Strong Approximation of Stochastic Differential Equations under - PowerPoint PPT Presentation

Strong Approximation of Stochastic Differential Equations under Non-Lipschitz Assumptions Andreas Neuenkirch U Mannheim & TU Kaiserslautern 14.02.2012 Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 1/26

Outline Part I: Introduction Part II: Euler Schemes under Non-Lipschitz Assumptions Part III: Strong Approximation of Square-root Diffusions Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 2/26

Stochastic Differential Equations Continuous time random dynamics in R d m � b ( j ) ( X t ) dW ( j ) ( SDE ) dX t = a ( X t ) dt + , t ∈ [0 , T ] t j =1 X 0 = x 0 ∈ R d where • a : R d → R d drift coefficient • b = ( b (1) , . . . , b ( m ) ) with b ( j ) : R d → R d diffusion coeff. • W = ( W (1) , . . . , W ( m ) ) ′ m -dimensional Brownian motion Assumption (SDE) has unique strong solution X = Φ a , b , x 0 ( W ) Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 3/26

Computational SDEs X = Φ a , b , x 0 ( W ) Problems (i) Approximate Itˆ o map Φ a , b , x 0 strong / pathwise approximation (ii) Approximate law P X weak approximation (iii) Compute expectation E f ( X ) for f : C ([0 , T ]; R d ) → R quadrature (iv) ... Maruyama (1955) ... Milstein (1974) ... Kloeden, Platen (1992) ... Classically: a , b globally Lipschitz, i.e. there exists L > 0 s.th. x , y ∈ R d (Lip) | a ( x ) − a ( y ) | + | b ( x ) − b ( y ) | ≤ L · | x − y | , Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 4/26

Euler Scheme Equidistant discretization t i = i ∆ where ∆ = T / n X (∆) � = x 0 0 X (∆) X (∆) X (∆) X (∆) � t i +1 = � + a ( � )∆ + b ( � ) ∆ i W , i = 0 , . . . , n − 1 t i t i t i with ∆ i W = W t i +1 − W t i Extension to [0 , T ] by piecewise linear interpolation, i.e. = t i +1 − t + t − t i X (∆) X (∆) X (∆) � � � t i +1 , t ∈ [ t i , t i +1 ] t t i ∆ ∆ Theorem (strong error) Faure (1992); ... Under (Lip) � X t | 2 � 1 / 2 ≤ c ( a , b , x 0 ) · (∆ | log(∆) | ) 1 / 2 t ∈ [0 , T ] | X t − � E max Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 5/26

Two SDEs from Mathematical Finance Heston model � �� � 1 − ρ 2 dW (1) + ρ dW (2) dS t = µ S t dt + | V t | S t , s 0 > 0 t t � | V t | dW (2) dV t = κ ( λ − V t ) dt + θ , v 0 > 0 t where ρ ∈ ( − 1 , 1), κ, λ, θ > 0, µ ∈ R ( V t ) t ∈ [0 , T ] : Cox-Ingersoll-Ross process (CIR) 3/2-model � �� � 1 − ρ 2 dW (1) + ρ dW (2) dS t = µ S t dt + | V t | S t , s 0 > 0 t t dV t = c 1 V t ( c 2 − V t ) dt + c 3 | V t | 3 / 2 dW (2) , v 0 > 0 t with c 1 , c 2 , c 3 > 0 Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 6/26

Properties (1) SDEs take values in subsets D ⊂ R 2 only: • Heston model D = (0 , ∞ ) × [0 , ∞ ) • 3/2-model D = (0 , ∞ ) × (0 , ∞ ) (2) Coefficients not globally Lipschitz on D (3) Coefficients smooth on interior of D Standard theory does not apply! Pioneering works on stochastic Euler schemes under non-standard assumptions: I. Gy¨ ongy (1998); D. Higham, X. Mao, A. Stuart (2002) Many contributions since then; several talks here at MCQMC 2012 on numerics of SDEs under non-standard assumptions Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 7/26

Part II: Euler Schemes under Non-Lipschitz Assumptions Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 8/26

Euler Scheme for SDEs on Domains dX t = a ( X t ) dt + b ( X t ) dW t , X 0 = x 0 SDE with values in a domain D , i.e. D ⊂ R d open and (S) P ( X t ∈ D , t ≥ 0) = 1 Euler scheme X t i +1 = � � X t i + a ( � X t i ) ∆ + b ( � � X t i ) ∆ i W , X 0 = x 0 Extension to [0 , T ] by piecewise linear interpolation Theorem (pathwise error) Gy¨ ongy (1998) If (S) , a ∈ C 1 ( D ; R d ), b ∈ C 1 ( D ; R d , m ), then for all ε > 0 t ∈ [0 , T ] | X t ( ω ) − � X t ( ω ) | ≤ C ε ( ω ) · ∆ 1 / 2 − ε max for almost all ω ∈ Ω, where C ε almost surely finite random variable Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 9/26

