Continuous-time random walks and fractional calculus: Theory and applications
description
Transcript of Continuous-time random walks and fractional calculus: Theory and applications
![Page 1: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/1.jpg)
Continuous-time random walks and fractional calculus:
Theory and applicationsEnrico Scalas (DISTA East-Piedmont University)
www.econophysics.org
DIFI Genoa (IT) 20 October 2004
![Page 2: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/2.jpg)
Summary
• Introduction to CTRW and applications to Finance
• Applications to Physics
• Conclusions
![Page 3: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/3.jpg)
Introduction to CTRW and applications to Finance
![Page 4: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/4.jpg)
1999-2004: Five years of continuous-time random walks in
Econophysics
Enrico Scalas (DISTA East-Piedmont University)
www.econophysics.org
WEHIA 2004 - Kyoto (JP) 27-29 May 2004
![Page 5: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/5.jpg)
Summary• Continuous-time random walks as models of
market price dynamics
• Limit theorem
• Link to other models
• Some applications
![Page 6: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/6.jpg)
Tick-by-tick price dynamics
0 20 40 60 80 100
12,0
12,2
12,4
12,6
12,8
13,0
Price variations as a function of time
S
t
Pric
e
Time
![Page 7: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/7.jpg)
Theory (I)Continuous-time random walk in finance
(basic quantities)
tS : price of an asset at time t
tStx log : log price
, : joint probability density of jumps and of waiting times
iii txtx 1 iii tt 1
txp , : probability density function of finding the log price x at time t
![Page 8: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/8.jpg)
Theory (II): Master equation
0
, d
, d
Marginal jump pdf
Marginal waiting-time pdf
,
Permanence in x,t Jump into x,t
In case of independence:
0
' '1Pr d Survival probability
' ' ','',' ,0
dxdttxpttxxtxtxpt
![Page 9: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/9.jpg)
This is the characteristic
function of the log-price process
subordinated to a generalised
Poisson process.
Theory (III): Limit theorem, uncoupled case (I)
0 1
1
n
nn
n
ttE
(Scalas, Mainardi, Gorenflo, PRE, 69, 011107, 2004)
Mittag-Leffler function
10
0 !
,n
nn
n
xtEn
ttxp
1ˆ,ˆ tEtp
Subordination: see Clark, Econometrica, 41, 135-156 (1973).
![Page 10: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/10.jpg)
Theory (IV): Limit theorem, uncoupled case (II)(Scalas, Gorenflo, Mainardi, PRE, 69, 011107, 2004)
1ˆ,ˆ ,
hr
tEtp rh
rssrttr r ~~ , ,
hh hˆˆ ,
1 ,1ˆ 0,0 rhh
r
hhh
tEtp rh
rh
,ˆlim ,
0,
This is the characteristic function for the Greenfunction of the fractionaldiffusion equation.
20
Scaling of probabilitydensity functions
Asymptotic behaviour
![Page 11: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/11.jpg)
Theory (V): Fractional diffusion(Scalas, Gorenflo, Mainardi, PRE, 69, 011107, 2004)
tEtu ,ˆ
txWt
txu ,
1,
tuyidyW ,ˆexp2
1,
1,~ˆ,
~ˆ ssussu
Green function of the pseudo-differential equation (fractional diffusion equation):
Normal diffusion
for =2, =1.
![Page 12: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/12.jpg)
Continuous-time random walks (CTRWs)
CTRWs
Cràmer-Lundberg ruin theory for
insurance companies
Compound Poisson processesas models of high-frequency
financial data
(Scalas, Gorenflo, Luckock, Mainardi, Mantelli, Raberto QF, submitted, preliminary version cond-mat/0310305, or preprint:
www.maths.usyd.edu.au:8000/u/pubs/publist/publist.html?preprints/2004/scalas-14.html)
Normal and anomalous
diffusion in physicalsystems
Subordinatedprocesses
Fractionalcalculus
Diffusionprocesses
Mathematics
PhysicsFinance andEconomics
![Page 13: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/13.jpg)
Example: The normal compound Poisson process (=1)
0
00 !
