Quadratic BSDE systems and applications

Transcription

1 Quadratic BSDE systems and applications Hao Xing London School of Economics joint work with Gordan Žitković Stochastic Analysis and Mathematical Finance-A Fruitful Partnership 25 May, / 23

2 Backward stochastic differential equation Given f : [0, T ] Ω R n R n d R n, consider dy t = f (t, Y t, Z t )dt + Z t db t, Y T = ξ. n 1 f Lipschitz in z: [Pardoux-Peng 90] n = 1 f continuous in z: [Lepeltier-San Martin 97] f quadratic in z: [Kobylanski 00], [Briand-Hu 06], [Barrieu-El Karoui 13] f superquadratic in z: [Delbaen-Hu-Bao 11] 2 / 23

3 Backward stochastic differential equation Given f : [0, T ] Ω R n R n d R n, consider dy t = f (t, Y t, Z t )dt + Z t db t, Y T = ξ. n 1 f Lipschitz in z: [Pardoux-Peng 90] n = 1 f continuous in z: [Lepeltier-San Martin 97] f quadratic in z: [Kobylanski 00], [Briand-Hu 06], [Barrieu-El Karoui 13] f superquadratic in z: [Delbaen-Hu-Bao 11] Our goal: n 1, f quadratic growth in z 2 / 23

4 Backward stochastic differential equation Given f : [0, T ] Ω R n R n d R n, consider dy t = f (t, Y t, Z t )dt + Z t db t, Y T = ξ. n 1 f Lipschitz in z: [Pardoux-Peng 90] n = 1 f continuous in z: [Lepeltier-San Martin 97] f quadratic in z: [Kobylanski 00], [Briand-Hu 06], [Barrieu-El Karoui 13] f superquadratic in z: [Delbaen-Hu-Bao 11] Our goal: n 1, f quadratic growth in z = quadratic BSDE system. 2 / 23

5 Motivation 1: stochastic equilibria in incomplete markets Agent: for i = 1,..., n, 1. utility: U i (x) = e x/δ i, δ i > 0, 2. random endowment: E i L (F T ). 3 / 23

6 Motivation 1: stochastic equilibria in incomplete markets Agent: for i = 1,..., n, 1. utility: U i (x) = e x/δ i, δ i > 0, 2. random endowment: E i L (F T ). Market: a single risky asset with return db λ t = λ t dt + db t, 3 / 23

7 Motivation 1: stochastic equilibria in incomplete markets Agent: for i = 1,..., d, 1. utility: U i (x) = e x/δ i, δ i > 0, 2. random endowment: E i L (F B,W T ). Market: a single risky asset with return db λ t = λ t dt + db t, W B. 3 / 23

8 Motivation 1: stochastic equilibria in incomplete markets Agent: for i = 1,..., d, 1. utility: U i (x) = e x/δ i, δ i > 0, 2. random endowment: E i L (F B,W T ). Market: a single risky asset with return db λ t = λ t dt + db t, W B. Equilibrium: λ, (π i ) 1 i d, 1. Utility maximization: E [ U i (π i B λ T + E i ) ] Max; 2. Market clearing: d i=1 π i = 0. 3 / 23

9 Quadratic BSDE system { [ T bmo = µ : sup Eτ τ Certainty-equivalent process τ } L µ u du] 2 <. U i (Y i t) = ess sup π E t [U i (π B λ T π B λ t + E i )], t [0, T ]. 4 / 23

10 Quadratic BSDE system { [ T bmo = µ : sup Eτ τ Certainty-equivalent process τ } L µ u du] 2 <. U i (Y i t) = ess sup π E t [U i (π B λ T π B λ t + E i )], t [0, T ]. Theorem For λ bmo, the following are equivalent: 1. λ is an equilibrium; 2. λ = A[µ] = i αi µ i for some solution (Y i, µ i, ν i ) i of the BSDE system ( 1 dyt i = µ i tdb t + νtdw i t + 2 (νi t) 2 1 ) 2 λ2 t + λ t µ i t dt, Y i T = E i /δ i, i {1, 2,..., n}, and (µ i, ν i ) bmo for all i. 4 / 23

