{"title":"On Being an Academic Side Chick: Tales of Two Adjunct Faculty in the Academy That Trained Them","authors":"LaWanda M. Simpkins, D. Tafari","doi":"10.31390/taboo.18.1.05","DOIUrl":"https://doi.org/10.31390/taboo.18.1.05","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45018297","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}
{"title":"Taboo 18:1 - Full Issue","authors":"Taboo Journal","doi":"10.31390/taboo.18.1.12","DOIUrl":"https://doi.org/10.31390/taboo.18.1.12","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48152401","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 study on epidemic models plays an important role in mathematical biology and mathematical epidemiology. There has been much effort devoted to epidemic models using ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). Much study has been carried out and substantial progress has been made. In contrast to the development, this work presents an effort from a different angle, namely, modeling and analysis using stochastic partial differential equations (SPDEs). Specifically, we consider dynamic systems featuring SIS (Susceptible-Infected-Susceptible) epidemic models. Our emphasis is on spatial dependent variations and environmental noise. First, a new epidemic model is proposed. Then existence and uniqueness of solutions of the underlying SPDEs are examined. In addition, stochastic partial differential equation models with Markov switching are examined. Our analysis is based on the use of mild solution. Our hope is that this paper will open up windows for investigation of epidemic models from a new angle.
{"title":"Stochastic Partial Differential Equation SIS Epidemic Models: Modeling and Analysis","authors":"N. Nguyen, G. Yin","doi":"10.31390/cosa.13.3.08","DOIUrl":"https://doi.org/10.31390/cosa.13.3.08","url":null,"abstract":"The study on epidemic models plays an important role in mathematical biology and mathematical epidemiology. There has been much effort devoted to epidemic models using ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). Much study has been carried out and substantial progress has been made. In contrast to the development, this work presents an effort from a different angle, namely, modeling and analysis using stochastic partial differential equations (SPDEs). Specifically, we consider dynamic systems featuring SIS (Susceptible-Infected-Susceptible) epidemic models. Our emphasis is on spatial dependent variations and environmental noise. First, a new epidemic model is proposed. Then existence and uniqueness of solutions of the underlying SPDEs are examined. In addition, stochastic partial differential equation models with Markov switching are examined. Our analysis is based on the use of mild solution. Our hope is that this paper will open up windows for investigation of epidemic models from a new angle.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41977832","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 main goal of this paper is to apply the machinery of variational analysis and generalized differentiation to study infinite horizon stochastic dynamic programming (DP) with discrete time in the Banach space setting without convexity assumptions. Unlike to standard stochastic DP with stationary Markov processes, we investigate here stochastic DP in $L^p$ spaces to deal with nonstationary stochastic processes, which describe a more flexible learning procedure for the decision-maker. Our main concern is to calculate generalized subgradients of the corresponding value function and to derive necessary conditions for optimality in terms of the stochastic Euler inclusion under appropriate Lipschitzian assumptions. The usage of the subdifferential formula for integral functionals on $L^p$ spaces allows us, in particular, to find verifiable conditions to ensure smoothness of the value function without any convexity and/or interiority assumptions.
{"title":"Subdifferentials of Value Functions in Nonconvex Dynamic Programming for Nonstationary Stochastic Processes","authors":"B. Mordukhovich, Nobusumi Sagara","doi":"10.31390/COSA.13.3.05","DOIUrl":"https://doi.org/10.31390/COSA.13.3.05","url":null,"abstract":"The main goal of this paper is to apply the machinery of variational analysis and generalized differentiation to study infinite horizon stochastic dynamic programming (DP) with discrete time in the Banach space setting without convexity assumptions. Unlike to standard stochastic DP with stationary Markov processes, we investigate here stochastic DP in $L^p$ spaces to deal with nonstationary stochastic processes, which describe a more flexible learning procedure for the decision-maker. Our main concern is to calculate generalized subgradients of the corresponding value function and to derive necessary conditions for optimality in terms of the stochastic Euler inclusion under appropriate Lipschitzian assumptions. The usage of the subdifferential formula for integral functionals on $L^p$ spaces allows us, in particular, to find verifiable conditions to ensure smoothness of the value function without any convexity and/or interiority assumptions.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44032946","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}
