- Comprehend the construction and properties of the Riemann-Stieltjes-Integral;
- Know the path properties of the Wiener process and independently prove elementary properties;
- Understand the construction of the Ito integral;
- Understand the properties if the Ito integral and prove them indepndently in simple situations;
- Know and apply Ito's lemma;
- Comprehend the construction of stochastic differential equations and understand the proof of existence and uniqueness;
- Know standard examples of solutions of stochastic differential equations, e.g., the geometrical Brownian motion and the Ornstein-Uhlenbeck process;
- Apply transformation methods for stochastic differential equations, e.g., Lamperti's transform and Girsanov's theorem;
- Know the construction of the Stratonovich integral and its relationship to the Ito integral;
- Comprehend the definition of Markov and diffusion processes and apply them to stochastic differential equations;
- Understand the relationship between stochastic differential equations and partial differential equations, in particular determine Kolmogorov's forward and backward equations or the Fokker-Planck equation, respectively, from the coefficients of a stochastic differential equation.
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Riemann-Stieltjes integral, Construction and properties of the Ito integral, Ito's Lemma, stochastic differential equations, geometric Brownsian motion and Ornstein-Uhlenbeck process, transformation methods for SDEs, Girsanov's theorem, Stratonovich integral, Markov- and Diffusion processes, Kolmogorov's forward and backward equation, Fokker-Planck equation, connection to partial differential equations
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