Optimal control of diffusion coefficients via decoupling fields

Transcription

1 Opimal conrol of diffusion coefficiens via decoupling fields Sefan Ankirchner, Alexander Fromm To cie his version: Sefan Ankirchner, Alexander Fromm. Opimal conrol of diffusion coefficiens via decoupling fields <hal > HAL Id: hal hps://hal.archives-ouveres.fr/hal Submied on 11 Oc 217 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 Opimal conrol of diffusion coefficiens via decoupling fields Sefan Ankirchner 1 and Alexander Fromm 1 1 Insiue for Mahemaics, Universiy of Jena, Erns-Abbe-Plaz 2, 7743 Jena, Germany Ocober 11, 217 Absrac We consider a diffusion conrol problem, where he conroller oally deermines he sae s diffusion coefficien bu has no influence on he sae s drif rae. By using he Ponryagin maximum principle we characerize an opimal conrol in erms of he adjoin forward-backward sochasic differenial equaion (FBSDE), urning ou o be fully coupled. We use he mehod of decoupling fields for proving ha he adjoin FBSDE possesses a soluion. 21 Mahemaics Subjec Classificaion. 93E2, 6H3, 49J55. Keywords. Opimal sochasic conrol, diffusion coefficien, forward-backward sochasic differenial equaion, decoupling field. Inroducion Le (M α ) be a sochasic process wih conrolled dynamics of he form dm α = µ(, M α )d + α dw, where µ is affine linear in is second argumen and W is a one-dimensional Brownian moion. In his aricle we consider he conrol problem ha consiss in minimizing, among a suiable class of conrols α, he arge funcional [ T ] E f(, M α, α )d + g(mt α ), (1) where f and g are nice funcions, in paricular convex in M and α. This conrol problem arises in siuaions where one can conrol a sae s flucuaion inensiy bu no is drif, and where one aims a seering he sae ino a arge area. To give an explici example, M α may describe he posiion of a paricle in a medium wih emperaure α. By heaing or cooling he medium he paricle s flucuaions increase or decrease respecively. The funcion f reflecs he coss involved by any emperaure change. Diffusion conrol problems arise also in porfolio opimizaion. In his conex M α can be inerpreed as a porfolio value process wih volailiy α. A reducion of he porfolio s s.ankirchner@uni-jena.de alexander.fromm@uni-jena.de 1

3 volailiy involves hedging coss f. The funcion g can be aken o be a uiliy funcion. The opimal conrol of diffusion coefficiens appear also in oher fields of applicaions, see e.g. [16] for examples arising in biology. Assuming a Markovian framework, one can choose a Hamilon-Jacobi-Bellman (HJB) approach and characerize he value funcion in erms of he HJB equaion. In a non-markovian framework he problem of minimizing (1) seems o be unsolved, o he bes of our knowledge. Our soluion mehod is based on he maximum principle, leading o a probabilisic represenaion of he value funcion s sensiiviy w.r.. he conrolled space variable. More precisely, we reduce he conrol problem o an adjoin forward-backward sochasic differenial equaion (FBSDE). The adjoin FBSDE urns ou o be fully coupled, and hence i is a priori no clear wheher a soluion exiss. Our idea is o use he so-called mehod of decoupling fields for proving ha he fully coupled sysem possesses indeed a soluion. In order o make our idea work, we firs ransform he adjoin FBSDE so as o reduce he dependence of he forward diffusion coefficien on he conrol. The decoupling field of he ransformed FBSDE can be conrolled by using he convexiy properies of he cos funcions. More precisely, we show ha he space derivaive of he decoupling field is bounded and bounded away from zero. This allows o conclude exisence of a soluion of he adjoin equaion and hence o obain an opimal conrol for he problem of minimizing (1). We remark ha he convexiy assumpions also guaranee ha he maximum principle applies in he firs place. We now explain how our soluion mehod compares o oher approaches. One can srive o obain a probabilisic represenaion of he value funcion iself (and no only of is derivaive). In he Markovian case he value funcion saisfies a fully non-linear PDE, allowing a represenaion in erms of a 2nd order BSDE (see [21]). Our approach focuses on he derivaive of he value funcion, which saisfies a semi-linear PDE and hence has a represenaion in erms of a sandard FBSDE. A difficuly of he HJB approach arises from he fac ha he diffusion coefficien is unbounded in he conrol. In order o circumven his difficuly, [4] inroduce a ailor-made modificaion of he viscosiy soluion concep. The conrol problem considered in [4] is more general han he problem of he presen aricle. In conras o [4], our approach does no require ha he funcion g has sricly smaller growh in x han he funcion f in a. Finally, a furher advanage of our mehod compared o he HJB approach is ha i allows for a non-markovian seing, i.e. he funcions f and g can addiionally depend, in a progressively measurable way, on he Brownian pahs. The lieraure provides many examples of problems involving he opimal conrol of a diffusion coefficien. We do no srive o give an overview on his classical ype of conrol problem, bu selec some aricles ha seem closes o he problem we sudy in he curren aricle. A simple problem version wih infinie ime horizon is discussed in Example 7.6, [2]. Specific problems wih a finie ime horizon have been sudied in [19], [9] and [1]. A reverse diffusion conrol problem is solved by McNamara [15]: he deermines he reward funcions for which a given bang-bang diffusion conrol is opimal. A diffusion conrol problem wihin an exponenial maringale model is sudied in [2]. As menioned earlier, our approach is rooed in reducing he iniial conrol problem o an FBSDE. I is a longsanding challenge o find condiions guaraneeing ha a given fully coupled FBSDE possesses a soluion. Sufficien condiions are provided e.g. in [11], [17], [14], [18], [5], [12] (see also references herein). The mehod of decoupling fields, developped in [6] (see also he precursor aricles [13], [7] and [12]), is pracically useful for deermining wheher a soluion exiss. A decoupling field describes he funcional dependence of he 2

