Conditioned Martingales

Transcription

1 Condiioned Maringales Nicolas Perkowski Insiu für Mahemaik Humbold-Universiä zu Berlin Johannes Ruf Oxford-Man Insiue of Quaniaive Finance Universiy of Oxford This draf: Ocober 8, 202 Absrac I is well known ha upward condiioned Brownian moion is a hree-dimensional Bessel process, and ha a downward condiioned Bessel process is a Brownian moion. We give a simple proof for his resul, which generalizes o any coninuous local maringale and clarifies he role of finie versus infinie ime in his seing. As a consequence, we can describe he law of regular diffusions ha are condiioned upward or downward. Keywords: Doob s h-ransform, change of measure; upward condiioning; downward condiioning; local maringale; diffusion; nullse; Bessel process. AMS MSC 200: 60G44; 60H99; 60J60. Inroducion We sudy he law Q of a coninuous nonnegaive P -local maringale X, if condiioned never o hi zero. The key sep in our analysis is he simple observaion ha he condiional measure Q, on he corresponding σ-algebra, is given by X T /X 0 dp, where T denoes he firs hiing ime of eiher 0 or anoher value y > X 0. This observaion relaes he change of measure over an infinie ime horizon hrough a condiioning argumen o he change of measure in finie ime via he Radon-Nikodym derivaive X T. Under he condiional measure Q, he process X diverges o, and /X is a local maringale. This insigh allows us o condiion X downwards, which corresponds o condiioning /X upwards and can herefore be reaed wih our previously developed argumens. In he case of a diffusion i is possible o wrie down he dynamics of he upward condiioned process explicily, defined via is scale funcion, - and similarly for downward condiioned diffusion. For example, if X is a P -Brownian moion sopped in 0, hen X is a Q-hree-dimensional Bessel process. This connecion of Brownian moion and Bessel process has been well known, a leas since he work of McKean [] building on Doob [4]. Following McKean, several differen proofs were given for his resul, mosly embedding his saemen in a more general resul such as he one abou pah decomposiions in Williams [9]. Mos of hese proofs are analyical and rely srongly on he Markov propery of Brownian moion and Bessel process - or even on he fac ha he ransiion densiies are known for hese processes. As he sudy of he law of upward and downward condiioned processes has usually no been he main focus of hese papers, resuls have, o he bes of our knowledge, no been proven in he full generaliy of his paper, and he underlying argumens were ofen only indirec. Our proof uses only elemenary argumens, i is probabilisic, and works for coninuous local maringales We hank Peer Carr, Peer Imkeller, and Kosas Kardaras for heir commens and suggesions. We are very graeful o Ioannis Karazas for careful reading an earlier version of his paper. We hank an anonymous referee for heir supporive remarks. N. P. is suppored by a Ph.D. scholarship of he Berlin Mahemaical School. Perkowsk@mahemaik.hu-berlin.de Johannes.Ruf@oxford-man.ox.ac.uk

2 and cerain jump processes. We show ha in finie ime i is no possible o obain a Bessel process by condiioning a Brownian moion no o hi zero and we poin ou ha condiioning a Brownian moion upward and condiioning a Bessel process downward can be undersood using he same resul. In Subsecion 2. we rea he case of upward condiioning of local maringales and in Subsecion 2.2 he case of downward condiioning. In Secion 3 we sudy he implicaions of hese resuls for diffusions. In Appendix A we illusrae ha condiioning on a nullse such as he Brownian moion never hiing zero is highly sensiive wih respec o he approximaing sequence of ses. Appendix B conains he slighly echnical proof of Proposiion 3.2, which describes he change of dynamics