Existence and stability results for strong solutions of lipschitzian quantum stochastic differential equations

Ayoola, E. O.

doi:10.1081/sap-120003434

Cited by 2 publications

(8 citation statements)

References 12 publications

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“…Several authors have studied the weak solutions of the following quantum stochastic differential equation introduced by Hudson and Parthasarathy:

\begin{matrix} right d z (t) = & left U (t, z (t)) d \land_{π} (t) + V (t, z (t)) d A_{g} (t) & right \\ right & left + W (t, z (t)) d A_{f^{+}} (t) + H (t, z (t)) d t, \\ right z (t_{0}) = & left z_{0}, t \in I = [t_{0}, T], \end{matrix}

where the coefficients U , V , W , H are stochastic processes. The gauge, creation, and annihilation processes

{normalΛ}_{normalΠ}, A_{f^{+}}, A_{g}

and the Lebesgue measure t are well defined in Ayoola z∈truescriptB˜, t ∈[ t 0 , T ]= I .…”

Section: Introductionmentioning

confidence: 99%

“…The study of the qualitative and topological properties of solutions of have received appreciable attention by several authors. See . Ekhaguere, studied existence of solutions of and established a relationship between the solutions of and those of its convexification.…”

Section: Introductionmentioning

confidence: 99%

“…Bishop et al established some results on mild solutions of impulsive quantum stochastic differential equations with nonlocal conditions using the fixed point approach. In the works of Ayoola, existence, uniqueness, and stability of solution of Equation were studied under the strong topology. The results in Ayoola are extensions of some existing results concerning classical stochastic differential equations.…”

Section: Introductionmentioning

confidence: 99%

“…In the works of Ayoola, existence, uniqueness, and stability of solution of Equation were studied under the strong topology. The results in Ayoola are extensions of some existing results concerning classical stochastic differential equations. In the theory of classical ordinary differential equations, differential equations with delay and stochastic differential equations with nonlocal conditions have attracted the attention of several authors who have dealt with the qualitative and quantitative properties of solutions of these class of equations.…”

Section: Introductionmentioning

confidence: 99%

“…Notable among these are the works where results on different types of evolution equations with nonlocal conditions were considered. Motivated by the simplicity of the method of successive approximations used in Ayoola and Ekhaguere, we introduce and study mild solutions of evolution equation under the strong topology with nonlocal conditions that are continuous. The results here improve several existing results of under the weak topology.…”

Section: Introductionmentioning

confidence: 99%

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Mild solutions of evolution quantum stochastic differential equations with nonlocal conditions

Bishop

Okeke

Eke

2020

Math Methods in App Sciences

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We study existence of a unique mild solution of evolution quantum stochastic differential equations with nonlocal conditions under the strong topology. Using the method of successive approximations, we do not need to transform the nonlocal problem to a fixed point form. The evolution operator A generates a family of semigroup that are continuous. Nonlocal conditions allow additional measurements of certain phenomena that cannot be captured by the traditional initial conditions. We show that under some given conditions, the mild solution is unique and also stable. The method applied here is much easier when compared with previous methods used in literature.

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“…Several authors have studied the weak solutions of the following quantum stochastic differential equation introduced by Hudson and Parthasarathy:

\begin{matrix} right d z (t) = & left U (t, z (t)) d \land_{π} (t) + V (t, z (t)) d A_{g} (t) & right \\ right & left + W (t, z (t)) d A_{f^{+}} (t) + H (t, z (t)) d t, \\ right z (t_{0}) = & left z_{0}, t \in I = [t_{0}, T], \end{matrix}

where the coefficients U , V , W , H are stochastic processes. The gauge, creation, and annihilation processes

{normalΛ}_{normalΠ}, A_{f^{+}}, A_{g}

and the Lebesgue measure t are well defined in Ayoola z∈truescriptB˜, t ∈[ t 0 , T ]= I .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Mild solutions of evolution quantum stochastic differential equations with nonlocal conditions

Bishop

Okeke

Eke

2020

Math Methods in App Sciences

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show abstract

Further results on the existence, uniqueness and stability of strong solutions of quantum stochastic differential equations

Ayoola

Gbolagade

2005

Applied Mathematics Letters

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Existence and stability results for strong solutions of lipschitzian quantum stochastic differential equations

Cited by 2 publications

References 12 publications

Mild solutions of evolution quantum stochastic differential equations with nonlocal conditions

Mild solutions of evolution quantum stochastic differential equations with nonlocal conditions

Further results on the existence, uniqueness and stability of strong solutions of quantum stochastic differential equations

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