Continuous-time stochastic processes
In this chapter we introduce the elements of the theory of stochastic processes that we will use in continuous-time financial models. After a general presentation, we define the one-dimensional Brownian motion and we discuss some equivalence notions among stochastic processes. The most substantial part of the chapter is devoted to the study of the first and the second variation of a process: such a concept is introduced at first in the framework of the classical function theory and for Riemann-Stieltjes integration. Afterwards, we extend our analysis to the Brownian motion by determining its quadratic-variation process.
KeywordsStochastic Process Brownian Motion Risky Asset Markov Property Measurable Random Variable
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