Functional Itô calculus and Applications. David-Antoine FOURNIE. Supervised by Professor Rama CONT

Transcription

1 Functonal Itô calculus and Applcatons Davd-Antone FOURNIE Supervsed by Professor Rama CONT Submtted n partal fulfllment of the requrements for the degree of Doctor of Phlosophy n the Graduate School of Arts and Scences COLUMBIA UNIVERSITY 21

2 c 21 Davd-Antone FOURNIE All Rghts Reserved

3 ABSTRACT Functonal Itô calculus and Applcatons Davd-Antone FOURNIE Ths thess studes extensons of the Itô calculus to a functonal settng, usng analytcal and probablstc methods, and applcatons to the prcng and hedgng of dervatve securtes. The frst chapter develops a non-antcpatve pathwse calculus for functonals of two cadlag paths, wth a predctable dependence n the second one. Ths calculus s a functonal generalzaton of Follmer s analytcal approach to Itô calculus. An Itô-type change of varable formula s obtaned for non-antcpatve functonals on the space of rght-contnuous paths wth left lmts, usng purely analytcal methods. The man tool s the Dupre dervatve, a Gateaux dervatve for non-antcpatve functonals on the space of rght-contnuous paths wth left lmts. Our framework mples as a specal case a pathwse functonal Itô calculus for cadlag semmartngales and Drchlet processes. It s shown how ths analytcal Itô formula mples a probablstc Itô formula for general cadlag semmartngales. In the second chapter, a functonal extenson of the Itô formula s derved usng stochastc analyss tools and used to obtan a constructve martngale representaton theorem for a class of contnuous martngales verfyng a regularty property. By contrast wth the Clark- Haussmann-Ocone formula, ths representaton nvolves non-antcpatve quanttes whch can be computed pathwse. These results are used to construct a weak dervatve actng on square-ntegrable martngales, whch s shown to be the nverse of the Itô ntegral, and derve an ntegraton by parts formula for Itô stochastc ntegrals. We show that ths weak dervatve may be vewed as a non-antcpatve lftng of the Mallavn dervatve. Regular functonals of an Itô martngale whch have the local martngale property are characterzed as solutons of a functonal dfferental equaton, for whch a unqueness result s gven.

4 It s also shown how a smple verfcaton theorem based on a functonal verson of the Hamlton-Jacob-Bellman equaton can be stated for a class of path-dependent stochastc control problems. In the thrd chapter, a generalzaton of the martngale representaton theorem s gven for functonals satsfyng the regularty assumptons for the functonal Itô formula only n a local sense, and a suffcent condton takng the form of a functonal dfferental equaton s gven for such locally regular functonal to have the local martngale property. Examples are gven to llustrate that the noton of local regularty s necessary to handle processes arsng as the prces of fnancal dervatves n computatonal fnance. In the fnal chapter, functonal Itô calculus for locally regular functonals s appled to the senstvty analyss of path-dependent dervatve securtes, followng an dea of Dupre. A general valuaton functonal dfferental equaton s gven, and many examples show that all usual optons n local volatlty model are actually prced by ths equaton. A defnton s gven for the usual senstvtes of a dervatve, and a rgorous expresson of the concept of Γ Θ tradeoff s gven. Ths expresson s used together wth a perturbaton result for stochastc dfferental equatons to gve an expresson for the Vega bucket exposure of a path-dependent dervatve, as well as ts of Black-Scholes Delta and Delta at a gven skew stckness rato. An effcent numercal algorthm s proposed to compute these senstvtes n a local volatlty model.

5 Table of Contents 1 Synopss 1 2 Pathwse calculus for non-antcpatve functonals Motvaton Non-antcpatve functonals on spaces of paths Horzontal and vertcal perturbaton of a path Classes of non-antcpatve functonals Measurablty propertes Pathwse dervatves of non-antcpatve functonals Horzontal dervatve Vertcal dervatve Unqueness results for vertcal dervatves Change of varable formula for functonals of a contnuous path Change of varable formula for functonals of a cadlag path Functonals of Drchlet processes Functonals of semmartngales Cadlag semmartngales Contnuous semmartngales Functonal Itô calculus and applcatons Functonals representaton of non-antcpatve processes Functonal Itô calculus

6 3.2.1 Space of paths Obstructons to regularty Functonal Itô formula Intrnsc nature of the vertcal dervatve Martngale representaton formula Martngale representaton theorem Relaton wth the Mallavn dervatve Weak dervatves and ntegraton by parts for stochastc ntegrals Functonal equatons for martngales Functonal verfcaton theorem for a non-markovan stochastc control problem Control problem Optmal control and functonal Hamlton-Jacob-Bellman equaton, frst verson Optmal control and functonal Hamlton-Jacob-Bellman equaton, second verson Localzaton Motvaton A local verson of the functonal Itô formula Spaces of contnuous and dfferentable functonals on optonal ntervals A local verson of functonal Itô formula Locally regular functonals Spaces of locally regular functonals A local unqueness result on vertcal dervatves Dervatves of a locally regular functonal Contnuty and measurablty propertes Martngale representaton theorem Functonal equaton for condtonal expectatons

7 5 Senstvty analyss of path-dependent dervatves Motvaton A short ntroducton to no-arbtrage prcng of dervatves securtes A short ntroducton to dervatves prcng and hedgng: a sell-sde trader s pont of vew Functonal valuaton equaton and greeks for exotc dervatve Valuaton equaton Delta, gamma and theta Θ Γ tradeoff Examples of the valuaton equaton Vanlla optons Contnuously montored barrer optons Contnuously montored Asan optons Contnuously montored varance swap Path-dependent optons wth dscrete montorng Optons on basket n a model wth an unobservable factor Senstvtes to market varables Drectonal dervatves wth respect to the volatlty functonal Senstvtes to market varables Multple Deltas of a dervatve Effcent numercal algorthm for the smultaneous computaton of Vega buckets and Deltas A Proof of theorems n chapter A.1 Some results on cadlag functons A.2 Proof of theorem A.3 Measure-theoretc lemmas used n the proof of theorem 2.4 and B Stochastc Dfferental Equatons wth functonal coeffcents 128 B.1 Stochastc dfferental equatons wth path dependent coeffcents B.1.1 Strong solutons

8 B.1.2 Contnuty n the ntal value B.1.3 Perturbaton of coeffcents v

9 Acknowledgments I thank frst and foremost my parents for ther contnuous support to pursue my dreams. 9 years of studes would never have been possble wthout ther moral, practcal and fnancal support. I thank a lot Amal Moussa for her great frendshp whch supported me throughout all my years of studes n Toulouse, Pars and New York. I wsh to thank deeply my advsor Professor Rama Cont for hs constant support and gudance durng my PhD studes at Columba Unversty, and for hs deep nvolvement n our jont work whch s compled n ths dssertaton. I am very grateful to Professor Ioanns Karatzas for my recrutment at Columba Unversty and hs support durng my PhD studes. Most of my knowledge and nterest n Probablty and Stochastc Analyss comes from hs books and hs lectures. I thank Bruno Dupre whose orgnal deas and thoughts opened the way to my research work. I thank Professor Ncole El Karou from the Ecole Polytechnque who encouraged me to go for a PhD at Columba Unversty, and my Mathematcs and Physcs Professors n Classes Préparatores Perre Gssot, Mchel Jmmy Gonnord and Jean-Mare Mercer to whom I owe my nterest n Mathematcs and Scence n general. Last but not least, all my grattude goes to the admnstratve team at the Mathematcs Department, and especally Terrance Cope, for ther great work and constant help n all ssues related to my studes here. v