of a diffusion afer a change of measure. In Appendix C we sudy a class of jump processes ha can be reaed wih our mehods. Review of exising lieraure The connecion of Brownian moion and he hree-dimensional Bessel process has been sudied in several imporan and celebraed papers. Mos of hese sudies have focused on more general saemens han his connecion only. To provide a complee lis of references is beyond his noe. In he following paragraphs, we ry o give an overview of some of he mos relevan and influenial work in his area. For a Markov process X, Doob [4] sudies is h-ransform, where h denoes an excessive funcion such ha, in paricular, hx is a supermaringale. Using hx/hx 0 as a Radon- Nikodym densiy, a new sub-probabiliy measure is consruced. Doob shows, among many oher resuls, ha, if h is harmonic and addiionally minimal, as defined herein, he process X converges under he new measure o he poins on he exended real line where h akes he value infiniy. In his sense, changing he measure corresponds o condiioning he process o he even ha X converges o hese poins. For example, if X is Brownian moion sared in, hen hx = x is harmonic and leads o a probabiliy measure, under which X, now disribued as a Bessel process, ends o infiniy. Our resuls also yield his observaion; furhermore hey conain he case of non-markovian processes X ha are nonnegaive local maringales only. An analyic proof of he fac ha upward condiioned Brownian moion is a hree-dimensional Bessel process is given in McKean s work [] on Brownian excursions. He shows ha if W is a Brownian moion sared in, if B F s, where F s is he σ-algebra generaed by W up o ime s for some s > 0, and if T 0 is he hiing ime of 0, hen P W B T 0 > P X B as, where X is a hree-dimensional Bessel process. The proof is based on echniques from parial differenial equaions. In ha aricle, also a pah decomposiion is given for excursions of Brownian moion in erms of wo Bessel processes, one run forward in ime, and he oher one run backward. McKean already generalizes all hese resuls o regular diffusions. Knigh [0] compues he dynamics of Brownian moion condiioned o say eiher in he inerval [ a, a] or, a] for some a > 0 and hus, derives also he Bessel dynamics. To obain hese resuls, Knigh uses a very asue argumen based on invering Brownian local ime. He moreover illusraes he complicaions arising from condiioning on nullses by providing an insighful example; we shall give anoher example based on a direc argumen, wihou he necessiy of any compuaions, in Appendix A o illusrae his poin furher. In his seminal paper on pah decomposiions, Williams [9] shows ha Brownian moion condiioned no o hi zero corresponds o he Bessel process. His resuls exend o diffusions and reach far beyond his observaion. For example, he shows ha siching a Brownian moion up o a cerain sopping ime and a hree-dimensional Bessel process ogeher yields anoher Bessel process. In Piman and Yor [5] his approach is generalized o killed diffusions. A diffusion process is killed wih consan rae and condiioned o hi infiniy before he killing ime. This allows he inerpreaion of a wo-parameer Bessel process as an upward condiioned one-parameer Bessel process. Piman [4] proves essenially Lemma 2. of his paper in he Brownian case. This is achieved 2

