Stochastic Calculation and Neural Networks

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Goals

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1. 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.

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

3. 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.

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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

• I don't know

• I don't know

• Course syllabus II

1. Introduction to mathematics of derived products

• Case study: one-period model

  • Simplified model

  • Lack of opportunity

  • Geometric and probabilistic interpretation

• A case with N periods

  • Relationship with martingales

  • Discreet Black-Scholes 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