Backward Stochastic Differential Equations and Applications in Finance

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1 Backward Sochaic Differenial Equaion and Applicaion in Finance Ying Hu Augu 1, Inroducion The aim of hi hor cae i o preen he baic heory of BSDE and o give ome applicaion in 2 differen domain: mahemaical finance and parial differenial equaion. Thi hor coure i divided ino 3 par: Par 1 give he baic heory of BSDE; Par 2 give he link beween BSDE and PDE and Par 3 give he recen developmen of quadraic BSDE. To olve a BSDE, i i o find a couple of adaped procee (Y, Z) aifying he following SDE: dy = f(, Y, Z )d + Z dw, T, wih he erminal condiion (ha why we call backward) Y T = ξ, where ξ i a quare-inegrable random variable. A SDE, uch equaion hould be underood in he ene of inegral, i.e. Y = ξ + f(, Y, Z )d Z dw, T. The BSDE were inroduced in 1973 by J.M. Bimu in he cae where f i linear w.r.. (Y, Z); and Pardoux and Peng developed he heory of BSDE where f i Lipchiz. Since hen, BSDE have been exenively udied becaue of heir profound applicaion o mahemaical finance a well a he link wih PDE. Le u fir give ome example. 1

2 In finance, an imporan queion i o deermine he price of an opion. The imple example i ha of he Black-Schole model and an European call. The price of hi financial produce V aifie he equaion: dv = (rv + θz )d + Z dw, where r i he hor erm inere rae and θ i he rik premium, wih he erminal condiion V T = (S T K) + where S i he price of he ock and K i a conan, which i fixed in advance. We ee ha i i a linear BSDE in hi imple model bu i can be non-linear in more complicaed financial model. Le u look a a econd example. Conider he following PDE: u(, x) x,xu(, x) + f(u(, x)) =, u(t, x) = g(x). Le u uppoe ha hi PDE admi a claical oluion u. Applying Io formula o u(, W ); we obain: du(, W ) = { u(, W ) x,xu(, W )}d + x u(, W )dw = f(u(, W ))d + x u(, W )dw. We obain ill one BSDE - which i nonlinear if f i - Puing Y = u(, W ) and Z = x u(, W ) becaue dy = f(y )d + Z dw, wih, Y T = g(w T ). 2 Par 1: BSDE: Lipchiz Cae The aim of hi ecion i o inroduce he definiion of BSDE, and o preciely give he noaion. Then we give he baic reul in Lipchiz cae: exience and uniquene; linear BSDE and comparion heorem; and ome example. 2.1 Vocabulary and noaion Preenaion of he problem. Le (Ω, F, (F ), P ) be a filered probabiliy pace and ξ an F T -meaurable random variable. One wan o olve he following ODE: dy d = f(y ),, T, wih, Y T = ξ, 2

3 moreover for any, Y doe no depend on he fuure afer, i.e. he proce Y i (F ) -adaped. Take he imple example f =. The naural candidae i Y = ξ which i no adaped if ξ i no deerminiic. The be approximaion ay in L 2 - adaped i he maringale Y = E(ξ F ). If we work wih he naural filraion of a BM, he Brownian maringale repreenaion heorem permi o conruc a quare-inegrable adaped proce Z uch ha One elemenary calculu give Y = ξ Y = E(ξ F ) = Eξ + Z dw. Z dw, i.e. dy = Z dw, wih, Y T = ξ. We ee ha he econd unknown (he proce Z) appear o le Y be adaped in hi imple example. A a generalizaion, we permi f o depend on he proce Z; he equaion become hen: Noaion. dy = f(, Y, Z )d + Z dw, wih, Y T = ξ. Le (Ω, F, P ) a complee probabiliy pace and W a d-dimenional BM on hi pace. We will denoe by (F ) he naural filraion of he BM W. We will work wih wo pace of procee: Denoe by S 2 (R k ) he vecor pace formed by he procee Y, progreively meaurable, wih value in R k, uch ha Y 2 S = E up Y 2 2 <, T and S 2 c he ubpace formed by he coninuou procee. Two indiinguihable procee will be equal and we will ue he ame noaion for he quoien pace. Denoe M 2 (R k d ) a he e of hoe procee Z, progreively meaurable, wih value in R k d, uch ha Z 2 M 2 = E 3 Z 2 d <,

4 where if z R k d, z 2 = race(zz ). Denoe M 2 (R k d ) a he e of equivalen clae of M 2 (R k d ). R k and R k d will be ofen omied; he pace S 2, S 2 c and M 2 are Banach pace for he defined norm. We denoe by B 2 he Banach pace S 2 (R k ) M 2 (R k d ). In hi chaper, we are given one random mapping defined on, T Ω R k R k d wih value in R k uch ha, for any (y, z) R k R k d, he proce {f(, y, z)} T i progreively meaurable. We conider alo a random variable ξ, F T -meaurable and wih value in R k. In hi par, we wan o olve he following BSDE: dy = f(, Y, Z )d + Z dw, T, Y T = ξ, Or, equivalenly, in inegral form: Y = ξ + f(r, Y r, Z r )dr Z r dw r, T. (1) The funcion f i called he generaor of he BSDE and ξ he erminal condiion. Now, we can give a precie definiion of he oluion of BSDE (1). Definiion 1 One oluion of he BSDE (1) i a couple of procee (Y, Z) aifying: 1. Y and Z are progreively meaurable wih value repecively in R k and R k d ; 2. P a.. ( f(r, Y r, Z r ) + Z r 2 )dr < ; 3. P a.., we have: Y = ξ + f(r, Y r, Z r )dr Z r dw r, T. Remark 2 I i imporan de noe he following wo poin: fir, he inegral of he equaion (1) are well defined, Y i a coninuou emi-maringale; hen, a he proce Y i progreively meaurable, i i adaped and hen in paricular Y i deerminiic. Before o give he fir heorem of exience and uniquene, we are going o how ha, under ome reaonable hypohei on he generaor f, he proce Y belong o S 2. 4

