Abstract
We consider a Stochastic Differential Equation (SDE) driven by a Wiener process and a Poisson measure. This latter measure is associated with a sequence of identically distributed jump amplitudes. Properties of the SDE solution are presented with respect to the associated Wiener and Poisson processes. An algorithm is provided allowing exact numerical simulations of such SDE and implementable within R environment. Statistical inference tools are presented and applied to hydrology data.
Highlights
IntroductionSome mathematicians use Stochastic Differential Equations (SDE) to model the random trajectories of these phenomena
In different fields, scientists are confronted with the study of random phenomena
Some mathematicians use Stochastic Differential Equations (SDE) to model the random trajectories of these phenomena. They are used in domains such as physics (Calif, 2012), population dynamics (Lungu and Oksendal, 1997), financial mathematics (Black and Scholes, 1973) and biology (Wilkinson, 2011)
Summary
Some mathematicians use Stochastic Differential Equations (SDE) to model the random trajectories of these phenomena. They are used in domains such as physics (Calif, 2012), population dynamics (Lungu and Oksendal, 1997), financial mathematics (Black and Scholes, 1973) and biology (Wilkinson, 2011). In financial mathematics, the Black-Scholes model (1973) is used to describe the volatility of certain options It is considered as a fundamental step forward for modern finance (Khaled and Samia, 2010). We consider a SDE with jumps driven by a Wiener process and a Poisson measure The solution of this SDE is a stochastic process following a Black-Scholes model with random jump amplitudes.
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