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El. knyga: Random Dynamical Systems in Finance

(University of Calgary, Alberta, Canada), (Tufts University, USA)
  • Formatas: 357 pages
  • Išleidimo metai: 19-Apr-2016
  • Leidėjas: Chapman & Hall/CRC
  • Kalba: eng
  • ISBN-13: 9781439867198
Kitos knygos pagal šią temą:
Random Dynamical Systems in FinanceDidesnis paveikslėlis
  • Formatas: 357 pages
  • Išleidimo metai: 19-Apr-2016
  • Leidėjas: Chapman & Hall/CRC
  • Kalba: eng
  • ISBN-13: 9781439867198
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The theory and applications of random dynamical systems (RDS) are at the cutting edge of research in mathematics and economics, particularly in modeling the long-run evolution of economic systems subject to exogenous random shocks. Despite this interest, there are no books available that solely focus on RDS in finance and economics. Exploring this emerging area, Random Dynamical Systems in Finance shows how to model RDS in financial applications.





Through numerous examples, the book explains how the theory of RDS can describe the asymptotic and qualitative behavior of systems of random and stochastic differential/difference equations in terms of stability, invariant manifolds, and attractors. The authors present many models of RDS and develop techniques for implementing RDS as approximations to financial models and option pricing formulas. For example, they approximate geometric Markov renewal processes in ergodic, merged, double-averaged, diffusion, normal deviation, and Poisson cases and apply the obtained results to option pricing formulas.





With references at the end of each chapter, this book provides a variety of RDS for approximating financial models, presents numerous option pricing formulas for these models, and studies the stability and optimal control of RDS. The book is useful for researchers, academics, and graduate students in RDS and mathematical finance as well as practitioners working in the financial industry.

Recenzijos

"... the timely publication of this book is very welcome and appreciated. There are not many books on RDS, and as far as this reviewer knows, this book is the only one that deals with the application of RDS in finance. ... There are many examples from finance, and it is very interesting to see that classical problems in finance, such as pricing European call-options, are interpreted in the viewpoint of RDS. Overall, this is a very useful book to both researchers of RDS and financial engineers ..." -Mathematical Reviews, November 2014

List of Figures
xiii
Preface xv
Acknowledgment xvii
1 Introduction
1(6)
2 Deterministic Dynamical Systems and Stochastic Perturbations
7(28)
2.1
Chapter overview
7(1)
2.2 Deterministic dynamical systems
7(11)
2.2.1 Ergodicity and Birkhoff individual ergodic theorem
9(1)
2.2.2 Stationary (invariant) measures and the Frobenius-Perron operator for deterministic dynamical systems
10(8)
2.3 Stochastic perturbations of deterministic dynamical systems
18(17)
2.3.1 Stochastic perturbations of deterministic systems and invariant measures
19(3)
2.3.2 A family of stochastic perturbations and invariant measures
22(1)
2.3.3 Matrix representation of PN
23(3)
2.3.4 Stability and convergence
26(2)
2.3.5 Examples
28(5)
References
33(2)
3 Random Dynamical Systems and Random Maps
35(50)
3.1
Chapter overview
35(1)
3.2 Random dynamical systems
35(1)
3.3 Skew products
36(1)
3.4 Random maps: Special structures of random dynamical systems
37(9)
3.4.1 Random maps with constant probabilities
38(1)
3.4.2 The Frobenius-Perron operator for random maps with constant probabilities
39(1)
3.4.3 Properties of the Frobenius-Perron operator
39(2)
3.4.4 Representation of the Frobenius-Perron operator
41(2)
3.4.5 Existence of invariant measures for random maps with constant probabilities
43(1)
