Let X be a two parameter smooth semimartingale and (?) be its process of the product variation. It is proved that (?) can be approximated as D_∞-limit of sums of its discrete product variations as the mesh of divisio...Let X be a two parameter smooth semimartingale and (?) be its process of the product variation. It is proved that (?) can be approximated as D_∞-limit of sums of its discrete product variations as the mesh of division tends to zero. Moreover, this result can be strengthen to yield the quasi sure convergence of sums by estimating the speed of the convergence.展开更多
We prove that two parameter smooth continuous martingales have ∞-modification and establish a Doob’s inequality in terms of(p,r)-capacity for two parameter smooth martingales.
We obtained a number of inequalities and laws of large numbers for two parameter vector valued martingales.In the other direction we characterized p smoothness and q convexity of Banach spaces by using the...We obtained a number of inequalities and laws of large numbers for two parameter vector valued martingales.In the other direction we characterized p smoothness and q convexity of Banach spaces by using these inequalities and laws of large numbers for two parameter vector valued martingales.展开更多
Let {Y (t);t= (t 1,t 2)≥0}={X k(t 1,t 2);t 1≥0,t 2≥0} ∞ k=1 be a sequence of two parameter Ornstein Uhlenbeck processes (OUP 2) with coefficients α k>0,β k>0 . A Fernique type in...Let {Y (t);t= (t 1,t 2)≥0}={X k(t 1,t 2);t 1≥0,t 2≥0} ∞ k=1 be a sequence of two parameter Ornstein Uhlenbeck processes (OUP 2) with coefficients α k>0,β k>0 . A Fernique type inequality is established and the sufficient condition for a.s. l 2 continuity of Y(·) is studied by means of the inequality.展开更多
In part I and II of this series, experimental investigation in both EPFM and LEFM had been discussed. In this part, further theoretical analysis is given. The theoretical development of Two Parameter Fracture Mechanic...In part I and II of this series, experimental investigation in both EPFM and LEFM had been discussed. In this part, further theoretical analysis is given. The theoretical development of Two Parameter Fracture Mechanics by Hancock etc, has rationalized our experimental results. This method can be applied to engineering practice, and will allow the advantage of enhanced toughness for specimens with low levels of constraint to be taken into account for defect assessment.展开更多
Kenyan insurance firms have introduced insurance policies of chronic illnesses like cancer</span><span style="font-family:Verdana;">;</span><span style="font-family:Verdana;"&g...Kenyan insurance firms have introduced insurance policies of chronic illnesses like cancer</span><span style="font-family:Verdana;">;</span><span style="font-family:Verdana;"> however</span><span style="font-family:Verdana;">,</span><span style="font-family:Verdana;"> they have faced a huge challenge in the pricing of these policies as cancer can transit into different stages</span><span style="font-family:Verdana;">,</span><span style="font-family:Verdana;"> which consequently leads to variation in the cost of treatment. This has made the estimation of aggregate losses of diseases which have multiple stages of transitions such as cancer</span><span style="font-family:Verdana;">,</span><span style="font-family:""><span style="font-family:Verdana;"> an area of interest of many insurance firms. Mixture phase type distributions can be used to solve this setback as they can in-cooperate the transition in the estimation of claim frequency while also in-cooperating the he</span><span style="font-family:Verdana;">terogeneity aspect of claim data. In this paper</span></span><span style="font-family:Verdana;">,</span><span style="font-family:Verdana;"> we estimate the aggregate losses</span><span style="font-family:""><span style="font-family:Verdana;"> of secondary cancer cases in Kenya using mixture phase type Poisson Lindley distributions. Phase type (PH) distributions for one and two parameter Poisson Lindley are developed as well their compound distributions. The matrix parameters of the PH distributions are estimated using continuous Chapman Kolmogorov equations as the disease process of cancer is continuous while severity is modeled using Pareto, Generalized Pareto and Weibull distributions. This study shows that aggregate losses for Kenyan data are best estimated using PH-OPPL-Weibull model in the case of PH-OPPL distribution models and PH-TPPL-Generalized Pareto model in the case of PH-TPPL distribution models. Comparing the two best models, PH-OPPL-Weibull model provided the best fit for secondary cancer cases in Kenya. This model is also </span><span style="font-family:Verdana;">recommended for different diseases which are dynamic in nature like cancer.展开更多
文摘Let X be a two parameter smooth semimartingale and (?) be its process of the product variation. It is proved that (?) can be approximated as D_∞-limit of sums of its discrete product variations as the mesh of division tends to zero. Moreover, this result can be strengthen to yield the quasi sure convergence of sums by estimating the speed of the convergence.
