Strong uniqueness of solutions of stochastic differential equations with jumps and non-Lipschitz random coefficients

In the paper we establish strong uniqueness of solution of a system of stochastic differential equations with random non-Lipschitz coefficients that involve both the square integrable continuous vector martingales and centered and non-centered Poisson measures.

In the present paper we generalize the result of Kulinich [4] on a strong uniqueness of the solution  for the following system of stochastic differential equations A strong solution of equation ( 1) is ℑ -progressively measurable stochastic process () = ( (),  = 1, ), which is continuous on the right and without discontinuities of the second kind and whose stochastic differential takes form (1) (see [2]).
The problem of a strong uniqueness of a solution of the equation (1) with Lipschitz coefficients was considered in a number of works, among which we mention only monograph [2] that contains both an extended bibliography and analysis of the state of art.The problem of a strong uniqueness of the solution in the case of non-Lipschitz coefficients was examined in a small number of papers and only for one-dimensional equations.We point out the papers [5,6,8], in which the new original methods of investigation of a strong uniqueness of solution for one-dimensional equations of form (1) without jumps and with  () =  () are proposed.Then in [3,7,9] these methods were extended to wider classes of one-dimensional equations of form (1) but also with  () =  ().
The most complete study of the problem under consideration is presented in [2], but there the conditions on the smoothness of the coefficients  (, ) in the multidimensional case are stronger than in our paper.However, it is important to note that we can significantly weaken the conditions on the coefficients  (, ) of equation ( 1) only for a special class of equations, in particular, for the equations containing Hölder's coefficients with the index  ⩾ 1 2 .For example, the coefficients  1 (, ) = (√| 1 |, sin ),  2 (, ) = (cos , √| 2 |) satisfy conditions of our theorem but were inappropriate for any previous theorems.We apply the method proposed in [8] to prove the main result.

Strong uniqueness
Now we state and prove the main result.

Also let
Then the strong solution  of equation ( 1) is strongly unique.
Remark.If additionally to the assumptions of the theorem the following conditions hold true with probability 1: (1) there exists a constant  > 0 such that (2) for arbitrary then equation ( 1) has a weak solution (see [2]: Theorem 3, §2, Chap.4).
Corollary.If the assumptions of the theorem and the conditions of the remark are satisfied, then equation ( 1) has a unique strong solution (see [2]: Theorem 9, §3, Chap.6).

Conclusion
In this work we investigate the problem of strong uniqueness of a solution for a system of stochastic differential equations with random coefficients and with differentials with respect to martingales, which may be continuous with probability 1 or discontinuous with jumps of Poisson type.Substantial generalizations of previously known results are got for the systems of the special type.For example, coefficients corresponding to the differentials of continuous martingales can be Hölder with index  ⩾ 1 2 .Such systems are of particular interest, for instance, in connection with applications in finance for modeling instantaneous interest rate [1], in investigation of limiting behavior of unstable components of solutions of stochastic differential equations [3].