Research On Dynamic Properties Of Some Stochastic Models

In this academic dissertation,we study the dynamic properties of three kinds of stochastic models and achieve some results.They are the stochastic Fisher-KPP equation,the Kaldor macroeconomic model with macro shock and the pricing model for credit rating migration risk.In Chapter 2,traveling wave for a Fisher-KPP equation with stochastic advection and stochastic environmental capacity is investigated.Some conditions are imposed on the reaction rate and noise intensities such that the stochastic transition front exists.Following the results on stochastic transition front,the existence of stochastic traveling waves for the equation is established.Explicit relation between the wave speed and noise attributes including noise intensities and correlation is shown,which can realize the noise effects.It is found that noises reduce the wave speed.In addition,the positive correlation of noises may complement this reduction in a way.But the negative correlation of noises will further aggravate this reduction.There exists a threshold value on the noise correlation making the traveling wave wandering.If the correlation is larger than this threshold value,the wave travels with a forward tendency.Otherwise,the wave travels with a backward tendency.Bifurcations for asymptotic behaviors of the equation induced by the noise intensities and correlation are presented.In Chapter 3,we develop a Kaldor macroeconomic model with shock.The shock is due to the investment uncertainty.We then provide an approach for macroeconomic control by calibrating the evolvement of the shocked Kaldor macroeconomic model with some expected benchmark process.The calibration is realized through the setting for investment.The benchmark process is usually the reflection of decisions or policies.An optimal investment setting associated with a five-dimensional nonlinear system of ordi-nary differential equations is presented.Through a logical modification for the boundary conditions,the nonlinear system is simplified to be linear and a completely explicit for-mula for the optimal investment setting is achieved.To cope with the systematic risk caused by the macro shock,we define a dynamic Value-at-Risk(VaR)as the risk measure capturing the risk level of the shocked Kaldor macroeconomic model and introduce a risk constraint into the programming of calibration.Then a constrained investment setting is presented.Finally,we carry out an application of the theoretical results by calibrat-ing the evolvement of the shocked Kaldor macroeconomic model with the business cycle generated from the classical Kaldor model through the investment setting.In Chapter 4,a pricing model for corporate bond with credit rating migration risk is proposed.Multiple credit ratings are considered in the model.The credit rating migration is captured through the dynamic proportion of the debt and value of the firm.The solution of the model satisfies a free boundary problem of partial differential equation.Then the existence,uniqueness and regularity of the solution are obtained together with some interesting properties,which theoretically supports the rationality of the pricing model.In Chapter 5,we introduce the risk discount rate to the model in Chapter 4.Except the existence and uniqueness of the solution,by inductive method,we establish the existence of traveling wave in the free boundary problem as the risk discount rate is between the volatility in the highest credit rating and the volatility in the lowest credit rating.Furthermore,we prove the solution of the free boundary problem converges to the traveling wave,which provides an intuition for the price volatility of the corporate bond such that we can better understand it.