10 To my parents v

11 CHAPTER 1. SYNOPSIS 1 Chapter 1 Synopss Itô s stochastc calculus [36, 37, 18, 49, 45, 54] has proven to be a powerful and useful tool n analyzng phenomena nvolvng random, rregular evoluton n tme. Two characterstcs dstngush the Itô calculus from other approaches to ntegraton, whch may also apply to stochastc processes. Frst s the possblty of dealng wth processes, such as Brownan moton, whch have non-smooth trajectores wth nfnte varaton. Second s the non-antcpatve nature of the quanttes nvolved: vewed as a functonal on the space of paths ndexed by tme, a non-antcpatve quantty may only depend on the underlyng path up to the current tme. Ths noton, frst formalzed by Doob [21] n the 195s va the concept of a fltered the dea of causalty. probablty space, s the mathematcal counterpart to Two pllars of stochastc calculus are the theory of stochastc ntegraton, whch allows to defne ntegrals T Y dx for of a large class of non-antcpatve ntegrands Y wth respect to a semmartngale X = (X(t), t [, T ]), and the Itô formula [36, 37, 49] whch allows to represent smooth functons Y (t) = f(t, X(t)) of a semmartngale n terms of such stochastc ntegrals. A central concept n both cases s the noton of quadratc varaton [X] of a semmartngale, whch dfferentates Itô calculus from the calculus of smooth functons. Whereas the class of ntegrands Y covers a wde range of non-antcpatve path-dependent functonals of X, the Itô formula s lmted to functons of the current value of X. Gven that n many applcatons such as statstcs of processes, physcs or mathematcal fnance, one s naturally led to consder functonals of a semmartngale X and ts quadratc

12 CHAPTER 1. SYNOPSIS 2 varaton process [X] such as: t g(t, X t )d[x](t), G(t, X t, [X] t ), or E[G(T, X(T ), [X](T )) F t ] (1.1) (where X(t) denotes the value at tme t and X t = (X(u), u [, t]) the path up to tme t) there has been a sustaned nterest n extendng the framework of stochastc calculus to such path-dependent functonals. In ths context, the Mallavn calculus [7, 9, 5, 48, 51, 56, 59] has proven to be a powerful tool for nvestgatng varous propertes of Brownan functonals, n partcular the smoothness of ther denstes. Yet the constructon of Mallavn dervatve, whch s a weak dervatve for functonals on Wener space, does not refer to the underlyng fltraton F t. Hence, t naturally leads to representatons of functonals n terms of antcpatve processes [9, 34, 51], whereas n applcatons t s more natural to consder non-antcpatve, or causal, versons of such representatons. In a recent nsghtful work, B. Dupre [23] has proposed a method to extend the Itô formula to a functonal settng n a non-antcpatve manner. Buldng on ths nsght, we develop hereafter a non-antcpatve calculus for a class of functonals -ncludng the above examples- whch may be represented as Y (t) = F t ({X(u), u t}, {A(u), u t}) = F t (X t, A t ) (1.2) where A s the local quadratc varaton defned by [X](t) = t A(u)du and the functonal F t : D([, t], R d ) D([, t], S + d ) R represents the dependence of Y on the underlyng path and ts quadratc varaton. For such functonals, we defne an approprate noton of regularty (Secton 2.2.2) and a nonantcpatve noton of pathwse dervatve (Secton 2.3). Introducng A t as addtonal varable allows us to control the dependence of Y wth respect to the quadratc varaton [X] by requrng smoothness propertes of F t wth respect to the varable A t n the supremum norm, wthout resortng to p-varaton norms as n rough path theory [46]. Ths allows to consder a wder range of functonals, as n (1.1).

13 CHAPTER 1. SYNOPSIS 3 Usng these pathwse dervatves, we derve n chapter 2 a purely analytcal Itô formula for functonals of two varable (x, v) whch are cadlag paths, and the functonal has a predctable dependence n the second varable. Ths formula (Theorems 2.4, 2.5) s preceded by other useful analytcal results on the class of functonals that we consder. Our method follows the sprt of H. Föllmer s [29] pathwse approach to Itô calculus, where the term takng the place of the stochastc ntegral n stochastc calculus s defnng as a dagonal lmt of dscretzed ntegrals wth ntegrand evaluated on dscretzed paths. It s then show n secton 2.6 that ths analytcal formula allows to defne a pathwse noton of stochastc ntegral for functonals of a cadlag Drchlet processes, for whch a functonal Itô formula s derved, and n secton 2.7 t s shown that t mples an Itô formula for functonals of cadlag semmartngales. In the case of contnuous semmartngales, we provde an alternatve dervaton (2.4) under dfferent regularty assumptons. In chapter 3, a drect probablstc dervaton of the functonal Itô formula s provded for contnuous semmartngales. We then use ths functonal Itô formula to derve a constructve verson of the martngale representaton theorem (Secton 3.3), whch can be seen as a non-antcpatve form of the Clark-Haussmann-Ocone formula [9, 33, 34, 51]. The martngale representaton formula allows to obtan an ntegraton by parts formula for Itô stochastc ntegrals (Theorem 3.4), whch enables n turn to defne a weak functonal dervatve, for a class of stochastc ntegrals (Secton 3.4). We argue that ths weak dervatve may be vewed as a non-antcpatve lftng of the Mallavn dervatve (Theorem 3.6). We then show that regular functonals of an Itô martngale whch have the local martngale property are characterzed as solutons of a functonal analogue of Kolmogorov s backward equaton (Secton 3.5), for whch a unqueness result s gven (Theorem 3.8). Fnally, n secton 3.6, we present as a potental drecton for further research a settng of path-dependent stochastc control problem, wth dependence of the coeffcents of the dffuson as well as the objectve and cost functons on the whole path of the controlled process, and eventually on the path of ts quadratc varaton. We are able to prove two versons of a verfcaton theorem based on a functonal verson of the Hamlton-Jacob-Bellman equaton, theorems 3.9 and 3.1, dependng on whether or not there s explct dependence on the quadratc

