Reduced Basis Methods Karsten Urban for (some particular) Ulm - PowerPoint PPT Presentation

Reduced Basis Methods Karsten Urban for (some particular) Ulm University (Germany) HJB equations Institute for Numerical Mathematics

page 1/27 RBM for HJB | RICAM 2016 | Karsten Urban | Acknowledgements Acknowledgements ◮ joint work with ◮ R¨ udiger Kiesel (Duisburg-Essen) ◮ Silke Glas, Sebastian Steck (Ulm) ◮ Funding: ◮ Deutsche Forschungsgemeinschaft (DFG: GrK1100, Ur-63/9, SPP1324) ◮ Federal Ministry of Economy (BMWT)

page 2/27 RBM for HJB | RICAM 2016 | Karsten Urban | Acknowledgements Outline 1 “Particular”HJB: The EU-ETS (A very short) Introduction to RBM 2 RBM for the EU-ETS-HJB 3 Conclusions and outlook 4

page 3/27 RBM for HJB | RICAM 2016 | Karsten Urban | “Particular” HJB: The EU-ETS 1 “Particular”HJB: The EU-ETS (A very short) Introduction to RBM 2 RBM for the EU-ETS-HJB 3 Conclusions and outlook 4

page 4/27 RBM for HJB | RICAM 2016 | Karsten Urban | “Particular” HJB: The EU-ETS EU-ETS: European Union Emission Trading System ◮ anthropogenic global warming ◮ Kyoto protocol: limit of CO 2 -emissions ◮ according amount of emission permits are issued

page 4/27 RBM for HJB | RICAM 2016 | Karsten Urban | “Particular” HJB: The EU-ETS EU-ETS: European Union Emission Trading System ◮ anthropogenic global warming ◮ Kyoto protocol: limit of CO 2 -emissions ◮ according amount of emission permits are issued ◮ permits are traded at the exchange: EU-ETS ◮ penalty for emissions not covered by permits

page 4/27 RBM for HJB | RICAM 2016 | Karsten Urban | “Particular” HJB: The EU-ETS EU-ETS: European Union Emission Trading System ◮ anthropogenic global warming ◮ Kyoto protocol: limit of CO 2 -emissions ◮ according amount of emission permits are issued ◮ permits are traded at the exchange: EU-ETS ◮ penalty for emissions not covered by permits ◮ goal here: public control of EU-ETS: abate 5% of emissions

page 5/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System Modeling EU-ETS ◮ trading periods: [0 , T ] ◮ equlibirium ≡ sum of costs of all market participants is minimal [1] [1] Camora, Fehr, Hinz: Optimal Stochastic Control and Carbon Price Formation, 2009

page 5/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System Modeling EU-ETS ◮ trading periods: [0 , T ] ◮ equlibirium ≡ sum of costs of all market participants is minimal [1] ◮ state Y τ ∈ R d , τ ∈ [0 , T ]: amount of uncovered emissions ( d : # companies) [1] Camora, Fehr, Hinz: Optimal Stochastic Control and Carbon Price Formation, 2009

page 5/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System Modeling EU-ETS ◮ trading periods: [0 , T ] ◮ equlibirium ≡ sum of costs of all market participants is minimal [1] ◮ state Y τ ∈ R d , τ ∈ [0 , T ]: amount of uncovered emissions ( d : # companies) ◮ control π τ ∈ R d : additional abatement compared to business as usual [1] Camora, Fehr, Hinz: Optimal Stochastic Control and Carbon Price Formation, 2009

page 5/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System Modeling EU-ETS ◮ trading periods: [0 , T ] ◮ equlibirium ≡ sum of costs of all market participants is minimal [1] ◮ state Y τ ∈ R d , τ ∈ [0 , T ]: amount of uncovered emissions ( d : # companies) ◮ control π τ ∈ R d : additional abatement compared to business as usual ◮ optimal abatement strategy π = ( π τ ) τ ∈ [0 , T ] : should minimize the expected abatement costs (cost functional) �� T � f π ( τ, Y τ ) d τ + h ( Y T ) J ( π ) := E 0 ◮ f π : running abatement cost using strategy π ◮ h : penalty at the end of the trading period [1] Camora, Fehr, Hinz: Optimal Stochastic Control and Carbon Price Formation, 2009

page 5/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System Modeling EU-ETS ◮ trading periods: [0 , T ] ◮ equlibirium ≡ sum of costs of all market participants is minimal [1] ◮ state Y τ ∈ R d , τ ∈ [0 , T ]: amount of uncovered emissions ( d : # companies) ◮ control π τ ∈ R d : additional abatement compared to business as usual ◮ optimal abatement strategy π = ( π τ ) τ ∈ [0 , T ] : should minimize the expected abatement costs (cost functional) �� T � f π ( τ, Y τ ) d τ + h ( Y T ) J ( π ) := E 0 ◮ f π : running abatement cost using strategy π ◮ h : penalty at the end of the trading period ◮ stochastic model for Y τ : dY τ = b π ( τ, Y τ ) d τ + σ π ( τ, Y τ ) dW τ , τ ∈ (0 , T ] , Y 0 = y 0 ◮ W τ : a d -dimensional Wiener process ◮ b π , σ π : drift and volatility coefficients, b π , σ π ( σ π ) T linear in π . [1] Camora, Fehr, Hinz: Optimal Stochastic Control and Carbon Price Formation, 2009

