47 results on '"Semimartingale"'
Search Results
2. Tool Degradation Prediction Based on Semimartingale Approximation of Linear Fractional Alpha-Stable Motion and Multi-Feature Fusion.
- Author
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Yuan, Yuchen, Chen, Jianxue, Rong, Jin, Cattani, Piercarlo, Kudreyko, Aleksey, and Villecco, Francesco
- Subjects
- *
LYAPUNOV exponents , *PRINCIPAL components analysis , *BROWNIAN motion , *FORECASTING - Abstract
Tool wear will reduce workpieces' quality and accuracy. In this paper, the vibration signals of the milling process were analyzed, and it was found that historical fluctuations still have an impact on the existing state. First of all, the linear fractional alpha-stable motion (LFSM) was investigated, along with a differential iterative model with it as the noise term is constructed according to the fractional-order Ito formula; the general solution of this model is derived by semimartingale approximation. After that, for the chaotic features of the vibration signal, the time-frequency domain characteristics were extracted using principal component analysis (PCA), and the relationship between the variation of the generalized Hurst exponent and tool wear was established using multifractal detrended fluctuation analysis (MDFA). Then, the maximum prediction length was obtained by the maximum Lyapunov exponent (MLE), which allows for analysis of the vibration signal. Finally, tool condition diagnosis was carried out by the evolving connectionist system (ECoS). The results show that the LFSM iterative model with semimartingale approximation combined with PCA and MDFA are effective for the prediction of vibration trends and tool condition. Further, the monitoring of tool condition using ECoS is also effective. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
3. SDEs with two reflecting barriers driven by semimartingales and processes with bounded [formula omitted]-variation.
- Author
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Falkowski, Adrian and Słomiński, Leszek
- Subjects
- *
WIENER processes , *STOCHASTIC differential equations , *BROWNIAN motion - Abstract
We study the existence, uniqueness and approximation of solutions of general stochastic differential equations (SDEs) with two time-dependent reflecting barriers driven by semimartingales and processes with bounded p -variation, p ∈ [ 1 , 2). We do not assume that the barriers have to be completely separated. Applications to currency option pricing in financial models with fractional Brownian motion and standard Brownian motion are given. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
4. Girsanov Theorem
- Author
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Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.
- Published
- 2018
- Full Text
- View/download PDF
5. Dynamic risk measure for BSVIE with jumps and semimartingale issues.
- Author
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Agram, Nacira
- Subjects
- *
STOCHASTIC integrals , *INTEGRAL equations , *INSURANCE rates , *INSURANCE , *LIFE insurance , *VOLTERRA equations - Abstract
Risk measure is a fundamental concept in finance and in the insurance industry. It is used to adjust life insurance rates. In this article, we will study dynamic risk measures by means of backward stochastic Volterra integral equations (BSVIEs) with jumps. We prove a comparison theorem for such a type of equations. Since the solution of a BSVIEs is not a semimartingale in general, we will discuss some particular semimartingale issues. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
6. On the semimartingale property of Brownian bridges on complete manifolds.
- Author
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Güneysu, Batu
- Subjects
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BROWNIAN motion , *RIEMANNIAN manifolds , *SEMIMARTINGALES (Mathematics) , *GEOMETRY , *WIENER processes - Abstract
It is shown that every adapted Brownian bridge on a geodesically complete connected Riemannian manifold is a semimartingale including its terminal time, without any further assumptions on the geometry. In particular, it follows that every such process can be horizontally lifted to a smooth principal fiber bundle with connection, including its terminal time. The proof is based on a localized Hamilton-type gradient estimate. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
7. Higher-order small time asymptotic expansion of Itô semimartingale characteristic function with application to estimation of leverage from options
- Author
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Viktor Todorov
- Subjects
Statistics and Probability ,Leverage (finance) ,Semimartingale ,Characteristic function (probability theory) ,Applied Mathematics ,Modeling and Simulation ,Jump ,Estimator ,Applied mathematics ,Volatility (finance) ,Asymptotic expansion ,Brownian motion ,Mathematics - Abstract
In this paper, we derive a higher-order asymptotic expansion of characteristic functions of an Ito semimartingale over asymptotically shrinking time intervals. The leading term in the expansion is determined by the value of the diffusive coefficient at the beginning of the interval. The higher-order terms are determined by the jump compensator as well as the coefficients appearing in the diffusion dynamics. The result is applied to develop a nearly rate-efficient estimator of the leverage coefficient of an asset price, i.e., the coefficient in its volatility dynamics that appears in front of the Brownian motion that drives also the asset price.
