Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging

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1 Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging SAMUEL DRAPEAU Humboldt-Universität zu Berlin BSDEs, Numerics and Finance Oxford July 2, 2012 joint work with GREGOR HEYNE and MICHAEL KUPPER Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 1

2 Supersolutions of BSDE Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 2

3 Motivation T Y 0 + Z uds u ξ 0 }{{} trading gains Superhedging problem Y 0 = superhedging price of ξ Z = superhedging strategy Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 3

4 Motivation T Y t + Z uds u ξ t }{{} trading gains Superhedging problem Y t= superhedging price of ξ at t Z = superhedging strategy Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 3

5 Motivation Supersolutions of Backward Stochastic Differential Equations Definition (Y, Z ) is a supersolution of the Backward Stochastic Differential Equation with driver g and terminal condition ξ if T T Y t g(y u, Z u)du + Z udw u ξ t [0, T ] t t }{{}}{{} drift part martingale part Y = value process Z = control process Equality instead of inequality: (Y, Z ) is solution of the BSDE. Extensively studied: Bismut, Pardoux, Peng, Ma, Protter, Yong, Briand, Hu, Kobylanski, Touzi, Delbaen, Imkeller, El Karoui,... Applications in utility maximization, stochastic games, stochastic equilibria,... Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 4

6 Motivation Supersolutions of Backward Stochastic Differential Equations Supersolutions are typically not unique. Find a minimal supersolution (Y min, Z min )! That is Y min Y for any other supersolution (Y, Z ). Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 5

7 Minimal Supersolutions Filtrated probability space (Ω, F, (F t), P), filtration generated by a Brownian motion W satisfying the usual conditions. t t Y s g(y u, Z u)du + Z udw u Y t, 0 s t T s s (0.1) Y T ξ 1 ξ is F T -measurable. 2 Y is (F t)-adapted and càdlàg S 3 Z is (F t)-progressive, such that T 0 Z 2 u du < + and! Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 6

8 Minimal Supersolutions Filtrated probability space (Ω, F, (F t), P), filtration generated by a Brownian motion W satisfying the usual conditions. t t Y s g(y u, Z u)du + Z udw u Y t, 0 s t T s s (0.1) Y T ξ 1 ξ is F T -measurable. 2 Y is (F t)-adapted and càdlàg S 3 Z is (F t)-progressive, such that T Z 2 0 u du < + and Z is admissible, i.e. ZdW is a supermartingale ( Dudley and Harrison/Pliska) L The set of supersolutions with driver g and terminal condition ξ A := {(Y, Z ) S L : (Y, Z ) fulfills (0.1)} Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 6

9 Minimal Supersolutions A generator is a lower semicontinuous function g : R R d ], ]. Additional properties: (Pos) g (y, z) [0, + ] for all (y, z). (Conv) z g (y, z) is convex. (Mon) g (y, z) g (y, z) for all y y. (Mon ) g (y, z) g (y, z) for all y y. Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 7

10 Minimal Supersolutions A natural candidate for the value process of a minimal supersolution: Ê t = ess inf {Y t : (Y, Z ) A}, t [0, T ] Question: Does there exist a càdlàg modification E of Ê and a control process Z L such that (E, Z ) is a supersolution? Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 8

11 Minimal Supersolutions A natural candidate for the value process of a minimal supersolution: Ê t = ess inf {Y t : (Y, Z ) A}, t [0, T ] Theorem: Assume (Pos), (Conv) and either (Mon) or (Mon ). Suppose ξ L 1 and A. Then E t := Ê + t = lim s t,s Q Ê s is the value process of the unique minimal supersolution, that is, there exists a unique control process Z such that (E, Z ) A. Compactness (DELBAEN and SCHACHERMAYER) versus fixpoint. Drop positivity for (Pos ) g(y, z) az + b. (utility maximization) Gregor Heyne, Michael Kupper and Christoph Mainberger, drop convexity in z for g(y, 0) = 0. (BARLOW and PROTTER). Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 8

12 Minimal Supersolutions Compactness Any sequence (x n) in R d such that sup n N x n < has a subsequence (x nk ) converging to some x R d. Let (X n) be a sequence of random variables in L 2 (Ω, F, P) such that sup n N E[X 2 n ] <. Then there exists a sequence Y n conv(x n, X n+1,... ) such that Y n Y in L 2 (Ω, F, P). (Delbaen/Schachermayer) Let ( H n dw ) be a H 1 -bounded sequence of martingales. Then there exist K n conv{h n, H n+1,... } and a localizing sequence of stopping times (τ n ) such that ( K n dw ) τ n KdW in H 1. Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 9

13 Minimal Supersolutions Idea of the proof 1) Paste strategies between stopping times construct (Y n, Z n ) A with, Ê t n k Y n n t k n 1/n, and Yt Y n+1 t, 2) Y = lim n Y n and Ê are supermartingales E := Ê + = Y +. 3) Show that Ê t E t. 4) There is a localizing sequence (σ k ) such that ( ) σk Z n dw is bounded in H 1. 5) DELBAEN and SCHACHERMAYER convex combinations such that t 0 Z n s dw s n + t 0 Z sdw s. 6) Verification with (E, Z ) is based on Helly s theorem and Fatou s lemma. Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 10

