Stochastic Differention Equations

1
Index
Stochastic Differention Equations
Computing and Simulation
Email: [email protected]
Since December 24, 2007
Prologue
This lecture is designed for the persons who want to develop professional skill in stochastic calculus and its application to problems in finance but don’t involve in too much mathematics. As
well-known, the more experience we have with mathematical skills (especially real analysis and
probability), the more piece we can get. Nevertheless, the course will move quickly and students
can expect to acquire tools that are deep enough and rich enough to be relied upon throughout
their professional careers.
Course Plan: The course begins with basic concepts and theory of probability space,
needed in this lecture. This material is used to motivate the theory of continuous time
stochastic processes, especially Brownian motion. The construction of Brownian motion is given
in detail, and enough material on the subtle properties of Brownian paths is developed so that
the student should evolve a good sense of when intuition can be trusted and when it cannot.
The course then takes up the Itô integral and aims to provide a development that is honest and
complete without being pedantic. With the Itô integral in hand, the course focuses more on
models. Stochastic processes of importance in Finance and Economics are developed in concert
with the tool. The financial notion of replication is developed, and the Black-Scholes PDE will
be introduced with its importance. At the last part of this course, we introduce enough of the
theory of the diffusion equation to be able to solve the Black-Scholes PDE and prove the
uniqueness of the solution.
Besides the theoretical lectures introduced, some computer algrba systems (CAS) will be
introduced too, including Maxima, Axiom and Python language, especially visual module.
Maxima and Axiom are used to solve general classical mathematical problems, differentiation,
integration,function plotting, ODE solver etc. Python will be used to simulate the paths of the
Brownian motion partical in 2D plane and plot the data of simulayion for Langevin equation.
Also a symbolic SDE’s solver of Axiom [14] will be introduced. Since codes will be embedded
into the documentation directly by the session functionality, welcome to test the code by changing the parameters and share your results!
Texts: Many not to be listed here
Grading: Grades are based on
1. regular homework (40%)
2. mid-semester report (30%)
3. final quiz (30%)
3
4
Prologue
Brownian Motion
-
•
•
(Stochastic Processes)
Itô Integral
TEXMAC S
(Stochastic Calculus)
Table of contents
Prologue
.. ... .. .. .. .. .. .. ... .. .. .. .. .. .. .. ... .. .. .. .. .. .. ... . 3
Table of contents
. .. .. .. .. .. ... .. .. .. .. .. .. .. ... .. .. .. .. .. .. ... . 7
List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1 Basic Probability Space
1.1
1.2
1.3
1.4
1.5
1.6
1.7
.. .. ... .. .. .. .. .. .. .. ... .. .. .. .. .. .. ... . 9
Measurable spaces and functions
Random Variables . . . . . . . . .
Convergences . . . . . . . . . . . .
Characteristic Functions . . . . .
Conditional expectations . . . . .
Laws of Large Number . . . . . .
Central Limit Theorem (CLT) . .
2 Martingales
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. . 9
. 14
. 18
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. 25
. .. .. .. .. .. .. ... .. .. .. .. .. .. .. ... .. .. .. .. .. .. ...
2.1 Discrete time Martingales . . . .
2.2 Continuous Time Martingales . .
Examples of Martingales . . . . . .
Problems to Work for Understanding
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29
34
38
39
3 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Key Concepts . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Consistency Condition . . . . . . . . . . . . .
3.2 Simulation of Brownian Motion . . . . . . . . . . .
3.3 Characterization of Brownian Motion . . . . . . .
3.4 Brownian Bridge . . . . . . . . . . . . . . . . . . . .
3.5 Brownian Motion and Martingales . . . . . . . . .
3.6 Non-differentiability of Brownian’s Sample Path
3.7 LIL for Brownian Motion . . . . . . . . . . . . . . .
3.8 Strong Markov Property of B.M. . . . . . . . . . .
3.9 Brownian Hitting Time . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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66
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77
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79
4.1 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Itô’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Brownian Local Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
85
89
5 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4 Stochastic Integrals
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5.1 Existence and Uniqueness of SDE . .
5.1.1 Transformation of SDE . . . . . .
5.2 Existence of Solution . . . . . . . . . .
5.3 Uniqueness of SDE . . . . . . . . . . . .
6 Diffusion Process
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29
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. 97
. 97
. 98
. 104
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5
6
6.1
6.2
6.3
6.4
Table of contents
Kolmogorov’s Backward/Forward Equation
Feynman-Kac’s Formula . . . . . . . . . . . .
Examples . . . . . . . . . . . . . . . . . . . . . .
P.D.E. with Respect to Brownian Motion .
