{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T17:09:50Z","timestamp":1785431390106,"version":"3.56.0"},"reference-count":39,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T00:00:00Z","timestamp":1762214400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We study backward stochastic difference equations (BS\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100454_inline1.png\">\n                          <jats:alt-text content-type=\"machine-generated\">normal upper Delta<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi mathvariant=\"normal\">\u0394<\/mml:mi>\n                        <\/mml:math>\n                        <jats:tex-math>$\\Delta$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    Es) driven by a\n                    <jats:italic>d<\/jats:italic>\n                    -dimensional stochastic process on a lattice, whose increments take only\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100454_inline2.png\">\n                          <jats:alt-text content-type=\"machine-generated\">d plus 1<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi>d<\/mml:mi>\n                          <mml:mo>+<\/mml:mo>\n                          <mml:mn>1<\/mml:mn>\n                        <\/mml:math>\n                        <jats:tex-math>$d+1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    possible values that generate the lattice. Interpreting the driving process as a\n                    <jats:italic>d<\/jats:italic>\n                    -dimensional asset price process, we provide applications to an optimal investment problem and to a market equilibrium analysis, where utility functionals are defined via BS\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100454_inline3.png\">\n                          <jats:alt-text content-type=\"machine-generated\">normal upper Delta<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi mathvariant=\"normal\">\u0394<\/mml:mi>\n                        <\/mml:math>\n                        <jats:tex-math>$\\Delta$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    Es.\n                  <\/jats:p>","DOI":"10.1017\/jpr.2025.10045","type":"journal-article","created":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T08:47:40Z","timestamp":1762246060000},"page":"580-606","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Backward stochastic difference equations on lattices with application to market equilibrium analysis"],"prefix":"10.1017","volume":"63","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0848-5001","authenticated-orcid":false,"given":"Masaaki","family":"Fukasawa","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/035t8zc32","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Osaka"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Takashi","family":"Sato","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01hcx6992","id-type":"ROR","asserted-by":"publisher"}],"name":"Humboldt University of Berlin"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jun","family":"Sekine","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/035t8zc32","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Osaka"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2025,11,4]]},"reference":[{"key":"S0021900225100454_ref26","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-937X.2007.00439.x"},{"key":"S0021900225100454_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/s004400100172"},{"key":"S0021900225100454_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4149(01)00131-4"},{"key":"S0021900225100454_ref12","first-page":"1047","article-title":"BS","volume":"19","author":"Cheridito","year":"2013","journal-title":"Bernoulli"},{"key":"S0021900225100454_ref24","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-021-00449-4"},{"key":"S0021900225100454_ref23","doi-asserted-by":"publisher","DOI":"10.1093\/rfs\/5.3.593"},{"key":"S0021900225100454_ref29","first-page":"302","article-title":"Numerical method for backward stochastic differential equations","volume":"12","author":"Ma","year":"2002","journal-title":"Ann. 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