4 backward par Y on he forward componen X. If he coefficiens of a fully coupled FBSDE saisfy a Lipschiz condiion, hen here exiss a maximal non-vanishing inerval possessing a soluion riple (X, Y, Z) and a decoupling field wih nice regulariy properies. The mehod of decoupling fields consiss in analyzing he dynamics of he decoupling field s gradien in order o deermine wheher he FBSDE has a soluion on he whole ime inerval [, T ]. The mehod can be successfully applied o various problems involving coupled FBSDE: In [8] soluions o a quadraic srongly coupled FBSDE wih a wo-dimensional forward equaion are consruced o obain soluions o he Skorokhod embedding problem for Gaussian processes wih nonlinear drif. In Chaper 5 of [6] he problem of uiliy maximizaion in incomplee markes is reaed for a general class of uiliy funcions via consrucion of soluions o he associaed coupled FBSDE. In he more recen work [3], he mehod is used o obain soluions o he problem of opimal posiion argeing for general cos funcionals. The paper is organized as follows. In Secion 1 we rigorously describe he problem and is mahemaical se-up. In Secion 2 we apply he maximum principle o reduce he conrol problem o he adjoin FBSDE. Secion 3 provides a brief inroducion ino he mehod of decoupling fields. In Secion 4 we firs ransform he adjoin FBSDE so as o dampen he dependence of he forward diffusion on he conrol. We hen prove exisence of a soluion by using he mehod of decoupling fields. In Secion 5 we illusrae he consrucion of a soluion wih an explici example. Finally, in Secion 6 we explain heurisically how he adjoin FBSDE is conneced o he conrol problem s HJB equaion. 1 Problem formulaion Le T > be a deerminisic finie ime horizon. Le W be a Brownian moion on a complee probabiliy space (Ω, F, P) and denoe by (F ) [,T ] he smalles filraion saisfying he usual condiions and conaining he filraion generaed by W. Le g : Ω R R be measurable and f : Ω [, T ] R R R be measurable such ha for all (m, a) R 2 he mapping (ω, ) f(ω,, m, a) is progressively measurable. We make he following addiional assumpions on f and g: (C) For every fixed pair (ω, ) Ω [, T ] he mappings (m, a) f(, m, a) and m g(m) are convex, wih f being sricly convex in a. Noe ha we follow he usual convenion and omi he funcion argumen ω. (C1) g( ) and f(,, ) are wice coninuously differeniable. Moreover, g, f m and f a are Lipschiz coninuous in he las wo componens and saisfy ( f m + f a )(,,, ) < and g () <. (C2) There exiss a posiive consan δ l > such ha f aa δ l everywhere. Throughou we assume ha δ l denoes he larges consan wih his propery. Noice ha (C1) implies ha here exiss a δ u [δ l, ) such ha for all (ω,, m, a) Ω [, T ] R R we have f aa (, m, a) δ u. (2) Moreover, (C1) implies ha f m and f a grow a mos linearly in m and a. 3

5 [ T Le A be he se of all progressively measurable α : Ω [, T ] R such ha E. For all m R and α A we define M m,α = m + (b s + B s M m,α s ) ds + α2 s ds ] < α s dw s, (3) where b, B : Ω [, T ] R are progressively measurable and bounded processes. As oulined in he inroducion, our aim is o solve he following problem: [ T Minimize J(m, α) := E f (s, Ms m,α, α s ) ds + g ( M m,α ) ] T over all α A. (4) For simpliciy we someimes wrie M α or jus M insead of M m,α. In oher words, for given m, he goal is o choose α from he se A of admissible conrols in such a way ha J is minimized. 2 Reducing he problem o an FBSDE The so-called Hamilonian of he conrol problem (4) is defined by H(, m, a, y, z) := (b + B m)y + az + f(, m, a), for [, T ] and (m, a, z) R R R. Noice ha min H(, m, a, y, z) = (b + B m)y f (, m, z), (5) a R where f (, m, ) is he convex conjugae of f(, m, ). Observe ha condiion (C2) guaranees ha f assumes real values only. For he following observaion we need boh (C1) and (C2). The minimum in (5) is aained a a = f3 (, m, z), where f 3 denoes he parial derivaive of f wih respec o he las componen. This parial derivaive exiss since f is differeniable and, by (C2), he image of f a (, m, ) is he whole real line. More precisely, using Ferma s heorem applied o he minimizaion problem min a R H(, m, a, y, z) = min a R H(, m, a, y, z) one can deduce a = f3 (, m, z) = fa 1 (, m, z), where fa 1 (, m, ) denoes he inverse of he funcion f a (, m, ), which is sricly increasing. The so-called adjoin forward-backward sochasic differenial equaion (FBSDE) for he conrol problem (4) is given by Ms m = m + s (b r + B r Mr m ) dr + s f 3 (r, M r m, Zr m )dw r, Ys m = g (MT m) T s Zm r dw r + T s (B ry r + f m (r, Mr m, f3 (r, M r m, Zr m )))dr for all s [, T ]. To simplify he noaions, when here is no ambiguiy, (M m, Y m, Z m ) will be denoed by (M, Y, Z). In order o rigorously define wha we mean by a soluion of (6) we inroduce he following process spaces. For any [, T ) we denoe by H 2,T he se of all (F )-progressively measurable processes (X ) [,T ] such ha E T X 2 d <. We define HT 2 := H2,T. A soluion o (6) is a riple (M, Y, Z) = (M m, Y m, Z m ) of progressively measurable processes such ha 4 (6)

