Lara Trussardi - A kinetic equation modelling irrationality and ......A kinetic equation modelling...
Transcript of 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
www.itn-strike.eu
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Index
1 Introduction
2 Main mathematical results
3 Numerical results
4 Outlook
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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)
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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)
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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
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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
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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).
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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
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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
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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 .
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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
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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.
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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.
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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+
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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
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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
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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
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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
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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
www.itn-strike.eu
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