Theorem Gy¨ ongy (1998) If (S) , a ∈ C 1 ( D ; R d ), b ∈ C 1 ( D ; R d , m ), then for all ε > 0 t ∈ [0 , T ] | X t ( ω ) − � X t ( ω ) | ≤ C ε ( ω ) · ∆ 1 / 2 − ε max for almost all ω ∈ Ω Remarks • Proof uses localization strategy • Applies to Heston model if 2 κλ ≥ θ 2 and to 3 / 2-model • For D � = R d : use suitable modification of the coefficients outside D for better numerical stability, e.g. x + instead of | x | • Above result can be extended to general Itˆ o-Taylor schemes Jentzen, Kloeden, N (2009) Strong convergence of Euler scheme? Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 10/26

Strong Convergence Theorem Higham, Mao, Stuart (2002) If (S) , a ∈ C 1 ( D ; R d ), b ∈ C 1 ( D ; R d , m ) and t ∈ [0 , T ] | X t | p < ∞ , | p < ∞ X (∆) t ∈ [0 , T ] | � (M1) E max (M2) sup E max t ∆ > 0 for some p > 2, then | 2 → 0 t ∈ [0 , T ] | X t − � X (∆) E max for ∆ → 0 t Proof previous Theorem and integration to the limit using (M) Remarks • Original proof did not use Gy¨ ongy’s result • Applies to CIR if 2 κλ ≥ θ 2 (strictly positive sample paths) • Euler scheme strongly convergent for CIR also for 2 κλ < θ 2 Higham, Mao (2005) Condition (M2) ’technical nuisance’? Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 11/26

Volatility process in 3/2-model dV t = 1 . 2 V t (0 . 8 − V t ) dt + | V t | 3 / 2 dW t , v 0 = 0 . 5 Euler based Monte-Carlo estimator � N 1 V ( i ) | � 4 | N i =1 V ( i ) for E | V 4 | = 0 . 5662 ... where � iid copies of � V 4 4 ∆ = 2 0 2 − 2 2 − 4 2 − 6 2 − 8 2 − 10 repetitions / stepsize N = 10 3 6.3272 Inf Inf 0.5502 0.5535 0.5551 10 4 6.8947 Inf Inf Inf 0.5627 0.5634 10 5 7.4306 Inf Inf Inf 0.5662 0.5671 10 6 7.2274 Inf Inf Inf Inf 0.5658 10 7 7.2792 Inf Inf Inf Inf Inf Empirical first moment explodes! Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 12/26

Moment Explosions Here m = d = 1 Superlinearly growing coefficients: let c ≥ 1, β > α > 1 s.th. max {| a ( x ) | , | b ( x ) |} ≥ 1 c · | x | β , min {| a ( x ) | , | b ( x ) |} ≤ c · | x | α (G) for | x | ≥ c . Theorem Hutzenthaler, Jentzen, Kloeden (2011) Let p > 1 s.th. sup t ∈ [0 , T ] E | X t | p < ∞ and let b ( x 0 ) � = 0. If (G) , then | p = ∞ X (∆) ∆ → 0 E | � lim T Remarks • Moment explosion caused by very large increments of Brownian motion (rare events) • 3/2-model: β = 2, α = 3 / 2. Can modification of coefficients outside (0 , ∞ ) prevent moment explosion? Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 13/26

Drift-implicit Euler Scheme Higham, Mao, Stuart (2002): Drift-implicitness provides numerical stability for SDEs with one-sided Lipschitz drift coefficients, i.e. � x − y , a ( x ) − a ( y ) � ≤ L | x − y | 2 , x , y ∈ R d (one-sided Lip) Example: a ( x ) = x − x 3 Drift-implicit Euler scheme X t i +1 = X t i + a ( X t i + 1 ) ∆ + b ( X t i ) ∆ i W , X 0 = x 0 Extension to [0 , T ] by piecewise linear interpolation Note: implicit equations of the form y − a ( y )∆ = c with c ∈ R d have to be solved Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 14/26

Theorem Szpruch, Mao (2012) If a ∈ C 1 pol ( R d ; R d ), b ∈ C 1 pol ( R d ; R d , m ), a one-sided Lipschitz and � x , a ( x ) � + 1 2 | b ( x ) | 2 ≤ α + β | x | 2 , x ∈ R d (monotone) for some α, β > 0, then | p = 0 (∆) ∆ → 0 E max lim t ∈ [0 , T ] | X t − X t for all p ∈ [1 , 2) Remarks • Similar for Ait-Sahalia interest model Szpruch et al. (2011) • Standard L 2 -convergence rate (∆ | log(∆) | ) 1 / 2 recovered if b additionally globally Lipschitz Higham, Mao, Stuart (2002) • Solving implicit equations avoided by tamed Euler scheme Hutzenthaler, Jentzen, Kloeden (2012) Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 15/26

Summary of Part II dX t = a ( X t ) dt + b ( X t ) dW t , X 0 = x 0 with C 1 -coefficients Euler scheme X t i +1 = � � X t i + a ( � X t i ) ∆ + b ( � � X t i ) ∆ i W , X 0 = x 0 • pathwise convergence • strong convergence under a moment condition • moment explosions possible Drift-implicit Euler scheme X t i +1 = X t i + a ( X t i +1 ) ∆ + b ( X t i ) ∆ i W , X 0 = x 0 • monotone condition + drift one-sided Lipschitz + ... : strong convergence Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 16/26

Part III: Strong Approximation of Square-root Diffusions Andreas Neuenkirch Strong Approximation of SDEs under Non-Lipschitz Assumptions 17/26