Contents Introduction Main properties of Bessel processes : : : : : : : : : : : : : : : First hitting times of radial Ornstein{Uhlenbeck processes 7 B

Transcription

1 Some Generalizations of Bessel Processes Anja Going, ETH Zurich Risklab Report April 997

2 Contents Introduction Main properties of Bessel processes : : : : : : : : : : : : : : : First hitting times of radial Ornstein{Uhlenbeck processes 7 BESQ processes with negative dimensions and etensions 4 3 Time reversal 3. Doob's h-transform : : : : : : : : : : : : : : : : : : : : : : : : 5 3. Checking a time reversal theorem in Elworthy{Li{Yor : : : : The three-parameters-family of processes with law P ; : : : : 3 Bibliography 38 i

3 Introduction Bessel processes are a one-parameter family of diusion processes that appear in many nancial problems and have remarkable properties. Following this introduction we recall the denition of Bessel processes and their properties in detail (see I. { I.6). One important property of Bessel processes is, that the transition densities are known eplicitly (see I.); apart from the case of Bessel processes they are essentially only known for Ornstein{Uhlenbeck processes and Brownian motion with drift. Further important properties are the scaling property (see I.4) and the additivity property of squared Bessel processes (see I.3), which allow us to reduce many problems to simpler ones. As for the importance of Bessel processes in nancial mathematics, let us rst mention their contribution to the problem of pricing Asian options with arithmetic asset average. Using the fact that the eponential of Brownian motion with drift, i.e. geometric Brownian motion, is a time-changed Bessel process (see representation (.4) in I.4), all moments of the arithmetic asset average can be calculated and an epression of the Laplace transform of the Asian option price is obtained, see Geman{Yor [7]. Furthermore, for the Co{Ingersoll{Ross (CIR) model for interest rates [4], see (.6), Bessel processes are playing an important role on whichwe will concentrate here. The CIR processes are time-space-transformed squared Bessel processes and can also be transformed by a time change to squared radial Ornstein-Uhlenbeck processes (for details see I.5 and I.6). Therefore, in order to obtain results for the CIR processes, we will consider (squared) Bessel processes respectively (squared) radial Ornstein-Uhlenbeck processes. In nancial mathematics, diusion processes are often studied until they reach a certain level for the rst time, as for instance in the case of barrier options (see e.g. Geman{Yor [8], Chesney{Geman{Jeanblanc-Picque{Yor [3]). Also for statistical reasons, rst hitting times are very interesting, e.g. if we

4 want to condition a process on never hitting a given barrier. In order to obtain results about rst hitting times of the CIR process, in the rst chapter we are interested in (the law of) the rst time a squared radial Ornstein{Uhlenbeck process hits an arbitrary level, especially level zero. In Chapter we introduce Bessel processes with negative dimensions. For instance, as can be seen from representation (.4), these processes arise quite naturally when the eponential of Brownian motion with a negative drift is considered. We study their properties and etend the results to a wider class of processes. In the last chapter we more deeply investigate time reversal which was an important tool in the previous chapters. We review general results for timereversed diusions and as an application we check a time reversal result for radial Ornstein{Uhlenbeck processes by Elworthy{Li{Yor [6]. Furthermore, we introduce a three-parameters-family of processes. Acknowledgements: I take great pleasure in thanking Professor Marc Yor for introducing me to Bessel process theory and guiding me through the literature and various technical diculties. Main properties of Bessel processes Now we give the denition of Bessel processes and state some of their properties. As for the study of Bessel processes we refer to Revuz{Yor [3] and Pitman{Yor [7, 9]. I. For every and, the solution to the equation q X t = + t+ jx s j dw s is unique and strong. In the case =; =,the solution X t is identically zero and applying the comparison theorem we conclude X t for all. Denition (BESQ ) For every and the unique strong solution to the equation q X t = + t+ X s dw s is called the square of a -dimensional Bessel process started at and is denoted by BESQ ().

5 Denote the law of BESQ () onc(ir + ; IR) by Q.We call the number the dimension of BESQ. This notation arises from the fact that a BESQ process X t can be represented by the square of the Euklidian norm of -dimensional Brownian motion B t : X t = jb t j. The number =, is called the inde of the process BESQ. Denition (BES ) The square root of BESQ (a ), ; a is called the Bessel process of dimension started at a and is denoted by BES (a). Denote the law ofbes (a) by P a. In the case BES (a), a >, will never reach. For > a BES (a) process satises E[ R t (ds=z s )] < and is the solution to the equation = a +, Z s ds + W t : For the situation is less simple. For = we have with It^o-Tanaka's formula = jw t j = ~W t + L t ; where ~W t 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{ Yor [3], Chapter VI. For <we have = a +, where the principal value is dened as p.v. ds Z s p.v. ds Z s + W t ; (.), (L t, L t )d and the family of local times (L t ; ) is dened as '(Z s )ds = '()L t, d for all Borel functions ' :IR +! IR +, see Yor [4], Chapter. The decomposition (.) was obtained using the fact that a power of a Bessel process is another Bessel process time-changed, see (.5). I. Transition densities Bessel (squared) processes are Markov processes and their transition densities are known eplicitly. For >, the transition density for BESQ is equal to q t (; y) = y ep, t + y p! y I ; (.) t t 3

6 where t>; >;, and I is the modied Bessel function of the rst kind of inde. For =we have q t (;y)=(t),,(=), y (=), ep, y : (.3) t The transition density for BES is obtained from (.) resp. (.3) and is equal to p t (; y) = y ) y ep (, t + y y I ; t t with t>; >, and p t (;y)=, t,(+),( +), y + ep (, y I.3 Additivity property of Bessel squared processes An important and well-known property of BESQ processes with is the following additivity property. Theorem (Shiga{Watanabe [35]) For every ; and ; : Q Q = Q+ + ; where Q Q denotes the convolution of Q and Q. For a proof see Shiga{Watanabe [35] or Revuz{Yor[3], p. 4. I.4 Scaling property and representations BES processes have the Brownian scaling property, i.e. if X is a BES (), then the process c, X c t is a BES (=c) for any c>. BESQ processes have the following scaling property: if X is a BESQ (), then the process c, X ct is a BES (=c). An important result by Lamperti [9], see also Williams [38], is that the eponential of Brownian motion with drift can be represented as a timechanged Bessel process: ep(b t + t)=x () ep (B s + s) ds ; t ; (.4) where (X () (u);u ) is a Bessel process with inde. We remark that because of the Brownian scaling property this result can be etended to a representation of ep(ab t + t). There is also a representation of ep(ab t + 4 t ) :