3 by approximaing he coninuous processes by random walks, which can be couned. For he coninuous case, he saemen hen follows by a weak convergence argumen. The main resul of ha aricle is Piman s famous heorem ha 2W W is a Bessel process if W is a Brownian moion and W is running maximum. Baudoin [] akes a differen approach. Given a Brownian moion, a funcional Y of is pah and a disribuion ν, Baudoin consrucs a probabiliy measure under which Y is disribued as ν. The recen monograph by Roynee and Yor [7] sudies penalizaions of Brownian pahs, which can be undersood as a generalizaion of condiioned Brownian moion. Under he penalized measure, he coordinae process can have radically differen behavior han under he Wiener measure. In our example i does no hi zero. In Roynee and Yor [7] here is an example of a penalized measure under which he supremum process says almos surely bounded. 2 General case: coninuous local maringales Le Ω = C abs := C abs R +, [0, ] be he space of [0, ]-valued funcions ω ha are absorbed in 0 and, and ha are coninuous on [0, T ω, where T ω denoes he firs hiing ime of { } by ω, o be specified below. Le X be he coordinae process, ha is, X ω = ω. Define, for sake of noaional simpliciy, X := lim sup X lim inf X wih 0 :=. Denoe he canonical filraion by F 0 wih F = σx s : s, and wrie F = 0 F. For all a [0, ], define T a as he firs hiing ime of {a}, o wi, T a = inf{ [0, ] : X = a} 2. wih inf := T, represening a ime beyond infiniy. The inroducion of T allows for a unified approach o rea examples like geomeric Brownian moion. We shall exend he naural ordering o [0, ] {T} by < T for all [0, ]. For all sopping imes τ, define he σ-algebras F τ as F τ = {A F : A {τ } F [0, } = σx τ s : s < = σx τ T 0 s : s <, where X τ X τ T 0 is he process X sopped a he sopping ime τ. Le P be a probabiliy measure on Ω, F, such ha X is a nonnegaive local maringale wih P X 0 = =. 2. Upward condiioning In his secion, we sudy he law of he local maringale X condiioned never o hi zero. This even can be expressed as {T 0 = T} = {T a T 0 } {T a T 0 = T}. 2.2 a [0, a 0, ] The core of his aricle is he following simple observaion: Lemma 2. Upward condiioning. If P T a T 0 < T = for some a,, we have ha Proof. Noe ha X Ta which implies ha, for all A F, dp T a T 0 = X Ta dp. is bounded and hus a uniformly inegrable maringale. In paricular, P A T a T 0 = P A {T a T 0 } P T a T 0 yielding he saemen. = E P X Ta = ap T a T 0 + 0, = P A {T a T 0 } a = E P X T a A, The definiion of X is no furher relevan as X converges or diverges o infiniy almos surely under all measures ha we shall consider. We chose his definiion of X since i commues wih aking he reciprocal /X. 3

4 Three differen probabiliy measures Consider hree possible probabiliy measures:. The local maringale X inroduces an h-ransform Q of P. This is he unique probabiliy measure Q on Ω, F ha saisfies dq Fτ = X τ dp Fτ for all sopping imes τ for which X τ is a uniformly inegrable maringale. The probabiliy measure Q is called he Föllmer measure of X, see Föllmer [6] and Meyer [2]. 2 Noe ha he consrucion of his measure does no require he densiy process X o be he canonical process on Ω - he exension only relies on he opological srucure of Ω = C abs. This will be imporan laer, when we consider diffusions. We remark ha, in he case of X being a P -maringale, we could also use a sandard exension heorem, such as Theorem.3.5 in Sroock and Varadhan [8]. 2. If P T 0 = T = 0, Lemma 2. in conjuncion wih 2.2 direcly yields he consisency of he family of probabiliy measures {P T a T 0 } a> on he filraion F Ta a>. By Föllmer s consrucion again, here exiss a unique probabiliy measure Q on Ω, F, such ha Q FTa = P T a T 0 FTa. 3. If P T 0 = T > 0, we can define he probabiliy measure Q = P T 0 = T via he Radon-Nikodym derivaive {T0 =T}/P T 0 = T. Since in he case P T 0 = T = 0, we have {T a T 0 } = P a.s. {T a < T 0 } for all a 0, ], he measure Q is also called upward condiioned measure since i is consruced by ieraively condiioning he process X o hi any level a before hiing 0. Relaionship of probabiliy measures We are now ready o relae he hree probabiliy measures consruced above: Theorem 2.2 Ideniy of measures. Se b := P T 0 = T = P X > 0. If b = 0, hen Q = Q. If b > 0, hen Q = Q if