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Title: Chaos Modelling and Applications in Financial Engineering


1
2010????????????????? Chaos Modelling and
Applications in Financial Engineering ????????????
????????? ??? ?? ????
2
Chaos Modelling and Applications in Financial
Engineering
ZENGQIANG CHEN
Department of Automation Nankai University Email
chenzq_at_nankai.edu.cn
3
?? ??????????? ??????????? ????? ?????????????
4
Outline
  • Introduction to Chaos
  • Topological Horseshoe Theory
  • Chaos in Economics
  • The Analysis of two Economic Systems

5
Introduction to Chaos

What is Chaos?
Chaos exists in nonlinear dynamical systems
6
Introduction to Chaos

Basic properties of Chaos
  • sensitive dependence on initial conditions

.506127
.506
1961, Lorenzs experiment of weather prediction
7
Introduction to Chaos
  • The trajectory is bounded
  • and never repeats
  • Self-similar

8
Introduction to Chaos
  • Unpredictability

Chaos is aperiodic long-term behavior in a
deterministic system that exhibits sensitive
dependence on initial conditions 1.   1
Nonlinear Dynamics and chaos, Strogatz, S. H.,
Addison-Wesley Publishing Company, Boston, 1994.
9
Introduction to Chaos

Classical Chaotic attractors
Lorenz attractor
10
Introduction to Chaos

Classical Chaotic attractors
Rössler attractor
11
Introduction to Chaos

Classical Chaotic attractors
Chen attractor
12
Introduction to Chaos

How to determine chaos
  • Lyapunov exponents
  • Topological entropy
  • Bifurcation, such as period-doubling route to
    chaos
  • Melnikov method
  • Šilnikov method
  • Topological horseshoe theory and symbolic
    dynamics

13
Outline
  • Introduction to Chaos
  • Topological Horseshoe Theory
  • Chaos in Economics
  • The Analysis of two Economic Systems

14
Topological Horseshoe Theory
  • Smale horseshoe map pioneering work

Smale horseshoe map is the prototypical map
possessing a chaotic invariant set
Theorem. There is a closed invariant set
for which is conjugate to a
two-sided 2-shift.
15
Topological Horseshoe Theory
  • Topological horseshoe J. Kennedy and J.A.
    Yorkes work 2

Assumptions (1) is a separable
metric space (2) is locally
connected and compact (3) The map
is continuous (4) The set
and are disjoint and
compact, and each component of
intersects both and (5)
has crossing number
Theorem. There is a closed invariant set
for which is semi-conjugate to a
one-sided M-shift. (If f is homeomorphism, then
is two-sided).
Remark. (1) This theorem applies to invariant
set with only one expanding direction
(2) Core concept , The crossing number,
is useless in practical point of view
2 J. Kenney, and J. A. Yorke, Topological
horseshoes, Trans. Amer. Math Soc., vol. 353,
pp. 2513-2530, Feb. 2001.
16
Topological Horseshoe Theory
  • Topological horseshoe Yang Xiao-Songs work

Proposing a recent famous topological horseshoe
theorem
  • Applicable for continuous system, piecewise
    continuous system,
  • discrete system
  • Applicable for invariant set with multiple
    expanding direction
  • Combines with computer numerical simulations

17
Topological Horseshoe Theory
  • Topological horseshoe Yang Xiao-Songs work
    3

Definition f-connected family
Let be a metric space, is a compact
subset of , and is map
satisfying the assumption that there exist m
mutually disjoint subsets and
of , the restriction of to each
, i.e., is continuous.
Definition. Let be a compact subset of
, such that for each
is nonempty and compact, then is
called a connection with respect to
. Let be a family of connections
s with respect to
satisfying the following property Then
is said to be a -connected family with
respect to .
3 X. S. Yang, and Y. Tang, Horseshoe in
piecewise continuous maps, Chaos Solitons
Fractals, vol. 19, pp. 841845, Apr. 2004.
18
Topological Horseshoe Theory
  • Theorem

Theorem. Suppose that there exists a f-connected
family with respect to and
. Then there exists a compact invariant set
, such that is semi-conjugate to
m-shift.
Topological entropylogm
Remark. The semi-conjugacy is defined as follows.
If there exists a continuous and onto map Such
that , then is
said to be semi-conjugate to .

An important fact is the following statement.
Lemma. Consider two dynamical systems
and . If is
semi-conjugate to , then the
topological entropy of is not less than
that of , i.e.

.
19
Topological Horseshoe Theory
  • Important Comment

(1)
Topological horseshoe theorem
Continuous time system
Poincaré Map
Topological horseshoe theorem the
computer-assisted computation
  • more applicable, can be applied to many systems
  • provides a geometrical method to find the
    topological horseshoe

(2) Every statement about existence of
horseshoe can tolerate some fixed
bounded errors , because of inevitable of errors
in computer computation
20
Topological Horseshoe Theory
  • Steps for applying the Theorem

-- Continuous case
  • Construct Poincaré cross-section and the proper
    Poincaré map
  • Find an invariant set, such that the Poincaré map
    is semi-conjugate
  • to a m-shift map.

