AMS 523, Mathematics of High Frequency Finance
The course explores Elements of real and complex linear spaces. Fourier series and transforms, the Laplace transform and z-transform. Elements of complex analysis including Cauchy theory, residue calculus, conformal mapping and Mobius transformations. Introduction to convex sets and analysis in finite dimensions, the Legendre transform and duality. Examples are given in terms of applications to high frequency finance.

Required Textbook:

"Linear Algebra in Action (Graduate Studies in Mathematics, Volume 232)" by Harry Dym; 3rd edition; 2023; American Mathematical Society; ISBN #978-1-4704-7418-8 (eBook)

 

Recommended Textbook:

"Conformal Mapping" by Zeev Nehari, Dover Publications, ISBN #9780486611372

"Complex Variables - Harmonic & Analytic Functions" by Francis J. Flanigan; 1983, Dover Publications; ISBN:  978-0486613888 (softcover)

FALL 3 credits, ABCF grading

Learning Outcomes:

1) Demonstrate mastery of basic concepts of linear spaces:
      * algebraic properties of scalar fields and linear spaces;
      * subspaces, span, independence, basis, linear transformations and similarities;
      * linear transformations with respect to bases for spaces;
      * Matrix multiplication as a group;
      * Triangular factorization and elimination. Rank. Schur complements, partitioned matrices.

2) Demonstrate mastery of invariant subspaces, characteristic polynomial and Cayley-Hamilton theorem.
      * One dimensional invariant subspaces (eigenvectors) and their eigenvalues; 
      * Existence of eigenvectors in finite dimensions, possible nonexistence in infinite dimensions;
      * Cyclic vectors.

3) Demonstrate mastery of transpositions and permutations, determinants, transpose:
      * Algebra of determinants;
      * Relation of determinants to singularity;
      * The value in theory of some determinant techniques (e.g. Cramer's rule) as opposed to computational limitations of those techniques.

4) Demonstrate mastery of topology concepts:
      * Basic topology in first countable spaces; 
      * Open and closed sets, interior, closure, convergence, compactness, Cauchy sequences and completeness;
      * Convex combinations and algebra of convex sets;
      * Metric spaces, norms, Banach spaces;
      * Connection between norms and convexity;
      * Duality in the context of linear programming.

5) Demonstrate mastery of unitary (inner product) spaces:
      * Unitary matrices. Normal matrices;
      * Householder reflections, the QR decomposition, Schur triangularization, the singular value decomposition and polar decomposition;
      * Positive definite matrices. Cholesky decomposition;
      * Diagonalization of normal matrices and the special cases for Hermitian and skew-Hermitian matrices;
      * Solution of systems of equations via unitary factorization;
      * Sensitivity to perturbation as exposed by the singular value decomposition;
       * The unitary structure of a space is determined by the norm (polarization identity);

6)  Demonstrate mastery of matrix norms:
      * Submultiplicative and unitarily invariant norms, estimates for singular values;
      * Condition numbers.

7) Demonstrate mastery of basic complex analysis:
      * Analyticity as opposed to real differentiability;
      * Cauchy-Riemann equations, harmonic functions;
      * Contour integrals, Cauchy theory;
      * Power series;
      * Understand the Hardy Hilbert norm both as square summable power series coefficients and as an integral mean.