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Is a Gaussian process with covariance $C''(0)=0$ allowed?

Let $y'(t)$ be the derivative of $y(t)$ in the mean-square sense. Note that $$Cov(y'(s), y'(t)) = \frac{d}{ds}\frac{d}{dt}Cov(y(s), y(t)) = -C''(s - t).$$ Hence $$C''(0) = -Var(y'(t)).$$ If $C''(0) = ...
Mason's user avatar
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Expectation of Solution to SDE, logarithmic extension of Vasicek

You will need the generating function of a standard Gaussian: $$ \langle e^{tX}\rangle = e^{t^2/2} $$ This generalises to independent standard gaussians: $$ \left\langle \exp\left(\sum t_iX_i\right)\...
LPZ's user avatar
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Expectation of Solution to SDE, logarithmic extension of Vasicek

$r(t)$ is of the form $$\exp\bigg(g(t)+\sigma e^{-at}\int_{0}^{t}e^{as}dW(s)\bigg)$$ Now note that $e^{as}$ is a deterministic function in $s$ and hence $\int_{0}^{t}e^{as}dW_{s}$ is a Gaussian ...
Mr.Gandalf Sauron's user avatar

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