由马氏链驱动的正倒向随机微分方程比较定理
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摘要
为了研究由马氏链驱动的完全耦合的正倒向随机微分方程的比较定理,采用在研究通常的完全耦合的正倒向随机微分方程时常用的连续性方法,通过运用半鞅的Ito乘积法则与Lebesgue控制收敛定理,得到由马氏链驱动的完全耦合的正倒向随机微分方程的2个关于初值的比较定理。结果表明,对于2个结构相同、仅过程X_t的初值X_0不同的由马氏链驱动的完全耦合的正倒向随机微分方程,在正向过程X_t与倒向过程Y_t这2个过程的取值都是一维实数值,并且在2个正倒向随机微分方程都满足单调性条件,从而解都存在且唯一的条件下,X_0越大,则过程Y_t的初值Y_0越大。
To study the comparison theorems of fully coupled forward-backward stochastic differential equations on Markov chains,using the method of continuation usually used to study fully coupled forward-backward stochastic differential equations,the Ito product rule of Semimartingales,and the Lebesgue control convergence theorem,two comparison theorems of the fully coupled forward-backward stochastic differential equations on Markov chains for solutions of initial values were obtained. The results show that for the two forward-backward stochastic differential equations on Markov chains with the same structure and different initial values,if all of the values of X_t and Y_t are real numbers,and the two forwardbackward stochastic differential equations satisfy the monotonicity conditions so that each forward-backward stochastic differential equation has a unique solution,then the greater X_0 is,the greater the initial value Y_0 of the process Y_t is.
To study the comparison theorems of fully coupled forward-backward stochastic differential equations on Markov chains,using the method of continuation usually used to study fully coupled forward-backward stochastic differential equations,the Ito product rule of Semimartingales,and the Lebesgue control convergence theorem,two comparison theorems of the fully coupled forward-backward stochastic differential equations on Markov chains for solutions of initial values were obtained. The results show that for the two forward-backward stochastic differential equations on Markov chains with the same structure and different initial values,if all of the values of X_t and Y_t are real numbers,and the two forwardbackward stochastic differential equations satisfy the monotonicity conditions so that each forward-backward stochastic differential equation has a unique solution,then the greater X_0 is,the greater the initial value Y_0 of the process Y_t is.
引文
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[2]COHEN S,ELLIOTT R.Solutions of backward stochastic differential equations on markov chains[J].Communications on Stochastic Analysis,2008,2(2):251-262.
[3]COHEN S,ELLIOTT R.Comparisons for backward stochastic differential equations on markov chains and related no-arbitrage conditions[J].The Annals of Applied Probability,2010,20(1):267-311.
[4]COHEN S,ELLIOTT R,CHARLES E.A general comparison theorem for backward stochastic differential equations[J].Advances in Applied Probability,2010,42(3):878-898.
[5]COHEN S.Representing filtration consistent nonlinear expectations as g-expectations in general probability spaces[J].Stochastic Processes and their Applications,2012,122(4):1601-1626.
[6]HU Y,PENG S G.Solutions of forward-backward stochastic differential equations[J].Probability Theory and Related Fields,1995,103(2):273-283.
[7]PENG S G,WU Z.Fully coupled forward-backward stochastic differential equations[J].SIAM Journal on Control and Optimization,1999,37(3):825-843.
[8]WU Z.The comparison theorem of FBSDE[J].Statistics and Probability Letters,1999,44(1):1-6.
[9]WU Z.Forward-backward stochastic differential equations with brownian motion and poisson process[J].Acta Mathematicae Applicatae Sinica,1999,15(4):433-443.
[10]JI S L,LIU H D,XIAO X L.Fully coupled forward-backward stochastic differential equations on markov chains[J].Advances in Difference Equations,2016,2016(133):1-18.