is strong, i.e. adapted with respect to the natural ltration of (B t ), and takes values in [; 1). If a = and =, the solution of (1) is X t, and from

Transcription

1 A Survey and Some Generalizations of Bessel Processes Anja Göing-Jaeschke Λ Marc Yor y ' z Abstract Bessel processes play an important role in nancial mathematics because of their strong relation to nancial models like geometric Brownian motion or CIR processes. We are interested in the rst time Bessel processes and more generally, radial OrnsteinUhlenbeck processes hit a given barrier. We give eplicit epressions of the Laplace transforms of rst hitting times by (squared) radial OrnsteinUhlenbeck processes, i. e., CIR processes. As a natural etension we study squared Bessel processes and squared OrnsteinUhlenbeck processes with negative dimensions or negative starting points and derive their properties. Keywords: First hitting times; CIR processes; Bessel processes; radial Ornstein Uhlenbeck processes; Bessel processes with negative dimensions 1 Introduction Bessel processes have come to play a distinguished role in nancial mathematics for at least two reasons, which have a lot to do with the models being usually considered. One of these models is the CoIngersollRoss (CIR) family of diffusions, also known as square-root diffusions, which solve dx t =(a + bx t ) dt + c p jx t j db t ; (1) with X =, a, b R, c> and (B t ) standard Brownian motion. For every given value, equation (1) admits a unique solution; this solution Λ Baloise Insurance Group, Aeschengraben 1, CH-4 Basel, anja.goeingjaeschke@basler.ch; formerly: ETH Zurich, Department of Mathematics, CH-89 Zurich. The author gratefully acknowledges support from RiskLab, Switzerland. y Université Pierre et Marie Curie, Laboratoire de Probabilités, 4 Place Jussieu, F-755 Paris Cede 5. z Wewould like to thank an anonymous referee whose suggestions were decisive to treat the case < in Section. 1

2 is strong, i.e. adapted with respect to the natural ltration of (B t ), and takes values in [; 1). If a = and =, the solution of (1) is X t, and from the comparison theorem for one-dimensional diffusion processes (Revuz and Yor [59], Theorem IX.(3.7)), we deduce X t for a,. Hence, in this case, the absolute value in (1) may be omitted a posteriori. Co, Ingersoll and Ross [1] proposed this family of diffusions for modelling short term interest rates. In recent nancial literature they are often studied from different points of view, or serve as important reference processes, see for instance Chen and Scott [8], Deelstra and Parker [1], Delbaen [13], Dufe and Singleton [17], Frydman [3] and Leblanc [4, 39]; they are used for modeling stochastic volatility, see e. g. Ball [3], Genotte and Marsh [5] and Heston [9]. The other even more fundamental model is geometric Brownian motion, standardly used as a model for stock prices S t = S ep(νt+ ffb t ) ; () with ν; ff R, (B t ) standard Brownian motion. In both cases, X and S can be represented in terms of (squares of) Bessel processes. We recall the denition of (squares of) Bessel processes (Revuz and Yor [59], Chapter XI). Denition 1 For every f and the unique strong solution to the equation p X t = + ft+ jxs j db s (3) is called thesquare of a f-dimensional Bessel process started at and is denoted by BESQ f. Clearly, equation (3) is a particular case of equation (1), with a = f, b =, c =.We call the number f the dimension of BESQ f. This terminology arises from the fact that, in the case f N, a BESQ f process X t can be represented by the square of the Euclidean norm of f-dimensional Brownian motion B t : X t = jb t j. The number ν f= 1 is called the inde of the process BESQ f. For f, BESQ f processes will never reach for t>, and for» f< they reach almost surely. Denition The square root of BESQ f y, f ; y is called the Bessel process of dimension f started aty and is denoted bybes f y. For a study of Bessel processes we refer to Revuz and Yor [59] and Pitman and Yor [53, 54], see also Appendi A. We now show how a general CIR process (1) may be represented in terms of a BESQ process. The relation c X t = e bt Y 4b (1 ebt ) ; (4) where Y denotes a squared Bessel process with dimension f = 4a c, clearly establishes a correspondance between the two families of processes. We remark that

3 this relation is used e. g. in Delbaen and Shirakawa [14] and Szatzschneider [67]. For geometric Brownian motion (), there is the Lamperti relation where ρ ( ν ff ) (u);u Z S t = ρ ( ν ff ) t ff S s ds ; t ; (5) denotes a Bessel process with inde ν ff, with ρ ( ν ff ) () = S, see Lamperti [37] and also Williams [7]. The Lamperti representation (5) has been very useful e. g. in connection with Asian option pricing, see Geman and Yor [4] and Yor [75], Chapter 6. For a multivariate etension of the Lamperti R relation we refer to Jacobsen [3]. It may be helpful to indicate that A t = ff t S s ds admits as its inverse process u! Z u dh ff ρ ( ν ff ) h We now make some remarks about the range of the values of the parameters a; b; c; ff and ν which appear in (1), (), (3), (4) and (5). The signs of c and ff are irrelevant since B (d) = B. We assume c; ff >. In fact, by scaling we may and will restrict ourselves to c = and ff =1.For c =,CIRprocesses dened by (1) coincide with squared radial OrnsteinUhlenbeck processes which are studied in detail in Section, see (7), and in the following we will use either one or the other terminology. The sign of b influences the behaviour of a CIR process X, since in the case a> there eists a unique stationary density of X only if b<; we note that stationary CIR processes also enjoy the ergodic property. In the case a c =, a CIR process starting in stays strictly positive fort>; for a<, a CIR process X t starting in hits with probability p ]; 1[ if b> and almost surely if b». Note that in the case a> the boundary is instantaneously reflecting, whereas in the case a =, as soon as a CIR process X t hits it is etinct, i.e. it remains at. Until now, studies of square-root diffusions dened by (1) have always assumed a and. Under these assumptions, we have seen how to deduce X t for all t from the comparison theorem for one-dimensional diffusion processes and the absolute value in the square root term can be omitted. But it seems to be natural also to consider a< or to start the process in <. In such cases one should be already careful about the formulation of the square root term; for the introduction we consider the case a< and > with the formulation (1). But see also Section 3 for a more general substitution, i. e., we replace ff() c p jj by ~ff() c p ff + +, ff;. The process (X t ), t, dened by (1) with a<, b R, c> and starting in > remains in R + until T = infft >jx t =g. Then, since Y t X T+t satises dy t =(a by t ) dt + c p jy t j d ~ Bt ; 3