文摘We prove that two parameter smooth continuous martingales have ∞-modification and establish a Doob’s inequality in terms of(p,r)-capacity for two parameter smooth martingales.
文摘We obtained a number of inequalities and laws of large numbers for two parameter vector valued martingales.In the other direction we characterized p smoothness and q convexity of Banach spaces by using these inequalities and laws of large numbers for two parameter vector valued martingales.
基金Research supported by National Natural Science Foundation of China(1 0 0 71 0 2 7)
文摘Let {Y (t);t= (t 1,t 2)≥0}={X k(t 1,t 2);t 1≥0,t 2≥0} ∞ k=1 be a sequence of two parameter Ornstein Uhlenbeck processes (OUP 2) with coefficients α k>0,β k>0 . A Fernique type inequality is established and the sufficient condition for a.s. l 2 continuity of Y(·) is studied by means of the inequality.
文摘In part I and II of this series, experimental investigation in both EPFM and LEFM had been discussed. In this part, further theoretical analysis is given. The theoretical development of Two Parameter Fracture Mechanics by Hancock etc, has rationalized our experimental results. This method can be applied to engineering practice, and will allow the advantage of enhanced toughness for specimens with low levels of constraint to be taken into account for defect assessment.
文摘Kenyan insurance firms have introduced insurance policies of chronic illnesses like cancer</span><span style="font-family:Verdana;">;</span><span style="font-family:Verdana;"> however</span><span style="font-family:Verdana;">,</span><span style="font-family:Verdana;"> they have faced a huge challenge in the pricing of these policies as cancer can transit into different stages</span><span style="font-family:Verdana;">,</span><span style="font-family:Verdana;"> which consequently leads to variation in the cost of treatment. This has made the estimation of aggregate losses of diseases which have multiple stages of transitions such as cancer</span><span style="font-family:Verdana;">,</span><span style="font-family:""><span style="font-family:Verdana;"> an area of interest of many insurance firms. Mixture phase type distributions can be used to solve this setback as they can in-cooperate the transition in the estimation of claim frequency while also in-cooperating the he</span><span style="font-family:Verdana;">terogeneity aspect of claim data. In this paper</span></span><span style="font-family:Verdana;">,</span><span style="font-family:Verdana;"> we estimate the aggregate losses</span><span style="font-family:""><span style="font-family:Verdana;"> of secondary cancer cases in Kenya using mixture phase type Poisson Lindley distributions. Phase type (PH) distributions for one and two parameter Poisson Lindley are developed as well their compound distributions. The matrix parameters of the PH distributions are estimated using continuous Chapman Kolmogorov equations as the disease process of cancer is continuous while severity is modeled using Pareto, Generalized Pareto and Weibull distributions. This study shows that aggregate losses for Kenyan data are best estimated using PH-OPPL-Weibull model in the case of PH-OPPL distribution models and PH-TPPL-Generalized Pareto model in the case of PH-TPPL distribution models. Comparing the two best models, PH-OPPL-Weibull model provided the best fit for secondary cancer cases in Kenya. This model is also </span><span style="font-family:Verdana;">recommended for different diseases which are dynamic in nature like cancer.
