| Fractional Brownian motion is a continuous Gaussian process.Neither it is Markov process nor semimartingale.When Hurst index is greater than 1/2,it has long-term memory and the increments are positively correlated.While when Hurst index is small-er than 1/2,its increments are negatively correlated.The stochastic analysis theory related to fractional Brownian motion has been widely studied and applied to many fields,such as climatology,hydrology and financial Mathematics.This article mainly studies the option pricing problems described by the fractional Brownian motion and the optimal control problems of discrete stochastic control system driven by fractional noise.Since the famous Black-Scholes formula of European option pricing was put for-ward in the 1970s,there has been a qualitative leap in the study of financial market.However,although this theory is a milestone in option pricing,its pricing result is somewhat different from the actual situation.Therefore,constant volatility can not explain the existing price volatility.Then many scholars began to study the stochastic volatility model,among which Heston model,Hull-White model and Stein-Stein model are widely used.Compared with the standard normal distribution curve,the real yield distribution curve is inclined,peaked,long-term memory and incremental correlation.In order to describe the law of the price of the real financial market more appropriately,the standard Brownian motion was replaced by the fractional Brownian motion in the volatility equation.According to the above statement,two fractional stochastic volatility models are considered in this article.In order to better describe the actual market price trend,we adopt the volatility equations driven by both fractional Brownian motion and s-tandard Brownian motion.The introduction of the fractional Brownian motion also increases the difficulty of solving the problems.We have to apply Ito's formula for fractional Brownian motion and Malliavin calculus to deal with the corresponding s-tochastic volatility problems.We first consider a general style of Hull-White stochastic volatility model.The volatility is not a mean reversion process and it increases expo-nentially.For increasing the flexibility of financial products and avoiding the risks to investors caused by excessive volatility,we add barriers to the volatility.Furthermore,we also consider another stochastic volatility model which satisfies fractional Ornstein-Uhlenbeck process.Because the volatility equation satisfy mean reversion,the price fluctuation is more stable and it also effectively reduces investors'risks.In these two models,we use the option replication strategy,Ito's formula and Malliavin calculus to obtain the partial differential equation that the price of European call option meets.Then we solve the partial differential equation and obtain a explicit solution of the price of European call option.The numerical simulations for the results are also made for proving the rationality.Next,we give a brief introduction of the main results of the option pricing under the above two fractional stochastic volatility models.We consider option pricing under fractional Hull-White stochastic volatility model in chapter 3:where St is risky asset price process and vt is instantaneous volatility.Here,?is the drift rate of risk asset price process and ? is the drift rate of the volatility process.?1 and ?2 are volatilities of volatility.BtS and Btv are two standard Brownian motion on(?,F,P)and their correlation coefficient is ?.BtH is fractional Brownian motion on(?,F,P)with Hurst parameter H>1/2 and independent with BtS and Btv.The explicit solution of the volatility is shown as follows:and it increases exponentially.So we add upper and lower barriers denoted by B and A.The partial differential equation of European call option price is obtained by using Malliavin calculus,Ito's formula and option replication strategy.The equation is introduced as follows:where ??T-t and T is the due date of European call option.We solve the above partial differential equation by introducing the Green's function and using the method of indeterminate coefficient.A closed-form solution is obtained asThe fractional Ornstein-Uhlenbeck process is considered to describe the stochastic volatility for pricing European option in chapter 4.The volatility is a mean reversion process and no longer increases exponentially.The stochastic volatility is written as where St is risky as,set price process and vt is instantaneous volatility.Here,? is the drift rate of risk asset price process and ? is the average recurrent rate of the volatility process.?1 and ?2 are volatilities of volatility.BtsandBtv are two mutually independent standard Brownian motion on(?,F,P).BtH is fractional Brownian motion on(?,F,P)with Hurst parameter H>1/2 and independent with BtS and Btv.The partial differential equation of the price of European call option is obtained by using the option replicat,ion strategy.The equation is shown as follows:where ?=T-t,?=?+?,????/?+?andWe solve the above partial differential equation and obtain an closed-form solution as where A(?,k),B(?,k)and C(?,k)are derived as Here,As a necessary condition to solve the optimal control problems,the Pontryagin's maximum principle was proposed by Pontryagin and his team in the 1950s.One al-ways obtains the discrete data in practical applications.Therefore,the discrete optimal control problems are more meaningful than the continuous case.However,the meth-ods to study optimal control problems in continuous case can not be directly applied to discrete cases.Some scholars considered to add some convex condition to obtain the maximum principle of discrete controlled system.The stochastic optimal control problems were studied extensively.A classical result was introduced by Peng[98]and he obtained a maximum principle of stochastic control system,where the control vari-ables were included in the diffusion term and the domain of the admissible control need not be convex.Compared to standard Brownian motion,the general fractional Brownian motion can better describe the actual phenomena and laws because of its long-term memory and self similarity.Some scholars gave results for discrete controlled system driven by white noise recently.According to the above statement,we consider maximum principle of the discrete stochastic optimal control problems driven by both fractional noise and white noise.Let{BtkH}k=0,1,2,…,N-1 be a sequence of d-dimunsion fractional Brownian motion that satisfies the following conditions:(?){BtkH}k=0,1,2,…,N is Fk-1-measurable.(?)The increments of(?)are stationary and need not be independent.(?)For every BtkH=(BtkH,1,BtkH,2,…,BtkH,d)T,BtkH,1,BtkH,2…,BtkH,dare indepen-dent R-valued Gaussian random variables.(?)(?)E[BtkH]=0,E[BtiH,lBtjH,l]=1/2(ti2H-|ti-tj|2H),wherel=1,2,…,d.Let gtk=Btk+1H-BtkHbe the fractional noise.Let D be a bounded domain and the space of admissible controls is defined asUad={Uk}(?)k=0N-1(?){vtk?Uk?Rn:??D|vtkisFk-1-mcasurable,E[vtkTvtk]<+?,k=0.1,…N-1?.The discrete controlled stochastic system in chapter 5 is described as with the cost functional as Our stochastic optimal control problem is to minimize the cost functional J(v)over Uad.Namely,to find the optimal u*?Uad satisfyingLet u*(·)be the optimal control and(a*(·),v*(·))be the optimal pair.A necessary condition which optimal control satisfies is derived by using classical variational method and Malliavin calculus and the main result is introduced as where DH(·)and D(·)are Malliavin derivatives with respect to fractional Brownian motion and standard Brownian motion.?tk satisfies the following equation:... |
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