14 CHAPTER 1. SYNOPSIS 4 varaton of the controlled process. In chapter 4, we present a local verson, usng optonal tmes, of some results from chapter 3, especally the martngale representaton theorem 4.6, n the sense that they apply to functonals defned on contnuous paths whch can be extended only locally to regular functons of cadlag paths. A suffcent condton, takng the form of a functonal dfferental equaton, s gven on such functonals for defnng local martngales (Theorem 4.7). Ths extenson s motvated by examples of functonals whch defne martngales and do satsfy a functonal dfferental equaton, but fal to satsfy the regularty assumptons of chapter 3. The choce of the examples come from processes tradtonally encountered n Mathematcal Fnance, and hence show that chapter 4 s necessary n order to use the functonal settng for the senstvty analyss of path-dependent dervatves (Chapter 5). In the fnal chapter, buldng on Dupre s orgnal nsght [23], we show how the settng of functonal Itô calculus s a natural formalsm for the hedgng of path-dependent dervatves, emphaszng the noton of senstvty of the opton prce to the underlyngs and to market varables. A valuaton functonal dfferental equaton (Theorem 5.1) s derved, and shows that the theoretcal replcaton portfolo of the dervatve s the portfolo hedgng the drectonal senstvty. Ths theorem also extends the classcal relatonshp between the senstvtes of vanlla optons to general path-dependent payoffs, and hence gves a precse meanng to the concept of Theta - Gamma tradeoff whch s famlar to dervatves traders (Theorem 5.2). We then show that the valuaton functonal equaton apples to most payoffs encountered n the markets; n partcular the dfferent classcal PDEs satsfed by the prces of Vanlla, Barrer, Asan, Varance Swap optons are actually shown to be partcular cases of ths unversal functonal equaton, whch can therefore be seen as a unfed descrpton for dervatves prcng. We then buld on the functonal settng to gve expressons for the senstvtes of the dervatve to observable market varables, such as the Vega bucket exposure (Secton 5.4.2), whch are the man tool for a volatlty trader to understand hs exposure. We also provde wth expressons for the Black-Scholes Delta and Delta at a gven skew stckness rato (Secton 5.4.3), whch are used by practton-

15 CHAPTER 1. SYNOPSIS 5 ers to hedge ther portfolo. We then proceed to suggest an effcent numercal algorthm for the computaton of these senstvtes. Ths fnal chapter may be seen as an attempt to formalze concepts used n dervatves tradng from the pont of vew of a sell-sde traders. Appendx A contans numerous techncal lemmas used n the proofs n chapter 2, more precsely results on the approxmaton of cadlag functons by pecewse-constant functons and measure-theoretc results. Appendx B s a self-contaned dgresson on strong solutons for stochastc dfferental equatons wth functonal coeffcents, whch s the settng we need for the secton on stochastc control (Secton 3.6). In partcular, t defnes a concept of strong soluton startng from a gven ntal value, n the case where the coeffcent have an explct dependence on the path of the quadratc varaton (Defnton B.2). Contnuty propertes of the soluton n the ntal value s also nvestgated (Secton B.1.2), as well as perturbaton of the coeffcent (Secton B.1.3). Ths last perturbaton result has applcaton n the computaton of Vega bucket exposure and Deltas n sectons and

16 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 6 Chapter 2 Pathwse calculus for non-antcpatve functonals 2.1 Motvaton In hs semnal paper Calcul d Ito sans probabltés [29], Hans Föllmer proposed a nonprobablstc verson of the Ito formula [36]: Föllmer showed that f a real-valued cadlag (rght contnuous wth left lmts) functon x has fnte quadratc varaton along a sequence π n = (t n k ) k=..n of subdvsons of [, T ] wth step sze decreasng to zero, n the sense that the sequence of dscrete measures n 1 x(t n k+1 ) x(tn k ) 2 δ t n k k= converges vaguely to a Radon measure wth Lebesgue decomposton ξ + t [,T ] x(t) 2 δ t then for f C 1 (R) one can defne the pathwse ntegral T n 1 f(x(t))d π x = lm f(x(t n n )).(x(t n +1) x(t n )) (2.1) = as a lmt of Remann sums along the subdvson π = (π n ) n 1. In partcular f X = (X t ) t [,T ] s a semmartngale [18, 49, 54], whch s the classcal settng for stochastc calculus, the paths of X have almost surely fnte quadratc varaton along such subsequences: when appled to the paths of X, Föllmer s ntegral (2.1) then concdes, wth probablty

17 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 7 one, wth the Ito stochastc ntegral T f(x)dx wth respect to the semmartngale X. Ths constructon may n fact be carred out for a more general class of processes, ncludng the class of Drchlet processes [12, 29, 3, 47]. Of course, the Ito stochastc ntegral wth respect to a semmartngale X may be defned for a much larger class of ntegrands: n partcular, for a cadlag process Y defned as a nonantcpatve functonal Y (t) = F t (X(u), u t) of X, the stochastc ntegral T Y dx may be defned as a lmt of non-antcpatve Remann sums [54]. Usng a noton of drectonal dervatve for functonals proposed by Dupre [23], we extend Föllmer s pathwse change of varable formula to non-antcpatve functonals on the space D([, T ], R d ) of cadlag paths (Theorem 2.4). The requrement on the functonals s to possess certan drectonal dervatves whch may be computed pathwse. Our constructon allows to defne a pathwse ntegral F t (x)dx, defned as a lmt of Remann sums, for a class of functonals F of a cadlag path x wth fnte quadratc varaton. Our results lead to functonal extensons of the Ito formula for semmartngales (Secton 2.7) and Drchlet processes (Secton 2.6). In partcular, we show the stablty of the the class of semmartngales under functonal transformatons verfyng a regularty condton. These results yeld a non-probablstc proof for functonal Ito formulas obtaned n [23] usng probablstc methods and extend them to the case of dscontnuous semmartngales. Notaton For a path x D([, T ], R d ), denote by x(t) the value of x at t and by x t = (x(u), u t) the restrcton of x to [, t]. Thus x t D([, t], R d ). For a stochastc process X we shall smlarly denote X(t) ts value at t and X t = (X(u), u t) ts path on [, t]. 2.2 Non-antcpatve functonals on spaces of paths Let T >, and U R d be an open subset of R d and S R m be a Borel subset of R m. We call U-valued cadlag functon a rght-contnuous functon f : [, T ] U wth left lmts such that for each t ], T ], f(t ) U. Denote by U t = D([, t], U) (resp. S t = D([, t], S) the space of U-valued cadlag functons (resp. S), and C ([, t], U) the set of contnuous functons wth values n U.

18 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 8 When dealng wth functonals of a path x(t) ndexed by tme, an mportant class s formed by those whch are non-antcpatve, n the sense that they only depend on the past values of x. A famly Y : [, T ] U T R of functonals s sad to be non-antcpatve f, for all (t, x) [, T ] U T, Y (t, x) = Y (t, x t ) where x t = x [,t] denotes the restrcton of the path x to [, t]. A non-antcpatve functonal may thus be represented as Y (t, x) = F t (x t ) where (F t ) t [,T ] s a famly of maps F t : U t R. Ths motvates the followng defnton: Defnton 2.1 (Non-antcpatve functonals on path space). A non-antcpatve functonal on U T s a famly F = (F t ) t [,T ] of maps F t : U t R Y s sad to be predctable 1 f, for all (t, x) [, T ] U T, Y (t, x) = Y (t, x t ) where x t denotes the functon defned on [, t] by x t (u) = x(u) u [, t[ x t (t) = x(t ) Typcal examples of predctable functonals are ntegral functonals, e.g. Y (t, x) = t G s (x s )ds where G s a non-antcpatve, locally ntegrable, functonal. If Y s predctable then Y s non-antcpatve, but predctablty s a stronger property. Note that x t s cadlag and should not be confused wth the caglad path u x(u ). We consder throughout ths work non-antcpatve functonals F = (F t ) t [,T ] F t : U t S t R where F has a predctable dependence wth respect to the second argument: t T, (x, v) U t S t, F t (x t, v t ) = F t (x t, v t ) (2.2) F can be vewed as a functonal on the vector bundle Υ = t [,T ] U t S t. We wll also consder non-antcpatve functonals F = (F t ) t [,T [ ndexed by [, T [. 1 Ths noton concdes wth the usual defnton of predctable process when the path space U T s endowed wth the fltraton of the canoncal process, see Dellachere & Meyer [18, Vol. I].

