MAT4760 - Spring 2011

Teaching plan

DateTeacherPlaceTopicLecture notes / comments
14.04.2011GdN? ? Dynamic risk measures and risk measures derived from g-conditional expectations? Definition of dynamic risk measures, properties of these measures, discussion about time consistency. Examples of dynamic risk measures. Contextualized to the example of coherent risk measures, discussion about time consistency and m-stability.

Dynamic risk measures derived from g-conditional expectation: definition and properties.

Discussion on the result describing the existence and uniqueness of a driver g and a corresponding g-conditional expectation derived from a given dynamic risk measure.

Reference: [RG] Section 4.?

07.04.2011GdN? ? Static risk measures derived from g-expectations ? Continuation of the topic. Representation of risk measures derived from g-expectations. Discussion of the properties of monotonicity.

Example: risk measures, BSDEs, and pricing (Black and Scholes classical set-up).

Reference: [RG] Section 2 and 3. ?

31.03.2011GdN? ? Static risk measures derived from g-expectations? Definition of risk measure derived from g-expectation. Analysis of the properties.

Reference: [RG] Section 2. ?

24.03.2011GdN? ? g-expectations ? Monotonic limit theorem for BSDEs

Reference: [Peng] p. 199?

17.03.2011GdN? ? g-expectations ? Continuation on g-expectations and its properties.

Reference: [Peng] page 191, Proposition 3.7, continuity in L2, characterization of a norm in L2 given by g-expectations of a certain type. definition 3.5, proposition 3.8 and 3.9.

Definition of g-martingales (submartingales, supermartingales) and covergence result.

Reference: [Peng] Definition 3.4, Theorem 3.7 (no proof)?

10.03.2011GdN? ? BSDEs and g-expectations? Linear BSDEs and comparison theorem for BSDEs.

Reference: [Pham] Section 6.2.2, 6.2.3 pages 141-143.

Definition of g-expectation and discussion on the properties that make this concept be regarded as a generalization of an expectation.

Reference: [Peng] Definition 3.2, Proposition 3.6, Theorem 3.4, Lemma 3.2.?

03.03.2011GdN? ? Backward stochastic differential equations? Revision of basic concepts on various forms of measurability of stochastic processes with respect to a filtration,stopping times, local martingales, stochastic Ito integral in the classical L_2 setting for Brownian motion, generalized Ito integral for Brownian motion (local martingale).

References: [Pham] Sections 1.1, 1.2

Books by:

  • B. ?ksendal: Stochastic differential equations,Springer

  • I. Karatzas and S.E. Shreve: Brownian motion and stochastic calculus, Springer

Introduction to the topic of BSDEs. Definition, existence and uniqueness of the solution

[Pham] pages 139-141?

24.02.2011GdN? ? Putting order in risk measures? Risk measures in L\infty and Lp.

Spaces of financial positions and their dual in the Lp and L\infty settings. Revision of representation theorems of linear functionals (functional analysis).

General point of view on representations for risk measures (the importance of lower semicontinuity)

Reference: M. Frittelli and E. Rosazza Gianin: Putting order in risk measures. Journal of banking and finance (2002), vol. 26, pages 1473-1486.?

17.02.2011GdN? ? Convex risk measures on L_\infty? Representation theorems, definition of relevance. entropic risk measures

[FS] pages 172-174, 122.

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10.02.2011GdN? ? Representation of convex risk measures. Convex risk measures on L_\infty? Introduction to the Fenchel-Legendre transform and its relationship to risk measures representations

[FS] pages 164, 402, 430-431

Representation of convex risk measures in terms of sets of probability measures

[FS] pages 166-167 (no proof of Lemma 4.22, no Example 4.24)

Convex risk measures on L_\infty. Equivalence relationship

[FS] pages 171-172.

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03.02.2011GdN? ? Spaces of measures that are finitely additive. Representation theorems for convex risk measures? Spaces of measures that are finitely additive

[FS] pages 426-427 (from Appendix A.6)

Introduction to representationof convex risk measures. Definition of penalty function

[FS] page 161

Representation theorem for convex risk measures (use of the separating hyperplane theorem)

[FS] pages 162-163.

Representation of coherent risk measures

[FS] pages 164-165. (no Prop. 4.19)?

27.01.2011GdN? ? ? Given a class of acceptable positions, one can define a monetary risk measure.

[FS] pages 156-157 (Prop. 4.7)

Examples of risk measures: [FS] pages 157-159 (Examples 4.8, 4.11) page 178 (Example 4.41)?

20.01.2011GdN? ? ? Each monetary risk measure induces a class of acceptable positions - acceptance set

[FS] pages 155-156 (Prop. 4.6)?

17.01.2011Giulia diNunno? ? Monetary risk measures - static setting? Quantification of the risk of financial positions. Motivations. Definition of monetary risk measure in a static setting.

[FS = Follmer and Schied] pages 152-155?

Published Jan. 24, 2011 3:05 PM - Last modified Apr. 14, 2011 6:04 PM
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