4 with ( ~ Bt ) standard Brownian motion, we know that Y t for all t, thus (X t ;t T ) ranges in R and (Y t ) is a CIR process (1) with parameters a >; b R and c>. Consider formula (5) with ν<. We know from Dufresne [18] (see also Pollack and Siegmund [58] and Yor [76], Théor eme 1) ep (B s + νs)ds (d) = 1 Z (ν) ; where Z μ, μ>, is a Gamma variable with inde μ, i.e., P (Z μ dt) = tμ1 e t (μ) R Since S t!, as t! 1, then ρ (ν) 1 u! as u converges to ep (B s + νs)ds, and T ~ R infftjρ (ν) 1 t = g = ep (B s + νs)ds. It seems natural to consider (ρ T ~ +u ;u ). The case ν < 1 corresponds to a Bessel process ρ(ν) with dimension f = (ν +1) <. In Section 3, we will derive and discuss properties of negative-dimensional squared Bessel processes, and also squared radial OrnsteinUhlenbeck processes with f<, with starting points in R. The above discussion shows how rst hitting times of squared radial Ornstein Uhlenbeck processes, i.e. CIR processes, may arise. In Section, we present a survey of the eplicit computations of the Laplace transform of rst hitting times of (squared) radial OrnsteinUhlenbeck processes, by eploiting the relation between radial OrnsteinUhlenbeck processes and Bessel processes. Note that because of the previous discussions about the sign and behaviour of X in the negative dimensional case we will have no difculties in computing the Laplace transforms of rst hitting times T y of a negative dimensional process X starting in in all possible cases, say > >y. From a nancial point of view we are interested in the quantity E a [1 (T<t)(R t k) + ] ; with a radial OrnsteinUhlenbeck process (R t ) starting in a, see Section, and k R +. We remark that the quantity E[1 (T<t)(S t k) + ], epressing values of barrier options with underlying stock price process (S t ) as in (), is investigated in Chesney et al. [9] by considering Laplace transforms with respect to time. The Laplace transform of E a [1 (T<t)(R t k) + ] with respect to time is e fft E a [1 (T<t)(R t k) + ]dt = E a [ T dt : e fft (R t k) + dt] = E a [e fft e ffu (R T+u k) + du] : 4

5 Using the strong Markov property this equals E a [e fft ] E [ e ffu (R u k) + du] ; (6) which gives a clear motivation for our net computations of the Laplace transforms of rst hitting times of radial OrnsteinUhlenbeck processes, as well as to that of their resolvents, here applied to the function r! (r k) + ; as for the resolvent we refer to Remark 6 in Section. To close up this introduction let us mention that we have preferred, in this paper, the use of stochastic arguments, i.e. It^o's formula, Doob's h-transform, time reversal, etc. to that of differential equations arguments which nonetheless play an important role throughout. First hitting times of radial Ornstein Uhlenbeck processes As motivated in the introduction, we are interested in the law of rst hitting times of (squared) radial OrnsteinUhlenbeck processes. For general discussions of rst hitting times of diffusions we refer to Arbib [], Breiman [7], Horowitz [3], Kent [35], Nobile et al. [48], Novikov [49, 5, 51], Pitman and Yor [56, 57], Ricciardi and Sato [6], Rogers [61], Salminen [6], Shepp [64], Siegert [66], Truman and Williams [69] and R Yor [73]. More general discussions T of inverse local times and occupation times 1 (X s»y)ds, whenx is a diffusion and T a particular stopping time, are dealt with in Hawkes and Truman [8], Truman [68] and Truman et al. [7], with particular emphasis on the Ornstein Uhlenbeck case. First we recall the denition of (squared) radial OrnsteinUhlenbeck processes. We will use another notation than in (1) which is related to the notation in Denition 1. Let (W t ) be a one-dimensional Brownian motion, R, f and z. The solution to the equation p = z + (f Z s ) ds + jzs j dw s (7) is unique and strong (Revuz and Yor [59], Chapter IX 3); as in the discussion of equation (1) we deduce. It is called a squared f-dimensional radial OrnsteinUhlenbeck process with parameter and its law onc(r + ; R) is denoted by Q f z. It is a Markov process; hence, the square root of this process is also a Markov process and is called a f-dimensional radial OrnsteinUhlenbeck process with parameter. Denote its law onc(r + ; R) by P f where = p z. The following application of Girsanov's theorem relates P f to P f, the law of a BES f () process, hence obviously Q f z to Q f z, the law of a squared BES f (z) process. 5

6 Proposition 1 For every R and, the following relationship holds: Z P f Ft =ep ρ t ff [Rt ft] Rs ds P f Ft : (8) Proof: The only care needed to justify (8) is that the Girsanov local martingale which appears in (8) is in fact a martingale. But this follows from the fact that both diffusions with laws P f and P f are non-eplosive (see e.g. McKean's presentation of Girsanov's theorem in the case of eplosion [44], Section 3.7). This closes the proof. Remark 1 More generally, we can replace in (8) the restrictions to the ff-elds F t by restrictions to F T (T < 1), for any (F t )-stopping time T. Corollary 1 For f (resp. f<), and >, (R t ;t ) does not visit a.s. (resp. visits a.s.) under P f. Proof: The result for = is well-known, and it etends to any R, with the help of (8). This closes the proof. The following consequence of Proposition 1 will be helpful later to transfer results valid for P f, with >, to P f. Corollary For every R and every stopping time T with respect to (F t ), there is the absolute continuity relationship: P f = FT (T<1) ep((r T ft)) P f FT (9) (T<1) Because R t = p reaches a.s. for f <, we need some care to use It^o's formula and epress (R t ) as the solution of some stochastic equation: For f>1, it is the solution to the equation dr t = f 1 R t R t dt + dw t ; R = = p z: For f = 1 we have with It^o-Tanaka's formula jr t j = jj jr s j ds + ~ Wt + L t ; where (R t )isa P 1 process, ~ Wt R t sgn(r s)dw s is standard Brownian motion and L t is the local time of Brownian motion. For f<1we obtain from (48) R t = + f 1 p.v. ds R s R s ds + ^Wt ; where ( ^Wt ) is standard Brownian motion under P f principal value. and p.v. denotes the 6

7 Our aim is to nd the law of T!y infft j R t = yg ; (1) the rst time a radial OrnsteinUhlenbeck process (R t ) with parameter starting in hits the level y. We distinguish between the cases» y< and» <y.for f<, we have P f (T! < 1) >, that is, the process (R t )mayreach;iff< and >, then P f (T! < 1) = 1, that is, (R t ) reaches almost surely and hence every y almost surely,» y<.for f, we have P f (T!y < 1) = 1 almost surely for every y. Call P f the law ofaf-dimensional radial OrnsteinUhlenbeck process with parameter. The density p f (t) of the rst hitting time of of a radial OrnsteinUhlenbeck process is calculated in Elworthy et al. [1], Corollary 3.1, by using a time reversal argument from T!, that is, for y = the problem is already solved: p f (t) = f ν (ν) ep (ft + (1 coth(t))) Λ» sinh(t) where f<, >, > and ν = 4f 1. We have for» y» T! (law) = T!y + T y! ; 4f ; (11) where T!y and T y! may be assumed to be independent because of the strong Markov property. Hence, with (11) we obtain for the Laplace transform (LT) of T!y E f [ep (μt!y )] = f (μ) f y (μ) ; (1) where, still in the case >, we nd f (μ) = E f [ep (μt! )] = ep(μt) p f (t) dt: Results for the case < will be derived from those for > with the help of Corollary. Our main results are Theorem 1,, 3, 4 and Corollary 3 and 4, where we derive eplicit epressions of the LTs of rst hitting times by radial OrnsteinUhlenbeck processes with arbitrary ; y. But let us rst concentrate on the case =, i.e. Bessel processes..1 First hitting times of Bessel processes. In order to nd an epression of the law and/or the LT of certain rst hitting times of Bessel processes, it is convenient to consider time reversed Bessel processes. More generally, for time reversed diffusion processes, see Appendi B. 7

