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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 …
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.
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 $P(X_t = 0 \ \ \forall t) = 1$.