7 t) in terms of BESQ processes (see Geman{Yor [7]), and in terms of arbitrary powers of BES processes. This result is closely related to the fact that a power of a Bessel process is another Bessel process time-changed q=q (t) =Z q ds Z =p (s)! ; (.5) where + =; >,, see e.g. Revuz{Yor [3], Proposition.(.). p q q I.5 Transformation: CIR process { squared Bessel process Co, Ingersoll and Ross [4] have proposed an interest rate model in which the process for the short term interest rate is the solution to the equation q X t = + (a + bx s ) ds + X s dw s ; (.6) for t ; a ; >. In the nance literature this process is known as the Co{Ingersoll{Ross (CIR) process. For a treatment of interest rate models including the CIR model we refer to Lamberton{Lapeyre [8], 6. We remark that this process is also used in a population growth model, see Karlin{Taylor [5], p The CIR process is a space-time transformed BESQ process, more eplicitly: A BESQ (y) process Y can be transformed to the CIR process X by where 4a=. X t = e bt Y 4b (, e,bt ) I.6 Transformation: CIR process { squared radial Ornstein{Uhlenbeck process A squared radial Ornstein{Uhlenbeck process is the solution to the equation q Y t = y + (, Y s ) ds + Y s dw s for t ;. Radial Ornstein{Uhlenbeck processes are studied in detail in the rst chapter, see (.). A squared radial Ornstein{Uhlenbeck process can be transformed by the time transformation g(t) = t=4 to the CIR process X, where 4a=,,b=, with the notation in (.6). The transformations in I.5 and I.6 show the strong relation between CIR processes and squared Bessel respectively squared radial Ornstein{Uhlenbeck 5! ;

8 processes. In Chapter we are considering rst hitting times of squared radial Ornstein{Uhlenbeck processes, and by means of the transformation in I.6 we see immediately that, once the law of the rst hitting time of a squared radial Ornstein{Uhlenbeck process is found, we know the law of the rst hitting time of the CIR process. 6

9 Chapter First hitting times of radial Ornstein{Uhlenbeck processes As motivated in the introduction (see I.6) we are interested in nding the law of rst hitting times of squared radial Ornstein{Uhlenbeck processes. As for the subject of rst hitting times we refer to Breiman [], Novikov [4, 5, 6], Shepp [34] and Yor [39]. First we recall the denition of (squared) radial Ornstein{Uhlenbeck processes. Let fw t g be a one-dimensional Brownian motion, IR, and z. The solution to the equation q = z + (, Z s ) ds + jz s j dw s is unique and strong (see Revuz{Yor [3] Chapter IX 3). Since in the case =;z =, the solution is, we deduce from the comparison theorem for all, and hence the absolute value can be omitted; the solution of q = z + (, Z s ) ds + Z s dw s (.) is called a squared -dimensional radial Ornstein{Uhlenbeck process with parameter,. It is a Markov process; hence, the square root of this process is also a Markov process and is called a -dimensional radial Ornstein{ Uhlenbeck process with parameter,. For and z >, it almost surely does not hit zero. For >itisthesolution to the equation dr t =,!, R t dt + dw t ; R = = p z: R t 7

10 Our aim is to nd the law of T y = infft j R t = yg ; (.) the rst time a radial Ornstein{Uhlenbeck process fr t g with parameter, starting in > hits the level y,<y<.for <, wehave, P (T < ) >, that is, the process fr t g may reach ; if < and >, then, P (T < ) =, that is, fr t g reaches a.s. and hence every y a.s., <y<. We remark, that since: T y = infft j R t = yg = infft j R t = y g ; once we found the law of (.) we have solved our original problem: nding the law of the rst time a squared radial Ornstein{Uhlenbeck process with < starting in a (= ) hits b (= y ), <b<a. Call, P the law ofa-dimensional radial Ornstein{Uhlenbeck process with parameter,. The density, p (t) of the rst hitting time of of a radial Ornstein{Uhlenbeck process is calculated in Elworthy{Li{Yor [6] (see Cor. 3.) by using a time reversal argument, that is, for y = the problem is solved: h i " #, p (t) =,,() ep (t + 4, (, coth(t))) ; (.3) sinh(t) where <, >, > and = 4,,. In the following, let T!y denote the rst time a radial Ornstein{Uhlenbeck process with dimension <, starting in >, hits y, y <. Why calculating the density of T!y is more complicated than calculating the density of T! will be eplained later. We have T! (law) = T!y + T y! ; where T!y and T y! are independent because of the strong Markov property. Hence, we have for the Laplace transforms (LTs), E [ep (,T! )] =, E [ep (,T!y )], E y [ep (,T y! )] ; or equivalently,, E [ep (,T!y )] = () y () ; (.4) 8

11 where () =, E [ep (,T! )] = ep(,t), p (t) dt: We are interested in giving an eplicit epression of the LT in (.4). So far, we are not able to do this in general, but we can nd eplicit epressions of the LT in (.4) for some eamples in the case =, that is for Bessel processes. For that it is convenient to consider Bessel processes time-reversed. We refer to Chapter 3 where time reversal is studied in detail. Let fr t g be a Bessel process with dimension <. Then the time reversed process ^R u R T,u ; u ; is a ^-dimensional Bessel process starting in, with ^ (4, ). The process ( ^R u ) is transient since ^ >. We know by the time reversal theorem (3.7) that: where (R T,u; u T ) (law) = ^Ru ; u ^L ; (.5) ^L = supft j ^R t = g is the last eit time of by the process f ^R t g. In particular, as remarked in Getoor{Sharpe [9], Sharpe [33], ^L under P ^ has the same law as T under P. We know (see Getoor [], Pitman{Yor [7]): ^L (law) = Z^ ; (.6) where Z^ is a Gamma variable with parameter ^ ^,, i.e.: and hence t^, e,t P (Z^ dt) = dt ; (.7),(^) P (T dt) =P ^ (^L dt) = t,(^)!^ e, t dt : (.8) t Let ^L! resp. ^L!y denote the last eit time of resp. y by a^-dimensional Bessel process ( ^R u ) starting in. With ^L y% we denote the last eit time of by (^R u ) starting in y at the last eit time of y, i.e. we consider the process 9

