapted ...
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### Ito Lemma for Poisson Process

I'm new to stochastic calculus on jump processes and encountered a difficulty. I would appreciate some clarification from the community on the following question. Let $g_t$ be a $\mathcal{F_t}$-...
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### Brownian Bridge general case

The SDE for the Brownian bridge is the following: $dY_t=\frac{b-Y(t)}{1-t}dt+dW(t)$ with solution: $Y(t)=Y(0)(1-t)+bt+(1-t)\int_0^t \dfrac{dW(s)}{1-s}$ Can someone help me on proving that \lim_{t\...
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Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\... 3 votes 0 answers 120 views ### MGF of Generalised Itô Integral The following derivation produces a moment closure problem - I would appreciate any insight. It may seem trivial at first glance, but the key aspect is the integrand dependence on$t$. Consider$W_t$... 3 votes 1 answer 368 views ### Bergomi Volatility Model I was studying on the Bergomi volatility model(using forward variance represented as$\xi_{t}^{T}$).However I don't understand how the author passes from the sde to the first step by only integrating ... 4 votes 1 answer 356 views ### Weak solution of a SDE$\text { Consider the } \operatorname{SDE} d X_{t}=\operatorname{sign}\left(X_{t}\right) d t+d B_{t} \text { on } 0 \leq t \leq T, \text { where } \operatorname{sign}(x)=1\\ \text { for } x>0 \text ...
$\text{ Let } X_{t}=1+t+B_{t}, \text { and } T=\inf \left\{t: X_{t}=0\right\} . \text { Define } G(t)=\int_{0}^{t \wedge T} \frac{d s}{X_{s}}.$ \$\text { Let }\ \tau_{t}=G^{-1}(t) \text { be the ...