Multi-level stochastic approximation algorithms Noufel Frikha - PowerPoint PPT Presentation

Introduction Analysis of the SA scheme Multi-level stochastic approximation algorithms Numerical results Multi-level stochastic approximation algorithms Noufel Frikha Universit´ e Paris Diderot, LPMA 28th October, 2013. Stochastic processes and their statistics in Finance in Okinawa Noufel Frikha Multi-level stochastic approximation algorithms

Introduction Multi-level Monte Carlo method Analysis of the SA scheme Toward Multi-level stochastic approximation algorithms Multi-level stochastic approximation algorithms A short analysis of the different steps Numerical results Outline of the presentation Introduction 1 Multi-level Monte Carlo method Toward Multi-level stochastic approximation algorithms A short analysis of the different steps Analysis of the SA scheme 2 On the implicit discretization error Optimal tradeoff between implicit discretization and statistical errors Multi-level stochastic approximation algorithms 3 Statistical Romberg SA : a two-level SA scheme Multi-level stochastic approximation algorithm Numerical results 4 Noufel Frikha Multi-level stochastic approximation algorithms

Introduction Multi-level Monte Carlo method Analysis of the SA scheme Toward Multi-level stochastic approximation algorithms Multi-level stochastic approximation algorithms A short analysis of the different steps Numerical results Introduction ⊲ Multi-level Monte Carlo paradigm was originally introduced for the computation of : E x [ f ( X T )] where f : R q → R and ( X t ) t ∈ [ 0 , T ] is a q -dimensional process satisfying � t � t ∀ t ∈ [ 0 , T ] , X t = x + b ( X s ) ds + σ ( X s ) dW s . ( SDE b ,σ ) 0 0 When no closed formula is available, one proceeds in two steps : ⊲ Step 1 : Discretization scheme of ( SDE b ,σ ) by � t � t X n b ( X n σ ( X n φ n ( s ) ) dW s , φ n ( s ) = sup { t i : t i ≤ s } . t = x + φ n ( s ) ) ds + 0 0 with time step ∆ = T / n and regular points t i = i ∆ , i = 0 , · · · , n . Noufel Frikha Multi-level stochastic approximation algorithms

Introduction Multi-level Monte Carlo method Analysis of the SA scheme Toward Multi-level stochastic approximation algorithms Multi-level stochastic approximation algorithms A short analysis of the different steps Numerical results This step induces a weak error E D ( f , n , T , b , σ ) = E x [ f ( X T )] − E x [ f ( X n T )] ≈ ∆ see Talay & Tubaro (90), Bally & Talay (96), ... T )] by M − 1 × � M ⊲ Step 2 : Estimation of E x [ f ( X n j = 1 f (( X n T ) j ) induces a statistical error : M T )] − 1 � E S ( M , f , n , T , b , σ ) = E x [ f ( X n f (( X n T ) j ) M j = 1 The global error associated to the computation of E x [ f ( X T )] writes : M E Glob ( M , n ) := E x [ f ( X T )] − 1 � f (( X n T ) j ) M j = 1 = E D ( f , n , T , b , σ ) + E S ( M , f , n , T , b , σ ) . Noufel Frikha Multi-level stochastic approximation algorithms

Introduction Multi-level Monte Carlo method Analysis of the SA scheme Toward Multi-level stochastic approximation algorithms Multi-level stochastic approximation algorithms A short analysis of the different steps Numerical results Complexity analysis Optimal complexity : How to balance M w.r.t n to achieve a global error of order ǫ ? ⊲ Duffie & Glynn (95) : If the weak discretization error of order n − α , i.e. ∃ α ∈ ( 0 , 1 ] , n α ( E x [ f ( X T )] − E x [ f ( X n T )]) → C ( α, f , b , σ, T ) , n → + ∞ then,   n 2 α  1 � n α  = f (( X n T ) j ) − E x [ f ( X T )] ⇒ N ( C ( α, f , b , σ, T ) , Var ( f ( X T ))) . n 2 α j = 1 ⊲ It is optimal to set M = n 2 α to achieve an error of order ǫ = n − α : C MC = C × M × n = C × n 2 α + 1 . Noufel Frikha Multi-level stochastic approximation algorithms

Introduction Multi-level Monte Carlo method Analysis of the SA scheme Toward Multi-level stochastic approximation algorithms Multi-level stochastic approximation algorithms A short analysis of the different steps Numerical results Statistical Romberg Monte Carlo scheme ⊲ To reduce the complexity, Kebaier (05) proposed a two-level Monte Carlo scheme to approximate E x [ f ( X T )] by : n γ 1 n γ 2 T M ( γ 1 , γ 2 , β ) := 1 � 1 � ˆ f ((ˆ X n β T ) j ) − f (( X n β T ) j ) + f (( X n T ) j ) n γ 1 n γ 2 T j = 1 j = 1 ((ˆ X n β T ) j , X n β T ) j ) j ∈ [ ] and ((( X n T ) j ) j ∈ [ ] are independent. [ 1 , n γ 1 ] [ 1 , n γ 2 T ] T , X n β ( X n T ) are computed with the same path but with different time steps. ⊲ Main result : If n α ( E x [ f ( X T )] − E x [ f ( X n T )]) → C ( α, f , b , σ, T ) , then n α � � ˆ M ( 2 α, 2 α − β, β ) − E x [ f ( X T )] = ⇒ N ( C ( α, f , b , σ, T ) , Var ( f ( X T )) + Var ( ∇ f ( X T ) U T ⊲ Optimal Complexity to achieve an error of order n − α : 2 , β = 1 C SR − MC = C × ( n β n 2 α + ( n β + n ) n 2 α − β ) ≈ C × n 2 α + 1 2 . Noufel Frikha Multi-level stochastic approximation algorithms