11 Results with smallness type conditions [Choi-Larsen 14], [Kardaras-X.-Zitkovic 15]: smallness-type assumption E i L /δ i is small for all i, or T is small 5 / 23

12 Results with smallness type conditions [Choi-Larsen 14], [Kardaras-X.-Zitkovic 15]: smallness-type assumption E i L /δ i is small for all i, or T is small Numeric result for Markovian case 5 / 23

13 Results with smallness type conditions [Choi-Larsen 14], [Kardaras-X.-Zitkovic 15]: smallness-type assumption E i L /δ i is small for all i, or T is small Numeric result for Markovian case Question: global existence, uniqueness??? 5 / 23

14 Motivation 2: Γ-martingale 6 / 23

15 Motivation 2: Γ-martingale In Euclidean space, (Hessf ) ij (y) = D ij f (y). dy = Z t dw t then f (Y ) 1 2 is a local martingale for each smooth f. 0 Hess f (Y u )(Z u, Z u )du (1) 6 / 23

16 Motivation 2: Γ-martingale In Euclidean space, (Hessf ) ij (y) = D ij f (y). dy = Z t dw t then f (Y ) Hess f (Y u )(Z u, Z u )du (1) is a local martingale for each smooth f. On manifold, Hess f is the covariant Hessian of f (Hess f ) ij (y) = D ij f (y) Γ k ij(y)d k f (y) Y is a Γ-martingale if (1) is a local martingale for each smooth f [Émery 89]. 6 / 23

17 Motivation 3: Γ-martingale In local coordinate, dy k t = 1 2 d Γ k i,j(y t )(Zt i ) Zt j dt + Zt k dw t, k = 1,..., n, i,j=1 with terminal condition Y = ξ; [Darling 95], [Blache 05, 06] This is probabilistic analogue of harmonic map. 7 / 23

18 Motivation 3: Γ-martingale In local coordinate, dy k t = 1 2 d Γ k i,j(y t )(Zt i ) Zt j dt + Zt k dw t, k = 1,..., n, i,j=1 with terminal condition Y = ξ; [Darling 95], [Blache 05, 06] This is probabilistic analogue of harmonic map. Question: Does Γ-martingale exist? 7 / 23

19 System of quadratic BSDEs Open problem: [Peng 99] [Tang 03]: Riccati system [Tevzadze 08]: existence when terminal condition is small [Frei-dos Reis 11]: counter example [Cheridito-Nam 14]: generator f + z g, f and g are Lipschitz [Hu-Tang 14]: diagonally quadratic [Jamneshan-Kupper-Luo 15]: cases not covered by [Tevzadze 08] Applications: Stochastic game: [Bensoussan-Frehse 00], [El Karoui-Hamadène 03] Relative performance: [Espinosa-Touzi 13], [Frei-dos Reis 11], [Frei 14] Equilibrium pricing: [Cheridito-Horst-Kupper-Pirvu 12] Market making: [Kramkov-Pulido 14] 8 / 23

20 [Kobylanski 00] revisited 1. Approximation: (Y m, Z m ) 2. Monotone convergence: Y m Y m+1 3. Verification: Y m strong convergence = Z m strong convergence 9 / 23

21 [Kobylanski 00] revisited 1. Approximation: (Y m, Z m ) 2. Monotone convergence: Y m Y m+1 3. Verification: Y m strong convergence = Z m strong convergence Comparison theorem fails for systems [Hu-Peng 06] 9 / 23

22 [Kobylanski 00] revisited 1. Approximation: (Y m, Z m ) 2. Monotone convergence: Y m Y m+1 3. Verification: Y m strong convergence = Z m strong convergence Comparison theorem fails for systems [Hu-Peng 06] Compact set in S??? 9 / 23

23 [Kobylanski 00] revisited 1. Approximation: (Y m, Z m ) 2. Monotone convergence: Y m Y m+1 3. Verification: Y m strong convergence = Z m strong convergence Comparison theorem fails for systems [Hu-Peng 06] Compact set in S??? Markovian case: X satisfies dx t = b(t, X t )dt + σ(t, X t )dw t Y m t = v m (t, X t ) 9 / 23

