MATHEMATICAL AND PHYSICAL TABLES
CONTENTS BY SUCH FATE OF SIGNS OPTIMAL CONTROL OF FORWARD-BACKWARD STOCHASTIC VOLTERRA EQUATIONS A UNIFIED APPROACH TO INFINITE DIMENSIONAL INTEGRALS OF PROBABILISTIC AND OSCILLATORY TYPE WITH APPLICATIONS TO FEYNMAN PATH INTEGRALS THE NUMBERS LEAD A DANCE COMPENSATED COMPACTNESS IN BANACH SPACES AND WEAK RIGIDITY OF ISOMETRIC IMMERSIONS OF MANIFOLDS THE INITIAL-BOUNDARY-VALUE PROBLEM FOR AN OSTROVSKY–HUNTER TYPE EQUATION MODELING CROWD DYNAMICS THROUGH HYPERBOLIC – ELLIPTIC EQUATIONS ON THE WELL-POSEDNESS OF SOLUTIONS WITH FINITE ENERGY FOR NONLOCAL EQUATIONS OF POROUS MEDIUM TYPE ON THE SPECTRUM OF LEAKY SURFACES WITH A POTENTIAL BIAS ON THE DECAY OF ALMOST PERIODIC SOLUTIONS OF ANISOTROPIC DEGENERATE PARABOLIC-HYPERBOLIC EQUATIONS FACTORIZATIONS AND HARDY–RELLICH-TYPE INEQUALITIES SYMMETRIES AND MULTIPEAKON SOLUTIONS FOR THE MODIFIED TWO-COMPONENT CAMASSA–HOLM SYSTEM VANISHING VISCOSITY SOLUTIONS OF RIEMANN PROBLEMS FOR MODELS OF POLYMER FLOODING EFFICIENT COMPUTATION OF ALL SPEED FLOWS USING AN ENTROPY STABLE SHOCK-CAPTURING SPACE-TIME DISCONTINUOUS GALERKIN METHOD DISPERSION ESTIMATES FOR SPHERICAL SCHRÖDINGER EQUATIONS WITH CRITICAL ANGULAR MOMENTUM SIXTY YEARS OF MOMENTS FOR RANDOM MATRICES BOUND STATES OF SCHRÖDINGER TYPE OPERATORS WITH HEISENBERG SUB-LAPLACIAN ON HOLDEN’S SEVEN GUIDELINES FOR SCIENTIFIC COMPUTING AND DEVELOPMENT OF OPEN-SOURCE COMMUNITY SOFTWARE SHARP UNIQUENESS RESULTS FOR DISCRETE EVOLUTIONS SPATIAL ANALYTICITY OF SOLUTIONS TO NONLINEAR DISPERSIVE PDE BY SUCH FATE OF SIGNS A SESTINA FOR HELGE BY SUCH FATE OF SIGNS, THAT OFT HYPERBOLIC TURMOIL OF ELEGANT, BRITTLE EQUATIONS, IS FORGED IN THE RUGGED CLIMATE OF NORWAY QUIET, YET SOLID, AND TRULY DISTINGUISHED A SINGULAR LEADER, A TRUSTED COMPANION, RIDDLES SURRENDER, RIDDLES RESOLVE TIME UNFOLDS IN ITS STEDFAST RESOLVE, CAUSAL IN STRUCTURES, A FLOW HYPERBOLIC GAZING IN MIRRORS, WE SEE A COMPANION, EXPERIENCE-LADEN, ANCIENT EQUATIONS, LESS NOW ‘TWEEN ‘I’ AND ‘THEM’ DISTINGUISH AGE COMES QUIETLY, EVEN IN NORWAY ON BEAUTIFUL FJORDS, ON MOUNTAINS OF NORWAY, A PEOPLE SO FUSED BY COMMON RESOLVE, TO BE - THERE - AN EQUAL, AND ALSO DISTINGUISHED TO LIVE SUCH A LIFE, AT ONCE HYPERBOLIC, BUT ALSO IN SERVICE, TO ANCIENT EQUATIONS, THAT NORDIC-LIKE SORROW A CONSTANT COMPANION TO BE SO, TO LIFE, A FAITHFUL COMPANION, ROAMING THE GLOBE, YET ROOTED IN NORWAY, TO BALANCE IN EVERYTHING WEIGHTED EQUATIONS THIS TAKES A SPECIAL KIND OF RESOLVE NEVER IN LANGUAGE, NOR DEED, HYPERBOLIC, BUT EVER THE COOL AND QUIET DISTINGUISHED