8 Let (X t ) be a Bessel process with dimension f< starting in >. The time reversed process (X (T!)u ; u» T! ) enjoys the following relationship with ( ^Xu ; u ), a ^f-dimensional Bessel process starting in, with ^f (4 f), see Appendi B: X(T!)u; u» T! (law) = ^Xu ; u» ^L! ; where ^L! = supft j ^Xt = g; which is nite since f ^Xu ;u g is transient (which follows from ^f > ). In particular, as remarked in Getoor and Sharpe [7], Sharpe [63], ^L! under P ^f has the same law as T! under P.We f know (Getoor [6], Pitman and Yor [53]): ^L! (law) = Z^ν ; (13) where Z^ν is a Gamma variable with parameter ^ν ^f 1, i.e.: and hence P (Z^ν dt) = t^ν1 e t (^ν) P f (T! dt) =P ^f (^L! dt) = 1 t(^ν) dt ; (14) t ^ν e t dt : (15) We note that (11) was obtained in Elworthy et al. [1] in an analogous way, additionally using Girsanov's transformation. Since, for y 6=,y<, the rst hitting time T!y is not distributed as ^Ly!, the last eit time of by ( ^Xu ) starting in y, we cannot use the same time reversal argument asabove in order to nd an epression for its LT. T!y has the same law as ^Ly% where ^Ly% denotes the last eit time of by ( ^Xu ) starting in y at the last eit time of y, i.e., we consider the process starting in y conditioned on never visiting y again. We have ^L! (law) = ^L!y + ^Ly% ; (16) where ^L!y and ^Ly% are independent because of the strong Markov property (or path decomposition) at last eit times (for a survey see Millar [45]). Figure 1 illustrates the situation. Our aim is to nd an epression of the LT of^ly% : h i E ^f ep ff ^L y% = hep E ^f h ep E ^f ff ^L! i ff ^L!y i : 8

9 X t y T y 4 T 6 8 t ˆL y ˆL y Figure 1: A simulated path of a BES f process, f<, starting in =5. With (15), and with the Sommerfeld integral representation of the Macdonald function K^ν = K ν,alsoknown as the modied Bessel function of the third kind of order ^ν = ν, (see e.g. Lebedev [38] p. 119, (5.1.5)) K ν (z) = 1 we obtain the LT of^l!y : E ^f z ν h i ep ff ^L!y z t e 4t t ν1 dt ; (17) (ffy)^ν K^ν (ffy) = : ^ν1 (^ν) Thus (see also Getoor [6], Getoor and Sharpe [7], Kent [34] (3.7) and Pitman and Yor [53]) for y<, y 6= : h i h i E f ep ff T!y = E ^f ep ff ^L y% = y ^ν K^ν (ff) K^ν (ffy) = y ν Kν (ff) K ν (ffy) : This formula might also have beenobtained by looking for a function f such that f(x t ) ep( ff t) is a local martingale. Likewise, in the case <<ywe have, see Kent [34] (3.8) and Pitman and 9

10 Yor [53]): E f h i y ν ep ff T Iν (ff)!y = I ν (ffy) ; with I ν the modied Bessel function of the rst kind of order ν. Since I ν (μ") μ ν lim "# " ν = 1 (ν +1) (18) we obtain for =;y> h E f ep ff T!y i ff ν = 1 y ν (ν +1) I ν (ffy) : We remark that for Brownian motion, that is, in the case f =1,wehave T! (law) = T y! +(~ T(y)! ); where > y and ( ~ T(y)! ) denotes an independent copy of T!y. We know the density oft! for Brownian motion ff (t) = p e t ; ßt 3 i. e., we know the density oft (y)! and hence the density oft!y.notethat h i E 1 ep ff T! = e ff t ff (t) dt = e ff :. First hitting times of radial OrnsteinUhlenbeck processes. Our aim is to nd an eplicit epression of the LT (1) of T!y of a radial OrnsteinUhlenbeck process R. We pursue an idea we got by studying Breiman [7] and eploit the relation between CIR and BES processes. Breiman considers an OrnsteinUhlenbeck process Y (u) =e u B( 1 eu )(our 1 corrects a misprint in[7]) with B standard Brownian motion, B(1) =, and gives the LT of the rst hitting time T c for Y from the boundaries ±c where E(e fftc )= ~ D 1 ff (c) ; ~D ff (c) = 1ff= ( ff ) e t t ff1 cosh(ct) dt ; see also Borodin and Salminen [6], II.7...1, p. 49. First hitting times of OrnsteinUhlenbeck processes are also studied by Ricciardi and Sato [6] and Horowitz [3]. 1

11 A f-dimensional radial OrnsteinUhlenbeck process R t with parameter, R =, can be written as e R t = e t t 1 X ; (19) where X is a BES f process, X =.We are interested in the rst time T, R t hits the level >. Although some of our net arguments may work for <, we restrict ourselves for now to the case >. Later, to deal with the case <, we shall use Corollary 4. Analogously to Breiman [7] we assume that D ff is a function such that fd ff (R t )e fft, t g is a martingale with respect to the. Equivalently, fd ff ( 1 ltration of X e t 1 p u+1 X u )(u+1) ff ;u g is a martingale with respect to the ltration of X u. This means via It^o's formula that r fd ff ( p u+1 )(u +1) ff ;r ; u g solves 1 d dr + f1 d r dr + d du =, i. e., it is a space-time harmonic function with respect to 1 d dr + f1 d r dr + d du.further we assume that (D ff (r);r» ρ) is bounded for every ρ. Then fd ff (R t )e fft ;t<t g is bounded. Applying the optional stopping theorem (see e. g. Revuz and Yor [59], II.3) we have E[D f ff (R T )e fft ]=D ff () ; thus E f [e fft ]= D ff() D ff () : Now we are motivated to nd the function D ff eplicitly. First, we derive the space-time harmonic functions for Bessel processes. We know by Widder [71], Theorem , that the most general R + -valued space-time harmonic function h(; t) with R n ;t, with respect to 1 + d dt is of the form h(; t) = Z R n ep ο jοj t m(dο); () where m is a positive measure. We assume h is not identically zero. From () we obtain the general positive space-time harmonic function (~ h(r;t); r;tr+ ) with respect to 1 d dr + n1 d r dr + d dt via ~h(r;t)= Z S n1 h( r;t) ff(d ) = ZRn Z ep(ο r jοj t ) ff(d ) m(dο); (1) S n1 where ff is the uniform probability measure on the unit sphere S n1.weknow the following integral representation for the Bessel function I ν I ν (j j) = j jν (ν +1) Z S n1 ep( ) ff(d ) ; 11