24 [Kobylanski 00] revisited 1. Approximation: (Y m, Z m ) 2. Monotone convergence: Y m Y m+1 3. Verification: Y m strong convergence = Z m strong convergence Comparison theorem fails for systems [Hu-Peng 06] Compact set in S??? Markovian case: X satisfies dx t = b(t, X t )dt + σ(t, X t )dw t Y m t = v m (t, X t ) (v m ) has a convergent subsequence, if (v m ) is uniformly bounded; (v m ) is equi-continuous. 9 / 23

25 [Kobylanski 00] revisited 1. Approximation: (Y m, Z m ) 2. Monotone convergence: Y m Y m+1 3. Verification: Y m strong convergence = Z m strong convergence Comparison theorem fails for systems [Hu-Peng 06] Compact set in S??? Markovian case: X satisfies dx t = b(t, X t )dt + σ(t, X t )dw t Y m t = v m (t, X t ) (v m ) has a convergent subsequence, if (v m ) is uniformly bounded; (v m ) is equi-continuous. Partial regularity: Uniform Hölder estimate for (v m ); [Struwe 81], [Bensoussan-Frehse 02] 9 / 23

26 Lyapunov pair A pair (h, k) of nonnegative functions with h : R n R, h(0) = 0, Dh(0) = 0, is a c-lyapunov pair for f if 1 2 D2 h(y) : z, z σ 2 Dh(y)f (t, x, y, z) z 2 k(t, x), for all (t, x, z) [0, T ] R n R n d and y c. 10 / 23

27 Lyapunov pair A pair (h, k) of nonnegative functions with h : R n R, h(0) = 0, Dh(0) = 0, is a c-lyapunov pair for f if 1 2 D2 h(y) : z, z σ 2 Dh(y)f (t, x, y, z) z 2 k(t, x), for all (t, x, z) [0, T ] R n R n d and y c. h(y ) is essentially a submartingale 10 / 23

28 Lyapunov pair A pair (h, k) of nonnegative functions with h : R n R, h(0) = 0, Dh(0) = 0, is a c-lyapunov pair for f if 1 2 D2 h(y) : z, z σ 2 Dh(y)f (t, x, y, z) z 2 k(t, x), for all (t, x, z) [0, T ] R n R n d and y c. h(y ) is essentially a submartingale When n = 1 and f has quadratic growth, h = exp(αy) for sufficiently large α 10 / 23

29 Uniform estimate: abstract form Theorem Let (f m ) and (g m ) be sequences of functions such that dy t = f m (t, X t, Y t, Z t )dt + Z t σ(t, X t )dw t, Y T = g m (X T ), admits a Markovian solution (v m, w m ). 11 / 23

30 Uniform estimate: abstract form Theorem Let (f m ) and (g m ) be sequences of functions such that dy t = f m (t, X t, Y t, Z t )dt + Z t σ(t, X t )dw t, Y T = g m (X T ), admits a Markovian solution (v m, w m ). There exist constants α, M and c such that 1. (g m ) is uniformly bounded in C α ; 11 / 23

31 Uniform estimate: abstract form Theorem Let (f m ) and (g m ) be sequences of functions such that dy t = f m (t, X t, Y t, Z t )dt + Z t σ(t, X t )dw t, Y T = g m (X T ), admits a Markovian solution (v m, w m ). There exist constants α, M and c such that 1. (g m ) is uniformly bounded in C α ; 2. each v m is continuous, v m (t, x) c, for all (t, x) [0, T ] R n ; 11 / 23

32 Uniform estimate: abstract form Theorem Let (f m ) and (g m ) be sequences of functions such that dy t = f m (t, X t, Y t, Z t )dt + Z t σ(t, X t )dw t, Y T = g m (X T ), admits a Markovian solution (v m, w m ). There exist constants α, M and c such that 1. (g m ) is uniformly bounded in C α ; 2. each v m is continuous, v m (t, x) c, for all (t, x) [0, T ] R n ; 3. f m (t, x, y, z) M( z 2 + k(t, x)) for all y c; 11 / 23