5 Propoiion 3 Suppoe ha here exi a proce f M 2 (R) and a conan λ uch ha: (, y, z), T R k R k d, f(, y, z) f + λ( y + z ). If (Y, Z) i a oluion of he BSDE (1) uch ha Z M 2, hen Y belong o S 2 c. Proof: The reul follow from Gronwall inequaliy and he fac ha Y i deerminiic. In fac, we have, for any, T, Y = Y and hen, uing he hypohei on f, Seing Y Y + ζ = Y + f(r, Y r, Z r )dr + (f r + λ Z r )dr + up T (f r + λ Z r )dr + up T Z r dw r, Z r dw r + λ Z r dw r. Y r dr. From he hypohei, Z belong o M 2 and hen, via Doob inequaliy, he hird erm i quare-inegrable ; i i he ame for f, and Y i deerminiic hence quare-inegrable; we deduce ha ζ i a quare-inegrable random variable. A Y i a coninuou proce - ee he la remark, Gronwall inequaliy give he inequaliy up T Y ζe λt which how ha Y belong o S 2. Remark 4 The reul remain rue when f 1 i a quare-inegrable random variable. We finih by a reul of inegrabiliy which will be ued everal ime. { } Lemma 5 Le Y S 2 (R k ) and Z M 2 (R k d ). Then Y Z dw,, T i a uniformly inegrable maringale. Proof: The Burkholder-Davi-Gundy inequaliie give ( T ) 1/2 E up Y Z dw CE Y r 2 Z r 2 dr T ( ) 1/2 CE up T Y Z r 2 dr, 5

6 and hen, a ab a 2 /2 + b 2 /2, E Y Z dw C (E up T up Y 2 + E T Z r dr ) 2. Bu he la quaniy i finie by he hypohei; we ge he reul. 2.2 Lipchiz cae Pardoux-Peng reul In hi ubecion, we will how he fir reul of exience and uniquene. Thi reul i due o E. Pardoux and S. Peng; hi i he fir reul of exience and uniquene for he BSDE when he generaor i non-linear. Recall for he la ime ha f i defined on, T Ω R k R k d wih value in R k uch ha, for any (y, z) R k R k d, he proce {f(, y, z)} T i progreively meaurable. We conider alo ξ a random variable, F T - meaurable, wih valued in R k. Here are he hypohee under which we will udy: (L) There exi a conan λ uch ha P -a.., 1. Condiion of Lipchiz w.r.. (y, z): for any, y, y, z, z, f(, y, z) f(, y, z ) λ( y y + z z ); 2. Condiion of inegrabiliy: E ξ 2 + f(r,, ) 2 dr <. We begin wih a very imple cae when f doe depend neiher on y nor on z, i.e. we will be given quare-inegrable ξ and a proce F in M 2 (R k ) and we wan o find a oluion of he BSDE Y = ξ + F r dr Z r dw r, T. (2) Lemma 6 Le ξ L 2 (F T ) and {F } T M 2 (R k ). The BSDE (2) admi a unique oluion (Y, Z) uch ha Z M 2. Proof: Le u fir uppoe ha (Y, Z) i a oluion aifying Z M 2. If we ake he condiional expecaion w.r.. F, we deduce ( ) Y = E ξ + F r dr F. 6

7 We define hen Y wih he help of he la formula, and i remain o find Z. Noe ha becaue of Fubini heorem, a F i progreively meaurable, F rdr i an (F ),T adaped proce; in fac in Sc 2 a F i quare-inegrable inegrable. We have hen, for any, T, ( Y = E ξ + ) F r dr F F r dr = M F r dr. M i a Brownian maringale. By he Brownian maringale repreenaion heorem, we conruc a proce Z belonging o M 2 uch ha Y = M F r dr = M + Z r dw r F r dr. We check eaily ha (Y, Z) conruced i a oluion of he BSDE becaue Y T = ξ, Y ξ = M + = F r dr Z r dw r Z r dw r. ( ) F r dr M + Z r dw r F r dr The uniquene i eviden for he oluion aifying Z M 2. We how now he Pardoux-Peng Theorem. Theorem 7 Pardoux-Peng 9 Under he hypohei (L), he BSDE (1) admi a unique oluion (Y, Z) uch ha Z M 2. Proof: We ue a fixed poin argumen in he Banach pace B 2 by conrucing an applicaion Ψ from B 2 in ielf uch ha (Y, Z) B 2 i oluion of he BSDE (1) if and only if i i a fixed poin of Ψ. For (U, V ) elemen of B 2, we define (Y, Z) = Ψ(U, V ) a being he oluion of he BSDE: Y = ξ + f(r, U r, V r )dr Z r dw r, T. Noe ha hi la BSDE admi a unique oluion which i in B 2. In fac, e F r = f(r, U r, V r ). Thi proce belong o M 2 becaue, f being Lipchiz, F r f(r,, ) + λ U r + λ V r, 7

8 and he la hree procee are quare-inegrable. The, we can apply he Lemma 6 o obain a unique oluion (Y, Z) B 2. The mapping Ψ from B 2 in ielf i hen well defined. Le (U, V ) and (U, V ) wo elemen in B 2 and (Y, Z) = Ψ(U, V ), (Y, Z ) = Ψ(U, V ). Denoe y = Y Y and z = Z Z. We have, y T = and dy = {f(, U, V ) f(, U, V )}d + z dw. We apply Iô formula o e α y 2 o obain d(e α y 2 ) = αe α y 2 d 2e α y {f(, U, V ) f(, U, V )}d+2e α y z dw +e α z 2 d. A a conequence, inegraing from o T, we obain e α y 2 + e αr z r 2 dr = e αr ( α y r 2 + 2y r {f(r, U r, V r ) f(r, U r, V r )}dr 2e αr y r z r dw r, and, a f i Lipchiz, i follow, denoing by u and v for U U and V V repecively, e α y 2 + e αr z r 2 dr e αr ( α y r 2 +2λ y r u r +2λ y r v r )dr 2e αr y r z r dw r. For any ɛ >, we have 2ab a 2 /ɛ + ɛb 2, and hen, he la inequaliy give e α y 2 + e αr z r 2 dr e αr ( α + 2λ 2 /ɛ) y r 2 dr +ɛ e αr ( u r 2 + v r 2 )dr, 2e αr y r z r dw r and puing α = 2λ 2 /ɛ, we have, denoing R ɛ = ɛ eαr ( u r 2 + v r 2 )dr,, T, e α y 2 + e αr z r 2 dr R ɛ 2 e αr y r z r dw r. (3) From he Lemma 5, he local maringale { eαr y r z r dw r },T i in fac a real maringale null a a Y, Y belong o S 2 and Z, Z belong M 2. 8