12 R n ;ν= n 1, see Lebedev [38], 5.1, and (5.c) in Pitman and Yor [53], and inserting in (1) we have ~h(r;t)=(ν +1) ν ZR n I ν (jοjr) ep jοj t (jοjr) ν m(dο) : Substituting jοj by u R + we obtain a space time harmonic function h ~ with respect to a Bessel process with inde ν ~h(r;t)=(ν +1) ν I ν (ur)e u t (ur) ν ~m(du) : We believe this formula is true for all dimensions, although a simple proof still eludes us. In fact, we only need to nd a function D ff such that r D ff p (t +1) t +1 ff = h(r;t) ~ =(ν +1) ν I ν (ur)e u t r ν f(u) du with an L 1 -function f : R +! R + ; then fd ff (R t )e fft ;t» T g is a bounded martingale and the optional stopping theorem applies. Let us rst look for D ff such that ν r D ff p (t +1) t +1 ff =(ν +1) I ν (ru) e u t f(u) du ; r equivalently, D ff (y) =(t +1) ff ν (ν +1) y =(t +1) ff ν+1 (ν +1) y ν ν I ν ( p t +1yu) e u t f(u) du I ν ( y) e 4 g p d ;() t +1 with g(v) =e v 4 f(v). The righthandside of () should not depend on t, hence we maychoose g(v) c v ff ν1, with a constant c >, or f(v) c v ff ν1 e v 4 ; (3) ( L 1 ). We see that D ff ( r p t )(t) ff = ~ h(r;t), and certainly also ~ h(r;t+ c) with a constant c, is a space-time harmonic function with respect to 1 f1 d r dr + d dt.thus r D ff p (t +1) t +1 ff ν =(ν +1) I ν (ru) e u (t+ ) 1 f(u) du r 1 d dr +

13 with f in (3) and we nd With (18) we see D ff (y) =(ν +1) D ff () = ν I ν ( y) e ff 4 ν1 d : (4) y e 4 ff 1 d = ff 1 ff ( ff ) : Recalling the denition of the confluent hypergeometric function f where b 6= ; 1; ;::: and (r) =1; (r) j = f(a; b; z) = (r + j) (r) 1X j= (a) j (b) j z j j! ; = r(r +1):::(r + j 1); j =1; ;:::, see e.g. Lebedev [38] (9.9.1), and of the Whittaker's functions M k;μ (z) =z μ+ 1 e 1 z f(μ k + 1 ; μ +1;z); see e.g. Lebedev [38] ( ), nally, we obtain together with the Bateman Manuscript Project [4], 4.16.(), the following theorem. Theorem 1 The LT of the rst time T = infft j R t = g a f-dimensional radial OrnsteinUhlenbeck process R t starting in with parameter hits the level is For <y<we have E(e f ff ν1 ν ( ff fft )= ) ff (ν +1) I ν ( ) e ff 4 ν1 d = (p ) ν+1 e = M 1 ( ff +ν+1); ν ( ) 1 f( ff ;ν+1; ) : T! = T!y + T y! where T!y and T y! are independent because of the strong Markov property. We deduce 13

14 Corollary 3 The LT of the rst time T = infft j R t = g a f-dimensional radial OrnsteinUhlenbeck process R t starting in y, <y<, with parameter hits the level is ν+1 M e (y 1 ) ( ff +ν+1); (y ) ν E f y(e fft )= y = f( ff ;ν+1;y ) f( ff ;ν+1; ) : M 1 ( ff +ν+1); ( ) ν Analogously, we obtain the LT of T in the case < < y. Note that here we have to use the modied Bessel functions of the third kind K ν instead of the modied Bessel functions of the rst kind I ν such that the required uniform integrability assumption is satised. With the confluent hypergeometric function of the second kind ψ ψ(a; b; z) = (1 b) 1) f(a; b; z)+(b z 1b f(1 + a b; b; z); (1 + a b) (a) see e.g. Lebedev [38], (9.1.3), and with Whittaker's functions W k;μ (z) =z μ+ 1 e 1 z ψ(μ k + 1 ;μ +1;z); see e.g. Lebedev [38], ( ), we obtain with Bateman Manuscript Project [4], 4.16.(37), the following theorem. Theorem The LT of the rst time T = infft j R t = g a f-dimensional radial OrnsteinUhlenbeck process R t starting in y, <<y, with parameter hits the level is E f y(e fft )= = y y ν R 1 K ν ( y)e 4 ff ν1 d R 1 K ν ( )e 4 ff ν1 d ν+1 W e (y 1 ) ( ff +ν+1); (y ) ν W 1 ( ff +ν+1); ( ) ν = ψ( ff ;ν+1;y ) ψ( ff ;ν+1; ) : With K ν () ο (ν) for! andν<, we obtain immediately ν+1 ν 14

15 Corollary 4 The LT of the rst time T = infft j R t = g a f-dimensional radial OrnsteinUhlenbeck process R t starting in y with f < with parameter hits is (ν) Ey(e f fft )= ν ff + y ν ff ff =4( p y) ν1 e y =4 ( ff ν) (ν) K ν ( y) e 4 ff ν1 d ( ff ν) (ν) ψ( ff ;ν+1;y ) W 1 ( ff +ν+1); ν (y ) For f we have T = 1 almost surely, i.e., E f y(e fft )=. Remark In the case treated in Corollary 4 we know the density of T (11). by Remark 3 The Laplace transforms of rst hitting times of squared fdimensional Bessel or radial OrnsteinUhlenbeck processes are obtained immediately. Moreover, from the investigation of the behaviour of squared Bessel or radial OrnsteinUhlenbeck processes with negative dimensions or negative starting points, see the introduction and Section 3, we have no difculties in etending the results above to, e. g., the case > >y. Remark 4 In disguised versions the formulae above also appear in Pitman and Yor [53], Chapter 1, and Eisenbaum [19]. We now show how to deduce the laws of rst hitting times in the case <for our previous formulae. Theorem 3 ( < ): Let μ = >. The LT of the rst time T = infft j R t = g a f-dimensional radial OrnsteinUhlenbeck process R t starting in with parameter μ hits the level is μ E f (e fft )= e μ f( ff+fμ μ ; f ; μ ) : Theorem 4 ( < ): Let μ = >. The LT of the rst time T = infft j R t = g a f-dimensional radial OrnsteinUhlenbeck process R t starting in y, <<y,withparameter μ hits the level is ff+fμ ψ( μ Ey(e f fft )=e μ( y ) μ ; f ; μy ) ψ( ff+fμ μ ; f ; μ ) : 15

