Backward stochastic differential equations in dynamics of life insurance solvency risk

In this thesis we describe the dynamics of solvency level in life insurance contracts. We do this by representing the underlying sources of risk and the solvency level as the solution to a forward-backward stochastic differential equation system. We start by introducing Brownian motion, stochastic i...

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Päätekijä: Hinkkanen, Onni
Muut tekijät: Matemaattis-luonnontieteellinen tiedekunta, Faculty of Sciences, Matematiikan ja tilastotieteen laitos, Department of Mathematics and Statistics, Jyväskylän yliopisto, University of Jyväskylä
Aineistotyyppi: Pro gradu
Kieli:eng
Julkaistu: 2022
Aiheet:
Linkit: https://jyx.jyu.fi/handle/123456789/84222
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author Hinkkanen, Onni
author2 Matemaattis-luonnontieteellinen tiedekunta Faculty of Sciences Matematiikan ja tilastotieteen laitos Department of Mathematics and Statistics Jyväskylän yliopisto University of Jyväskylä
author_facet Hinkkanen, Onni Matemaattis-luonnontieteellinen tiedekunta Faculty of Sciences Matematiikan ja tilastotieteen laitos Department of Mathematics and Statistics Jyväskylän yliopisto University of Jyväskylä Hinkkanen, Onni Matemaattis-luonnontieteellinen tiedekunta Faculty of Sciences Matematiikan ja tilastotieteen laitos Department of Mathematics and Statistics Jyväskylän yliopisto University of Jyväskylä
author_sort Hinkkanen, Onni
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description In this thesis we describe the dynamics of solvency level in life insurance contracts. We do this by representing the underlying sources of risk and the solvency level as the solution to a forward-backward stochastic differential equation system. We start by introducing Brownian motion, stochastic integration, stochastic differential equations, and backward stochastic differential equations. With these notions described we can start constructing the model for solvency risk. Afterwards we also give a link to partial differential equation theory and a Monte Carlo example for obtaining explicit representations for the processes involved. We will denote the net value of the contract by a process N, which will depend on underlying economic and demographic variables. We say that the contract is solvent at time t if Nt ≥ 0. We can express the change in solvency probability at the expiry time T as P(NT ≥ 0|Ft) − P(NT ≥ 0|F0) = Z t 0 U ⊤ r dMX r = Z t 0 Z ⊤ r dBr, where the filtration (Ft)t≥0 describes the information available at time t, MX r is the martingale part from Doob’s 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 driver B has on the solvency level, respectively. More technically, the forward-backward system we study is of the form ( d(Xs, V − s ) ⊤ = ˜µ(s, Xs, V − s )ds + ˜σ(s, Xs)dBs, (Xt , V − t ) ⊤ = (v, x) ⊤ −dYs = −Z ⊤ s dBs, YT = Ψ X (t,x) T , V −(t,x,v) T , where ˜µ and ˜σ are used in defining the process X and contain the information on actuarial assumptions, V − is the retrospective reserve, which describes the present value of assets that belong to the insurance contract at each time t, and Ψ 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 (Xs, V − s , Ys, Zs) s∈[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 and the other studying the contributions of the Brownian driver
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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. 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spellingShingle Hinkkanen, Onni Backward stochastic differential equations in dynamics of life insurance solvency risk backward stochastic differential equations stochastic analysis Stokastiikka ja todennäköisyysteoria Stochastics and Probability 4041 stokastiset prosessit vakuutusmatematiikka henkivakuutus matemaattiset mallit stochastic processes insurance mathematics life insurance mathematical models
title Backward stochastic differential equations in dynamics of life insurance solvency risk
title_full Backward stochastic differential equations in dynamics of life insurance solvency risk
title_fullStr Backward stochastic differential equations in dynamics of life insurance solvency risk Backward stochastic differential equations in dynamics of life insurance solvency risk
title_full_unstemmed Backward stochastic differential equations in dynamics of life insurance solvency risk Backward stochastic differential equations in dynamics of life insurance solvency risk
title_short Backward stochastic differential equations in dynamics of life insurance solvency risk
title_sort backward stochastic differential equations in dynamics of life insurance solvency risk
title_txtP Backward stochastic differential equations in dynamics of life insurance solvency risk
topic backward stochastic differential equations stochastic analysis Stokastiikka ja todennäköisyysteoria Stochastics and Probability 4041 stokastiset prosessit vakuutusmatematiikka henkivakuutus matemaattiset mallit stochastic processes insurance mathematics life insurance mathematical models
topic_facet 4041 Stochastics and Probability Stokastiikka ja todennäköisyysteoria backward stochastic differential equations henkivakuutus insurance mathematics life insurance matemaattiset mallit mathematical models stochastic analysis stochastic processes stokastiset prosessit vakuutusmatematiikka
url https://jyx.jyu.fi/handle/123456789/84222 http://www.urn.fi/URN:NBN:fi:jyu-202212085484
work_keys_str_mv AT hinkkanenonni backwardstochasticdifferentialequationsindynamicsoflifeinsurancesolvencyrisk