6 M, Y and Z are processes in H 2 T, M and Y are coninuous processes, he wo equaions (6) are saisfied a.s. for every fixed s [, T ]. Consrucing soluions o he above FBSDE is imporan for he following reason: Proposiion 2.1. If here exiss a soluion (M, Y, Z) of (6), hen an opimal conrol for problem (4) is given by α s = f 3 (s, M s, Z s ), s [, T ]. Proof. We adap he proof of Theorem 5.2 in [22] o our seing. For m R and ᾱ A le M = M m,ᾱ be he associaed sae process. Le us define δm := M M and δα := ᾱ α. Noe ha M = M = m, such ha δm =. Since g is convex we have a.s. Y T δm T Y δm = Y T δm T = g (M T )δm T g( M T ) g(m T ). (7) A he same ime Iô s formula proves ha Y δm Y δm = = = (δm s ) dy s for all [, T ]. Now noe ha Y s d(δm s ) + (δm s ) (B s Y s + f m (s, M s, α s )) ds + Y s B s (δm s ) ds + Y s (δα s ) dw s + (δα s )Z s ds ((δα s )Z s (δm s )f m (s, M s, α s )) ds + (δm s )Z s dw s (δα s )Z s ds (Y s (δα s ) + (δm s )Z s ) dw s, (8) H a (s, M s, α s, Y s, Z s ) = Z s + f a (s, M s, f 3 (s, M s, Z s )) = Z s Z s =. Togeher wih he convexiy of H his implies H(s, M s, ᾱ s, Y s, Z s ) H(s, M s, α s, Y s, Z s ) H m (s, M s, α s, Y s, Z s )(δm s ) Thus, due o he definiion of H +H a (s, M s, α s, Y s, Z s )(δα s ) = (B s Y s + f m (s, M s, α s )) (δm s ). f(s, M s, ᾱ s ) f(s, M s, α s ) = H(s, M s, ᾱ s, Y s, Z s ) H(s, M s, α s, Y s, Z s ) B s (δm s )Y s Z s (δα s ) f m (s, M s, α s )(δm s ) Z s (δα s ). 5

7 Le τ be a [, T ]-valued sopping ime such ha M, M and Y are bounded on [, τ]. Formula (8), ogeher wih inequaliy (9), implies [ τ ( E [Y τ δm τ Y δm ] E f(s, Ms, α s ) f(s, M s, ᾱ s ) ) ] ds, where all inegrals are well-defined, since M, M, Y are bounded on [, τ], α and ᾱ are square inegrable and f is a mos quadraic in a. Noe ha sup M s, sup M s and sup Y s s [,T ] s [,T ] s [,T ] are square inegrable. Choosing an appropriae localizing sequence τ n T of sopping imes we can pass o he limi using dominaed convergence and obain [ T ( E [Y T δm T Y δm ] E f(s, Ms, α s ) f(s, M s, ᾱ s ) ) ] ds. (9) Combining (9) wih (7) we obain This leads o which shows opimaliy of α. E [ g( M T ) g(m T ) ] [ T ( E f(s, Ms, α s ) f(s, M s, ᾱ s ) ) ] ds. J(m, ᾱ) J(m, α), 3 The mehod of decoupling fields As menioned above, solving (6) is crucial in consrucing opimal conrols. As a key resul of his paper we prove in Secion 4 he solvabiliy of (6). Noe ha even under our Lipschiz assumpions, i is no rivial o show well-posedness of (6) due o is coupled naure. I is necessary o ake more suble srucural properies ino accoun o conduc he proof. Our argumenaion will be based on he so-called mehod of decoupling fields which we will briefly sum up in his secion. For a fixed finie ime horizon T >, we consider a complee filered probabiliy space (Ω, F, (F ) [,T ], P), where F consiss of all null ses, (W ) [,T ] is a 1-dimensional Brownian moion and F := σ(f, (W s ) s [,] ) wih F := F T. The dynamics of an FBSDE is given by X s = X + s Y = ξ(x T ) µ(r, X r, Y r, Z r )dr + T s f(r, X r, Y r, Z r )dr σ(r, X r, Y r, Z r )dw r, T Z r dw r, for s, [, T ] and X R, where (ξ, (µ, σ, f)) are measurable funcions such ha ξ : Ω R R, µ: [, T ] Ω R R R R, σ : [, T ] Ω R R R R, f : [, T ] Ω R R R R, 6

