A Branching Process Approach to Power Markets Simone Scotti - PowerPoint PPT Presentation

A Branching Process Approach to Power Markets A Branching Process Approach to Power Markets Simone Scotti Universit´ e Paris-Diderot Joint work with : Ying Jiao , ISFA, University of Lyon Chunhua Ma , Nankai University Carlo Sgarra , Politecnico di Milano S´ eminaire chaire FDD et IdR FiME Paris, 16 juin 2017 Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets Self-Exciting structure F IGURE : Benth et al. A critical empirical study of three electricity spot price models . Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets Power Price Modeling We will assume then the spot price process S ( t ) to evolve according to the basic dynamics : S ( t ) = B ( t ) + Y ( t ) , where B ( t ) is a seasonality function of deterministic type and the process Y ( t ) is a superposition of the factors X i ( t ) : � Y ( t ) = X i ( t ) , The main objective is to propose new candidates for the evolution of the factors X including self-exciting structure. We propose to look at the class of continuous state branching processes with immigration. Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets Branching Processes in continuous time Branching property Branching property : A process X has the Branching Property if for any t and x , y in the state space of X , X x + y is t t and X y equal in law to the independent sum of X x t . If a process X can be decomposed as X = X ( 1 ) + X ( 2 ) where for i = 1 , 2, X ( i ) satisfying the same SDE with X 0 = X ( 1 ) + X ( 2 ) , then the process is said a branching process . 0 0 We have the following result, see Kawazu and Watanabe (1971). Generator Markov process X with state space R + with Branching mechanism : � ∞ Ψ( q ) = β q + 1 2 σ 2 q 2 + ( e − q ζ − 1 + q ζ ) π ( d ζ ) , 0 � ∞ ( ζ ∧ ζ 2 ) π ( d ζ ) < ∞ . with σ ≥ 0, β ∈ R and π being a L´ evy measure such that 0 The CBI process X has as generator the operator L acting on C 2 0 ( R + ) as � ∞ L f ( x ) = σ 2 � � 2 x f ′′ ( x ) − β x f ′ ( x ) + x f ( x + ζ ) − f ( x ) − ζ f ′ ( x ) π ( d ζ ) . 0 Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets Branching Processes in continuous time Dawson Li (2006) representation Integral representation The previous generator admits the following semigroup (Hille-Yosida theorem). � t � X s � t � X s X t = − a du ds + σ W ( ds , du ) 0 0 0 0 � t � X s − � R + ζ � + N ( ds , du , d ζ ) , 0 0 W ( ds , du ) : white noise on R 2 + with intensity ds dv , � N ( ds , du , d ζ ) : compensated Poisson random measure on R 3 + with intensity ds du π ( d ζ ) , Besides, W and N are independent of each other. Main problem : the process converges to 0 if a > 0 or to ∞ otherwise. As a consequence it is not ergodic. Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets Branching Processes in continuous time Continuous state branching process with immigration Integral representation � t � t � X s X t = a ( b − X s ) ds + σ W ( ds , du ) 0 0 0 � t � X s − � � t � R + ζ � + γ N ( ds , du , d ζ ) + γ R + ζ M ( ds , d ζ ) , 0 0 0 M ( ds , d ζ ) : compensated Poisson random measure on R 2 + with intensity ds π ( d ζ ) , The process will be exponential ergodic if a > 0. � ∞ a ( b − x ) f ′ ( x ) σ 2 � � 2 x f ′′ ( x ) + x f ( x + ζ ) − f ( x ) − ζ f ′ ( x ) L f ( x ) = π ( d ζ ) 0 � ∞ � � + f ( x + ζ ) − f ( x ) π ( d ζ ) 0 Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets Branching Processes in continuous time Continuous state branching process with immigration (CBI) CBI (Kawazu & Watanabe 1971) of branching mechanism Ψ( · ) and immigration rate Φ( · ) : Markov process X with state space R + verifying � � t � E x � e − p X t � � � = exp − x v ( t , p ) − Φ v ( s , p ) , ds 0 where v : R + × R + → R satisfies ∂ v ( t , p ) = − Ψ( v ( t , p )) , v ( 0 , p ) = p ∂ t , and Ψ and Φ are functions on R + given by � ∞ Ψ( q ) = a q + 1 2 σ 2 q 2 + ( e − q ζ − 1 + q ζ ) π ( d ζ ) , 0 � ∞ ( 1 − e − q ζ ) π ( d ζ ) , Φ( q ) = ab q + 0 � ∞ ( ζ ∧ ζ 2 ) π ( d ζ ) < ∞ with σ, ab ≥ 0, a ∈ R and π , π being two L´ evy measures such that � ∞ 0 and ( 1 ∧ ζ ) π ( d ζ ) < ∞ . 