THUS HE HIMSELF FROM OTHERS DISTINGUISHED, REMAINS TO HIS FELLOWS A LOYAL COMPANION, THE LAWS CONSERVING, THE LAWS HYPERBOLIC, HAVE MADE HIM KNOWN, NOT ONLY IN NORWAY AS ONE WHO COULD TWISTED QUESTIONS RESOLVE INTO TRANSPARENTLY CLEAR BLUE EQUATIONS HIS LIFELONG COMPANION, THOSE EAGER EQUATIONS, DISCOVERED IN NORWAY, ARE CLEARLY DISTINGUISHED, NO LESS HYPERBOLIC THAN HELGE’S RESOLVE A COLLECTION OF MATHEMATICAL AND PHYSICAL TABLES FROM A COURSE OF EXPERIMENTS OPTIMAL CONTROL OF FORWARD-BACKWARD STOCHASTIC VOLTERRA EQUATIONS DEDICATED TO HELGE HOLDEN ON THE OCCASION OF HIS 60TH BIRTHDAY ABSTRACT WE STUDY THE PROBLEM OF OPTIMAL CONTROL OF A COUPLED SYSTEM OF FORWARD-BACK- WARD STOCHASTIC VOLTERRA EQUATIONS WE USE HIDA–MALLIAVIN CALCULUS TO PROVE A SUFFICIENT AND A NECESSARY MAXIMUM PRINCIPLE FOR THE OPTIMAL CONTROL OF SUCH SYSTEMS EXISTENCE AND UNIQUENESS OF BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS ARE PROVED AS AN APPLICATION OF OUR METHODS, WE SOLVE A RECURSIVE UTILITY OPTIMISATION PROBLEM IN A FINANCIAL MODEL WITH MEMORY 1 INTRODUCTION THE PURPOSE OF THIS PAPER IS TO ESTABLISH SOLUTION TECHNIQUES FOR OPTIMAL CONTROL OF COUPLED SYSTEMS OF STOCHASTIC VOLTERRA EQUATIONS STOCHASTIC VOLTERRA EQUATIONS APPEAR IN MODELS FOR DYNAMIC SYSTEMS WITH NOISE AND MEMORY AS A MOTIVATING EXAMPLE, CONSIDER THE FOLLOWING VOLTERRA EQUATION, MODELING A STOCHASTIC CASH FLOW ??(??) = ????(??) SUBJECT TO A CONSUMPTION RATE ??(??) AT TIME ??: ??(??) = ??(??) + ? ?? (??(??, ??) ???(??))??(??) ????+ ? ?? ??(??, ??)??(??) ????(??) + ? ?? BE A GIVEN RIGHT-CONTINUOUS COMPLETE SUBFILTRATION OF ??, IN THE SENSE THAT ????????FOR ALL ???[0, ??] THE SIGMA-ALGEBRA ????REPRESENTS THE INFORMATION AVAILABLE TO THE CONSUMER AT TIME ?? LET ??(??) BE THE ??-ALGEBRA OF ??-PREDICTABLE SUBSETS OF ? × ?+, I E , THE ??-ALGEBRA GENERATED BY THE LEFT CONTINUOUS ??-ADAPTED PROCESSES THE FORWARD STOCHASTIC VOLTERRA INTEGRAL EQUATION (FSVIE) (1 1) CAN BE WRITTEN IN DIFFERENTIAL FORM AS ????(??) = ???(??) ????+ (??(??, ??) ???(??))??(??) ????+ (? ?? ???? ????(??, ??)??(??) ????) ???? + ??(??, ??)??(??) ????(??) + (? ?? ???? ????(??, ??)??(??) ????(??)) ???? + ? ? ??(??, ??, ??)??(??) ˜??(????, ????) + (? ? ? ?? ???? ????(??, ??, ??)??(??) ˜??(????, ????)) ????, ???[0, ??] (1 2) FROM (1 2) WE SEE THAT THE DYNAMICS OF ??(??) CONTAINS HISTORY OR MEMORY TERMS REPRESENTED BY THE ????-INTEGRALS FOLLOWING A SUGGESTION OF DUFFIE AND EPSTEIN [5] WE NOW MODEL THE TOTAL UTILITY OF THE CONSUMPTION RATE ??(??) BY A RECURSIVE UTILITY PROCESS ??(??) = ????