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      A stochastic process is a collection of random variables usually indexed by a totally ordered set.

      7
      votes
      6answers
      Suppose $X$ is a non-explosive diffusion with dynamics $dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$, where $W$ is a standard Brownian motion. My intuition about $X$ is that if $\mu$ and $\sigma$ are suffi …
      asked Jan 24 '11 by Simon Lyons
      26
      votes
      4answers
      Can anyone give an example of a Markov process which is not a strong Markov process? The Markov property and strong Markov property are typically introduced as distinct concepts (for example in Oksend …
      asked Oct 27 '10 by Simon Lyons
      2
      votes
      0answers
      Hi there, Suppose I have a diffusion process $dX_t = a(X_t)dt + b(X_t)dW_t$. Is there a straightforward method for approximating the first few moments of $X_T$ for some time $T$? Clearly, one could …
      asked Aug 11 '11 by Simon Lyons
      4
      votes
      2answers
      Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to adaptedne …
      asked Feb 7 '13 by Simon Lyons
      7
      votes
      1answer
      I recently noticed something about the covariance function of a Brownian motion that I don't quite understand, and I was wondering if anyone could help me. Suppose $W$ is a Brownian motion, and we ha …
      asked May 23 '18 by Simon Lyons
      10
      votes
      1answer
      The Karhunen–Loève theorem says that Brownian motion on the interval [0,1] can be represented as follows: $W_t = \sum_{n=1}^\infty Z_n \frac{\sin((n-1/2)\pi t)}{(n-1/2)\pi},$ where $Z_n \sim \mathcal …
      asked Mar 23 '11 by Simon Lyons
      2
      votes
      2answers
      I'm working on a PhD project that involves parameter estimation for diffusion processes. I'm based in a machine learning research group, and the emphasis here is strongly on "practical" research. I' …
      asked Aug 22 '11 by Simon Lyons
      5
      votes
      2answers
      Is there a known connection between Weierstrass' function $W_\alpha (x) = \sum_{n=0}^\infty b^{- n \alpha} \cos(b^n x)$ and Brownian motion? Specifically, when $\alpha = 1/2$, the Weierstrass fun …
      asked Mar 21 '11 by Simon Lyons
      14
      votes
      8answers
      Is it reasonable to view the Brownian bridge as a kind of Brownian motion indexed by points on the circle? The Brownian bridge has some strange connections with the Riemann zeta function (see Williams …
      asked Nov 22 '10 by Simon Lyons
      4
      votes
      4answers
      I was wondering if someone could recommend a reference that deals with time integrals of diffusion processes. Suppose $X$ is an Ito diffusion process with dynamics $dX_t = \mu(X_t)dt + \sigma(X_t)dW_ …
      asked Jan 19 '11 by Simon Lyons

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