Semilinear perturbations of Kolmogorov operators, obstacle problems, - PowerPoint PPT Presentation

Semilinear perturbations of Kolmogorov operators, obstacle problems, and optimal stopping Carlo Marinelli Hausdorff Institute for Mathematics Universit¨ at Bonn and Facolt` a di Economia Universit` a di Bolzano http://www.uni-bonn.de/ ∼ cm788 Based on joint work with Viorel Barbu and Zeev Sobol

Outline 1. Motivation and history of the problem ◮ Classical: the price of an American option is (often) the solution of an obstacle problem ◮ Not so classical (Kholodnyi NA ’97): the price of an American option is expected to “solve” a semilinear PDE with a discontinuous reaction term ◮ Q: Is there a general connection among optimal stopping, obstacle problems, variational inequalities, semilinear PDEs? 2. Solving obstacle problems via semilinear PDEs ◮ Solution of a semilinear PDE is the solution of the obstacle problem ◮ (Nonlinear) monotone operator techniques give a natural concept of solution ◮ Equation is globally well-posed 3. Back to optimal stopping ◮ Solution of the semilinear PDE is also (often) the value function of the original optimal stopping problem

Starting point: 1D Black-Scholes heuristics Assume Black-Scholes dynamics dS t = ( r − d ) S t dt + σ S t dW t and an American option with payoff g : R + → R + , with price E t , x e − r ( τ − t ) g ( S τ ) v ( t , x ) = sup τ ∈ [ t , T ] (Bensoussan AAM ’84, Karatzas AMO ’88) By heuristic (clever!) arguments, Kholodnyi “showed” that v solves the PDE v t + 1 2 σ 2 x 2 v xx + ( r − d ) xv x − rv = q ( x , v ) , v ( T , x ) = g ( x ) , where � − d ( x ) , v ≤ g ( x ) , q ( x , v ) = 0 , v > g ( x ) , − 1 � + . 2 σ 2 x 2 g xx − ( r − d ) xg x + rg � d ( x ) =

Issues with Kholodnyi’s equation ★ What do we mean by “solution”? The reaction term q is discontinuous (if at all defined) ★ The space domain is unbounded, coefficients are unbounded ★ d is in general defined only as a distribution in D ′ ( R ) As far as we know, no known technique applies in the general case: ✘ PDE with discontinuous terms: does not handle unbounded coefficients ✘ PDE with growing coefficients: does not handle discontinuous terms ✘ PDE with measure data: does not handle “rough” equations ✔ Viscosity solutions approach for call/put options on a single BS asset works! (Benth et al. F&S ’03)

Semilinear PDEs vs. Variational Inequalities Classical analytic approach: solve the obstacle problem � v t + L 0 v − cv ≤ f , v t + L 0 v − cv = f on { v ( t , x ) > g ( x ) } v ( T , x ) = g ( x ) e.g. turning it into a VI dv dt + L 0 v − cv − N g ( v ) ∋ f , v ( T ) = g ( T ) . Why another approach? ◮ Much easier to do numerical analysis on a PDE rather than on a free boundary problem/VI (Benth et al. IFB ’04) ◮ Nonlinear discontinuous PDEs have an intrinsic mathematical interest ◮ New way to solve obstacle problems without a variational setting ◮ Can one do better than call/put options on one BS asset?

An abstract general framework Let X be a right Markov process on a Hilbert space H , with semigroup P t , and consider the optimal stopping problem R τ t c ( X s ) ds g ( X τ ) E t , x e − v ( t , x ) = sup τ ∈ [ t , T ] Goal: characterization of the value function v in terms of the solution of a suitable semilinear equation. Plan: ◮ Construct a state space E ◮ Formulate abstract semilinear (evolution) eq. on E ◮ Specify the concept of solution ◮ Prove well-posedness ◮ Prove that the solution coincides with the solution of the obstacle problem ◮ Prove that the solution coincides with the value function

State space: L p ( H , ν ), p ≥ 1 Let ν be an excessive probability measure for P t with full topological support, i.e. such that � � P t ϕ d ν ≤ e ω t ∀ ϕ ∈ C b ( H ) + , ϕ d ν H H and ν ( U ) > 0 for any nonempty open set U ⊆ H . Such a measure ν always exists! (R¨ ockner-Trutnau IDAQP ’07) Set A = − N p + cI , where − N p is the generator of P t on L p ( H , ν ). Then − N p (hence also A ) is ω - m -accretive, i.e. (let ω = 0 for simplicity) (i) � x , J ( y ) � ≥ 0 ∀ [ x , y ] ∈ A ; (ii) R ( λ I + A ) = L p ( E , ν ) ∀ λ > 0.

The semilinear PDE Given g ∈ L p ( H , ν ) and d ∈ L p ( H , ν ) + define the nonlinear multivalued operator B d on L p ( H , ν ) by  − d ( x ) , y ( x ) < g ( x ) ,   [ B d y ]( x ) = [ − d ( x ) , 0] , y ( x ) = g ( x ) ,  0 , y ( x ) > g ( x ) .  We are going to establish well-posedness in L p ( E , ν ) of the evolution equation du dt ( t ) + Au ( t ) + B d u ( t ) ∋ 0 , u (0) = g . (1) We are to “use” several concepts of solutions: ◮ Strong: (1) satisfied a.e. on (0 , T ) ◮ Generalized: SOLA ◮ Mild in the sense of Crandall-Liggett: limit of a discrete scheme ◮ Mild in the sense of Duhamel’s principle

Well-posedness Theorem. The equation du dt ( t ) + Au ( t ) + B d u ( t ) ∋ 0 , u (0) = g admits a unique CL-mild solution in L p ( E , ν ), p ≥ 1 (which is SOLA for p > 1). Moreover, if g ∈ D ( A ) and p > 1, then it admits a unique strong solution u ∈ W 1 , ∞ ([0 , T ] → L p ( H , ν )) ∩ L ∞ ([0 , T ] → D ( A )) which is also right-differentiable. Proof. Have to show that A + B d is ω - m -accretive...