19 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS Horzontal and vertcal perturbaton of a path Consder a path x D([, T ]), U) and denote by x t U t ts restrcton to [, t] for t < T. For h, the horzontal extenson x t,h D([, t + h], R d ) of x t to [, t + h] s defned as x t,h (u) = x(u) u [, t] ; x t,h (u) = x(t) u ]t, t + h] (2.3) For h R d small enough, we defne the vertcal perturbaton x h t obtaned by shftng the endpont by h: of x t as the cadlag path x h t (u) = x t (u) u [, t[ x h t (t) = x(t) + h (2.4) or n other words x h t (u) = x t (u) + h1 t=u. By conventon, x u t,h = (xu t ) t,h, e the vertcal perturbaton precedes the horzontal extenson. We now defne a dstance between two paths, not necessarly defned on the same tme nterval. For T t = t + h t, (x, v) U t S + t defne and (x, v ) D([, t + h], R d ) S t+h d ( (x, v), (x, v ) ) = sup x t,h (u) x (u) + sup v t,h (u) v (u) + h (2.5) u [,t+h] u [,t+h] If the paths (x, v), (x, v ) are defned on the same tme nterval, then d ((x, v), (x, v )) s smply the dstance n supremum norm Classes of non-antcpatve functonals Usng the dstance d defned above, we now ntroduce varous notons of contnuty for non-antcpatve functonals. Defnton 2.2 (Contnuty at fxed tmes). A non-antcpatve functonal F = (F t ) t [,T ] s sad to be contnuous at fxed tmes f for any t T, F t : U t S t R s contnuous for the supremum norm. Defnton 2.3 (Left-contnuous functonals). Defne F l as the set of functonals F = (F t, t [, T ]) whch satsfy: t [, T ], ɛ >, (x, v) U t S t, η >, h [, t], (x, v ) U t h S t h, d ((x, v), (x, v )) < η F t (x, v) F t h (x, v ) < ɛ (2.6)

20 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 1 Defnton 2.4 (Rght-contnuous functonals). Defne F r as the set of functonals F = (F t, t [, T [) whch satsfy: t [, T ], ɛ >, (x, v) U t S t, η >, h [, T t], (x, v ) U t+h S t+h, d ((x, v), (x, v )) < η F t (x, v) F t+h (x, v ) < ɛ (2.7) paths: We denote F = F r F l the set of contnuous non-antcpatve functonals. We call a functonal boundedness preservng f t s bounded on each bounded set of Defnton 2.5 ( Boundedness-preservng functonals). Defne B as the set of non-antcpatve functonals F such that for every compact subset K of U, every R >, there exsts a constant C K,R such that: t T, (x, v) D([, t], K) S t, sup v(s) < R F t (x, v) < C K,R (2.8) s [,t] In partcular f F B, t s locally bounded n the neghborhood of any gven path.e. (x, v) U T S T, C >, η >, t [, T ], (x, v ) U t S t, d ((x t, v t ), (x, v )) < η t [, T ], F t (x, v ) C (2.9) The followng lemma shows that a contnuous functonal also satsfes the local boundedness property (2.9). Lemma 2.1. If F F, then t satsfes the property of local boundedness 2.9. Proof. Let F F and (x, v) U T S T. For each t < T, there exsts η(t) such that, for all t < T, (x, v ) U t S t : d ((x t, v t ), (x, v )) < η(t) F t (x t, v t ) F t (x, v ) < 1 d ((x t, v t ), (x, v )) < η(t) F t (x t, v t ) F t (x, v ) < 1 (2.1)

21 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 11 Snce (x, v) has cadlag trajectores, for each t < T, there exsts ɛ(t) such that: t > t, t t < ɛ(t) d ((x t, v t ), (x t, v t )) < η(t) 2 t < t, t t < ɛ(t) d ((x t, v t ), (x t, v t )) < η(t) 2 (2.11) Therefore one can extract a fnte coverng of compact set [, T ] by such ntervals N [, T ] (x tj ɛ(t j ), x tj + ɛ(t j )) (2.12) j=1 Let t < T and (x, v ) U t S t. Assume that: d ((x t, v t ), (x, v )) < η(t j ) mn 1 j N 2 (2.13) t (x tj ɛ(t j ), x tj + ɛ(t j )) for some j N. If t < t j, then: d ((x tj, v tj ), (x, v )) < d ((x t, v t ), (x, v )) + d ((x t, v t ), (x tj, v tj )) (2.14) where both terms n the sum are less than η(t j) 2, so that: F t (x, v ) < F tj (x tj, v tj ) + 1 (2.15) If t t j, then: d ((x tj, v tj ), (x, v )) < d ((x t, v t ), (x, v )) + d ((x t, v t ), (x tj, v tj )) (2.16) where both terms n the sum are less than η(t j) 2, so that: F t (x, v ) < F tj (x tj, v tj ) + 1 (2.17) so that n any case: F t (x, v ) < max 1 j N max( F t j (x tj, v tj ), F tj (x tj, v tj ) ) + 1 (2.18) The followng result descrbes the behavor of paths generated by the functonals n the above classes:

22 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 12 Proposton 2.1 (Pathwse regularty). 1. If F F l then for any (x, v) U T S T, the path t F t (x t, v t ) s left-contnuous. 2. If F F r then for any (x, v) U T S T, the path t F t (x t, v t ) s rght-contnuous. 3. If F F then for any (x, v) U T S T, the path t F t (x t, v t ) s cadlag and contnuous at all ponts where x and v are contnuous. 4. If F F further verfes (2.2) then for any (x, v) U T S T, the path t F t (x t, v t ) s cadlag and contnuous at all ponts where x s contnuous. 5. If F B, then for any (x, v) U T S T, the path t F t (x t, v t ) s bounded. Proof. 1. Let F F l and t [, T ). For h > suffcently small, d ((x t h, v t h ), (x t, v t )) = sup u (t h,t) x(u) x(t ) + sup u (t h,t) Snce x and v are cadlag, ths quantty converges to as h +, so F t h (x t h, v t h ) F t (x t, v t ) h + v(u) v(t ) + h (2.19) so t F t (x t, v t ) s left-contnuous. 2. Let F F r and t [, T ). For h > suffcently small, d ((x t+h, v t+h ), (x t, v t )) = sup u [t,t+h) x(u) x(t) + sup u [t,t+h) Snce x and v are cadlag, ths quantty converges to as h +, so F t+h (x t+h, v t+h ) F t (x t, v t ) h + v(u) v(t) + h (2.2) so t F t (x t, v t ) s rght-contnuous. 3. Assume now that F s n F and let t ], T ]. Denote ( x(t), v(t)) the jump of (x, v) at tme t. Then d ((x t h, v t h ), x x(t) t, v v(t) t )) = sup x(u) x(t) + u [t h,t) sup u [t h,t) v(u) v(t) + h

