Viscosity Solutions of Path-dependent Integro-Differential Equations

Transcription

1 Viscosity Solutions of Path-dependent Integro-Differential Equations Christian Keller University of Southern California Conference on Stochastic Asymptotics & Applications Joint with 6th Western Conference in Mathematical Finance Christian Keller (USC) PPDEs 1 / 14

2 Outline 1 Introduction 2 PPDEs 3 Path-dependent integro-differential equations Christian Keller (USC) PPDEs 2 / 14

3 Standard PDEs vs. PPDEs Second-order parabolic standard PDEs/ integro-differential equations: t u+g(t, x, u, x u, 2 xxu, (I α,β u) α,β ) = 0, (t, x) [0, T ) R. u = u(t, x): Object whose evolution at given time depends on its current state. Markovian case. Second-order parabolic path-dependent PDEs (PPDEs)/ integro-differential equations: t u+g(t, ω, u, ω u, 2 ωωu, (I α,β u) α,β ) = 0, (t, ω) [0, T ) Ω, where Ω is a path space u non-anticipating: u(t, ω) = u(t, ω t ). u = u(t, ω): Object whose evolution at given time depends on its past. Non-markovian case. Christian Keller (USC) PPDEs 3 / 14

4 Some related work Peng (2010): PPDEs Dupire (2009): Functional Itô calculus Cont & Fournié (JFA, 2010): " " Viscosity solutions of PPDEs: Lukoyanov (Diff.Urav. 2007): Fully nonlinear 1st order Ekren, K., Touzi, Zhang (AoP, 2014): Semilinear 2nd order Ekren, Touzi, Zhang (2012ab): Fully Nonlinear 2nd order Pham & Zhang (SICON, 2014): Isaacs equations Ekren (2013): Obstacle equations Ren (2014): Elliptic equations K. (2014): Semilinear 2nd order integro-differential equations Christian Keller (USC) PPDEs 4 / 14

5 Applications Path-dependent options Control of stochastic delay equations Convergence of numerical schemes Non-markovian problems in general In particular: Stochastic differential games: Pham & Zhang (SICON, 2014) Numerics for BSDEs: Henry-Labordère, Tan, Touzi (SPA, 2014) Numerics for PPDEs: Zhang & Zhuo (JFinEng, 2014) Martingale problems: Costantini & Kurtz (2014) Christian Keller (USC) PPDEs 5 / 14

6 Path-dependent heat equation and setup t u ωωu = 0 Ω := C([0, T ]), Λ := [0, T ) Ω, Λ := [0, T ] Ω pseudometric on Λ: d ((t, ω), (t, ω )) := t t + ω t ω t Cb 0( Λ): bounded functionals Λ R cont. under d Note: u Cb 0( Λ) u non-anticipating X canonical process: X t (ω) = ω t F 0 = (Ft 0 ) filtration generated by X P t,ω probability measure such that, P t,ω -a.s., X = ω on [0, t] X = ωt + W W t on (t, T ] where W Wiener process. Christian Keller (USC) PPDEs 6 / 14

7 Derivatives Time derivative as in Dupire (2009): C 1,2 b 1 t u(t, ω) := lim h 0 h [u(t + h, ω t u(t, ω)] (Λ): all u C0 b ( Λ) such that: t u Cb 0(Λ) there exist ω u, ωωu 2 Cb 0 (Λ) such that: (s, ω): ( du = t u + 1 ) 2 2 ωωu dt + ω u dx on [s, T ], P s,ω -a.s. Christian Keller (USC) PPDEs 7 / 14

8 Viscosity solutions Standard heat equation: Lu := t u xxu = 0 Au(t, x): all ϕ C 1,2 ([0, T ) R) such that, for some ε > 0, 0 = (ϕ u)(t, x) = min (ϕ u)(s, y) (s,y) O ε(t,x) u C([0, T ] R) is viscosity subsolution if (t, x) [0, T ) R, ϕ Au(t, x) Lϕ(t, x) 0 Main problem: R locally compact, but Ω is not! Change pointwise minimum to minimum in mean Optimal stopping Christian Keller (USC) PPDEs 8 / 14

9 Viscosity solutions Standard heat equation: Lu := t u xxu = 0 Au(t, x): all ϕ C 1,2 ([0, T ) R) such that, for some ε > 0, 0 = (ϕ u)(t, x) = min (ϕ u)(s, y) (s,y) O ε(t,x) u C([0, T ] R) is viscosity subsolution if (t, x) [0, T ) R, ϕ Au(t, x) Lϕ(t, x) 0 Main problem: R locally compact, but Ω is not! Change pointwise minimum to minimum in mean Optimal stopping Path-dependent heat equation: Lu := t u ωωu = 0 Au(t, ω): all ϕ C 1,2 (Λ) such that, for some hitting time H > t, 0 = (ϕ u)(t, ω) = min E t,ω [(ϕ u) τ H ] τ F 0 -stopp.time,τ t u C( Λ) is viscosity subsolution if (t, ω) Λ, ϕ Au(t, ω) Lϕ(t, ω) 0 Christian Keller (USC) PPDEs 8 / 14