12 starting in y but never visiting y again; distinguish carefully between ^L y% and ^L y!, the last eit time of by ( ^R u ) starting in y. Then we have ^L! (law) = ^L!y + ^L y% ; where ^L!y and ^L y% are independent because of the strong Markov property of last eit times (for a survey see Millar []). Since ^L y% has the same law as T!y, our aim is to nd an eplicit epression of the LT of ^L y% : h i ep, ^L! h E ^ ep, ^L y% i = E ^ E ^ h ep, ^L!y i : Now we can eplain the dierence between calculating the density of T! and the density of T!y for y 6= of a Bessel process. The density of T! is obtained by considering the time reversed process started in and using (law) (.6) and that T! = ^L!. The density of T! of a radial Ornstein{ Uhlenbeck process (.3) was obtained in a similar way, additionally using a Girsanov transformation. As for the density oft!y we cannot use the same time reversal argument since we are interested in the time reversed process started at the last eit time of y, and this is the time reversed process started in y conditioned on never hitting y again. With (.8) and with an integral representation of K^, i.e. the modied Bessel function of the second kind of order ^, (see e.g. Lebedev [] p.9, (5..5)) we obtain the LT of ^L!y : h E ^ ep, i ^L (y)^ K^ (y)!y = : (.9) ^,,(^) Thus (see also Getoor [], Getoor{Sharpe [9], Kent [6]): E h ep, T y i h i = E ^ ep, ^L y% =!^ K^ () y K^ (y) : Now we want to nd a tractable form of K^ ()=K^ (y) to obtain the density eplicitly.we cannot solve this problem in general but we have found eplicit forms in some special cases. a) First, we consider Brownian motion, that is, the case = (equivalently ^ =3or^ = ). In this case we have ^L! (law) = T! (law) = T y! +(~T (,y)! );

13 where ( ~T (,y)! ) denotes an independent copy. Moreover, we know the density of T! for Brownian motion: q (t) = p e, t ; (.) t 3 that is, we know the density of T (,y)! and therefore the density of ^L y%. Note that E h ep, T! i = e, t q (t) dt = e, : (.) b) In the cases =,;,3;,5;:::, or equivalently ^ = 3; 5; 7 ;:::, we need to look precisely at We know where K n+ () (y) ; K n+ n IN: r K n+ (z) = P n(z)e,z z,(n+ ) ; (.) P n (z) = nx k= (n + k)! z n,k (n, k)! k k! ; n IN ; see Ismail{Kelker [], p. 8. From (.) we deduce K n+ () K n+ (y) = y Hence, the LT of ^L y% in is h i E ^ ep, ^L y% = in the case ^ = n +. n+ e,(,y) P n() P n (y) :!^ K^ () y K^ (y) = P n() e,(,y) P n (y) (.3) Let us look at the case n =. For the right term in (.3) we have the following equalities: e,(,y) P () P (y),(,y) + = e (.4) y + = y + e,(,y) + y + e,(,y) (.5) =! (y +), e,(,y) + y y + y + e,(,y) (.6) = y e,(,y) {z } A +, y y + e,(,y) {z } B : (.7)

14 We know that A is the LT in of q y,y(t), see (.) and (.). As for the second term B we have B =, y e,(,y) =, y which is the LT in of, y And since ^q(t) y q,y(t)+, y ep[,( + y) u] du (.8) e,u e,(yu+,y) du; (.9) e,u q (yu+,y) (t) du: is a conve combination of densities we have that e,(,y) P () P (y) e,u q (yu+,y) (t) du is the LT in of ^q(t). As mentioned, so far we are not able to invert the LT of T!y of a - dimensional radial Ornstein{Uhlenbeck process R with < in general. But considering the time reversed radial Ornstein{Uhlenbeck process ^R started in, we can write the process after ^L!y, the last eit of y, as a diusion, that is, as the solution of a stochastic dierential equation. This we obtain by using a specic technique, the \enlargement of ltration\. For a treatment see e.g. Jeulin [4] and Yor [4]. Heuristically spoken, we enlarge the original ltration progressively, so that the last eit time ^L!y becomes a stopping time. Applying Theorem.4 in Yor [4] we obtain Z u ~R ^R ( ^L = y +!y b( ~R +u) v )dv, Z u s ( ~R v ) s(y), s( ~R v ) ( ~ R v>y) dv + ~W u ; (.) where u, b is the drift and s is the scale function of the transient diusion ^R. Thus we have written the process after ^L!y, that is, the time reversed process started in y conditioned on never hitting y again, as a diusion ~ R. As an illustration consider the process ^R to be a transient BES process ^X, i.e. a BES process with inde >, started in. Its scale function may be chosen as s() =,, and we obtain from (.) ~X u ^X ( ^L!y +u) = y + Z u ( + ) ~X v +(, )y dv + ~W u : (.) ~X v ~X v, y

15 For a BES 3 () process ^X (.) reduces to ~X u ^X ( ^L!y +u) = y + Z u dv ~X v, y + ~W u : Hence for a BES 3 () process X we have (X Ly+u, y; u ) (law) = (X u ; u ) : In general, the transition density ~p of the diusion ~R in (.) 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) =, s() ~p t(y; ) dt ; (.) see Borodin{Salminen [] IV.43, Revuz{Yor [3] VII.(4.6), where R ~ and s is the scale function of R ~ with lim a# s(a) =, and s() =. = y 3

