Stochastic Calculation and Neural Networks
Goals

Study the theoretical and technical bases of stochastic processes that allow understanding the ideas and scope of Stochastic Calculus with an emphasis on financial theory.

Invite the student to the methods of deep learning through the training algorithms related to the stochastic approach: stochastic gradient.

Study some of the main hypotheses in finance related to the Brownian Movement as well as its interaction with Ito's formal definition of integral.
Course syllabus I
1. Martingales

Basic notions of probability spaces and random variables

Conditional hope and leaks

Basic definitions of Martingales

First examples: discrete case

Doob inequality and its financial interpretation

Law of large numbers for Martingales

Relationship with Markov chains and ergodic theorems

Polya Urns
2. Neural networks and stochastic approach

Classic Perceptron algorithm

Stochastic descending gradient algorithm for the Perceptron

Neural networks in general

Stochastic gradient in general

Relationship with Martingales and Stochastic Approach Algorithms

General Stochastic Approach
3. Brownian Movement

Formal definition

Gaussian processes and random beds

Markov property

Wiener integral and relation to stochastic integral

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Course syllabus II
1. Introduction to mathematics of derived products

Case study: oneperiod model

Simplified model

Lack of opportunity

Geometric and probabilistic interpretation


A case with N periods

Relationship with martingales

Discreet BlackScholes formula

2. Stochastic integral

Formal definition

Properties and first examples

Local Martingale

Ito processes

Ito's fundamental motto
3. Applications and relationships with deep learning