Bessel process

A Bessel process of dimension $d$ is the radial part of $d$-dimensional Brownian motion. Away from zero it solves the stochastic differential equationThis equation also defines noninteger dimensions through its weak solutions.

A dimension-two Bessel process starting at $r>0$ has the displayed marginal law, where $G_1,G_2$ are independent standard normal random variables. Take the norm of a planar Brownian motion starting at $(r,0)$. Planar Brownian motion avoids a fixed point, so the norm stays positive; Itô formula gives drift $1/(2R)$ and a martingale term of bracket $t$. The Lévy characterization of Brownian motion identifies that term as its driving Brownian motion. The squared radius divided by $t$ has a noncentral chi-squared distribution with two degrees of freedom and noncentrality $r^2/t$.

A squared Bessel process of dimension $\delta \ge 0$ is a nonnegative continuous solution of the displayed stochastic differential equation. It extends the squared radial part of integer-dimensional Brownian motion to arbitrary nonnegative dimensions. Before hitting zero its square root solves $dR=dB+(\delta -1)(2R{)}^{-1}dt$. At dimension zero, a process started at zero remains there. The coefficients are not Lipschitz at zero, so positivity or uniqueness must not be justified by a global Lipschitz theorem.

For independently driven nonnegative squared Bessel processes $X,Y$, put $Z=X+Y$. Where $Z>0$, use Brownian weights $\sqrt{X/Z}$ and $\sqrt{Y/Z}$. Where $Z=0$, fill the weights with $(1,0)$. Their squared norm is always one and their cross-variation is zero, so their stochastic integral is Brownian motion by the Lévy characterization of Brownian motion. Multiplying by $2\sqrt{Z}$ recovers the sum's martingale part even at zero, and the drifts add to $\alpha +\beta$.

For $0<a<1/2$, couple $dX=dB+a{X}^{-1}dt$ and $dY=-dB+a{Y}^{-1}dt$ from positive $x,y$, killing each at zero. Their sum has no Brownian term and increases, so simultaneous extinction is impossible. The ratio $U=Y/(X+Y)$ has time-changed generator $\frac{1}{2}{\partial }_u^2+a(1-2u)[u(1-u){]}^{-1}{\partial }_u$. Its scale density is $u^{-2a}(1-u{)}^{-2a}$, integrable at both endpoints. Optional stopping of its normalized scale function at the first extinction gives the displayed regularized incomplete beta function. At $a=1/3$ this is Cardy's formula.

For $0<\epsilon <x$, the drift $g_{\epsilon }(y)=a/(y\vee \epsilon )$ is globally Lipschitz and bounded. Solve its equation strongly for the prescribed Brownian motion, and stop at the first hit of $\epsilon$. Local uniqueness makes these stopped solutions agree up to the larger threshold; their stopping times increase as the threshold decreases. With $\nu =2a-1>0$, Itô's formula for $X^{-\nu }$ gives $\mathbb{P}(T_{\epsilon }\le t)\le (\epsilon /x{)}^{\nu }$ on every finite horizon. Hence the patched lifetime is infinite. Localization away from zero proves pathwise uniqueness of the positive global solution. Uniqueness before a stopping time refers to the stopped solution, not to arbitrary unconstrained continuations afterward.

The radial part of three-dimensional Brownian motion started away from the origin is a Bessel process of dimension three. The radial Laplacian is $\Delta |x|=2/|x|$, so the Itô formula gives the displayed equation. The radial martingale has unit quadratic variation rate and is Brownian motion by the Lévy characterization of Brownian motion. The reciprocal $1/R$ is a local martingale. Stopping it on $(\epsilon ,M)$ gives ${\mathbb{P}}_x(T_{\epsilon }<T_M)=\epsilon (M-x)/(x(M-\epsilon ))$. Letting $\epsilon \downarrow 0$ shows that zero is not hit.

For $W$ started at a point of radius one, its radial process has the law of the inverse-clock process $e^{B_{J_t}+J_t/2}$, where $J$ inverts $H_s={\int }_0^s{e}^{2B_r+r}dr$. The logarithmic growth of an exponential Brownian clock with positive drift gives $\log H_s/s\to 1$, whereas $\log (e^{B_s+s/2})/s\to 1/2$. Substituting $s=J_t$ proves the displayed almost sure escape rate and divergence of the radius. Independently, Brownian scaling gives the same rate in probability; if an almost sure logarithmic rate is already known to exist, that identifies its value.