7 Symbolic Stochastic Integration
7.1 Basics of Itovsn3 . . . . . . . .
7.1.1 Installation . . . . . . . .
7.1.2 Basics of itovsn3 module
7.2 Examples . . . . . . . . . . . . .
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107
109
113
117
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121
121
121
121
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
List of figures
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n
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Buffon’s needle problem . . . . .
r.v and probability measure . .
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Jensen’s Inequality . . . . . . . .
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Counterexample for DCT . . . .
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Convgence in Lp but not convergence a.e.
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Upcrossing stopping time . . . . . . . . . .
pre-τ σ-field is larger
.. .. .. .. .. . .
Strong Markov Property . . . . . . . . . . .
Versions with Path Continuity. . . . . . . .
Markov Property . . . . . . . . . . . . . . .
Navigate zero consecutively . . . . . . . . .
Brownian bridge . . . . . . . . . . . . . . .
Markov property of B.M. . . . . . . . . . .
Refinement Estimation . . . . . . . . . . .
Partitions for different numbers . . . . . . .
LIL of B.M. . . . . . . . . . . . . . . . . .
The graph of h(x) near 0 . . . . . . . . . .
Partition points nearest to t1 and t2 . . . .
Estimation for the series sum . . . . . . . .
Step-wise Markov time . . . . . . . . . . . .
Displacement of B.M. during each time step .
Reflection principle of Brownian motion . . .
Jointly p.d.f. of B.M. and its maximum . . .
Unbounded set for hitting time at x = 0 . . .
Inverve of composite of functions
Decreasing Events for limsup Xn
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Correspondence between hitting times of two B.M.s
Mapping Relation for Stochastic Integrals . . . . .
Simple functions . . . . . . . . . . . . . . . . . . .
Itô’s integrals for simple function . . . . . . . . . .
New clock of B.M. . . . . . . . . . . . . . . . . . .
Increasing decomposition . . . . . . . . . . . . . .
Decompositon of Reflected B.M. . . . . . . . . . .
Reflected B.M. with new clock τt . . . . . . . . . .
Brownian bridge between X0 = x and XT = y . . .
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11
12
14
15
16
17
22
31
36
37
41
42
49
52
54
55
58
61
61
65
66
67
70
72
74
75
76
80
82
83
87
94
95
95
103
Index
Xn converges to X almost surely . . . . . . . . . 18
sure step function . . . . . . . . . . . . . . . . . 79
(, δ)-criteria . . . . . . . . . . . . . . . . . . . . 82
σ-algebra generated by U . . . . . . . . . . . . . 10
absolutely continuous . . . . . . . . . . . . . . 110
adapted . . . . . . . . . . . . . . . . . . . . . . . 29
σ-algebra . . . . . . . . . . . . . . . . . . . . . . . 9
Black-Scholes equation . . . . . . . . . . . . . . 100
Blumenthal’s 0 − 1 Law . . . . . . . . . . . . . . 68
Borel σ-algebra . . . . . . . . . . . . . . . . . . . 10
Borel Cantelli Lemma . . . . . . . . . . . . . 15, 19
Brownian bridge
solution of . . . . . . . . . . . . . . . . . . 101
Brownian motion . . . . . . . . . . . . . . . . . . 42
Brownian Motion . . . . . . . . . . . . . . . 46, 53
2D . . . . . . . . . . . . . . . . . . . . . . . . 42
Brownian bridge . . . . . . . . . . . . . . . . 52
Brownian bridge . . . . . . . . . . . . . . . 111
expansion . . . . . . . . . . . . . . . . . . . . 52
hitting time . . . . . . . . . . . . . . . . . . . 71
p.d.f. . . . . . . . . . . . . . . . . . . . . . 73
joint moment generating function of brownians
motions . . . . . . . . . . . . . . . . . . . . 113
jointly probability distribution . . . . . . . . 49
killed . . . . . . . . . . . . . . . . . . . . . . 43
local time . . . . . . . . . . . . . . . . . . . . 93
non-differentiability . . . . . . . . . . . . . . 54
reflected . . . . . . . . . . . . . . . . . . . . . 93
reflected
. . . . . . . . . . . . . . . . . . . . 43
reflection principle . . . . . . . . . . . . . . . 72
Second variation . . . . . . . . . . . . . . . . 56
standard . . . . . . . . . . . . . . . . . . . . 71
Strong Markov Property . . . . . . . . . . . . 67
Brownian Scaling . . . . . . . . . . . . . . . . . 53
Buffon’s needle problem . . . . . . . . . . . . . . 13
Cauchy-Schwarz theorem . . . . . . . . . . . . . 82
Central Limit Theorem . . . . . . . . . . . . . . 26
Chapman-Kolmogorov equation . . . . . . . . . 107
characteristic function . . . . . . . . . . . . . . . 23
Chebyshev’s inequality . . . . . . . . . . . . . . . 16
complement . . . . . . . . . . . . . . . . . . . . . 9
complete metric space . . . . . . . . . . . . . 81, 82
complete orthonormal basis . . . . . . . . . . . . 80
complete probability space . . . . . . . . . . . . 14
conditional expectation . . . . . . . . . . . . . . 25
continuous orthonomal basis . . . . . . . . . . . 51
convergent in distribution . . . . . . . . . . . . . 21
converges to X in L p . . . . . . . . . . . . . . . 20
converges to X in probability . . . . . . . . . . . 18
covariance function . . . . . . . . . . . . . . . . . 52
diagonal argument . . . . . . . . . . . . . . . . . 82
diffusion equation
solution of . . . . . . . . . . . . . . . . . . 113
diffusion process . . . . . . . . . . . . . . . . . 107
diffusion coefficient . . . . . . . . . . . . . . 107
displacement . . . . . . . . . . . .