23 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 13 and ths quantty goes to because x and v have left lmts. left lmt F t (x x(t) t F t (x t, v t ). Hence the path has, v v(t) ) at t. A smlar reasonng proves that t has rght-lmt t 4. If F F verfes (2.2), for t ], T ] the path t F t (x t, v t ) has left-lmt F t (x x(t) t, v v(t) t ) at t, but (2.2) mpled that ths left-lmt equals F t (x x(t) t, v t ) Measurablty propertes Consder, on the path space U T S T, endowed wth the supremum norm and ts Borel σ-algebra, the fltraton (F t ) generated by the canoncal process (X, V ) : U T S T [, T ] U S (x, v), t (X, V )((x, v), t) = (x(t), v(t)) (2.21) F t s the smallest sgma-algebra on U T S T such that all coordnate maps (X(., s), V (., s)), s [, t] are F t -measurable. The optonal sgma-algebra O on U T S T [, T ] s the sgma-algebra on U T S T [, T ] generated by all mappngs f : U T S T [, T ] R the set nto whch, for every ω U T S T, are rght contnuous n t, have lmts from the left and are adapted to (F t ) t [,T ]. The predctable sgma-algebra P s the sgma-algebra on U T S T [, T ] generated by all mappngs f : U T S T [, T ] R the set nto whch, for every ω U T S T, are leftcontnuous n t and are adapted to (F t ) t [,T ]. A postve map τ : U T S T [, [ s called an optonal tme f {ω U T S T, τ(ω) < t} F t for every t [, T ]. The followng result, proved n Appendx A.2, clarfes the measurablty propertes of processes defned by functonals n F l, F r : Theorem 2.1. If F s contnuous at fxed tme, then the process Y defned by Y ((x, v), t) = F t (x t, v t ) s F t -adapted. If F F l or F F r, then: 1. the process Y defned by Y ((x, v), t) = F t (x t, v t ) s optonal.e. O-measurable. 2. the process Z defned by Z((x, v), t) = F t (x t, v t ) s predctable.e. P-measurable.

24 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS Pathwse dervatves of non-antcpatve functonals Horzontal dervatve We now defne a pathwse dervatve for a non-antcpatve functonal F = (F t ) t [,T ], whch may be seen as a Lagrangan dervatve along the path x. Defnton 2.6 (Horzontal dervatve). The horzontal dervatve at (x, v) U t S t of a non-antcpatve functonal F = (F t ) t [,T [ s defned as D t F (x, v) = lm h + F t+h (x t,h, v t,h ) F t (x, v) h (2.22) f the correspondng lmt exsts. If (2.22) s defned for all (x, v) Υ the map D t F : U t S t R d (x, v) D t F (x, v) (2.23) defnes a non-antcpatve functonal DF = (D t F ) t [,T [, the horzontal dervatve of F. We wll occasonally use the followng local Lpschtz property that s weaker than horzontal dfferentablty: Defnton 2.7. A non-antcpatve functonal F s sad to have the horzontal local Lpschtz property f and only f: (x, v) U T S T, C >, η >, t 1 < t 2 T, (x, v ) U t1 S t1, d ((x t1, v t1 ), (x, v )) < η F t2 (x t 1,t 2 t 1, v t 1,t 2 t 1 ) F t1 ((x t 1, v t 1 )) < C(t 2 t 1 ) (2.24) Vertcal dervatve Dupre [23] ntroduced a pathwse spatal dervatve for non-antcpatve functonals, whch we now ntroduce. Denote (e, = 1..d) the canoncal bass n R d. Defnton 2.8. A non-antcpatve functonal F = (F t ) t [,T ] s sad to be vertcally dfferentable at (x, v) D([, t]), R d ) D([, t], S + d ) f R d R e F t (x e t, v t )

25 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 15 s dfferentable at. Its gradent at F t (x he t, v) F t (x, v) x F t (x, v) = ( F t (x, v), = 1..d) where F t (x, v) = lm (2.25) h h s called the vertcal dervatve of F t at (x, v). If (2.25) s defned for all (x, v) Υ, the vertcal dervatve x F : U t S t R d (x, v) x F t (x, v) (2.26) defne a non-antcpatve functonal x F = ( x F t ) t [,T ] wth values n R d. Remark 2.1. If a vertcally dfferentable functonal s predctable wth respect to the second varable (F t (x t, v t ) = F t (x t, v t )), so s ts vertcal dervatve. Remark 2.2. F t (x, v) s smply the drectonal dervatve of F t n drecton (1 {t} e, ). Note that ths nvolves examnng cadlag perturbatons of the path x, even f x s contnuous. Remark 2.3. If F t (x, v) = f(t, x(t)) wth f C 1,1 ([, T [ R d ) then we retreve the usual partal dervatves: D t F (x, v) = t f(t, x(t)) x F t (x t, v t ) = x f(t, x(t)). Remark 2.4. Note that the assumpton (2.2) that F s predctable wth respect to the second varable entals that for any t [, T ], F t (x t, vt e ) = F t (x t, v t ) so an analogous noton of dervatve wth respect to v would be dentcally zero under assumpton (2.2). If F admts a horzontal (resp. vertcal) dervatve DF (resp. x F ) we may terate the operatons descrbed above and defne hgher order horzontal and vertcal dervatves. Defnton 2.9. Defne C j,k as the set of functonals F whch are contnuous at fxed tmes, admt j horzontal dervatves and k vertcal dervatves at all (x, v) U t S t, t [, T [ D m F, m j, n xf, n k are contnuous at fxed tmes.