9 In paricular, aking he expecaion he ochaic inegral diappear via he la remark we obain eaily, for =, ha E e αr z r 2 dr ER ɛ. (4) Coming back o he inequaliy (3), he BDG inequaliie provide wih C univeral ( ) 1/2 E up e α y 2 ER ɛ + CE e 2αr y r 2 z r 2 dr T ER ɛ + CE hen, a ab a 2 /2 + b 2 /2, E up e α y 2 ER ɛ + 1 T 2 E up e α/2 y T ( up e α y 2 + C2 T 2 E Taking ino conideraion he inequaliy (4), we obain finally E up e α y 2 + T e αr z r 2 dr and hen, coming back o he definiion of R ɛ, E up e α y 2 + T e αr z r 2 dr ɛ(3+c 2 )(1 T )E ) 1/2 e αr z r 2 dr, (3 + C 2 )ER ɛ, e αr z r 2 dr. up e α u 2 + T Taking ɛ uch ha ɛ(3 + C 2 )(1 T ) = 1/2, he mapping Ψ i a ric conracion from B 2 o ielf if we equipped i wih he norm (U, V ) α = E up e α U 2 + T e αr V r 2 dr 1/2, which i a Banach pace - hi la norm being equivalen o he uual norm correponding o he cae when α =. Ψ ha hen a unique fixed poin, which enure he exience and he uniquene of a oluion of he BSDE (1) in B 2. We obain hen a unique oluion aifying Z M 2 becaue he Propoiion 3 implie ha uch a oluion belong o B 2. Remark 8 From now on, he expreion he oluion of he BSDE ignifie he oluion of he BSDE aifying Z M 2. e αr v r 2 dr. 9

10 2.2.2 The role of Z We are going o ee ha he role of Z, more preciely ha of erm Z r dw r i o render he proce Y adaped and when hi i no neceary, Z i zero. Propoiion 9 Le (Y, Z) he oluion of he (1) and le τ a opping ime up bounded by T. We uppoe, beide he hypohei (L), ha ξ i F τ - meaurable and ha f(, y, z) = a oon a τ. Then Y = Y τ and Z = if τ. Proof: We have, P-a.., Y = ξ + f(r, Y r, Z r )dr Z r dw r, T, and hen, for = τ, a f(, y, z) = a oon a τ, Y τ = ξ + f(r, Y r, Z r )dr Z r dw r = ξ Z r dw r. τ τ τ A a conequence Y τ = E(ξ F τ ) = ξ and hen τ Z rdw r = from which we deduce ha ( ) 2 E Z r dw r = E Z r 2 dr =, τ τ and finally ha Z r 1 r τ =. I follow immediaely ha, if τ, Y = Y τ, a from he hypohei, Y τ = Y + τ f(r, Y r, Z r )dr τ Z r dw r = Y, which conclude he proof. Noe ha when ξ and f are deerminiic, Z i zero and Y i he oluion of he ODE dy d = f(, Y, ), Y T = ξ A priori eimae We finih hi ecion by giving a fir eimae on he BSDE: i i in fac o udy he dependance of he oluion of he BSDE w.r.. he parameer ξ and he proce {f(,, )} T. 1

11 Propoiion 1 Suppoe ha (ξ, f) aifie (L). Le (Y, Z) he oluion of he BSDE (1) uch ha Z M 2. Then, here exi a conan C u univeral uch ha, for any β 1 + 2λ + 2λ 2, E up e β Y 2 + T e β Z 2 d Proof: We apply Iô formula o e β Y 2 o obain: e β Y 2 + e βr Z r 2 dr = e βt ξ 2 + C u E e βt ξ 2 + e β f(,, ) 2 d. A f i λ-lipchiz, we have, for any (, y, z), e βr ( β Y r 2 +2Y r f(r, Y r, Z r ))dr 2y f(, y, z) 2 y f(,, ) + 2λ y 2 + 2λ y z, 2e βr Y r Z r dw r. and hen uing he fac ha 2ab ɛa 2 + b2 ɛ for ɛ = 1 and ɛ = 2, we have 2y f(, y, z) (1 + 2λ + 2λ 2 ) y 2 + f(,, ) 2 + z 2 /2. For β 1 + 2λ + 2λ 2, we obain, for any, T, e β Y e βr Z r 2 dr e βt ξ 2 + e βr f(r,, ) 2 dr 2 e βr Y r Z r dw r. 2 (5) The local maringale { eβr Y r Z r dw r,, T } i a maringale - cf. Lemma 5. In paricular, aking he expecaion, we obain eaily, for =, E e βr Z r 2 d 2E e βt ξ 2 + e βr f(r,, ) 2 dr. Coming back o he inequaliy (5), he BDG inequaliie provide - wih C univeral -, E up e β Y 2 E T ( ) 1/2 e βt ξ 2 + e βr f(r,, ) 2 dr +CE e 2βr Y r 2 Z r 2 dr. On he oher hand, ( ) 1/2 CE e 2βr Y r 2 Z r 2 dr CE 1 2 E up e β/2 Y T ( up e β Y 2 + C2 T 2 ) 1/2 e βr Z r 2 dr e βr Z r 2 dr. 11

