On continuous time contract theory

Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218

Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs

(Static) Principal-Agent Problem Principal delegates management of output process X, only observes X pays salary defined by contract ξ(x ) Agent devotes effort a = X a, chooses optimal effort by ( V A (ξ) := max E U A ξ(x a ) c(a) ) = â(ξ) a Principal chooses optimal contract by solving ( max E U P X â(ξ) ξ(x â(ξ) ) ) under constraint V A (ξ) ρ ξ Non-zero sum Stackelberg game

(Static) Principal-Agent Problem ==> Continuous time Principal delegates management of output process X, only observes X pays salary defined by contract ξ(x ) Agent devotes effort a = X a, chooses optimal effort by ( V A (ξ) := max E U A ξ(x a ) c(a) ) = â(ξ) a Principal chooses optimal contract by solving ( max E U P X â(ξ) ξ(x â(ξ) ) ) under constraint V A (ξ) ρ ξ = Non-zero sum Stackelberg game

Semimartingale distributions on the paths space Ω := { ω C (R +, R d ) : ω() = } X : canonical process, i.e. X t (ω) := ω(t) F t := σ(x s, s t), F := {F t, t } P W : collection of all semimartingale measures P such that dx t = b t (X )dt + σ t (X )dw t, P a.s. for some F processes b and σ, and P Brownian motion W

Quadratic variation process X : quadratic variation process (defined on R + Ω) X t := Xt 2 t 2X sdx s = P lim π n 1 Xt t π n X t t π 2 n 1 for all P P W, and set ˆσ 2 t := lim h X t+h X t h Example : (d = 1) Let P 1 :=Wiener measure, i.e. X is a P 1 BM, and define P 2 := P 1 (2X ) 1. Then P 1 and P 2 are singular ˆσ t = 1, P 1 a.s. and ˆσ t = 2, P 2 a.s.

Principal-Agent problem formulation Agent problem : V A (ξ) := sup E P[ ξ(x ) P P ] c t (ν t )dt P P : weak solution of Output process for some ν valued in U : dx t = b t (X, ν t )dt + σ t (X, ν t )dw P t P a.s. Given solution P (ξ), Principal solves the optimization problem V P := sup ξ Ξ ρ E P (ξ) [ U ( l(x ) ξ(x ) )] where Ξ ρ := { ξ(x. ) : V A (ξ) ρ } Possible extensions : random (possibly ) horizon, heterogeneous agents with possibly mean field interaction, competing Principals...

Principal-Agent problem formulation : non-degeneracy Agent problem : V A (ξ) := sup E P[ ξ(x ) P P ] c t (ν t )dt P P : weak solution of Output process for some ν valued in U : dx t = σ t (X, β t ) [ λ t (X, α t )dt + dwt P ] P a.s. Given solution P (ξ), Principal solves the optimization problem V P := sup ξ Ξ ρ E P (ξ) [ U ( l(x ) ξ(x ) )] where Ξ ρ := { ξ(x. ) : V A (ξ) ρ } Possible extensions : random (possibly ) horizon, heterogeneous agents with possibly mean field interaction, competing Principals...

GENERAL SOLUTION APPROACH

A subset of revealing contracts Path-dependent Hamiltonian for the Agent problem : { H t (ω, z, γ) := sup bt (ω, u) z + 1 u U 2 σ tσt (ω, u):γ c t (ω, u) } For Y R, Z, Γ F X prog meas, define P a.s. for all P P t Yt Z,Γ = Y + Z s dx s + 1 2 Γ s : d X s H s (X, Z s, Γ s )ds ( Proposition V A Y Z,Γ) T = Y. Moreover P is optimal iff ν t = Argmax u U H t(z t, Γ t ) = ˆν(Z t, Γ t )

Proof : classical verification argument! For all P P, denote J A (ξ, P) := E P[ ξ cν t dt ]. Then J A ( Y Z,Γ T, P) = E P[ Y + Z t dx t + 1 2 Γ t:d X t H t (Z t, Γ t )dt ] ct ν dt { = Y +E P bt ν Z t + 1 } 2 σσ :Γ t ct ν H t (Z t, Γ t ) dt+z t σt ν dwt P Y by definition of H with equality iff ν = ν maximizes the Hamiltonian

Principal problem restricted to revealing contracts Dynamics of the pair (X, Y ) under optimal response ( ( ) dx t = b t X, ˆν(Y Z,Γ t, Z t, Γ t ) )dt + σ t X, ˆν(Y t Z,Γ, Z t, Γ t ) dw t dy Z,Γ t = Z t dx t + 1 2 Γ t : d X t H t (X, Yt Z,Γ, Z t, Γ t )dt is a (1 state augmented) controlled SDE with controls (Z, Γ) = Principal s value function under revealing contracts : [ V P V (X, Y ) := sup E U ( l(x ) Y Z,Γ ) ] T, for all Y ρ (Z,Γ) V { where V := (Z, Γ) : Z H 2 (P) and P ( Y Z,Γ ) } T

Theorem (Cvitanić, Possamaï & NT 15) Assume V. Then V P = sup V (X, Y ) Y ρ Given maximizer Y, the corresponding optimal controls (Z, Γ ) induce an optimal contract ξ = Y + Zt dx t + 1 2 Γ t : d X t H t (X, Y Z,Γ t, Zt, Γ t )dt

