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

**7**

votes

**6**answers

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

**4**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**8**answers

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

**4**answers

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