8 Throughou he whole secion µ, σ and f are assumed o be progressively measurable wih respec o (F ) [,T ]. A decoupling field comes wih an even richer srucure han jus a classical soluion (X, Y, Z). Definiion 3.1. Le [, T ]. A funcion u: [, T ] Ω R R wih u(t, ) = ξ a.e. is called decoupling field for (ξ, (µ, σ, f)) on [, T ] if for all 1, 2 [, T ] wih 1 2 and any F 1 -measurable X 1 : Ω R here exis progressively measurable processes (X, Y, Z) on [ 1, 2 ] such ha X s = X 1 + s µ(r, X r, Y r, Z r )dr + 1 s σ(r, X r, Y r, Z r )dw r, 1 Y s = Y 2 2 s Y s = u(s, X s ), f(r, X r, Y r, Z r )dr 2 s Z r dw r, a.s. for all s [ 1, 2 ]. In paricular, we wan all inegrals o be well-defined. Some remarks abou his definiion are in place. The firs equaion in (1) is called he forward equaion, he second he backward equaion and he hird will be referred o as he decoupling condiion. Noe ha, if 2 = T, we ge Y T = ξ(x T ) a.s. as a consequence of he decoupling condiion ogeher wih u(t, ) = ξ. A he same ime Y T = ξ(x T ), ogeher wih he decoupling condiion, implies u(t, ) = ξ a.e. If 2 = T we can say ha a riple (X, Y, Z) solves he FBSDE, meaning ha i saisfies he forward and he backward equaion, ogeher wih Y T = ξ(x T ). This relaionship Y T = ξ(x T ) is referred o as he erminal condiion. For he following we need o inroduce furher noaion. Le I [, T ] be an inerval and u : I Ω R R a map such ha u(s, ) is measurable for every s I. We define L u,x := sup inf{l for a.a. ω Ω : u(s, ω, x) u(s, ω, x ) L x x for all x, x R}, s I where inf :=. We also se L u,x := if u(s, ) is no measurable for every s I. One can show ha L u,x < is equivalen o u having a modificaion which is ruly Lipschiz coninuous in x R. We denoe by L σ,z he Lipschiz consan of σ w.r.. he dependence on he las componen z. We se L σ,z = if σ is no Lipschiz coninuous in z. if L σ,z > and oherwise. For an inegrable real valued random variable F he expression E [F ] refers o E[F F ], while E, [F ] refers o ess sup E[F F ], which migh be, bu is always well defined as he infimum of all consans c [, ] such ha E[F F ] c a.s. Addiionally, we wrie F for he essenial supremum of F. In pracice i is imporan o have explici knowledge abou he regulariy of (X, Y, Z). For insance, i is imporan o know in which spaces he processes live, and how hey reac o changes in he iniial value. By L 1 σ,z = 1 L σ,z we mean 1 L σ,z 7 (1)

9 Definiion 3.2. Le u: [, T ] Ω R R be a decoupling field o (ξ, (µ, σ, f)). 1. We say u o be weakly regular if L u,x < L 1 σ,z and sup s [,T ] u(s,, ) <. 2. A weakly regular decoupling field u is called srongly regular if for all fixed 1, 2 [, T ], 1 2, he processes (X, Y, Z) arising in (1) are a.e unique and saisfy sup E 1, [ X s 2 ] + s [ 1, 2 ] sup s [ 1, 2 ] E 1, [ Y s 2 ] + E 1, [2 1 ] Z s 2 ds <, (11) for each consan iniial value X 1 = x R. In addiion hey are required o be measurable as funcions of (x, s, ω) and even weakly differeniable w.r.. x R n such ha for every s [ 1, 2 ] he mappings X s and Y s are measurable funcions of (x, ω) and even weakly differeniable w.r.. x such ha [ ess sup x R sup E 1, x X s 2] <, ess sup x R s [ 1, 2 ] ess sup x R E 1, sup E 1, s [ 1, 2 ] [2 1 [ x Y s 2] <, ] x Z s 2 ds <. (12) 3. We say ha a decoupling field on [, T ] is srongly regular on a subinerval [ 1, 2 ] [, T ] if u resriced o [ 1, 2 ] is a srongly regular decoupling field for (u( 2, ), (µ, σ, f)). Under suiable condiions a rich exisence, uniqueness and regulariy heory for decoupling fields can be developed. Assumpion (SLC): (ξ, (µ, σ, f)) saisfies sandard Lipschiz condiions (SLC) if 1. (µ, σ, f) are Lipschiz coninuous in (x, y, z) wih Lipschiz consan L, 2. ( µ + f + σ ) (,,,, ) <, 3. ξ : Ω R R is measurable such ha ξ(, ) < and L ξ,x < L 1 σ,z. In order o have a noion of global exisence we need he following definiion: Definiion 3.3. We define he maximal inerval I max [, T ] of he problem given by (ξ, (µ, σ, f)) as he union of all inervals [, T ] [, T ], such ha here exiss a weakly regular decoupling field u on [, T ]. Noe ha he maximal inerval migh be open o he lef. Also, le us remark ha we define a decoupling field on such an inerval as a mapping which is a decoupling field on every compac subinerval conaining T. Similarly we can define weakly and srongly regular decoupling fields as mappings which resriced o an arbirary compac subinerval conaining T are weakly (or srongly) regular decoupling fields in he sense of he definiions given above. Finally, we have global exisence and uniqueness on he maximal inerval: 8

10 Theorem 3.4 ([6], Theorem , Lemma and Corollary 2.5.5). Le (ξ, (µ, σ, f)) saisfy SLC. Then here exiss a unique srongly regular decoupling field u on I max. Furhermore, eiher I max = [, T ] or I max = ( min, T ], where min < T. In he laer case we have lim L u(, ),x = L 1 σ,z. (13) min Moreover, for any I max and any iniial condiion X = x R here is a unique soluion (X, Y, Z) of he FBSDE on [, T ] saisfying [ T ] sup E[ X s 2 ] + sup E[ Y s 2 ] + E Z s 2 ds <. s [,T ] s [,T ] Equaliy (13) allows o verify global exisence, i.e. I max = [, T ], via conradicion. We refer o his approach as he mehod of decoupling fields. 4 Transforming he FBSDE The aim of his secion is o prove ha he adjoin FBSDE (6) has a soluion on he whole inerval [, T ]. To his end we firs ransform he FBSDE in a way ha reduces he dependence of he forward diffusion on he conrol. We hen apply he mehod of decoupling fields o he ransformed sysem. Le γ = 1 δ u. We consider he auxiliary FBSDE Xs x = x + s (b r + B r Mr x B r (Xr x Mr x ) γf m (r, Mr x, Z r x ))dr + s ( Z r x γf a (r, Mr x, Z r x ))dw r Ms x = (Id + γg ) 1 (XT x) T s Z r x dw r T s (b r + B r Mr x ) dr, for all s [, T ]. We firs show ha he parameers of he FBSDE (14) saisfy he sandard Lipschiz condiions. Noice ha γg is non-decreasing in x since g is convex. Hence Id+γg is sricly increasing wih a derivaive of a leas 1. This implies ha he inverse in x, denoed by ξ = (Id + γg ) 1, exiss and ha he inverse is Lipschiz coninuous in x wih a Lipschiz consan smaller han or equal o 1. Since g () is essenially bounded, also ξ() = (Id + γg ) 1 () is essenially bounded. Indeed, ξ() = ξ() ξ((id + γg )()) (Id + γg )() γ g (). Le µ(, x, m, z) = (b + B m) B (x m) γf m (, m, z) and σ(, m, z) = z γf a (, m, z) be he drif and he diffusion coefficien of he forward equaion in (14), respecively. Condiion (C1) enails ha µ and σ are Lipschiz coninuous and ha ( σ + µ )(,,, ) is bounded. Noice ha σ(, m, z) is differeniable in z and ha he derivaive akes values only in [, 1 δ l δ u ]. Hence he diffusion coefficien is Lipschiz coninuous in z wih a Lipschiz consan equal o δu δ l. In paricular, he Lipschiz consan of (Id + γg ) 1 is sricly smaller han L 1 σ,z = δ u δu δ u δ l. Noe ha L σ,z = if and only if δ l = δ u. To sum up, we have verified ha he parameers of he FBSDE (14) saisfy he sandard Lipschiz condiions (SLC). The benefi of considering (14) comes from he following observaion. 9 (14)

