In this paper we establish the existence and the uniqueness of the solution of a special class of BSDEs for L'{e}vy processes in the case of a Lipschitz generator of sublinear growth. We then study a related problem of logarithmic utility maximization of the terminal wealth in the filtration generated by an arbitrary L'{e}vy process.
{"title":"BSDEs and log-utility maximization for Lévy processes","authors":"P. D. Tella, H. Engelbert","doi":"10.15559/19-VMSTA144","DOIUrl":"https://doi.org/10.15559/19-VMSTA144","url":null,"abstract":"In this paper we establish the existence and the uniqueness of the solution of a special class of BSDEs for L'{e}vy processes in the case of a Lipschitz generator of sublinear growth. We then study a related problem of logarithmic utility maximization of the terminal wealth in the filtration generated by an arbitrary L'{e}vy process.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"9 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88493371","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}
A general jackknife estimator for the asymptotic covariance of moment estimators is considered in the case when the sample is taken from a mixture with varying concentrations of components. Consistency of the estimator is demonstrated. A fast algorithm for its calculation is described. The estimator is applied to construction of confidence sets for regression parameters in the linear regression with errors in variables. An application to sociological data analysis is considered.
{"title":"Jackknife covariance matrix estimation for observations from mixture","authors":"R. Maiboroda, O. Sugakova","doi":"10.15559/19-VMSTA145","DOIUrl":"https://doi.org/10.15559/19-VMSTA145","url":null,"abstract":"A general jackknife estimator for the asymptotic covariance of moment estimators is considered in the case when the sample is taken from a mixture with varying concentrations of components. Consistency of the estimator is demonstrated. A fast algorithm for its calculation is described. The estimator is applied to construction of confidence sets for regression parameters in the linear regression with errors in variables. An application to sociological data analysis is considered.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75243684","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 need to model a Markov renewal on-off process with multiple off-states arise in many applications such as economics, physics, and engineering. Characterization of the occupation time of one specific off-state marginally or two off-states jointly is crucial to understand such processes. The exact marginal and joint distributions of the off-state occupation times are derived. The theoretical results are confirmed numerically in a simulation study. A special case when all holding times have Lévy distribution is considered for the possibility of simplification of the formulas.
{"title":"On occupation time for on-off processes with multiple off-states","authors":"Chaoran Hu, V. Pozdnyakov, Jun Yan","doi":"10.15559/22-vmsta210","DOIUrl":"https://doi.org/10.15559/22-vmsta210","url":null,"abstract":"The need to model a Markov renewal on-off process with multiple off-states arise in many applications such as economics, physics, and engineering. Characterization of the occupation time of one specific off-state marginally or two off-states jointly is crucial to understand such processes. The exact marginal and joint distributions of the off-state occupation times are derived. The theoretical results are confirmed numerically in a simulation study. A special case when all holding times have Lévy distribution is considered for the possibility of simplification of the formulas.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"21 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74828962","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 investigate the fractional Vasicek model described by the stochastic differential equation $dX_t=(alpha -beta X_t),dt+gamma ,dB^H_t$, $X_0=x_0$, driven by the fractional Brownian motion $B^H$ with the known Hurst parameter $Hin (1/2,1)$. We study the maximum likelihood estimators for unknown parameters $alpha$ and $beta$ in the non-ergodic case (when $beta <0$) for arbitrary $x_0in mathbb{R}$, generalizing the result of Tanaka, Xiao and Yu (2019) for particular $x_0=alpha /beta$, derive their asymptotic distributions and prove their asymptotic independence.
{"title":"Maximum likelihood estimation in the non-ergodic fractional Vasicek model","authors":"S. Lohvinenko, K. Ralchenko","doi":"10.15559/19-VMSTA140","DOIUrl":"https://doi.org/10.15559/19-VMSTA140","url":null,"abstract":"We investigate the fractional Vasicek model described by the stochastic differential equation $dX_t=(alpha -beta X_t),dt+gamma ,dB^H_t$, $X_0=x_0$, driven by the fractional Brownian motion $B^H$ with the known Hurst parameter $Hin (1/2,1)$. We study the maximum likelihood estimators for unknown parameters $alpha$ and $beta$ in the non-ergodic case (when $beta <0$) for arbitrary $x_0in mathbb{R}$, generalizing the result of Tanaka, Xiao and Yu (2019) for particular $x_0=alpha /beta$, derive their asymptotic distributions and prove their asymptotic independence.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"7 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87898356","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 we present a numerical scheme for stochastic differential equations based upon the Wiener chaos expansion. The approximation of a square integrable stochastic differential equation is obtained by cutting off the infinite chaos expansion in chaos order and in number of basis elements. We derive an explicit upper bound for the $L^2$ approximation error associated with our method. The proofs are based upon an application of Malliavin calculus.
{"title":"The asymptotic error of chaos expansion approximations for stochastic differential equations","authors":"T. Huschto, M. Podolskij, S. Sager","doi":"10.15559/19-VMSTA133","DOIUrl":"https://doi.org/10.15559/19-VMSTA133","url":null,"abstract":"In this paper we present a numerical scheme for stochastic differential equations based upon the Wiener chaos expansion. The approximation of a square integrable stochastic differential equation is obtained by cutting off the infinite chaos expansion in chaos order and in number of basis elements. We derive an explicit upper bound for the $L^2$ approximation error associated with our method. The proofs are based upon an application of Malliavin calculus.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83012413","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}
A bivariate integer-valued autoregressive process of order 1 (BINAR(1)) with copula-joint innovations is studied. Different parameter estimation methods are analyzed and compared via Monte Carlo simulations with emphasis on estimation of the copula dependence parameter. An empirical application on defaulted and non-defaulted loan data is carried out using different combinations of copula functions and marginal distribution functions covering the cases where both marginal distributions are from the same family, as well as the case where they are from different distribution families.