33 Uniform estimate: abstract form Theorem Let (f m ) and (g m ) be sequences of functions such that dy t = f m (t, X t, Y t, Z t )dt + Z t σ(t, X t )dw t, Y T = g m (X T ), admits a Markovian solution (v m, w m ). There exist constants α, M and c such that 1. (g m ) is uniformly bounded in C α ; 2. each v m is continuous, v m (t, x) c, for all (t, x) [0, T ] R n ; 3. f m (t, x, y, z) M( z 2 + k(t, x)) for all y c; 4. there exists a 2c-Lyapunov pair (h, k) for (f m ). 11 / 23

34 Uniform estimate: abstract form Theorem Let (f m ) and (g m ) be sequences of functions such that dy t = f m (t, X t, Y t, Z t )dt + Z t σ(t, X t )dw t, Y T = g m (X T ), admits a Markovian solution (v m, w m ). There exist constants α, M and c such that 1. (g m ) is uniformly bounded in C α ; 2. each v m is continuous, v m (t, x) c, for all (t, x) [0, T ] R n ; 3. f m (t, x, y, z) M( z 2 + k(t, x)) for all y c; 4. there exists a 2c-Lyapunov pair (h, k) for (f m ). Then (v m ) is uniformly bounded in C α for some α (0, 1]. 11 / 23

35 Uniform estimate: abstract form Theorem Let (f m ) and (g m ) be sequences of functions such that dy t = f m (t, X t, Y t, Z t )dt + Z t σ(t, X t )dw t, Y T = g m (X T ), admits a Markovian solution (v m, w m ). There exist constants α, M and c such that 1. (g m ) is uniformly bounded in C α ; 2. each v m is continuous, v m (t, x) c, for all (t, x) [0, T ] R n ; 3. f m (t, x, y, z) M( z 2 + k(t, x)) for all y c; 4. there exists a 2c-Lyapunov pair (h, k) for (f m ). Then (v m ) is uniformly bounded in C α for some α (0, 1]. Remark: Any bounded continuous solution v is also Hölder continuous. 11 / 23

36 Uniform estimate: abstract form Theorem Let (f m ) and (g m ) be sequences of functions such that dy t = f m (t, X t, Y t, Z t )dt + Z t σ(t, X t )dw t, Y T = g m (X T ), admits a Markovian solution (v m, w m ). There exist constants α, M and c such that 1. (g m ) is uniformly bounded in C α ; 2. each v m is continuous, v m (t, x) c, for all (t, x) [0, T ] R n ; 3. f m (t, x, y, z) M( z 2 + k(t, x)) for all y c; 4. there exists a 2c-Lyapunov pair (h, k) for (f m ). Then (v m ) is uniformly bounded in C α for some α (0, 1]. Remark: Any bounded continuous solution v is also Hölder continuous. A local version is available, where x is restricted to B N. 11 / 23

37 Existence Theorem Let f and g be Borel functions. Assume that there exist sequence (f m ) and (g m ) satisfying assumptions of the previous theorem and lim m f m (t, x, y m, z m ) = f (t, x, y, z) lim m g m (x) = g(x). 12 / 23

38 Existence Theorem Let f and g be Borel functions. Assume that there exist sequence (f m ) and (g m ) satisfying assumptions of the previous theorem and lim m f m (t, x, y m, z m ) = f (t, x, y, z) lim m g m (x) = g(x). Then the system dy t = f (t, X t, Y t, Z t )dt + Z t σ(t, X t )dw t, Y T = g(x T ) admits a locally Hölder solution (v, w) such that v is locally uniform limit of a subsequence of (v m ) and w is the weak Jacobian of v. 12 / 23

39 Uniqueness and bmo property Theorem Given a constant c, there exists a c-lyapunov pair (h, k) with k L ; 13 / 23

40 Uniqueness and bmo property Theorem Given a constant c, there exists a c-lyapunov pair (h, k) with k L ; g C α and f (t, x, y, z) M(1 + z 2 ) for all y c. 13 / 23