0 Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets Branching Processes in continuous time Link to Hawkes process When σ = γ = 0 and π ( d ζ ) = δ 1 ( d ζ ) , then X is given by � t � t � X s − � � a + π ( R + ) X t = X 0 − X s ds + N ( ds , du ) (1) 0 0 0 � t � X s − which is the intensity of Hawkes process N ( ds , du ) , N being the Poisson random 0 0 measure with intensity ds du . � � X ( n ) Consider a sequence , t ≥ 0 defined by (1) with parameters ( a / n , nb ) . Then t X ( n ) L nt / n − → Y t in D ( R + ) , where D ( R + ) is the Skorokhod space of c` adl` ag processes and � t � t � Y s Y t = a ( b − Y s ) ds + W ( ds , du ) . 0 0 0 See Jiao et al. (2016). Jaisson and Rosenbaum (2015) : nearly unstable Hawkes process converges, after suitable scaling, to a CIR process. Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets α -CIR model The α -CIR model setup : Integral representation (Dawson-Li) Integral form by using the random fields � t � t � X s X t = X 0 + a ( b − X s ) ds + σ W ( ds , du ) 0 0 0 (2) � t � X s − � R + ζ � + σ Z N ( ds , du , d ζ ) , 0 0 W ( ds , du ) : white noise on R 2 + with intensity dsdu , � N ( ds , du , d ζ ) : compensated Poisson random measure on R 3 + with intensity ds du µ ( d ζ ) , � ∞ ( ζ ∧ ζ 2 ) µ ( d ζ ) < ∞ . µ ( d ζ ) : a L´ evy measure satisfying 0 We choose the L´ evy measure to be 1 { ζ> 0 } d ζ µ ( d ζ ) = − cos ( πα/ 2 ) Γ( − α ) ζ 1 + α , 1 < α < 2 , For existence and uniqueness of the solution see Dawson and Li (2012), Theorem 3.1 and Li and Ma (2015) Theorem 2.1. Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets α -CIR model The α -CIR model setup We consider the following usual SDE � t � t � t √ X 1 /α X t = X 0 + a ( b − X s ) ds + σ X s dB s + σ Z s − dZ s (3) 0 0 0 B = ( B t , t ≥ 0 ) a Brownian motion Z = ( Z t , t ≥ 0 ) a spectrally positive α -stable compensate L´ evy process with parameter α ∈ ( 1 , 2 ] with � � t q α � e − q Z t � E = exp − , q ≥ 0 . cos ( π α/ 2 ) B and Z are independent. Z t follows the α -stable distribution S α ( t 1 /α , 1 , 0 ) with scale parameter t 1 /α , skewness parameter 1 and zero drift. The existence of a unique strong solution for the SDE (3) follows from Fu and Li (Theorem 5.3, 2010). Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets α -CIR model Simulation of processes Z and X with different α α −stable process Z: r 0 =0.1, a=0.1, b=0.3, σ =0.1, σ Z =0.3 α −CIR process: r 0 =0.1, a=0.1, b=0.3, σ =0.1, σ Z =0.3 4 0.35 α =2 α =2 α =1.5 α =1.5 3 0.3 α =1.2 α =1.2 2 0.25 1 0.2 Z t r t 0 0.15 −1 0.1 −2 0.05 −3 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 t t F IGURE : Three parameters of α : 2 (blue), 1.5 (green) and 1.2 (black). Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets α -CIR model Similar properties with CIR model I Boundary condition : The point 0 is an inaccessible boundary if and only if 2 a b ≥ σ 2 . In particular, a pure jump α -CIR process with ab > 0 never reaches 0 since σ = 0. Ergodic law : The process is exponentially ergodic, the limit distribution denoted by r ∞ satisfies   � p   � e − pX ∞ � abq = exp  −  . E cos ( πα/ 2 ) q α dq σ α aq + σ 2 2 q 2 − Z 0 Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets α -CIR model Similar properties with CIR model II Branching property : r can be decomposed as X = X ( 1 ) + X ( 2 ) where for i = 1 , 2, X ( i ) is an α -CIR ( a , b ( i ) , σ, σ Z , α ) process such that X 0 = X ( 1 ) + X ( 2 ) and b = b ( 1 ) + b ( 2 ) . 0 0 See Dawson and Li (2006). This property is a direct consequence of linearity of integrals, homogeneity of measures. Simone Scotti (Paris Diderot) Paris, 16 juin 2017

A Branching Process Approach to Power Markets α -CIR model Equivalence of two representations Then the root representation (3) and the integral representation (2) are equivalent in the following sense : The solutions of the two equations have the same probability law. On an extended probability space, they are equal almost surely. See Theorem 9.32 in Li (2011). The equivalence is useful since we have two ways to study the properties of the model. Simone Scotti (Paris Diderot) Paris, 16 juin 2017