lara trussardi - a kinetic equation modelling irrationality and ......a kinetic equation modelling...

A kinetic equation modelling irrationality and herding ofagents

Bertram During1 Ansgar Jungel2 Lara Trussardi2

1 University of Sussex, United Kingdom 2 Vienna University of Technology, Austria

Lyon - July 7, 2015

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B.During, A.Jungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 1 / 16

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Index

1 Introduction

2 Main mathematical results

3 Numerical results

4 Outlook


Herding

Herd behavior: a large numberof people acting in the sameway at the same time

Stock market: greed in frenziedbuying (named bubbles) andfear in selling (named crash)


Irrationality and aim

Goal

To describe the evolution of the distribution of the value of a givenproduct (w ∈ R+) in a large market by means of microscopic interactions

among individuals in a society


Model

Background literature:

1 G. Toscani, Kinetic models of opinion formation (2006)

2 M. Levy, H. Levy, S. Solomon, Microscopic simulation of FinancialMarket (2000)

3 M. Delitala, T. Lorenzi, A mathematical model for value estimationwith public information and herding (2014)

The model is based on binary interactions.

It describes two aspects of the opinion formation:I interaction with the public information (rational investor)I effect of herding and imitation phenomena (irrational investor)

We also have a drift term: process which modifies the rationality ofthe agents (x ∈ R).


Public information - microscopic view

Fixed background W which represents the fair asset value.

Interaction rule

w∗ = w − αP(|w −W |)(w −W ) + ηD(|w2|)w −→ •

I−→ w∗

w∗: asset value after exchanging information with the background W

α ∈ (0, 1) mesures the attitude of agents in the market to changetheir 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 σ2I


Herding - microscopic view

Interaction rule

w∗ = w − βγ(w , v)(w − v) + η1D(|w |)

v∗ = v − βγ(w , v)(v − w) + η2D(|w |)

•v

w

v∗

w∗

The function γ describes a socio-economic scenario where the agentsare over-confident in the product.

β ∈ (0, 1/2) mesures the attitude of agents in the market to changetheir mind

D(·) describes the local diffusion for a given value

η1, η2 is a random variable with mean zero and variance σ2H


Boltzmann like collision operator

Let f = f (x ,w , t) : R× R+ × R+ → R: number of individuals withrationality x and asset value w at time t.

1 Collision I (public information):

w∗ = w − αP(|w −W |)(w −W ) + ηD(|w2|)

〈QI (f , f ), φ〉 =∫∫∫

f (x ,w , t)[φ(w∗)− φ(w)]M(W )dxdwdW

2 Collision H (herding):w∗ = w − βγ(w , v)(w − v) + η1D(|w |)v∗ = v − βγ(w , v)(v − w) + η2D(|v |)

〈QH(f , f ), φ〉 =∫∫∫

f (x ,w , t)f (y , v , t)[φ(w∗)− φ(w)]dxdwdv

Both collision operators can be seen as a balance between a gain anda loss of agents with asset value w .


Boltzmann equation

Evolution law of the unknown f = f (x ,w , t):

Boltzmann equation

∂

∂tf (x ,w , t) +

∂

∂x

[Φ(x ,w)f (x ,w , t)

]=

1

τI (x)QI (f , f ) +

1

τH(x)QH(f , f )

where Φ describes how the drift changes with time.

Let R > 0 constant which represents the range within which bubblesand crashes do not occur;

Φ(x ,w) =

−δκ, |w −W | ≤ Rκ, |w −W | > R

κ > 0, δ > 0

Aim

Analysis of moments

Diffusion limit

Numerical experiments


Properties

Recall the Boltzmann equation and the collision kernels:∂∂t f (x ,w , t) + ∂

∂x [Φ(x ,w)f (x ,w , t)] = 1τIQI (f , f ) + 1

τHQH(f , f )

〈QI (f , f ), φ〉 =∫∫∫

f (x ,w , t)[φ(w∗)− φ(w)]M(W )dxdwdW

〈QH(f , f ), φ〉 =∫∫∫

f (x ,w , t)f (y , v , t)[φ(w∗)− φ(w)]dxdwdv

1 The mass is conserved: ‖f (·, ·, t)‖L1(Ω) =∥∥f in∥∥

L1(Ω)for a.e. t ≥ 0.

