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Oggetto:

ADVANCED STOCHASTIC CALCULUS WITH APPLICATIONS

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ADVANCED STOCHASTIC CALCULUS WITH APPLICATIONS

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Academic year 2024/2025

Course ID
SEM0229
Teachers
Tiziano De Angelis (Lecturer)
Luca Regis (Lecturer)
Degree course
Generic
Year
2nd year
Teaching period
Second semester
Type
Elective
Credits/Recognition
6
Course disciplinary sector (SSD)
SECS-S/06 - mathematical methods of economy, finance and actuarial sciences
Delivery
Formal authority
Language
English
Attendance
Optional
Type of examination
Written
Prerequisites
Knowledge of measure theoretic foundations of probability, Brownian motion, stochastic differential equations and Ito formula.
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Sommario del corso

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Course objectives

This course illustrates advanced methods and techniques from stochastic calculus which are needed to solve various stochastic control problems. Applications of stochastic control are presented in selected examples from economics, finance and insurance. 

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Results of learning outcomes

At the end of the course students will have developed an understanding of stochastic control and its applications in economics, finance and insurance. Students will be capable of:

  • Formulating a stochastic control problem
  • Deriving the associated Hamilton-Jacobi-Bellman equation
  • Obtaining a verification theorem with a candidate optimal control 

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Program

  • Stochastic differential equations
  • Feynman-Kac formula
  • Introduction to optimal stopping
  • Introduction to stochastic control
  • Applications of optimal stopping to American option pricing
  • Applications of stochastic control to portfolio optimisation problems and insurance risk management

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Course delivery

 

The course is articulated in 48 hours of formal in-class lecture time and in at least as many hours of at-home work solving practical exercises.

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Learning assessment methods

The course grade is determined solely on the basis of a written test. The test evaluates the student's ability to do the following:

  1. Present briefly the main ideas, concepts and results developed in the course, also explaining intuitively the meaning and scope of the definitions and the arguments behind the validity of the results
  2. Use effectively the concepts and the results to answer questions around stochastic calculus, optimal stopping and stochastic control.

The above is accomplished by asking the student to answer open questions (2-4 questions). Questions can be essay questions or exercises. The minimum exam grade is 18/30, the maximum grade is 30/30 cum laude. More details on the exam can be found on Moodle. 

The exam is an open-book exam lasting 2 hours. Use of calculators is permitted.

Suggested readings and bibliography



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Book
Title:  
Stochastic calculus
Year of publication:  
2017
Publisher:  
Springer
Author:  
Paolo Baldi
ISBN  
Required:  
Yes


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Book
Title:  
Continuous-time stochastic control and optimization with financial applications
Year of publication:  
2009
Publisher:  
Springer
Author:  
Huyen Pham
ISBN  
Required:  
Yes


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Book
Title:  
Optimal stopping and free boundary problems
Year of publication:  
2006
Publisher:  
Springer
Author:  
Goran Peskir and Albert Shiryaev
ISBN  
Required:  
Yes


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Book
Title:  
Arbitrage theory in continuous time
Year of publication:  
2009
Publisher:  
Oxford University Press
Author:  
Tomas Bjork
ISBN  
Required:  
Yes


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