and only if X is a uniformly inegrable maringale wih P X {0, /b} =. Proof. Firs, consider he case b = 0. Boh Q and Q saisfy, for all a >, d Q FTa = X Ta dp FTa = dq FTa. Thus Q and Q agree on a> F Ta = a> σx Ta : 0 = F. Nex, consider he case b > 0. Then, Q = Q and d Q/dP F /b imply ha X /b, yielding ha X is a uniformly inegrable maringale wih X = dq/dp {0, /b}. For he reverse direcion, observe ha X = {T0 =T}/b. This observaion ogeher wih is uniform inegrabiliy complees he proof. This heorem implies, in paricular, ha in finie ime he hree-dimensional Bessel process canno be obained from condiioning Brownian moion no o hi zero. However, over finie ime-horizons, a Bessel-process can be consruced via he h-ransform X T dp, when X is P - Brownian moion sared in and sopped in 0. Over infinie ime-horizons, one has wo choices; he firs one is using an exension heorem for he h-ransforms, he second one is condiioning X no o hi 0 by approximaing his nullse by he sequence of evens ha X his any a > 0 before i his 0. 2 See also Delbaen and Schachermayer [3] for a discussion of his measure, Pal and Proer [3] for he exension o infinie ime horizons and Carr, Fisher, and Ruf [2] for allowing nonnegaive local maringales. 4

5 Remark 2.3 Condiioning on nullses. We remark ha he inerpreaion of he measure Q as P condiioned on a nullse requires specifying an approximaing sequence of ha nullse. In Appendix A we illusrae his suble bu imporan poin. Remark 2.4 The rans-infinie ime T. The inroducion of T in his subsecion allows us o inroduce he upward-condiioned measure Q and o show is equivalence o he h-ransform Q if X converges o zero bu no necessarily his zero in finie ime, such as P -geomeric Brownian moion. If one is only ineresed in processes as, say, sopped Brownian moion, hen one could formulae all resuls in his subsecion in he sandard way when inf := in 2.. One would hen need o exchange T by hroughou his subsecion; in paricular, one would have o assume in Lemma 2. ha P T a T 0 < = and replace he condiion P T 0 = T = 0 by P T 0 = = 0 for he consrucion of he upward-condiioned measure Q. We noe ha he argumens of his secion can be exended o cerain jump processes. In Appendix C we rea a simple random walk example o illusrae his observaion. 2.2 Downward condiioning In his subsecion, we consider he converse case of condiioning X downward insead of upward. Towards his end, we firs provide a well-known resul; see for example [2]. For sake of compleeness, we provide a direc proof: Lemma 2.5 Local maringaliy of /X. Under he h-ransformed measure Q, he process /X is a nonnegaive local maringale and QT = T = E P [X ]. Proof. Observe ha fe Q A X T /n +s = lim E Q = lim E P = lim E P = lim E Q = E Q A X T /n A {Tm>} A {Tm>} A {Tm>} A {Tm>} X T /n T m +s X T /n T m +s X T /n T m X T /n T m + E Q X Tm +s X Tm + E Q + E Q + E Q A {T } X T /n +s A {T } X T /n A {T } X T /n A {T } X T /n for all A F and s, 0, where in he hird equaliy we considered he wo evens {T /n } and {T /n > } separaely and used he P -maringaliy of X Tm afer condiioning on F and F T/n, respecively - noe ha A {T m > } {T /n > } F T/n. The local maringaliy of /X hen follows from Q lim T /n < = lim Q lim T /n < T m = lim E P {limn T n n /n <T m}x Tm = 0. Therefore, /X converges Q-almos surely o some random variable /X. We observe ha QT = T = lim QT m < = lim E P {Tm< }X Tm = lim E P {Tm }X = E P X, where we use ha X converges P -almos surely. 5