41 Uniqueness and bmo property Theorem Given a constant c, there exists a c-lyapunov pair (h, k) with k L ; g C α and f (t, x, y, z) M(1 + z 2 ) for all y c. Then 1. Any continuous solution (v, w) with v L c has bmo martingale part; 13 / 23

42 Uniqueness and bmo property Theorem Given a constant c, there exists a c-lyapunov pair (h, k) with k L ; g C α and f (t, x, y, z) M(1 + z 2 ) for all y c. Then 1. Any continuous solution (v, w) with v L c has bmo martingale part; 2. When, additionally, f is continuous, does not depend on y, and f (t, x, z) f (t, x, z ) M( z + z ) z z. 13 / 23

43 Uniqueness and bmo property Theorem Given a constant c, there exists a c-lyapunov pair (h, k) with k L ; g C α and f (t, x, y, z) M(1 + z 2 ) for all y c. Then 1. Any continuous solution (v, w) with v L c has bmo martingale part; 2. When, additionally, f is continuous, does not depend on y, and f (t, x, z) f (t, x, z ) M( z + z ) z z. Then the system admits at most one continuous solution (v, w) with v L c. 13 / 23

44 Sufficient condition for Lyapunov pair Bensoussan-Frehse condition: f is continuous and admits a decomposition f i (t, x, z) = z i I i (t, x, z)) + q i (t, x, z) + s i (t, x, z) + k i (t, x), I i (t, x, z) C(1 + z ), (quadratic linear) q i (t, x, z) C ( i 1 + z j 2), (quadratic-triangular) j=1 lim z si (t, x, z) / z 2 = 0, k i L, (subquadratic) (z-independent) 14 / 23

45 Sufficient condition for Lyapunov pair Proposition When f satisfies (BF), then for any c > 0, there exists a Lyapunov pair (h, k) for f. Consider α(u) = e u + e u 2. Define the map H : R n R n via H n (y) = exp(α(γ n y n )), H i (y) = exp(α(γ i y i ) + H i+1 (y)), i = 1,, n 1. h(y) = H 1 (y) for suitable choice of (γ i ). 15 / 23

46 Approximately (BF) (BF) holds approximately if Lyapunov pair also exists. f diag(zi ) q s k ɛ z / 23

47 Approximately (BF) (BF) holds approximately if Lyapunov pair also exists. f diag(zi ) q s k ɛ z 2. Special example: [Struwe 81] I = q = s = k = 0. h = 1 2 y 2 is a Lyapunov pair when 1 ɛ <. 4Λ v L 16 / 23

48 Sufficient condition for L bound Positively spanning set: A set of non-zero vectors a 1,..., a k in R n positive span R n, if for each a R n, λ 1 a λ k a k = a, for λ,..., λ k / 23

49 Sufficient condition for L bound Positively spanning set: A set of non-zero vectors a 1,..., a k in R n positive span R n, if for each a R n, λ 1 a λ k a k = a, for λ,..., λ k 0. f satisfies the condition (wab) if there exists a positively spanning set a 1,..., a k, l L 1 [0, T ], L k (t, x, z) C(1 + z ), k = 1,..., k, such that a k f (t, x, z) l(t) a k z 2 + a k z L k (t, x, z). Comparison theorem can be applied after projecting f to 1-dim subspace. 17 / 23

50 Motivation 1: revisited Consider dx t = b(t, X t )dt + σ(t, X t )dw t, where b, σ bounded, Lipschitz, and σ is elliptic. Theorem Suppose the terminal condition is of the form G = g(x T ) for some g C α L. Then the system admit a unique bounded continuous solution. An incomplete equilibrium exists and is unique in the class of equilibria in which each agent s certainty equivalence is a continuous function. 18 / 23

51 Motivation 2: revisited Theorem Assume that the terminal condition is of the form g(w T ), g takes value in M 0, where M 0 = φ 1 ((, 0]) for some convex function. Moreover, (covariant) Hessian of φ is strictly positive on M 0. Then there exists a Γ-martingale, it takes values in M / 23