12 I follow ha, E up e β Y 2 2E T and finally we obain E up e β Y 2 + T e βt ξ 2 + e βr f(r,, ) 2 dr +C 2 E e βr Z r 2 dr, e β Z 2 d 2(2+C 2 )E e βt ξ 2 + e β f(,, ) 2 d, which conclude he proof of he propoiion by aking C u = 2(2 + C 2 ). 2.3 Linear BSDE and comparion heorem linear BSDE In hi ubecion, we udy paricular cae of linear BSDE for which we give ome explici formulae. Le k = 1; Y i real and Z i a row vecor of dimenion d. Propoiion 11 Le (a, b ) be a proce wih value in R R d, progreively meaurable and bounded. Le {c } be an elemen in M 2 (R) and ξ a random variable, F T -meaurable, quare-inegrable wih real value. Then he linear BSDE Y = ξ + {a r Y r + Z r b r + c r }dr Z r dw r, admi a unique oluion: wih, for any, T, ( ) Y = Γ 1 E ξγ T + c r Γ r dr F, { Γ = exp b r dw r 1 2 b r 2 dr + Proof: We begin by noing ha he proce Γ aifie: dγ = Γ (a d + b dw ), Γ = 1. } a r dr. On he oher hand, a b i bounded, Doob inequaliy how ha Γ belong o S 2. 12

13 Furhermore, he hypohee of hi propoiion enure he exience of a unique oluion (Y, Z) o he linear BSDE; i uffice o e f(, y, z) = a y + zb + c and o check ha (L) i aified. Y belong o S 2 from he Propoiion (3). The inegraion by par formula give dγ Y = Γ dy + Y dγ + d Γ, Y = Γ c d + Γ Z dw + Γ Y b dw, which how ha he proce Γ Y + c Γ d i a local maringale which i in fac a maringale a c M 2 and Γ, Y are in S 2. I follow ha Γ Y + c r Γ r dr = E which give he announced formula. ( Γ T Y T + c r Γ r dr F ), Remark 12 Noe ha if ξ and c hen he oluion of he linear BSDE verifie Y. Thi remark permi u o obain he comparion heorem in he nex ubecion. To illurae hi reul aking he cae when a and c are zero. We have hen ( { Y = E exp b r dw r 1 } ) b r 2 dr ξ F = E (ξ F ), 2 where P i he deniy meaure w.r.. P { L T = exp b r dw r 1 2 } b r 2 dr. Anoher mehod o ee hi, by he idea of rik-neural probabiliy, i o udy he BSDE under P. In fac, under P, B = W b rdr i a BM by Giranov heorem. Then he equaion can be wrien dy = Z b d + Z dw = Z db, Y T = ξ. Then, under P, Y i a maringale, which how alo he formula Comparion heorem Thi ubecion i devoed o he comparion heorem which permi o compare he oluion of wo BSDE (in R) a oon a one can compare he erminal condiion and he generaor. 13

14 Theorem 13 Suppoe ha k = 1 and ha (ξ, f), (ξ, f ) verify he hypohei (L). We denoe (Y, Z) and (Y, Z ) he oluion of correponding BSDE. We uppoe alo ha P a.. ξ ξ and ha f(, Y, Z ) f (, Y, Z ) m P a.e. (m i he Lebegue meaure). Then P a..,, T, Y Y. Moreover, if Y = Y, hen P a.., Y = Y, T and f(, Y, Z ) = f (, Y, Z ) m P a.e. In paricular, a oon a P (ξ < ξ ) > or f(, Y, Z ) < f (, Y, Z ) on a e wih poiive m P -meaure hen Y < Y Proof: We ue he echnique of liberalizaion which permi u o he linear BSDE cae. We look for an equaion aified by U = Y Y. Denoing V = Z Z and ζ = ξ ξ, U = ζ + (f (r, Y r, Z r) f(r, Y r, Z r ))dr We divide he incremen of f ino hree par by wriing V r dw r. f (r, Y r, Z r) f(r, Y r, Z r ) = f (r, Y r, Z r) f (r, Y r, Z r) + f (r, Y r, Z r) f (r, Y r, Z r ) +f (r, Y r, Z r ) f(r, Y r, Z r )(which i non-negaive). We inroduce wo procee a and b: a i wih real value and b i a column vecor of dimenion d. We pu and a r = f (r, Y r, Z r) f (r, Y r, Z r) U r 1 Ur >, b r = (f (r, Y r, Z r) f (r, Y r, Z r ))Vr V r 2 1 Vr >. We noe ha, a f i Lipchiz, hee wo procee are progreively meaurable and bounded. Wih hee noaion, we have: U = ζ + (a r U r + V r b r + c r )dr V r dw r, where c r = f (r, Y r, Z r ) f(r, Y r, Z r ). From he hypohei, we have ζ and c r. Applying he explici formula for he linear BSDE - Propoiion 11)-, we have, for, T, U = Γ 1 E ( ) ζγ T + c r Γ r dr F 14.

15 where, for r T, { r Γ r = exp b u dw u 1 2 r r } b u 2 du + a u du. A already menioned in he remark following he Propoiion 11, hi formula how ha U, a oon a ζ and c r. For he econd par of he reul (ric comparion), if moreover, U = we have ( ) = E ζγ T + c r Γ r dr F, which conclude he proof Black-Schole model The Black-Schole model i an example of linear BSDE. In a financial marke here exi one ock whoe price i given by he SDE ds = S (µd + σdw ), S = x, where µ R, σ > ; he parameer σ i called he volailiy. We have, for any, S = x exp{σw + (µ σ 2 /2)}. There exi alo a rik-free ae, whoe price i given by de = re d, E = y, i.e. E = ye r. The raegy i given by a couple of procee, (p, q ), which are adaped procee. p i he number of ock, and q he number of rik-free ae. If we conider only he elf-financing raegy, hen he value of he porfolio will be given by dv = q de + p ds = rq E d + p S (µd + σdw ). A q E = V p S, and hen, we have by noing π = p S (he amoun of money inveed in riky ae), dv = rv d + π σ(µ r)/σd + π σdw. Puing Z = π σ and θ = (µ r)/σ (he rik premium), dv = rv d + θz d + Z dw. 15