A + B d is ω - m -accretive in L p ( H , ν ) Lemma. B d is m -accretive in L p ( H , ν ). Proof. B d is accretive because y �→ d ( x )( H ( y − g ( x )) − 1) is a maximal monotone graph in R × R for each x ∈ H . Maximality: equation y + By = f , f ∈ L p ( H , ν ) easily admits a solution. Theorem. A + B d is ω - m -accretive in L p ( H , ν ). Proof. Three different cases: ◮ p = 2: follows by Rockafellar’s criterion: D ( A ) ∩ int D ( B d ) � = ∅ . ◮ p > 1: solve u λ + A λ u λ + B d u λ ∋ f , get a priori estimates on u λ , let λ → 0 (reflexivity of L p is crucial) ◮ p = 1: solve u ε + Au ε + B d ,ε u ε = f , get monotonicity for u ε , let ε → 0.

Solution of the PDE is a solution of the obstacle problem Have to choose d first! Two cases: Theorem. ◮ If g ∈ D ( A ), let d := ( Ag ) + . ◮ If g ∈ L p ( H , ν ) = D ( A ), assume that ( A λ g ) + is weakly compact in L p ( H , ν ), and let d be such that ( A λ g ) + ⇀ d . Then u ( t ) ≥ g ν -a.s.. Proof. Prove that S ( t ) := e − t ( A + B d ) leaves invariant � ϕ ∈ L p ( H , ν ) : ϕ ≥ g � K g := ν -a.e. . Enough to prove that ( I + λ A + λ B d ) − 1 K g ⊆ K g for all λ ∈ ]0 , ω − 1 [. Use sub-Markovianity of A and definition of B d . Key observation: By definition of B d , u ( t ) ≥ g ν -a.e. implies that u is the solution of the obstacle problem!

Solution as value function of the optimal stopping problem Theorem 1. Assume that P t is strong Feller (or L ( X t ) ≪ ν ). Then u ( T − t ) = v ( t ) ν -a.e. ∀ t ≤ T Proof. Two steps: 1. Establish a Duhamel representation for CL-mild solutions. 2. Refine the proof in Barbu-M (AMO ’08) Remark. Still true for Markov processes that are limits of strong Feller processes (hence always true for solutions of SDEs on R d ) Q: ◮ Can one approximate any right Markov process by a strong Feller Markov process? ◮ Counterexamples?

Further properties ◮ Under very mild assumptions v is continuous, hence u has a continuous ν -modification. Moreover, recall that ν has full topological support! ◮ Convexity assumptions on g are needed also for the 1D viscosity solutions approach. ◮ Finite-difference schemes converge to the mild solution (Crandall-Liggett theorem) ◮ If dim H < ∞ one expects further regularity for u ◮ Infinite-horizon optimal stopping problems ( � elliptic PDEs) are automatically included

An example Let X be the solution of an SDE on H � s � s X s = x + b ( X r ) dr + σ ( X r ) dW ( r ) t t with Kolmogorov operator − N 0 φ = 1 2 Tr[( σ Q 1 / 2 )( σ Q 1 / 2 ) ∗ D 2 φ ] + � b ( x ) , D φ � H , φ ∈ C 2 b ( H ) . Consider an optimal stopping problem as before. Let P t φ ( x ) := E x φ ( X ( t )), φ ∈ C b ( H ). Let P ∗ t ν ≤ e ω t ν , extend P t to L 2 ( H , ν ), and set − N 2 φ := lim h ↓ 0 h − 1 ( P h φ − φ ) in L 2 ( H , ν ). Lemma. Let b ∈ C 2 ( H ) ∩ L 2 ( H , ν ), σ ∈ C 2 ( H , L ( H , H )), and | Db ( x ) | H + | D σ ( x ) | L ( H , H ) ≤ C ∀ x ∈ H . Then − N 0 is ω -accretive and − N 2 is the closure in L 2 ( H , ν ) of − N 0 defined on D ( N 0 ) = C 2 b ( H ).

(contd.) Proof. − N 0 is ω -accretive, hence closable. Fix f ∈ C 2 b ( H ) and consider the eq. ( λ I + N 0 ) ϕ = f . Candidate solution is � ∞ e − λ t f ( X x ϕ ( x ) = E t ) dt , λ > ω. 0 By second order differentiability of x �→ X x t and Itˆ o’s formula, we see that φ actually is the solution. Then R ( I + λ N 0 ) ⊂ L 2 ( H , ν ) densely, so − N 0 is ω - m -accretive by Lumer-Phillips theorem, so it must be N 2 = N 0 because − N 2 is also ω - m -accretive. Remarks. ◮ Similar results for Kolmogorov operators go through under much weaker assumptions, and also for equations of the type � s � s � s � X s = x + b ( X r ) dr + σ ( X r ) dW ( r )+ g ( z , X r − ) ¯ µ ( dz , dr ) t t t Z (M-Pr´ evˆ ot-R¨ ockner JFA ’10) ◮ Enough to take N ∗ 0 ν ≤ ων .