16 Chapter BESQ processes with negative dimensions and etensions Bessel processes with nonnegative dimension and starting point are well-studied, see Introduction I.-I.4 and e.g. Revuz{Yor [3], Chapter XI. It seems to be quite natural also to consider Bessel processes with nonnegative dimension and negative starting point. As will be developed later this case is strongly related to Bessel processes with negative dimensions. Therefore we are motivated to etend the denition of BESQ processes (see I.) to BESQ processes with arbitrary ; IR. Denition 3 The solution to the stochastic dierential equation q dx t = dt+ jx t j dw t ; X = ; (.) where fw t g is a one-dimensional Brownian motion, IR and IR, is called the square of a -dimensional Bessel process, starting in, and is denoted by BESQ. Equation (.) has a unique strong solution (see Revuz{Yor [3], Chapter IX 3). Denote its law by Q. First, we want to investigate the behaviour of a BESQ process, starting in > with dimension. In the case = the process reaches in nite time and stays there. As for the case <, we deduce from the comparison theorem that this process is smaller than the process with =, hence, that is also reached in nite time. Let us consider the behaviour of a BESQ process fx t g with <and > after it reached ; we nd: Z T +u q ~X u X T +u = u+ jx s j dw s ; u ; (.) T 4

17 where T denotes the rst time the process fx t g hits T = infft j X t =g : With the notation, we obtain from (.) Z u q, ~X u = u+ j ~X s j d ~W s ; u ; where ~W s,(w s+t, W T ), that is, after the BESQ,, it behaves as a {BESQ, >. process fx t g hits From the above discussion we deduce that a BESQ process with < and behaves as a {BESQ,,, especially it never becomes positive. Now we return to the case which motivated us to study Bessel processes with arbitrary ; IR: For a BESQ process with dimension and starting point, we obtain with the same argument as above, that it behaves as a {BESQ,, process; this means, until it hits for the rst time it behaves as a {BESQ,, process, and after that it behaves as a BESQ. An important and well-known property of squared Bessel processes with nonnegative dimensions is the additivity property, see Introduction I.3. We show that the additivity property is no longer true for BESQ processes with IR arbitrary. Consider the BESQ process Z and the BESQ ~ y process Z ~, where > ; ~, < with, and y. Assuming the additivity property holds, would yield: Z + Z ~ (law) = Z, Z (law) = Z, ; since. Z and Z are independent processes and for the following simple reason Z Z cannot be true: Let X, Y be two independent realvalued random variables with F Y () < for all IR. Then, if =P (X Y ) = E[F Y (X)], we have F Y (X) = a.s. and hence, X has to be innite, which is a contradiction. Now we apply the time reversal result (3.7) to a BESQ, process fx t g with,, (X t ; t T ) (law) = ( X ~ ; t ~ L ~,t L ); (.3) where ( ~X t ; t ) denotes a BESQ 4+ process and ~L = supft j ~X t = g. Our aim is to nd the semigroup of a BESQ, process fx t g with,. We have to decompose the process (X t ) before and after it hits for the rst time separately: E, {z } A [f(x t )] = E, [f(x t ) (t<t )] 5 + E, [f(x t ) (tt )] : {z } B

18 By means of (.3) we obtain for the rst term A A = E 4+ [f(x ~ L,t ) (t< ~ L ) ] ; where X is a BESQ 4+ process. Since we know (see (.6)) ~L (law) = Z ; where Z is a Gamma variable with parameter 4+,, see (.7), we have Q 4+ (~L dt) = e, t dt q (t) dt : (.4) t,() t We obtain E 4+ [f(x ~ L,t ) (t< ~ L ) ] = = E 4+ t t [f(x s,t )] q (s) ds f(y) p 4+ s,t (;y) dy q (s) ds ; where p t (;y), with >, is the semigroup density p t (; y) in y of BESQ for = p,y t(;y)= t,( ) y, ep : t Analogously, for the second term B we have since (,X t,t )isabesq B = Putting A and B together we obtain E, [f(x t )] = t + f(,y) p t,s(;y) dy f(y) p 4+ s,t (;y) dy q (s) ds : f(,y) p t,s(;y) dy q (s) ds q (s) ds ; and hence, the semigroup density ^p (,) t in y of a BESQ, process is equal to ^p (,) t (; y) = t p 4+ s,t (;y) q (s) ds ; for y>, and ^p (,) t (; y) = p t,s(; jyj) q (s) ds ; (.5) 6

19 for y <. We want to nd the semigroup density ^p (,) t Consider the case y>: ^p (,) t (; y) = g(; y; ) = g(; y; ) t (s(s, t)),(+ ) ep, y t,3, (u(u, )) (+ ) ep, more eplicitly. ds s, t + s y t (u, ) + tu!! du ; where With g(; y; ) =,, + (y) + (+) ( + ) : we obtain h(; y; ; t) g(; y; ) t,3, ; m + ; a y t ; b t ^p (,) t (; y) = h(; y; ; t) = h(; y; ; t) du (u(u, )) ep, a m s (m,) (, s) ep,bs, m and hence we nd a more eplicit formula for ^p (,) t : ^p (,) t (; y) =h(; y; ; t) e a,b Analogously we obtain in the case y<: where and ^p (,) t (; y) =k(; y; ; t)e,a,b k(; y; ; t),, (, w) (m,) ep w m (w +) m, w m u, + b u as, s ds ;!! bw, a dw : w ep,bw, a dw ; w + jyj, t,, ; m ; a jyj t ; b t : As an application of the time reversal result (.3) we make some computations concerning the law Q, of a BESQ, process ( ) t with ; >. Since 7