Couple $dX=dB+a{X}^{-1}dt$ and $dY=dB+a{Y}^{-1}dt$ from $0<x<y$, with $1/4<a<1/2$. Suppose their finite lifetimes end at zero and ${\int }_0^{{\sigma }_y}{Y}_t^{-2}dt=\infty$. Their difference is positive and has derivative $-a(Y-X)/(XY)$, so ${\sigma }_x\le {\sigma }_y$. For $\Theta =(Y-X)/Y$, the Itô formula givesThe normalized scale function of a one-dimensional diffusion $I_{\Theta }(4a-1,1-2a)$ is a bounded local martingale. Its finite quadratic variation and the divergent $Y^{-2}$ clock force $\Theta \to 0$ on simultaneous extinction; on strict extinction $\Theta \to 1$. Its terminal value is therefore the strict-extinction indicator, proving the formula by optional sampling theorem. For SLE boundary images, $a=2/\kappa$; this compares strict and simultaneous boundary swallowing times for $4<\kappa <8$.

For a three-dimensional Bessel process started at one, $R$ stays strictly positive and satisfies $dR=dW+R^{-1}dt$. The Itô formula makes its reciprocal a local martingale, but $\mathbb{E}[R_t^{-1}]=2\Phi (1/\sqrt{t})-1<1$ for $t>0$. Thus the reciprocal is a strict local martingale. It can deflate the market with constant bank account and stock price $R$ without yielding a true pricing density for the bank account.

For a Bessel process of dimension $\delta$, started at $x>0$, the Itô formula giveson its stochastic interval before hitting zero. Thus the power is a local martingale after localization away from zero. For $\delta \ge 2$ this interval is the whole lifetime; at $\delta =2$ the power is constant. The unqualified assertion fails for a reflecting Bessel process of dimension one, which acquires local time of a semimartingale at zero.

For Brownian motion started at radius $x>0$, the displayed process is a continuous local martingale. It is a bounded true martingale when $0<a\le x$, but a strict local martingale when $a=0$ or $a>x$. In the latter case the boundary is outside the starting point, so stopping does not remove the singularity at the origin. Brownian scaling gives $\mathbb{E}\Vert B_t{\Vert }^{2-d}=O(t^{-(d-2)/2})$; when $a>x$, the stopped expectation tends to $a^{2-d}<x^{2-d}$.

For a Bessel process of dimension $\delta >2$ started at $x>0$, its scale function of a one-dimensional diffusion gives ${\mathbb{P}}_x({\tau }_r<\infty )=(r/x{)}^{\delta -2}$ for $0<r<x$. In particular, ${\inf }_{t\ge 0}{X}_t>0$ almost surely. The minimum is attained because the process tends to infinity; the hitting formula alone already implies positivity of the infimum.

A Bessel process of dimension $d>0$ started away from zero hits zero almost surely exactly when $d<2$. For $d\ne 2$, its scale function of a one-dimensional diffusion is $s(x)=x^{2-d}$; for $d=2$ it is $s(x)=\log x$. The boundary hitting probability from a diffusion scale function then gives the classification by taking the inner boundary to zero and the outer boundary to infinity.

For $dX=dB+a{X}^{-1}dt$, put $\gamma =1-2a\in (0,1)$ and $v_R(x)=(R^{2-\gamma }{x}^{\gamma }-x^2)/(1+2a)$. It is nonnegative on $[0,R]$, vanishes at both endpoints and solves $\frac{1}{2}{v}_R^{\prime \prime }+(a/x)v_R^{\prime }=-1$ on $(0,R)$. Optional stopping at $T_{\epsilon }\wedge T_R$ gives $\mathbb{E}(T_{\epsilon }\wedge T_R)\le v_R(x)$ uniformly as $\epsilon \downarrow 0$. Thus the limiting exit to zero or $R$ occurs in finite time, not merely as an asymptotic approach to zero. Scale probabilities give ${\mathbb{P}}_x(T_0<T_R)=1-(x/R{)}^{\gamma }$; letting $R\to \infty$ gives almost sure finite hitting of zero.

If $X$ is a Bessel process of dimension $d$ started at $x$ and $r>0$, thenis a Bessel process of dimension $d$ started at $rx$. This follows from Brownian scaling in the stochastic differential equation and uniqueness in law.

For $Z_t=e^{B_t+at}$, use the clock $C_t={\int }_0^t{Z}_s^{2}ds$. After the inverse time change, the Dambis-Dubins-Schwarz theorem givesso the time-changed process is a Bessel process of dimension $d=2a+2$. Its lifetime is $C_{\infty }$, which is finite almost surely exactly when $d<2$.

The strong law for Brownian motion gives $B_s/s\to 0$. For every $0<\epsilon <\mu$, eventually $e^{(2\mu -2\epsilon )s}\le e^{2B_s+2\mu s}\le e^{(2\mu +2\epsilon )s}$. Integrating these bounds, taking logarithms and dividing by $t$ squeezes the growth rate between $2\mu -2\epsilon$ and $2\mu +2\epsilon$. Let $\epsilon \downarrow 0$. The corresponding inverse clock grows like $(\log t)/(2\mu )$.

If $X$ is a Bessel process of dimension $d$ and $\alpha >0$, then $X^{\alpha }$, after the clock ${\alpha }^2\int X_s^{2\alpha -2}ds$, is a Bessel process of dimension