16 The statements of the <version of Corollary 3 and 4 are left to the reader. Proof of Theorems 3 and 4: From Corollary we know that for any f : R + R +! R, f(r t ;t)isa P f -martingale if and only if f(r t ;t) ep((r t ft)) is a μ P f -martingale. In other terms, a function H : R +! R + is such that: μ P f H(r)e fft is space time harmonic wrt. the P f -family, (5) if and only if, H(r)e μr e (ff+fμ)t is space time harmonic with respect to the -family. Consequently, iff (r)e t is space time harmonic with respect to μ P f, then: H ff (r) =F ff+fμ (r)e μr μ satises (5). Now, Theorems 3 and 4 follow from Theorems 1 and applied with the parameter μ(= ) >. This closes the proof. Remark 5 We mention another approach to obtain results concerning rst hitting times of radial OrnsteinUhlenbeck processes. Consider a f-dimensional radial OrnsteinUhlenbeck process (R t ), starting in >, f <. We are interested in T y = infftjr t = yg, > y. In analogy to the case of BES processes in Section.1 we consider the process (R t ) time reversed, starting from T =infftjr t =g, see Appendi B. Denote the time reversed process by ( ^Rt ). With the same notation as in (16) we are interested in ^L%y,i.e.,inthe process ( ^Rt ) after ^L!y. By using the technique of enlargement of ltration, we can write ( ^Rt ) after ^L!y as a diffusion. For a treatment see e.g. Jeulin [33] and Yor [78], 1. Heuristically spoken, we enlarge the original ltration progressively, so that the last eit time ^L!y becomes a stopping time. Applying Theorem 1.4 in Yor [78] we obtain Proposition For the diffusion process ~ Ru ^R( ^L!y+u) we have ~R u = y + Z u b( ~ Rv )dv Z u s ( ~ Rv ) s(y) s( ~ Rv ) 1 ( ~ R v>y) dv + ~ Wu ; (6) where u, b is the drift and s is the scale function of the transient diffusion ^R. As an illustration consider the process ^R to be a transient BES process ^X, i.e. a BES process with inde ν>, started in. Its scale function may bechosen as s() = ν and we obtain from (6) ~X u ^X( ^L!y+u) = y + Z u 1 (ν + 1) X ~ ν v +(ν 1 )yν ~X v ~X v ν dv + Wu ~ : (7) yν 16

17 For a BES 3 () process ^X (7) reduces to Z u ~X u ^X( ^L = y +!y+u) Hence for a BES 3 () process X we have dv ~X v y + ~ Wu : (X Ly+u y; u ) (law) = (X u ;u ) : In general, the transition density ~p of the diffusion ~ R in (6) is unknown. Note that if it were known, we would obtain the density of the last hitting time of by the process ~ R immediately from the formula P y (~ L dt) = 1 s() ~p t(y; ) dt ; see Borodin and Salminen [6] IV.43, Revuz and Yor [59] VII.(4.16), where ~ R = y and s is the scale function of ~ R with lima# s(a) =1 and s(1) =. Remark 6 As motivated in the end of Section 1 we are interested in the resolvent ofradial OrnsteinUhlenbeck processes. We obtain for the resolvent (see (6)) with f(r) (r k) + E [ e fft f(r t )dt] = f(r) e fft p t (; r) dt dr with R p t the transition density of R t, leaving us with the computation of 1 e fft p t (; y) dt. First, we consider BES f processes and obtain with Appendi A. e fft 1 y ν t y ep + y t where and z 1 = p ff min(; y) z = p ff ma(; y); y y ν I ν dt = yiν (z 1 ) K ν (z ); t see Oberhettinger and Badii [5] (I ). For a treatment of the resolventfor Bessel processes we refer to It^o and McKean [31], and also to De Schepper et al. [11] and Alili and Gruet [1]. Using relation (19) we obtain for a f-dimensional radial OrnsteinUhlenbeck process with parameter e fft y et e t ν y e t ep + e t y 1 e t 1 y ν (u +1) = y 1 (ν ff ) ep +(u +1)y u u I ν ye t e t 1 dt I ν y p u +1 u du: 17

18 3 BESQ processes with negative dimensions and etensions Bessel processes with nonnegative dimension f and starting point are well-studied, e.g. Revuz and Yor [59], Chapter XI. As already pointed out in the introduction it seems to be quite natural also to consider Bessel processes with negative dimensions or negative starting points. Therefore we are motivated to etend the denition of BESQ f processes and to introduce the class of BESQ f processes with arbitrary f; R. Denition 3 The solution to the stochastic differential equation dx t = fdt+ p jx t j dw t ; X = ; (8) where fw t g is a one-dimensional Brownian motion, f R and R, is called the square of a f-dimensional Bessel process, starting in, and is denoted by BESQ f. Moreover, we generalize the denition of a CIR process, i.e., a squared fdimensional radial Ornstein-Uhlenbeck process with parameter in (7), by allowing the starting point and f to be in R. Denition 4 The solution to the stochastic differential equation dx t =(f +X t ) dt + p jx t j dw t ; X = ; (9) where f; ; R and fw t g is a one-dimensional Brownian motion, is called a squared f-dimensional radial OrnsteinUhlenbeck process with parameter. First, we will derive and discuss properties of BESQ f processes (8) with f; R, and as an etension we study CIR processes (9) in Section 3.1. As mentioned in the introduction equation (8) is not the only possible wayof dening BESQ f processes with f R. We will discuss this point after investigating the behaviour of BESQ f processes with f R dened by (8). Equation (8) has a unique strong solution (Revuz and Yor [59], Chapter IX 3). Denote its law by Q f. First, we want toinvestigate the behaviour of a BESQ f process, starting in > with dimension f». In the case f = the process reaches in nite time and stays there. As for the case f <, we deduce from the comparison theorem that is also reached in nite time. Let us consider the behaviour of a BESQ f process fx t g with f < and> after it reached ; we nd: Z T+u p ~X u X T+u = fu+ jxs j dw s ; u ; (3) T where T = infft j X t =g denotes the rst time the process fx t g hits. With the notation fl f we obtain from (3) Z u q Xu ~ = flu+ j Xs ~ j d Ws ~ ; u ; 18

19 X t t Figure : A simulated path of a BES fl process with fl ; =5. X t t Figure 3: A simulated path of a BES fl process with fl<; =5. where Ws ~ (W s+t W T ), that is, after the BESQ fl process fx t g hits, it behaves as a BESQ fl process. Two simulated paths of a BESfl process with fl, respectively fl<, are shown in Figures and 3. From the above discussion we deduce that a BESQ f process with f < and» behaves as [BESQ f ], especially it never becomes positive. For a BESQ f process with dimension f and starting point», we obtain with the same 19

20 argument, that it behaves as a BESQ f process; this means, until it hits for the rst time it behaves as a BESQ f process, and after that it behaves as a BESQ f process. Let us now pursue another way of etending Denition 1 to BESQ f processes with p f R. Instead of a BESQ f process in (8) with diffusion coefcient ff() = jj we consider the process q X t = ft + ffx s + + X s db s ; where ff;, + = ma(; ), = ma(; ), and as before > ; f>. For t» T,wehave p X t = ft + ffxs db s : (31) After T the process evolves as X T+t = ft + and with Y t X T+t we have Y t = ft + Z T+t T q ffx + s + X s db s ; q ffy s + Y + s d ~ Bs : This admits only one solution which is the positive process p Y t = ft + Ys d Bs ~ : (3) Note that the parameter in (3), as well as the parameter ff in (31), can be thought of as coming from a time transformation of a BESQ fl process Z p = flt + Zs db s ; since Z c t = flc t + p c Z c s d μ Bs : After this remark let us now continue to investigate properties of processes dened by (8). An important and well-known property of squared Bessel processes with nonnegative dimensions is their additivity property, see Appendi A.4. The additivity property is no longer true for BESQ f processes with f R arbitrary. Consider the BESQ process Z and the BESQ ~ y process Z ~,where