20 (,Z T +t;t ) is a BESQ process, independent of the past of ( ) up to T,we have Q, ep(, f(u) Z u du) = Q, f(u) Z u du) T = t Q ep( ep (, f(t + v) X v du) Q, [T dt] ; with a BESQ process (X v ) v and a Borel function f.from the time reversal result (.3) we obtain Q, ep(, Q 4+ and with the equalities ep (, f(u) Z u du) T = t = Q 4+ = Q 4+ = Q 4+ nally, we have with (.4) ep(, Q, = Q 4+ ep(, f(t, u)x u du) L = t ep (, ep (, ep (, f(u) Z u du) Q = ep( As an interesting eample consider f(t, u)x u du) X t = f(t, u)x t,u du) X t = f(t,u)x u du) L = t ; f(u)x u du) X t = Q 4+ ep (, f(t + v) X v dv) f(u) [;a](u) ; a; > : We know (see Pitman{Yor [9], p. 43 (.m)) Z Q 4+ "ep(, t # X u du) X t = = " t sinh(t) f(u)x u du) X t = q (t) dt : # 4+ ep, t (t coth(t), ) ; 8

21 and hence we have "ep(, Q, Z a Q + # dt) = "ep( a Q 4+ If in addition (a, t) <, "ep( Q Z a " Z a,t t sinh(t) # 4+ ep, t (t coth(t), ) # X v dv) q (t) dt Z "ep(, a X u du) X t = Z a,t # X v dv) = cos((a, t)), ; # q (t) dt : and we have (with = +) Z Q, "ep(, a # dt) Z = + a ep, cos((a, t)),+,() coth(t) dt sinh(t) + Z + Q 4+ "ep(, a # X u du) e,,() a X t t = dt : + t Etension Now we etend the results for squares of Bessel processes with negative dimensions to a wider class of processes. We generalize the denition of a squared -dimensional radial Ornstein-Uhlenbeck process with parameter, and starting point z in (.), by considering arbitrary ; z IR. Denition 4 The solution to the stochastic dierential equation q dx t =(a + bx t ) dt + jx t j dw t ; X = ; (.6) where a; b; IR and fw t g is a one-dimensional Brownian motion, is called a squared radial Ornstein{Uhlenbeck process. In the following, we consider the case a< and > which corresponds for b = to a BESQ a process with negative dimension a. We call b Q a the law on C(IR + ; IR) and for b =, as before, simply Q a. Via Girsanov transformation we obtain the relationship 9

22 Lemma b Q a where () = " b # = ep (X s ) dw s, b jx s j ds 8 q jj sgn (). Proof: Consider the Q a -process dx t = adt+ q jx t j dw t ; X = : R t Q a j ; (.7) Via Girsanov's theorem ~W q t = W t, b sgn (X s) jx s j ds is a Brownian motion under the absolutely continuous probability with density " b q Z D t ep sgn (X s ) jx s j dw s, b t # jx s j ds ; 8 with respect to Q a, from which the result follows. We may also write the stochastic integral R t sgn (X s) q jx s j dw s in a simpler form, since we have from It^o's formula: jx t j = jj + sgn (X s )(ads+ q 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: L t (X) = lim "# " 4 = lim "# " Hence we have sgn (X s ) lim 4 "# [;"] (X s ) dhx; Xi s ([3], p., Cor..9) jx s j [;"] (X s ) ds [;"] (X s ) ds = : q jx s j dw s = jx t j,jj,a sgn (X s ) ds : Thus, (.7) takes the form b Q a " b ep 4 Q a jx t j,jj,a sgn (X s ) ds, b 8 # jx s j ds : (.8)

23 Now we can apply formula (.8) to obtain some conditional epectation formula:! b b q a t (; y) = q a t (; y) ep 4 (jyj,jj) Q a " ep, ab 4 sgn(x s ) ds, b 8 # jx s j ds! X t = y :(.9) The case y> corresponds to t<t, and y to t T. For y>, (.9) reduces to b q a t (; y) = q a t (; y) ep Q a "ep, b b abt (y, ), 4 4 8!! # X s ds X t = y Using the time-space transformation from a BESQ a process (X a t ) to a squared radial Ornstein{Uhlenbeck process ( b X a t ) : b X a t = e bt X a,e,bt ; b we also have together with the relationship (.9) the following b q a t (; y) =e,bt q a,e,bt (; e,bt y) ; (.) from which b q a t (; y) is obtained since we know q a t (; y), see (.5). Hence we know the conditional epectation Q a [:::jx t = y] in formula (.9). b

24 Chapter 3 Time reversal In the previous chapters we used results on time reversed diusions as an essential tool, e.g. in the rst chapter as for the distribution of rst hitting times by Bessel processes, or in Chapter as for the transition densities of BESQ processes with negative dimensions. In this chapter we investigate these time reversal results more deeply. As for time reversal we refer to Nagasawa [, 3] and Revuz{Yor [3], Chapter VII. First, some results on time reversed diusions in general are reviewed. In section 3. we study Doob's h- transform and consider some applications. Using these results, in section 3. we check a time reversal theorem by Elworthy{Li{Yor [6], and based on this theorem, we introduce a three-parameters-family of processes in section 3.3. Consider a transient diusion (X t ), living on IR +, 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 a.s., and for < a, L a > a.s. We want to identify the time reversed process ( ~X t ), where ~X t (!) ( XLa(!),t(!) ; if <t<l a (!) if L a (!) t or L a (!) = ; (3.) denotes the cemetary, and ~X (!) = X La(!)(!), if < L a (!) <, else ~X (!) We remark that a diusion cannot only be time reversed at a last eit time, but more generally at a cooptional time, see Nagasawa [, 3], Revuz{Yor [3], Chapter VII.4. For our purposes it is reasonable to restrict ourselves to last eit times. We will need the following (see also It^o{McKean [3], p. 49) Lemma Let (p t ) be the transition density of a diusion (X t ) with respect to the speed measure m. Then we have for all ; y the symmetry p t (; y) =p t (y; ) :

