Pub Date : 2014-11-21DOI: 10.1080/17442508.2014.991324
Boualem Djehiche, Ali Hamdi
We formulate and solve a finite horizon full balance sheet of a two-mode optimal switching problem related to trade-off strategies between expected profit and cost yields. Given the current mode, this model allows for either a switch to the other mode or termination of the project, and this happens for both sides of the balance sheet. A novelty in this model is that the related obstacles are nonlinear in the underlying yields, whereas, they are linear in the standard optimal switching problem. The optimal switching problem is formulated in terms of a system of Snell envelopes for the profit and cost yields which act as obstacles to each other. We prove the existence of a continuous minimal solution of this system using an approximation scheme and fully characterize the optimal switching strategy.
{"title":"A full balance sheet two-mode optimal switching problem","authors":"Boualem Djehiche, Ali Hamdi","doi":"10.1080/17442508.2014.991324","DOIUrl":"https://doi.org/10.1080/17442508.2014.991324","url":null,"abstract":"We formulate and solve a finite horizon full balance sheet of a two-mode optimal switching problem related to trade-off strategies between expected profit and cost yields. Given the current mode, this model allows for either a switch to the other mode or termination of the project, and this happens for both sides of the balance sheet. A novelty in this model is that the related obstacles are nonlinear in the underlying yields, whereas, they are linear in the standard optimal switching problem. The optimal switching problem is formulated in terms of a system of Snell envelopes for the profit and cost yields which act as obstacles to each other. We prove the existence of a continuous minimal solution of this system using an approximation scheme and fully characterize the optimal switching strategy.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"1 1","pages":"604 - 622"},"PeriodicalIF":0.9,"publicationDate":"2014-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91112916","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 : 2014-10-24DOI: 10.1080/17442508.2014.895357
A. Berkaoui
First we consider a set of probabilities and denote by , the associated dynamic sublinear expectation, defined by for and a fixed filtration . We prove that for a positive -supermartingale X, there exits an increasing adapted process C such that is a local -martingale. Second we apply such a result to incomplete market under model misspecification, generalizing the results of Kramkov [D.O. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Prob. Theor. Relat. Field. 15 (1996), pp. 459–479] and Riedel [F. Riedel, On optimal stopping under Ambiguity, Econometrica. 77 (2009), pp. 857–908].
{"title":"On a generalized optional decomposition theorem","authors":"A. Berkaoui","doi":"10.1080/17442508.2014.895357","DOIUrl":"https://doi.org/10.1080/17442508.2014.895357","url":null,"abstract":"First we consider a set of probabilities and denote by , the associated dynamic sublinear expectation, defined by for and a fixed filtration . We prove that for a positive -supermartingale X, there exits an increasing adapted process C such that is a local -martingale. Second we apply such a result to incomplete market under model misspecification, generalizing the results of Kramkov [D.O. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Prob. Theor. Relat. Field. 15 (1996), pp. 459–479] and Riedel [F. Riedel, On optimal stopping under Ambiguity, Econometrica. 77 (2009), pp. 857–908].","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"193 1","pages":"906 - 921"},"PeriodicalIF":0.9,"publicationDate":"2014-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73679487","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 : 2014-10-24DOI: 10.1080/17442508.2013.865132
B. Buonaguidi, P. Muliere
We study the connection between the martingale and free-boundary approaches in sequential detection problems for the drift of a Brownian motion, under the assumption of exponential penalty for the delay. By means of the solution of a suitable free-boundary problem, we show that the reward process can be decomposed into the product between a gain function of the boundary point and a positive martingale inside the continuation region.
{"title":"On the martingale and free-boundary approaches in sequential detection problems with exponential penalty for delay","authors":"B. Buonaguidi, P. Muliere","doi":"10.1080/17442508.2013.865132","DOIUrl":"https://doi.org/10.1080/17442508.2013.865132","url":null,"abstract":"We study the connection between the martingale and free-boundary approaches in sequential detection problems for the drift of a Brownian motion, under the assumption of exponential penalty for the delay. By means of the solution of a suitable free-boundary problem, we show that the reward process can be decomposed into the product between a gain function of the boundary point and a positive martingale inside the continuation region.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"12 1","pages":"865 - 869"},"PeriodicalIF":0.9,"publicationDate":"2014-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88181144","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 : 2014-10-24DOI: 10.1080/17442508.2014.899600
H. Engelbert, G. Peskir
We study (i) the stochastic differential equation (SDE) systemfor Brownian motion X in sticky at 0, and (ii) the SDE systemfor reflecting Brownian motion X in sticky at 0, where X starts at x in the state space, is a given constant, is a local time of X at 0 and B is a standard Brownian motion. We prove that both systems (i) have a jointly unique weak solution and (ii) have no strong solution. The latter fact verifies Skorokhod's conjecture on sticky Brownian motion and provides alternative arguments to those given in the literature.