drift coefficient . . . . . . . . . . .
generator of . . . . . . . . . . . . .
distribution . . . . . . . . . . . . . . .
Dominated Convergence Theorem . .
counterexample . . . . . . . . . . .
Doob’s Decomposition . . . . . . . . .
Doob’s Inequality . . . . . . . . . . .
Dvoretsky . . . . . . . . . . . . . . . .
Erdös . . . . . . . . . . . . . . . . . .
expectation . . . . . . . . . . . . . . .
Fatou’s Lemma . . . . . . . . . . . . .
Feynman-Kac’s formula . . . . . . . .
σ-field
pre-τ σ-field . . . . . . . . . . . . .
filtration . . . . . . . . . . . . . . . .
adapted to . . . . . . . . . . . . .
Gaussian . . . . . . . . . . . . . . . .
Gaussian Process . . . . . . . . . . . .
covariance function . . . . . . . . .
mean function . . . . . . . . . . .
stationary . . . . . . . . . . . . . .
Growall’s inequality . . . . . . . . . .
Guido van Rossum . . . . . . . . . . .
Hölder’s inequality . . . . . . . . . . .
heat equation . . . . . . . . . . . . . .
Hincˇin . . . . . . . . . . . . . . . . .
independent . . . . . . . . . . . . . .
independent and stationary increments
Independent Stationary Increment . .
Itô’s formula . . . . . . . . . . . . . .
Itô’s integral . . . . . . . . . . . . . .
Itô’s process . . . . . . . . . . . . . .
Jensen’s Inequality . . . . . . . . . . .
Kakutani . . . . . . . . . . . . . . . .
Karhunem-Loève’s expansion . . . . .
Kolmogorov’s backward equation . . .
Kolmogorov’s forward equation . . . .
Kolmogorov’s Inequality . . . . . . . .
Lévy martingale . . . . . . . . . . . .
Lévy modulus of continuity . . . . . .
Law of Iterated Logarithm . . . . . .
Laws of Large Number
SLLN . . . . . . . . . . . . . . . .
WLLN . . . . . . . . . . . . . . .
Levy . . . . . . . . . . . . . . . . . .
liminf . . . . . . . . . . . . . . . . . .
limsup . . . . . . . . . . . . . . . . .
Lipschitz continuous . . . . . . . . . .
Markov . . . . . . . . . . . . . . . . .
Markov process . . . . . . . . . . . . .
Markov process
strong Markov process . . . . . . .
Markov time . . . . . . . . . . . . . .
127
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..
35
29, 34
. . 34
. . 49
. . 81
. . 81
. . 81
. . 97
. . 99
. . 43
. . 16
. . 29
. . 60
13, 18
. . 46
. . 45
. . 85
. . 82
. . 85
. . 16
. . 54
. . 52
. 108
. 109
. . 34
. 109
. . 63
. . 58
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107
107
109
15
17
17
33
34
54
54
15
16
109
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25
25
94
12
12
54
42
35
. .. ...
. .. ...
36
35
128
martingale . . . . . . . . . . . .
continuous time martingale .
local martingale . . . . . . .
locally square integrable . . .
reversed martingale . . . . .
square integrable . . . . . . .
submartingale . . . . . . . .
supermartingale . . . . . . .
Martingale Convergent Theorem
Maxima . . . . . . . . . . . . . .
bc2() . . . . . . . . . . . . .
ode2() . . . . . . . . . . . . .
mean . . . . . . . . . . . . . . .
F -measurable . . . . . . . . . .
measurable process . . . . . . . .
measurable space . . . . . . . . .
measure . . . . . . . . . . . . . .
Mercer’s Theorem . . . . . . . .
metric . . . . . . . . . . . . . . .
Minkowski’s inequality . . . . . .
modifications . . . . . . . . . . .
Monotone Convergence Theorem
occupation density formula . . .
optional time . . . . . . . . . . .