3.4.6 Random maps of piecewise linear Markov transformations and the Frobenius-Perron operator
44(2)
3.5 Necessary and sufficient conditions for the existence of invariant measures for a general class of random maps with constant probabilities
46(7)
3.6 Support of invariant densities for random maps
53(9)
3.7 Smoothness of density functions for random maps
62(9)
3.8 Applications in finance
71(14)
3.8.1 One period binomial model for stock option
73(3)
3.8.2 The classical binomial interest rate models and bond prices
76(3)
3.8.3 Random maps with constant probabilities as useful alternative models for classical binomial models
79(2)
References
81(4)
4 Position Dependent Random Maps
85(46)
4.1
Chapter overview
85(1)
4.2 Random maps with position dependent probabilities
86(8)
4.2.1 The Frobenius-Perron operator
86(1)
4.2.2 Properties of the Frobenius-Perron operator
87(2)
4.2.3 Existence of invariant measures for position dependent random maps
89(1)
4.2.3.1 Existence results of Gora and Boyarsky
89(1)
4.2.3.2 Existence results of Bahsoun and Gora
90(4)
4.2.3.3 Necessary and sufficient conditions for the existence of invariant measures for a general class of position dependent random maps
94(1)
4.3 Markov switching position dependent random maps
94(6)
4.4 Higher dimensional Markov switching position dependent random maps
100(7)
4.4.1 Notations and review of some lemmas
100(2)
4.4.2 The existence of absolutely continuous invariant measures of Markov switching position dependent random maps in Rn
102(5)
4.5 Approximation of invariant measures for position dependent random maps
107(13)
4.5.1 Maximum entropy method for position dependent random maps
108(4)
4.5.1.1 Convergence of the maximum entropy method for random map
112(1)
4.5.2 Invariant measures of position dependent random maps via interpolation
113(7)
4.6 Applications in finance
120(11)
4.6.1 Generalized binomial model for stock prices
121(1)
4.6.2 Call option prices using one period generalized binomial models
121(4)
4.6.3 The multi-period generalized binomial models and valuation of call options
125(1)
4.6.4 The generalized binomial interest rate models using position dependent random maps and valuation of bond prices
126(3)
References
129(2)
5 Random Evolutions as Random Dynamical Systems
131(34)
5.1
Chapter overview
131(1)
5.2 Multiplicative operator functionals (MOF)
131(2)
5.3 Random evolutions
133(11)
5.3.1 Definition and classification of random evolutions
133(2)
5.3.2 Some examples of RE
135(2)
5.3.3 Martingale characterization of random evolutions
137(5)
5.3.4 Analogue of Dynkin's formula for RE
142(1)
5.3.5 Boundary value problems for RE
143(1)
5.4 Limit theorems for random evolutions
144(21)
5.4.1 Weak convergence of random evolutions
145(2)
5.4.2 Averaging of random evolutions
147(2)
5.4.3 Diffusion approximation of random evolutions
149(3)
5.4.4 Averaging of random evolutions in reducible phase space, merged random evolutions
152(3)
5.4.5 Diffusion approximation of random evolutions in reducible phase space
155(2)
5.4.6 Normal deviations of random evolutions
157(3)
5.4.7 Rates of convergence in the limit theorems for RE
160(3)
References
163(2)
6 Averaging of the Geometric Markov Renewal Processes (GMRP)
165(20)
6.1
Chapter overview
165(1)
6.2 Introduction
165(1)
6.3 Markov renewal processes and semi-Markov processes
166(1)
6.4 The geometric Markov renewal processes (GMRP)
167(3)
6.4.1 Jump semi-Markov random evolutions
167(1)
6.4.2 Infinitesimal operators of the GMRP
168(2)
6.4.3 Martingale property of the GMRP
170(1)
6.5 Averaged geometric Markov renewal processes
170(5)
6.5.1 Ergodic geometric Markov renewal processes
171(1)
6.5.1.1 Average scheme
172(1)
6.5.1.2 Martingale problem for the limit process St in average scheme
173(1)
6.5.1.3 Weak convergence of the processes STt in an average scheme
174(1)
6.5.1.4 Characterization of the limiting measure Q for QT as T → ∞
175(1)