6 The las lemma direcly implies he following observaion: Corollary 2.6 Muual singulariy. We have P X = 0 = if and only if QX = =. This observaion is consisen wih our undersanding ha eiher condiion implies ha he wo measures are suppored on wo disjoin ses. Corollary 2.6 is also consisen wih Theorem 2.2, which yields ha P X = 0 = implies he ideniy Q = Q, where Q denoes he upward condiioned measure. Lemma 2.5 indicaes ha we can condiion X downward under Q, corresponding o condiioning /X upward. The proof of he nex resul is exacly along he lines of he argumens in Subsecion 2.; however, now wih he Q-local maringale /X aking he place of he P -local maringale X: Theorem 2.7 Downward condiioning. If b of Thereom 2.2 saisfies b = 0, hen dq T /a T = X T/a dq for all a >. In paricular, here exiss a unique probabiliy measure P, such ha P FT/a = Q T /a < T; in fac, P = P. 3 Diffusions In his secion, we apply Theorems 2.2 and 2.7 o diffusions. 3. Definiion and h-ransform for diffusions We call diffusion any ime-homogeneous srong Markov process Y : C abs [0, [l, r] wih coninuous pahs in a possibly infinie inerval [l, r] wih l < r. Noe ha we explicily allow Y o ake he values l and r; we sop Y once i his he boundary of [l, r]. We define τ a for all a [l, r] as in 2. wih X replaced by Y. We denoe he probabiliy measure under which Y 0 = y [l, r] by P y. Since Y is Markovian i has an infiniesimal generaor see page 6 in Ehier and Kurz [5]. As we do no assume any regulariy of he semigroup of Y, we find i convenien o work wih he following exended infiniesimal generaor: A coninuous funcion f : [l, r] R {, } wih f R R is in he domain of he exended infiniesimal generaor L of Y if here exiss a coninuous funcion g : [l, r] R {, } wih g R R, and an increasing sequence of sopping imes {ρ n } n N, such ha P y lim n ρ n τ l τ r = and ρn fy ρn fy gy s ds 0 is a P y -maringale for all y l, r. In ha case we wrie f doml and Lf = g. Throughou his secion we shall work wih a regular diffusion Y ; ha is, for all y, z l, r we have ha P y τ z < > 0. In ha case here always exiss a coninuous, sricly increasing funcion s : l, r R {, }, uniquely deermined up o an affine ransformaion, such ha sy is a local maringale see Proposiions VII.3.2 and VII.3.5 in Revuz and Yor [6]. We call every such s a scale funcion for Y, and we exend is domain o [l, r] by aking limis. The nex resul summarizes Proposiion VII.3.2 in [6] and describes he relaionship of he scale funcion s and he limiing behaviour of Y : Lemma 3. Scale funcion. We have ha. P y τ l = T = 0 for one and hen for all y l, r if and only if sl R and sr = ; 6

7 2. P y τ r = T = 0 for one and hen for all y l, r if and only if sl = and sr R; 3. P y τ l τ r = T = 0 and P y τ l < T 0, for one and hen for all y l, r if and only if sl R and sr R. Throughou his secion, we shall work wih he sanding assumpion ha he scale funcion s saisfies sl > Assumpion L or sr < Assumpion R. Wihou loss of generaliy, we shall assume ha hen sl = 0 or sr = 0, respecively, and ha F = F τl τ r. Since by assumpion sy is a local maringale, i defines, under each P y, a Föllmer measure Q y as in Secion 2, where we would se X := sy /sy, for all y [l, r] wih 0/0 := / :=. The nex proposiion illusraes how he exended infiniesimal generaors of Y under P y and Q y are relaed: Proposiion 3.2 h-ransform for diffusions. The process Y is a regular diffusion under he probabiliy measures {Q y } y [l,r]. Is exended infiniesimal generaor L s under {Q y } y [l,r] is given by doml s = {ϕ : sϕ doml} and L s ϕy = sy L[sϕ]y. The proof of his proposiion is echnical and herefore posponed o Appendix B. The following observaion is a direc consequence of Lemma 2.5 and he fac ha Y is a regular diffusion under he probabiliy measures {Q y } y [l,r] : Lemma 3.3 Scale funcion for h-ransform. Under {Q y } y [l,r], he funcion s = /s is, wih he appropriae definiion of /0, a scale funcion for Y wih sl =, sr R under Assumpion L and wih sr =, sl R under Assumpion R. 3.2 Condiioned diffusions We now are ready o formulae and prove a version of he saemens of Secion 2 for diffusions: Corollary 3.4 