14 Minimal Supersolutions Nonlinear Expectations ( Peng s g-expectations) Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 11

15 Stability results For any nice generator g the mapping satisfies E g : ξ minimal supersolution with terminal condition ξ ξ E g 0 (ξ) E[ξ] = Ω ξ(ω)p(dω) (N) E g 0 (m) = m E [m] = m (T) E g 0 (ξ + m) = E g 0 (ξ) + m E[ξ + m] = E[ξ] + m (TC) E g 0 (ξ) = E g ( 0 E g t (ξ)) E[ξ] = E [E[ξ F t]] Linearity: E[λξ 1 + ξ 2 ] = λe[ξ 1 ] + E[ξ 2 ] nonlinear expectation Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 12

16 Stability results The nonlinear expectation E g 0 ( ) satisfies Monotone convergence: 0 ξ n ξ implies E g 0 (ξ) = limn E g 0 (ξn ) Fatou s lemma: E g 0 (lim infn ξn ) lim inf n E g 0 (ξn ) is σ(l 1, L )-lower semicontinuous. If g is independent of y, by convex duality: E g 0 (ξ) = sup {E Q [ξ] α min (Q)} Q P = representation of a convex risk measure Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 13

17 Model Uncertainty and Robust Hedging ( Peng s G-expectation) Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 14

18 The role of the probability measure P The probability P defines the dynamics of the process P probabilistic model, e.g. on C ([0, T ]; R) What is the probability measure P? Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 15

19 The role of the probability measure P P can only partially be identified by statistical methods Take into account a family P of probability measures (models) Model Uncertainty Remark: The probability measures are typically singular! P(A) := sup P[A] = capacity P P Denis, Martini, Peng, Hu, Bion-Nadal, Soner, Touzi, Zhang, Nutz,... E.g. dst(θ) S t(θ) = µdt + θdwt, θ θ θ Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 16

20 Minimal Supersolutions of Robust BSDEs Setting Θ is a set of volatility processes: θ : Ω [0, T ] R++ (S >0 d ). Our state space: Ω := Ω Θ Driving process: W : Ω [0, T ] R, where W (θ) = θdw, θ Θ Progressively learning about the volatility F t := σ( W s : s t), t [0, T ] In general not right-continuous. Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 17

21 Minimal Supersolutions of Robust BSDEs Capacity Let µ θ be the probability measure induced by W (θ) on C([0, T ], R d ) with the Borel σ-algebra. These probability measures are singular to each others. There is no dominating probability measures! P[A] := sup P [A(θ)], θ Θ A F T Properties (like equality and inequalities) holds quasi-surely if the event B, where they do not hold is a polar set, i.e., B F T with P[B] = 0. We assume that {µ θ : θ Θ} is weakly compact. Denis, Martini, Peng, Hu, Bion-Nadal, Soner, Touzi, Zhang, Nutz Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 18

22 Minimal Supersolutions of Robust BSDEs Supersolutions of Robust BSDE For all θ Θ, τ τ Y σ(θ) g u(y u(θ), Z u(θ))du + Z u(θ)d W u(θ) Y τ (θ), σ σ Y T (θ) ξ(θ) where σ, τ are (F t)-stopping times with 0 σ τ T. (0.2) 1 ξ is F T -measurable. 2 Y is làdlàg and Y t L 1 b( F t) C( F t), Y (θ) is optional S 3 Z is ( F t)-predictable, such that T 0 Z 2 u (θ)θ 2 udu < + and Z (θ)d W (θ) is a supermartingale for all θ Θ L 4 g : R R (, + ], such that g(θ) is a generator as before for all θ Θ. The set of supersolutions with driver g and terminal condition ξ { } A := (Y, Z ) S L : (Y, Z ) fulfills (0.2) Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 19

23 Minimal Supersolutions of Robust BSDEs Minimal Supersolution As a usual, our natural candidate for the minimal supersolution Ê t = ess inf {Y t : (Y, Z ) A} However, there is no reference probability measure! Bion-Nadal, Nutz,... We consider the infimum: Ê t = inf {Y t : (Y, Z ) A} Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 20

24 Minimal Supersolutions of Robust BSDEs Existence Result Our existence Theorem reads as follows Theorem: Assume (Pos), (Conv) and either (Mon) or (Mon ). Suppose ξ L 1 b( F T ), A and Ê t C( F t). Then there exists a làdlàg modification E of Ê, which is the value process of the unique minimal supersolution, that is, there exists a unique control process Z such that (E, Z ) A. We give conditions under which the Ê fulfills these assumptions (Markovian Setting). Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 21

25 Thank You! Samuel Drapeau Minimal Supersolutions of Backward Stochastic Differential Equations and Robust Hedging 22