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[{"key": "dc.contributor.advisor", "value": "Geiss, Christel", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.advisor", "value": "Geiss, Stefan", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Hinkkanen, Onni", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2022-12-08T07:58:00Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2022-12-08T07:58:00Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2022", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/84222", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "In this thesis we describe the dynamics of solvency level in life insurance contracts.\nWe do this by representing the underlying sources of risk and the solvency level as the\nsolution to a forward-backward stochastic differential equation system. We start by\nintroducing Brownian motion, stochastic integration, stochastic differential equations,\nand backward stochastic differential equations. With these notions described we can\nstart constructing the model for solvency risk. Afterwards we also give a link to\npartial differential equation theory and a Monte Carlo example for obtaining explicit\nrepresentations for the processes involved.\nWe will denote the net value of the contract by a process N, which will depend on\nunderlying economic and demographic variables. We say that the contract is solvent\nat time t if Nt \u2265 0. We can express the change in solvency probability at the expiry\ntime T as\nP(NT \u2265 0|Ft) \u2212 P(NT \u2265 0|F0) = Z t\n0\nU\n\u22a4\nr dMX\nr =\nZ t\n0\nZ\n\u22a4\nr dBr,\nwhere the filtration (Ft)t\u22650 describes the information available at time t, MX\nr\nis the\nmartingale part from Doob\u2019s decomposition of the process X. Furthermore, the pro gressively measurable processes U and Z represent the contributions of the aforemen tioned underlying variables to the overall solvency risk, and the effects the Brownian\ndriver B has on the solvency level, respectively.\nMore technically, the forward-backward system we study is of the form\n(\nd(Xs, V \u2212\ns\n)\n\u22a4 = \u02dc\u00b5(s, Xs, V \u2212\ns\n)ds + \u02dc\u03c3(s, Xs)dBs, (Xt\n, V \u2212\nt\n)\n\u22a4 = (v, x)\n\u22a4\n\u2212dYs = \u2212Z\n\u22a4\ns dBs, YT = \u03a8 \nX\n(t,x)\nT\n, V \u2212(t,x,v)\nT\n \n,\nwhere \u02dc\u00b5 and \u02dc\u03c3 are used in defining the process X and contain the information on\nactuarial assumptions, V\n\u2212 is the retrospective reserve, which describes the present\nvalue of assets that belong to the insurance contract at each time t, and \u03a8 is a ter minal condition, which in our case is not continuous. Under some Lipschitz, bound edness and continuity conditions it will yield a unique, square integrable solution\n(Xs, V \u2212\ns\n, Ys, Zs)\ns\u2208[t,T] which we use for the description of solvency level in two differ ent viewpoints; one considering the effects of the underlying demographic variables\nand the other studying the contributions of the Brownian driver", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2022-12-08T07:58:00Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2022-12-08T07:58:00Z (GMT). No. of bitstreams: 0\n Previous issue date: 2022", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "49", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "eng", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "backward stochastic differential equations", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "stochastic analysis", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Backward stochastic differential equations in dynamics of life insurance solvency risk", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202212085484", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Stokastiikka ja todenn\u00e4k\u00f6isyysteoria", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Stochastics and Probability", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "stokastiset prosessit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "vakuutusmatematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "henkivakuutus", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matemaattiset mallit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "stochastic processes", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "insurance mathematics", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "life insurance", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "mathematical models", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", 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