{"title":"A copula-based bivariate integer-valued autoregressive process with application","authors":"A. Buteikis, R. Leipus","doi":"10.15559/19-VMSTA130","DOIUrl":"https://doi.org/10.15559/19-VMSTA130","url":null,"abstract":"A bivariate integer-valued autoregressive process of order 1 (BINAR(1)) with copula-joint innovations is studied. Different parameter estimation methods are analyzed and compared via Monte Carlo simulations with emphasis on estimation of the copula dependence parameter. An empirical application on defaulted and non-defaulted loan data is carried out using different combinations of copula functions and marginal distribution functions covering the cases where both marginal distributions are from the same family, as well as the case where they are from different distribution families.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"50 14","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72489085","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 the fractional Cox-Ingersoll-Ross process on $mathbb{R}_+$ for $H 0}}+varepsilon}-a Y_{varepsilon}(t))dt+sigma dB^H(t)$, as $varepsilondownarrow0$. Properties of such limit process are considered. SDE for both the limit process and the fractional Cox-Ingersoll-Ross process are obtained.
{"title":"Fractional Cox–Ingersoll–Ross process with small Hurst indices","authors":"Y. Mishura, Anton Yurchenko-Tytarenko","doi":"10.15559/18-VMSTA126","DOIUrl":"https://doi.org/10.15559/18-VMSTA126","url":null,"abstract":"In this paper the fractional Cox-Ingersoll-Ross process on $mathbb{R}_+$ for $H 0}}+varepsilon}-a Y_{varepsilon}(t))dt+sigma dB^H(t)$, as $varepsilondownarrow0$. Properties of such limit process are considered. SDE for both the limit process and the fractional Cox-Ingersoll-Ross process are obtained.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"251 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79409417","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 we consider a telegraph equation with time-dependent coefficients, governing the persistent random walk of a particle moving on the line with a time-varying velocity $c(t)$ and changing direction at instants distributed according to a non-stationary Poisson distribution with rate $lambda(t)$. We show that, under suitable assumptions, we are able to find the exact form of the probability distribution. We also consider the space-fractional counterpart of this model, finding the characteristic function of the related process. A conclusive discussion is devoted to the potential applications to run-and-tumble models.
{"title":"Probability distributions for the run-and-tumble models with variable speed and tumbling rate","authors":"L. Angelani, R. Garra","doi":"10.15559/18-VMSTA127","DOIUrl":"https://doi.org/10.15559/18-VMSTA127","url":null,"abstract":"In this paper we consider a telegraph equation with time-dependent coefficients, governing the persistent random walk of a particle moving on the line with a time-varying velocity $c(t)$ and changing direction at instants distributed according to a non-stationary Poisson distribution with rate $lambda(t)$. We show that, under suitable assumptions, we are able to find the exact form of the probability distribution. We also consider the space-fractional counterpart of this model, finding the characteristic function of the related process. A conclusive discussion is devoted to the potential applications to run-and-tumble models.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"6 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78562404","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}
For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $Dsubset mathbb {R}^d$ and driven by an $L^2(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new result on existence and uniqueness of a mild solution is established. Compared to the existing results, the uniqueness in a fully nonlinear case is shown, not assuming the coefficient in front of the noise to be affine. Additionally, the existence of moments for the solution is established.
对于定义在有界开子集$D子集mathbb {R}^ D $上的一类非自治抛物型随机偏微分方程，由具有Hurst指标$H>1/2$的$L^2(D)$值分数阶布朗运动驱动，得到了一类温和解的存在唯一性的新结果。与已有结果相比，在不假设噪声前系数为仿射的情况下，证明了完全非线性情况下的唯一性。此外，还证明了解的矩的存在性。
{"title":"Existence and uniqueness of mild solution to fractional stochastic heat equation","authors":"K. Ralchenko, G. Shevchenko","doi":"10.15559/18-VMSTA122","DOIUrl":"https://doi.org/10.15559/18-VMSTA122","url":null,"abstract":"For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $Dsubset mathbb {R}^d$ and driven by an $L^2(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new result on existence and uniqueness of a mild solution is established. Compared to the existing results, the uniqueness in a fully nonlinear case is shown, not assuming the coefficient in front of the noise to be affine. Additionally, the existence of moments for the solution is established.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"46 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83647804","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 problem of European-style option pricing in time-changed L'{e}vy models in the presence of compound Poisson jumps is considered. These jumps relate to sudden large drops in stock prices induced by political or economical hits. As the time-changed L'{e}vy models, the variance-gamma and the normal-inverse Gaussian models are discussed. Exact formulas are given for the price of digital asset-or-nothing call option on extra asset in foreign currency. The prices of simpler options can be derived as corollaries of our results and examples are presented. Various types of dependencies between stock prices are mentioned.
{"title":"Option pricing in time-changed Lévy models with compound Poisson jumps","authors":"R. Ivanov, K. Ano","doi":"10.15559/18-VMSTA124","DOIUrl":"https://doi.org/10.15559/18-VMSTA124","url":null,"abstract":"The problem of European-style option pricing in time-changed L'{e}vy models in the presence of compound Poisson jumps is considered. These jumps relate to sudden large drops in stock prices induced by political or economical hits. As the time-changed L'{e}vy models, the variance-gamma and the normal-inverse Gaussian models are discussed. Exact formulas are given for the price of digital asset-or-nothing call option on extra asset in foreign currency. The prices of simpler options can be derived as corollaries of our results and examples are presented. Various types of dependencies between stock prices are mentioned.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"93 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85696818","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}
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