10 Justification of new formulation Consider Lu = 0 (standard heat equation). u classical solution, ϕ Au(t, x). (ϕ u)(t, x) (ϕ u)(s, x) (s, x) O ε (t, x) [ ] (ϕ u)(t, x) E (ϕ u)(τ H ε ), W t,x τ F 0 -stopp.time τ Hε Here: W t,x Wiener process starting at x at time t H ε := inf{s t : Wr t,x O ε (t, x)} Christian Keller (USC) PPDEs 9 / 14

11 Justification of new formulation Consider Lu = 0 (standard heat equation). u classical solution, ϕ Au(t, x). (ϕ u)(t, x) (ϕ u)(s, x) (s, x) O ε (t, x) [ ] (ϕ u)(t, x) E (ϕ u)(τ H ε ), W t,x τ F 0 -stopp.time τ Hε Here: W t,x Wiener process starting at x at time t H ε := inf{s t : Wr t,x O ε (t, x)} [ τ H ] ε 0 E L(ϕ u)(s, Ws t,x ) ds t by Itô s formula 0 Lϕ(t, x) since Lu = 0 Christian Keller (USC) PPDEs 9 / 14

12 Justification of new formulation Consider Lu = 0 (standard heat equation). u classical solution, ϕ Au(t, x). (ϕ u)(t, x) (ϕ u)(s, x) (s, x) O ε (t, x) [ ] (ϕ u)(t, x) E (ϕ u)(τ H ε ), W t,x τ F 0 -stopp.time τ Hε Here: W t,x Wiener process starting at x at time t H ε := inf{s t : Wr t,x O ε (t, x)} [ τ H ] ε 0 E L(ϕ u)(s, Ws t,x ) ds t by Itô s formula 0 Lϕ(t, x) since Lu = 0 Note: To derive viscosity property, we just used (ϕ u)(t, x) = min τ E t,x [(ϕ u) τ H ε] Christian Keller (USC) PPDEs 9 / 14

13 A linear path-dependent integro-differential equation t u ωωu + u(t, ω + 1 [t,t ] ) u(t, ω) ω u = 0 Ω := {ω : [0, T ] R càdlàg}, Λ, Λ, d as before C 0 b ( Λ): all bounded u : Λ R such that, for every x R, (t, ω) u(t, ω + x.1 [t,t ] ) is cont. under d X canonical process P t,ω probability measure such that, P t,ω -a.s., X = ω on [0, t] X = ωt + W W t + Ñ Ñt on (t, T ) where Ñ compensated Poisson process with intensity 1. Christian Keller (USC) PPDEs 10 / 14

14 Filtrations and derivatives F 0 + = (F 0 t+ ) F = (F t ), where F t = t u defined as before C 1,2 b du = (s,ω) Ft+ 0 P s,ω (Λ): all u C0 b ( Λ) such that: t u Cb 0(Λ) there exist ω u, ωωu 2 Cb 0 (Λ) such that: (s, ω): ( t u + 1 ) 2 2 ωωu + u(t, X + 1 [t,t ] ) u ω u dt + ω u dw + ( u(t, X + 1 [t,t ] ) u ) d Ñ on [s, T ], P s,ω -a.s. Christian Keller (USC) PPDEs 11 / 14

15 Viscosity solutions Equation: Lu := t u ωωu + u(t, ω + 1 [t,t ] ) u ω u = 0 Au(t, ω): all ϕ C 0 b ( Λ) C 1,2 (Λ) such that, for some hitting time H > t, 0 = (ϕ u)(t, ω) = min τ F-stopp.time,τ t E t,ω [(ϕ u) τ H ] u C( Λ) is viscosity subsolution if (t, ω) Λ, ϕ Au(t, ω) Lϕ(t, ω) 0 Christian Keller (USC) PPDEs 12 / 14

16 Difficulties in jump case Hitting times of closed sets are totally inaccessible and not F 0 -stopping times = Need F instead of F 0 Careful with regular conditional probabilities: Ft+ 0 not countably generated Use of piecewise Markovian" functionals & approximation with classical solutions of standard integro-differential equations more complicated as with standard PDEs Skorohod topologies are used: M1 instead of J1 Christian Keller (USC) PPDEs 13 / 14

17 Main result (2nd order semilinear equations) Operators on C 1,2 (Λ): Lu(t, ω) = b ω u(t, ω) c 2 ωωu(t, ω) + [u(t, ω + 1[t,T ] ) u(t, ω) z ω u(t, ω) ] K (dz) Iu(t, ω) = [u(t, ω + 1[t,T ] ) u(t, ω) ] η t K (dz) Theorem If ξ and f are nice, then t u + Lu + f (t, ω, u, ω u, Iu) = 0 u(t, ) = ξ in Λ on Ω has a unique viscosity solution u. Moreover, u has a stochastic representation in terms of BSDEs with jumps. Christian Keller (USC) PPDEs 14 / 14