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E-raamat: Optional Processes: Theory and Applications [Taylor & Francis e-raamat]

, (University of Alberta, Edmonton, Canada)
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Optional Processes: Theory and ApplicationsSuurem pilt
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It is well-known that modern stochastic calculus has been exhaustively developed under usual conditions. Despite such a well-developed theory, there is evidence to suggest that these very convenient technical conditions cannot necessarily be fulfilled in real-world applications.

Optional Processes: Theory and Applications seeks to delve into the existing theory, new developments and applications of optional processes on "unusual" probability spaces. The development of stochastic calculus of optional processes marks the beginning of a new and more general form of stochastic analysis.

This book aims to provide an accessible, comprehensive and up-to-date exposition of optional processes and their numerous properties. Furthermore, the book presents not only current theory of optional processes, but it also contains a spectrum of applications to stochastic differential equations, filtering theory and mathematical finance.

Features











Suitable for graduate students and researchers in mathematical finance, actuarial science, applied mathematics and related areas











Compiles almost all essential results on the calculus of optional processes in unusual probability spaces





Contains many advanced analytical results for stochastic differential equations and statistics pertaining to the calculus of optional processes





Develops new methods in finance based on optional processes such as a new portfolio theory, defaultable claim pricing mechanism, etc.
Preface ix
Introduction xi
1 Spaces, Laws and Limits
1(28)
1.1 Foundation
1(2)
1.2 Measurable Spaces, Random Variables and Laws
3(12)
1.2.1 Measurable spaces
3(1)
1.2.2 Measurable functions
4(1)
1.2.3 Atoms and separable fields
5(1)
1.2.4 The case of real-valued random variables
5(1)
1.2.5 Monotone class theorem
6(1)
1.2.6 Probability and expectation
7(2)
1.2.6.1 Convergence of random variables
9(1)
1.2.6.2 Fubini's Theorem
9(1)
1.2.6.3 Uniform integrability
10(1)
1.2.6.4 Completion of probability spaces
11(1)
1.2.6.5 Independence
12(1)
1.2.6.6 Conditional expectation
12(3)
1.3 Analytic Set Theory
15(14)
1.3.1 Paving and analytic sets
15(2)
1.3.2 Separable sets
17(2)
1.3.3 Lusin and Souslin spaces
19(2)
1.3.3.1 Souslin-Lusin Theorem
21(1)
1.3.4 Capacities and Choquet's Theorem
21(3)
1.3.4.1 Constructing capacities
24(2)
1.3.5 Theorem of cross-section
26(3)
2 Stochastic Processes
29(60)
2.1 Construction
29(5)
2.1.1 Time law
31(1)
2.1.2 Canonical process
32(2)
2.2 Processes on Filtrations
34(2)
2.2.1 Adapted processes
34(1)
2.2.2 Progressive measurability
35(1)
2.3 Paths Properties
36(23)
2.3.1 Processes on dense sets
37(2)
2.3.2 Upcrossings and downcrossings
39(3)
2.3.3 Separability
42(3)
2.3.3.1 Doob's separability theorems
45(2)
2.3.4 Progressive processes of random sets
47(2)
2.3.5 Almost equivalence
49(5)
2.3.5.1 Pseudo-Paths
54(5)
2.4 Random Times
59(27)
2.4.1 Stopping times
60(3)
2.4.2 Basic properties of stopping times
63(3)
2.4.3 Stochastic intervals
66(1)
2.4.4 Optional and predictable σ-fields
66(5)
2.4.5 Predictable stopping times
71(6)
2.4.6 Classification of stopping times
77(2)
2.4.7 Quasi-left-continuous filtrations
79(1)
2.4.8 Optional and predictable cross-sections
80(6)
2.5 Optional and Predictable Processes
86(3)
3 Martingales
89(50)
3.1 Discrete Parameter Martingales
89(20)
3.1.1 Basic properties
90(1)
3.1.2 Right and left closed supermartingales
91(1)
3.1.3 Doob's stopping theorem
92(2)
3.1.3.1 Extension to unbounded stopping times
94(3)
3.1.4 Fundamental inequalities
97(1)
3.1.4.1 Maximal lemma
97(2)
3.1.4.2 Domination in Lp
99(1)
3.1.4.3 Martingales upcrossings and downcrossings
100(2)
3.1.5 Convergence and decomposition theorems
102(1)
3.1.5.1 Almost sure convergence of supermartingales
103(1)
3.1.5.2 Uniform integrability and martingale convergence
104(1)
3.1.5.3 Riesz decompositions of supermartingales
105(2)
3.1.5.4 Krickeberg decomposition of martingales
107(1)
3.1.6 Some applications of convergence theorems
108(1)
3.2 Continuous Parameter Martingales
109(30)
3.2.1 Supermartingales on countable sets
109(1)
3.2.1.1 Fundamental inequalities
110(1)
3.2.1.2 Existence of right and left limits
111(3)