11 Lemma 4.1. If (X, M, Z ) is a soluion of (14), hen (M, 1 γ (X M ), f a (, M, Z )) is a soluion of (6). Proof. Le (X, M, Z ) be a soluion of (14). In paricular, all hree processes are in HT 2. The linear growh condiion on f a implies ha also (f a (, M, Z )) is in HT 2. Thus each process in he new riple (M, 1 γ (X M ), f a (, M, Z )) is in HT 2. A sraighforward calculaion shows ha he hree processes of he new riple saisfy he dynamics (6). We use he mehod of decoupling fields for proving ha here exiss a soluion of (14) on [, T ]. Since he parameers of (14) saisfy he (SLC), here exiss a maximal inerval I max wih a weakly regular decoupling field u (see Theorem 3.4). In he following fix I max. Le (X, M, Z) = (X,x, M,x, Z,x ) be he soluion of (14) on [, T ] wih iniial value x R such ha M = u(, X ) a.s. for all (, x) [, T ] R. According o srong regulariy u is weakly differeniable w.r.. he iniial value x R. In he following we denoe by u x a version of he weak derivaive of u w.r.. x such ha i coincides wih he classical derivaive a all poins for which i exiss and wih everywhere else. Moreover, he processes (X, M, Z) are weakly differeniable w.r.. x. We can formally differeniae he forward and he backward equaion in (14). One can verify ha one can inerchange differeniaion and inegraion and ha a chain rule for weak derivaives applies (see Secions A.2 and A.3 in [6]). We hus obain ha for every version ( x X, x M, x Z) = ( x X,x, x M,x, x Z,x ) of he weak derivaive, such ha for every s [, T ] ( x X s, x M s ) is a weak derivaive of (X s, M s ), we have for every [, T ]: and x X =1 + + B s (2 x M s x X s ) ds γ(f mm (s, M s, Z s ) x M s + f ma (s, M s, Z s ) x Zs )ds ( x Zs γf ma (s, M s, Z s ) x M s γf aa (s, M s, Z s ) x Zs )dw s (15) T T x M = ξ (X T ) x X T + B s x M s ds + x Zs dw s, (16) for P λ - almos all (ω, x) Ω R. By redefining ( x X, x M) as he righ-hand-sides of (15) and (16) respecively, we obain processes ( x X, x M) ha are coninuous in ime for all (ω, x) bu remain weak derivaives of X, M w.r.. x. From now on, we always assume ha x X and x M are coninuous in ime. We also assume ha for fixed [, T ] he mappings x X and x M are weak derivaives of X and M w.r.. x R. In paricular x X = 1 a.s. for almos all x R. In order o obain bounds on he weak derivaive u x, we sudy he process V := u x (, X ), [, T ]. Recall ha M = u(, X ) a.s. for all (, x) [, T ] R. Therefore, for fixed [, T ], he weak derivaives of he wo sides of he equaion w.r.. x R mus coincide up o a P λ - null se. The chain rule for weak derivaives (see Corollary 3.2 in [1] or Lemma A.3.1. in [6]) implies, for any fixed [, T ], ha we have for P λ - almos all (ω, x) x M 1 { xx >} = u x (, X ) x X 1 { xx >} = V x X 1 { xx >}. (17) Now, choose a fixed x R such ha x X = 1 a.s., (17), (15), (16) are saisfied for almos all (ω, ) [, T ] Ω and, in addiion, (17) is saisfied for =, P - almos surely. Noe 1

