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Lecture 6.2: Levy-Processes and non-Gaussian Ornstein-Uhlenbeck processes

Simulation of Poisson and compound Poisson processes. Simulation of subordinated Brownian motion and non-Gaussian Ornstein-Uhlenbeck processes.

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Lecture 2 - Simulation of basic processes

Examples: simulation from the multidimensional Gaussian law. Cholesky decomposition, PCA. Examples: generation of the skeleton of the multidimensional Brownian motion and the Ornstein-Uhlenbeck…

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Ph.D. course "The mathematics of energy markets" - Lecture 4

Brownian motion. Nondifferentiability of paths. The stochastic integral and the Ito isometry.

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Lecture 6: Functional limit theorems for financial markets with long-range dependence

* General conditions of weak convergence - Convergence of sums - Multiplicative scheme* Monotonicity along the diagonal of a triangular matrix in the Cholesky decomposition of a positive definite…

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Lecture 2: Equations with unbounded coefficients and H > 1/2

1. Cox-Ingersoll-Ross equations with fBm. 2. Equations with unbounded drift. 3. Reflected Ornstein-Uhlenbeck equations.

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Lecture 1: Standard stochastic differential equations with fBm

1. Integration with respect to fBm. 2. Elements of fractional and fractional stochastic calculus. 3. Stochastic differential equations with fBm and smooth coefficients.

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Lecture 62: martingale representation theorem

The martingale representation theorem in the case of a multidimensional Brownian motion: representation of L^2 martingales and local martingales.

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Lecture 58: the Girsanov theorem

The Girsanov theorem in the multidimensional case. Novikov's condition.

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Lecture 57: Levy's characterization

Stochastic integral of complex-valued processes. Levy's characterization of a multidimensional Brownian motion.

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Lecture 45: consequences of the Ito-Doeblin formula

A note on the integrability in the Ito-Doeblin formula. The two-dimensional case.

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Lecture 32: cross-variation

Cross-variation: definition. Cross-variation of independent and correlated Brownian motions.

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Lecture 31: the Cholesky decomposition

The Cholesky decomposition

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Lecture 30: multidimensional Brownian motion

Definitions of multidimensional standard and correlated Brownian motions: correlation matrix.

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Lectures 27-28: the Girsanov theorem and the martingale representation theorem

Levy's characterization of Brownian motion. Drift transformation in the Brownian motion and the Girsanov theorem. The martingale representation theorem. Representation of densities of…

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Lecture 16: the Ito-Doeblin formula

Stochastic integral with respect to an Ito process. The Ito-Doeblin formula. Quadratic variation of an Ito process.

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Lecture 12: examples of stochastic integrals

Stochastic integrals of simple processes. Stochastic integral of Brownian motion.

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