# Tagged Questions

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### Monte Carlo by point or by interval

Say I compute monte carlo output from input scenarios. Input are discrete time series. I choose time series as an example to make the problem more obvious - this could be also any curve. Computation ...
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### Given a covarince matrix, generate a Gaussian random variable

Given a $M \times  M$ desired covariance, $R$, and a desired number of sample vectors, $N$ calculate a $N \times M$ Gaussian random vector, $X$. Not really sure what to do here. You can calculate ...
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### Weighted integral of random variables

Given a random zero-mean gaussian random variable $X(t)$ with parameter $t$, such that $E [X(t) X(t^\prime)] = \sigma^2 (t) \delta_{tt^\prime}$, is it possible to produce a single gaussian random ...
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### Characterization of the law of a stochastic process by its finite dimensional distributions

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space. Let $(X_t)_{t \in [0,T]}$, $(Y_t)_{t \in [0,T]}$ (real-valued) centered Gaussian processes such that the finite dimensional distributions ...
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### Standard Brownian Motion

Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$. Can anyone give me some hint to start the ...
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### Does an independent-increment Gaussian process necessarily have Gaussian increments?

Suppose a stochastic process is both independent-increment and Gaussian. Are all its increments Gaussian distributed? Thanks!
### Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion
Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...