12 ha, since x X, x M are coninuous in ime, (15) and (16) in fac hold for all [, T ], P - almos surely. Observe ha V is bounded since u x (, x) is bounded. Furhermore, according o he definiion of he maximal inerval, if L σ,z >, hen here exiss ε > such ha for all (, x) [, T ] R we have u x (, x) (1 ε) δu δ u δ l. A priori, ε depends on. We will see below ha i can be chosen independenly of. In he case L σ,z = here exiss a consan K such ha for all (, x) [, T ] R we have u x (, x) K. We will show below ha K can be chosen independenly of. We now urn o he dynamics of V. Lemma 4.2. The process (V ) [,T ] has a ime-coninuous version which is an Iô process. Moreover, here exiss Ẑ H2,T V = ξ (X T ) such ha (V, Ẑ) is he unique soluion of he BSDE T Ẑ s dw s where ρ : Ω [, T ] R R R is defined by wih T ρ(s, V s, Ẑs)ds, ρ(, v, z) =v[2b (1 v) + γf mm (, M, Z )v + γf ma (, M, Z )h(, v, z)] z[ γf ma (, M, Z )v + (1 γf aa (, M, Z ))h(, v, z)], h(, v, z) = z γf ma(, M, Z )v 2 1 v[1 γf aa (, M, Z )]. [ Furhermore, Ẑ BMO(P), i.e. sup T [,T ] E Ẑs 2 ds ] F <. Proof. Le τ n = T inf{ : x X 1 n }. On [ 1, τ n ] we have V = x M xx, a.e. Hence V has a version which is an Iô process on [, τ n ]. We denoe he Iô process decomposiion by V = u x (, x) + The produc formula yields, on [, τ n ], Ẑ s dw s + κ s ds, [, τ n ]. d(v x X ) =V (B (2 x M x X ) γ(f mm (, M, Z ) x M + f ma (, M, Z ) x Z ))d + x X κ d + Ẑ( x Z γf ma (, M, Z ) x M γf aa (, M, Z ) x Z )d ( + x X Ẑ + V [ x Z γf ma (, M, Z ) x M γf aa (, M, Z ) ) x Z ] dw. The drif and diffusion coefficiens coincide wih he coefficiens in (16). This implies, using sraighforward ransformaions, he definiion of h and he propery V x X = x M, and κ = 1 x X [B x M x Z = x X h(, V, Ẑ) V (B (2 x M x X ) γ(f mm (, M, Z ) x M + f ma (, M, Z ) x Z )) Ẑ( x Z γf ma (, M, Z ) x M γf aa (, M, Z ] ) x Z ) =ρ(, V, Ẑ). 11

13 We nex show ha he denominaor of h is bounded away from zero. Assume firs L σ,z >. Then for all v wih v (1 ε) δu δ u δ l we have v(1 γf aa (s, M s, Z s )) v δu δ l δ u (1 ε), and hence he denominaor of h(, V, Ẑ) is greaer han or equal o ε. In he case where L σ,z =, he denominaor is equal o 1. I remains o show ha τ := lim n τ n = T a.s. To his end noe ha x X saisfies, on [, τ), he linear SDE where and Consequenly, d x X = α x X d + β x X dw, α := B (2V 1) γ(f mm (, M, Z )V + f ma (, M, Z )h(, V, Ẑ)) β := h(, V, Ẑ) γf ma (, M, Z )V γf aa (, M, Z )h(, V, Ẑ). ( τn ( x X τn = exp α s 1 ) τn ) 2 β2 s ds + β s dw s. Noe ha α and β are boh bounded by C(1 + Ẑ ) for some sufficienly large C >. We claim ha Ẑ has a bounded BMO-norm wih a bound which does no depend on n. Indeed, he pair (V, Ẑ) can be inerpreed as a soluion of a quadraic BSDE on [, τ n]. Since V is bounded, sandard resuls imply ha Ẑ has a bounded BMO-norm (see e.g. Theorem A in [6] for deails). ( ) τ Now if (lim n x X τn ) (ω) = for some ω, hen Ẑ 2 d (ω) = mus hold for he same ω. This, however, is false for almos all ω, due o Ẑ being a BMO-process on [, τ). In oher words, he coninuous process x X does no reach wih probabiliy 1 and, herefore, lim n τ n = T a.s. 1 In paricular V = x M xx a.e. and V has a ime-coninuous version. In he following we assume ha V refers o he ime-coninuous version of Lemma 4.2. Lemma 4.3. There exiss a P-equivalen probabiliy measure Q and a Q-BM W Q such ha (V, Ẑ) is he unique soluion of he BSDE T V =ξ (X T ) Ẑ s dws Q (18) ( T 2B s V s (1 V s ) + γf mm (s, M s, Z s )Vs 2 γ 2 fma(s, 2 M s, Z ) s ) 1 V (1 γf aa (s, M s, Z s )) V 3 ds. Proof. Noe ha where ρ(, v, z) =2B v(1 v) + γf mm (, M, Z )v 2 γ 2 f 2 ma(s, M s, Z s ) 1 v(1 γf aa (s, M s, Z s )) v3 zψ(, v, z), f ma (s, M s, ψ(, v, z) = γv Z s ) 1 v(1 γf aa (s, M s, Z s )) γf ma(, M, Z )v + (1 γf aa (, M, Z ))h(, v, z). 12

