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# 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|IEMS468 Application of coloured noise as a driving force in the stochastic differential equations

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## 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|Introduction

Monte carlo simulation is gaining popularity in areas such as oceanographic, atmospheric as well as electricity spot pricing applications. White noise is often used as important process in many of these applications which involve some error prediction as in $\mathrm{A}$. W. Heemink (1990); H.B. Fischer et al. (1979); J. R. Hunter et al. (1993); J.W. Stijnen et al. (2003). In these types of applications usually the deterministic models in the form of partial differential equations are available and employed. The solution is in most cases obtained by discretising the partial differential equations as in G.S. Stelling (1983). Processes such as transport of pollutants and sediments can be described by employing partial differential equations(PDEs). These well known PDEs are called advection diffusion equations. In particular when applied in shallow water e.g River, Lakes and Oceans, such effects of turbulence might be considered. However when this happens, it results into a set of partial differential equations. These complicated set of PDEs are difficult to solve and in most cases not easy to get a closed solution. In this chapter we explore the application coloured noise a a driving force to a set of stochastic differential equations(SDEs). These stochastic differential equations are sometimes called Random flight models. They are used for prediction of the dispersion of pollutants in atmosphere or in shallow waters e.g Lake, Rivers J. R. Hunter et al. (1993); R.W.Barber et al. (2005). Usually the advection and diffusion of pollutants in shallow waters use the well known partial differential equations called Advection diffusion equations(ADEs). These are consistent with the stochastic differential equations which are driven by Wiener processes as in C.W. Gardiner (2004); P.E. Kloeden et al. (2003). The stochastic differential equations which are driven by Wiener processes are called particle models. When the Kolmogorov’s forward partial differential equations(Fokker-Planck equation) is interpreted as an advection diffusion equation, the associated with this set of stochastic differential equations called particle model are derived and are exactly consistent with the advection-diffusion equation as in W. M. Charles et al. (2009). Still, neither the advection-diffusion equation nor the related traditional particle model accurately takes into account the short term spreading behaviour of particles. This is due to the fact that the driving forces are Wiener processes and these have independent increment. To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this article. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection-diffusion equation.

## 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|Coloured noise processes

In this part coloured noise forces are introduced and represent the stochastic velocities of the particles, induced by turbulent fluid flow. It is assumed that this turbulence is isotropic and that the coloured noise processes are stationary and completely described by their zero mean and Lagrangian auto covariance functionH.M. Taylor et al. (1998); W. M. Charles et al. (2009).

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