21 >; ~ fl< with fl, and y». Assuming that the additivity property holds, would yield: Z + Z ~ (law) = Z fl (law) Z = Z fl ; since fl, which cannot be true because of the independence of Z and Z fl. Our aim is to nd the semigroup of a BESQ fl process fx t g with fl,. Our result is the following Proposition 3 The semigroup of a BESQ fl process, fl >,, is given by: Q fl (X t dy) =q fl t (; y)dy ; with q fl t (; y) =q 4+fl t (y; )1 (y>) + (fl +)q 4+fl s (;) q fl ts(; y)1 (y<) ds : Proof: a) We decompose the process (X t ) before and after T, its rst hitting time of ; we obtain E fl [f(x t )] = A + B; where A = E fl [f(x t )1 (t<t)] ; B = E fl [f(x t )1 (T<t)] =E fl = Q fl (T ds) Z 1 [f(x T+(tT ))1 (T<t)] f(y) q fl ts(; y)dy ; since (X (T+u); u ) is distributed as a BESQ fl process. b) We shall compute A with either of the following arguments: (b.1) We use the absolute continuity relationship: Q fl Ft (t<t ) = Xt ν Q 4+fl Ft ; where 4 + fl = (1 + ν), and we note, from formula (49), that: y ν q 4+fl t (; y) =q 4+fl t (y; ) : Thus, we obtain: A = f(y) q 4+fl t (y; ) dy : (33) 1

22 (b.) The following time reversal argument shall conrm formula (33). First, we note E fl [f(x t )1 (t<t)] = E 4+fl [f(x Lt)1 (t<l)] = t q (s) E 4+fl [f(x st )j X s = ] ds ; (34) where q (s) ds Q 4+fl (L ds). Then, standard Markovian computations show that: E 4+fl [f(x st )j X s = ] = Z f(y) q4+fl st (;y) q 4+fl (;) q 4+fl s Moreover, we nd, by comparison of (15) and (5), say, that: t (y; ) dy : (35) q (s) =(+fl) q 4+fl s (;) : (36) Putting together (34), (35) and (36), we obtain: E fl [f(x t )1 (t<t)] = t ( + fl) Z f(y) q 4+fl st (;y) q 4+fl t (y; ) dy ds : Using Fubini's theorem, integrating in (ds), and using (36) with y instead of, we obtain nally: that is, formula (33). A = f(y)q 4+fl t (y; ) dy ; c) The computation of B is done with the formula Q fl (T ds) = q s (s)ds, followed by the use of formula (36). This closes the proof. In order to make the semigroup formula q fl t (; y) more eplicit, we need to compute q 4+fl s (;) q fl ts(; μy) ds ; where μy = y, for y <. Elementary computations yield: with q (fl) t (; y) =k(; y; fl; t)e ff k(; y; fl; t) fl (w +1) m ep fl w m w ff w fl fl +1 jyj fl 1 t fl1 ; dw ; (37)

23 and m fl We epand formula (37) for fl N as follows: mx m q (fl) t (; y) = k(; y; fl; t) e ff k = k(; y; fl; t) e ff ; ff jyj t ; t : k= mx k= m k ff w mk ep w ff dw w 1 (km1) K km1 ( p ff) ; where K ν (z) denotes the Macdonald function with inde ν, see formula (17), taken from Lebedev [38], (5.1.5). As a test on our above description of the law Q fl of a BESQ fl process ( ) t with fl; >, we now make some computations involving linear functionals of this process; see also Revuz and Yor [59] (Eercise (XI.1.34)), and for an improvement of this eercise see Föllmer et al. []. Since (Z T+t;t ) is a BESQ fl process, independent of the past of ()uptot,wehave Q fl» ep( Z u f(u) du) = Q fl Q fl» ep(» ep ( Z u f(u) du) T = t X v f(t + v) du) Q fl [T dt]; with a BESQ fl process (X v) v and a positive Borel function f. In the following, negative dimensional BESQ processes are denoted by ( ) and positive dimensional ones by (X t ). We obtain Q fl» ep( and with the equalities Z u f(u) du) T = t Q 4+fl = Q 4+fl = Q 4+fl = Q 4+fl» ep (» ep (» ep (» ep ( = Q 4+fl» ep( X u f(t u) du) L = t X u f(t u) du) X t = X tu f(t u) du) X t = X u f(tu) du) L = t X u f(u) du) X t = ; 3

24 nally, wehave» ep( Q fl Our aim is to determine and Z u f(u) du) = Q 4+fl» ep ( Q fl» ep( Q fl Q 4+fl» ep(» ep ( X u f(u) du) X t = X v f(t + v) dv) X u f(u) du) X t = X v f(t + v) dv) q (t) dt :(38) more eplicitly. Using the well-known fact that for a function h : R +! R in L 1 with h()» c for all R + and continuous in a neighbourhood of : we deduce that the ratio lim!1 e y h(y)dy = h() ; Q f (ep(x t R t X u f(u) du)) Q f (ep(x t )) = R 1 R e y qt f (; y)q f t (ep( X u f(u) du)jx t = y) dy e y qt f ; (39) (; y) dy R 1 as tends to innity, converges to ep( Q f X u f(u) du)jx t = : Let us consider the numerator and denominator of ratio (39) separately. Via Pitman and Yor [55] (formula (1.h), and Revuz and Yor [59] (Theorem (XI.1.7)) we know Lemma 1 Consider apositive function f : R +! R + all n. Then» ep( X u f(u)du X t) Q f =(ψ f (t)+ψ f (t)) f= ep " R n with f(s)ds < 1 for f f () (f f + f f )(t) (ψ f + ψ f )(t) # ;(4) 4

25 where f f is the unique solution of the Sturm-Liouville equation 1 f f (s) =f(s)f f (s); with s (; 1), f f () = 1, which is positive and non increasing, and f f () is the right derivative in, and Furthermore Q f ep( ψ f (t) f f (t) ds f f (s) : where f f (1) [; 1] is the limit at innity of f f (s). X u f(u)du) = f f (1) ep f= f f () ; (41) From the formula following Corollary (XI.1.3) in Revuz and Yor [59] we know Q f (ep( X t)) = (1 + t) f ep ψ 1+t and together with (4) we obtain Q f (ep( X u f(u) du)jx t =) f= t ep ψ f (t) =! (f f () f f (t) ψ f (t) + 1 t ) : (4) Now we will determine Q f (ep( R 1 X s f(t + s)ds)) more eplicitly. Note that the function f t (s) f f (t + s) f f (t) solves 1 f t (s) =f(t + s)f t (s); and that formula (41) leads to Q f ep X s f(t + s)ds) = f t (1) ep f= f t() = f= ff (1) ep f f (t) ; f f (t) f f (t) : 5

26 Lemma For a positive decreasing function f : R +! R +, such that ep X s f(s)ds < 1; (43) Q f there eists a function ~ f which is on (; 1) the unique solution of with f() ~ = 1 such that ep X u f(t + u)du Q f 1 ~ f (s) =f(s) ~ f(s); = ψ ~f(1) ~f(t)! f= epψ ~f (t) Proof: Concerning verications of condition (43) we refer to McGill [43]. We want to determine functions ff(t) and (t) with ep X u f(t + u)du =((t)) f= ep(ff(t)): (44) Q f Using the Markov property wehave ep X u f(u)dujf t Denote Q f = ep X u f(u)du Q fxt ep h(; t;!) ((t)) f= ep(ff(t)) ep ~f(t)! X u f(t + u)du : X u f(u)du Since Q f ep R 1 X u f(u) dujf t is a martingale, with It^o's formula we seethat the functions ff and have to fulll the following condition in terms of h This @ h(; t)+ h(; t)+f h(; @ ff (t)+f(t)+ff (t) =; ff(t)+ (t) (t) =: ff(t) = (t) (t) ; f(t) (t) = (t) (t) : (t) Consider the function ~ f : R +! R + with ~f(s) () (s) : 6 : :

