Quadratic and Superquadratic BSDEs and Related PDEs

Transcription

1 Quadraic and Superquadraic BSDEs and Relaed PDEs Ying Hu IRMAR, Universié Rennes 1, FRANCE hp://perso.univ-rennes1.fr/ying.hu/ ITN Marie Curie Workshop "Sochasic Conrol and Finance" Roscoff, March 21 Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 1/43

2 I. Uiliy maximizaion The financial marke consiss of one bond wih ineres rae zero and d m socks. In case d < m we face an incomplee marke. The price process of sock i evolves according o he equaion ds i S i = b i d + σ i db, i = 1,..., d, (1) where b i (resp. σ i ) is a R valued (resp. R 1 m valued) sochasic process. The lines of he d m marix σ are given by he vecor σ i, i = 1,..., d. The volailiy marix σ = (σ i ) i=1,...,d has full rank ( i.e. σσ r is inverible P-a.s. ) The predicable R m valued process ( called he risk premium ) is defined by: θ = σ r (σ σ r ) 1 b, [, T]. A d dimensional F predicable process π = (π ) T is called rading sraegy if π ds S is well defined, e.g. π σ 2 d < P a.s. For 1 i d, he process π i describes he amoun of money invesed in sock i a ime. The number of shares is πi. S i Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 2/43

3 The wealh process X π of a rading sraegy π wih iniial capial x saisfies he equaion X π = x + d i=1 π i,u S i,u ds i,u = x + Suppose our invesor has a liabiliy F a ime T. π u σ u (db u + θ u du). Le us recall ha for α > he exponenial uiliy funcion is defined as U (x) = exp( αx), x R. We allow consrains on he rading sraegies. Formally, hey are supposed o ake heir values in a closed se, i.e. π (ω) C, wih C R 1 d, and C. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 3/43

4 Definiion [Admissible Sraegies wih consrains ] Le C be a closed se in R 1 d and C. The se of admissible rading sraegies A D consiss of all d dimensional predicable processes π = (π ) T which saisfy π σ 2 d < and π C P-a.s., as well as {exp( αx π τ ) : τ sopping ime wih values in [, T]} is a uniformly inegrable family. So he invesor wans o solve he maximizaion problem [ ( ( V (x) := sup π A D E exp α x + π ds S where x is he iniial wealh. V is called value funcion. F ))], (2) Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 4/43

5 This problem has been sudied by many auhors, bu hey suppose ha he consrain is convex in order o apply convex dualiy. Our saring poin of he work is he paper [Hu-Imkeller-Muller, AAP 25] where boh he risk premium θ and he liabiliy F are bounded. The main mehod can be described as follows. In order o find he value funcion and an opimal sraegy one consrucs a family of sochasic processes R (π) wih he following properies: R (π) T = exp( α(x π T F)) for all π A D; R (π) = R is consan for all π A D ; R (π) is a supermaringale for all π A D and here exiss a π A D such ha R (π ) is a maringale. The process R (π) and is iniial value R depend of course on he iniial capial x. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 5/43

6 Given processes possessing hese properies we can compare he expeced uiliies of he sraegies 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), (3) whence π is he desired opimal sraegy. Consrucion of R (π) : R (π) := exp( α(x (π) Y )), [, T], π A D, where (Y, Z) is a soluion of he BSDE Y = F Z s db s + f (s, Z s )ds, [, T]. In hese erms one is bound o choose a funcion f for which R (π) is a supermaringale for all π A D and here exiss a π A D such ha R (π ) is a maringale. This funcion f also depends on he consrain se (C) where (π ) akes is values. One ges hen V (x) = R (π,x) = exp( α(x Y )), for all π A D. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 6/43

7 In order o saisfy he supermaringale and he maringale properies, one finds f (, z) = α 2 min πσ (z + 1 π C α θ ) 2 zθ 1 2α θ 2. The funcion f is well defined because i only depends on he disance beween a poin and a closed se. Imporan: he generaor f is of quadraic growh wih respec o z! Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 7/43