3.2.2 Right-continuous supermartingale
114(3)
3.2.3 Projections theorems
117(10)
3.2.4 Decomposition of supermartingales
127(1)
3.2.4.1 Functional analytic decomposition theorem
127(4)
3.2.4.2 Extension to non-positive functionals
131(1)
3.2.4.3 Decomposition of positive supermartingale of class D
132(4)
3.2.4.4 The general case of Doob decomposition
136(3)
4 Strong Supermartingales
139(20)
4.1 Introduction
139(4)
4.2 Projection Theorems
143(6)
4.3 Special Inequalities
149(5)
4.4 Mertens Decomposition
154(1)
4.5 Snell Envelope
155(4)
5 Optional Martingales
159(42)
5.1 Introduction
159(4)
5.2 Existence and Uniqueness
163(3)
5.3 Increasing and Finite Variation Processes
166(5)
5.3.1 Integration with respect to increasing and finite variation processes
167(2)
5.3.2 Dual projections
169(2)
5.4 Decomposition Results
171(13)
5.4.1 Decomposition of elementary processes
172(5)
5.4.2 Decomposition of optional martingales
177(7)
5.5 Quadratic Variation
184(5)
5.5.1 Predictable and optional
184(3)
5.5.2 Kunita-Watanabe inequalities
187(2)
5.6 Optional Stochastic Integral
189(12)
5.6.1 Integral with respect to square integrable martingales
189(3)
5.6.2 Integral with respect to martingales with integrable variation
192(2)
5.6.3 Integration with respect to local optional martingales
194(7)
6 Optional Supermartingales Decomposition
201(14)
6.1 Introduction
201(1)
6.2 Riesz Decomposition
202(2)
6.3 Doob-Meyer-Galchuk Decomposition
204(11)
6.3.1 Decomposition of DL class
214(1)
7 Calculus of Optional Semimartingales
215(28)
7.1 Integral with Respect to Optional Semimartingales
215(2)
7.2 Formula for Change of Variables
217(6)
7.3 Stochastic Integrals of Random Measures
223(6)
7.4 Semimartingales and Their Characteristics
229(7)
7.4.1 Canonical representation
229(3)
7.4.2 Component representation
232(4)
7.5 Uniform Doob-Meyer Decompositions
236(7)
7.5.1 Supporting lemmas
241(2)
8 Optional Stochastic Equations
243(38)
8.1 Linear Equations, Exponentials and Logarithms
243(9)
8.1.1 Stochastic exponential
244(3)
8.1.2 Stochastic logarithm
247(2)
8.1.3 Nonhomogeneous linear equation
249(2)
8.1.4 Gronwall lemma
251(1)
8.2 Existence and Uniqueness of Solutions of Optional Stochastic Equations
252(17)
8.2.1 Stochastic equation with monotonicity condition
253(2)
8.2.2 Existence and uniqueness results
255(2)
8.2.2.1 Uniqueness
257(2)
8.2.2.2 Existence
259(9)
8.2.3 Remarks and applications
268(1)
8.3 Comparison of Solutions of Optional Stochastic Equations
269(12)
8.3.1 Comparison theorem
270(9)
8.3.2 Remarks and applications
279(2)
9 Optional Financial Markets
281(26)
9.1 Introduction
281(1)
9.2 Market Model
282(1)
9.3 Martingale Deflators
283(8)
9.3.1 The case of stochastic exponentials
284(4)
9.3.2 The case of stochastic logarithms
288(3)
9.4 Pricing and Hedging
291(2)
9.5 Absence of Arbitrage
293(4)
9.6 Examples of Special Cases
297(10)
9.6.1 Ladlag jumps diffusion model
297(1)
9.6.1.1 Computing a local martingale deflator
298(1)
9.6.1.2 Pricing of a European call option
299(3)
9.6.1.3 Hedging of a European call option
302(1)
9.6.2 Basket of stocks
303(1)
9.6.3 Defaultable bond and a stock
304(3)
10 Defaultable Markets on Unusual Space
307(18)
10.1 Introduction
307(3)
10.2 Optional Default
310(1)
10.3 Defaultable Cash-Flow
311(4)
10.3.1 Portfolio with default
313(2)
10.4 Probability of Default
315(2)
10.5 Valuation of Defaultable Cash-Flow and Examples
317(8)
11 Filtering of Optional Semimartingales
325(42)
11.1 The Filtering Problem
325(1)
11.2 The Usual Case of Optimal Filtering
326(32)
11.2.1 Auxiliary results
326(9)
11.2.2 Martingales' integral representation
335(13)
11.2.3 Filtering of cadlag semimartingales
348(10)
11.3 The Unusual case of Optimal Filtering
358(4)
11.3.1 Filtering on unusual stochastic basis
358(2)
11.3.2 Filtering on mixed stochastic basis
360(2)
11.4 Filtering in Finance
362(5)
Bibliography 367(10)
Index 377
Mohamed Abdelghani completed his PhD in Mathematical Finance from the University of Alberta. He is currently working as a V.P. in quantitative finance and machine learning at Morgan Stanley, New York, USA.

Alexander Melnikov is a Professor in Mathematical Finance at the University of Alberta, Edmonton, Canada. His research interests belong to the area of contemporary stochastic analysis and its numerous applications in Mathematical Finance, Statistics and Actuarial Science. He has written six books as well as over one hundred research papers in leading academic journals.
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