Proceedings of the International scientific and practical conference ―Science, technology and art in global context‖ (October 14-16, 2025) / Publisher website: www.naukainfo.com. – Warsaw, Poland, 2025. - 72 p.

64 ( ) are mutually independent Poisson measures defined on ( ) independent on ( ) . On the probability space ( ) we consider an increasing, right continuous family of complete sub- -algebras * + , where { ( ) ( ) } Assumption 1. It is assumed that ( ) ( ) ( ) ( ) ( ) ( ) , ( ) , ( ) are bounded, continuous on functions, ( ) ( ) * + , ∫ ( ) ( ) Theorem 1. If Assumption 1 holds, then there exists a unique global solution ( ) to the system (1) for any initial value ( ) , and * ( ) + . Proof. Let us consider the system of stochastic differential equations ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) ( ) ( ) ∫ ( ( )) ̃ ( ) ∫ ( ( )) ̃ ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) ( ) ( ) ∫ ( ( )) ̃ ( ) ∫ ( ( )) ̃ ( ) ( ) The coefficients of system (2) are locally Lipschitz continuous. So, for any initial value ( ) there exists a unique local solution ( ) ( ( ) ( )) on , ) , where | ( )| , [3, p.246]. Therefore, from the Itô’s formula we derive that the process ( ) ( * ( )+ * ( )+) is a unique, positive local solution to system (2). To show this solution is global, we need to show that