8 Lemma Suppose ha boh he risk premium θ and he liabiliy F are bounded. Then, he value funcion of he opimizaion problem (2) is given by V (x) = exp( α(x Y )), where Y is defined by he unique soluion (Y, Z) of he BSDE wih Y = F Z s db s + f (s, Z s )ds, [, T], (4) f (, z) = α 2 min π C πσ (z + 1 α θ) 2 zθ 1 2α θ 2. There exiss an opimal rading sraegy π A D wih π argmin{ πσ (Z + 1 α θ ), π C}, [, T], P a.s. (5) Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 8/43

9 II. Dynamic g Risk Measures (Barrieu-El Karoui, arxiv 27) Definiion Assume ha (ξ T, g) saisfies: (1) g is a P B(R) B(R d )-measurable generaor saisfying z g(, z) is convex, and g(, z) g(, ) + k 2 z 2, g(, ) 1 2 M ; (2) ξ T is an F T -measurable bounded random variable. Define R g (ξ T ) as he unique soluion of he BSDE ( ξ T, g). Proposiion (Barrieu-El Karoui) R g is a dynamic risk measure. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 9/43

10 Inf-convoluion In he case of bounded ξ, B-E esablished Proposiion R ga g B (ξ T ) = R ga R gb (ξ T ). Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 1/43

11 Backward Sochasic Differenial Equaion Y = ξ + f (s, Y s, Z s ) ds Z s db s (E ξ,f ) ξ is he erminal value : F T measurable f is he generaor (Y, Z) is he unknown (Y, Z) has o be adaped o F Pardoux Peng, 9 If f is Lipschiz w.r.. (y, z) and [ E ξ 2 + ] f (s,, ) 2 ds < + (E ξ,f ) has a unique square inegrable soluion. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 11/43

12 Nonlinear Feynman-Kac Formula Semilinear PDE (P) u(, x) + Lu(, x) + f (, x, u(, x), ( x u) r σ(, x)) =, u(t,.) = g, Lu(, x) = 1 2 race(σσ 2 xu(, x)) + b(, x) x u(, x). Linear par = SDE X,x = x + b ( ) s, Xs,x ds + σ ( ) s, Xs,x dbs Nonlinear par = BSDE (B) Y,x ( = g X,x T ) + f ( s, X,x s, Y,x s ) T, Zs,x ds Zs,x db s Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 12/43

13 Nonlinear Feynman-Kac Formula If u is smooh soluion o (P) ( ( u, X,x ) (, ( x u) r σ, X,x )) solves he BSDE (B) Feynman-Kac s Formula u(, x) := Y,x is a (viscosiy) soluion o (P). Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 13/43

14 Quadraic BSDEs A real valued BSDE Y = ξ + f (s, Y s, Z s ) ds Z s db s (E ξ,f ) B is a Brownian moion in R d ; ξ is F T measurable; he generaor f : [, T] Ω R R d R is measurable and (y, z) f (, y, z) is coninuous f is quadraic wih respec o z: f (, y, z) α() + β y + γ 2 z 2 where β, γ > and α is a nonnegaive process. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 14/43

15 The bounded case If ξ and α or more generally α 1 := Exisence Uniqueness, Comparison Theorem Sabiliy α(s) ds are bounded References: M. Kobylanski (AP 2); M.-A. Morlais (non Brownian seing, Ph. D 27) These resuls yield The Nonlinear Feynman-Kac Formula Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 15/43

16 Applicaions of bounded case Uiliy maximizaion: El Karoui & Rouge (MF 2), Hu, Imkeller & Muller (AAP 25) (wih closed consrain), Mania & Schweizer (AAP 25), Becherer (AAP 26), Morlais (Ph D 27) Sochasic linear quadraic conrol: Bismu ( ), Peng, Kohlman & Tang, Hu & Zhou (wih cone consrain SICON 25), Schweizer e al. Quadraic g risk measure: Barrieu & El Karoui Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 16/43

17 The unbounded case Boundedness of ξ and α is no necessary o consruc a soluion; Exponenial momen is enough! Theorem Exisence of Soluion [Briand & Hu, PTRF 26] Le ζ := ξ + us assume ha E [ exp ( γe βt ζ )] < +. Then, (E ξ,f ) has a leas a soluion such ha α(s) ds and le Y 1 ( ) ) (exp γ log E γe βt ζ F. (6) Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 17/43

