Lecture

Stochastic Calculus: Itô's Formula

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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 Calculus, focusing on Itô's Formula and Stochastic Differential Equations. It explains the core formula of stochastic calculus, the sketch of proof for Itô's Formula, the stochastic exponential, and the covariation revisited. The lecture also delves into the uniqueness lemma, martingale properties, and the geometric Brownian motion. Furthermore, it discusses stochastic differential equations, the solution methods, and the Black-Scholes model for option pricing. Linear SDEs, Ornstein-Uhlenbeck processes, mean-reversion, the Gauss property, and stationarity are also explored.

Instructors (2)

Semyon Malamud

Teaching & PhD PhD Students Johannes Schwab, Onur Demiray, Giuseppe Matera Past EPFL PhD Students Rémy Praz, Evgeny Petrov, Yuan Zhang, Julien Arsène Blatt, Erik Hapnes, Teng Andrea Xu, Boris Kuznetsov Courses Big Data and Machine Learning for Financial Economics FIN-622 This class is an introduction to Machine Learning and High Dimensional Statistics in Finance. We start with purely empirical approach, focusing first on high dimensional regressions then moving to kernel methods and deep l

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.

Official source

Ontological neighbourhood

Mathematics

Analysis: Differential anaysis

Related concepts (32)

Stochastic calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created and started by the Japanese mathematician Kiyosi Itô during World War II. The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces.

Stochastic process

In probability theory and related fields, a stochastic (stəˈkæstɪk) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.

Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Semimartingale

In probability theory, a real valued stochastic process X is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itô integral and the Stratonovich integral can be defined. The class of semimartingales is quite large (including, for example, all continuously differentiable processes, Brownian motion and Poisson processes).

Stratonovich integral

In stochastic processes, the Stratonovich integral or Fisk–Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral. Although the Itô integral is the usual choice in applied mathematics, the Stratonovich integral is frequently used in physics. In some circumstances, integrals in the Stratonovich definition are easier to manipulate. Unlike the Itô calculus, Stratonovich integrals are defined such that the chain rule of ordinary calculus holds.

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