- Published
- 2021
8. Stationary distributions for two-dimensional sticky Brownian motions: Exact tail asymptotics and extreme value distributions
- Author
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Hongshuai Dai and Yiqiang Q. Zhao
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Semimartingale ,Kernel method ,Mathematics::Probability ,Joint probability distribution ,General Mathematics ,Mathematical finance ,Statistical physics ,Marginal distribution ,Extreme value theory ,Brownian motion ,Mathematics ,Copula (probability theory) - Abstract
Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions, which find applications in many areas including queueing theory and mathematical finance. In this paper, we focus on stationary distributions for sticky Brownian motions. Main results obtained here include tail asymptotic properties in the marginal distributions and joint distributions. The kernel method, copula concept and extreme value theory are the main tools used in our analysis.
- Published
- 2021
9. Approximation of the Rosenblatt process by semimartingales.
- Author
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Yan, Litan, Li, Yumiao, and Wu, Di
- Subjects
- *
APPROXIMATION theory , *SEMIMARTINGALES (Mathematics) , *INTEGRABLE functions , *BROWNIAN motion , *DETERMINISTIC algorithms - Abstract
In this paper, we consider the optimal approximation of a Rosenblatt process based on semimartingales of the formwhere (y1,y2)↦a(y1,y2) is a square integrable process andBis a standard Brownian motion. We show that there exists a unique semimartingale closest to Rosenblatt process ifais deterministic. [ABSTRACT FROM PUBLISHER]
- Published
- 2017
- Full Text
- View/download PDF
10. A new family of positive recurrent semimartingale reflecting Brownian motions in an orthant
- Author
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Abdelhak Yaacoubi
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Statistics and Probability ,010104 statistics & probability ,Pure mathematics ,Semimartingale ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,0101 mathematics ,01 natural sciences ,Analysis ,Brownian motion ,010305 fluids & plasmas ,Orthant ,Mathematics - Abstract
Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes, which arise as approximations for open d-station queueing networks of various kinds. The data for such a process are a drift vector θ, a nonsingular d × d {d\times d} covariance matrix Δ, and a d × d {d\times d} reflection matrix R. The state space is the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motions, and that reflect against the boundary in a specified manner. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions are formulated for some classes of reflection matrices and in two- and three-dimensional cases, but not more. In this work, we identify a new family of reflection matrices R for which the process is positive recurrent if and only if the drift θ ∈ Γ ̊ {\theta\in\mathring{\Gamma}} , where Γ ̊ {\mathring{\Gamma}} is the interior of the convex wedge generated by the opposite column vectors of R.
- Published
- 2020
11. LOCAL MARTINGALES WITH TWO REFLECTING BARRIERS.
- Author
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PIHLSGÅRD, MATS
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MARTINGALES (Mathematics) ,QUADRATIC forms ,PROBLEM solving ,WIENER processes ,SEMIMARTINGALES (Mathematics) - Abstract
We give an account of the characteristics that result from reflecting a drifting local martingale (i.e. the sum of a local martingale and a multiple of its quadratic variation process) in 0 and b > 0. We present conditions which guarantee the existence of finite moments of what is required to keep the reflected process within its boundaries. Also, we derive an associated law of large numbers and a central limit theorem which apply when the input is continuous. Similar results for integrals of the paths of the reflected process are also presented. These results are in close agreement to what has previously been shown for Brownian motion. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
12. Stochastic process-based degradation modeling and RUL prediction: from Brownian motion to fractional Brownian motion
- Author
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Maoyin Chen, Youxian Sun, Hanwen Zhang, Chunjie Yang, and Jun Shang
- Subjects
symbols.namesake ,Semimartingale ,Fractional Brownian motion ,General Computer Science ,Computer science ,Stochastic process ,symbols ,Markov process ,Statistical physics ,First-hitting-time model ,Degradation process ,Brownian motion ,Degradation (telecommunications) - Abstract
Brownian motion (BM) has been widely used for degradation modeling and remaining useful life (RUL) prediction, but it is essentially Markovian. This implies that the future state in a BM-based degradation process relies only on its current state, independent of the past states. However, some practical industrial devices such as Li-ion batteries, ball bearings, turbofans, and blast furnace walls show degradations with long-range dependence (LRD), where the future degradation states depend on both the current and past degradation states. This type of degradation naturally brings two interesting problems, that is, how to model the degradations and how to predict their RULs. Recently, in contrast to the work that uses only BM, fractional Brownian motion (FBM) is introduced to model practical degradations. The most important feature of the FBM-based degradation models is the ability to characterize the non-Markovian degradations with LRD. Although FBM is an extension of BM, it is neither a Markovian process nor a semimartingale. Therefore, how to obtain the first passage time of an FBM-based degradation process has become a challenging task. In this paper, a review of the transition of RUL prediction from BM to FBM is provided. The peculiarities of FBM when addressing the LRD inherent in some practical degradations are discussed. We first review the BM-based degradation models of the past few decades and then give details regarding the evolution of FBM-based research. Interestingly, the existing BM-based models scarcely consider the effect of LRD on the prediction of RULs. Two practical cases illustrate that the newly developed FBM-based models are more generalized and suitable for predicting RULs than the BM-based models, especially for degradations with LRD. Along with the direction of FBM-based RUL prediction, we also introduce some important and interesting problems that require further study.