Recall the subclass of contracts Y Z,Γ t = Y + t Z s dx s + 1 2 Γ s :d X s H s (X, Ys Z,Γ, Z s, Γ s )ds P a.s. for all P P To prove the main result, it suffices to prove the representation for all ξ?? (Y, Z, Γ) s.t. ξ = Y Z,Γ, P a.s. for all P P OR, weaker sufficient condition : T for all ξ?? (Y n, Z n, Γ n ) s.t. Y Z n,γ n T ξ

Connexion with nonlinear parabolic PDEs Consider the Markov case ξ = g(x T ), required representation is : Y T = g(x T ), and dy t = Z t dx t + 1 2 Γ t :d X t H t (X t, Y t, Z t, Γ t )dt P a.s. for all P P Intuitively, Y t = v(t, X t ) with decomposition (Itô s formula) dy t = t v(t, X t )dt + Dv(t, X t ) dx t + 1 2 D2 v(t, X t ):d X t By direct identification : Z t = Dv(t, X t ), Γ t = D 2 v(t, X t ), and t v + H(., v, Dv, D 2 v) =, with boundary cond.v t=t = g Representation path-dependent nonlinear parabolic PDE

Outline Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs

Nonlinear expectation operators P : subset of local martingale measures, i.e. Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs dx t = σ t dw t, P a.s. for all P P = Nonlinear expectation E := sup P P E P Similarly, P L : subset of measures Q λ such that dx t = σ t (λ t dt + dw t ), Q a.s. for some λ, F adapted, λ L = Another nonlinear expectation E L := sup P P L E Q E and E L will play the role of Sobolev norms...

Random horizon 2 nd order backward SDE Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs For a stop. time τ, and F τ measurable ξ : Y t τ = ξ + τ t τ F s (Y s, Z s, ˆσ s )ds τ t τ Z s dx s + τ t τ K non-decreasing, K =, and minimal in the sense t τ ] inf EP[ dk r =, for all s t P P s τ dk s, P q.s. P q.s. MEANS P a.s. for all P P

Nonlinearity Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs Assumptions F : R + ω R R d S d + R satisfies (C1 L ) Lipschitz in (y, σz) : F (., y, z, σ) F (., y, z, σ) L ( y y + σ(z z ) ) (C2 µ ) Monotone in y : (y y ) [F (., y,.) F (., y,.) ] µ y y 2 Denote f t := F t (,, σt ) Remark Deterministic finite horizon τ = T : (C2) µ not needed Soner, NT & Zhang 14 and Possamaï, Tan & Zhou 16

Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs Wellposedness of random horizon 2 nd order backward SDE Y t τ = ξ + τ t τ F s (Y s, Z s, ˆσ s )ds τ t τ Z s dx s + τ t τ dk s, K non-decreasing, and inf P P E P[ t τ s τ dk r ] =, s t P q.s. Theorem (Y. Lin, Z. Ren, NT & J. Yang 17) Assume ρ> µ, q >1 : E L[ e ρτ ξ q ] + E L [( τ e ρt f 2 ds ) q ] 2 < Then, Random horizon 2BSDE has a unique solution (Y, Z) with Y D p η,τ (P L ), Z H p η,τ (P L ) for all η [ µ, ρ), p [1, q) Y p D p η,τ (P L ) := EL[ sup e ηt ] Y t p, Z p τ H p t τ η,τ (P) := EL[( e ηt σ t T ) p Z t 2 dt t 2 ]

Back to Principal-Agent problem Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs Recall the subclass of contracts Y Z,Γ t = Y + t Z s dx s + 1 2 Γ s :d X s H s (X, Ys Z,Γ, Z s, Γ s )ds, P q.s. To prove the main result, it suffices to prove the representation for all ξ?? (Y, Z, Γ) s.t. ξ = Y Z,Γ T, P q.s. OR, weaker sufficient condition : for all ξ?? (Y n, Z n, Γ n ) s.t. Y Z n,γ n T ξ

Reduction to second order BSDE Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs H t (ω, y, z, γ) non-decreasing and convex in γ, Then { 1 H t (ω, y, z, γ) = sup σ 2 σ2 : γ Ht (ω, y, z, σ)} Denote k t := H t (Y t, Z t, Γ t ) 1 2 ˆσ2 t : Γ t + H t (Y t, Z t, ˆσ t ) Then, required representation ξ = Y Z,Γ, P q.s. is equivalent to ξ = Y + T Z s dx s + 1 2 Γ s :d X s H s (X, Ys Z,Γ, Z s, Γ s )ds compare with 2BSDE ξ = Y F s (Y s, Z s, ˆσ s )ds + Z s dx s dk s, P q.s.

Reduction to second order BSDE Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs H t (ω, y, z, γ) non-decreasing and convex in γ, Then { 1 H t (ω, y, z, γ) = sup σ 2 σ2 : γ Ht (ω, y, z, σ)} Denote k t := H t (Y t, Z t, Γ t ) 1 2 ˆσ2 t : Γ t + H t (Y t, Z t, ˆσ t ) Then, required representation ξ = Y Z,Γ, P q.s. is equivalent to ξ = Y + Z t dx t + H t (Y t, Z t, ˆσ t )dt T k t dt, P q.s. compare with 2BSDE ξ = Y F s (Y s, Z s, ˆσ s )ds + Z s dx s dk s, P q.s.