{"title":"Stochastic differential equations for sticky Brownian motion","authors":"H. Engelbert, G. Peskir","doi":"10.1080/17442508.2014.899600","DOIUrl":"https://doi.org/10.1080/17442508.2014.899600","url":null,"abstract":"We study (i) the stochastic differential equation (SDE) systemfor Brownian motion X in sticky at 0, and (ii) the SDE systemfor reflecting Brownian motion X in sticky at 0, where X starts at x in the state space, is a given constant, is a local time of X at 0 and B is a standard Brownian motion. We prove that both systems (i) have a jointly unique weak solution and (ii) have no strong solution. The latter fact verifies Skorokhod's conjecture on sticky Brownian motion and provides alternative arguments to those given in the literature.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"61 1","pages":"1021 - 993"},"PeriodicalIF":0.9,"publicationDate":"2014-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86163371","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 : 2014-10-24DOI: 10.1080/17442508.2014.883078
Deli Li, A. Spǎtaru
Let be a sequence of independent and identically distributed B-valued random variables, and set . Let , and q>0, and putWe strengthen the convergence rate for the Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers in Banach space, by showing that , if and only if and
{"title":"Refinement of convergence rate for the strong law of large numbers in Banach space","authors":"Deli Li, A. Spǎtaru","doi":"10.1080/17442508.2014.883078","DOIUrl":"https://doi.org/10.1080/17442508.2014.883078","url":null,"abstract":"Let be a sequence of independent and identically distributed B-valued random variables, and set . Let , and q>0, and putWe strengthen the convergence rate for the Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers in Banach space, by showing that , if and only if and","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"14 1","pages":"882 - 888"},"PeriodicalIF":0.9,"publicationDate":"2014-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87361717","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 : 2014-10-24DOI: 10.1080/17442508.2014.895356
Wanmo Kang, Kiseop Lee
We study a hedging problem in a market where traders have various levels of information. The exclusive information available only to informed traders is modelled by a diffusion process rather than discrete arrivals of new information. The asset price follows a jump–diffusion process and an information process affects jump sizes of the asset price. We find the local risk minimization hedging strategy of informed traders. Numerical examples as well as their comparison with the Black–Scholes strategy are provided via Monte Carlo.
{"title":"Information on jump sizes and hedging","authors":"Wanmo Kang, Kiseop Lee","doi":"10.1080/17442508.2014.895356","DOIUrl":"https://doi.org/10.1080/17442508.2014.895356","url":null,"abstract":"We study a hedging problem in a market where traders have various levels of information. The exclusive information available only to informed traders is modelled by a diffusion process rather than discrete arrivals of new information. The asset price follows a jump–diffusion process and an information process affects jump sizes of the asset price. We find the local risk minimization hedging strategy of informed traders. Numerical examples as well as their comparison with the Black–Scholes strategy are provided via Monte Carlo.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"5 1","pages":"889 - 905"},"PeriodicalIF":0.9,"publicationDate":"2014-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72660826","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 : 2014-10-24DOI: 10.1080/17442508.2014.895360
D. Nualart, Fangjun Xu
The purpose of this note is to extend a second order limit law for one dimensional Cauchy process obtained in Kasahara (Y. Kasahara, Limit theorems for occupation times of Markov processes, Publ. RIMS, Kyoto Univ. 12 (1977), pp. 801–818), using the method of moments and some kind of chaining argument.