16 One problem in finance i o give a price o opion. A European buyopion (call opion) wih mauriy T and exercie price K i a conrac which give he righ bu no obligaion o hi holder o buy one hare of he ock a he exercie price K. Equivalenly, he eller of he opion ha o pay (S T K) + o i holder, which repreen he profi ha permi he exercie of he opion. More generally we can imagine a claim whoe profi i ju a non-negaive random variable ξ depending on S. A which price v hould one ell he opion? To find v, he fundamenal idea i he replicaion, he eller ell he opion a he price v and inve hi um in he marke following he raegy Z o be found. The value of hi porfolio i governed by he SDE: dv = rv d + θz d + Z dw, V = v. The problem i hen o find v and Z uch ha he oluion of he previou SDE verifie V T = ξ; we ay ha in hi cae v i he fair price. i.e., can we find adaped (V, Z) uch ha dv = rv d + θz d + Z dw, V T = ξ. In hi cae i uffice o ell he opion a he price v = V. pricing i o olve he BSDE which i linear. Hence he Suppoe now ha he regulaor of he marke wan o avoid hor-elling of he ock. I can diuade hi kind of ranacion by penalizing inveor by a proporional co βπ = γz (γ > ). In hi cae, replicaing a claim i o olve he following BSDE dv = (rv + θz γz )d + Z dw, V T = ξ. Thi BSDE i no linear bu i verifie ill he hypohei (L). Anoher example of BSDE nonlinear encounered in finance i he following dv = (rv + θz )d + Z dw (R r)(v Z /σ) d, V T = ξ. We hould olve hi la BSDE o replicae a claim when he borrowing rae R i bigger han he lending rae r. We remark ha he raegie are admiible (V ). Thi reul follow from he comparion heorem a ξ and f(,, ). 16

17 3 Par 2: BSDE and PDE We conider here he Markovian BSDE: i i a very pecial cae when he he erminal condiion and he generaor on ω i given via he oluion of a SDE. We will ee ha he Markov propery hold for he BSDE a oon a i hold for he SDE, ha i why we call i a Markovian cae. 3.1 Elemenary properie of flow We will work here for SDE wih deerminiic iniial condiion which permi u o ake a filraion he naural filraion of BM {F W }. We conider wo coninuou funcion b :, T R n R n and σ :, T R n R n d. We uppoe ha here exi one conan K > uch ha, for any, for any x, x in R n, 1. b(, x) b(, x ) + σ(, x) σ(, x ) K x x ; 2. b(, x) + σ(, x) K(1 + x ). Under hee hypohee, we can conruc, given, T and a random variable θ L 2 (F ), {X,θ } u T a he oluion of he SDE: X,θ u = θ + u u b(r, X,θ r )dr + u σ(r, X,θ r )dw r, u T ; (6) a a convenion, if u, Xu,θ = E(θ F u ). In he deerminiic cae, if σ =, he flow X,x, denoed φ,x, have many properie. In paricular: 1.φ,x i Lipchiz w.r.. (, x, ); 2. for r, φ r,x = φ,φr,x ; 3. if, x φ,x i a homeomorphim of R n Coninuiy To how he coninuiy properie of he flow, we will apply he crierion of Kolmogorov. In order o do i, we hould eablih ome eimae on he momen of X,x. The proof are quie involved bu no very difficul: i uffice o apply Holder inequaliy and Gronwall lemma. Propoiion 14 Le p 1. There exi a conan C, depending on T and p only, uch ha:, T, x R n, E up X,x p C(1 + x p ) (7) T 17

18 We know ha he oluion of he SDE ha he momen of all order. By claical reul, we alo have an eimae of he ame ype for he momen of incremen of X. Propoiion 15 Le 2 p <. There exi a conan C uch ha, for any (, x), (, x ) belonging o, T R n, E up X,x X,x p C( x x p + p/2 (1 + x p )). (8) T Corollary 16 There exi a modificaion of he proce X uch ha he mapping from, T R n wih value in Sc 2, (, x) ( X,x ) i coninuou. In paricular, (, x, ) X,x i coninuou. Proof: Thi i a direc applicaion of he previou eimae which i rue for any p 2 and of he crierion of Kolmogorov. Remark 17 I i imporan o noe ha he previou corollary implie in paricular ha, P-a.., for any (, x, ), he equaion (6) i rue. We end hi ubecion by preciely indicaing he regulariy of he flow generaed by he SDE. Propoiion 18 Le 2 p <. There exi a conan C uch ha, for any (, x, ), (, x, ), E X,x X,x p C( x x p + (1 + x p )( p/2 + p/2 )). (9) In paricular, he rajecorie (, x, ) X,x are Holder-coninuou (locally in x) in of order β, in x of order α, and in of order β, for any β < 1/2 and α < 1. Proof: The regulariy of he rajecorie follow from he crierion of Kolmogorov and he eimae Markov Propery We will eablih he Markov propery for he oluion of he SDE (6) a a conequence of he propery of he flow. Propoiion 19 For any x R n and any r, we have X r,x = X,Xr,x, P a.. 18

19 Remark 2 In fac, by coninuiy, he inequaliy X r,x = X,Xr,x for any x and for any r T excep P -null e. happen We will now eablih he Markov propery. Recall ha a proce X i Markov if i doe no depend on he pa only given he preen. We have he following mahemaical definiion: Definiion 21 Markov propery Le X a progreively meaurable proce w.r.. he filraion {F }. We ay ha X ha he Markov propery w.r.. {F } if, for any bounded Borel funcion f, and for any, E(f(X ) F ) = E(f(X ) X ), P a.. Le u how he Markov propery for he oluion of he SDE (6) w.r.. he σ-algebra of he BM W. Theorem 22 Le x R n and, T. (X,x ) T i a Markov proce w.r.. he filraion of he BM. If f i meaurable and bounded, hen for any r, E(f(X,x ) F r ) = Λ(Xr,x ) P a.. wih Λ(y) = Ef(X r,y ). Proof: Le r T. From he propery of he flow, we have X,x = X r,x,x r. Moreover, we will how ha X r,y i meaurable w.r.. he σ-algebra of he incremen of he BM σ{w r+u W r, u, r}. Hence, X r,y where Φ i meaurable. I follow ha, X,x = X r,x,x r = Φ(y, W r+u W r ; u r), = Φ(Xr,x, W r+u W r ; u r). To conclude, we hould noe ha Xr,x i F r -meaurable and ha F r i independen of he σ-algebra σ{w r+u W r, u, r}. We have hen E(f(X,x ) F r ) = E(f{Φ(Xr,x, W r+u W r ; u r)} F r ) = E f{φ(y, W r+u W r ; u r)} y=x,x r = E f(x r,y ) y=x,x r = Λ(Xr,x ). 19

