Markov diffusion

A Markov diffusion is a continuous-path Markov process whose local evolution is described by a drift vector and diffusion matrix. For sufficiently regular coefficients, its infinitesimal generator is a second-order differential operator and its conditional expectations satisfy a Kolmogorov backward equation.

A driftless square-root diffusion process is the nonnegative solution of $d{X}_t=\sqrt{X_t}d{W}_t$, with zero an absorbing boundary. For elapsed time $h>0$, starting from $x\ge 0$, its Laplace transform isTo prove this, fix the final time $h$ and set $F(t,x)=\exp [-xu/(1+u(h-t)/2)]$. Differentiation gives $F_t+\frac{1}{2}x{F}_{xx}=0$, while $F(h,x)=e^{-ux}$. By Itô formula, $F(t,X_t)$ is a local martingale; it takes values in $[0,1]$, so localization and bounded convergence make it a true martingale. Taking expectations at the two endpoints proves the formula. The Markov property gives the same formula conditionally at any starting time.

The elapsed-time $h>0$ transition distribution of a driftless square-root diffusion starting at $x$ is the compound Poisson distributionwhere all variables on the right are independent and the empty sum is zero. Here the exponential distribution parameter is its rate. The Laplace transform of this sum iswhich is the Laplace transform of the driftless square-root diffusion; uniqueness of the Laplace transform proves the distributional identity. In particular, for $0\le v<2/h$, the exponential moment isFor $x>0$ this exponential moment is infinite when $v\ge 2/h$, since the event $N=1$ has positive probability and one exponential jump already has an infinite moment there. At $x=0$ the process remains zero and every such moment is one.

A diffusion limit rescales a sequence of stochastic processes so that its limit is a diffusion process. For suitably rescaled independent small jumps, the first two jump moments determine the limiting drift and diffusion coefficient. Brownian motion arises when the limiting drift is zero and the variance grows linearly in time.

For a diffusion with generator $({\sigma }^2/2){\partial }_{xx}+b{\partial }_x$ and increasing scale function of a one-dimensional diffusion $s$, its speed density is $m=2/({\sigma }^2{s}^{\prime })$. Scale records hitting probabilities; speed records expected occupation and exit times. Multiplying scale by a positive constant divides speed by the same constant, leaving their Green-kernel product unchanged.

The expected exit time from $(\ell ,r)$ for a one-dimensional diffusion is ${\int }_{\ell }^{r}K(x,v)m(v)dv$ when finite. Differentiating this expression on each side of $v=x$ gives the equation $\mathcal{L}u=-1$ with zero endpoint values. For the SLE two-boundary-point ratio diffusion, $s(v)-s(1)$ behaves as $(v-1{)}^{1-4/\kappa }$ and the speed density behaves as $(v-1{)}^{4/\kappa }$, so the product is integrable at $1$ for $\kappa >4$.