# 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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### Given ${\rm d}X_t=v_t(X_t){\rm d}t+\text{white noise}$, obtain a SDE for $v$ in a Hilbert space $H\subseteq L^2$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $U,H\subseteq L^2(\Omega,\mathbb R^d)$ be separable $\mathbb R$-Hilbert spaces Let $t\mapsto X_t\in\mathbb R^d$ be the trajectory of a ...
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### Probabilistic Method/Model for Traffic Flow

Context: Given a network system or a traffic system with some value related to the system. Question: Which probabilistic methods, model, distributions are used frequently to predict a event (for ...
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### Solving an Itô Integral

Can someone please show me how to solve this Itô Integral? \begin{align}\int_{1}^{t}\frac{dB_s}{B_s^2 + B_s^4} && \end{align}
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### Finding the limit $\lim_{t\to ∞} \mathbb{E}[R_t]$ of an SDE

I have the SDE $$dR_t = (1 - \beta R_t)dt + \sigma dB_t$$ In this equation, $R_0 = r$ in which $r > 0$ Can someone please help me find the $\lim_{t\to ∞} \mathbb{E}[R_t]$? Thus far I have ...
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### Integration with respect to a Poisson random measure

Let $N$ be a Poisson random measure (PRM) on a Polish space, $\left(X,\mathcal{B}(X)\right)$, and let $\tilde{\nu}$ be its mean measure. Then, let $f$ be any non negative and bounded function on $X$. ...
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### For a one-dimensional Brownian motion $B_t$ $Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}.$

A one-dimensional Brownian motion $B_t$ has exponential moments of all orders, i.e. $$Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}. (2.6)$$ This is given as a corollary to ...
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### Time-changed Brownian Motion

Let $B_t$ be a standard Brownian motion and let $\tau_{-1}= \inf \{ t \geq 0: B_t(\omega) = -1\}$. By the Continuous Time Stopping Theorem we know that \begin{align} Z_t = B_{t \wedge \tau_{-1}} \...
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### Sharpen Doob's Maximal Inequality

Let $B_t$ be a Brownian motion, $B_T^* = \sup_{0\leq t \leq T} B_t$ and $\lambda > 0$. Applying Doob's maximal inequality gives: \begin{align} P(B_T^* \geq \lambda)\leq \frac{\mathbb{E}[B_T^p]}{\...
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