Acta Math Vietnam (2013) 38:317–338 DOI 10.1007/s40306-013-0022-3 STOCHASTIC DYNAMIC EQUATIONS ON TIME SCALES Nguyen Huu Du · Nguyen Thanh Dieu Received: 20 December 2011 / Published online: 14 May 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013 Abstract The aim of this paper is to consider the ∇-stochastic dynamic equation on time scales. We give a condition for the existence and uniqueness of solutions, study Markovian property of the solutions concerning its time-dependent generator. This work can be consid- ered as a unification and a generalization of similar results in random difference equation and stochastic differential one. Keywords Itô’s formula · Markov process · Semimartingale · Stochastic integration · Time-dependent operator · Time scales Mathematics Subject Classification (2010) 37B25 · 39A50 · 60H05 · 76M35 · 93D05 · 93E35 1 Introduction The theory of stochastic integration, introduced by K. Itô in 1944, has drawn a lot attention in literature. Kunita and Wantanabe in 1967 [10] proposed a generalization of what is now called Itô’s integral. After these results, making use of P.A. Meyer’s decomposition theorem for supermartingale, one expanded this theory to the stochastic integration for locally square integrable martingale and studied the stochastic dynamic equations. Besides, stochastic difference equations might define simplest dynamical systems, but they play an important role in the investigation of a stochastic dynamical system. The differ- ence equations arise naturally when we want to study the evolution of biological population or economic models on a fixed period of time. They can also be illustrated as discretization of stochastic differential equations in computing process. N.H. Du (B ) Faculty of Mathematics, Mechanics, and Informatics University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam e-mail: dunh@vnu.edu.vn N.T. Dieu Department of Mathematics, Vinh University, 182 Le Dan, Vinh, Nghe An, Vietnam e-mail: dieunguyen2008@gmail.com