Introduction Multi-level Monte Carlo method Analysis of the SA scheme Toward Multi-level stochastic approximation algorithms Multi-level stochastic approximation algorithms A short analysis of the different steps Numerical results Multi-level Monte Carlo scheme ⊲ Generalizing Kebaier’s approach, Giles (08) proposed a multi-level Monte Carlo scheme to approximate E x [ f ( X T )] by : N 0 L N ℓ M ( n ) := 1 1 � � � f (( X m ℓ T ) j ) − f (( X m ℓ − 1 ˆ f (( X 1 T ) j ) + ) j ) T N 0 N ℓ j = 1 ℓ = 1 j = 1 L + 1 independent empirical mean sequences. Euler schemes with geometric sequence of time steps, m L = n . L L M ( n )) = 1 1 � � Var ( f ( X m ℓ T ) − f ( X m ℓ − 1 Var ( ˆ N − 1 ℓ m − ℓ Var ( f ( X 1 T )) + )) ≤ C T N 0 N ℓ ℓ = 1 ℓ = 0 ⊲ Optimal Complexity to achieve an error of order n − α : C ML-MC = C × n 2 α ( log ( n )) 2 , for N ℓ := 2 c 2 n 2 α ( L + 1 ) T / m ℓ . ⊲ See also the recent work of Kebaier & Ben Alaya (12). Noufel Frikha Multi-level stochastic approximation algorithms

Introduction Multi-level Monte Carlo method Analysis of the SA scheme Toward Multi-level stochastic approximation algorithms Multi-level stochastic approximation algorithms A short analysis of the different steps Numerical results Stochastic approximation algorithm ⊲ Aim : Extend the scope of the ML-MC method to stochastic optimization by means of stochastic approximation (SA). ⊲ Introduced by H.Robbins & S.Monro (1951). It is a recursive simulation-based algorithm to estimate θ ∗ solution of h ( θ ) := E [ H ( θ, U )] = 0 , H : R d × R q → R d , U ∼ µ ⊲ Behind and implicitly assumed : Computation of h is costly compared to the computation of H and to the simulation of U . ⊲ Devise the following scheme p ∈ N , θ 0 ∈ R d θ p + 1 = θ p − γ p + 1 H ( θ p , U p + 1 ) = θ p − γ p + 1 ( h ( θ p ) + ∆ M p + 1 ) � �� � Corrupted observations of h ( θ p ) with ( U p ) p ≥ 1 i.i.d. R q -valued r.v. with law µ and � � γ 2 γ p = + ∞ , p < + ∞ , p ≥ 1 p ≥ 1 to take advantage of an averaging effect along the scheme. Noufel Frikha Multi-level stochastic approximation algorithms

Introduction Multi-level Monte Carlo method Analysis of the SA scheme Toward Multi-level stochastic approximation algorithms Multi-level stochastic approximation algorithms A short analysis of the different steps Numerical results Asymptotic properties of ( θ p ) p ≥ 1 a . s . convergence and convergence rate ⊲ a . s . convergence : Robbins-Monro Theorem mean-reverting assumption ∀ θ ∈ R d , θ � = θ ∗ , � θ − θ ∗ , h ( θ ) � > 0 , domination assumption ∀ θ ∈ R d , | h ( θ ) | 2 ≤ E | H ( θ, U ) | 2 ≤ C ( 1 + | θ − θ ∗ | 2 ) . Then, one has : a . s . → θ ∗ , p → + ∞ . θ p − ⊲ Weak convergence rate : under mild assumptions, in “standard cases”, one has � γ − 1 p ( θ p − θ ∗ ) = ⇒ N ( 0 , Σ ∗ ) , p → + ∞ . Noufel Frikha Multi-level stochastic approximation algorithms

Introduction Multi-level Monte Carlo method Analysis of the SA scheme Toward Multi-level stochastic approximation algorithms Multi-level stochastic approximation algorithms A short analysis of the different steps Numerical results Some applications in computational finance ⊲ In many applications, notably in computational finance, we are interested in estimating the zero θ ∗ of h ( θ ) = E x [ H ( θ, X T )] . ⊲ Some examples among others : Implied volatility : σ ∈ R + s . t . E x [( X T ( σ ) − K ) + ] = P market . Implied correlation between X 1 T and X 2 T : ρ ∈ ( − 1 , 1 ) s . t . E x [( max ( X 1 T , X 2 T ( ρ )) − K ) + ] = P market . VaR and CVaR of a financial portfolio : 1 ( ξ, C ) s . t . P x ( F ( X T ) ≤ ξ ) = α, C = VaR α + 1 − α E x [( F ( X T ) − VaR α ) + ] Portfolio optimization : sup θ ∈ R q E x [ U ( F ( X T ) − θ. ( X T − x ))] . Noufel Frikha Multi-level stochastic approximation algorithms