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A comparison theorem for solutions of stochastic differential equations and its applications


Author: Zhi Yuan Huang
Journal: Proc. Amer. Math. Soc. 91 (1984), 611-617
MSC: Primary 60H10; Secondary 34F05
DOI: https://doi.org/10.1090/S0002-9939-1984-0746100-9
MathSciNet review: 746100
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Abstract: A new kind of comparison theorem in which an SDE is compared with two deterministic ODEs is established by means of the generalized sample solutions of SDEs. Using this theorem, we can compare solutions of two SDEs with different diffusion coefficients and obtain some asymptotic estimations for the paths of diffusion processes.


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  • [1] A. V. Skorohod, Studies in the theory of random processes, Addison-Wesley, Reading, Mass., 1965. MR 0185620 (32:3082b)
  • [2] A. F. Fillipov, Differential equations with discontinuous right hand side, Mat. Sb. (N.S.) 51 (1960), 99-128. (Russian) MR 0114016 (22:4846)
  • [3] T. Yamada, On a comparison theorem for solutions of stochastic differential equations and its applications, J. Math. Kyoto Univ. 13 (1973), 497-512. MR 0339334 (49:4093)
  • [4] N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math. 14 (1977), 619-633. MR 0471082 (57:10822)
  • [5] H. Doss, Liens entre équations différentielles stochastique et ordinaires, Ann. Inst. H. Poincaré Sect. A (N.S.) 13 (1977), 99-125. MR 0451404 (56:9690)
  • [6] H. Doss and E. Lenglart, Sur l'existence, l'unicité et le comportement asymptotique des solutions d'équations différentielles stochastique, Ann. Inst. H. Poincaré Sect. A (N.S.) 14 (1978), 189-214. MR 507733 (80c:60076)
  • [7] P. Malliavin, Géométrie différentielle stochastique, Les Presses de l'Université de Montréal, Montréal, 1978. MR 540035 (81d:60077)
  • [8] G. L. O'Brien, A new comparison theorem for solutions of stochastic differential equations, Stochastics 3 (1980), 245-249. MR 581039 (82f:60139)
  • [9] T. Yamada and Y. Ogura, On the strong comparison theorem for solutions of stochastic differential equations, Z. Wahrsch. Verw. Gebiete 56 (1981), 3-19. MR 612157 (82f:60142)
  • [10] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland, Kodansha, 1981. MR 1011252 (90m:60069)
  • [11] Z. Huang, M. Xu and Z. Hu, On the generalized sample solutions of stochastic differential equations, Wuhan Univ. J. (Natural Sci. Ed.) 2 (1981), 11-21. (Chinese) MR 651340 (83e:60059)
  • [12] P. Protter, On the existence, uniqueness, convergence and explosions of solutions of systems of stochastic differential equations, Ann. Probab. 5 (1977), 243-261. MR 0431380 (55:4380)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0746100-9
Keywords: Stochastic differential equation, comparison theorem, generalized sample solution, diffusion coefficient, Carathéodory conditions, explosion time, smooth approximation
Article copyright: © Copyright 1984 American Mathematical Society

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