0000000000009342

AUTHOR

Marek T. Malinowski

0000-0002-3585-1265

showing 4 related works from this author

Set-valued and fuzzy stochastic differential equations driven by semimartingales

2013

Abstract In the paper we present set-valued and fuzzy stochastic integrals with respect to semimartingale integrators as well as their main properties. Then we study the existence of solutions to set-valued and fuzzy-set-valued stochastic differential equations driven by semimartingales. The stability of solutions is also established.

Stratonovich integralApplied MathematicsMathematical analysisStochastic calculusStability (learning theory)Fuzzy logicSet (abstract data type)Stochastic partial differential equationStochastic differential equationSemimartingaleMathematics::ProbabilityApplied mathematicsAnalysisMathematicsNonlinear Analysis-Theory Methods & Applications
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The interrelation between stochastic differential inclusions and set-valued stochastic differential equations

2013

Abstract In this paper we connect the well established theory of stochastic differential inclusions with a new theory of set-valued stochastic differential equations. Solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty, bounded, convex and closed subsets of the space L 2 consisting of square integrable random vectors. We show that for the solution X to a set-valued stochastic differential equation corresponding to a stochastic differential inclusion, there exists a solution x for this inclusion that is a ‖ ⋅ ‖ L 2 -continuous selection of X . This result enables us to draw inferences about the reachable sets of solutio…

Continuous-time stochastic processApplied MathematicsMathematical analysisStochastic calculusMalliavin calculusStochastic partial differential equationsymbols.namesakeStochastic differential equationDifferential inclusionRunge–Kutta methodsymbolsApplied mathematicsAnalysisMathematicsAlgebraic differential equationJournal of Mathematical Analysis and Applications
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Set-valued stochastic integral equations driven by martingales

2012

Abstract We consider a notion of set-valued stochastic Lebesgue–Stieltjes trajectory integral and a notion of set-valued stochastic trajectory integral with respect to martingale. Then we use these integrals in a formulation of set-valued stochastic integral equations. The existence and uniqueness of the solution to such the equations is proven. As a generalization of set-valued case results we consider the fuzzy stochastic trajectory integrals and investigate the fuzzy stochastic integral equations driven by bounded variation processes and martingales.

Stratonovich integralContinuous-time stochastic processApplied MathematicsMathematical analysisMathematicsofComputing_NUMERICALANALYSISStochastic calculusRiemann–Stieltjes integralRiemann integralsymbols.namesakeQuantum stochastic calculusImproper integralsymbolsDaniell integralAnalysisMathematicsJournal of Mathematical Analysis and Applications
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Fuzzy Stochastic Integral Equations Driven by Martingales

2011

Exploiting the properties of set-valued stochastic trajectory integrals we consider a notion of fuzzy stochastic Lebesgue–Stieltjes trajectory integral and a notion of fuzzy stochastic trajectory integral with respect to martingale. Then we use these integrals in a formulation of fuzzy stochastic integral equations. We investigate the existence and uniqueness of solution to such the equations.

Stratonovich integralMathematical analysisMathematicsofComputing_NUMERICALANALYSISApplied mathematicsUniquenessMartingale (probability theory)Fuzzy logicStochastic integralMathematics
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