{"title":"Preface","authors":"H. Kuo, G. Yin","doi":"10.31390/cosa.13.3.01","DOIUrl":"https://doi.org/10.31390/cosa.13.3.01","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47751091","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}
Exponential processes in the Itô theory of stochastic integration can be viewed in three aspects: multiplicative renormalization, martingales, and stochastic differential equations. In this paper we initiate the study of anticipating exponential processes from these aspects viewpoints. The analogue of martingale property for anticipating stochastic integrals is the near-martingale property. We use examples to illustrate essential ideas and techniques in dealing with anticipating exponential processes and stochastic differential equations. The situation is very different from the Itô theory. 1. Exponential Processes Let B(t), 0 ≤ t ≤ T, be a fixed Brownian motion. Suppose {Ft; 0 ≤ t ≤ T} is the filtration given by this Brownian motion, i.e., Ft = σ{B(s); 0 ≤ s ≤ t} for each t ∈ [0, T ]. Take an {Ft}-adapted stochastic process h(t), 0 ≤ t ≤ T, satisfying the Novikov condition, i.e., E exp [1 2 ∫ T 0 h(t) dt ] <∞. (1.1) The exponential process given by h(t) is defined to be the stochastic process Eh(t) = exp [ ∫ t 0 h(s) dB(s)− 1 2 ∫ t 0 h(s) ds ] , 0 ≤ t ≤ T. (1.2) Note that under the Novikov condition in equation (1.1) we have ∫ T 0 h(t) dt <∞ almost surely so that the Itô integral ∫ t 0 h(s) dB(s) is defined for each t ∈ [0, T ] (see Chapter 5 of the book [7].) The exponential process Eh(t) plays a very important role in the Itô theory of stochastic integration and is widely used in the mathematical finance. It can be viewed and understood in the following three aspects. (1) Multiplicative renormalization: The multiplicative renormalization of a random variable X with nonzero expectation is defined to be the random variable X/EX. Suppose h(t) is a deterministic function in L[0, T ]. Received 2019-10-13; Accepted 2019-10-14; Communicated by guest editor George Yin. 2010 Mathematics Subject Classification. Primary 60H05; Secondary 60H20.
Itô随机积分理论中的指数过程可以从三个方面来看待:乘法重整化、鞅和随机微分方程。本文从这几个方面的观点出发，对预测指数过程进行了研究。预测随机积分的鞅性质的类似物是近似鞅性质。我们用例子来说明处理预测指数过程和随机微分方程的基本思想和技术。情况与Itô理论大不相同。1. 设B(t)为固定布朗运动，0≤t≤t。假设{英尺;0≤t≤t}为该布朗运动给出的滤波，即Ft = σ{B(s);对于每个t∈[0,t]， 0≤s≤t}。取一个{Ft}自适应随机过程h(t)， 0≤t≤t，满足Novikov条件，即E exp[1 2∫t0 h(t) dt] <∞。(1.1)的指数过程h (t)被定义为随机过程呃(t) = exp(∫t 0 h (s) dB (s)−1 2∫t 0 h (s) ds), 0≤t≤t(1.2)注意,诺维科夫先生条件下在方程(1.1)我们∫t 0 h (t) dt <∞几乎肯定,伊藤积分∫t 0 h (s) dB (s)被定义为每个t∈[0,t](见第五章书的[7])。指数过程Eh(t)在Itô随机积分理论中占有非常重要的地位，在数学金融中有着广泛的应用。可以从以下三个方面来看待和理解。(1)乘性重整化:定义非零期望随机变量X的乘性重整化为随机变量X/EX。设h(t)是L[0, t]中的确定性函数。收到2019-10-13;接受2019-10-14;2010年数学学科分类。主要60 h05;二次60净水。
{"title":"Anticipating Exponential Processes and Stochastic Differential Equations","authors":"C. Hwang, H. Kuo, Kimiaki Saitô","doi":"10.31390/cosa.13.3.09","DOIUrl":"https://doi.org/10.31390/cosa.13.3.09","url":null,"abstract":"Exponential processes in the Itô theory of stochastic integration can be viewed in three aspects: multiplicative renormalization, martingales, and stochastic differential equations. In this paper we initiate the study of anticipating exponential processes from these aspects viewpoints. The analogue of martingale property for anticipating stochastic integrals is the near-martingale property. We use examples to illustrate essential ideas and techniques in dealing with anticipating exponential processes and stochastic differential equations. The situation is very different from the Itô theory. 1. Exponential Processes Let B(t), 0 ≤ t ≤ T, be a fixed Brownian motion. Suppose {Ft; 0 ≤ t ≤ T} is the filtration given by this Brownian motion, i.e., Ft = σ{B(s); 0 ≤ s ≤ t} for each t ∈ [0, T ]. Take an {Ft}-adapted stochastic process h(t), 0 ≤ t ≤ T, satisfying the Novikov condition, i.e., E exp [1 2 ∫ T 0 h(t) dt ] <∞. (1.1) The exponential process given by h(t) is defined to be the stochastic process Eh(t) = exp [ ∫ t 0 h(s) dB(s)− 1 2 ∫ t 0 h(s) ds ] , 0 ≤ t ≤ T. (1.2) Note that under the Novikov condition in equation (1.1) we have ∫ T 0 h(t) dt <∞ almost surely so that the Itô integral ∫ t 0 h(s) dB(s) is defined for each t ∈ [0, T ] (see Chapter 5 of the book [7].) The exponential process Eh(t) plays a very important role in the Itô theory of stochastic integration and is widely used in the mathematical finance. It can be viewed and understood in the following three aspects. (1) Multiplicative renormalization: The multiplicative renormalization of a random variable X with nonzero expectation is defined to be the random variable X/EX. Suppose h(t) is a deterministic function in L[0, T ]. Received 2019-10-13; Accepted 2019-10-14; Communicated by guest editor George Yin. 2010 Mathematics Subject Classification. Primary 60H05; Secondary 60H20.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47222559","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}