18 Consrucion : f, ξ (Y n, Z n ) minimal soluion Y n = ξ n + σ n = inf 1 s σn f (s, Ys n, Zs n ) ds { : } α(s) ds n Z n s db s Sep 1: a priori esimae Y n 1 γ E (exp [γe βt ( ξ + α(s) ds)] F ) Sep 2: aking he limi in n: Difficul sep: Localizaion procedure Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 18/43

19 The localizaion procedure Main Idea: Work on he inerval [, τ k ] where ( ( { : [γe 1γ E βt τ k = inf exp ξ + ) } α(s) ds)] F k T Se Y n k () = Y n τ k, Z n k () = 1 τ k Z n τk τk Yk n () = Yτ n k + 1 s σn f (s, Yk n (s), Zk n (s)) ds τ k Zk n () db s τ k For fixed k, (Y n k ) n N is nondecreasing, Y n k () k Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 19/43

20 The localizaion procedure k fixed, lim n + τk τk Y k () = ξ k + f (s, Y k (s), Z k (s)) ds τ k Z k (s) db s, τ k ξ k = sup Yτ n k n 1 By consrucion Define (Y, Z) by Y k () = Y k+1 ( τ k ), Z k () = 1 τk Z k+1 () Y = Y k (), Z = Z k () if τ k k + gives a soluion τk τk Y = ξ k + f (s, Y s, Z s ) ds τ k Z s db s τ k Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 2/43

21 Remarks Quesions Uniqueness? Sabiliy? Feynman-Kac formula? Answer When f is convex (or concave) w.r.. z Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 21/43

22 Moivaion: Sochasic Conrol Problem (Fuhrman, Hu & Tessiore SICON 26) Conrolled diffusion process X = x + b(s, X s ) ds + where u akes is values in a nonempy closed se C. σ(s, X s )[dw s + r(u s ) ds] Minimize he cos funcional [ J(u) = E g(x T ) + over all he admissible conrols u. ] G(, X, u ) d Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 22/43

23 Moivaion Associaed BSDE Y = g(x T ) + f (s, X s, Z s ) ds Z s db s X = x + b(s, X s ) ds + σ(s, X s ) db s f (, x, z) = inf {G(, x, u) + r(u)z : u C} Imporan feaure of he generaor z f (, x, z) is concave Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 23/43

24 Assumpions (H) There exis β, γ and a nonnegaive process α s.. P a.s. f is Lipschiz w.r.. y: for any, z, f (, y, z) f (, y, z) β y y ; quadraic growh in z: f (, y, z) α() + β y + γ 2 z 2 ; for any, y, z f (, y, z) is a convex funcion; ξ is F T measurable and [ ( [ ])] λ >, E exp λ ξ + α(s) ds < +. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 24/43

25 Some esimaes Proposiion (E ξ,f ) has a soluion (Y, Z) s.. [ ( p 1, E sup [,T] e p Y + ] ) p/2 Z s 2 ds C where C depends only on p, T and he exponenial momens of ξ + α 1. The esimae for Y comes direcly from Y 1 γ log E ( exp ( γe βt ( ξ + α 1 ) ) F ) For Z, sandard compuaion saring from Iô s formula o 1 ( ) γ 2 e γ Y 1 γ Y Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 25/43

26 Comparison heorem Theorem Uniqueness of Soluion [Briand & Hu, PTRF 28] Le (Y, Z) and (Y, Z ) be soluion o (E ξ,f ) and (E ξ,f ) where (ξ, f ) saisfies (H) and Y, Y belongs o E (E := exponenial momen of all order). If ξ ξ and f f hen [, T], Y Y If moreover, Y = Y hen ( P ξ = ξ, f (s, Y s, Z s) ds = In paricular, (E ξ,f ) has a unique soluion in he class E. ) f (s, Y s, Z s) ds >. Main idea: Esimae of Y µy for µ (, 1). Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 26/43