25 Proof: Let A denote the innitesimal generator of the diusion and let the functions and be the fundamental solutions of the generalized dierential equation Au = u; > : Call f + the right derivative of a function f with respect to the scale function s of the diusion, f( + h), f() f + () = lim h# s( + h), s() : Then the so called Wronskian w is dened as and is independent of. The function w = + () (), () + () g (; y) is called the Green function and e,t p t (; y) dt (3.) ( w, g (; y) = () (y); y; w, (y) (); y: (3.3) We obtain g (; y) =g (y; ), and hence, from the injectivity of the Laplace transform the result follows. Furthermore, some assumptions and notations are needed. Call (P t ) the semigroup of (X t ). The time reversed process ( ~X t ) is a Markov process (see Nagasawa [] and [3], III.3., p. 66) and we call its semigroup ( ~P t ). The resolvent G of order, >, of the semigroup (P t ), and its analog ~G for ( ~P t ), is G (f)() = e,t P t f() dt ; for all positive Borel functions f. The semigroup (P t ) has a transition density (p t ) with respect to the speed measure m: P t (; dy) =p t (; y) m(dy) (see It^o{ McKean [3], p. 49). We have Z G (f)() = e,t P t f() dt = e,t p t (; y)f(y) m(dy) dt = Z g (; y)f(y) m(dy); for all positive Borel functions f, with g in (3.). G is the potential kernel of X, and we assume that there is a probability measure such that the 3

26 potential = G is a Radon measure, and additionally, that the resolvents ~G and G are in duality with respect to, that is hg f;gi = hf; ~G gi ; for all > and for all positive Borel functions f and g. Then we have the duality between the semigroups (P t ) and ( ~P t ): hp t f;gi = hf; ~P t gi (3.4) for all positive Borel functions f and g, see Nagasawa [3], 3., Revuz{Yor [3], Th. 4.5, p. 9. From G (A) = Z G (; A)(d) = Z A Z g (; y)(d) m(dy); for every Borel set A, we obtain = G m. With the notation h(y) d(g Z ) dm (y) = g (; y) (d) (3.5) for m-almost all y we have hp t f;gi = = = Z Z f(y) p t (; y) m(dy) g()h()m(d) Z! g()h() p t (; y)m(d) f(y)h(y) m(dy) h(y) Z h(y) P t(gh)(y) f(y)h(y) m(dy) ; by Lemma, and from duality (3.4) we deduce for m-almost all. ~P t g() = h() P t(gh)() (3.6) We state a time reversal result proven with the results above (see Sharpe [33], Getoor{Sharpe [9]), which was a useful tool in the previous chapters. Theorem For X atransient diusion, living on IR +, starting at, ~X the time reversed process (3.), starting at a, and T inffuj ~X u =g we have fx u ; u L a g (law) = f ~X T,u; u T g : (3.7) 4

27 This implies that the law ofl a for the process X starting at is identical to the law oft for the process ~X starting at a. We want to derive a more eplicit formula for h in (3.5). We know, see (.): P (L a dt) =, s(a) p t( ;a) dt ; (3.8) where X =, and s is the scale function with lim y# s(y) =, and s() =. Consider the two cases a and >a. If a, then, s(a) p t ( ;a) dt =; hence, s(a) =g ( ;a). If >a, then the law ofl a has mass at hence, P (L a =)=P (T a = ) =, s( ) s(a) ; P ( <L a )=, s(a) and, s( )=g ( ;a). We conclude for ;a IR + arbitrary. p t ( ;a) dt = s( ) s(a) ; g ( ;a)=, s( _a) (3.9) Now, assume =,that is, the transient process (X t ) starts at and is the Dirac measure at. Then from (3.5) and (3.9) we obtain h(a) =g (;a)=, s(a): (3.) As we will see in the net section, the process ( ~X t ) is the Doob's h-transform of (X t ). 3. Doob's h-transform Consider a one-dimensional diusion X, with sample space ; F c ), where I [,; denotes the cemetary and := f! :[; ) 7! I [f@gg, F c := f!(t)j t g. Denition 5 A non-negative measurable function h : I 7! IR[fg is called -ecessive,, for X, if 5

28 a) e,t E (h(x t )) h(), for all I; t, b) e,t E (h(x t ))! h(); for all I as t #. A -ecessive function is simply called ecessive. Let h be an -ecessive function for a diusion X. The life time of a path! is dened by (!) := infft j! t construct a new probability measure P h by P h,t h(!(t)) = e P j h() ; (3.) for t < and I. The process under the new measure P h is a regular diusion and is called Doob's h-transform of X. Eamples of -ecessive functions are and, see Lemma, and g (; ), see (3.). We remark, that all -ecessive functions h (ecept for h orh ) can be epressed in terms of a linear combination of these three functions. As an application we consider a Doob's h-transform (3.) with the ecessive function h in (3.) for the transient diusion X. As seen in (3.) the factor, cancels, so we simply consider h s, with s the scale function of X. Let us denote the probability measure associated to X by P ("). We obtain: The Doob's h-transform of the transient diusion X with h s the scale function of X, that is, the process under the new measure P h, is a process which reaches almost surely. For clarity we write P () P h. With this notation we have Note that we obtain and hence, P (") s (X t ) s() = P (") s (Xt^T ) P () : (3.) () s (t<t ) P () ; Q () t f() s() E [(fs)(x t )] s() P (") t (fs)() ; (3.3) is the semigroup of the process under P () killed when it reaches. In other words, we have the following time reversal result: the Doob's h-transform of the transient process under P (") with h = s is the process under P () killed at T ; this is the process ~X in our former notation. 6

29 From formula (3.) we can obtain eplicit formulae of the diusion processes via Girsanov's theorem. Assume, a process (X t ) under P () has the form X t = + b(x s ) ds + Via Girsanov's theorem we have with (3.) B t = s (Xu ) s (Xu ) du + ^B t = a(x s ) db s ; t T :!, s (X u ) du + ^B t ; s where ( ^B t ) is Brownian motion under P ("). Hence, the process under P (") has the form X t = + b,!! Z a s t (X u ) du + a(x u ) d s ^B u : Note, that if we assume, a process (X t ) under P (") has the form X t = + (X u ) du + (X u )d ^B u ; then analogously we obtain that the process under P () X t = + + s s has the form!! (X u ) du + (X u ) db u ; t T : Let A () resp. A (") denote the innitesimal generator of the process under P () resp. P ("). If we want to avoid a Girsanov argument, we can compute the innitesimal generator of the time-reversed process from (3.3) A () (f) = h A(") (hf) ; with h the scale function of X ("). The term A(hf) can be written as A(hf) =ha(f)+fa(h)+,(h; f) ; where, is the 'Operateur Carre du Champ', see Kunita [7] and Revuz{ Yor [3], p. 36. For instance, if the innitesimal generator takes the form A(f) = f + bf, then,(f;g) =f g for f;g C. 7