(??) DEFINED BY THE EQUATION ??(??) = ??[?? ?? ?? {??(??)??(??) + LN(??(??)??(??))} ???? | | | ???] ; ???[0, ??] BY THE MARTINGALE REPRESENTATION THEOREM WE SEE THAT THERE EXIST PROCESSES ??(??), ??(??, ??) SUCH THAT THE TRIPLE (??, ??, ??) SOLVES THE BACKWARD STOCHASTIC DIFFERENTIAL EQUATION (BSDE) ?? ?? ? ????(??) = ?[??(??)??(??) + LN(??(??)??(??))] ????+ ??(??) ????(??) + ? ? ??(??, ??) ˜??(????, ????); ???[0, ??] ??(??) = (1 3) WE NOW CONSIDER THE OPTIMAL RECURSIVE UTILITY PROBLEM TO MAXIMISE THE TOTAL RECURSIVE UTILITY OF THE CONSUMPTION IN OTHER WORDS, WE WANT TO FIND AN OPTIMAL CONSUMPTION RATE ????????SUCH THAT SUP ??????? ????(0) = ?????(0), (1 4) A COLLECTION OF MATHEMATICAL AND PHYSICAL TABLES FROM A COURSE OF EXPERIMENTS WHERE ????IS A GIVEN SET OF ADMISSIBLE ??-ADAPTED CONSUMPTION PROCESSES THIS IS A PROBLEM OF OPTIMAL CONTROL OF A COUPLED SYSTEM CONSISTING OF THE FORWARD STOCHASTIC VOLTERRA EQUATION (1 1) AND THE BSDE (1 3) IN THE FOLLOWING SECTIONS WE WILL PRESENT SOLUTION METHODS FOR GENERAL OPTIMAL CONTROL FOR SYSTEMS OF FORWARD-BACKWARD STOCHASTIC VOLTERRA EQUATIONS THEN IN THE LAST SECTION WE WILL APPLY THE METHODS TO SOLVE THE OPTIMAL RECURSIVE UTILITY CONSUMPTION PROBLEM ABOVE THERE HAS BEEN A LOT OF RESEARCH ACTIVITY RECENTLY WITHIN STOCHASTIC VOLTERRA INTEGRAL EQUATIONS (SVIES) RECENTLY, BOTH OF FORWARD AND BACKWARD TYPE SEE, E G , [2, 7, 11, 12, 14, 13, 15, 16, 17, 18] PERHAPS THE PAPER CLOSEST TO OUR PAPER IS [13] HOWEVER, THAT PAPER HAS A DIFFERENT APPROACH THAN OURS, DOES NOT HAVE A SUFFICIENT MAXIMUM PRINCIPLE, AND DOES NOT DEAL WITH JUMPS AND PARTIAL INFORMATION, AS WE DO 2 STOCHASTIC MAXIMUM PRINCIPLE FOR FBSVE THIS SECTION IS AN EXTENSION TO FORWARD-BACKWARD SYSTEMS OF THE RESULTS OBTAINED IN [2] WE CONSIDER A SYSTEM GOVERNED BY A COUPLED SYSTEM OF CONTROLLED FORWARD- BACKWARD STOCHASTIC VOLTERRA EQUATIONS (FBSVE) OF THE FORM: ??(??) = ??(??) + ? ?? ??(??, ??, ??(??), ??(??)) ????+ ? ?? ??(??, ??, ??(??), ??(??)) ????(??) + ? ?? ? ? ??(??, ??, ??(??), ??(??), ??) ˜??(????, ????), ???[0, ??], (2 1) ??(??) = ??(??(??)) + ? ?? ?? ??(??, ??, ??(??), ??(??), ??(??, ??), ??(??, ??, ?), ??(??)) ???? ?? ?? ?? ??(??, ??) ????(??) ?? ?? ?? ? ? ??(??, ??, ??) ˜??(????, ????), ???[0, ??] (2 2) THE QUADRUPLE (??, ??, ??, ??) IS SAID TO BE A SOLUTION OF (2 1)–(2 2) IF IT SATISFIES BOTH EQUATIONS TO THE BEST OF OUR KNOWLEDGE, RESULTS ABOUT EXISTENCE AND UNIQUE- NESS OF SOLUTIONS FOR SUCH GENERAL SYSTEMS ARE NOT KNOWN CONDITIONS UNDER WHICH THERE EXISTS A UNIQUE SOLUTION (??, ??, ??) OF (2 2) ARE STUDIED IN SECTION 3 IN THE ABOVE THE FUNCTIONS ??, ??ARE ASSUMED TO BE DETERMINISTIC AND ??1, WHILE A COLLECTION OF MATHEMATICAL AND PHYSICAL TABLES FROM A COURSE OF EXPERIMENTS THE FUNCTIONS ??