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In his PhD dissertation, Bachelier (1900) tried, for the first time in history, to model the asset prices on the Paris stock exchange through stochastic processes. In particular, he used the so-called Brownian motions (or Wiener processes) simply because they proved themselves very useful for describing many natural phenomena (like the heat transfer). Finance, nowadays, heavily relies on Wiener processes (also called diffusion processes) for describing the dynamic behaviour of asset prices. More recently, and mainly because of the big financial crisis which burst in 2007/2008, also so-called jump processes have become relevant in finance: they describe the behaviour of a stochastic variable which may take a finite variation in an infinitesimal time interval (i.e. a so-called jump). In this book we will present the main theoretical properties of diffusion and jump processes together with numerical applications written in R.
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