Nour El Houda Bouaicha, F. Chighoub, I. Alia, A. Sohail
The paper presents a characterization of equilibrium in a game-theoretic description of discounting conditional stochastic linear-quadratic (LQ for short) optimal control problem, in which the controlled state process evolves according to a multidimensional linear stochastic differential equation, when the noise is driven by a Poisson process and an independent Brownian motion under the effect of a Markovian regime-switching. The running and the terminal costs in the objective functional are explicitly dependent on several quadratic terms of the conditional expectation of the state process as well as on a nonexponential discount function, which create the time-inconsistency of the considered model. Open-loop Nash equilibrium controls are described through some necessary and sufficient equilibrium conditions. A state feedback equilibrium strategy is achieved via certain differential-difference system of ODEs. As an application, we study an investment–consumption and equilibrium reinsurance/new business strategies for mean-variance utility for insurers when the risk aversion is a function of current wealth level. The financial market consists of one riskless asset and one risky asset whose price process is modeled by geometric Lévy processes and the surplus of the insurers is assumed to follow a jump-diffusion model, where the values of parameters change according to continuous-time Markov chain. A numerical example is provided to demonstrate the efficacy of theoretical results.
{"title":"Conditional LQ time-inconsistent Markov-switching stochastic optimal control problem for diffusion with jumps","authors":"Nour El Houda Bouaicha, F. Chighoub, I. Alia, A. Sohail","doi":"10.15559/22-vmsta199","DOIUrl":"https://doi.org/10.15559/22-vmsta199","url":null,"abstract":"The paper presents a characterization of equilibrium in a game-theoretic description of discounting conditional stochastic linear-quadratic (LQ for short) optimal control problem, in which the controlled state process evolves according to a multidimensional linear stochastic differential equation, when the noise is driven by a Poisson process and an independent Brownian motion under the effect of a Markovian regime-switching. The running and the terminal costs in the objective functional are explicitly dependent on several quadratic terms of the conditional expectation of the state process as well as on a nonexponential discount function, which create the time-inconsistency of the considered model. Open-loop Nash equilibrium controls are described through some necessary and sufficient equilibrium conditions. A state feedback equilibrium strategy is achieved via certain differential-difference system of ODEs. As an application, we study an investment–consumption and equilibrium reinsurance/new business strategies for mean-variance utility for insurers when the risk aversion is a function of current wealth level. The financial market consists of one riskless asset and one risky asset whose price process is modeled by geometric Lévy processes and the surplus of the insurers is assumed to follow a jump-diffusion model, where the values of parameters change according to continuous-time Markov chain. A numerical example is provided to demonstrate the efficacy of theoretical results.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"26 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84824888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a sample estimator of the tangency portfolio (TP) weights is considered. The focus is on the situation where the number of observations is smaller than the number of assets in the portfolio and the returns are i.i.d. normally distributed. Under these assumptions, the sample covariance matrix follows a singular Wishart distribution and, therefore, the regular inverse cannot be taken. In the paper, bounds and approximations for the first two moments of the estimated TP weights are derived, as well as exact results are obtained when the population covariance matrix is equal to the identity matrix, employing the Moore–Penrose inverse. Moreover, exact moments based on the reflexive generalized inverse are provided. The properties of the bounds are investigated in a simulation study, where they are compared to the sample moments. The difference between the moments based on the reflexive generalized inverse and the sample moments based on the Moore–Penrose inverse is also studied.
{"title":"On the mean and variance of the estimated tangency portfolio weights for small samples","authors":"Gustav Alfelt, S. Mazur","doi":"10.15559/22-vmsta212","DOIUrl":"https://doi.org/10.15559/22-vmsta212","url":null,"abstract":"In this paper, a sample estimator of the tangency portfolio (TP) weights is considered. The focus is on the situation where the number of observations is smaller than the number of assets in the portfolio and the returns are i.i.d. normally distributed. Under these assumptions, the sample covariance matrix follows a singular Wishart distribution and, therefore, the regular inverse cannot be taken. In the paper, bounds and approximations for the first two moments of the estimated TP weights are derived, as well as exact results are obtained when the population covariance matrix is equal to the identity matrix, employing the Moore–Penrose inverse. Moreover, exact moments based on the reflexive generalized inverse are provided. The properties of the bounds are investigated in a simulation study, where they are compared to the sample moments. The difference between the moments based on the reflexive generalized inverse and the sample moments based on the Moore–Penrose inverse is also studied.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"14 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82416539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The factorial moments of any Markov branching process describe the behaviour of its probability generating function $F(t,s)$ in the neighbourhood of the point $s=1$. They are applied to solve the forward Kolmogorov equation for the critical Markov branching process with geometric reproduction of particles. The solution includes quickly convergent recurrent iterations of polynomials. The obtained results on factorial moments enable computation of statistical measures as shape and skewness. They are also applicable to the comparison between critical geometric branching and linear birth-death processes.
