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Lecture

Stochastic Optimal Control: Martingale Theorem

In course

FIN-415: Probability and stochastic calculus

This course gives an introduction to probability theory and stochastic calculus in discrete and continuous time. The fundamental notions and techniques introduced in this course have many applications in finance, for example for option pricing, risk mana ...

Description

This lecture covers Stochastic Optimal Control, focusing on Optimal Consumption and Investment, the Martingale Representation Theorem, and the Verification Theorem. It delves into dynamic programming, the HJB Equation, and the Linear Regulator. The lecture also discusses the optimal investment/consumption law, the Bernoulli Equation, and the State-Price Deflator Property.

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Instructors (2)

Damir Filipovic

Damir Filipovic holds the Swissquote Chair in Quantitative Finance and is Swiss Finance Institute Professor at the Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland. Prior to this, he was head of the Vienna Institute of Finance and professor at the University of Vienna. He previously held the chair of financial and insurance mathematics at the University of Munich, and he was on the faculty of Princeton University. He received his Ph.D. in mathematics from ETH Zurich in 2000. Damir Filipovic worked as a scientific consultant for the Swiss Federal Office of Private Insurance from 2003 to 2004. There he co-developed the Swiss Solvency Test, which defines the regulatory capital requirement for all Swiss based insurance companies and groups. He is on the editorial board of several academic journals. His research interests include the term structure of interest rates, credit and volatility risk, quantitative methods in risk management, and stochastic processes. His papers have been published in a variety of academic journals including the Journal of Financial Economics, Mathematical Finance, Finance and Stochastics, and the Annals of Applied Probability. He is the author of a textbook titled Term-Structure Models.

Semyon Malamud

Official source

Ontological neighbourhood

Economics

Microeconomics: Financial economics

Related concepts (32)

Bellman equation

A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. It writes the "value" of a decision problem at a certain point in time in terms of the payoff from some initial choices and the "value" of the remaining decision problem that results from those initial choices. This breaks a dynamic optimization problem into a sequence of simpler subproblems, as Bellman's “principle of optimality" prescribes.

Intertemporal portfolio choice

Intertemporal portfolio choice is the process of allocating one's investable wealth to various assets, especially financial assets, repeatedly over time, in such a way as to optimize some criterion. The set of asset proportions at any time defines a portfolio. Since the returns on almost all assets are not fully predictable, the criterion has to take financial risk into account. Typically the criterion is the expected value of some concave function of the value of the portfolio after a certain number of time periods—that is, the expected utility of final wealth.

Modern portfolio theory

Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return.

Optimal control

Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure.

Isoelastic utility

In economics, the isoelastic function for utility, also known as the isoelastic utility function, or power utility function, is used to express utility in terms of consumption or some other economic variable that a decision-maker is concerned with. The isoelastic utility function is a special case of hyperbolic absolute risk aversion and at the same time is the only class of utility functions with constant relative risk aversion, which is why it is also called the CRRA utility function.

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