IN 2015 THE TERRESTRIAL LIGO GRAVITATIONAL-WAVE DETECTORS OBSERVED THE FIRST GRAVITATIONAL-WAVE SIGNAL GENERATED IN A MERGER OF TWO BLACK HOLES ABOUT 1.3 BILLION LIGHTYEARS AWAY. THIS EVENT HAS MARKED THE BEGINNING OF THE NEW FIELD OF GRAVITATIONAL-WAVE ASTRONOMY. GRAVITATIONAL-WAVE DETECTORS HAVE SINCE STARTED TO ROUTINELY OBSERVE OBJECTS AND EVENTS IN THE UNIVERSE THAT ARE NOT ACCESSIBLE TO THE TRADITIONAL ELECTROMAGNETIC OBSERVATIONS. FURTHERMORE IN 2016-2017 THE SUCCESSFUL LISA PATHFINDER MISSION DEMONSTRATED SOME OF THE KEY TECHNOLOGIES NEEDED FOR DEVELOPING A SPACE-BORNE GRAVITATIONAL-WAVE DETECTOR. COMING ON THE HEELS OF THESE REMARKABLE BREAKTHROUGHS THE EUROPEAN SPACE AGENCY (ESA) RECENTLY SELECTED THE LASER INTERFEROMETER SPACE ANTENNA (LISA) AS THE THIRD LARGE-CLASS MISSION IN ESA S SCIENCE PROGRAMME. LISA IS EXPECTED TO BE LAUNCHED IN 2034 AND WITH A SIGNIFICANT CONTRIBUTION FROM NASA IT WILL BE THE FIRST SPACE-BORNE GRAVITATIONAL-WAVE OBSERVATORY. THIS PROPOSAL FOCUSES ON THE STOCHASTIC GRAVITATIONAL-WAVE BACKGROUND (SGWB) WHICH IS ONE OF THE SCIENCE TARGETS OF THE LISA MISSION. THE SGWB IS EXPECTED TO ARISE AS A SUPERPOSITION (SUM) OF MANY INCOHERENT SOURCES OF GRAVITATIONAL WAVES. IT COULD BE OF ASTROPHYSICAL ORIGIN FOR EXAMPLE DUE TO CONTRIBUTIONS OF NUMEROUS BINARY SYSTEMS IN THE MILKY WAY OR THROUGHOUT THE UNIVERSE. IT COULD ALSO BE OF COSMOLOGICAL ORIGIN GENERATED BY VERY ENERGETIC PROCESSES IN THE EARLY UNIVERSE. DETECTION OF A COSMOLOGICAL SGWB WOULD PROVIDE UNIQUE INFORMATION ABOUT THE FUNDAMENTAL PHYSICAL LAWS THAT APPLY AT VERY HIGH ENERGY SCALES INACCESSIBLE TO STANDARD LABORATORY EXPERIMENTS. DETECTION OF AN ASTROPHYSICAL SGWB WOULD PROVIDE UNIQUE INFORMATION ABOUT PROPERTIES AND EVOLUTION OF STRUCTURE WE OBSERVE TODAY IN THE UNIVERSE INCLUDING OBJECTS SUCH AS NEUTRON STARS AND BLACK HOLES. THIS PROPOSAL WILL DEVELOP A NEW METHOD FOR MEASURING THE PROPERTIES OF THE SGWB USING LISA DATA DRAWING FROM SIMILAR METHODS DEVELOPED IN THE CONTEXT OF TERRESTRIAL GRAVITATIONAL-WAVE DETECTORS AND OF PULSAR TIMING EXPERIMENTS. THE NEW METHOD KNOWN AS THE PHASE-COHERENT MAPPING APPROACH WILL ENABLE DIRECT ESTIMATES OF THE FREQUENCY DIRECTIONALITY AND POLARIZATION CONTENT OF THE SGWB. IT IS ALSO COMPATIBLE WITH THE GLOBAL-SOLUTION APPROACH TO LISA DATA ANALYSIS WHICH ADVOCATES DOING A GLOBAL FIT TO ALL OF THE GRAVITATIONAL-WAVE SIGNAL AND NOISE COMPONENTS IN THE DATA. FURTHERMORE A STATISTICAL FRAMEWORK WILL BE DEVELOPED TO PERFORM THE SGWB MODEL SELECTION AND PARAMETER ESTIMATION FOR LISA. BY LEVERAGING DIFFERENCES IN DIRECTIONAL POLARIZATION AND FREQUENCY STRUCTURE IN DIFFERENT SGWB MODELS THIS FRAMEWORK WILL ENABLE IDENTIFICATION OF CONTRIBUTIONS FROM DIFFERENT SGWB SOURCES HENCE SEPARATING THE COSMOLOGICAL SGWB FROM THE ASTROPHYSICAL FOREGROUNDS . THE FRAMEWORK WILL THEREFORE COMBINE THE DEVELOPMENT OF SGWB MODELS AND SGWB SEARCH TECHNIQUES TO FULLY EXPLOIT THE SCIENCE POTENTIAL OF SGWB SEARCHES WITH LISA.