On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem

On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem Koichiro TAKAOKA Dept of Applied Physics, Tokyo Institute of Technology Abstract M Yor constructed a family of one-dimensional continuous martingales in connection with Saisho and Tanemura s extension of Pitman s theorem This paper reveals some properties of these martingales and the corresponding stochastic differential equations In particular, this implies that the pathwise uniqueness theorem by Yamada and Watanabe cannot be generalized to a non-diffusion case 1 Introduction M Yor has recently showed the following property based on Saisho and Tanemura s generalization [5] of Pitman s theorem [] Theorem 11 Yor [9], Corollary 151 Let R t be an - t [, dimensional Bessel process starting from the origin on a certain probability space Ω, F,P Define X t = min s [t, R +s R +t for t [, Then, for each >, X t is an F X -martingale, where F X = t [, F X t denotes the filtration generated by X t Remarks i As shown in Revuz-Yor [4] Theorem VI35, we see from Theorem 11 with = 1 and from Lévy s characterization theorem that X 1 t is a onedimensional Brownian motion Therefore, { X ; > } is a family of R-valued continuous martingales that includes one-dimensional Brownian motion Oh-okayama Meguro-ku Tokyo 15, Japan E-mail:takaoka@neptuneaptitechacjp 1

ii As known from the literature eg the above cited book of Revuz-Yor, it holds that max X s = min s [,t] s [t, R +s, t [,, as, and hence Pitman s theorem is equivalent to the above mentioned fact that X1 t is a one-dimensional Brownian motion Thus, Theorem 11 can be viewed as an extension of Pitman s theorem Yor [9] has actually proved that Theorem 11 holds for a larger class of diffusions; an even further extension is done recently by Rauscher [3] It should also be mentioned that in several works, different generalizations of Pitman s theorem were studied; eg Bertoin [1] and Tanaka [6] [7] iii Theorem 11 is proved in Yor s book by using the enlargement of filtration technique first introduced by T Jeulin Note here that the filtration F X is strictly larger than F R + : t, F X t = F R + t σ min R +s s [t, The aim of the present paper is to investigate which properties of onedimensional { Brownian motion hold for other members of our martingale family X ; > } and which do not Among others, the following two properties will be shown: 1 The stochastic differential equations henceforth SDEs satisfied by X t, 1, are of non-diffusion type and do not fulfill the Lipschitz condition If 1, then pathwise uniqueness holds for our SDE On the other hand, if >1, even uniqueness in law fails; in particular, for, our SDEs are counterexamples showing that the famous Yamada-Watanabe pathwise uniqueness theorem for one-dimensional diffusion-type SDEs cannot be extended to non-diffusion cases Theorem 4 For each fixed t, the random variable X t is symmetrically distributed with respect to the origin, while the processes X t and X t do not have the same law if 1 Proposition and Theorem 3 This paper is organized as follows In Section we state our results The proofs of these properties will be given in Section 3 Throughout this paper, we frequently cite the book of Revuz-Yor [4] as the basic reference Acknowledgements A stimulating conversation with Professor T Shiga has improved Theorem 4iii; sincere thanks are due to him The author also wishes to thank Prof V Vinogradov and Dr J Akahori for their helpful comments Statement of the results: some properties of the martingales X t As mentioned above in the Introduction, the proofs of all the properties listed in this section will be given in Section 3

Proposition 1 i If 1, then X t is not a Markov process, while the R -valued process X t, max s s [,t] t [, is Markov for any > ii For each >, X t is self-similar in the sense that c >, c / X ct t [, d = X t t [, iii For each >, X t is a divergent martingale: lim t [X ] t = as, where [X ] t t [, denotes the quadratic variation process of X t The next proposition generalizes well-known results for one-dimensional Brownian motion Proposition Fix > and t> i The distribution of X t is symmetric with respect to the origin: X t d = X t In more detail, we have P [ X t dx ] = 1 t / Γ exp x / dx, t x R ii The following four random variables are all identically distributed: a max s = min s [,t] s [t, R +s ; b max s X t = R+t min s [,t] s [t, R +s ; c X t = min s [t, R +s R+t ; d Rt The two questions which arise naturally from Proposition are as follows: Is X t, as a process, symmetric with respect to the origin? Do b, c and d of Proposition ii have, as processes, the same law? since it is well known that the answer is yes for both of them if =1 The next theorem, however, answers these questions in the negative for 1 3

