A kinetic equation modelling irrationality and herding of agents - PowerPoint PPT Presentation

A kinetic equation modelling irrationality and herding of agents uring 1 ungel 2 Lara Trussardi 2 Bertram D¨ Ansgar J¨ 1 University of Sussex, United Kingdom 2 Vienna University of Technology, Austria Lyon - July 7, 2015 www.itn-strike.eu B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 1 / 16

Index Introduction 1 Main mathematical results 2 Numerical results 3 Outlook 4 B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 2 / 16

Herding Herd behavior: a large number Stock market: greed in frenzied of people acting in the same buying (named bubbles) and way at the same time fear in selling (named crash) B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 3 / 16

Herding Herd behavior: a large number Stock market: greed in frenzied of people acting in the same buying (named bubbles) and way at the same time fear in selling (named crash) B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 3 / 16

Irrationality and aim B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 4 / 16

Irrationality and aim Goal To describe the evolution of the distribution of the value of a given product ( w ∈ R + ) in a large market by means of microscopic interactions among individuals in a society B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 4 / 16

Model Background literature: 1 G. Toscani, Kinetic models of opinion formation (2006) 2 M. Levy, H. Levy, S. Solomon, Microscopic simulation of Financial Market (2000) 3 M. Delitala, T. Lorenzi, A mathematical model for value estimation with public information and herding (2014) The model is based on binary interactions. It describes two aspects of the opinion formation: ◮ interaction with the public information (rational investor) ◮ effect of herding and imitation phenomena (irrational investor) We also have a drift term: process which modifies the rationality of the agents ( x ∈ R ). B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 5 / 16

Public information - microscopic view Fixed background W which represents the fair asset value. Interaction rule → w ∗ w − → • I − w ∗ = w − α P ( | w − W | )( w − W ) + η D ( | w 2 | ) w ∗ : asset value after exchanging information with the background W α ∈ (0 , 1) mesures the attitude of agents in the market to change their mind P ( · ) describes the local relevance of the compromise D ( · ) describes the local diffusion for a given value η is a random variable with mean zero and variance σ 2 I B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 6 / 16

Herding - microscopic view Interaction rule w ∗ w w ∗ = w − βγ ( w , v )( w − v ) + η 1 D ( | w | ) • v ∗ = v − βγ ( w , v )( v − w ) + η 2 D ( | w | ) v v ∗ The function γ describes a socio-economic scenario where the agents are over-confident in the product. β ∈ (0 , 1 / 2) mesures the attitude of agents in the market to change their mind D ( · ) describes the local diffusion for a given value η 1 , η 2 is a random variable with mean zero and variance σ 2 H B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 7 / 16

Boltzmann like collision operator Let f = f ( x , w , t ) : R × R + × R + → R : number of individuals with rationality x and asset value w at time t . Collision I (public information): 1 w ∗ = w − α P ( | w − W | )( w − W ) + η D ( | w 2 | ) ��� � Q I ( f , f ) , φ � = f ( x , w , t )[ φ ( w ∗ ) − φ ( w )] M ( W )d x d w d W Collision H (herding): 2 � w ∗ = w − βγ ( w , v )( w − v ) + η 1 D ( | w | ) v ∗ = v − βγ ( w , v )( v − w ) + η 2 D ( | v | ) ��� f ( x , w , t ) f ( y , v , t )[ φ ( w ∗ ) − φ ( w )]d x d w d v � Q H ( f , f ) , φ � = Both collision operators can be seen as a balance between a gain and a loss of agents with asset value w . B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 8 / 16

Boltzmann equation Evolution law of the unknown f = f ( x , w , t ): Boltzmann equation ∂ t f ( x , w , t ) + ∂ ∂ � � 1 1 Φ( x , w ) f ( x , w , t ) = τ I ( x ) Q I ( f , f ) + τ H ( x ) Q H ( f , f ) ∂ x where Φ describes how the drift changes with time. Let R > 0 constant which represents the range within which bubbles and crashes do not occur; � − δκ, Aim | w − W | ≤ R Φ( x , w ) = Analysis of moments κ, | w − W | > R Diffusion limit κ > 0 , δ > 0 Numerical experiments B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 9 / 16