{"title":"Hybrid Models and Switching Control with Constraints","authors":"J. Menaldi, M. Robin","doi":"10.31390/cosa.13.3.03","DOIUrl":"https://doi.org/10.31390/cosa.13.3.03","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41248210","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 the optimal marginal value of discrete-time optimal multiple stopping problems and find that it can be formulated as a single optimal stopping optimization as well. Based on this result propose a marginal-value-based lower bound method to achieve a small bound on the iterative error. We further introduce a non-nested upper bound method. The convergence of both methods is analysed. The implementation details and enhancement techniques are discussed as well. Overall, our methods make a good trade-off between the time-efficiency and the tightness in dual bounds.
{"title":"Non-Nested Monte Carlo Dual Bounds for Multi-Exercisable Options","authors":"Xiang Cheng, Z. Jin","doi":"10.31390/COSA.13.3.02","DOIUrl":"https://doi.org/10.31390/COSA.13.3.02","url":null,"abstract":"We study the optimal marginal value of discrete-time optimal multiple stopping problems and find that it can be formulated as a single optimal stopping optimization as well. Based on this result propose a marginal-value-based lower bound method to achieve a small bound on the iterative error. We further introduce a non-nested upper bound method. The convergence of both methods is analysed. The implementation details and enhancement techniques are discussed as well. Overall, our methods make a good trade-off between the time-efficiency and the tightness in dual bounds.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48632191","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}
. Stochastic calculus has played an important role in the development of the financial markets in the past 40+ years. The Black-Sholes option pricing model published in 1973 revolutionized the derivatives market. The advances in volatility estimate such as GARCH helped to improve the risk measures and risk management process. Other developments might have contributed to the onsite of the great financial crisis (GFC). In celebrating Professor Chow’s successful career, I would like to share some of the applications of stochastic calculus in the financial engineering, and the role it played in the financial market development.
{"title":"Stochastic Process and its Role in The Development of the Financial Market: Celebrating Professor Chow's Long and Successful Career","authors":"Xisuo L. Liu","doi":"10.31390/cosa.13.3.07","DOIUrl":"https://doi.org/10.31390/cosa.13.3.07","url":null,"abstract":". Stochastic calculus has played an important role in the development of the financial markets in the past 40+ years. The Black-Sholes option pricing model published in 1973 revolutionized the derivatives market. The advances in volatility estimate such as GARCH helped to improve the risk measures and risk management process. Other developments might have contributed to the onsite of the great financial crisis (GFC). In celebrating Professor Chow’s successful career, I would like to share some of the applications of stochastic calculus in the financial engineering, and the role it played in the financial market development.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46645691","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 develop Euler-Maruyama scheme for a wideranging class of stochastic differential equations with regime switching under such conditions that allow drift and diffusion coefficients being Hölder continuous. The strong convergence of the numerical method is proved. In addition, the rate of convergence is obtained under similar conditions to the case of usual diffusions. Some numerical examples are provided to illustrate the results.
{"title":"Euler-Maruyama Method for Regime Switching Stochastic Differential Equations with Hölder Coefficients","authors":"D. Nguyen, S. L. Nguyen","doi":"10.31390/cosa.13.3.04","DOIUrl":"https://doi.org/10.31390/cosa.13.3.04","url":null,"abstract":"In this paper, we develop Euler-Maruyama scheme for a wideranging class of stochastic differential equations with regime switching under such conditions that allow drift and diffusion coefficients being Hölder continuous. The strong convergence of the numerical method is proved. In addition, the rate of convergence is obtained under similar conditions to the case of usual diffusions. Some numerical examples are provided to illustrate the results.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48453833","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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