14 Since V is bounded, here exiss a consan C > such ha ψ(, V, Ẑ) C(1 + Ẑ ) for all [, T ]. Consequenly ψ(, V, Ẑ) BMO(P). In paricular here exiss a probabiliy measure Q P such ha ( dq T dp = exp ψ(, V, Ẑ) dw 1 T ) ψ 2 (, V, 2 Ẑ) d. By Girsanov s heorem W Q := W ψ(s, V s, Ẑs) ds, [, T ], is a Brownian moion w.r.. Q. Finally, observe ha V saisfies (18). Lemma 4.4. For all [, T ] we have a.s. q V 1, where q := exp ( T (2 B + γ f mm )) 1 + γ g (, 1). Proof. Le α(s, y) = 2B y(1 y) γf mm (s, M s, Z s )y 2 + γ 2 fma(s,m 2 s, Z s) 1 y(1 γf aa(s,m s, Z y3 be he s)) generaor of he BSDE (18). In he case L σ,z > one can modify α o a Lipschiz coninuous generaor by replacing y wih ( ( )) δ u δ u y (1 ε) (1 ε). δ u δ l δ u δ l Indeed, (V, Ẑ) solves also he modified BSDE since V s (1 ε) δu δ u δ l. Noice ha for all v wih v (1 ε) δu δ u δ l we have v(1 γf aa (s, M s, Z s )) v δu δ l δ u (1 ε), which furher implies ha v 1 1 v(1 γf aa (s, M s, Z s )) akes values only in [1, 1 ε ] and is Lipschiz coninuous. In he case L σ,z = he generaor α becomes Lipschiz coninuous by replacing y wih (y K) K. Now consider he cu-off funcion c(v) = ((v ) 1) and se Observe ha ˇα(s, v) = 2B c(v)(1 c(v)) γf mm (s, M s, Z s )c(v) 2 + γ 2 f 2 ma(s, M s, Z s ) 1 c(v)(1 γf aa (s, M s, Z s )) c(v)3. ˇα(s, v) 2B c(v)(1 c(v)) γf mm (s, M s, Z s )c(v) 2 + γ f 2 ma(s, M s, Z s ) f aa (s, M s, Z s )) c(v)2 2B c(v)(1 c(v)) γc(v) 2 1 f aa (s, M s, Z s ) de(d2 f)(s, M s, Z s ) 2B c(v)(1 c(v)). Le ( ˇV, Ž) be he soluion of he BSDE wih parameers (ξ (X T ), ˇα). The comparison heorem, applied o ( ˇV, Ž) and he BSDE wih parameers (1, 2B c(1 c)), implies ˇV 1. 13

15 Nex, we esimae ˇV from below. Noice ha ˇα(s, v) 2 B c(v)(1 c(v)) γ f mm c(v) 2 (2 B + γ f mm ) c(v). Moreover, ξ 1 (x) = 1+γg (ξ(x)) 1 1+γ g. A comparison wih he BSDE wih parameers ( ) γ g, (2 B + γ f mm ) c yields ˇV q. Noice ha his furher enails ha ( ˇV, Ž) is also a soluion of he BSDE (18). Uniqueness implies ha ˇV = V. Recall ha he diffusion coefficien in he forward par of (14), given by σ(, m, z) = z γf a (, m, z), is Lipschiz coninuous in z wih Lipschiz consan δu δ l δ u. Lemma 4.4 implies ha < q u x (, x) = V 1 < δu δ u δ l. Noe ha we have chosen a version of V such ha V = xm xx for all [, T ]. Moreover, for =, we have V = xm 1 = u x (, x), a.s. Since x was chosen arbirarily ouside a λ - null se, we have ha he u x (, ) is essenially bounded by 1. Since he bound does no depend on, by Theorem 3.4 i mus hold ha I max = [, T ]. Moreover, he following holds rue: Proposiion 4.5. There exiss a unique weakly regular decoupling field u for (14) on [, T ]. Moreover, for every [, T ] we have q u x (, x) 1 for almos all (ω, x) Ω R. We can now formulae he main resul of he secion. Theorem 4.6. Le m R. Then 1. for all [, T ] he funcion u(, ) has a Lipschiz coninuous inverse u 1 (, ), P -a.s., 2. here exiss a soluion (M, Y, Z) of (6), 3. he funcion ν(, m) := 1 γ (u 1 (, m) m) is a decoupling field for (6), i.e. Y = ν(, M ) for all [, T ], 4. an opimal conrol of problem (4) is given by α = Z, where Z is par of he soluion riple of (14) wih inial value x = u 1 (, m). Proof. Since u x (, ) is bounded away from zero and bounded from above by 1, u(, ) : R R is a bijecion. Therefore, for every m R here is an x R such ha M = u(, x) = m. For his x R here exiss according o Theorem 3.4 a unique riple (X, M, Z) such ha [ T ] sup E[ X s 2 ] + sup E[ M s 2 ] + E Z s 2 ds < s [,T ] s [,T ] and such ha FBSDE (14) is saisfied. Finally, using Lemma 4.1 we obain a soluion o FBSDE (6). A sraighforward calculaion shows ha for all [, T ] we have ν(, M ) = Y. Finally, using Proposiion 2.1 we obain ha Z = fa 1 (, M, Z ) is an opimal conrol for problem (4). 14

16 5 An illusraing example In general one can no expec he FBSDE (6) o possess a soluion in closed form. For some examples, however, one can calculae he process V explicily. This allows hen o obain a candidae for he decoupling field of FBSDE (14) and hence o derive a soluion of (6) in closed form. In he following we illusrae such an explici consrucion for a paricular choice of f and g. Le ā be a bounded and progressively measurable process, l > and L. Suppose ha f(, m, a) = lm 2 + (a ā ) 2. Furhermore, le b =, B R and g(m) = Lm 2. We can choose δ u = δ l = 2, and hence γ = 1 2. In his example we inerpre M as he sae of a paricle in a medium wih emperaure α a ime. ā is he naural emperaure process. Any cooling or heaing enails quadraic coss. Problem (4) corresponds o he aim of seering he paricle as close as possible o zero while keeping he coss for a emperaure conrol low. Noice ha ξ(x) = x 1+L and ξ (x) = 1 1+L. Moreover, he funcion ρ defined in Lemma 4.2 is given by ρ(, v) = 2Bv + (l 2B)v 2. Therefore, he process (V ) is he soluion of he Riccai equaion V () = 2BV () + (l 2B)V 2 (), V (T ) = L. By solving he Riccai equaion we obain explici expressions for V. 1. case: B =. 2. case: B. V = V = 1 l(t ) L e 2B(T ) 1 + L + l 2B 2B (1 e 2B(T ) ) Noe ha V does no depend on he spaial variable x. In paricular, he decoupling field u of (14) is affine linear in x. This implies ha M is affine linear in X. Noe ha in his example he FBSDE (14) akes he form X = x + ((2B l)m s BX s )ds + āsdw s, M = X T 1+L T Z s dw s T BM s ds. Assuming M = V X, he drif of he forward equaion is linear in X, which allows o solve he equaion explicily. Proposiion 5.1. The soluion of (19) is given by ( ) X = x + Hs 1 ā s dw s H, M = V X, Z = ā V, (19) 15