(??, ??, ??, ??)? [0, ??]2 × ?× ??× ? ??, ??(??, ??, ??, ??)? [0, ??]2 × ?× ??× ? ??, ??(??, ??, ??, ??, ??, ??( ?), ??)? [0, ??]2 × ?3 × ??2(??) × ??× ? ??, ??(??, ??, ??, ??, ??)? [0, ??]2 × ?× ??× ?0 × ? ??, ARE ASSUMED TO BE CONTINUOUSLY DIFFERENTIABLE WITH RESPECT TO THEIR FIRST VARI- ABLES, AND FOR ALL ??, ??, ??, ??, ??, ??, ??THE PROCESSES ?????(??, ??, ??, ??), ?????(??, ??, ??, ??), ?????(??, ??, ??, ??, ??, ??( ?), ??), ?????(??, ??, ??, ??, ??) ARE ???-MEASURABLE FOR ALL ????? WE ASSUME THAT ?????(??, ??) AND ?????(??, ??, ?) ARE ??1 FOR ALL ??, ??, ??AND THAT ??[? ?? ? ?? (???? ????(??, ??)) ????????+ ? ?? ? ? (???? ????(??, ??, ??)) ??(????) ????????] < ? (2 3) IT IS KNOWN THAT (2 3) HOLDS FOR SOME LINEAR SYSTEMS SEE [6] LET ??BE A GIVEN OPEN CONVEX SUBSET OF ?AND LET ??= ????BE A GIVEN FAMILY OF ADMISSIBLE CONTROLS, REQUIRED TO BE ??-PREDICTABLE, WHERE, AS BEFORE, ??= {????}???0 IS A GIVEN SUBFILTRATION OF ??= {???}???0, IN THE SENSE THAT ????????FOR ALL ?? WE ASSOCIATE TO THE SYSTEM (2 1)–(2 2) THE FOLLOWING PERFORMANCE FUNCTIONAL: ??(??) = ??[? ?? ??(??, ??(??), ??(??), ??(??)) ????+ ??(??(??)) + ??(??(0))] , FOR GIVEN FUNCTIONS ???[0, ??] × ?2 × ??× ? ??, ??????, ?????? THE FUNCTIONS ??, ??ARE ASSUMED TO BE ??1, WHILE ??(??, ??, ??, ??) IS ASSUMED TO BE ??- ADAPTED WITH RESPECT TO ??AND ??1 WITH RESPECT TO ??, ??, ??FOR EACH S WE REMARK HERE THAT OUR PERFORMANCE FUNCTIONAL IS NOT OF VOLTERRA TYPE OUR OPTIMISATION CONTROL PROBLEM IS TO FIND ????????SUCH THAT SUP ???? ??(??) = ??(???) (2 4) LET ?BE THE SET OF ALL ??-ADAPTED STOCHASTIC PROCESSES, AND LET ?DENOTE THE SET OF ALL FUNCTIONS ?????? DEFINE THE HAMILTONIAN FUNCTIONAL: ?(??, ??, ??, ??, ??( ?), ??, ??, ??( ?), ??, ??, ??( ?), ??( ?)) ???0(??, ??, ??, ??, ??( ?), ??, ??, ??, ??, ??( ?)) + ??1(??, ??, ??, ??, ??( ?), ??, ??( ?), ??( ?)), A COLLECTION OF MATHEMATICAL AND PHYSICAL TABLES FROM A COURSE OF EXPERIMENTS WHERE ??0 ?[0, ??] × ?3 × ?× ??× ?3 × ??? IS DEFINED BY ??0(??, ??, ??, ??, ??( ?), ??, ??, ??, ??, ??( ?)) ???(??, ??, ??, ??) + ??(??, ??, ??, ??)??+ ??(??, ??, ??, ??)?? + ? ? ??(??, ??, ??, ??)??(??, ??)??(????) + ??(??, ??, ??, ??, ??, ??( ?), ??)?? AND ??1 ?[0, ??] × ?3 × ?× ??× ?× ??? IS DEFINED BY ??1(??, ??, ??, ??, ??( ?), ??, ??( ?), ??( ?)) ?? ?? ?? ???? ????(??, ??, ??, ??)??(??) ????+ ? ?? ?? ???? ????(??, ??, ??, ??)