- Published
- 2021
13. Jump-robust volatility estimation using dynamic dual-domain integration method
- Author
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Yan-Yong Zhao and Xu-Guo Ye
- Subjects
Statistics and Probability ,Statistics::Theory ,021103 operations research ,Dual domain ,0211 other engineering and technologies ,Nonparametric statistics ,Estimator ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,Semimartingale ,Mathematics::Probability ,Jump ,Applied mathematics ,Time domain ,0101 mathematics ,Volatility (finance) ,Brownian motion ,Mathematics - Abstract
In this paper, we propose a nonparametric procedure to estimate the volatility when the underlying price process is governed by Brownian semimartingale with jumps. The estimator combines th...
- Published
- 2019
14. Model-adaptive optimal discretization of stochastic integrals
- Author
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Uladzislau Stazhynski and Emmanuel Gobet
- Subjects
Statistics and Probability ,Semimartingale ,Mathematics::Probability ,Discretization ,Modeling and Simulation ,Applied mathematics ,Discretization error ,Upper and lower bounds ,Brownian motion ,Mathematics - Abstract
We study the optimal discretization error of stochastic integrals driven by a multidimensional continuous Brownian semimartingale. In the previous works a pathwise lower bound for the renor...
- Published
- 2018
15. Quadratic covariations for the solution to a stochastic heat equation with space-time white noise
- Author
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Xichao Sun, Xianye Yu, and Litan Yan
- Subjects
Local time ,Algebra and Number Theory ,Partial differential equation ,Functional analysis ,lcsh:Mathematics ,Applied Mathematics ,010102 general mathematics ,lcsh:QA1-939 ,Fractional Brownian motion ,01 natural sciences ,Quadratic variation ,Combinatorics ,010104 statistics & probability ,Semimartingale ,Quadratic covariation ,Itô formula ,Ordinary differential equation ,Heat equation ,0101 mathematics ,Analysis ,Brownian motion ,Stochastic heat equation ,Mathematics ,Real number - Abstract
Let $u(t,x)$u(t,x) be the solution to a stochastic heat equation $$ \frac{\partial }{\partial t}u=\frac{1}{2} \frac{\partial ^{2}}{\partial x^{2}}u+ \frac{\partial ^{2}}{\partial t\,\partial x}X(t,x),\quad t\geq 0, x\in { \mathbb{R}} $$∂∂tu=12∂2∂x2u+∂2∂t∂xX(t,x),t≥0,x∈R with initial condition $u(0,x)\equiv 0$u(0,x)≡0, where Ẋ is a space-time white noise. This paper is an attempt to study stochastic analysis questions of the solution $u(t,x)$u(t,x). In fact, it is well known that the solution is a Gaussian process such that the process $t\mapsto u(t,x)$t↦u(t,x) is a bi-fractional Brownian motion with Hurst indices $H=K=\frac{1}{2}$H=K=12 for every real number x. However, the many properties of the process $x\mapsto u(\cdot ,x)$x↦u(⋅,x) are unknown. In this paper we consider the generalized quadratic covariations of the two processes $x\mapsto u(\cdot ,x),t\mapsto u(t,\cdot )$x↦u(⋅,x),t↦u(t,⋅). We show that $x\mapsto u(\cdot ,x)$x↦u(⋅,x) admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with “Itô’s integral”, but it is not a semimartingale. Moreover, some generalized Itô formulas and Bouleau–Yor identities are introduced.