Condiioning of diffusions. Fix y l, r and make Assumpion L.. Suppose ha P y τ l = T = 0, which is equivalen o sr =. Then he family of probabiliy measures {P y τ a τ l Fτa } y<a<r is consisen and hus has an exension Q y on F. Moreover, he exension saisfies Q y = Q y. 2. Suppose ha P y τ l = T > 0, which is equivalen o sr <, and define Q y = P y τ l = T. Then Q y saisfies Q y = Q y. Furhermore, provided ha sr =, he family of probabiliy measures {Q y τ a τ r Fτa } l<a<y is consisen. Is unique exension is P y. Under Assumpion R, all saemens sill hold wih r exchanged by l and, implicily, y < a < r exchanged by l < a < y. Proof. We only consider he case of Assumpion L, as Assumpion R requires he same seps. We wrie X = sy /sy. The hiing imes T a of X are defined as in 2.. Since s is sricly increasing, we have ha, for all y < a < r, {τ a τ l } = {T sa/sy T 0 }. Since X is a nonnegaive local maringale wih P y X 0 = =, he saemens in. and 2. follow immediaely from Theorem 2.2 and Lemma 3., which shows ha sy akes exacly wo values. The remaining asserions follow from Lemma 3.3 and Theorem

8 I is clear ha he measure Q under Assumpion L corresponds o he upward condiioned diffusion Y, while under Assumpion R i corresponds o he downward condiioned diffusion. Afer finishing his manuscrip we learned abou Kardaras [9]. Therein, by similar echniques i is shown ha Y under Q ends o infiniy if sr = ; see Secion 6.2 in [9]. In Secion 5 herein, a similar probabiliy measure is consruced for a Lévy process X ha drifs o. Afer a change of measure of he form sx for a harmonic funcion s, he process X under he new measure drifs now again o infiniy. 3.3 Explici generaors In his secion we formally derive he dynamics of upward condiioned and downward condiioned diffusions. For his purpose suppose ha Y is a diffusion wih exended infiniesimal generaor L, such ha doml C 2, where C 2 denoes he space of wice coninuously differeniable funcions on l, r, and Lϕy = byϕ y + 2 ayϕ y, ϕ C 2 for some locally bounded funcions b and a such ha ay > 0 for all y l, r. Finding he scale funcion hen a leas formally corresponds o solving he linear ordinary differenial equaion bys y + 2 ays y = This is for example done in Secion 5.5.B of Karazas and Shreve [8]. From now on, we coninue under eiher Assumpion L or Assumpion R wih s being eiher nonnegaive or nonposiive. We plug s ino he definiion of L s. Towards his end, le ϕ C 2. Then we have ha L s ϕy = sy Lsϕy = bysϕ y + sy 2 aysϕ y = bys yϕy + syϕ y + 2 ays yϕy + 2s yϕ y + syϕ y = sy by + ays y sy ϕ y + 2 ayϕ y since s = 2b/as due o 3.. Therefore, he upward or downward condiioned process has an addiional drif of as /s. This drif is always posiive or always negaive, as is o be expeced. Now, under Assumpion L upward condiioning wih l = 0, if b = 0, hen sy = y; herefore he addiional drif of he upward condiioned process is ay/y. Under Assumpion R downward condiioning wih r =, if by = ay/y, hen 3. yields sy = y and hus an addiional drif of ay/y = by. These observaions lead o he well-known fac: Corollary 3.5 Geomeric Brownian moion. A Brownian moion condiioned on hiing before hiing 0 is a hree-dimensional Bessel process. Vice versa, a hree-dimensional Bessel process condiioned o hi 0 is a Brownian moion. Moreover, a geomeric Brownian moion condiioned on hiing before hiing 0 is a geomeric Brownian moion wih uni drif. A Condiioning on nullses Before Theorem 2.2, we consruced a probabiliy measure Q by condiioning P on he nullse {T 0 = T} = a [0, {T a T 0 } using an exension heorem. I is imporan o poin ou ha he choice of he approximaing sequence of evens, necessary for his consrucion, is highly 8

