Pub Date : 2015-06-04DOI: 10.1080/17442508.2015.1019884
Chao Zhu, G. Yin, Nicholas A. Baran
This work develops Feynman–Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated with a general Lévy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables related to the regime-switching jump diffusion process converges in distribution to the arcsine law.
{"title":"Feynman–Kac formulas for regime-switching jump diffusions and their applications","authors":"Chao Zhu, G. Yin, Nicholas A. Baran","doi":"10.1080/17442508.2015.1019884","DOIUrl":"https://doi.org/10.1080/17442508.2015.1019884","url":null,"abstract":"This work develops Feynman–Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated with a general Lévy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables related to the regime-switching jump diffusion process converges in distribution to the arcsine law.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"21 1","pages":"1000 - 1032"},"PeriodicalIF":0.9,"publicationDate":"2015-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80847796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-06-04DOI: 10.1080/17442508.2015.1019881
A. Bibi, Ahmed Ghezal
This article investigates some probabilistic properties and statistical applications of general Markov-switching bilinear processes that offer remarkably rich dynamics and complex behaviour to model non-Gaussian data with structural changes. In these models, the parameters are allowed to depend on unobservable time-homogeneous and stationary Markov chain with finite state space. So, some basic issues concerning this class of models including necessary and sufficient conditions ensuring the existence of ergodic stationary (in some sense) solutions, existence of finite moments of any order and -mixing are studied. As a consequence, we observe that the local stationarity of the underlying process is neither sufficient nor necessary to obtain the global stationarity. Also, the covariance functions of the process and its power are evaluated and it is shown that the second (respectively, higher)-order structure is similar to some linear processes, and hence admit representation. We establish also sufficient conditions for the model to be mixing and geometrically ergodic. We then use these results to give sufficient conditions for mixing of a family of processes. A number of illustrative examples are given to clarify the theory and the variety of applications.
{"title":"On the Markov-switching bilinear processes: stationarity, higher-order moments and β-mixing","authors":"A. Bibi, Ahmed Ghezal","doi":"10.1080/17442508.2015.1019881","DOIUrl":"https://doi.org/10.1080/17442508.2015.1019881","url":null,"abstract":"This article investigates some probabilistic properties and statistical applications of general Markov-switching bilinear processes that offer remarkably rich dynamics and complex behaviour to model non-Gaussian data with structural changes. In these models, the parameters are allowed to depend on unobservable time-homogeneous and stationary Markov chain with finite state space. So, some basic issues concerning this class of models including necessary and sufficient conditions ensuring the existence of ergodic stationary (in some sense) solutions, existence of finite moments of any order and -mixing are studied. As a consequence, we observe that the local stationarity of the underlying process is neither sufficient nor necessary to obtain the global stationarity. Also, the covariance functions of the process and its power are evaluated and it is shown that the second (respectively, higher)-order structure is similar to some linear processes, and hence admit representation. We establish also sufficient conditions for the model to be mixing and geometrically ergodic. We then use these results to give sufficient conditions for mixing of a family of processes. A number of illustrative examples are given to clarify the theory and the variety of applications.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"11 1","pages":"919 - 945"},"PeriodicalIF":0.9,"publicationDate":"2015-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79471772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-06-04DOI: 10.1080/17442508.2015.1019880
Heidar Eyjolfsson
In their paper Barndorff-Nielsen et al. [4] employ so called ambit fields to model electricity spot-forward dynamics. We briefly introduce and discuss ambit fields, and introduce a novel method for approximating general ambit fields by a linear combination of ambit fields driven by exponential kernel functions (as has already been done in the null-spatial case of Lévy semistationary processes by Benth et al. [11]) by approximating the deterministic kernel function by a carefully selected finite sum. Moreover, we shall study examples, in the setting of modelling electricity forward markets, of ambit fields with singular kernel functions to illustrate the usefulness of our method for pricing purposes.
