Mathematical Sciences: Stochastic Integration and           Differential Equations
{'Code': '03040300', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'Stephen M. Samuels', 'PO_EMAI': None, 'PO_PHON': None}
The principal investigator will work on two projects in                    collaboration with many US and foreign scientist.  Both projects                relate to stochastic integrals and stochastic differential                      equations.  The many types of stochastic integrals that exist at                the moment have developed on an ad hoc basis, to fit the                        particular needs of the situation.  The principal investigator                  will develop a general unifying theory based on semi-martingale                 integrators.  It is hoped that in the process, the traditional                  assumption of asymptotic quasi-left continuity can be dropped.                  This will make the theory much more applicable.  A particular                   application of interest is in finance theory for the extension of               the Black-Scholes model.                                                             The second part of the project will deal with defining                     stochastic integrals with anticipating integrands.  In                          traditional definition of stochastic integrals, the integrand                   (non-anticipating) is assumed to be dependent only on the past of               the process with respect to which the integral is carried out.                  This makes the convergence properties of the integrals easy to                  prove.  However, in many applications, such as in Volterra                      equations in Physics, the integrand cannot be assumed to be non-                anticipating.  The work will use many technical results that the                principal investigator has developed in the past.
{'FirstName': 'Philip', 'LastName': 'Protter', 'PI_MID_INIT': 'E', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Philip E Protter', 'EmailAddress': 'pep2117@columbia.edu', 'NSF_ID': '000444357', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Purdue Research Foundation', 'CityName': 'WEST LAFAYETTE', 'ZipCode': '479064353', 'PhoneNumber': '3174946200', 'StreetAddress': '1281 WIN HENTSCHEL BLVD STE 2500', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Indiana', 'StateCode': 'IN', 'CONGRESSDISTRICT': '04', 'CONGRESS_DISTRICT_ORG': 'IN04', 'ORG_UEI_NUM': 'C1G1LGYJF5G5', 'ORG_LGL_BUS_NAME': 'PURDUE RESEARCH FOUNDATION', 'ORG_PRNT_UEI_NUM': 'YRXVL4JYCEF5'}
{'Name': None, 'CityName': None, 'StateCode': None, 'ZipCode': None, 'StreetAddress': None, 'CountryCode': None, 'CountryName': None, 'StateName': None, 'CountryFlag': None, 'CONGRESSDISTRICT': None, 'CONGRESS_DISTRICT_PERF': '""'}
{'Code': '126300', 'Text': 'PROBABILITY'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1988&All=true', 'xml': '8805595.xml'}