30 3. Checking a time reversal theorem in Elworthy{Li{Yor As an application of the previously stated results, we want to check Theorem 3.7 in Elworthy{Li{Yor [6]. Let, P denote the lawofa-dimensional radial Ornstein{Uhlenbeck process with parameter,, starting in. We have the following time reversal result: Theorem (Elworthy{Li{Yor, [6]): Let and >. For every bounded measurable function F on path space, E[F (Y La,t; t L a )] =, E (4,) a [F (Y t ; t T ) e,t ] ; (3.4) g(a) where g(a) =, E a (4,) (e,t ). Let (R t ) be a radial Ornstein{Uhlenbeck process with law P and denote by its scale function and by A its innitesimal generator. We know that the innitesimal generator of the time-reversed process is A(f) = [A(f)+fA()+,(;f)] = A(f)+,(;f)=A(f)+ f : (3.5) Let us check whether this is coherent with Theorem 3.7. From we obtain with E [F (Y La,t; t L a )] = ^P a, E (4,) a [F (Y t ; t T ) e,t ] g(a) ^E a [F ( ^Y t ; t ^T )] = D t, P (4,) a D t g(y t^t ) g(a) Writing the process (Y t ; t T ) simply as Y t = e,(t^t ) : b(y s )ds + B t ; where B is Brownian motion under, P a (4,) theorem Y t = b + g g 8 ; (3.6), then we obtain by Girsanov's! (Y s ) ds + ^B t ;

31 where ^B is Brownian motion under ^P a. Let A denote the innitesimal generator of the process (Y t ) under, P a (4,). We have for the new innitesimal generator ^A of the process ( ^Y t ) under ^P a : ^A(f) =A (f)+ g g f : Now, all we have to show is (see (3.5)) A(f)+ f = A (f)+ g g f ; that is, we have to identify the drift coecients:, r + r + = 3,, r + g r g : (3.7) The problem is that we do not know g eplicitly. But since D t is a martingale, the following two lemmas will help us. Lemma 3 g(y t ) e,t is a, P (4,) a martingale i g D(A) and A(g) = g. Proof: If g(y t )e,t is a martingale, then P h g(y t )e,(t+h),g(y t )e,t = for all h>, hence e,t P t g() =g() with Y = and we have d dt P tg()j t= = g(), that is g D(A). For g D(A), the process M g t = g(y t ), g(y ), Ag(Y s )ds is a martingale, see Revuz{Yor [3] Prop By It^o's lemma we obtain g(y t )e,t = e,s d(g(y s )), e,s g(y s )ds : With the notation: U V, U t, V t is a martingale, we obtain: g(y t )e,t e,s (Ag)(Y s )ds, e,s g(y s )ds ; and the claim follows. Lemma 4 ( g )(Y t)e,t is a martingale under, P (4,) a. 9

32 Proof: Consider the Doob's h-transform of the process under P with h = its scale function, or put dierently (see (3.)) where P = M t ^P a ; M t ( ^Y t^t ) (a) is a martingale (with respect to ^P a ). Together with (3.6) we know, M t is a martingale under ^P a if and only if M t D t is a martingale under, P a (4,), that is (( )g)(y t)e,t is a martingale under, P a (4,). We deduce A (g) =g and A ( g )=( g ). From g A = ga + A (g), g we obtain with A (g) = g = A ( g ) By It^o's lemma we have ga = g : (3.8) and hence A(f(u)) = f (u)(u ) + f (u)a(u) A Together with (3.8) we have = ( ) 3, A () :, A () = g g ; from which we obtain g(r) =c q (r) 3, r, e r ; (r) with a constant c>. Our aim is to derive (3.7), that is, A () = 3, r,, r, r ; 3

33 and since A () =( 3,, r r) +, it remains to show =,,, r : (3.9) r Remember, is the scale function of (R t ). Let b R denote the drift of (R t ) b R (r) =, r then b R + = and hence (3.9). + r ; 3.3 The three-parameters-family of processes with law P ; Based on Theorem 3.7 in Elworthy{Li{Yor [6] (see here p. 8) we introduce a three-parameters-family of processes. From (3.4) we deduce E [F (Y La,t; t L a ) e, La ]= g(a), E (4,) a [F (Y t ; t T ) e,(+ )T ] ; where ; > ; and g(a) =, E a (4,) (e,t ), and with the notation + we have g (a), E (4,) a [F (Y t ; t T ) e,t ]= g(a) E g (a) [F (Y La,t; t L a ) e, La ] ; where g (a) =, E a (4,) (e,t ). This motivates us to dene a three-parameters-family of diusion processes with laws P ; as P ; = (Y t) or more generally for arbitrary P ; = (Y t) () e, t P () e, t P ; for some increasing function and A =, C, where A is the innitesimal generator of P. (Y Indeed, with It^o's formula we see that t) () e, t is a martingale with respect to P and (F t ). 3

34 First, let us consider the case = where we simply write P ; instead of P ; and P, P ; = (Y t) () e, t P and P : (3.) By means of the optional stopping theorem (see Revuz{Yor [3].3, p. 65) we obtain (a) () E [e, Ta ]=: (3.) For =, >,; >a>> and > we know (see e.g. Kent [6] Th. 3., Pitman{Yor [7] Prop..3) E[e, a I () Ta ]= I (a) ; thus we have and (P ; (a) = I (a) a ; (3.) I (") () = lim =,, ( +): "# " ) is the family of laws of a diusion with innitesimal generator D + +, y Watanabe [36] calls a diusion process determined by such an innitesimal generator a Bessel diusion process in the wide sense with inde (; ). These diusions arise as the Euclidian norm of a -dimensional Brownian motion with vector drift starting in zero. Denoting the norm of -dimensional Brownian motion with drift starting at a nonzero vector by R t,we remark that R t is not a Markov process, whereas (R t ; R t ds ) has the Markovian property (see R s Pitman{Yor [7, 8] or Rogers{Pitman [3]). From (3.) we have For eample, in the case =3we obtain by (3.)! D (y) = I +(y) I (y) : (3.3) (a) () = sinh(a) a sinh() and 3 (a) () = sinh(a) : (3.4) a