52 Motivation 2: revisited Theorem Assume that the terminal condition is of the form g(w T ), g takes value in M 0, where M 0 = φ 1 ((, 0]) for some convex function. Moreover, (covariant) Hessian of φ is strictly positive on M 0. Then there exists a Γ-martingale, it takes values in M 0. Remark: (covariant) Hassian positive definite is called geodesically convex. Existence of φ is ensured by geometric argument [Kendall 90]: manifold with negative sectional curvature, or small ball on manifold with positive curvature. Partial positive answer to Conjecture 7.2 in [Darling 95]: doubly convex geometry is not needed. Smallness assumption is needed for manifold with positive curvature [Chang-Ding-Ye 92] 19 / 23

53 Campanato norm estimate Hole-filling technique by [Struwe 81] Proposition There exist a constant C and α 0 (0, 1) such that sup sup R d 2 2α0 v v 2 C. (t 0,x 0) R 1 20 / 23

54 Campanato norm estimate Hole-filling technique by [Struwe 81] Proposition There exist a constant C and α 0 (0, 1) such that sup sup R d 2 2α0 v v 2 C. (t 0,x 0) R 1 Campanato space: sup (t 0,x 0) sup R d 2 α v v 2 <, R Q δ,r (t 0,x 0) where Q δ,r (t 0, x 0 ) is a parabolic domain and v is the average of v on Q. 20 / 23

55 Campanato norm estimate Hole-filling technique by [Struwe 81] Proposition There exist a constant C and α 0 (0, 1) such that sup sup R d 2 2α0 v v 2 C. (t 0,x 0) R 1 Campanato space: sup (t 0,x 0) sup R d 2 α v v 2 <, R Q δ,r (t 0,x 0) where Q δ,r (t 0, x 0 ) is a parabolic domain and v is the average of v on Q. Campanato Hölder. 20 / 23

56 Campanato norm estimate Hole-filling technique by [Struwe 81] Proposition There exist a constant C and α 0 (0, 1) such that sup sup R d 2 2α0 v v 2 C. (t 0,x 0) R 1 Campanato space: sup (t 0,x 0) sup R d 2 α v v 2 <, R Q δ,r (t 0,x 0) where Q δ,r (t 0, x 0 ) is a parabolic domain and v is the average of v on Q. Campanato Hölder. Probabilistic analogue: [Weisz 90] 20 / 23

57 Sliceable in BMO For any Markovian soltuion (v, w), Lyapunuov pair implies E t,x[ h(v(t, X t )) h(v(t, x)) ] E t,x[ t t ] w(u, X u ) 2 du M(t t). 21 / 23

58 Sliceable in BMO For any Markovian soltuion (v, w), Lyapunuov pair implies E t,x[ h(v(t, X t )) h(v(t, x)) ] E t,x[ t Hölder estimate of v + norm estimate for X imply E t,x[ t t This means the martingale part is sliceable [Emery 78] and t ] w(u, X u ) 2 du M(t t). ] w(u, X u ) 2 du is small when t t is small. belongs to H BMO [Schachermayer 96]. 21 / 23

59 Sliceable in BMO For any Markovian soltuion (v, w), Lyapunuov pair implies E t,x[ h(v(t, X t )) h(v(t, x)) ] E t,x[ t Hölder estimate of v + norm estimate for X imply E t,x[ t t This means the martingale part is sliceable [Emery 78] and t ] w(u, X u ) 2 du M(t t). ] w(u, X u ) 2 du is small when t t is small. belongs to H BMO [Schachermayer 96]. Then uniqueness follows from [Frei 14], [Kramkov-Pulido 16] 21 / 23

60 Conclusion 1. An abstract framework for global existence + uniqueness 2. Easy to check sufficient conditions are provided 3. Applications in equilibrium, stochastic game, geometry 22 / 23

61 Conclusion 1. An abstract framework for global existence + uniqueness 2. Easy to check sufficient conditions are provided 3. Applications in equilibrium, stochastic game, geometry Current/future research: Non-Markovian, regularity on path space Numeric scheme Other applications 22 / 23

62 Thanks for your attention! 23 / 23