9 relevan. We remark ha his has been illusraed before by Knigh [0] wih anoher example, which, in our opinion, is slighly more involved han he one presened in he following. To illusrae he issue, consider he coninuous maringale X, defined as X = X + X {T3/4 } + 8 X 2 {T3/4 < T /4 }; he process X moves wice as much as X unil X his 3/4, hen i moves half as much as X unil X caches up, which occurs when X his /4. Wih his undersanding, i is clear ha X his zero exacly when X his zero. Therefore, we have ha {T 0 = T} = a [0, { T a T 0 }, where T a is defined exacly like T a wih X replaced by X in 2.. Now, i is easy o see ha P T a T 0 defines a consisen family of probabiliy measures on he filraion F T0 T a a> ; namely he one defined hrough he Radon-Nikodym derivaives X Ta. Since P X Ta X Ta > 0, he induced measure differs from he one in Theorem 2.2. Therefore, alhough in he limi we condiion on he same even, he induced probabiliy measures srongly depend on he approximaing sequence of evens. B Proof of Proposiion 3.2 We only discuss he case sl = 0 since he case sr = 0 follows in he same way. In order o show he Markov propery of Y under Q y, we need o prove ha E Qy fy ρ+ F ρ = E Qy fy ρ+ Y ρ for all 0, for all bounded and coninuous funcions f : [l, r] R, and for all finie sopping imes ρ. On he even {ρ τ r }, he equaliy holds rivially as Y ges absorbed in l and r. On he even {ρ < τ r }, observe ha E Qy fy ρ+ F ρ = lim E Qy fyρ+ τa F ρ = lim E Qy fy τa τa ρ+ Yρ = E Qy fy ρ+ Y ρ, a r a r where he second equaliy follows from he generalized Bayes formula in Proposiion C.2 in [2] and he Markov propery of Y τa under P y. Therefore, Y is srongly Markovian under Q y. Since Y is also ime-homogeneous under any of he measures Q y, we have shown ha Y is a diffusion under {Q y } y [l,r]. As for he regulariy, fix a l, y and b y, r. Observe ha Q y is equivalen o P y on F τa τb. This fac in conjuncion wih he regulariy of Y under P and Proposiion VII.3.2 in [6] yields ha Q y τ a < > 0 as well as Q y τ b < > 0. Denoe now he exended infiniesimal generaor of Y under {Q y } y [l,r] by G, le ϕ domg wih localizing sequence {ρ n } n N, and fix y l, r. Fix wo sequences {a n } n N and {b n } n N wih a n l and b n r as n. We may assume, wihou loss of generaliy, ha ρ n τ an τ bn. By definiion of he exended infiniesimal generaor, ϕy ρn ϕy ρn 0 GϕY s ds is a Q y -maringale. Since ϕ and Gϕ are bounded on [a n, b n ] his fac, in conjuncion wih Fubini s heorem, yields ha ρn ϕy ρn sy ρn ϕysy GϕYu ρn syu ρn du sy is a P y -maringale. Since {ρ n } n N converges P y -almos surely o τ l τ r for all y l, r his implies ha ϕs doml and L[sϕ]y = Gϕysy. The oher inclusion can be shown in he same manner, which complees he proof. 9 0

10 C Jumps Here we illusrae on a simple example ha our resuls abou upward condiioning can be exended o cerain jump processes. Towards his end, we consider he canonical space of pahs ω aking values in [0, ], geing absorbed in eiher 0 or, and being càdlàg on [0, T ω. The measure P is chosen in such a way ha he canonical process X is a purely disconinuous maringale saring in, whose semimaringale characerisics under he runcaion funcion hx = x x are given by 0, 0, ν. Here ν is a predicable random measure, he compensaor of he jump measure of X. We assume ha νω, ds, dx = ν in ω, ds 2 δ /N + δ /N dx, for some N N, where ν in denoes he jump inensiy, and ha ν in ω, ds X s ων in ω, ds; o wi, X only has jumps of size ±/N and ges absorbed when hiing 0. We furhermore assume ha ν is bounded away from and 0; ha is, ha for all 0 here exis wo nonnegaive funcions c and C ending o infiniy as increases such ha {X ω>0}c ν in ω, ds C. [0,] For example, X could be a compound Poisson process wih jumps of size ±/N, geing absorbed in 0. The condiions on X guaranee ha P T 0 < = since a one-dimensional random walk is recurren; furhermore, X saisfies P T n/n < > 0 for all n