26 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS Unqueness results for vertcal dervatves The analytcal Itô formula for contnuous paths (theorem 2.4), and ts probablstc counterpart (theorem 3.1), refer explctly to the vertcal dervatves of the functonal F, whch requres the functonal to be defned on cadlag path although ts argument x s contnuous. Snce a functonal defned on contnuous paths could be extended to Υ n multple ways, the vertcal dervatve and therefore Itô s formula seem to depend on the chosen extenson. The followng two theorems 2.2 and 2.3 show that ths s ndeed not the case, as the value of the vertcal dervatves on contnuous paths do not depend on the chosen extenson. Theorem 2.2. If F 1, F 2 C 1,1, wth F, x F F l and DF satsfyng the local boundedness assumpton 2.9 for = 1, 2, concde on contnuous paths: t ], T ] (x, v) UT c S T, Ft 1 (x t, v t ) = Ft 2 (x, v) then t ], T ], (x, v) UT c S T, x Ft 1 (x t, v t ) = x Ft 2 (x t, v t ) Proof. Let F = F 1 F 2 C 1,1 and (x, v) UT c S T. Then F t (x, v) = for all < t T. It s then obvous that D t F (x, v) s also on contnuous paths because the extenson (x t,h ) of x t s tself a contnuous path. Assume now that there exsts some (x, v) UT c S T such that for some 1 d and t ], T ], F t (x t, v t ) >. Let α = 1 2 F t (x t, v t ). By the left-contnuty of F and D t F at (x t, v t ), we may choose l < T t suffcently small such that, for any t [, t ], for any (x, v ) U t S t, d ((x t, v t ), (x, v )) < l F t (x, v ) > α and D t F (x, v ) < 1 (2.27) Choose t < t such that d ((x t, v t ), (x t, v t )) < l 2 x t to [, t + h], where h < l 4 (t t): and defne the followng extenson of z(u) = x(u), u t z j (u) = x j (t) + 1 =j (u t), t u t + h, 1 j d (2.28) Defne the followng sequence of pecewse constant approxmatons of z: z n j (u) = x j (t) + 1 =j h n n k=1 z n (u) = z(u), u t 1 kh u t, t u t + h, 1 j d (2.29) n

27 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 17 Snce d ((z, v t,h ), (z n, v t,h )) = h n, F t+h (z, v t,h ) F t+h (z n, v t,h ) n + We can now decompose F t+h (z n, v t,h ) F t (x, v) as F t+h (z n, v t,h ) F t (x, v) = + n k=1 n k=1 F t+ kh (z n n t+ kh n F t+ kh (z n n t+ kh, v t, kh n n, v t, kh n ) F t+ (k 1)h n ) F t+ kh (z n n t+ kh, v t, kh ) n n (z n t+ (k 1)h n, v (k 1)h t, ) (2.3) n where the frst sum corresponds to jumps of z n at tmes t + kh n extenson by a constant on [t + (k 1)h n, t + kh n [. and the second sum to ts where φ s defned as F t+ kh (z n n t+ kh n, v t, kh n ) F t+ kh (z n n t+ kh, v t, kh ) = φ( h ) φ() (2.31) n n n φ(u) = F t+ kh ((z n ) ue n t+ kh, v t, kh ) n n Snce F s vertcally dfferentable, φ s dfferentable and Snce φ (u) > α hence: On the other hand where n k=1 φ (u) = F t+ kh ((z n ) ue n t+ kh, v t, kh ) n n d ((x t, v t ), ((z n ) ue t+ kh, v t, kh )) h, n n F t+ kh (z n n t+ kh n F t+ kh (z n n t+ kh, v t, kh n n, v t, kh n ) F t+ (k 1)h n ψ(u) = F t+ (k 1)h+u n ) F t+ kh (z n n t+ kh, v t, kh ) > αh. n n (z n t+ (k 1)h n (z n t+ (k 1)h+u n, v (k 1)h t, ) = ψ( h n n ) ψ(), v (k 1)h+u t, ) n so that ψ s rght-dfferentable on ], h n [ wth rght-dervatve: ψ r(u) = D t+ (k 1)h+u n F (z n t+ (k 1)h+u n, v (k 1)h+u t, ) n

28 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 18 Snce F F l, ψ s also left-contnous contnuous by theorem 4 so n k=1 Notng that: we obtan that: F t+ kh (z n n t+ kh, v t, kh n n ) F t+ (k 1)h n (z n t+ (k 1)h n, v (k 1)h t, ) = n d ((z n t+u, v t,u ), (z t+u, v t,u )) h n h D t+u F (zt+u, n v t,u ) D t+u F (z t+u, v t,u ) = n + snce the path of z t+u s contnuous. Moreover D t+u F (z n t+u, v t,u )du D t F t+u (z n t+u, v t,u ) 1 snce d ((z n t+u, v t,u ), (x t, v t ) h, so by domnated convergence the ntegral goes to as n. Wrtng: F t+h (z, v t,h ) F t (x, v) = [F t+h (z, v t,h ) F t+h (z n, v t,h )] + [F t+h (z n, v t,h ) F t (x, v)] and takng the lmt on n leads to F t+h (z, v t,h ) F t (x, v) αh, a contradcton. The above result mples n partcular that, f x F C 1,1 ([, T ]), and F 1 (x, v) = F 2 (x, v) for any contnuous path x, then 2 xf 1 and 2 xf 2 must also concde on contnuous paths. We now show that the same result can be obtaned under the weaker assumpton that F C 1,2, usng a probablstc argument. Interestngly, whle the prevous result on the unqueness of the frst vertcal dervatve s based on the fundamental theorem of calculus, the proof of the followng theorem s based on ts stochastc equvalent, the Itô formula [36, 37]. Theorem 2.3. If F 1, F 2 C 1,2 wth F, x F, 2 xf F l and DF satsfyng the local boundedness assumpton 2.9 for = 1, 2, concde on contnuous paths:: (x, v) UT c S T, t ], T ], Ft 1 (x t, v t ) = Ft 2 (x, v) (2.32) then ther second vertcal dervatves also concde on contnuous paths: (x, v) U c T S T, t ], T ], 2 xf 1 t (x t, v t ) = 2 xf 2 t (x t, v t )

29 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 19 Proof. Let F = F 1 F 2. Assume now that there exsts some (x, v) UT c S T such that for some 1 d and t ], T ], and some drecton h R d, h = 1, t h 2 xf t (x t, v t ).h >, and denote α = 1 2 t h 2 xf t (x t, v t ).h. We wll show that ths leads to a contradcton. We already know that x F t (x t, v t ) = by theorem 2.2. Let η > be small enough so that: t t, (x, v ) U t S t, d ((x t, v t ), (x, v )) < η F t (x, v ) < F t (x t, v t ) + 1, x F t (x, v ) < 1, D t F (x, v ) < 1, t h 2 xf t (x, v ).h > α (2.33) Choose t < t such that d ((x t, v t ), (x t, v t )) < η 2 and denote ɛ = η 2 (t t). Let W be a one dmensonal Brownan moton on some probablty space ( Ω, B, P), (B s ) ts natural fltraton, and let τ = nf{s >, W (s) = ɛ 2 } (2.34) Defne, for t [, T ], U(t ) = x(t )1 t t + (x(t) + W ((t t) τ)h)1 t >t (2.35) and note that for all s < ɛ 2, d ((U t+s, v t,s ), (x t, v t )) < ɛ (2.36) Defne the followng pecewse constant approxmatons of the stopped process W τ : Denotng n 1 W n (s) = W ( ɛ 2n τ)1 s [ ɛ 2n,(+1) ɛ 2n [ + W ( ɛ 2 τ)1 s= ɛ, s ɛ 2 2n = (2.37) Z(s) = F t+s (U t+s, v t,s ), s [, T t] (2.38) U n (t ) = x(t )1 t t + (x(t) + W n ((t t) τ)h)1 t >t Z n (s) = F t+s (U n t+s, v t,s ) (2.39) we have the followng decomposton: Z( ɛ 2 ) Z() = Z( ɛ 2 ) Zn ( ɛ 2 ) + n + = =1 Z n ( ɛ 2n ) Zn ( ɛ 2n ) n 1 Z n (( + 1) ɛ 2n ) Zn ( ɛ 2n ) (2.4)