Properties Recall the Boltzmann equation and the collision kernels: ∂ t f ( x , w , t ) + ∂ ∂ ∂ x [Φ( x , w ) f ( x , w , t )] = 1 τ I Q I ( f , f ) + 1 τ H Q H ( f , f ) ��� f ( x , w , t )[ φ ( w ∗ ) − φ ( w )] M ( W )d x d w d W � Q I ( f , f ) , φ � = ��� f ( x , w , t ) f ( y , v , t )[ φ ( w ∗ ) − φ ( w )]d x d w d v � Q H ( f , f ) , φ � = � f in � � 1 The mass is conserved: � f ( · , · , t ) � L 1 (Ω) = L 1 (Ω) for a.e. t ≥ 0. � �� 2 The first moment converges toward � W � : wf d x d w → � W � . We compute it: �� �� = 1 + 1 ∂ t wf d x d w + Φ w ∂ x f d x d w �Q I , w � �Q H , w � τ I τ H � �� � � �� � � �� � =0 = − α � wf � + � W � =0 �� 3 The second moment converges toward 0: w 2 f d x d w → 0. B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 10 / 16

Fokker-Planck limit system Rescale: τ = α t , y = α x = ⇒ g ( y , w , τ ) = f ( x , w , t ) ∂ t f ( x , w , t )+Φ( x , w ) ∂ ∂ ∂ x f ( x , w , t ) = 1 1 ( f , f )+ 1 1 Q ( α ) Q ( α ) H ( f , f ) I α τ I α τ H Compute the limit α → 0, σ → 0 such that λ = σ 2 /α Fokker-Planck equation ∂ g ∂ t + ∂ ∂ x [Φ g ] = ( K [ g ] g ) w + ( H [ g ] g ) w + ( D ( w ) g ) ww � K ( w , τ ) = γ ( v , w )( w − v ) g ( v )d v , D ( w ) > 0 R + � H ( w , τ ) = ( w − W ) P ( | w − W | ) M ( W )d W R + Difficulties This equation is: non-linear, non-local, degenerate. B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 11 / 16

Existence theorem Theorem For x ∈ R , w ∈ R + let consider the problem g t + Φ g x = ( K [ g ] g ) w + ( H [ g ] g ) w + ( D ( w ) g ) ww (1) with g ( x , w , 0) = g 0 ( x , w ) and g | w =0 = 0. Then there exist a weak solution g ∈ L 2 (0 , T ; L 2 ( R × R + )) to (1). Idea of the proof: 1 reduction on bounded domain Q T = Ω × Ω ′ × (0 , T ) where Ω × Ω ′ ⊂ R × R + 2 approximate elliptic problem: for τ > 0 time discretisation and addition of the term ε g xx 3 Leray-Schauder fixed point theorem 4 estimates for ( τ, ε ) → 0 5 diagonal argument on the domain: Ω × Ω ′ � R × R + B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 12 / 16

Numerical model We implement the Boltzmann equation for f ( x , w , t ) (2D-model). We need to reduce the model to a bounded domain: for the rationality x ∈ [ − 1 , 1] and for the asset value w ∈ [0 , 1]. The scheme is divided into: ◮ drift: flux-limiters method (Lax-Wendroff scheme and up-wind scheme) ◮ collision with the public source & herding collision: slightly modified Bird method Goal 1 To check the analytical results regarding the asymptotic analysis 2 To understand the role of the parameters in the formation of bubbles and crashes B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 13 / 16

Moments Let fix W = 0 . 3. The mass: � � f in d x d w ρ = is conserved R R + 1 0.1 0.8 0.08 0.6 0.06 0.4 0.04 0.2 0.02 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 time time First moment w “converges” to Second moment w “converges” to 0 � W � = 0 . 3 B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 14 / 16

Role of α w ∗ = w − α P ( | w − W | )( w − W ) + η D ( | w 2 | ) 45 18 40 16 beta=0.5 beta=0.5 35 beta=0.05 14 beta=0.05 beta=0.005 beta=0.005 30 12 % bubble % crash 25 10 20 8 15 6 10 4 5 2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 alpha alpha B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 15 / 16

Role of α w ∗ = w − α P ( | w − W | )( w − W ) + η D ( | w 2 | ) 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 time time α = 0 . 05 α = 0 . 5 B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 15 / 16