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Successive approximations to solutions of stochastic differential equations. (English) Zbl 0744.34052
This paper studies the Itô stochastic differential equation $$dX(t)=f(t,X(t))+g(t,X(t))dw(t)\leqno(1)$$ with initial condition $X(0)=X\sb 0$ a.e., where the Brownian motion $w(t)$ and the random variable $X\sb 0$ are independent and $E(\vert X\sb 0\vert\sp 2)<\infty$. Under somewhat less restrictive conditions than previously used, the uniform convergence of the successive approximations $$X\sb n(t)=X\sb 0+\int\sp t\sb{t\sb 0}f(\tau,X\sb{n-1}(\tau))d\sp \tau+\int\sp t\sb{t\sb 0}g(\tau,X\sb{n+1} (\tau))dw(\tau)$$ to the solution $X(t)$ of initial value problem (1) is proved which also establishes the local existence and uniqueness of this solution. Then the paper concludes by proving global uniform convergence of the successive approximations and global existence-uniqueness of the solution of IVP (1) under appropriately amended conditions.

##### MSC:
 34F05 ODE with randomness 34A45 Theoretical approximation of solutions of ODE 60H10 Stochastic ordinary differential equations
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##### References:
 [1] Coddington, E. A.; Levinson, N.: Uniqueness and the convergence of successive approximations. J. indian math. Soc. 16, 75-81 (1952) · Zbl 0047.08305 [2] Coddington, E. A.; Levinson, N.: Theory of ordinary differential equations. (1955) · Zbl 0064.33002 [3] Friedman, A.: Stochastic differential equations and applications. 1 (1975) · Zbl 0323.60056 [4] Gard, T. C.: A general uniqueness theorem for stochastic differential equations. SIAM J. Control optim. 14, 445-457 (1976) · Zbl 0332.60037 [5] Gard, T. C.: Pathwise uniqueness for solutions of systems of stochastic differential equations. Stochastic process appl., 253-260 (1978) · Zbl 0373.60069 [6] Gard, T. C.: Introduction to stochastic differential equations. (1988) · Zbl 0628.60064 [7] Ikeda, N.; Watanabe, S.: A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14, 619-633 (1977) · Zbl 0376.60065 [8] Ikeda, N.; Watanabe, S.: Stochastic differential equations and diffusion processes. (1981) · Zbl 0495.60005 [9] Ito, K.: On stochastic differential equations. Mem. amer. Math. soc. 4 (1951) · Zbl 0054.05803 [10] Lakshmikantham, V.; Leela, S.: Differential and integral inequalities. 1 (1969) · Zbl 0177.12403 [11] Okabe, Y.; Shimizu, A.: On the pathwise uniqueness of solutions of stochastic differential equations. J. math. Kyoto univ. 11, 455-466 (1975) · Zbl 0353.60055 [12] Rodkina, A. E.: On existence and uniqueness of solution of stochastic differential equations with heredity. Stochastics monographs 12, 187-200 (1984) · Zbl 0568.60062 [13] Stroock, D. W.; Varadhan, S. R. S: Multidimensional diffusion processes. (1979) · Zbl 0426.60069 [14] Taniguchi, T.: On the estimate of solutions of perturbed linear differential equations. J. math. Anal. appl. 153, 288-300 (1990) · Zbl 0727.34040 [15] Taniguchi, T.: On sufficient conditions for non-explosion of solutions to stochastic differential equations. J. math. Anal. appl. 153, 549-561 (1990) · Zbl 0715.60072 [16] Tudor, C.: Successive approximations for solutions of stochastic integral equations of Volterra type. J. math. Anal. appl. 104, 27-37 (1984) · Zbl 0598.60064 [17] Veretennikov, A. Y.: On the strong solutions of stochastic differential equations. Theory probab. Appl. 24, 354-366 (1979) · Zbl 0434.60064 [18] Vidossich, G.: Global convergence of successive approximations. J. math. Anal. appl. 45, 285-292 (1974) · Zbl 0321.34054 [19] Watanabe, S.; Yamada, T.: On the uniqueness of solutions of stochastic differential equations, 2. J. math. Kyoto univ. 11, 553-563 (1971) · Zbl 0229.60039 [20] Yamada, T.: On the comparison theorem for solutions of stochastic differential equations and its applications. J. math. Kyoto univ. 13, 497-512 (1973) · Zbl 0277.60047 [21] Yamada, T.: Sur l’approximation des solutions d’équations differentielles stochastiques. Z. wahrschein. Ver. geb. 36, 153-164 (1976) · Zbl 0374.60081 [22] Yamada, T.: On the successive approximation of solutions of stochastic differential equations. J. math. Kyoto univ. 21, 501-515 (1981) · Zbl 0484.60053 [23] Yamada, T.; Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. math. Kyoto univ. 11, 155-167 (1971) · Zbl 0236.60037 [24] Yoshizawa, T.: Stability theory by Liapunov’s second method. (1966) · Zbl 0144.10802 [25] Xu, M. H.: J. Wuhan univ. Natur. sci. Ed.. (1985)