N. Therefore, he asserion of Lemma 2. holds for a = n/n for all n N wih n N; hence, he h-ransform Q, defined by dq F = X dp F, equals he upward condiioned measure Q, defined as he exension of he measures {P T n/n T 0 } n N. Girsanov s heorem Theorem III.3.24 in Jacod and Shiryaev [7] implies ha, under he probabiliy measure Q = Q, he process X has semimaringale characerisics 0, 0, ν, where ν ω, ds, dx = ν in ω, ds 2 Xs ω N X s ω δ /N + X s ω + N X s ω δ /N dx. These compuaions show ha we canno expec /X o be a Q-local maringale; indeed, in our example, he process /X is bounded by N and a rue Q-supermaringale. Thus, we canno obain P hrough condiioning X downward as we did for he coninuous case in Subsecion 2.2. Consider now he case of deerminisic jump imes wih ν in ω, ds = {Xs ω>0}δ nδ ds, n= where δ := /N 2. Wih a sligh misuse of noaion allowing X 0 o ake he value x = n/n for some n N, observe ha, for all C 2 -funcions f, δ E Q[fX δ X 0 = x] fx = [ E P fx δ X ] δ δ x X 0 = x fx [ = N 2 2 f x + x + /N + N x 2 f x ] x /N fx N x = [f 2 N 2 x + + f x ] 2fx N N 0

11 + x N [ f x + + f x ] 2 N N 2 f x + x f x. Using argumens based on he maringale problem, we obain he weak convergence of X under Q o a Bessel process as N ends o infiniy see Corollary in [5]. On he oher side, Donsker s heorem implies ha X converges weakly o a Brownian moion under P. We hus recover Piman s proof ha upward condiioned Brownian moion is a Bessel process; see Piman [4]. References [] Fabrice Baudoin, Condiioned sochasic differenial equaions: heory, examples and applicaion o finance, Sochasic Process. Appl , MR 9960 [2] Peer Carr, Travis Fisher, and Johannes Ruf, On he hedging of opions on exploding exchange raes, Preprin, arxiv: , 202. [3] Freddy Delbaen and Waler Schachermayer, Arbirage possibiliies in Bessel processes and heir relaions o local maringales, Probab. Theory Relaed Fields , no. 3, MR [4] Joseph L. Doob, Condiional Brownian moion and he boundary limis of harmonic funcions, Bull. Soc. Mah. France , MR [5] Sewar N. Ehier and Thomas G. Kurz, Markov processes: Characerizaion and convergence, John Wiley & Sons, Hoboken, NJ, 986. MR [6] Hans Föllmer, The exi measure of a supermaringale, Z. Wahrscheinlichkeisheorie und Verw. Gebiee 2 972, MR [7] Jean Jacod and Alber N. Shiryaev, Limi heorems for sochasic processes, 2nd ed., Springer, Berlin, MR [8] Ioannis Karazas and Seven E. Shreve, Brownian moion and sochasic calculus, 2nd ed., Springer, Berlin, 99. MR 2940 [9] Consaninos Kardaras, On he sochasic behavior of opional processes up o random imes, Preprin, arxiv:007.24, 202. [0] Frank B. Knigh, Brownian local imes and aboo processes, Trans. Amer. Mah. Soc , MR [] Henry P. McKean, Excursions of a non-singular diffusion, Z. Wahrscheinlichkeisheorie und Verw. Gebiee 963, MR [2] Paul A. Meyer, La mesure de H. Föllmer en héorie de surmaringales, Séminaire de Probabiliés, VI, Springer, Berlin, 972, pp MR [3] Soumik Pal and Philip E. Proer, Analysis of coninuous sric local maringales via h- ransforms, Sochasic Process. Appl , no. 8, MR [4] James W. Piman, One-dimensional Brownian moion and he hree-dimensional Bessel process, Adv. in Appl. Probab , no. 3, MR [5] Jim Piman and Marc Yor, Bessel processes and infiniely divisible laws, Sochasic Inegrals, LMS Durham Symposium 980, 98, pp MR

12 [6] Daniel Revuz and Marc Yor, Coninuous maringales and Brownian moion, 3rd ed., Springer, Berlin, 999. MR [7] Bernard Roynee and Marc Yor, Penalising Brownian pahs, Springer, Berlin, MR [8] Daniel W. Sroock and S. R. Srinivasa Varadhan, Mulidimensional Diffusion Processes, Springer, Berlin, Berlin, 2006, Reprin of he 997 ediion. MR [9] David Williams, Pah decomposiion and coninuiy of local ime for one-dimensional diffusions, I., Proc. Lond. Mah. Soc , MR