??[??????(??) | ???] ???? + ? ?? ?? ? ? ???? ????(??, ??, ??, ??)??[????,????(??) | ???]??(????) ???? + ? ?? ???? ????(??, ??, ??, ??, ??, ??( ?), ??)??(??) ???? + ? ?? ???? ????(??, ??, ??, ??, ??, ??( ?), ??)???? ????(??, ??)??(??) ???? + ? ?? ??????(??, ??, ??, ??, ??, ??( ?), ??), ???? ????(??, ??, ?)???(??) ???? HERE, AND IN THE FOLLOWING, ????AND ????,??DENOTE THE (GENERALISED) HIDA–MALLIAVIN DERIVATIVE AT ??AND AT (??, ??) WITH RESPECT TO ??AND ˜??, RESPECTIVELY, AND ???DENOTES THE FRÉCHET DERIVATIVE WITH RESPECT TO ?? WE REFER TO THE APPENDIX FOR MORE DETAILS THE ASSOCIATED FORWARD-BACKWARD SYSTEM FOR THE ADJOINT PROCESSES ??(??), (??(??), ??(??), ??(??, ?)) IS { ????(??) ???? ????(??) ????+ ??? ????(??) ????(??) + ? ? ?????? ???? (??) ˜??(????, ????), 0 ??????, ??(0) ???? (??(0)) , (2 5) AND A COLLECTION OF MATHEMATICAL AND PHYSICAL TABLES FROM A COURSE OF EXPERIMENTS ????(??) ????? ????(??) ????+ ??(??) ????(??) + ? ? ??(??, ??) ˜??(????, ????), 0 ??????, ??(??) ????(??(??)) + ??(??)???(??(??)), (2 6) WHERE WE HAVE USED THE SIMPLIFIED NOTATION ??? ????(??) = [??? ????(??, ??, ??(??), ??(??, ) , ??(??, ?), ??(??), ??(??), ??(??), ??(??), ??(??, ?))] ??=??(??) , AND SIMILARLY FOR ??? ????(??), ??? ????(??) AS IN [8] WE ASSUME THAT ??IS FRÉCHET DIFFERENTIABLE (??1) IN THE VARIABLES ??, ??, ??, ??, ??AND THAT THE FRÉCHET DERIVATIVE ?????OF ??WITH RESPECT TO ????AS A RANDOM MEASURE IS ABSOLUTELY CONTINUOUS WITH RESPECT TO ??, WITH RADON–NIKODYM DERIVATIVE ???????/???? THUS, IF ??????, ??DENOTES THE ACTION OF THE LINEAR OPERATOR ?????ON THE FUNCTION ???, WE HAVE ??????, ??= ? ? ?(??) ???????(??) = ? ? ?(??)???????(??) ????(??) ????(??) THE QUESTION OF EXISTENCE AND UNIQUENESS OF THE FORWARD-BACKWARD SYSTEM ABOVE WILL NOT BE STUDIED HERE IT IS A SUBJECT OF FUTURE RESEARCH SEE, HOWEVER OUR PARTIAL RESULT IN SECTION 3 2 1 A SUFFICIENT MAXIMUM PRINCIPLE IN THIS SUBSECTION, WE PROVE THAT UNDER SOME CONDITIONS SUCH AS THE CONCAVITY, A GIVEN CONTROL ???WHICH SATISFIES A MAXIMUM CONDITION OF THE HAMILTONIAN, IS AN OPTIMAL CONTROL FOR THE PROBLEM (2 4) FROM (2 1)–(2 2) WE CAN GET THE DIFFERENTIAL FORMS: ????(??) = ???(??)????+ ??(??, ??, ??(??), ??(??)) ????+ (? ?? ???? ????(??, ??, ??(??), ??(??)) ????) ???? + ??(??, ??, ??(??), ??(??)) ????(??) + (? ?? ???? ????(??, ??, ??(??), ??(??)) ????(??)) ???? + ? ? ??(??, ??, ??(??), ??(??), ??) ˜??(????, ????) + (? ?? ? ? ???? ????(??, ??, ??(??), ??(??), ??) ˜??(????, ????)) ????, (2 7) A COLLECTION OF MATHEMATICAL AND PHYSICAL TABLES FROM A COURSE OF EXPERIMENTS