30 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 2 The frst term n the rght-hand sde of (2.4) goes to almost surely snce The second term n (2.4) may be expressed as where: d ((U t+ ɛ, v t, ɛ ), (U n n 2 2 t+ ɛ, v t, ɛ )). (2.41) 2 2 Z n ( ɛ 2n ) Zn ( ɛ 2n ) = φ (W ( ɛ ɛ ) W (( 1) 2n 2n )) φ () (2.42) n,uh φ (u, ω) = F t+ ɛ (U 2n t+ ɛ (ω), v t, ɛ ) 2n 2n Note that φ (u, ω) s measurable wth respect to B ( 1)ɛ/2n whereas ts argument n (2.42) s ndependent wth respect to B ( 1)ɛ/2n. Let Ω 1 = {ω Ω, t W (t, ω) Then P(Ω 1 ) = 1 and for any ω Ω 1, φ (., ω) s C 2 wth: φ n,uh (u, ω) = x F t+ ɛ (U 2n t+ ɛ (ω), v t, ɛ )h 2n 2n contnuous}. φ (u, ω) = t h 2 n,uh xf t+ ɛ (U 2n t+ ɛ (ω), v t, ɛ ).h (2.43) 2n 2n So, usng the above arguments we can apply the Itô formula to (2.42). We therefore obtan, summng on and denotng (s) the ndex such that s [( 1) ɛ 2n, ɛ 2n ): n =1 Z n ( ɛ 2n ) Zn ( ɛ ɛ 2n ) = 2 n,uh x F t+(s) ɛ (U 2n t+(s) ɛ, v t,(s) ɛ )hdw (s) 2n 2n + ɛ 2 t h. 2 n,uh xf t+(s) ɛ (U 2n t+(s) ɛ, v t,(s) ɛ ).hds (2.44) 2n 2n Snce the frst dervatve s bounded by (2.33), the stochastc ntegral s a martngale, so takng expectaton leads to: E[ n =1 Z n ( ɛ 2n ) Zn ( ɛ 2n )] > α ɛ 2 (2.45) Now Z n (( + 1) ɛ 2n ) Zn ( ɛ 2n ) = ψ( ɛ ) ψ() (2.46) 2n where ψ(u) = F t+( 1) ɛ 2n +u (U n t+( 1) ɛ 2n,u, v t,( 1) ɛ 2n +u ) (2.47)

31 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 21 s rght-dfferentable wth rght dervatve: Snce F F l calculus yelds: = ψ (u) = D t F t+( 1) ɛ 2n +u (U n ( 1) ɛ 2n,u, v t,( 1) ɛ 2n +u ) (2.48) ([, T ]), ψ s left-contnuous by theorem 4 and the fundamental theorem of n 1 Z n (( + 1) ɛ 2n ) Zn ( ɛ ɛ 2n ) = 2 D t+s F (Ut+((s) 1) n ɛ +u, v t,s)ds (2.49) 2n The ntegrand converges to D t F t+s (U t+((s) 1) ɛ 2n +u, v t,s ) = snce D t F s zero whenever the frst argument s a contnuous path. Snce ths term s also bounded, by domnated convergence the ntegral converges almost surely to. It s obvous that Z( ɛ 2 ) = snce F (x, v) = whenever x s a contnuous path. On the other hand, snce all dervatves of F appearng n (2.4) are bounded, the domnated convergence theorem allows to take expectatons of both sdes n (2.4) wth respect to the Wener measure and obtan α ɛ 2 =, a contradcton. Remark 2.5. If a functonal s predctable n the second varable, so are ts vertcal dervatves hence we can state n the settng of theorems 2.2, 2.3 that x F 1 t (x t, v t ) = x F 2 t (x t, v t ), 2 xf 1 t (x t, v t ) = 2 xf 2 t (x t, v t ). Remark 2.6. Both results extend (replacng t ], T ] by t [, T [) f the vertcal dervatves (but not the functonal tself) are n F r nstead of F l, followng the same proof but extendng drectly the path of (x, v) from t rather than steppng back n tme frst. 2.4 Change of varable formula for functonals of a contnuous path We now state our frst man result, a functonal change of varable formula whch extends the Itô formula wthout probablty due to Föllmer [29] to functonals. We denote here S + d the set of postve symmetrc d d matrces. Defnton 2.1. Let π n = (t n,..., tn k(n) ), where = tn tn 1... tn k(n) = T, be a sequence of subdvsons of [, T ] wth step decreasng to as n. f C ([, T ], R) s

32 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 22 sad to have fnte quadratc varaton along (π n ) f the sequence of dscrete measures: ξ n = k(n) 1 = (f(t n +1) f(t n )) 2 δ t n (2.5) where δ t s the Drac measure at t, converge vaguely to a Radon measure ξ on [, T ] whose atomc part s null. The ncreasng functon [f] defned by [f](t) = ξ([, t]) s then called the quadratc varaton of f along the sequence (π n ). x C ([, T ], U) s sad to have fnte quadratc varaton along the sequence (π n ) f the functons x, 1 d and x + x j, 1 < j d do. The quadratc varaton of x along (π n ) s the S + d -valued functon x defned by: [x] = [x ], [x] j = 1 2 ([x + x j ] [x ] [x j ]), j (2.51) Theorem 2.4 (Change of varable formula for functonals of contnuous paths). Let (x, v) C ([, T ], U) S T such that x has fnte quadratc varaton along (π n ) and verfes sup t [,T ] πn v(t) v(t ). Denote: v n (t) = k(n) 1 = x n (t) = k(n) 1 = x(t +1 )1 [t,t +1 [(t) + x(t )1 {T } (t) v(t )1 [t,t +1 [(t) + v(t )1 {T } (t), h n = t n +1 t n (2.52) Then for any non-antcpatve functonal F C 1,2 such that: 1. F, x F, 2 xf F l 2. 2 xf, DF satsfy the local boundedness property (2.9) the followng lmt k(n) 1 lm n = exsts. Denotng ths lmt by T xf (x u, v u )d π x we have T + x F t n (x n t n, v n t n )(x(t n +1) x(t n )) (2.53) F T (x T, v T ) F (x, v ) = 1 2 tr (t 2 xf t (x u, v u )d[x](u) ) + T T D t F t (x u, v u )du (2.54) x F (x u, v u )d π x (2.55)