35 Hence, (P 3; ) is the family of laws of a diusion with innitesimal generator D + y D +( coth(y), y )D = D + coth(y)d: This diusion is the radial part of hyperbolic Brownian motion in dimension 3 (see Rogers{Pitman [3], Gruet []). As will be discussed below, see p. 34, the reversed process at L a is Brownian motion with drift (,) killed the rst time it reaches zero. Furthermore, we know, see Yor [4] p. 55, Theorem 3 Consider two transient diusions with laws P and Q such that Q j = D t P j : Then where h(a) = (a) (a) s(a) s (a) Q j FLa =(h(a)d La ) P j FLa ; (a) ; and s denote the respective scale functions for (a) Q and P and and are the respective diusion coecients for Q and P. Continuing our discussion we apply Theorem 3 to (3.). With D t (Y t) () e, t we deduce from the equality h(a) (a) () E (e, La )= (3.5) E ; [F (Y La,t; t L a )] = (E(e, La )), E[F (Y La,t; t L a ) e, La ] ; where for > arbitrary and for > E(e, a La )= I (( ^ a))k (( _ a)) ; see e.g. Pitman{Yor [7], p. 33, (7.~e), and where (see (.9)) E(e, La )= (a) K (a) :,,() 33

36 Note that by means of (3.5) we obtain for = a E a(e, La )=(h(a)), : For instance, for the case =3and =we have D t sinh(y! t) e, t ; Y t and obtain [F (Y La,t; t L a )] = e a E[F 3 (Y La,t; t L a ) e, La ] ; E 3; since, see also (.), E 3 (e, La )=e,a : Now let us look at the diusion process with law P 3; time-reversed at L a.as we remarked above (see p. 33) it will turn out that this is Brownian motion with drift. Let P denote the law of Brownian motion and P 3 the law of a 3-dimensional Bessel process starting in >. From Doob's h-transform (3.) we know P 3 = X t^t P ; and together with (3.) and (3.4) we have P 3; = sinh(x t^t ) sinh() e, (t^t ) P : (3.6) Let P ()() denote the law of Brownian motion with drift <. Its scale function s is s() = (, e, ) : Via Doob's h-transform (3.) we have P (")() = e,x t^t, e,, P ()() : The Doob's h-transform (3.) of Brownian motion with h(; t; ) e, t is identical in law to Brownian motion with drift, see e.g. Revuz{Yor [3], VIII.3, p. 37, or Borodin{Salminen [], IV.8, that is P ()() = ex t^t, (t^t ) P () e 34 : (3.7)

37 And nally, since sinh(x t^t ) sinh() e, (t^t ) = ex t^t, e, we obtain from (3.6) and from (3.7) with =, P 3; = ex t^t, e, P ()(,) e,x t^t, (t^t ) e, ; ; this means, a diusion process with law P 3; time-reversed is a Brownian motion with drift (,) killed the rst time it reaches zero. So far, we treated the case = in full generality and as an application looked at the case =3.Now let us consider the general case, i.e. we want to determine a P-martingale D t with P ; We know for = (see (3.)) P ; = (Y t) = D t P : () e, t P with in (3.) and via Girsanov's theorem we know P = ep ( Y s dw s, ; (3.8) Y s ds In order to obtain D t, rst we write the P-martingale in a dierent form. We will tackle this problem more generally. ) P : (3.9) (Y t) () e, t in (3.8) Let us assume, that H t is a continuous, strictly positive semimartingale, and A t is a continuous process with bounded variation such that H t e,at is a local martingale. We will write H t e,at in a dierent way. The continuous semimartingale H t can be decomposed uniquely H t = m t + a t ; (3.3) where m is a continuous local martingale and a a continuous adapted process of nite variation. With It^o's lemma we obtain H t e,at = epfln H t, A t g ( dh s = ep, H s 35 dhh; Hi s H s ), A t ;

38 where the process hh; Hi is the quadratic variation of H. With (3.3) and since hh; Hi = hm; mi we obtain H t e,at = ep ( dm s, dhm; mi ) s da s +, A H s Hs t : H s Since H t e,at is a martingale, or equivalently e,as da s, da s ds e,as H s ds = (see also the proof of Lemma 3), nally we have Denoting M t R t H t e,at = ep ( dm s, H s dhm; mi s H s ) : (3.3) dm s H s, equation (3.3) has the simple form H t e,at = ep M t, hm;mi t : (3.3) For eample in the case =3, =,we obtain sinh(y t ) Y t e, t = ep, ( coth(y s ), Y s )dw s ( coth(y s ), Y s ) ds With the notation N t R t Y s dw s, equation (3.9) can be written as = ep N t, hn;ni t P ; P and hence, together with (3.3) we have D t = ep M t + N t, hm + N;M + Ni t In the case =3, =,we obtain D t = ep, ( coth(y s ), Y s + Y s )dw s ( coth(y s ), Y s + Y s ) ds ; : : 36

39 and the innitesimal generator of the diusion process under P 3; is D +( coth(y)+y) D: For arbitrary > analogous calculations together with (3.3) lead to D t = ep ( ( I +(Y s ) I (Y s ) + Y s ) dw s, ( I +(Y s ) I (Y s ) + Y s ) ds ) ; and the innitesimal generator of the diusion process under P ; D + I +(y) +,! I (y) y + y D; is and hence, we described the three-parameters-family of diusion processes with law P ; in full generality. We remark that this family of diusion processes is closely related with Ornstein{Uhlenbeck processes built from Brownian motion with drift. 37

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