{"title":"A second order limit law for occupation times of the Cauchy process","authors":"D. Nualart, Fangjun Xu","doi":"10.1080/17442508.2014.895360","DOIUrl":"https://doi.org/10.1080/17442508.2014.895360","url":null,"abstract":"The purpose of this note is to extend a second order limit law for one dimensional Cauchy process obtained in Kasahara (Y. Kasahara, Limit theorems for occupation times of Markov processes, Publ. RIMS, Kyoto Univ. 12 (1977), pp. 801–818), using the method of moments and some kind of chaining argument.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"40 1","pages":"967 - 974"},"PeriodicalIF":0.9,"publicationDate":"2014-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87668508","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 : 2014-10-24DOI: 10.1080/17442508.2014.895359
A. Barth, F. Benth
In this paper an infinite-dimensional approach to model energy forward markets is introduced. Similar to the Heath–Jarrow–Morton framework in interest-rate modelling, a first-order hyperbolic stochastic partial differential equation models the dynamics of the forward price curves. These equations are analysed, and in particular regularity and no-arbitrage conditions in the general situation of stochastic partial differential equations driven by an infinite-dimensional martingale process are studied. Both arithmetic and geometric forward price dynamics are studied, as well as accounting for the delivery period of electricity forward contracts. A stable and convergent numerical approximation in the form of a finite element method for hyperbolic stochastic partial differential equations is introduced and applied to some examples with relevance to energy markets.
{"title":"The forward dynamics in energy markets – infinite-dimensional modelling and simulation","authors":"A. Barth, F. Benth","doi":"10.1080/17442508.2014.895359","DOIUrl":"https://doi.org/10.1080/17442508.2014.895359","url":null,"abstract":"In this paper an infinite-dimensional approach to model energy forward markets is introduced. Similar to the Heath–Jarrow–Morton framework in interest-rate modelling, a first-order hyperbolic stochastic partial differential equation models the dynamics of the forward price curves. These equations are analysed, and in particular regularity and no-arbitrage conditions in the general situation of stochastic partial differential equations driven by an infinite-dimensional martingale process are studied. Both arithmetic and geometric forward price dynamics are studied, as well as accounting for the delivery period of electricity forward contracts. A stable and convergent numerical approximation in the form of a finite element method for hyperbolic stochastic partial differential equations is introduced and applied to some examples with relevance to energy markets.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"66 1","pages":"932 - 966"},"PeriodicalIF":0.9,"publicationDate":"2014-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83936015","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 : 2014-10-24DOI: 10.1080/17442508.2014.899601
Bernard Wong
February 26, 2014 I would like to thank Dr Fontana for the critical analysis of our previous paper, in which we had aimed to provide an alternative proof (in a more limited setting) of the Fundamental Theorem of Asset Pricing (see, for example [1,2]). While naturally disappointed with the mistakes made in our paper, I am hopeful that the additional analysis provided by Dr Fontana [3] provides an enhanced understanding of the Fundamental Theorem for the readers of the journal.
{"title":"On ‘A note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing’","authors":"Bernard Wong","doi":"10.1080/17442508.2014.899601","DOIUrl":"https://doi.org/10.1080/17442508.2014.899601","url":null,"abstract":"February 26, 2014 I would like to thank Dr Fontana for the critical analysis of our previous paper, in which we had aimed to provide an alternative proof (in a more limited setting) of the Fundamental Theorem of Asset Pricing (see, for example [1,2]). While naturally disappointed with the mistakes made in our paper, I am hopeful that the additional analysis provided by Dr Fontana [3] provides an enhanced understanding of the Fundamental Theorem for the readers of the journal.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"34 3 1","pages":"1022 - 1022"},"PeriodicalIF":0.9,"publicationDate":"2014-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77464855","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 : 2014-08-05DOI: 10.1080/17442508.2013.879140
Xinghui Wang, Shuhe Hu
In this paper, we establish some weak laws of large numbers for arrays of dependent random variables satisfying the conditions of a kind of uniform integrability. Our results extend and improve the corresponding ones.
{"title":"Weak laws of large numbers for arrays of dependent random variables","authors":"Xinghui Wang, Shuhe Hu","doi":"10.1080/17442508.2013.879140","DOIUrl":"https://doi.org/10.1080/17442508.2013.879140","url":null,"abstract":"In this paper, we establish some weak laws of large numbers for arrays of dependent random variables satisfying the conditions of a kind of uniform integrability. Our results extend and improve the corresponding ones.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"1 1","pages":"759 - 775"},"PeriodicalIF":0.9,"publicationDate":"2014-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88775863","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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