33 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 23 Remark 2.7 (Föllmer ntegral). The lmt (2.53), whch we call the Föllmer ntegral, was defned n [29] for ntegrands of the form f(x(t)) where f C 1 (R d ). It depends a pror on the sequence π of subdvsons, hence the notaton T xf (x u, v u )d π x. We wll see n Secton 2.7 that when x s the sample path of a semmartngale, the lmt s n fact almost-surely ndependent of the choce of π. Remark 2.8. The regularty condtons on F are gven ndependently of (x, v) and of the sequence of subdvsons (π n ). Proof. Denote δx n = x(t n +1 ) x(tn ). Snce x s contnuous hence unformly contnuous on [, T ], and usng Lemma A.1 for v, the quantty η n = sup{ v(u) v(t n ) + x(u) x(t n ) + t n +1 t n, k(n) 1, u [t n, t n +1)}(2.56) converges to as n. Snce 2 xf, DF satsfy the local boundedness property (2.9), for n suffcently large there exsts C > such that t < T, (x, v ) U t S t, d ((x t, v t ), (x, v )) < η n D t F t (x, v ) C, 2 xf t (x, v ) C Denotng K = {x(u), s u t} whch s a compact subset of U, and U c = R U ts complement, one can also assume n suffcently large so that d(k, U c ) > η n. For k(n) 1, consder the decomposton: F t n +1 (x n t n +1, v n t n +1 ) F t n (x n t n, v n t n ) = F t n +1 (x n t n +1, v n t n,hn ) F t n (xn t n, vn t n ) + F t n (x n t n, vn t n ) F t n (x n t n, v n t n ) (2.57) where we have used property (2.2) to have F t n (x n t n, vn t n) = F t n(xn t n, vn t n ). The frst term can be wrtten ψ(h n ) ψ() where: ψ(u) = F t n +u(x n t n,u, v n t n,u ) (2.58) Snce F C 1,2 ([, T ]), ψ s rght-dfferentable, and moreover by lemma 4, ψ s leftcontnuous, so: t n F t n +1 (x n t n,,hn vn t n ) F,hn t n(xn t n, vn t n) = +1 t n D t n +uf (x n t n,u, vt n n,u )du (2.59)

34 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 24 The second term can be wrtten φ(δx n ) φ(), where: φ(u) = F t n (x n,u t n, vn t n ) (2.6) Snce F C 1,2 ([, T ]), φ s well-defned and C 2 on the convex set B(x(t n ), η n) U, wth: φ (u) = x F t n (x n,u t n, vn t n ) So a second order Taylor expanson of φ at u = yelds: φ (u) = 2 xf t n (x n,u t n, vn t n ) (2.61) F t n (x n t n, vn t n ) F t n (x n t n, vt n n ) = x F t n (x n t n, vt n n )δx n + 1 ) ( 2 tr 2 xf t n (x n t n, vt n n ) t δx n δx n + r n (2.62) where r n s bounded by K δx n 2 sup 2 xf t n x B(x(t n ),ηn) (x n,x x(tn t n, vt n n ) 2 xf t n (x n t n, vt n n ) (2.63) Denote n (t) the ndex such that t [t n n (t), tn n (t)+1). We now sum all the terms above from = to k(n) 1:. ) The left-hand sde of (2.57) yelds F T (x n T, vn T ) F (x, v ), whch converges to F T (x T, v T ) F (x, v ) by left-contnuty of F, and ths quantty equals F T (x T, v T ) F (x, v ) snce x s contnuous and F s predctable n the second varable. The frst lne n the rght-hand sde can be wrtten: T D u F (x n t n, n (u),u tn vn n t (u) n )du (2.64) n (u),u tn n (u) where the ntegrand converges to D u F (x u, v u ) and s bounded by C. domnated convergence theorem apples and (2.64) converges to: Hence the T D u F (x u, v u )du = snce v u = v u, du-almost everywhere. T D u F (x u, v u ) (2.65)

35 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 25 The second lne can be wrtten: k(n) 1 = k(n) 1 x F t n (x n t n, vt n n )(x t n +1 x t n ) + = 1 2 tr[ 2 xf t n (x n t n, v n t n )] t δx n δx n ](2.66) k(n) 1 + = r n (2.67) [ 2 xf t n (x n t n, vn t n )]1 t ]t n,tn +1 ] s bounded by C, and converges to 2 xf t (x t, v t ) by leftcontnuty of 2 xf, and the paths of both are left-contnuous by lemma 4. Snce x and the subdvson (π n ) are as n defnton 2.1, lemma A.5 n appendx A.3 apples and gves as lmt: T 1 2 tr[t 2 xf t (x u, v u )]d[x](u)] = T 1 2 tr[t 2 xf t (x u, v u )]d[x](u)] (2.68) snce 2 xf s predctable n the second varable.e. verfes (2.2). Usng the same lemma, snce r n s bounded by ɛn δxn 2 where ɛ n 2C, n (t) 1 = n (s)+1 rn converges to. Snce all other terms converge, the lmt: lm n k(n) 1 = exsts, and the result s establshed. converges to and s bounded by x F t n (x n t n, v n t n )(x(t n +1) x(t n )) (2.69) 2.5 Change of varable formula for functonals of a cadlag path We wll now extend the prevous result to functonals of cadlag paths. The followng defnton s taken from Föllmer [29]: Defnton Let π n = (t n,..., tn k(n) ), where = tn tn 1... tn k(n) = T be a sequence of subdvsons of [, T ] wth step decreasng to as n. f D([, T ], R) s sad to have fnte quadratc varaton along (π n ) f the sequence of dscrete measures: ξ n = k(n) 1 (f(t n +1) f(t n )) 2 δ t n (2.7) =

36 CHAPTER 2. PATHWISE CALCULUS FOR NON-ANTICIPATIVE FUNCTIONALS 26 where δ t s the Drac measure at t, converge vaguely to a Radon measure ξ on [, T ] such that [f](t) = ξ([, t]) = [f] c (t) + ( f(s)) 2 (2.71) <s t where [f] c s the contnuous part of [f]. [f] s called quadratc varaton of f along the sequence (π n ). x U T s sad to have fnte quadratc varaton along the sequence (π n ) f the functons x, 1 d and x + x j, 1 < j d do. The quadratc varaton of x along (π n ) s the S + d -valued functon x defned by: [x] = [x ], [x] j = 1 2 ([x + x j ] [x ] [x j ]), j (2.72) Theorem 2.5 (Change of varable formula for functonals of dscontnuous paths). Let (x, v) U T S T where x has fnte quadratc varaton along (π n ) and Denote v n (t) = sup x(t) x(t ) + v(t) v(t ) (2.73) t [,T ] π n k(n) 1 = x n (t) = k(n) 1 = x(t +1 )1 [t,t +1 )(t) + x(t )1 {T } (t) v(t )1 [t,t +1 )(t) + v(t )1 {T } (t), h n = t n +1 t n (2.74) Then for any non-antcpatve functonal F C 1,2 such that: 1. F s predctable n the second varable n the sense of (2.2) 2. 2 xf and DF have the local boundedness property (2.9) 3. F, x F, 2 xf F l 4. x F has the horzontal local Lpschtz property (2.24) the Föllmer ntegral, defned as the lmt ],T ] x F t (x t, v t )d π x := k(n) 1 lm n = x F t n (x n, x(tn ) t n, v n t n )(x(t n +1) x(t n )) (2.75)