# Tagged Questions

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### Is there a countably infinite Sigma-Algebra?. Why?

Assume $\,\mathcal{F}$ be a nonempty collection of subsets of $\Omega$. $\,\mathcal{F}$ is called a $\sigma$-Algebra whenever if $A\in\mathcal{F}$ then $A^c\in\mathcal{F}$, and if ...
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### Speed of mean reversion of an interest rate model

I would like to have a bit more of intuition about the concept of "speed of mean reversion" for an interest rate model, e.g. Vasicek or CIR. In particular, is a negative speed of mean reversion ...
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### Reflection Principle

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\{W_t ∶ t ≥ 0\}$ be a standard Wiener process. By setting $\tau$ as a stopping time and defining \begin{align} ...
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### Ito's formula for Jump process

Let $\{N_t\,|\,0\leq t\leq T\}$ be a Poisson process with intensity $\lambda>0$ defined on the probability space $(\Omega,\mathcal{F}_t,P)$ with respect to the filtration $\mathcal{F}_t$ and ...
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### Extended CIR and discretization

Did someone know how to discretize this process efficiently : $dX(t) = \kappa [\theta(t)-X(t)]dt + \sigma \sqrt{X(t)}dW(t)$ I am looking for something more sophisticated than the trivial Euler ...
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### Underlying Sample Space in Continuous Market Model

E.g., a model for $N$ stocks might have each follow a GBM $dS_i = \mu_i S_i dt + \sigma_i S_i dW_i$, where each $W_i$ is independent of the others. Letting $(\Omega, \mathcal{F}, P)$ be the ...
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### How can I calculate $Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)$

How can I calculate? \begin{align} Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right) \end{align} Thank you for your attention.
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### Normalized price process $Z(t)=\frac{\Pi(t)}{B(t)}$

If an interest rate model with the following $P$-dynamics for the short rate. $$dr(t)=\mu(t,r(t))dt+\sigma(t,r(t))d\bar{W}(t)$$ Now consider a $T$-claim of the form $\chi = \Phi(r(T))$ with ...
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### Probability distribution and Stock Price Movement [closed]

How can we use normal distribution for finding the probability of a stock price offer where current price offer depends upon the last price offer. The price offer on some day can go 10% above (at the ...
I'm trying to simulate stock prices using GBM. I am using the following formula, and MATLAB function, to determine the stock prices: $\nu = \mu - \frac{\sigma^{2}}{2}$; \$S = S0*\text{[ones(1,nsims); ...