{"title":"Approximating ambit fields via Fourier methods","authors":"Heidar Eyjolfsson","doi":"10.1080/17442508.2015.1019880","DOIUrl":"https://doi.org/10.1080/17442508.2015.1019880","url":null,"abstract":"In their paper Barndorff-Nielsen et al. [4] employ so called ambit fields to model electricity spot-forward dynamics. We briefly introduce and discuss ambit fields, and introduce a novel method for approximating general ambit fields by a linear combination of ambit fields driven by exponential kernel functions (as has already been done in the null-spatial case of Lévy semistationary processes by Benth et al. [11]) by approximating the deterministic kernel function by a carefully selected finite sum. Moreover, we shall study examples, in the setting of modelling electricity forward markets, of ambit fields with singular kernel functions to illustrate the usefulness of our method for pricing purposes.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"57 5","pages":"885 - 917"},"PeriodicalIF":0.9,"publicationDate":"2015-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/17442508.2015.1019880","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72538543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-05-12DOI: 10.1080/17442508.2015.1019883
Litan Yan, Xianye Yu
Let be a fractional Brownian motion taking values in with Hurst index . In this paper, we consider the self-intersection local time and its derivative in the spatial variable . In particular, we introduce the so-called integrated quadratic covariation and show that the Bouleau-Yor type identityholds for some suitable .
{"title":"Derivative for self-intersection local time of multidimensional fractional Brownian motion","authors":"Litan Yan, Xianye Yu","doi":"10.1080/17442508.2015.1019883","DOIUrl":"https://doi.org/10.1080/17442508.2015.1019883","url":null,"abstract":"Let be a fractional Brownian motion taking values in with Hurst index . In this paper, we consider the self-intersection local time and its derivative in the spatial variable . In particular, we introduce the so-called integrated quadratic covariation and show that the Bouleau-Yor type identityholds for some suitable .","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"45 1","pages":"966 - 999"},"PeriodicalIF":0.9,"publicationDate":"2015-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84764029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-05-04DOI: 10.1080/17442508.2014.959951
Yongfeng Wu, Jiangyan Peng, T. Hu
The authors study Lr-convergence, complete convergence and complete moment convergence for arrays of row-wise extended negatively dependent random variables under some appropriate conditions of h-integrability. The results in this paper extend and improve the results of Sung et al. [H.S. Sung, S. Lisawadi, A. Volodin, Weak laws of large numbers for arrays under a condition of uniform integrability, J. Korean Math. Soc. 45 (2008), pp. 289–300].
在适当的h-可积条件下，研究了行向扩展负相关随机变量阵列的lr收敛、完全收敛和完全矩收敛。本文的结果扩展和改进了Sung等人的研究结果宋，S. Lisawadi, a . Volodin，一致可积条件下阵列的弱数定律，J.数学。社会学报，45 (2008)，pp. 289-300]。
{"title":"Limiting behaviour for arrays of row-wise END random variables under conditions of h-integrability","authors":"Yongfeng Wu, Jiangyan Peng, T. Hu","doi":"10.1080/17442508.2014.959951","DOIUrl":"https://doi.org/10.1080/17442508.2014.959951","url":null,"abstract":"The authors study Lr-convergence, complete convergence and complete moment convergence for arrays of row-wise extended negatively dependent random variables under some appropriate conditions of h-integrability. The results in this paper extend and improve the results of Sung et al. [H.S. Sung, S. Lisawadi, A. Volodin, Weak laws of large numbers for arrays under a condition of uniform integrability, J. Korean Math. Soc. 45 (2008), pp. 289–300].","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"3 1","pages":"409 - 423"},"PeriodicalIF":0.9,"publicationDate":"2015-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75972097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-04-30DOI: 10.1080/17442508.2015.1026345
M. Diop, K. Ezzinbi, M. Mbaye
In this work, we establish a new concept of pseudo almost periodic processes in p-th mean sense using the measure theory. We use the μ-ergodic process to define the spaces of μ-pseudo almost periodic process in the p-th mean sense. We establish many interesting results on the functional space of such processes like completeness and composition theorems. The main objective of this paper is to use those results and some stochastic analysis approaches to study the existence, the uniqueness and the global attractiveness for a μ-pseudo almost periodic mild solution to a class of abstract stochastic evolution equations driven by fractional Brownian motion. We provide an example to illustrate our results.