27 Then ~ f is on (; 1) the unique solution of with ~ f() = 1. We obtain 1 ~ f (t) =f(t) ~ f(t); ff(t) = ~ f (t) ~ f(t) : From (44) we deduce by dominated convergence that lim t!1 (t) = 1, and hence ~ f(1) lim t!1 ~ f(t) =(). Altogether we obtain for (38) with (4) and Lemma Theorem 5 Q fl ep( 1 ψ f (t) Z u f(u)du) = fl + ψ ~ f(1) ~f(t) 1 ( fl +1)! fl= ep fl +1 with f f and ψ f from Lemma 1, and ~ f from Lemma. Eample: Consider f(u) 1 [;a](u) ; a; > : We know (Pitman and Yor [54], p. 43 (.m)) Q 4+fl and hence we have Q fl ep( f f () f f (t) ψ f (t) ep( X u du) X t = t 4+fl = ep sinh(t) t (t coth(t) 1) ; Z a dt) = We investigate more deeply Z a ep( Q 4+fl! (t) Z a t sinh(t) Q fl ep( + X u du) a Q 4+fl Q 4+fl 7 dt ; 4+fl ep t (t coth(t) 1) Z at ep( ep( X v dv) q (t) dt Z a Z a X u du) X t = X u du) X t = q (t) dt: ;

28 where a continuous process with law Q 4+fl is called the Squared Bessel Bridge! (t) from to over [;t], see e.g. Revuz and Yor [59], XI.3. For AF a wehave and Q 4+fl! (t) Q 4+fl X lim a y# Q 4+fl (A) = lim E 4+fl y# Q 4+fl E 4+fl = lim y# = lim E 4+fl y# (X ta [;y]) (X t [;y]) 1A 1 [;y] (X t ) (X t [;y]) 1A E 4+fl (1 [;y] (X t )jf a ) Q 4+fl ψ1 A Q 4+fl X a (X t [;y]) Q 4+fl (X ta [;y]) (X t [;y])! = q4+fl ta (X a ; ) q 4+fl ψ(a; X a ) ; t (; ) where q f t (; y) is the transition density of BESQ f, f>. From Yor [77] (proof of the theorem in.5) we obtain ta (X a ;y) = t (; y) q 4+fl lim y# q 4+fl t t a Formula (.k) in Pitman and Yor [54] gives us Z Q 4+fl ep X a a (t a) X u du = cosh(a)+ Hence we have Q 4+fl! (t) ep( 1 sinh(a) 4+fl (t a) Z a X u du) 4+fl ep X a ep : (t a) t t a = t If in addition (a t) < ß,we obtain Z at Q fl ep( X v dv) and nally, Q fl ep( Z a dt) ep ep@ t ; 1 1+ coth(a) (ta) A : coth(a)+ 1 (ta) cosh(a)+ 1t sinh(a) 4+fl 1 1+ coth(a) (ta) A : coth(a)+ 1 (ta) = cos((a t)) fl ;

29 1 = ( fl +1) + fl +1 " fl + Z a [cosh(a) u + 1 sinh(a)] fl ep ep cos((a t)) fl coth(t) sinh(t) fl + u+ coth(a) coth(a) u +1 dt du # : 3.1 Etension to squared radial OrnsteinUhlenbeck processes. As an etension to BESQ f processes in (8) with f; R we nowinvestigate squared f-dimensional radial OrnsteinUhlenbeck processes, i.e., CIR processes, dened by (9) with f; R. We consider the case f < and > ; for = this corresponds to a BESQ f process with negative dimension f. Wecall Q f the law onc(r + ; R) and, as before, we denoteq f Q f. Via Girsanov transformation we obtain the relationship Q f Ft = ep p jxs j sgn (X s ) dw s jx s j ds Q f Ft : (45) Note that because no eplosion occurs on both sides of this formula, the density is a p true martingale. We may also write the stochastic integral sgn (X s) jx s j dw s in a simpler form, since we have from It^o's formula: R t jx t j = jj + sgn (X s )(fds+ p jx s j dw s )+L t (X) ; where L t (X) is the semimartingale local time of X in. For L t (X) we obtain with Revuz and Yor [59], Corollary VI.1.9, 1 L t (X) =lim "# "» lim 4 "# Hence we have Z 4 t "# 1 [;"] (X s ) dhx; Xi s = lim " 1 [;"] (X s ) ds sgn (X s ) p jx s j dw s = 1 Thus, (45) takes the form Lemma 3 Q f Ft ep jx t jjjf = : jx s j 1 [;"] (X s ) ds jx t jjjf sgn (X s ) ds : sgn (X s ) ds jx s j ds Q f Ft : 9

30 Applying Lemma 3 we obtain the conditional epectation formula qt f (; y) =qt f (; y) ep (jyjjj) Q f ep f sgn(x s ) ds jx s j ds X t = y (46) where qt f denotes the semigroup density iny of a BESQ f process, f<, given by (37). Using the time-space transformation from a BESQ f process (Xt f ) to a squared radial OrnsteinUhlenbeck process ( Xt f ) Xt f = e t X f 1e ; t we also have together with the relationship (46) qt f (; y) =e t q f 1e (; e t y) ; t from which q f t (; y) is obtained since q f t (; y) is known, see (37). Hence we obtain from (46) Theorem 6 We have Z Q f ep f t sgn(x s ) ds = q f 1e (; e t y) ep t t with q f t (; y) given by (37). jx s j ds X t = y (jyjjj) =q f t (; y); 3

31 Appendices A Some properties of Bessel processes A.1 In addition to Denitions 1 and we give eplicit R epressions for Bessel processes. For f > 1 a BES f t process X t satises E[ (ds=x s)] < 1 and is the solution to the equation X t = + f 1 ds X s + W t : (47) For f» 1 the situation is less simple. For f = 1 the role of (47) is played by X t = jw t j = ~ Wt + L t ; where Wt ~ R t sgn(w s)dw s is a standard Brownian motion, and L t is the local time of Brownian motion. For a treatment of local times see e.g. Revuz and Yor [59], Chapter VI. For f<1wehave Z X t = + f 1 t ds p.v. + W t ; (48) X s where the principal value is dened as p.v. ds X s f (L t L t )d and the family of local times (L t ; ) is dened as '(X s )ds = '()L t f1 d for all Borel functions ' : R +! R +, see Bertoin [5]. The decomposition (48) was obtained using the fact that a power of a Bessel process is another Bessel process time-changed: ψ qxν 1=q (t) =X νq ds Xν =p (s) where 1 p + 1 q =1;ν > 1 q, see e.g. Revuz and Yor [59], Proposition (XI.1.11). A. Transition densities. (Squared) Bessel processes are Markov processes and their transition densities are known eplicitly. For f>, the transition density for BESQ f is equal to q f t (; y) = 1 t y ν ep ρ 31 + y t ff! ; I ν p y t ; (49)