Ornstein-Uhlenbeck process
solution of . . . . . . . . . .
Paley . . . . . . . . . . . . . . .
Index
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29
35
84
84
53
82
29
30
31
65
75
74
15
12
81
9, 11
. . 9
. 50
. 81
. 17
. 40
. 16
. 93
. 35
.. .. .. ..
.. .. .. ...
101
54
probabilistic measure . . . . . . . . . .
probability measure . . . . . . . . . . .
probability space . . . . . . . . . . . . .
product
.. ... .. .. .. .. .. ..
Python . . . . . . . . . . . . . . . . . .
pygame . . . . . . . . . . . . . . . .
random . . . . . . . . . . . . . . . .
scipy . . . . . . . . . . . . . . . . .
visual module . . . . . . . . . . . . .
random variable . . . . . . . . . . . . .
refinement . . . . . . . . . . . . . . . .
Riemann integral . . . . . . . . . . . . .
second order variation . . . . . . . . . .
simple . . . . . . . . . . . . . . . . . . .
Skorohod equation . . . . . . . . . . . .
standard Brownian motion . . . . . . .
stationary transition probability density
Stochastic differential equation . . . . .
SDE . . . . . . . . . . . . . . . . . .
Tanaka . . . . . . . . . . . . . . . . . .
transition probability . . . . . . . . . .
unbounded variation . . . . . . . . . . .
Upcrossing Inequality . . . . . . . . . .
versions . . . . . . . . . . . . . . . . . .
Wiener . . . . . . . . . . . . . . . . . .
Wiener process . . . . . . . . . . . . . .
Zygmund . . . . . . . . . . . . . . . . .
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. . 39
.. . 9
10, 14
. 13
. 42
. 44
. 44
. 43
. 43
. 14
37, 56
. . 50
. . 56
. . 81
. . 94
. . 45
. . 42
85, 97
. . 97
. . 93
. 107
. . 55
. . 31
. . 39
. . 54
. . 42
. . 54
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129
Index
References
1. K.L. Chung (1974), A course in probability theory, Academic Press.
2. K.L. Chung (1982), Lectures from Markov processes to Brownian motion, SpringerVerlag.
3. K.L. Chung and R.J. Williams (1990), Introduction to Stochastic Integration 2nd ,
Birhaüser.
4. K.L. Chung and Z.G. Zhao (1995), From Brownian motion to Schrödinger’s Equation,
Springer-Verlag.
5. R. Durrett (1984), Brownian motion and martingales in analysis, Wadsworth, Belmont,
CA
6. R. Durrett (1991), Probability: Theory and Examples,Brooks/Cole Publishing, Pacific
Grove, CA.
7. L.C. Evans, An Introduction to Stocastic Differential Equations, version 1.1, Department
of Mathematics, UC Berkeley.
8. G. Hara (2005), Lecture note 1: Stochastic Calculus.
9. M. Haugh (2005), Overview of Stochastic Calulus.
10. I.I. Gihman and A.V. Skorohod (1970), Stochastic Differential Equations, SpringerVerlag.
11. N. Ikeda and S. Watanabe (1989), Stochastic differential equations and diffusion processes, North-Holland Publishing.
12. K. Itô and H.P. McKean (1965), Diffusion processes and their sample paths, SpringerVerlag, NY.
13. Karatzas, I. and Shreve, S. E. (1987), Brownian Motion and Stochastic Calculus,
Springer.
14. W.S. Kendall (1999), Symbolic Itô Calculus in AXIOM: A nongoing story, Research
report n.327, University of Warwick.
15. F.C. Klebaner (1998), Introduction to Stochastic Calculus with Applications, Imperial
College Press.
16. R.S. Liptser and A.N. Shiryayev (1977), Statistics of Random Processes I, General
Theory, Springer-Verlag, New York Inc.
17. H.P. McKean (1969), Stochastic Integrals, Academic Press, NY.
18. Øksendal, B. (1995): Stochastic Differential Equations – An Introduction with Applications (4th edition). Springer-Verlag, Berlin.
19. D. Revuz and M. Yor (1991), Continuous Martingales and Brownian motion, SpringerVerlag.
20. S. Shreve (1997), Stochastic Calculus and Finance,
21. I.G. Sinai, Probability theory: an introductory course, Springer-Verlag.
22. J.M. Steele (2001), Stochastic Calculus and Financial Applications, Springer.
23. S.Taniguchi
(
)(2005), Lecture Note: Probability Theory,
24. S.Taniguchi
(
)(2005), Applied Mathematics III,
25. A.D. Wentzell (1981), A course in the theory of stochastic processes, McGraw-Hill Inc.
26. E. Wong and B. Hajek (1985), Stochastic processes in engineering systems, Springer,
130
Index
ISBN:957-41-5052-6