page 5/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System Modeling EU-ETS ◮ trading periods: [0 , T ] ◮ equlibirium ≡ sum of costs of all market participants is minimal [1] ◮ state Y τ ∈ R d , τ ∈ [0 , T ]: amount of uncovered emissions ( d : # companies) ◮ control π τ ∈ R d : additional abatement compared to business as usual ◮ optimal abatement strategy π = ( π τ ) τ ∈ [0 , T ] : should minimize the expected abatement costs (cost functional) �� T � f γ ( τ, Y τ ) d τ + h ( Y T ) J ( t , x ; γ ) := E t ◮ f π : running abatement cost using strategy π ◮ h : penalty at the end of the trading period ◮ stochastic model for Y τ : dY τ = b π ( τ, Y τ ) d τ + σ π ( τ, Y τ ) dW τ , τ ∈ ( t , T ] , Y t = x ◮ W τ : a d -dimensional Wiener process ◮ b π , σ π : drift and volatility coefficients, b π , σ π ( σ π ) T linear in π . [1] Camora, Fehr, Hinz: Optimal Stochastic Control and Carbon Price Formation, 2009

page 6/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System (Parameterized) Hamilton Jacobi Bellman Equation 1/3 ◮ value function ( x ∈ R d ) u ( t , x ) = inf γ ∈ Γ J ( t , x ; γ ) ∀ t ∈ [0 , T ) , u ( T , x ) = h ( x ) ◮ Γ ⊂ L ∞ ((0 , T ) × R d ; R d ): set of admissible controls [2] Yong, Zhou: Stochastic controls: Hamiltonian systems and HJB equations, 1999

page 6/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System (Parameterized) Hamilton Jacobi Bellman Equation 1/3 ◮ value function ( x ∈ R d ) u ( t , x ) = inf γ ∈ Γ J ( t , x ; γ ) ∀ t ∈ [0 , T ) , u ( T , x ) = h ( x ) ◮ Γ ⊂ L ∞ ((0 , T ) × R d ; R d ): set of admissible controls ◮ HJB [2] � 1 2tr( σ γ ( σ γ ) T ∇ 2 u ( t , x )) + b γ · ∇ u ( t , x ) − f γ ( t , x ) � ∂ t u ( t , x ) + sup = 0 γ ∈ Γ [2] Yong, Zhou: Stochastic controls: Hamiltonian systems and HJB equations, 1999

page 6/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System (Parameterized) Hamilton Jacobi Bellman Equation 1/3 ◮ value function ( x ∈ R d ) u ( t , x ) = inf γ ∈ Γ J ( t , x ; γ ) ∀ t ∈ [0 , T ) , u ( T , x ) = h ( x ) ◮ Γ ⊂ L ∞ ((0 , T ) × R d ; R d ): set of admissible controls ◮ HJB [2] � 1 2tr( σ γ ( σ γ ) T ∇ 2 u ( t , x )) + b γ · ∇ u ( t , x ) − f γ ( t , x ) � ∂ t u ( t , x ) + sup = 0 γ ∈ Γ ◮ parameters µ ∈ D ⊂ R P e.g. regulatory constraints, market values, etc. f γ ( µ ), b γ ( µ ), σ γ ( µ ), J ( µ ; t , x ; γ ) � u ( µ ) = u ( µ ; t , x ) [2] Yong, Zhou: Stochastic controls: Hamiltonian systems and HJB equations, 1999

page 6/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System (Parameterized) Hamilton Jacobi Bellman Equation 1/3 ◮ value function ( x ∈ R d ) u ( t , x ) = inf γ ∈ Γ J ( t , x ; γ ) ∀ t ∈ [0 , T ) , u ( T , x ) = h ( x ) ◮ Γ ⊂ L ∞ ((0 , T ) × R d ; R d ): set of admissible controls ◮ HJB [2] � 1 2tr( σ γ ( σ γ ) T ∇ 2 u ( t , x )) + b γ · ∇ u ( t , x ) − f γ ( t , x ) � ∂ t u ( t , x ) + sup = 0 γ ∈ Γ ◮ parameters µ ∈ D ⊂ R P e.g. regulatory constraints, market values, etc. f γ ( µ ), b γ ( µ ), σ γ ( µ ), J ( µ ; t , x ; γ ) � u ( µ ) = u ( µ ; t , x ) ◮ Goal: find“optimal”parameters (also in realtime) [2] Yong, Zhou: Stochastic controls: Hamiltonian systems and HJB equations, 1999

page 7/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System (Parameterized) Hamilton Jacobi Bellman Equation 2/3 ◮ parameterized coefficients: u �→ A γ ( µ ; u ) := − a γ ( µ ) ∆ u + b γ ( µ ) · ∇ u + c γ ( µ ) u , µ ∈ D .

page 7/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System (Parameterized) Hamilton Jacobi Bellman Equation 2/3 ◮ parameterized coefficients: u �→ A γ ( µ ; u ) := − a γ ( µ ) ∆ u + b γ ( µ ) · ∇ u + c γ ( µ ) u , µ ∈ D . ◮ parameterized Hamilton-type operator H ( µ ; u ) := sup { A γ ( µ ; u ) − f γ ( µ ) } γ ∈ Γ

page 7/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System (Parameterized) Hamilton Jacobi Bellman Equation 2/3 ◮ parameterized coefficients: u �→ A γ ( µ ; u ) := − a γ ( µ ) ∆ u + b γ ( µ ) · ∇ u + c γ ( µ ) u , µ ∈ D . ◮ parameterized Hamilton-type operator H ( µ ; u ) := sup { A γ ( µ ; u ) − f γ ( µ ) } γ ∈ Γ ◮ P-HJB ∂ t u + H ( µ ; u ) = 0 , in Ω T , (1a) ∂ ∂ nu = ψ, on ∂ Ω T = (0 , T ) × ∂ Ω , (1b) on ¯ u ( T ) = u T , Ω , (1c)