The purpose of this paper is to give a detailed proof of Yamada-Watanabe theorem for stochastic evolution equation driven by pure Poisson random measure.

We give an extension result of Watanabe's characterization for 2-dimensional Poisson processes. By using this result, the equivalence of uniqueness in law and joint uniqueness in law is proved for one-dimensional stochastic differential equations driven by Poisson processes. After that, we give a simplified Engelbert theorem for the stochastic differential equations of this type.

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We establish a Freidlin-Wentzell's large deviation principle for general stochastic evolution equations with Poisson jumps and small multiplicative noises by using weak convergence method.

In this paper, we consider the stochastic Lotka-Volterra model with additive jump noises. We show some desired properties of the solution such as existence and uniqueness of positive strong solution, unique stationary distribution, and exponential ergodicity. After that, we investigate the maximum likelihood estimation for the drift coefficients based on continuous time observations. The likelihood function and explicit estimator are derived by using semimartingale theory. In addition, consistency and asymptotic normality of the estimator are proved. Finally, computer simulations are presented to illustrate our results.