2 The first moment converges toward 〈W 〉:∫∫

wf dxdw → 〈W 〉.We compute it:

∂t

∫∫wf dxdw +

∫∫Φw∂x f dxdw︸︷︷︸

=0

=1

τI〈QI ,w〉︸︷︷︸

=0

+1

τH〈QH ,w〉︸︷︷︸

=−α〈wf 〉+〈W 〉

3 The second moment converges toward 0:∫∫

w2f dxdw → 0.


Fokker-Planck limit system

Rescale: τ = αt, y = αx =⇒ g(y ,w , τ) = f (x ,w , t)∂

∂tf (x ,w , t)+Φ(x ,w)

∂

∂xf (x ,w , t) =

1

α

1

τIQ(α)

I (f , f )+1

α

1

τHQ(α)

H (f , f )

Compute the limit α→ 0, σ → 0 such that λ = σ2/α

Fokker-Planck equation

∂g

∂t+

∂

∂x[Φg ] = (K[g ]g)w + (H[g ]g)w + (D(w)g)ww

K(w , τ) =

∫R+

γ(v ,w)(w − v)g(v)dv , D(w) > 0

H(w , τ) =

∫R+

(w −W )P(|w −W |)M(W )dW

Difficulties

This equation is: non-linear, non-local, degenerate.


Existence theorem

Theorem

For x ∈ R, w ∈ R+ let consider the problem

gt + Φgx = (K[g ]g)w + (H[g ]g)w + (D(w)g)ww (1)

with g(x ,w , 0) = g0(x ,w) and g |w=0 = 0.Then there exist a weak solution g ∈ L2(0,T ; L2(R× R+)) to (1).

Idea of the proof:

1 reduction on bounded domain QT = Ω× Ω′ × (0,T ) whereΩ× Ω′ ⊂ R× R+

2 approximate elliptic problem: for τ > 0 time discretisation andaddition of the term εgxx

3 Leray-Schauder fixed point theorem

4 estimates for (τ, ε)→ 0

5 diagonal argument on the domain: Ω× Ω′ R× R+


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 therationality x ∈ [−1, 1] and for the asset value w ∈ [0, 1].

The scheme is divided into:I drift: flux-limiters method (Lax-Wendroff scheme and up-wind scheme)I collision with the public source & herding collision: slightly modified

Bird method

Goal1 To check the analytical results regarding the asymptotic analysis

2 To understand the role of the parameters in the formation of bubblesand crashes


Moments

Let fix W = 0.3.The mass:

ρ =

∫R

∫R+

f indxdw is conserved

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

time

First moment w “converges” to〈W 〉 = 0.3

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.02

0.04

0.06

0.08

0.1

time

Second moment w “converges” to 0


Role of α

w∗ = w − αP(|w −W |)(w −W ) + ηD(|w2|)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

alpha

% b

ubble

beta=0.5

beta=0.05

beta=0.005

0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

14

16

18

alpha%

cra

sh

beta=0.5

beta=0.05

beta=0.005


Role of α

w∗ = w − αP(|w −W |)(w −W ) + ηD(|w2|)

500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time

α = 0.05

500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time

α = 0.5


Outlook

Summary:I importance of the reliability of public informationI herding promotes occurence of bubbles and crashes: may lead to

strong trends with low volatility of asset prices, but eventually also toabrupt corrections.

Furthes studiesI Fokker-Planck simulationI Investigate all the parameters and try to understand better their role

(counter action for herding)

Thanks for your attention

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lara trussardi - a kinetic equation modelling irrationality and ......a kinetic equation modelling...

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