20 which how he deired propery. We can alo how ha, if g i meaurable and for example bounded, hen where Γ(r, y) = E r ( E r g(u, X,x u )du F r ) = Γ(r, X,x r ), g(u, Xr,y u )du. Remark 23 If b and σ doe no depend on ime we ay ha he SDE i homogenou we can how ha he law of X,x i he ame a ha X,x. We have hen E(f(X,x ) F r ) = Ef(X,y r ) y=x,x r P a The model Hypohee and noaion Le u give a complee probabiliy pace (Ω, F, P ) on which i defined a d- dim BM W. The filraion {F } i he naural filraion of W augmened uch ha he uual hypohee are aified. We conider wo coninuou funcion b :, T R n R n and σ :, T R n R n d. We uppoe ha here exi a conan K > uch ha, for any and any x, x in R n, 1. b(, x) b(, x ) + σ(, x) σ(, x ) K x x ; 2. b(, x) + σ(, x) K(1 + x ). Under hee hypohee, we can conruc, given a real, T and a random variable θ L 2 (F ), {Xu,θ } u T a he oluion of he SDE X,θ u = θ + u b(r, X,θ r )dr + u σ(r, X,θ r )dw r, u T. (1) A a convenion, we define Xu,θ = E(θ F u ) for u. The properie {Xu,θ } u T have been udied in he la ubecion, in paricular in he cae θ = x R n. Conider alo wo coninuou funcion g : R k R k and f :, T R n R k R k d R k. We uppoe ha f and g verify he following hypohee: here exi wo real µ and p 1 uch ha, for any (, x, y, x, y, z, z ), 1. f(, x, y, z) f(, x, y, z ) µ y y ; 2. f(, x, y, z) f(, x, y, z ) K z z ; 2

21 3. g(x) + f(, x, y, z) K(1 + x p + y + z ). Under hee hypohee, if θ belong o L 2p (F ), we can olve he BSDE Y,θ u = g(x,θ T ) + u f(r, X,θ r, Y,θ r, Zr,θ )dr u Z,θ r dw r, u T. (11) In hi par, we will uppoe ha he previou hypohee on he coefficien b,σ, f and g are aified. Someime, we will uppoe a ronger hypohei on f and g. For example, we may ue he (Lip) hypohei, where g i K-Lipchiz and f i K-Lipchiz in x uniformly w.r.. (, y, z). We give wo elemenary properie of he BSDE (11). Propoiion 24 If θ L 2p (F ), hen he BSDE (11) ha a oluion (verifying Z M 2 ), {(Yu,θ, Zu,θ )} u T. Furhermore, here exi a conan C uch ha, for any, for any θ L 2p (F ), E up Yu,θ 2 + u T Zr,θ 2 dr C(1 + E θ 2p ). Remark 25 In paricular, when θ i a conan equal o x, we have E up Yu,x 2 + u T Zr,x 2 dr C(1 + x 2p ). Propoiion 26 If θ n converge o θ in L 2p (F ), hen E up Yu,θ Yu,θn 2 + u T Z,θ r Zr,θn 2 dr, if n. Moreover, under he hypohei (Lip), we have, if θ and θ are F -meaurable and quare-inegrable, E up Yu,θ Yu,θ 2 + u T Z,θ r 3.3 Markov Propery of BSDE Zr,θ 2 dr CE θ θ 2. In hi ubecion, we will eablih ha he Markov propery of he oluion of SDE ranlae o oluion of BSDE. Thi propery i imporan for he applicaion o PDE. We begin by how ha Y,x i a deerminiic quaniy. Inroduce a new noaion: if u, Fu = σ(n, W r W, r u). 21

22 Propoiion 27 If (, x), T R n, hen {(Xu,x, Yu,x )} u T i adaped w.r.. he filraion {Fu} u T. In paricular, Y,x i deerminiic. We can chooe a verion of {Zu,x } u T adaped w.r.. he filraion {Fu} u T. Since Y,x i deerminiic, we can define a funcion (, x), T R n, u(, x) := Y,x. Le u begin by udying he growh and he coninuiy of hi funcion. Propoiion 28 The funcion u i coninuou and wih polynomial growh, i.e., (, x), T R n, u(, x) C(1 + x p ). Moreover, under he hypohei (Lip), he funcion u verifie, for any (, x), (, x ), ha u(, x) u(, x ) C( x x + 1/2 (1 + x )). Wih he help of hi funcion, we can eablih he Markov propery for he BSDE. Theorem 29 Le, T and θ L 2p (F ). We have: P a.. Y,θ = u(, θ) = Y, θ. Corollary 3 Le, T and θ L 2p (F ). Then, we have P -a..,, T, Y,θ = u(, X,θ ). Remark 31 We ue ofen hi reul wih θ conanly equal o x. Thi Markov propery, Y,x = u(, X,x ), play an imporan role when we ry o conruc he oluion of he PDE wih he help of BSDE: we will ee i laely. 3.4 Feynman-Kac Formula We finih hi par wih he link beween he BSDE and he PDE. In he la ubecion, we have een ha Yr,x = u(r, Xr,x ) where u i a deerminiic funcion. We are going o ee ha u i oluion of a PDE. We will noe L he following econd order differenial operaor : for h regular Lh(, x) = 1 2 race(σσ (, x)d 2 h(, x)) + b(, x) h(, x). 22