32 where t>; >; ν f 1 and I ν is the modied Bessel function of the rst kind of inde ν. For =wehave q f t (;y)=(t) f (f=) 1 y f 1 ep n y o : (5) t For the case f =, the semi-group of BESQ is equal to n Q t (; ) = ep o " + Qt ~ (; ); (51) t where " is the Dirac measure in and Qt ~ (; ) has the density qt (; y) = 1 y 1 ff epρ + y p y I 1 : t t t The transition density for BES f is obtained from (49), (5) resp. (51) and is equal to p f t (; y) = 1 y ν ff t y ep ρ + y y I ν ; t t with t>; >, and p f t (;y)= ν t (ν+1) (ν +1) 1 y ν+1 ep for f>, and the semi-group for BES is equal to P t (; ) = ep ρ t ff " + ~ Pt (; ) ; ρ y where " is the Dirac measure in and Pt ~ (; ) has the density ρ ff p t (; y) = t ep + y y I 1 : t t A.3 Scaling property. BES f processes have the Brownian scaling property, i.e. if X is a BES f,thenthe process c 1 X c t is a BES f =c for any c>. BESQf processes have the following scaling property: if X is a BESQ f, then the process c 1 X ct is a BES f =c. A.4 Additivity property of squared Bessel processes. An important and well-known property of BESQ f processes with f isthe following additivity property. t ff ; 3

33 Theorem 7 (Shiga and Watanabe [65]) For every f; f and ; : Q f Λ Q f = Qf+f + ; where Q f Λ Q f denotes the convolution of Qf and Q f. For a proof see Shiga and Watanabe [65] or Revuz and Yor[59], Theorem (XI.1.). B Time reversal Consider a transient diffusion X, living on R +, with X =. Denote its last eit time of a by L a = supfu j X u = ag, where sup ; =.For a ed, L a is nite almost surely, and for <a, L a > almost surely. We consider the time reversed process ~ X, where ~X t (!) ρ XLa(!)t(!) ; if <t<l a (!) if L a (!)» t or L a (!) =1 ; (5) denotes the `cemetery`, and ~ X (!) = X La(!)(!), if < L a (!) < 1, else ~ X (!) =@. As a consequence we have the equality fx u ;u» L a g = f ~ XTu; u» T g ; (53) where we assume X starting in and T inffuj ~ Xu =g. We remark that a diffusion may be reversed at cooptional times, a more general class than last eit times, see Nagasawa [46, 47], Revuz and Yor [59], Chapter VII.4. However, for our purposes here it is reasonable to restrict ourselves to last eit times. In the following, we state a general time reversal result (see Nagasawa [46, 47], Sharpe [63], Getoor and Sharpe [7], Revuz and Yor [59]). Denote the semi-group of X by (P t ), the potential kernel of X by U, and let ~F t = ff( ~ Xs ;s» t) be the natural ltration of ~ X. Theorem 8 We assume that there is a probability measure μ such that the potential ν = μu isaradon measure. Further we assume that there isasecond semi-group on R +, denoted by ( ^Pt ), such that ^Pt f is right-continuous in t for every continuous function f with compact support on R + and such that the resolvents (U p ) and ( ^Up ) are in duality with respect to ν, i.e., Z U p f gdν = Z f ^Up gdν (54) for every p> and every positive Borel functions f and g. Equality (54) can also be written as hu p f;gi ν = hf; ^Up gi ν : 33

34 Then under P μ, the process ~ X is a Markov process with respect to ( ~ Ft ) with transition semi-group ( ^Pt ) and we have the duality hp f f;gi ν = hf; ^Pf gi ν ; for any positive Borel function f on R + where P f f() = R 1 f(t)p t f()dt. We will obtain eplicit formulae for time-reversed diffusions via Doob's h- transform. B.1 Doob's h transform. Consider a one-dimensional diffusion X, with sample space ; F c 1), where I [1; 1] and := f! :[; 1) 7! I [f@gg, F c 1 := fff!(t)j t g. Denition 5 A non-negative measurable function h : I 7! R [f1g is called ff-ecessive for X, ff, if a) e fft E (h(x t ))» h(), for all I; t, b) e fft E (h(x t ))! h(); for all I as t #. A -ecessive function is simply called ecessive. Let h be an ff-ecessive function for a diffusion X. The life time of a path! is dened by (!) := infft j! t We construct a new probability measure P h by P h fft h(!(t)) Ft = e P j h() Ft ; (55) for t< and I. The process under the new measure P h is a regular diffusion and is called Doob's h-transform of X. Asfor Doob's h-transform we refer to Doob [16] and Dellacherie and Meyer [15]; in presenting Doob's h-transform we followed Borodin and Salminen [6]. As an application let us consider a transient diffusion X under probability measure P with scale function s. Doob's h-transform of X with the ecessive function h s, that is, the process under the new measure P h, is a process which reaches almost surely. Wehave 1 s (Xt^T ) P j Ft = 1 P h Ft : (56) s () Note that we obtain and hence, s (X t ) s() P j Ft 1 (t<t) P h Ft ; Q h t f() 1 s() E [(fs)(x t )] 1 s() P t(fs)() ; (57) 34

35 is the semigroup of the process under P h killed when it reaches. In other words, we have the following time reversal result: Doob's h-transform of the transient process under P with h = s is the process under P h killed at T ; this is the process ~ X in our former notation. From formula (56) we can obtain eplicit formulae of the diffusion processes via Girsanov's theorem. Assume, a process (X t ) under P has the form X t = + (X u ) du + ff(x u ) db u ; then via Girsanov's theorem we obtain with (56) that the process under P h has the form X t = + + ff s (X u ) du + s ff(x u ) d ^Bu ; t» T : (58) References [1] Alili, L. and Gruet, J.C. An eplanation of a generalized Bougerol identity. In Eponential Functionals and Principal Values related to Brownian motion. Biblioteca de la Revista Matematica Iberoamericana, [] Arbib, M.A. Hitting and martingale characterizations of one-dimensional diffusions. Z. Wahrsch. Verw. Gebiete, 4:347, [3] Ball, C.A. A review of stochastic volatility models with application to option pricing. Financial Markets, Institutions and Instruments, (5):55 71, [4] Bateman Manuscript Project, A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi. Tables of Integral Transforms. Vol. I. McGraw-Hill Book Company, Inc., [5] Bertoin, J. Ecursions of a BES (d) and its drift term ( <d<1). Probab. Theory Related Fields, 84():315, 199. [6] Borodin, A.N. and Salminen, P. Handbook of Brownian Motion Facts and Formulae. Probability and Its Applications. Birkhäuser Verlag Basel Boston Berlin, [7] Breiman, L. First eit times from a square root boundary. Fifth Berkeley Symposium, ():916, [8] Chen, R. and Scott, L. Pricing interest rate options in a two-factor Co IngersollRoss model of the term structure. The Review of Financial Studies, 5(4):613636,