{"title":"Factorial moments of the critical Markov branching process with geometric reproduction of particles","authors":"Assen Tchorbadjieff, Penka Mayster","doi":"10.15559/22-vmsta201","DOIUrl":"https://doi.org/10.15559/22-vmsta201","url":null,"abstract":"The factorial moments of any Markov branching process describe the behaviour of its probability generating function $F(t,s)$ in the neighbourhood of the point $s=1$. They are applied to solve the forward Kolmogorov equation for the critical Markov branching process with geometric reproduction of particles. The solution includes quickly convergent recurrent iterations of polynomials. The obtained results on factorial moments enable computation of statistical measures as shape and skewness. They are also applicable to the comparison between critical geometric branching and linear birth-death processes.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"11 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73868147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In mixture with varying concentrations model (MVC) one deals with a nonhomogeneous sample which consists of subjects belonging to a fixed number of different populations (mixture components). The population which a subject belongs to is unknown, but the probabilities to belong to a given component are known and vary from observation to observation. The distribution of subjects’ observed features depends on the component which it belongs to. Generalized estimating equations (GEE) for Euclidean parameters in MVC models are considered. Under suitable assumptions the obtained estimators are asymptotically normal. A jackknife (JK) technique for the estimation of their asymptotic covariance matrices is described. Consistency of JK-estimators is demonstrated. An application to a model of mixture of nonlinear regressions and a real life example are presented.
{"title":"Jackknife for nonlinear estimating equations","authors":"R. Maiboroda, V. Miroshnychenko, O. Sugakova","doi":"10.15559/22-vmsta208","DOIUrl":"https://doi.org/10.15559/22-vmsta208","url":null,"abstract":"In mixture with varying concentrations model (MVC) one deals with a nonhomogeneous sample which consists of subjects belonging to a fixed number of different populations (mixture components). The population which a subject belongs to is unknown, but the probabilities to belong to a given component are known and vary from observation to observation. The distribution of subjects’ observed features depends on the component which it belongs to. Generalized estimating equations (GEE) for Euclidean parameters in MVC models are considered. Under suitable assumptions the obtained estimators are asymptotically normal. A jackknife (JK) technique for the estimation of their asymptotic covariance matrices is described. Consistency of JK-estimators is demonstrated. An application to a model of mixture of nonlinear regressions and a real life example are presented.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"8 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86414392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Reflected generalized backward stochastic differential equations (BSDEs) with one discontinuous barrier are investigated when the noise is driven by a Brownian motion and an independent Poisson measure. The existence and uniqueness of the solution are derived when the generators are monotone and the barrier is right-continuous with left limits (rcll). The link is established between this solution and a viscosity solution for an obstacle problem of integral-partial differential equations with nonlinear Neumann boundary conditions.
{"title":"Reflected generalized discontinuous BSDEs with rcll barrier and an obstacle problem of IPDE with nonlinear Neumann boundary conditions","authors":"Mohammed Elhachemy, M. El Otmani","doi":"10.15559/22-vmsta218","DOIUrl":"https://doi.org/10.15559/22-vmsta218","url":null,"abstract":"Reflected generalized backward stochastic differential equations (BSDEs) with one discontinuous barrier are investigated when the noise is driven by a Brownian motion and an independent Poisson measure. The existence and uniqueness of the solution are derived when the generators are monotone and the barrier is right-continuous with left limits (rcll). The link is established between this solution and a viscosity solution for an obstacle problem of integral-partial differential equations with nonlinear Neumann boundary conditions.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"44 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80248643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suitable families of random variables having power series distributions are considered, and their asymptotic behavior in terms of large (and moderate) deviations is studied. Two examples of fractional counting processes are presented, where the normalizations of the involved power series distributions can be expressed in terms of the Prabhakar function. The first example allows to consider the counting process in [Integral Transforms Spec. Funct. 27 (2016), 783–793], the second one is inspired by a model studied in [J. Appl. Probab. 52 (2015), 18–36].