6.6 Rates of convergence in ergodic averaging scheme
175(1)
6.7 Merged geometric Markov renewal processes
176(1)
6.8 Security markets and option prices using generalized binomial models induced by random maps
177(1)
6.9 Applications
177(8)
6.9.1 Two ergodic classes
177(1)
6.9.2 Algorithms of phase averaging with two ergodic classes
178(1)
6.9.3 Merging of S] in the case of two ergodic classes
178(1)
6.9.4 Examples for two states ergodic GMRP
179(1)
6.9.5 Examples for merged GMRP
179(3)
References
182(3)
7 Diffusion Approximations of the GMRP and Option Price Formulas
185(24)
7.1
Chapter overview
185(1)
7.2 Introduction
185(1)
7.3 Diffusion approximation of the geometric Markov renewal process (GMRP)
186(3)
7.3.1 Ergodic diffusion approximation
186(2)
7.3.2 Merged diffusion approximation
188(1)
7.3.3 Diffusion approximation under double averaging
189(1)
7.4 Proofs
189(6)
7.4.1 Diffusion approximation (DA)
189(1)
7.4.2 Martingale problem for the limiting problem GO(t) in DA
190(2)
7.4.3 Weak convergence of the processes GT(t) in DA
192(1)
7.4.4 Characterization of the limiting measure Q for QT as T → + ∞ in DA
192(1)
7.4.5 Calculation of the quadratic variation for GMRP
193(1)
7.4.6 Rates of convergence for GMRP
194(1)
7.5 Merged diffusion geometric Markov renewal process in the case of two ergodic classes
195(1)
7.5.1 Two ergodic classes
195(1)
7.5.2 Algorithms of phase averaging with two ergodic classes
195(1)
7.5.3 Merged diffusion approximation in the case of two ergodic classes
196(1)
7.6 European call option pricing formulas for diffusion GMRP
196(3)
7.6.1 Ergodic geometric Markov renewal process
196(2)
7.6.2 Double averaged diffusion GMRP
198(1)
7.6.3 European call option pricing formula for merged diffusion GMRP
198(1)
7.7 Applications
199(10)
7.7.1 Example of two state ergodic diffusion approximation ergodic diffusion approximation
199(1)
7.7.2 Example of merged diffusion approximation
200(5)
7.7.3 Call option pricing for ergodic GMRP
205(1)
7.7.4 Call option pricing formulas for double averaged GMRP
206(1)
References
206(3)
8 Normal Deviation of a Security Market by the GMRP
209(18)
8.1
Chapter overview
209(1)
8.2 Normal deviations of the geometric Markov renewal processes
209(4)
8.2.1 Ergodic normal deviations
209(1)
8.2.2 Reducible (merged) normal deviations
210(1)
8.2.3 Normal deviations under double averaging
211(2)
8.3 Applications
213(6)
8.3.1 Example of two state ergodic normal deviated GMRP
213(1)
8.3.2 Example of merged normal deviations in 2 classes
214(5)
8.4 European call option pricing formula for normal deviated GMRP
219(4)
8.4.1 Ergodic GMRP
219(2)
8.4.2 Double averaged normal deviated GMRP
221(1)
8.4.3 Call option pricing for ergodic GMRP
222(1)
8.4.4 Call option pricing formulas for double averaged GMRP
222(1)
8.5 Martingale property of GMRP
223(1)
8.6 Option pricing formulas for stock price modelled by GMRP
223(1)
8.7 Examples of option pricing formulas modelled by GMRP
224(3)
8.7.1 Example of two states in discrete time
224(1)
8.7.2 Generalized example in continuous time in Poisson case
225(1)
References
226(1)
9 Poisson Approximation of a Security Market by the Geometric Markov Renewal Processes
227(10)
9.1
Chapter overview
227(1)
9.2 Averaging in Poisson scheme
227(2)
9.3 Option pricing formula under Poisson scheme
229(1)
9.4 Application of Poisson approximation with a finite number of jump values
230(7)
9.4.1 Applications in finance
230(1)
9.4.1.1 Risk neutral measure
231(1)
9.4.1.2 On market incompleteness
232(1)
9.4.2 Example
233(2)
References
235(2)
10 Stochastic Stability of Fractional RDS in Finance
237(24)
10.1
Chapter overview
237(1)
10.2 Fractional Brownian motion as an integrator
238(2)
10.3 Stochastic stability of a fractional (B,S)-security market in Stratonovich scheme
240(5)
10.3.1 Definition of fractional Brownian market in Stratonovich scheme
240(1)
