The Estimation Of A Nonlinear Model Based On Risky And Riskless Assets

In the 21st century,the study of financial mathematics is more important.The estimationof parameters in the financial model is the main aim to be dealt with.In the financial market,there is much uncertainty and fluctuations in asset price.This uncertaintyand fluctuations in the financial market is called risk.The risk can bring not only the earnings,but a loss than expected.In practice,investors can not invest in one asset.As we all know,put eggs in one basket is very dangerous.Once missed the basket,the eggs are all broken.Investors should put eggs in different baskets.usc the principles of dispersion ,set up scientific and reasonable investmentportfolio.For example,investors can invest in a risky asset and a riskless asset.In this paper,arising from a portfolio of a riskless asset(bonds) and a risky as-set(stocks) in a complete market,using discrete-time pricing.we have the following linearmodel:The above model can be regarded as a special case of the following linear model:For more general situation,we have the following nonlinear model:whereThe parametric vectorθ(t) is time-depmdent,Y(X(t)),(?)(t) is observable,Z(X(t)) is unobservable,satisfyingIn the financial market,X(t),Y(X(t)),(?)(t) can be regarded respectively as the price in a risky investment,total wealth,average increment of wealth.For the linear models above ,there already exist some research. But for the nonlinearmodels above,little of work has been done.We are having a thorough new job.This paper mainly discusses the estimation of the above nonlinear models.First according to the conditions Z~2(X(t)) satisfying, using nonparametric methods,we give the estimator of the volatility function Z~2(X(t)) in the condition of△_t being small and large.We also give the asymptotic properties of the estimators of Z~2(X(t)),and the proof of them.Second we give the estimation methods ofθ(t) and we also obtain the asymptotic properties and give the proof. In order to verify the validity of our estimation methods, we also make the simulation in this paper.The simulation results show that our methods are effective.It can be verified that the above models' limit forms associated with their terminalconditions are related the forward-backward differential equations.However.to the best of our knowledge ,there is no systematic theory about the estimation of the backwarddifferential equations.There is little research in this area.This paper actually also provide a method of estimating forward-backward differential equations.