{"title":"Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion","authors":"M. Diop, K. Ezzinbi, M. Mbaye","doi":"10.1080/17442508.2015.1026345","DOIUrl":"https://doi.org/10.1080/17442508.2015.1026345","url":null,"abstract":"In this work, we establish a new concept of pseudo almost periodic processes in p-th mean sense using the measure theory. We use the μ-ergodic process to define the spaces of μ-pseudo almost periodic process in the p-th mean sense. We establish many interesting results on the functional space of such processes like completeness and composition theorems. The main objective of this paper is to use those results and some stochastic analysis approaches to study the existence, the uniqueness and the global attractiveness for a μ-pseudo almost periodic mild solution to a class of abstract stochastic evolution equations driven by fractional Brownian motion. We provide an example to illustrate our results.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"30 1 1","pages":"1061 - 1093"},"PeriodicalIF":0.9,"publicationDate":"2015-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82641572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-04-14DOI: 10.1080/17442508.2014.1000327
K. Nyström, Thomas Önskog
We consider the Skorohod problem for càdlàg functions, and the subsequent construction of solutions to normally reflected stochastic differential equations driven by Lévy processes, in the setting of non-smooth and time-dependent domains.
{"title":"Remarks on the Skorohod problem and reflected Lévy driven SDEs in time-dependent domains","authors":"K. Nyström, Thomas Önskog","doi":"10.1080/17442508.2014.1000327","DOIUrl":"https://doi.org/10.1080/17442508.2014.1000327","url":null,"abstract":"We consider the Skorohod problem for càdlàg functions, and the subsequent construction of solutions to normally reflected stochastic differential equations driven by Lévy processes, in the setting of non-smooth and time-dependent domains.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"124 1","pages":"747 - 765"},"PeriodicalIF":0.9,"publicationDate":"2015-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75807210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-04-10DOI: 10.1080/17442508.2014.1002785
C. Withers, S. Nadarajah
Cornish and Fisher gave expansions for the distribution and quantiles of asymptotically normal random variables whose cumulants behaved like those of a sample mean. This was extended by Hill and Davis to the case, where the asymptotic distribution need not be normal. Their results are cumbersome as they involve partition theory. We overcome this using Bell polynomials. The three basic expansions (for the distribution and its derivatives, for the inverse of the quantile, and for the quantile) involve three sets of polynomials. We give new ways of obtaining these from each other. The Edgeworth expansions for the distribution and density rest on the Charlier expansion. We give an elegant form of these as linear combinations of generalized Hermite polynomials, using Bell polynomials.
