Stichastic Calculus: Syllabus and Exercise
Narn-Rueih Shieh **Copyright Reserved**
§ This course is suitable for those who have taken Probability Theory (I);
some knowledge of Real Analysis is needed.
§ The content and exercise are adapted from
1. R. Durrett: Probability: Theory and Examples, Third Edition(2005, Duxbury) Chap- ters 4, 7.
2. B. Okensdal: Stochastic Differential Equations, Sixth Edition(2003, Springer) Chap-
ters 3,4,5,7,8.
§ Grading: Exams(mid-term and final): 80 percent, Homework/Report: 20 percent;
subject to change. Notice: classroom participation is noted.
§ Ex’s of I[1, Ch 4]: p220 1.2; p223 1.3,1.4;p225 1.6,1.7;p226 1.9,1.10(BP),1.11;p229 2.2;p235 2.5,2.6,2.8,2.12;p236 2.14;p241 3.7,3.8;p246 3.12;p251 4.8,4.9;p260 5.2,5.3;p261
5.6,5.7;p262 5.8;p263 6.1;p268 6.3;p271 7.1;p273 7.4,7.6,7.8
§ Ex’s of II[1, Ch 4]: p377 1.2(path non-differentiability), 1.3(quadratic varia- tion);p382 2.3;p385 3.2; p386 3.5;p393 read:example 4.4;p397 read:(5.8);p398 read:(5.9), 5.3;p399 5.7
I. Martingales, adapted from [1, Ch. 4]
§ Notice: In [1, ch. 4] the time index is discrete, while in this course we need to the continuous-time.
§ Review from PT I: conditional expectation, filtration, martingale (fair game process), submartingale, supermartingale
§ a Hilbert space viewpoint of conditional expectation
§ stopping (optional) times
§ predictable sequence
§ discrete-time “ stochastic integral” process (H · X)n
§ stopped rv, stopped process; the stopped process is still a martingale
§ upcrossings of a sequence over an interval [a, b]
§ a key property: the upper bound for EUn, Un: the upcrossings up to time n
§ martingale convergence theorem
§ Doob-Meyer decomposition theorem, the discrete-time case
§ an example from SRW
§ no “bounded infinite oscillations”
§ a conditional form of BC Lemma
§ martingale property of a Galton-Watson branching process
§ a theorem on EXN, N bounded by k
§ Doob’s inequality (1)
§ Doob’s inequality (2)
§ martingale Lp, 1 < p <∞, convergence theorem
§ uniform integrability and L1 convergence: from RA
§ martingale L1 convergence theorem
§ L´evy’s convergence theorem and 0–1 law
§ the stopped process is still u.i.
§ a theorem on EXN, N any stopping time
§ stopped σ-algebra FN; a monotone lemma
§ optional stopping theorem
§ backwards (reversed) martingale and its convergence theorem
§ continuous-time martingale, the ”cadlag” requirement
§ stopping times for continuous-time case
§ the continuous-time Doob-Meyer decomposition theorem, ?!
§ the space of square integrable martingales
II. Brownian Motions, adapted from [1, Ch. 7]
§ math def of BM: Wiener process
§ remarks on the sample paths
§ finite dim’l distributions of BM
§ translation and scaling invariance of BM
§ BM as a Gaussian process, inversion invariance
§ construction of BM on “canonical space”
§ Kolmogorov’s path-continuity criterion
§ Markov property
§ Blumenthal’s 0–1 law, two applications
§ hitting time of BM
§ strong Markov property
§ zero set and level sets
§ first passage time
§ reflection principle
§ non-differentiability of Brownian paths
§ quadratic variation of Brownian paths
§ U(t) = B2(t)− t is a martingale
§ V (t) = exp(λB(t) − (1/2)λ2t) is a martingale
§ from RW to BM (Donsker’s invariance principle)
§ martingale CLT
III, Basic Itˆo Calculus, ,adapted from [2, Ch. 3 and Ch. 4]
§ Ito integral: first step
§ Ito isometry
§ Ito integral: next step
§ two basic examples
§ basic properties
§ Ito integral process: path continuity
§ Ito integral process: martingale property
§ Ito integral: final step
§ multi-dim’l Ito integral
§ Stratonovich integral
§ Ito process
§ Ito formula, how to operate (dX)2
§ two basic examples
§ integration by parts formula
§ multi-dim’l case
§ f(Bt)
§ Bessel process
§ Ito representation theorem
§ martingale representation theorem
§ a dense lemma for L(dP )
§ exponential martingale
§ quadratic variation of Ito process
§ BM local time and Tanaka formula
IV. Stochastic Diffusion Equations (SDE) , adapted from [2, Ch. 5]
§ 1-dim’l SDE and geometric Brownian Motion
§ existence and uniqueness theorem for SDE solution, remind of ODE
§ proof of the uniqueness, based on and Gronwall inequality
§ proof of the existence, based on Picard’s approximation
§ strong solution and weak solution, why we need weak solution
§ Tanaka equation
§ geometric Brownian Motion
§ Ornstein-Uhlenbeck equation ( Langevin equation)
§ Brownian bridge
§ BM on the ellipse
V. Itˆo Diffusions, adapted from [2, Ch. 7 and Ch. 8]
§ (time-homogeneous) Ito diffusion
§ Markov property
§ strong Markov property
§ generator of Ito diffusion
§ basic example: generator for BM and space-time BM
§ Dynkin formula
§ Kolmogorov equation
§ semigroup of operators
§ resolvent operator
§ Feynman-Kac formula
§ martingale property of Ito diffusion
§ martingale problem
§ Ito process versus Ito diffusion
§ (random) time-change and the equivalence theorem
§ time-change of BM
§ L´evy characterization of BM
§ quadratic variation for multi-dim’l Ito process
§ absolute continuity and Random-Nykodym derivative, revisited
§ Girsanov Theorem (various version)
§ toward a beginning: Stochastic Finance ( Itˆo Legacy)