Theorem 3 Suppose 1 i The following three martingales have different laws from one another: X t ; t [, X t t [, ; sgn X s dx s t [, ii It also holds that b, c and d of Proposition ii have, as processes, different laws from one another We now turn our attention to the SDEs satisfied by our martingales Theorem 4 i For each >, X t is a weak solution to the onedimensional SDE 1 dx t = max X s X t dw t ; s [,t] 1 X = ; where W t is one-dimensional standard Brownian motion ii If >1, then the above SDE also has the trivial solution X, and so uniqueness in law fails Among the solutions of the SDE, the law of X t is characterized as follows: if a weak solution X t satisfies inf { t> X t } = as, then it is identical in law to X t iii If 1, then pathwise uniqueness holds; X t is the unique strong solution Remarks If, then 1 1 < 1 The first assertion of Theorem 4ii thus implies that uniqueness in law does not, in general, hold for the one-dimensional SDE dx t = σt, X dw t, where σt, X is a predictable functional and W t is one-dimensional Brownian motion, even if 3 1 η<1, K > such that σt, x σt, y K max x s y s η s [,t] In contrast, note that if σt, X depends only on t and X t, ie if σt, X σt, X t, and if η 1, K > such that σt, x σt, y K x y η, 4

then pathwise uniqueness follows from the Yamada-Watanabe theorem [8] It should also be mentioned that if η 1 instead of 1 η<1 in 3, then the Lipschitz condition is satisfied and pathwise uniqueness holds Finally, we deduce the following property from the proof of Theorem 4 Corollary 5 X t is a pure martingale, ie, F X = F β with β being the time-changed Brownian motion Consequently, X t has the martingale representation property 3 Proofs Proof of Proposition 1 i If X t were Markov for 1, then d[x ] t would depend only on the value of X t and not on the past history It holds, however, that d[x ] t = d[r +] t = R 1 + t dt = max s [,t] X s X t 1 dt The second assertion follows from the Markov property of the R -valued process R + t, min R +s s [t, t [, ii The scaling property of X t follows from that of R+t iii It follows from the scaling property of R + t that hence M >, t, [X ] t = 1 d = t P [ lim t [X ] t M ] = lim t P R 1 + s ds = lim t P = R 1 + s ds, [ [ 1 ] R 1 + s ds M R 1 + s ds M ] t The next trivial fact will be used in the proof of Proposition 5

Lemma 31 Let r> Suppose Y is a random variable uniformly distributed on the interval [,r] Then Y r is uniformly distributed on the interval [ r, r]; in particular, Y r d = Y r Furthermore, the three random variables Y, r Y and Y r are all identically distributed Proof of Proposition i Conditioned by F R + t, min s [t, R+ s is a random variable uniformly distributed on the interval [,R+t ], since the scale function of the diffusion R+t is sx = 1 x This and Lemma 31 imply that for each fixed t, X t and X t have the same distribution conditioned by F R + t, which yields the desired result The calculation of the density function follows along the same lines ii The same reasoning as in i leads to the equi-distribution property of a, b and c Revuz-Yor [4] Exercise XI118 shows that >, t, min R +s d = R t, s [t, so a and d have the same distribution We also need the following lemma to prove Theorem 3 Lemma 3 cf Revuz-Yor [4] Exercise VI3 Suppose M and N are divergent continuous local martingales starting from the origin Let β and γ denote the time-changed Brownian motions of M and N, respectively Then M t d =N t β t, [M] t d = γ t, [N] t Proof of Theorem 3 i We have already shown in Proposition 1iii that X t is divergent Let β be its time-changed Brownian motion Then by Lemma 3 we have X t d = X t d β t, [X ] t = β t, [X ] t β t d = β t conditioned by F [X ] Thus, to prove X t d X t it is sufficient to show that β t and β t have different laws conditioned by F [X ] This is a consequence of the following fact: F [X ] = F R 1 + = F R + since 1 = F X F β 6

Similarly, we can show that the third martingale in the statement of the theorem is not identical in law to the other two ii First, it is easy to see that the law of d is different from those of the other two processes, since only d is Markov among the three Furthermore, as stated in Revuz-Yor [4] Exercise VI3, X t d = max X s X t s [,t] sgn X s d dx s = X t, so by i we see that b and c, as processes, do not have the same law Proof of Theorem 4 i Straightforward ii We only have to prove the second assertion First observe that any weak solution of the SDE satisfying the additional condition is a divergent continuous local martingale Indeed, for almost all ω Ω, there exists some t = t ω > such that X t ω >, and hence Next, ine lim [X] tω = t t ω = τ t β t 1 max X sω X t ω dt s [,t] max s [,t] X sω X t ω 1 1 dt = inf{s > [X] s >t}; = Xτ t Clearly, β t is a one-dimensional Brownian motion starting from the origin By the inverse function theorem we have t >, dτ t dt dt 1 = max X u X τt u [,τ t ] 1 = max β u β t, u [,t] thus 1 t, τ t = max β u β s ds u [,s] Note that τ = as by the condition This implies that { X t = β inf s> s 1 max β v β u du > t} v [,u] 7

