Abstract
We introduce and analyze a new type of fuzzy stochastic differential equations. We consider equations with drift and diffusion terms occurring at both sides of equations. Therefore we call them the bipartite fuzzy stochastic differential equations. Under the Lipschitz and boundedness conditions imposed on drifts and diffusions coefficients we prove existence of a unique solution. Then, insensitivity of the solution under small changes of data of equation is examined. Finally, we mention that all results can be repeated for solutions to bipartite set-valued stochastic differential equations.
Highlights
Stochastic differential equations are often used in modelling dynamics of uncertain physical systems, where it is assumed that randomness and stochastic noises have an influence on a considered system
= x0 + ∫ f (s, x (s)) ds + ∫ g (s, x (s)) dB (s), t ∈ [0, T]. This way we introduce a new kind of fuzzy stochastic differential equations which are more general than those studied in our earlier works and mentioned above
In the paper we make an examination of initial value problem for fuzzy stochastic differential equations of a new form dx (t) + (−1) f (t, x (t)) dt + ⟨(−1) g (t, x (t)) dB (t)⟩
Summary
Stochastic differential equations are often used in modelling dynamics of uncertain physical systems, where it is assumed that randomness and stochastic noises have an influence on a considered system. We investigated the problem of existence of a unique solution, since it is almost impossible to find explicit forms of solutions to such equations This is very similar to the theory of crisp stochastic differential equations. Due to the new form of equations with integrals at both sides they will be called the bipartite fuzzy stochastic differential equations Solutions to such equations may lose property of monotonicity of fuzziness. It is shown that the solution is stable with respect to small changes of equation’s data; that is, the solution does not change much when the changes of drift and diffusion coefficients and initial value are small This shows that the theory introduced in the paper is well-posed. We prove existence and uniqueness of solution to such equations and study properties of solutions
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