Optimal control of Stochastic Fluid Programs [Elektronische Ressource] / Nicole Bäuerle

Optimal Control ofStochastic Fluid ProgramsHabilitationsschriftan der Fakult¨at fur¨ Mathematik undWirtschaftswissenschaftender Universit¨at UlmvorgelegtvonNicole B¨auerleUlm1999...meinem Mann Rolf,fur¨ seine Liebe und Geduld.List of SymbolsCommonly used SymbolsIN set of positive integersIN IN∪{0}0IR set of real numbersIR set of nonnegative real numbers+IR IR +{∞}+ +B(S) Borel-σ-algebra on S◦interior of SS1 (·) indicator function of set SSe i-th unit vectori1 vector of 1’s with dimension kk|h| max{h,−h}.k·k vector norm.x∧y componentwise minimum of vectors x and y.x∨y componentwise maximum of vectors x and y.∂ V(y,z) derivative w.r.t. y.∂yp˙ derivative w.r.t. time t.tI identity matrixδ Dirac measure.x⇒ weak convergence.<·> quadratic variation.N ND [0,∞) set of functions f : [0,∞)→IR which are rightcontinuous and have left-hand limits.Abbreviationsa.s. almost sure.DSFP Discretized Stochastic Fluid Program.i.i.d. independent and identically distributed.SFP Stochastic Fluid Program.w.l.o.g. without loss of generality.w.r.t. with respect to.Contents1 Introduction 12 β-Discounted Optimality 62.1 Continuous-time Definition . . . . . . . . . . . . . . . . . . . . . . . 62.2 Discrete-time Formulation . . . . . . . . . . . . . . . . . . . . . . . 82.3 A Relaxed Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 β-Discounted Cost Optimality Equation . . . . . . . . . . . . . . . 142.5 Properties of the Value Function.