10.3.2 Stability almost sure, in mean and mean square of fractional Brownian markets without jumps in Stratonovich scheme
240(2)
10.3.3 Stability almost sure, in mean and mean-square of fractional Brownian markets with jumps in Stratonovich scheme
242(3)
10.4 Stochastic stability of fractional (B, S)-security market in Hu and Oksendal scheme
245(5)
10.4.1 Definition of fractional Brownian market in Hu and Oksendal scheme
246(1)
10.4.2 Stability almost sure, in mean and mean square of fractional Brownian markets without jumps in Hu and Oksendal scheme
246(2)
10.4.3 Stability almost sure, in mean and mean square of fractional Brownian markets with jumps in Hu and Oksendal scheme
248(2)
10.5 Stochastic stability of fractional (B, S)-security market in Elliott and van der Hoek scheme
250(5)
10.5.1 Definition of fractional Brownian market in Elliott and van der Hoek Scheme
250(1)
10.5.2 Stability almost sure, in mean and mean square of fractional Brownian markets without jumps in Elliott and van der Hoek Scheme
251(2)
10.5.3 Stability almost sure, in mean and mean square of fractional Brownian markets with jumps in Elliott and van der Hoek scheme
253(2)
10.6 Appendix
255(6)
10.6.1 Definitions of Lyapunov indices and stability
256(1)
10.6.2 Asymptotic property of fractional Brownian motion
257(1)
References
258(3)
11 Stability of RDS with Jumps in Interest Rate Theory
261(16)
11.1
Chapter overview
261(1)
11.2 Introduction
261(1)
11.3 Definition of the stochastic stability
262(1)
11.4 The stability of the Black-Scholes model
263(1)
11.5 A model of (B, S)- securities market with jumps
264(3)
11.6 Vasicek model for the interest rate
267(1)
11.7 The Vasicek model of the interest rate with jumps
268(2)
11.8 Cox-Ingersoll-Ross interest rate model
270(2)
11.9 Cox-Ingersoll-Ross model with random jumps
272(1)
11.10 A generalized interest rate model
273(1)
11.11 A generalized model with random jumps
274(3)
References
275(2)
12 Stability of Delayed RDS with Jumps and Regime-Switching in Finance
277(12)
12.1
Chapter overview
277(1)
12.2 Stochastic differential delay equations with Poisson bifurcations
277(1)
12.3 Stability theorems
278(4)
12.3.1 Stability of delayed equations with linear Poisson jumps and Markovian switchings
280(2)
12.4 Application in finance
282(1)
12.5 Examples
283(6)
References
288(1)
13 Optimal Control of Delayed RDS with Applications in Economics
289(12)
13.1
Chapter overview
289(1)
13.2 Introduction
289(1)
13.3 Controlled stochastic differential delay equations
290(4)
13.3.1 Assumptions and existence of solutions
290(1)
13.3.2 Weak infinitesimal operator of Markov process (xt, x(t))
291(1)
13.3.3 Dynkin formula for SDDEs
292(1)
13.3.4 Solution of Dirichlet-Poisson problem for SDDEs
293(1)
13.3.5 Statement of the problem
293(1)
13.4 Hamilton-Jacobi-Bellman equation for SDDEs
294(3)
13.5 Economics model and its optimization
297(4)
13.5.1 Description of the model
297(1)
13.5.2 Optimization calculation
298(1)
References
299(2)
14 Optimal Control of Vector Delayed RDS with Applications in Finance and Economics
301(18)
14.1
Chapter overview
301(1)
14.2 Introduction
301(1)
14.3 Preliminaries and formulation of the problem
302(1)
14.4 Controlled stochastic differential delay equations
303(9)
14.5 Examples: optimal selection portfolio and Ramsey model
312(7)
14.5.1 An optimal portfolio selection problem
312(2)
14.5.2 Stochastic Ramsey model in economics
314(2)
References
316(3)
15 RDS in Option Pricing Theory with Delayed/Path-Dependent Information
319(14)
15.1
Chapter overview
319(1)
15.2 Introduction
319(3)
15.3 Stochastic delay differential equations
322(1)
15.4 General formulation
323(3)
15.5 A simplified problem
326(3)
15.5.1 Continuous time version of GARCH model
327(2)
15.6 Appendix
329(4)
References
330(3)
16 Epilogue
333(2)
Index 335
Swishchuk, Anatoliy ; Islam, Shafiqul
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