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M. Bicher:
"Agentenbasierte Modellbildung und Simulation auf Basis der Fokker-Planck-Gleichung";
Supervisor: F. Breitenecker; Institut für Analysis und Scientific Computing, 2013; final examination: 2013-06-03.

Motivation
Due to exponentially increasing performance of computers, nowadays more and more complex models can be simulated in shorter time with less efforts. Thus especially individual-based models,
so called microscopic models, requiring lots of memory and fast computation, are getting more and more popular. They pose a very well understandable modelling-concept especially to non-experts and are additionally very flexible regarding change of parameters or model structures.
Unfortunately modelling with these, often called agent-based models, is always subjected to a risk, because the behaviour of the models is hardly predictable and insufficiently studied.
Therefore it is often necessary to use reliable, less flexible, methods like differential equations instead, which have already been investigated for hundreds of years.
Content
At the end of the 20th century the Dutch physicist N.G. van Kampen published the basis of a theory,how the deterministic moments of stochastic agent-based models, in this case continuoustime
Markov-process based micro-models, can asymptotically be described by ordinary and partial differential equations. This method, sometimes known within physicists as "diffusion approximation", was formerly mainly used in quantum dynamics before its usage was extended
e.g. to economical models by M. Aoki in 2002.
Given N identical dynamic agents each described by a Markov-process with a finite number of states, also the system-vector consisting of the numbers of agents within the same state is described by a Markov-process. Thus the so called Master-equation,
dP
dt
(x(t) = i) =
X
j6=i
P(x(t) = j)!j;i 􀀀 P(x(t) = i)!i;j ;
holds. Taylor-approximation, in this case called Kramers-Moyal-decomposition, and certain substitutions on the one hand lead to an ordinary differential equation, solved by an approximation
of the mean value, and on the other hand to a special partial differential equation (Fokker- Planck-equation), solved by an approximation of the density function. The resulting curves
describe, neglecting an error O(N􀀀1 2 ), the temporal behaviour of mean value and variance.
It is important to mention, that the theory does not depend on, whether the agents are independently described by Markov-processes or are allowed to interact in a memoryless way!
Especially the last idea motivates the thought, that the theory can be extended from timecontinuous interacting Markov-processes to time-discrete interacting stochastic agent-based models,
which are much more commonly used. It shows that this assumption holds considering certain errors depending from the size of the transition rates.
Conclusion

Summarizing, formulas were created, how the deterministic system-variables of stochastic agentbased models can asymptotically be described by a system of differential equations. Thus a certain bijection between a small subspace of agent-based models and a subspace of the set of all differential equations is found which could be used to extend the fields of application for both
modelling-types.

http://publik.tuwien.ac.at/files/PubDat_224628.pdf