{"title":"Asymptotic results for families of random variables having power series distributions","authors":"C. Macci, B. Pacchiarotti, Elena Villa","doi":"10.15559/21-vmsta198","DOIUrl":"https://doi.org/10.15559/21-vmsta198","url":null,"abstract":"Suitable families of random variables having power series distributions are considered, and their asymptotic behavior in terms of large (and moderate) deviations is studied. Two examples of fractional counting processes are presented, where the normalizations of the involved power series distributions can be expressed in terms of the Prabhakar function. The first example allows to consider the counting process in [Integral Transforms Spec. Funct. 27 (2016), 783–793], the second one is inspired by a model studied in [J. Appl. Probab. 52 (2015), 18–36].","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"22 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84401669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The stochastic process of the form [ {X_{t}}={int _{0}^{t}}{s^{alpha }}left({int _{s}^{t}}{u^{beta }}{(u-s)^{gamma }}hspace{0.1667em}duright)hspace{0.1667em}d{W_{s}}] is considered, where W is a standard Wiener process, $alpha >-frac{1}{2}$, $gamma >-1$, and $alpha +beta +gamma >-frac{3}{2}$. It is proved that the process X is well-defined and continuous. The asymptotic properties of the variances and bounds for the variances of the increments of the process X are studied. It is also proved that the process X satisfies the single-point Hölder condition up to order $alpha +beta +gamma +frac{3}{2}$ at point 0, the “interval” Hölder condition up to order $min big(gamma +frac{3}{2},hspace{0.2222em}1big)$ on the interval $[{t_{0}},T]$ (where $0<{t_{0}}
{"title":"Gaussian Volterra processes with power-type kernels. Part I","authors":"Y. Mishura, S. Shklyar","doi":"10.15559/22-vmsta205","DOIUrl":"https://doi.org/10.15559/22-vmsta205","url":null,"abstract":"The stochastic process of the form [ {X_{t}}={int _{0}^{t}}{s^{alpha }}left({int _{s}^{t}}{u^{beta }}{(u-s)^{gamma }}hspace{0.1667em}duright)hspace{0.1667em}d{W_{s}}] is considered, where W is a standard Wiener process, $alpha >-frac{1}{2}$, $gamma >-1$, and $alpha +beta +gamma >-frac{3}{2}$. It is proved that the process X is well-defined and continuous. The asymptotic properties of the variances and bounds for the variances of the increments of the process X are studied. It is also proved that the process X satisfies the single-point Hölder condition up to order $alpha +beta +gamma +frac{3}{2}$ at point 0, the “interval” Hölder condition up to order $min big(gamma +frac{3}{2},hspace{0.2222em}1big)$ on the interval $[{t_{0}},T]$ (where $0<{t_{0}}<T$), and the Hölder condition up to order $min big(alpha +beta +gamma +frac{3}{2},hspace{0.2778em}gamma +frac{3}{2},hspace{0.2778em}1big)$ on the entire interval $[0,T]$.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"20 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75367945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered normal distribution. In this paper, some examples of classes of large deviation principles of this kind are presented, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.
{"title":"Some examples of noncentral moderate deviations for sequences of real random variables","authors":"R. Giuliano, C. Macci","doi":"10.15559/23-vmsta219","DOIUrl":"https://doi.org/10.15559/23-vmsta219","url":null,"abstract":"The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered normal distribution. In this paper, some examples of classes of large deviation principles of this kind are presented, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"142 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77370511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a class of Cannings models with population size N having a mixed multinomial offspring distribution with random success probabilities W1, . . . ,WN induced by independent and identically distributed positive random variables X1, X2, . . . via Wi := Xi/SN , i ∈ {1, . . . , N}, where SN := X1 + · · · + XN . The ancestral lineages are hence based on a sampling with replacement strategy from a random partition of the unit interval into N subintervals of lengths W1, . . . ,WN . Convergence results for the genealogy of these Cannings models are provided under regularly varying assumptions on the tail distribution of X1. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained in [15] for the case when X1 is Pareto distributed and complement those obtained in [37] for models where one samples without replacement from a supercritical branching process.
{"title":"Asymptotic genealogies for a class of generalized Wright–Fisher models","authors":"T. Huillet, M. Möhle","doi":"10.15559/21-vmsta196","DOIUrl":"https://doi.org/10.15559/21-vmsta196","url":null,"abstract":"We study a class of Cannings models with population size N having a mixed multinomial offspring distribution with random success probabilities W1, . . . ,WN induced by independent and identically distributed positive random variables X1, X2, . . . via Wi := Xi/SN , i ∈ {1, . . . , N}, where SN := X1 + · · · + XN . The ancestral lineages are hence based on a sampling with replacement strategy from a random partition of the unit interval into N subintervals of lengths W1, . . . ,WN . Convergence results for the genealogy of these Cannings models are provided under regularly varying assumptions on the tail distribution of X1. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained in [15] for the case when X1 is Pareto distributed and complement those obtained in [37] for models where one samples without replacement from a supercritical branching process.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"132 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80001738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
General models of random fields on the sphere associated with nonlocal equations in time and space are studied. The properties of the corresponding angular power spectrum are discussed and asymptotic results in terms of random time changes are found.
{"title":"Models of space-time random fields on the sphere","authors":"M. D’Ovidio, E. Orsingher, L. Sakhno","doi":"10.15559/22-vmsta200","DOIUrl":"https://doi.org/10.15559/22-vmsta200","url":null,"abstract":"General models of random fields on the sphere associated with nonlocal equations in time and space are studied. The properties of the corresponding angular power spectrum are discussed and asymptotic results in terms of random time changes are found.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"81 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90374411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：