Bessel processes

A family of continuous Markov processes (cf. Markov process) $ ( R _ {t} , t \geq 0 ) $ taking values in $ \mathbf R _ {+} $, parametrized by their dimension $ \delta $.

When $ \delta = d $ is an integer, $ ( R _ {t} ,t \geq 0 ) $ may be represented as the Euclidean norm of Brownian motion in $ \mathbf R ^ {d} $. Let $ Q _ {x} ^ {d} $ be the law of the square, starting from $ x \geq 0 $, of such a process $ ( R _ {t} ,t \geq 0 ) $, considered as a random variable taking values in $ \Omega = C ( \mathbf R _ {+} , \mathbf R _ {+} ) $. This law is infinitely divisible (cf. [a6] and Infinitely-divisible distribution). Hence, there exists a unique family $ ( Q _ {x} ^ \delta ;x \geq 0, \delta \geq 0 ) $ of laws on $ \Omega $ such that

$$ \tag{a1 } Q _ {x} ^ \delta * Q _ {x ^ \prime } ^ {\delta ^ \prime } = Q _ {x + x ^ \prime } ^ {\delta + \delta ^ \prime } \textrm{ for all } \delta, \delta ^ \prime ,x,x ^ \prime \geq 0 $$

( $ * $ indicates the convolution of probabilities on $ \Omega $), which coincides with the family $ ( Q _ {x} ^ {d} ,x \geq 0 ) $, for integer dimensions $ d $.

The process of coordinates $ ( X _ {t} ,t \geq 0 ) $ on $ \Omega $, under $ Q _ {x} ^ \delta $, satisfies the equation

$$ \tag{a2 } X _ {t} = x + 2 \int\limits _ { 0 } ^ { t } {\sqrt {X _ {s} } } {d \beta _ {s} } + \delta t, \quad t \geq 0, $$

with $ ( \beta _ {s} ,s \geq 0 ) $ a one-dimensional Brownian motion. Equation (a2) admits a unique strong solution, with values in $ \mathbf R _ {+} $. Call its square root a $ \delta $- dimensional Bessel process.

Bessel processes also appear naturally in the Lamperti representation of the process $ ( { \mathop{\rm exp} } ( B _ {t} + \nu t ) ,t \geq 0 ) $, where $ \nu \in \mathbf R $ and $ ( B _ {t} ,t \geq 0 ) $ denotes a one-dimensional Brownian motion. This representation is:

$$ \tag{a3 } { \mathop{\rm exp} } ( B _ {t} + \nu t ) = R _ {\int\limits _ { 0 } ^ { t } { { \mathop{\rm exp} } ( 2 ( B _ {s} + \nu s ) ) } {ds } } , t \geq 0, $$

where $ R $ is a $ \delta = 2 ( 1 + \nu ) $- dimensional Bessel process. This representation (a3) has a number of consequences, among which absolute continuity properties of the laws $ Q _ {x} ^ \delta $ as $ \delta $ varies and $ x >0 $ is fixed, and also the fact that a power of a Bessel process is another Bessel process, up to a time-change.

Special representations of Bessel processes of dimensions one and three, respectively, have been obtained by P. Lévy, as $ ( S _ {t} - B _ {t} ,t \geq 0 ) $, and by J. Pitman as $ ( 2S _ {t} - B _ {t} ,t \geq 0 ) $, where $ S _ {t} = \sup _ {s \leq t } B _ {s} $, and $ ( B _ {t} ,t \geq 0 ) $ is a one-dimensional Brownian motion.

Finally, the laws of the local times of $ ( B _ {t} ,t \geq 0 ) $ considered up to first hitting times, or inverse local times, can be expressed in terms of $ Q _ {0} ^ {2} $ and $ Q _ {x} ^ {0} $, respectively: this is the content of the celebrated Ray–Knight theorems (1963; [a1], [a5]) on Brownian local times. These theorems have been extended to a large class of processes, including real-valued diffusions.

References

[a1]	F.B. Knight, "Random walks and a sojourn density process of Brownian motion" Trans. Amer. Math. Soc. , 107 (1963) pp. 56–86
[a2]	J.W. Pitman, "One-dimensional Brownian motion and the three-dimensional Bessel process" Adv. Applied Probab. , 7 (1975) pp. 511–526
[a3]	J.W. Pitman, M. Yor, "Bessel processes and infinitely divisible laws" D. Williams (ed.) , Stochastic Integrals , Lecture Notes in Mathematics , 851 , Springer (1981)
[a4]	J.W. Pitman, M. Yor, "A decomposition of Bessel bridges" Z. Wahrscheinlichkeitsth. verw. Gebiete , 59 (1982) pp. 425–457
[a5]	D.B. Ray, "Sojourn times of a diffusion process" Ill. J. Math. , 7 (1963) pp. 615–630
[a6]	T. Shiga, S. Watanabe, "Bessel diffusions as a one-parameter family of one-dimensional diffusion processes" Z. Wahrscheinlichkeitsth. verw. Gebiete , 27 (1973) pp. 37–46
[a7]	D. Revuz, M. Yor, "Continuous martingales and Brownian motion" , Springer (1994) (Edition: Second)