23 Concerning he noaion, if h i a funcion of and x we denoe h or h he parial differenial in ime, h he gradien in pace ( column vecor) and Dh = ( h), D 2 h he marix of econd-order differenial. We uppoe alo ha he proce Y i real, i.e., k = 1. Z i a line vecor of dimenion d (ha of BM). To um up, f :, T R n R R 1 d R. The aim of hi par i o eablih he relaion beween {Yr,x, Zr,x } r T oluion of he BSDE Y,x r = g(x,x T ) + r f(u, X,x u, Y,x, Z,x )du u u r Z,x u dw u, r T, (12) where {Xr,x } r T i a oluion of he SDE - wih he convenion Xr,x = x if r - X,x r = x + r r b(u, Xu,x )du + σ(u, Xu,x )dw u, r T, (13) on he one hand and << he oluion >> v of he following parabolic PDE: h+lv(, x)+f(, x, v(, x), Dv(, x)σ(, x)) =, (, x), T R n, v(t, ) = g. (14) Propoiion 32 Suppoe ha he PDE (14) ha a oluion v, of cla C 1,2 (, T R n ), uch ha for any (, x), T R n, v(, x) C(1 + x q ). Then, for any (, x), T R n, he oluion of he BSDE (13), {Yr,x, Zr,x } r T, i given by he couple of procee {v(r, Xr,x ), Dvσ(r, Xr,x )} r T. In paricular, we obain he formula, u(, x) = Y,x = v(, x). The previou reul give a probabiliic repreenaion formula for he oluion of a nonlinear parabolic PDE - we ay emi-linear becaue he nonlineariy i no o rong -. Thi ype of formula, known under he name of Feynman-Kac - following he work of Richard Feynman and Mark Kac - applie from he origin o he linear problem. We obain a a corollary: Corollary 33 Take f(, x, y, z) = c(, x)y + h(, x), where c and h are wo bounded coninuou funcion. Under he previou hypohee, for any (, x), T R n, ( ) ( r ) v(, x) = E g(x,x T ) exp c(r, Xr,x )dr + h(r, Xr,x ) exp c(, X,x )d dr. 23

24 Proof: According o he previou propoiion, we know ha v(, x) = Y,x. Bu when he funcion f ake he form f(, x, y, z) = c(, x)y + h(, x), The BSDE which we hould olve i a linear BSDE. We ue hen he explici formula o conclude. Remark 34 In he la wo reul, we have uppoed he exience of a claical oluion v and we have deduced he oluion of BSDE and he formula v(, x) = Y,x. If he coefficien are regular, we can prove ha u which i defined by u(, x) = Y,x i a regular funcion which i a oluion of he PDE. (14). The previou mehod coni o udy he PDE, hen o deduce he oluion of he BSDE. Bu we can alo udy he BSDE and deduce he conrucion of he oluion of he PDE wihou auming he regular coefficien. For hi we apply he noion of he oluion of vicoiy of PDE. Le u recall he definiion of vicoiy oluion. Definiion 35 A coninuou funcion u, wih u(t, ) = g, i a vicoiy uboluion (uperoluion) if, whenever u φ ha a local maximum (minimum) a (, x) where φ i C 1,2, φ(, x) + Lφ(, x) + f(, x, u(, x), φσ(, x)), ( ) A oluion i boh a ub and a uperoluion. Theorem 36 The funcion u(, x) := Y,x PDE (14). i a oluion of vicoiy of he Proof: By conrucion u i coninuou and u(t, ) = g. Le u how ha u i a uboluion. Le (, x), T ) R n be a local maximum of u φ. Wihou lo of generaliy, we aume ha φ(, x) = u(, x). We have o prove ha φ(, x) + Lφ(, x) + f(, x, u(, x), φσ(, x)). If no, here exi δ > and < α T uch ha u(, y) φ(, y), φ(, y) + Lφ(, y) + f(, y, u(, y), φσ(, y)) δ a oon a + α and x y α. Conider he opping ime τ = inf{ : X,x 24 x α} ( + α).

25 We conider wo couple of procee. On he one hand, (Y, Z ) := (φ( τ, X τ,x ), 1 τ φσ(, X,x )) olve Y = φ(τ, X,x τ ) + +α 1 r τ { φ + Lφ}(r, Xr,x )dr On he oher hand, (Y τ,x, 1 τ Z,x ) olve he BSDE Y = Y +α + +α By he Markov propery, Y,x Y = u(τ, X,x τ ) + +α 1 r τ f(r, Xr,x, Y r, Z r )dr 1 r τ f(r, X,x r By he definiion of τ, u(τ, X,x τ f(, X,x = u(, X,x ) +α, u(r, X,x ), Z r )dr ) φ(τ, X,x ) and τ r +α Z dw r. +α Z dw r. Z dw r., u(, X,x ), x φσ(, X,x ) + { φ + Lφ}(, X,x ) δ. By he ric verion of comparion heorem, u(, x) = Y < Y = φ(, x) which i a conradicion. 4 Par 3: Quadraic BSDE and Uiliy Maximizaion 4.1 Problem of uiliy maximizaion The financial marke coni of one bond wih inere rae zero and d m ock. In cae d < m we face an incomplee marke. The price proce of ock i evolve according o he equaion ds i S i = b i d + σ i db, i = 1,..., d, (15) where b i (rep. σ i ) i a R valued (rep. R 1 m valued) ochaic proce. Denoe σ = ((σ 1 ),, (σ d ) ) a he volailiy marix whoe rank i d ( i.e. σσ r i inverible P-a.. ) The predicable R m valued proce ( called he rik premium ) i defined by: θ = σ r (σ σ r ) 1 b,, T. 25

26 A d dimenional F predicable proce π = (π ) T i called rading raegy if π ds S i well defined, e.g. π σ 2 d < P a.. For 1 i d, he proce π i decribe he amoun of money inveed in ock i a ime. The number of hare i πi. S i The wealh proce X π of a rading raegy π wih iniial capial x aifie he equaion X π = x + d i=1 π i,u S i,u ds i,u = x + π u σ u (db u + θ u du). Suppoe our inveor ha a liabiliy F a ime T. Le u recall ha for α > he exponenial uiliy funcion i defined a U(x) = exp( αx), x R. We allow conrain on he rading raegie. Formally, hey are uppoed o ake value in a cloed e, i.e. π (ω) C, wih C R 1 d, and C. Definiion 37 Admiible raegie wih conrain Le C be a cloed e in R 1 d and C. The e of admiible rading raegie A D coni of all d dimenional predicable procee π = (π ) T which aify π σ 2 d < and π C P-a.., a well a {exp( αx π τ ) : τ opping ime wih value in, T } i a uniformly inegrable family. The inveor wan o olve he maximizaion problem ( ( )) ds V (x) := up E exp α x + π F, (16) π A D S where x i he iniial wealh. V i called value funcion. Thi problem ha been udied by many auhor, bu hey uppoe ha he conrain i convex in order o apply convex dualiy. 4.2 Maringale mehod In order o find he value funcion and an opimal raegy one conruc a family of ochaic procee R (π) wih he following properie: 26