The law of a weak solution of 1 satisfying the condition is therefore uniquely determined iii The assertion is trivial if =1, so we assume <1in the sequel We divide the proof into three steps Step 1 We first show uniqueness in law Since <1, it is easy to see that any weak solution of this SDE must satisfy the condition Also, any weak solution is a divergent continuous local martingale Indeed, if a solution X t were not divergent, then, with positive probability, X t ω would converge to a real number as t Then max X uω X t ω would also converge to a real u [,t] number for such an ω as t and hence lim [X] tω = lim t t 1 max X uω X s ω ds =, u [,s] a contradiction The rest of the proof of uniqueness in law is exactly the same as in ii Step Suppose Ω, F, P,F t, W t, X t is a weak solution to the SDE 1, ie, Ω, F, P,F t is a filtered probability space, X t is a semimartingale on it, W t isanf t -Brownian motion starting from the origin, and they satisfy 1 Define R t Z t = max s [,t] X s X t 1/ ; = min s [t, R s R t It then follows from the Itô formula that Ω, F, P,F t, W t, Z t satisfies 31 Z t = W t + 1 ds max Z u Z s u [,s] For this equation, see also Revuz-Yor [4] Exercise XI19 and Yor [9] Corollary 151 Conversely, if Ω, F, P,F t, W t, Z t satisfies 31 and 3 then we ine inf { t> max u [,t] Z u > } = as, R t X t = max s [,t] Z s Z t ; = min s [t, R s R t ; 8

and it is not hard to see that X t satisfies 1 Therefore, pathwise uniqueness holds for 1 if and only if a solution of the SDE 31 satisfying the condition 3 is pathwisely unique We have already shown in Step 1 the uniqueness in law, so it suffices to show that if both Z 1 t and Z t satisfy 31 in the same set-up, then so does Y t = Z 1 t Z t Since Z 1 t Z t is a process with continuously differentiable trajectories, it is easy to verify that dy t = dz t + d Z 1 t Z + t 1 = 1 1 {Z dz t >Z t } t +1 1 {Z dz t Z t } t, and thus 33 Y t = W t + 1 { ds If Z s 1 ω >Z s ω then max 3, so we can rewrite 33 as 34 Y t = W t + 1 Note that max u [,s] Y u = 35 Y t = W t + 1 1 {Z 1 s >Z s } max u [,s] Z1 u Z s 1 u [,s] Z1 u ω max + u [,s] Z u { 1{Z 1 s Z s ds } max Y + u Y s u [,s] 1 {Z 1 s Z s } max u [,s] Z u Z s } ω, as we will see in Step 1 {Z 1 s =Z s } max u [,s] Z u Z s max u [,s] Z1 u max u [,s] Z u Similarly, we have { 1{Z 1 s Z s ds } max Y + u Y s u [,s] 1 {Z 1 s =Z s } max u [,s] Z1 u Z s 1 Comparing 34 and 35, we see that for almost all ω Ω: { µ t [, Z 1 t ω =Z t ω and max u ω max u [,t] Z1 where µ denotes the Lebesgue measure, and hence u [,t] Z u } } } ω =, 1 {Z 1 s =Z s } max u [,s] Z u Z s ds = 1 1 {Z s =Z s } max Y ds u Y s u [,s] This together with 34 implies that Y t = W t + 1 ds max Y u Y s u [,s] 9

Step 3 It remains to show that if Z 1 s ω >Z s ω then max max u [,s] Z u Then ω Let which implies that 1 s = sup { u [,s Z 1 u ω =Z u ω } 1 d du max u [,s ] Z1 u ω Z s 1 Z 1 u ω Z u ω u=s, ω 1 u [,s] Z1 u ω 1 max u [,s ] Z u ω Z s ω, max u [,s ] Z1 u ω max u [,s ] Z u ω since <1 Also Z u 1 ω >Z u ω for u s,s], thus we obtain the desired property Proof of Corollary 5 We have already shown in the proof of Theorem 4 that X t is pure Note that every pure local martingale has the martingale representation property; see, for instance, Revuz-Yor [4] V4 References [1] Bertoin, J, An extension of Pitman s theorem for spectrally positive Lévy processes, Ann Prob 1993, 1463 1483 [] Pitman, J, One-dimensional Brownian motion and the three-dimensional Bessel process, Adv Appl Prob 7 1975, 511 56 [3] Rauscher, B, Some remarks on Pitman s theorem, in this volume of the Séminaire de Probabilités [4] Revuz, D & Yor, M, Continuous martingales and Brownian motion, Second edition, Springer 1994 [5] Saisho, Y & Tanemura, H, Pitman type theorem for one-dimensional diffusion processes, Tokyo J Math 13 199, 49 44 [6] Tanaka, H, Time reversal of random walks in dimension one, Tokyo J Math 1 1989, 159 174 [7], Time reversal of random walks in R d, Tokyo J Math 13 199, 375 389 [8] Yamada, T & Watanabe, S, On the uniqueness of solutions of stochastic differential equations, J Math Kyoto Univ 11 1971, 155 167 [9] Yor, M, Some Aspects of Brownian Motion Part II: Some Recent Martingale Problems, ETH Lecture Notes in Mathematics, Birkhäuser to appear 1