27 Proof: f independen of y Se, for µ (, 1), U = Y µy, V = Z µz. U = U T + F s ds V s db s, F s = f (s, Z s ) µf (s, Z s) F = [f (, Z ) µf (, Z )] + µ [f (, Z ) f (, Z )] and δf () := f (, Z ) f (, Z ). Z = µz + (1 µ) Z µz 1 µ ( f (, Z ) = f, µz + (1 µ) Z µz ) 1 µ ( Convexiy µf (, Z ) + (1 µ)f, Z µz ) 1 µ Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 27/43

28 ( f (, Z ) µf (, Z ) (1 µ)f, F µδf () + (1 µ)α() + ) V (1 µ)α() + 1 µ γ 2(1 µ) V 2 γ 2(1 µ) V 2 (7) Second sep An exponenial change of variable o remove he quadraic erm P = e cu, Q = cp V, c c = γ 1 µ yields P = P T + c P s (F s c 2 V s 2) ds Q s db s Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 28/43

29 P E P E ( ( exp [ γ ( ) γ P T = exp 1 µ (ξ µξ ) exp [ γ ( ξ + ( α(s) + (1 µ) 1 µδf (s) ) ds ] P T F ) ( ( = exp γ ξ + µ )) 1 µ δξ ) ( µ α(s) ds + γ δξ + 1 µ In paricular, ( [ ( Y µy 1 µ log E exp γ ξ + γ α(s) ds δf (s) ds )] F ) )] F ) and sending µ o 1, we ge Y Y. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 29/43

30 Sric Comparison If Y = Y, hen P = e γy < E[P ] E Sending µ o 1, [ exp < E [ ( γ exp ( ( and ξ + γ [ ξ + ) ( µ α(s) ds + γ δξ + 1 µ α(s) ds ]) 1 δξ+ T δf (s) ds= ] δf (s) ds )]) Press if lae Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 3/43

31 The general case F = f (, Y, Z ) µf (, Y, Z ) = f (, Y, Z ) µf (, Y, Z ) + µδf () Convexiy f (, Y, Z ) µf (, Y, Z ) = f (, Y, Z ) µf (, Y, Z ) + µ (f (, Y, Z ) f (, Y, Z )). f (, Y, Z ) µf (, Y, Z ) (1 µ)(α() + β Y ) + γ 2(1 µ) V 2 Linearizaion: a() = (Y Y ) 1 (f (, Y, Z ) f (, Y, Z )) 1 Y Y µ (f (, Y, Z ) f (, Y, Z )) = µa() (Y Y ) a()u + (1 µ)β Y F µδf () + (1 µ)(α() + 2β Y ) + γ 2(1 µ) V 2 + a()u. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 31/43

32 The general case ( ) Se E = exp a(s) ds, Ũ = E U and Ṽ = E V. Then, Ũ = ŨT + F s ds Ṽ s db s wih, since a() β, F µe δf () + (1 µ)e (α() + 2β Y ) + γeβt Ṽ 2(1 µ) 2 This is he same inequaliy as before. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 32/43

33 Sabiliy Assume ha (ξ n, f n ) saisfies (H) wih α n, β, γ and λ >, sup n 1 E [exp {λ ( ξ n + α n 1 )}] < +. Theorem If ξ n ξ P p.s. and d dp a.e., (y, z), f n (, y, z) f (, y, z), hen [ ( ] T ) p/2 p 1, E sup [,T] Y Y n p + Z s Zs n 2 ds. Proof. Same mehod as in he proof of comparison heorem o Y µy n, Y n µy Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 33/43

34 Applicaion o PDEs Probabilisic represenaion for u(, x) + Lu(, x) + f (, x, u(, x), ( x u) r σ(, x)) =, u(t,.) = g, Lu(, x) = 1 2 race(σσ 2 xu(, x)) + b(, x) x u(, x). The SDE: X,x soluion o X = x + b(s, X s ) ds + The BSDE: (Y,x, Z,x ) soluion o ( Y = g X,x T ) + σ(s, X s ) db s f ( ) T s, Xs,x, Y s, Z s ds Z s db s Nonlinear Feynman-Kac formula: u(, x) := Y,x is a viscosiy soluion Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 34/43