27 R (π) T = exp( α(x π T F )) for all π A D; R (π) = R i conan for all π A D ; R (π) i a upermaringale for all π A D and here exi a π A D uch ha R (π ) i a maringale. The proce R (π) and i iniial value R depend of coure on he iniial capial x. Given procee poeing hee properie we can compare he expeced uiliie of he raegie π A D and π A D by E exp( α(x π T F )) R (x) = E exp( α(x π T F )) = V (x), (17) hence π i he deired opimal raegy. Conrucion of R (π) : R (π) := exp( α(x (π) Y )),, T, π A D, where (Y, Z) i a oluion of he BSDE Y = F Z db + f(, Z )d,, T. In hee erm one i bound o chooe a funcion f for which R (π) i a upermaringale for all π A D and here exi a π A D uch ha R (π ) i a maringale. Thi funcion f alo depend on he conrain e (C) where (π ) ake i value. One ge hen V (x) = R (π,x) = exp( α(x Y )), for all π A D. In order o aify he upermaringale and he maringale properie, one find f(, z) = α 2 min π C πσ (z + 1 α θ ) 2 zθ 1 2α θ 2. The funcion f i well defined becaue i only depend on he diance beween a poin and a cloed e. Imporan: he generaor f i of quadraic growh wih repec o z! 27

28 Lemma 38 Suppoe ha boh he rik premium θ and he liabiliy F are bounded. Then, he value funcion of he opimizaion problem (16) i given by V (x) = exp( α(x Y )), where Y i defined by he unique oluion (Y, Z) of he BSDE Y = F Z db + f(, Z )d,, T, (18) wih f(, z) = α 2 min πσ (z + 1 π C α θ) 2 zθ 1 2α θ 2. There exi an opimal rading raegy π A D wih π argmin{ πσ (Z + 1 α θ ), π C},, T, P a.. (19) 4.3 The imple example Conider Y = ξ Z 2 d Z dw (2) Propoiion 39 (2) admi a oluion if and only if Ee ξ <. In hi cae, he oluion i given by Y = ln Ee ξ F. Propoiion 4 If Ee ξ <, hen (2) admi a unique oluion uch ha e Y i in cla (D). 4.4 The bounded cae We conider he following real valued BSDE Y = ξ + f(, Y, Z )d Z dw, T, (21) where he generaor f :, T R R d R i coninuou in (y, z). We call he equaion (21) a a Quadraic BSDE if f(, y, z) α + β y z 2, where α, β, γ are nonnegaive real number. Theorem 41 M. Kobylanki, 2 If ξ i bounded, hen he BSDE (21) ha a bounded oluion. 28

29 4.5 Unbounded Quadraic BSDE Boundedne of ξ i no neceary o conruc a oluion. In fac, ome finie exponenial momen of ξ i enough! Theorem 42 Exience of Soluion, Briand & Hu, 26 Le u aume ha E exp ( γe βt ξ ) < +. Then, (21) ha a lea a oluion uch ha Y αt e βt + 1 ( ) ) (exp γ log E γe βt ξ F. (22) Theorem 43 (Briand and Hu, 28) Le u uppoe moreover ha f i Lipchiz w.r.. y and convex in z. Then for any F T -meaurable ξ wih Eexp(λ ξ ) < for any λ >, (21) admi a unique oluion in he cla E = {Y. : Ee λ up Y < + λ > }. Theorem 44 (Delbaen, Hu and Richou, 211) The uniquene hold for (Y, Z) uch ha here exi p > γ, Ee py <. Theorem 45 (Delbaen, Hu and richou, 213) Aume moreover ha f i ricly convex and independen of y. Then uniquene hold wihin he cla (Y, Z) uch ha e γy i in cla (D). 4.6 Superquadraic BSDE Le u conider he following BSDE: Y = ξ g(z )d + Z db, (23) where g i convex wih g() =, and i uperquadraic, i.e. lim up z g(z) z 2 = ; and ξ i a bounded F T -meaurable random variable. The goal here i o look for a oluion (Y, Z) uch ha Y i a bounded proce. Differen from BSDE wih quadraic growh, he bounded oluion o he BSDE wih uperquadraic growh doe no alway exi. 29

30 Theorem 46 (Non-exience) There exi η L (F T ) uch ha BSDE (23) wih up-quadraic growh ha no bounded oluion. Even if he BSDE ha a bounded oluion, he oluion are no unique. The main reaon i ha he generaor g i uperquadraic which make g(z r)dr grow much faer ha Z rdb r wih repec o Z. Following hi obervaion, we can conruc oher oluion. Theorem 47 (Non-uniquene) If he BSDE (g, ξ) wih uperquadraic growh ha a bounded oluion Y for ome ξ L (F T ), hen for each y < Y, here are infiniely many bounded oluion X wih X = y. The BSDE wih uperquadraic growh i ill-poed. However, in he paricular Markovian cae, oluion o BSDE exi. Define he diffuion proce X,x be he oluion o he SDE: X = x + b(r, X r )dr + σdb r, T, (24) where b i Lipchiz wih repec o x, and σ i a conan (marix). Le u conider he BSDE (23) wih ξ = Φ(X,x T ): Y = Φ(X,x T ) g(z r )dr + Z r db r. Theorem 48 Le u uppoe ha Φ i bounded and coninuou. Then here exi a oluion (Y, Z) o Markovian BSDE. Reference 1 Briand, P. and Hu, Y., BSDE wih quadraic growh and unbounded erminal value. Probab. Theory Relaed Field 136 (26), no. 4, 64C El Karoui, N., Peng, S., and Quenez, M. C., Backward ochaic differenial equaion in finance. Mah. Finance 7 (1997), no. 1, 1C71. 3 Kobylanki, M., Backward ochaic differenial equaion and parial differenial equaion wih quadraic growh. Ann. Probab. 28 (2), no. 2, 558C62. 4 Pardoux, E. and Peng, S. G., Adaped oluion of a backward ochaic differenial equaion. Syem Conrol Le. 14 (199), no. 1, 55C61. 3