-- Discrete case
Find a proper map which is semi-conjugate to a
m-shift map.
21
????????
????
? ????????????????????????????
? ??,??????,???????????Šilnikov?????????????????
??
? ???????????????????????????????????????????????
????
22
????????
  • ?????????????

(1) ?????Smale????4
Smale??????,???????,?????? ??????,??????,?????,???
??
(3) Zgliczynski?Gidead???????6
??????????????????,??????
4 Wiggins S. New York Springer-Verlag,
1990 5 Kennedy J, Yorke J. Tran. Amer. Manth.
Soc., 2001, 353 25132530 6 Zgliczynski P,
Gidea M. J. Differential Equations, 2004, 202
3258
23
3.1????????
  • Yang?????????7

???????????????? ???????,(??)?????????,?????
????,?????
7 Yang X S, Tang Y. Chaos Solitons Fractals,
2004, 19(4) 841845
24
????????
  • ??????????????

??????
???? Rössler??????Chen???Lorenz???
Hopfield?????
????? ????????,?????????????????
25
Outline
  • Introduction to Chaos
  • Topological Horseshoe Theory
  • Chaos in Economics
  • The Analysis of two Economic Systems

26
Chaos in Economics

Chaotic economics (Nonlinear economics)
Day is among the pioneers of chaotic research in
economics as this field was becoming
increasingly popular in the early 1980s.
  • Ref. 4 Wandering growth cycles Chaos emerge
  • Nowadays, chaotic economics includes almost
    every fields of economics
  • Economic cycle, Monetary, Finance, Stock
    market, Firm supply and demand

4 Day, R., Irregular Growth Cycles, American
Economic Review, 72, 406-414, 1982.
27
Chaos in Economics
  • Topics on chaotic economics
  • Investigating real economic data to find
    evidence of chaos
  • Analyzing nonlinear dynamics of some economic
    behaviors
  • Explaining the intrinsic mechanism and reasons
    of economic behaviors
  • Predicting economic behavior
  • Modeling and analyzing economic behavior

28
Chaos in Economics

Istanbul stock exchange 5
ISE system has very high chaotic phenomena
  • Phase space reconstruction The embedding
    dimension of ISE time series is very high,

  • and the strange attractor dimension is 0.15.

Time series of ISE index
3D phase space of ISE time series
5 Muge Iseri,Hikmet Caglar, Nazan Caglar . A
model proposal for the chaotic structure of
Istanbul stock exchange . Chaos, Solitons and
Fractals 36 (2008) 13921398
29
Chaos in Economics

The C/US exchange rate 6
chaotic structure
Time series of daily exchange rate
data 14/02/1973---29/03/2003
Lyapunov exponent of the time series
  • Chaos also exists in daily data for the Swedish
    Krona against Deutsche Mark,
  • the ECU, the US Dollar and the Yen exchange
    rates.7

6 R. Weston, The chaotic structure of the
C/US exchange rate, International Business
Economics Research Journal, 2007,
619-28. 7 Mikael Bask.A positive Lyapunov
exponent in Swedish exchange rate? Chaos,
Solitons and Fractals 14(2002) 1295-1304.
30
Chaos in Economics

Economic prediction8
  • Several economic time series are tested by
    using a deterministic predictive
  • technique is introduced, which is based on
    the embedding theorem by Takens
  • and the recently developed wavelet networks
  • Based on phase space reconstruction technique,
    the predicted values correspond
  • quite well with the actual values.

Chinese microeconomic time series National
financial expenditure
Gross output value of industry
8 LG Cao, YG Hong, HZ Zhao and SH Deng,
Predicting economic time series using a nonlinear
deterministic technique, Computational
Economics, 1996, 9149-178.
31
Chaos in Economics

Economic Modeling
  • Lots of economic models are presented to study
    the rich nolinear dynamical behavior.

Such as cobweb price adjustment processes,
optimal growth models, overlapping
generations models, Keynesian business cycle
models, Kaldor and Goodwin
growth cycle models, demand models with adaptive
preferences, models of
productivity growth, duopoly models, and others..
  • Researchers analyze the chaotic properties of
    these models
  • Equilibrium, Lyapunov
    exponents, bifurcation diagram

32
Outline
  • Introduction to Chaos
  • Topological Horseshoe Theory
  • Chaos in Economics
  • The Analysis of two Economic Systems

33
The Analysis of two Economic Systems
  • The Cournot duopoly Kopel economic Model
  • A Business cycle model
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