35 Assumpions b, σ, f and g are coninuous; b, σ Lipschiz w.r.. x b(, x) b(, x ) + σ(, x) σ(, x ) β x x ; resricion: σ is bounded; f is Lipschiz w.r.. y f (, x, y, z) f (, x, y, z) β y y ; z f (, x, y, z) is convex; p < 2 s.. g(x) + f (, x, y, z) C ( 1 + x p + y + z 2). Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 35/43

36 Firs properies Proposiion u(, x) := Y,x is coninuous and u(, x) C (1 + x p ). Proof. Since σ is bounded, λ >, [ ( E exp λ sup [,T] X,x p)] e C(1+ x p ) Sabiliy = Coninuiy a priori esimae u(, x) C (1 + x p ) Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 36/43

37 u is a viscosiy soluion Definiion A coninuous funcion u s.. u(t, ) = g is a viscosiy subsoluion (supersoluion) if, whenever u ϕ has a local maximum (minimum) a (, x ) where ϕ is C 1,2, ϕ(, x ) + Lϕ(, x ) + f (, x, u(, x ), ( x u) r σ(, x )), ( ) Soluion = Subsoluion + Supersoluion Proposiion u(, x) := Y,x is a viscosiy soluion o he PDE. Proof. Markov propery : u(, X,x ) = Y,x, and Comparison heorem Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 37/43

38 Exension and Open Quesions Weaken he inegrabiliy assumpions Delbaen, Hu and Richou (Arxiv 29): Uniqueness holds among soluions which admi some given exponenial momens. These exponenial momens are naural as hey are given by he exisence heorem. Open Quesion 1: Prove uniqueness and sabiliy wihou convexiy f (, y, z) f (, y, z ) C z z (1 + z + z ) Open Quesion 2: Muli-dimensional quadraic BSDEs and sysem of quadraic PDEs. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 38/43

39 Superquadraic BSDEs (join work wih Delbaen and Bao) (Arxiv 29, o appear in PTRF). Le us consider he following BSDE: Y = ξ g(z s )ds + Z s db s, (8) where g is convex wih g() =, and is superquadraic, i.e. lim sup z g(z) z 2 = ; and ξ is a bounded F T -measurable random variable. The goal here is o look for a soluion (Y, Z) such ha Y is a bounded process. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 39/43

40 Non-exisence of soluion Differen from BSDEs wih quadraic growh, he bounded soluion o he BSDE wih superquadraic growh does no always exis. Theorem (Non-exisence) There exiss η L (F T ) such ha BSDE (8) wih sup-quadraic growh has no bounded soluion. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 4/43

41 Non-uniqueness of soluion Even if he BSDE has a bounded soluion, he soluions are no unique. The main reason is ha he generaor g is superquadraic which makes g(z r)dr grow much faser ha Z rdb r wih respec o Z. Following his observaion, we can consruc oher soluions. Theorem (Non-uniqueness) If he BSDE (g, ξ) wih superquadraic growh has a bounded soluion Y for some ξ L (F T ), hen for each y < Y, here are infiniely many bounded soluions X wih X = y. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 41/43

42 Exisence of soluion o BSDE in Markovian case The BSDE wih superquadraic growh is ill-posed. However, in he paricular Markovian case, soluions o BSDE exis. Define he diffusion process X,x be he soluion o he SDE: X s = x + s b(r, X r )dr + s σdb r, s T, (9) where b is Lipschiz wih respec o x, and σ is a consan (marix). Le us consider he BSDE (8) wih ξ = Φ(X,x T ): Y s = Φ(X,x T ) g(z r )dr + Z r db r. s s Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 42/43

43 Exisence of Soluion Theorem Le us suppose ha Φ is bounded and coninuous. Then here exiss a soluion (Y, Z) o Markovian BSDE. Main ool: if Φ is smooh, we can ge an esimae in he spiri of Gilding e al. by use of some maringale mehod: Z C Φ (T ) 1 2. Finally, we can prove ha u(, x) = Y,x corresponding PDE. is a viscosiy soluion of he Remark: Cheridio and Sadje (Arxiv 21): No discree convergence for quadraic BSDEs. Richou (Ph D 21): numerical simulaion applying he above esimae. Ying Hu, Univ. Rennes 1 Quadraic and Superquadraic BSDEs Roscoff, March 21 43/43