exp,n
n
n
xn
tttxp
22 2exp2
1
nnxn
xn Convolution of n Gaussians
ttn
tnkktxttxx
1
)(
txpxp ,
0 txE 0
22var tx
The distribution of x is leptokurtic
04224
4 36 tK
(S.J. Press, Journal of Business, 40, 317-335, 1967)
![Page 14: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/14.jpg)
Generalisations
Perturbations of the NCPP:
• general waiting-time and log-return densities;(with R. Gorenflo, Berlin, Germany and F. Mainardi, Bologna, Italy, PRE, 69, 011107, 2004);
• variable trading activity (spectrum of rates);(with H.Luckock, Sydney, Australia, QF submitted);
• link to ACE;(with S. Cincotti, S.M. Focardi, L. Ponta and M. Raberto, Genova, Italy, WEHIA 2004!);
• dependence between waiting times and log-returns;(with M. Meerschaert, Reno, USA, in preparation, but see P. Repetowicz and P. Richmond,xxx.lanl.gov/abs/cond-mat/0310351);
• other forms of dependence (autoregressive conditional duration models, continuous-time Markov models);(work in progress in connection to bioinformatics activity).
![Page 15: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/15.jpg)
Applications
• Portfolio management: simulation of a synthetic market(E. Scalas et al.: www.mfn.unipmn.it/scalas/~wehia2003.html).
• VaR estimates: e.g. speculative intra-day option pricing. If g(x,T) is the payoff of a European option with delivery time T:
dy
dxTtygpTyq
Txgy
,,,
,
1
(E. Scalas, communication submitted to FDA ‘04).
• Large scale simulations of synthetic markets with supercomputers are envisaged.
![Page 16: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/16.jpg)
Empirical results on the waiting-time survival function and their relevance for market models (Anderson-Darling test) (I)
Interval 1 (9-11): 16063 data; 0 = 7 sInterval 2 (11-14): 20214 data; 0 = 11.3 sInterval 3 (14-17): 19372 data; 0 =7.9 s
nnn
iA iin
n
i
6.011lnln12
11
2
where 1 2 … n A1
2= 352; A22= 285; A3
2= 446 >> 1.957 (1% significance)
![Page 17: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/17.jpg)
• Non-exponential waiting-time survival function now observed by many groups in many different markets (Mainardi et al. (LIFFE) Sabatelli et al. (Irish market and ), K. Kim & S.-M. Yoon (Korean Future Exchange)), but see also Kaizoji and Kaizoji (cond-mat/0312560)
• Why should we bother? This has to do both with the market price formation mechanism and with the bid-ask process. If the bid-ask process is modelled by means of a Poisson distribution (exponential survival function), its random thinning should yield another Poisson distribution. This is not the case!
• A clear discussion can be found in a recent contribution by the GASM group.
• Possible explanation related to variable daily activity!
Empirical results on the waiting-time survival function and their relevance for market models (Anderson-Darling test) (II)
![Page 18: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/18.jpg)
Applications to Physics
![Page 19: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/19.jpg)
![Page 20: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/20.jpg)
Problem
• Understanding the scaling of transport with domain size has become the critical issue in the design of fusion reactors.
• It is a challenging task due to the overwhelming complexity of magnetically confined plasmas that are typically in a turbulent state.
• Diffusive models have been used since the beginning.
![Page 21: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/21.jpg)
Focus and method
• Tracer transport in pressure-gradient-driven plasma turbulence.
• Variations in pressure gradient trigger instabilities leading to intermittent and avalanchelike transport.
• Non-linear equations for the motion of tracers are numerically solved.
• The pdf of tracer position is non-Gaussian with algebraic decaying tails.
![Page 22: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/22.jpg)
![Page 23: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/23.jpg)
Solution I• There is tracer trapping due to turbulent eddies.• There are large jumps due to avalanchelike events.• These two effects are the source of anomalous diffusion.
![Page 24: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/24.jpg)
Solution II• Fat tails (nearly three decades)
![Page 25: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/25.jpg)
Fractional diffusion model
![Page 26: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/26.jpg)
Model I
![Page 27: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/27.jpg)
Model II
![Page 28: Continuous-time random walks and fractional calculus: Theory and applications](https://reader035.fdocuments.us/reader035/viewer/2022062322/568148c6550346895db5e32c/html5/thumbnails/28.jpg)
Conclusions
• CTRWs are suitable as phenomenological models for high-frequency market dynamics.
• They are related to and generalise many models already used in econometrics.
• They are suitable phenomenological models of anomalous diffusion.