17 where H = e [(2B l)vs B]ds. In paricular, he decoupling field for (19) is given by u(, x) = V x. Moreover an opimal conrol of problem (4) is given by α = ā V (noe ha he opimal conrol does no depend on he iniial condiion). Proof. Le X, M and Z be defined as in Proposiion 5.1. A sraighforward calculaion shows ha X saisfies he dynamics dx = [(2B l)v B]X d + ā dw = ((2B l)m BX )d + ā dw. We now verify ha M saisfies he backward equaion in (19). Firs noice ha M saisfies he erminal condiion M T = X 1+L. The produc formula implies dm = V dx + X dv = V [(2B l)v B]X d + V ā dw + X (2BV + (l 2B)V 2 )d = BM d + Z dw. Hence (X, M, Z) saisfies (19). 6 Linking he decoupling field o he HJB equaion In his secion we ry o explain he link beween he FBSDE and he HJB approach for solving (4). Our argumens are mainly heurisic. Le us assume ha b, B, f, g do no depend on ω. Consider (6) and assume ha here exiss a differeniable funcion w : [, T ] R R such ha w(, M m ) = Y m a.s. for all iniial values m R and all [, T ]. Assuming ha w is coninuously differeniable in ime and wice coninuously differeniable in space, we can apply he Iô formula o w(, M ): Y = w(, M ) =w(, m) + + w (s, M s ) ds + w m (s, M s )(b s + B s M s ) ds A he same ime he backward equaion in (6) yields Y = Y + Z s dw s w m (s, M s )f 3 (s, M s, Z s ) dw s w mm (s, M s ) (f 3 (s, M s, Z s )) 2 ds. (B s Y s + f m (s, M s, f 3 (s, M s, Z s ))) ds. Comparing he maringale and he finie variaion pars leads o Z s = w m (s, M s )f 3 (s, M s, Z s ), w (s, M s ) + w m (s, M s )(b s + B s M s ) w mm(s, M s ) (f 3 (s, M s, Z s )) 2 = (B s w(s, M s ) + f m (s, M s, f 3 (s, M s, Z s ))). 16

18 The firs equaion is equivalen o w m (s, M s )α s +f a (s, M s, α s ) =, where α s := f 3 (s, M s, Z s ). Assume ha w m (s, M s ) and le ϕ : [, T ] R [, ) R be he funcion such ha α s = ϕ(s, M s, w m (s, M s )). Then we obain w (s, M s ) + w m (s, M s )(b s + B s M s ) w mm(s, M s ) (ϕ(s, M s, w m (s, M s ))) 2 = B s w(s, M s ) f m (s, M s, ϕ(s, M s, w m (s, M s ))). Thus, i is naural o expec w o saisfy he PDE w (s, m) + w m (s, m)(b s + B s m) w mm(s, m) (ϕ(s, m, w m (s, m))) 2 = B s w(s, m) f m (s, m, ϕ(s, m, w m (s, m))). Now le us look a he classical HJB approach o problem (4): The associaed HJB equaion for he value funcion v : [, T ] R R has he form [ v (s, m) + (b s + B s m)v m (s, m) sup 1 ] a R 2 a2 v mm (s, m) f(s, m, a) =. Assuming v mm, he opimal a in he above expression is given by ϕ(s, m, v mm (s, m)), which is sraighforward o verify using Ferma s heorem. We can rewrie he HJB equaion as v (s, m) + (b s + B s m)v m (s, m) v mm(s, m) (ϕ(s, m, v mm (s, m))) 2 = f(s, m, ϕ(s, m, v mm (s, m))). Noe ha w(t, ) = g and v(t, ) = g. I is, herefore, naural o conjecure ha w = v m holds rue. We verify his by assuming ha v is hree imes coninuously differeniable and by differeniaing he HJB equaion for v w.r.. m, hus obaining an equaion for v m =: w: w (s, m) + w m (s, m)(b s + B s m) w mm(s, m) (ϕ(s, m, w m (s, m))) 2 + B s w(s, m) + w m (s, m)ϕ(s, m, w m (s, m)) (ϕ m (s, m, w m (s, m)) + ϕ 3 (s, m, w m (s, m)) w mm (s, m)) = f m (s, m, ϕ(s, m, w m (s, m))) f a (s, m, ϕ(s, m, w m (s, m))) (ϕ m (s, m, w m (s, m)) + ϕ 3 (s, m, w m (s, m)) w mm (s, m)), where ϕ 3 is he derivaive w.r.. he las componen. Due o he definiion of ϕ he above simplifies o w (s, m) + w m (s, m)(b s + B s m) w mm(s, m) (ϕ(s, m, w m (s, m))) 2 = B s w(s, m) f m (s, m, ϕ(s, m, w m (s, m))). Indeed, w and w saisfy he same PDE, which makes i plausible o believe ha i is in fac he same objec. Now noe ha while v saisfies a fully non-linear PDE, he PDE saisfied by is spaial derivaive w is quasi-linear and hence is easier o analyze. In paricular, he quasi-lineariy of w allows o reduce he problem o he FBSDE (6). Alhough he FBSDE is srongly coupled, i sill crucially simplifies he problem. Indeed, as we have seen in Secion 4, he mehod of decoupling fields allows o prove exisence, uniqueness and regulariy. 17

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