On the stochastic differential equation for a two-dimensional Brownian motion with boundary condition

Masaaki TSUCHIYA

doi:10.2969/jmsj/03220233

Masaaki TSUCHIYA

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JOURNAL FREE ACCESS

1980 Volume 32 Issue 2 Pages 233-249

DOI https://doi.org/10.2969/jmsj/03220233

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Published: 1980 Received: January 09, 1978 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) E. D. Conway, Stochastic equations with discontinuous drift I, II, Trans. Amer. Math. Soc., 157 (1971), 235-245 and Indiana Univ. Math. J., 22 (1972), 91-99.
2) K. L. Chung and J. L. Doob, Fields, optionality and measurability, Amer. J. Math., 87 (1966), 397-424.
3) C. Dellacherie, Capacités et processus stochastiques, Springer-Verlag, Berlin-Heiderberg-New York, 1972.
4) C. Dellacherie and P. A. Meyer, Probabilités et potentiel, Hermann, Paris, 1975.
5) N. Ikeda, On the construction of two dimensional diffusion processes satisfying Wentzell's boundary conditions and its application to boundary value problems, Mem. Coll. Sci. Univ. Kyoto, Ser. A, 33 (1961), 367-427.
6) N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633.
7) T. Komatsu, Markov processes associated with certain integro-differential operators, Osaka J. Math., 10 (1973), 271-303.
8) J. -P. Lepeltier and B. Marchal, Problème des martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel, Ann. Inst. H. Poincaré, Sect. B, 12 (1976), 43-103.
9) P. A. Meyer, Probabilités et potentiel, Hermann, Paris, 1966.
10) M. Motoo, Brownian motions in the half plane with singular inclined periodic boundary conditions, Topics in Probability Theory, edited by D. W. Stroock and S. R. S. Varadhan, New York Univ., 1973, 163-179.
11) S. Nakao, On the existence of solutions of stochastic differential equations with boundary conditions, J. Math. Kyoto Univ., 12 (1972), 151-178.
12) S. Nakao and T. Shiga, On the uniqueness of solutions of stochastic differential equations with boundary conditions, J. Math. Kyoto Univ., 12 (1972), 451-478.
13) D. W. Stroock, Diffusion processes associated with Levy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32 (1975), 209-244.
14) H. Tanaka, M. Tsuchiya and S. Watanabe, Perturbation of drift-type for Lévy processes, J. Math. Kyoto Univ., 14 (1974), 73-92.
15) M. Tsuchiya, On the stochastic differential equation for a Brownian motion with oblique reflection on the half plane, Proc. Intern. Symp. Stoch. Diff. Eq., edited by K. Ito, Kinokuniya, Tokyo, 1978, 427-436.
16) E. E. Viktrovskii, On a generalization of the concept of integral curves for a discontinuous field of directions, Mat. Sb., 34(76) (1954), 213-248 (in Russian).
17) S. Watanabe, On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions I, II, J. Math. Kyoto Univ., 11 (1971), 169-180 and 545-551.
18) S. Watanabe, Stochastic differential equations, Sangyo Tosho, Tokyo, 1975 (in Japanese).
19) T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11 (1971), 155-167.

Right : [1] E. D. Conway, Stochastic equations with discontinuous drift I, II, Trans. Amer. Math. Soc., 157 (1971), 235-245 and Indiana Univ. Math. J., 22 (1972), 91-99.
[2] K. L. Chung and J. L. Doob, Fields, optionality and measurability, Amer. J. Math., 87 (1966), 397-424.
[3] C. Dellacherie, Capacités et processus stochastiques, Springer-Verlag, Berlin-Heiderberg-New York, 1972.
[4] C. Dellacherie and P. A. Meyer, Probabilités et potentiel, Hermann, Paris, 1975.
[5] N. Ikeda, On the construction of two dimensional diffusion processes satisfying Wentzell's boundary conditions and its application to boundary value problems, Mem. Coll. Sci. Univ. Kyoto, Ser. A, 33 (1961), 367-427.
[6] N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633.
[7] T. Komatsu, Markov processes associated with certain integro-differential operators, Osaka J. Math., 10 (1973), 271-303.
[8] J. -P. Lepeltier and B. Marchal, Problème des martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel, Ann. Inst. H. Poincaré, Sect. B, 12 (1976), 43-103.
[9] P. A. Meyer, Probabilités et potentiel, Hermann, Paris, 1966.
[10] M. Motoo, Brownian motions in the half plane with singular inclined periodic boundary conditions, Topics in Probability Theory, edited by D. W. Stroock and S. R. S. Varadhan, New York Univ., 1973, 163-179.
[11] S. Nakao, On the existence of solutions of stochastic differential equations with boundary conditions, J. Math. Kyoto Univ., 12 (1972), 151-178.
[12] S. Nakao and T. Shiga, On the uniqueness of solutions of stochastic differential equations with boundary conditions, J. Math. Kyoto Univ., 12 (1972), 451-478.
[13] D. W. Stroock, Diffusion processes associated with Lévy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32 (1975), 209-244.
[14] H. Tanaka, M. Tsuchiya and S. Watanabe, Perturbation of drift-type for Lévy processes, J. Math. Kyoto Univ., 14 (1974), 73-92.
[15] M. Tsuchiya, On the stochastic differential equation for a Brownian motion with oblique reflection on the half plane, Proc. Intern. Symp. Stoch. Diff. Eq., edited by K. Itô, Kinokuniya, Tokyo, 1978, 427-436.
[16] E. E. Viktrovskii, On a generalization of the concept of integral curves for a discontinuous field of directions, Mat. Sb., 34 (76) (1954), 213-248 (in Russian).
[17] S. Watanabe, On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions I, II, J. Math. Kyoto Univ., 11 (1971), 169-180 and 545-551.
[18] S. Watanabe, Stochastic differential equations, Sangyo Tosho, Tokyo, 1975 (in Japanese).
[19] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11 (1971), 155-167.

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