Search Results

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 …

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 …

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' …

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 …

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 …

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 …

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 …

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 …

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_ …

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 …

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 …

If X is a continuous martingale of finite variation such that $X_0 = 0$, then $P(X_t = 0 \ \ \forall t) = 1$.

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.