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

33 votes
4 answers
9k views

A Markov process which is not a strong markov process?

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 …
7 votes
4 answers
2k views

Time integrals of diffusion processes

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_ …
16 votes
8 answers
4k views

Brownian bridge interpreted as Brownian motion on the circle

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 …
4 votes

Exact simulation of SDE

Beskos and Roberts' whole approach relies on being able to transform the SDE to one with unit diffusion coefficient. If you can't do that, then the bridged process law isn't equivalent to the law of a …
Martin Sleziak's user avatar
4 votes
3 answers
1k views

Converse to Girsanov's theorem?

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 …
10 votes
2 answers
1k views

Covariance function of Brownian motion and the second derivative operator

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 …
8 votes
6 answers
753 views

Diffusion sample paths as deformed Brownian sample paths

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 …
2 votes

Brownian motion, martingales, Markov Chains - Rosetta Stone

Levy's characterisation of Brownian motion: If $X$ is a continuous martingale and $X$ has quadratic variation process $[ X ]_t = t$ then $X$ is a standard Brownian motion.
Leo's user avatar
  • 101
12 votes
1 answer
3k views

Karhunen–Loève approximation of Brownian motion and diffusions

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 …
2 votes
2 answers
349 views

Diffusion processes in probabilistic modelling

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' …
2 votes
0 answers
228 views

Estimating moments of diffusion processes

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 …
6 votes
2 answers
2k views

Weierstrass' function and Brownian motion

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 …
0 votes

Brownian motion, martingales, Markov Chains - Rosetta Stone

If X is a continuous martingale of finite variation such that $X_0 = 0$, then $P(X_t = 0 \ \ \forall t) = 1$.
Simon Lyons's user avatar
  • 1,666