{"title":"Edgeworth–Cornish–Fisher–Hill–Davis expansions for normal and non-normal limits via Bell polynomials","authors":"C. Withers, S. Nadarajah","doi":"10.1080/17442508.2014.1002785","DOIUrl":"https://doi.org/10.1080/17442508.2014.1002785","url":null,"abstract":"Cornish and Fisher gave expansions for the distribution and quantiles of asymptotically normal random variables whose cumulants behaved like those of a sample mean. This was extended by Hill and Davis to the case, where the asymptotic distribution need not be normal. Their results are cumbersome as they involve partition theory. We overcome this using Bell polynomials. The three basic expansions (for the distribution and its derivatives, for the inverse of the quantile, and for the quantile) involve three sets of polynomials. We give new ways of obtaining these from each other. The Edgeworth expansions for the distribution and density rest on the Charlier expansion. We give an elegant form of these as linear combinations of generalized Hermite polynomials, using Bell polynomials.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"177 1","pages":"794 - 805"},"PeriodicalIF":0.9,"publicationDate":"2015-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72685594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-04-10DOI: 10.1080/17442508.2014.995660
Robert W. Chen, I. Grigorescu, Min Kang
An (m,p) urn contains m balls of value − 1 and p balls of value +1. A player starts with fortune k and in each game draws a ball without replacement with the fortune increasing by one unit if the ball is positive and decreasing by one unit if the ball is negative, having to stop when k = 0 (risk aversion). Let V(m,p,k) be the expected value of the game. We are studying the question of the minimum k such that the net gain function of the game V(m,p,k) − k is positive, in both the discrete and the continuous (Brownian bridge) settings. Monotonicity in various parameters m, p, k is established for both the value and the net gain functions of the game. For the cut-off value k, since the case m − p < 0 is trivial, for p → ∞, either , when the gain function cannot be positive, or , when it is sufficient to have , where α is a constant. We also determine an approximate optimal strategy with exponentially small probability of failure in terms of p. The problem goes back to Shepp [8], who determined the constant α in the unrestricted case when the net gain does not depend on k. A new proof of his result is given in the continuous setting.
{"title":"Optimal stopping for Shepp's urn with risk aversion","authors":"Robert W. Chen, I. Grigorescu, Min Kang","doi":"10.1080/17442508.2014.995660","DOIUrl":"https://doi.org/10.1080/17442508.2014.995660","url":null,"abstract":"An (m,p) urn contains m balls of value − 1 and p balls of value +1. A player starts with fortune k and in each game draws a ball without replacement with the fortune increasing by one unit if the ball is positive and decreasing by one unit if the ball is negative, having to stop when k = 0 (risk aversion). Let V(m,p,k) be the expected value of the game. We are studying the question of the minimum k such that the net gain function of the game V(m,p,k) − k is positive, in both the discrete and the continuous (Brownian bridge) settings. Monotonicity in various parameters m, p, k is established for both the value and the net gain functions of the game. For the cut-off value k, since the case m − p < 0 is trivial, for p → ∞, either , when the gain function cannot be positive, or , when it is sufficient to have , where α is a constant. We also determine an approximate optimal strategy with exponentially small probability of failure in terms of p. The problem goes back to Shepp [8], who determined the constant α in the unrestricted case when the net gain does not depend on k. A new proof of his result is given in the continuous setting.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"120 1","pages":"702 - 722"},"PeriodicalIF":0.9,"publicationDate":"2015-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80158304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-04-10DOI: 10.1080/17442508.2014.989525
Andrea Cosso, D. Marazzina, C. Sgarra
In the present paper we analyse the American option valuation problem in a stochastic volatility model when transaction costs are taken into account. We shall show that it can be formulated as a singular stochastic optimal control problem, proving the existence and uniqueness of the viscosity solution for the associated Hamilton–Jacobi–Bellman partial differential equation. Moreover, after performing a dimensionality reduction through a suitable choice of the utility function, we shall provide a numerical example illustrating how American options prices can be computed in the present modelling framework.
{"title":"American option valuation in a stochastic volatility model with transaction costs","authors":"Andrea Cosso, D. Marazzina, C. Sgarra","doi":"10.1080/17442508.2014.989525","DOIUrl":"https://doi.org/10.1080/17442508.2014.989525","url":null,"abstract":"In the present paper we analyse the American option valuation problem in a stochastic volatility model when transaction costs are taken into account. We shall show that it can be formulated as a singular stochastic optimal control problem, proving the existence and uniqueness of the viscosity solution for the associated Hamilton–Jacobi–Bellman partial differential equation. Moreover, after performing a dimensionality reduction through a suitable choice of the utility function, we shall provide a numerical example illustrating how American options prices can be computed in the present modelling framework.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"23 1","pages":"518 - 536"},"PeriodicalIF":0.9,"publicationDate":"2015-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87274115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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