- Published
- 2020
16. Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization
- Author
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Julio Backhoff-Veraguas, Anastasiia Zalashko, and Beatrice Acciaio
- Subjects
Statistics and Probability ,01 natural sciences ,91G80, 60G44, 90C08 ,Value of information ,010104 statistics & probability ,Mathematics::Probability ,FOS: Mathematics ,Applied mathematics ,Optimal stopping ,HA Statistics ,QA Mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Brownian motion ,Mathematics ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Utility maximization ,Semimartingale ,Optimization and Control (math.OC) ,Modeling and Simulation ,Stochastic optimization ,Minification ,Martingale (probability theory) ,Mathematics - Probability - Abstract
The martingale part in the semimartingale decomposition of a Brownian motion with respect to an enlargement of its filtration, is an anticipative mapping of the given Brownian motion. In analogy to optimal transport theory, we define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforementioned non-adapted mappings. We provide a necessary and sufficient condition for a Brownian motion to remain a semimartingale in an enlarged filtration, in terms of certain minimization problems over sets of causal transport plans. The latter are also used in order to give robust transport-based estimates for the value of having additional information, as well as model sensitivity with respect to the reference measure, for the classical stochastic optimization problems of utility maximization and optimal stopping. Our results have natural extensions to the case of general multidimensional continuous semimartingales., 33 pages
- Published
- 2020
17. Estimating Jump Activity Using Multipower Variation
- Author
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Aleksey Kolokolov
- Subjects
Statistics and Probability ,Economics and Econometrics ,Multipower variation ,bitcoin ,Inference ,Jumps ,01 natural sciences ,010104 statistics & probability ,Jump activity ,Simple (abstract algebra) ,Component (UML) ,0502 economics and business ,Applied mathematics ,Statistical physics ,0101 mathematics ,Brownian motion ,050205 econometrics ,Mathematics ,High-frequency data ,05 social sciences ,Nonparametric statistics ,Estimator ,Variation (linguistics) ,Semimartingale ,Jump ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Jump process ,Social Sciences (miscellaneous) - Abstract
Realized multipower variation, originally introduced to eliminate jumps, can be extremely useful for inference in pure-jump models. The paper shows how to build a simple and precise estimator of the jump activity index of a semimartingale observed at a high frequency by comparing different multipowers. The novel methodology allows to infer whether a discretely observed process contains a continuous martingale component. The empirical part of the paper undertakes a nonparametric analysis of jump activity of bitcoin. The implementation of the new jump activity estimator indicates that bitcoin is a pure jump process with high jump activity, which is critically different from conventional currencies that include a Brownian motion component.
- Published
- 2020
- Full Text
- View/download PDF
18. Comment on: Limit of Random Measures Associated with the Increments of a Brownian Semimartingale*
- Author
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Xinghua Zheng and Yingying Li
- Subjects
Economics and Econometrics ,05 social sciences ,01 natural sciences ,010104 statistics & probability ,Semimartingale ,0502 economics and business ,Statistics ,Limit (mathematics) ,Statistical physics ,0101 mathematics ,Finance ,Brownian motion ,050205 econometrics ,Mathematics - Published
- 2017
19. Remaining Useful Life Prediction for Degradation Processes With Long-Range Dependence
- Author
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Maoyin Chen, Donghua Zhou, Hanwen Zhang, and Xiaopeng Xi
- Subjects
0209 industrial biotechnology ,021103 operations research ,Fractional Brownian motion ,Weak convergence ,Stochastic process ,0211 other engineering and technologies ,Markov process ,Wavelet transform ,02 engineering and technology ,symbols.namesake ,020901 industrial engineering & automation ,Semimartingale ,symbols ,Applied mathematics ,Electrical and Electronic Engineering ,First-hitting-time model ,Safety, Risk, Reliability and Quality ,Brownian motion ,Mathematics - Abstract
A prerequisite for the existing remaining useful life prediction methods based on stochastic processes is the assumption of independent increments. However, this is in sharp contrast to some practical systems including batteries and blast furnace walls, in which the degradation processes have the property of long-range dependence. Based on the fractional Brownian motion, we adopt a degradation process with long-range dependence to predict the remaining useful life of the above systems. Because the degradation process with long-range dependence is neither a Markovian process nor a semimartingale, the exact analytical first passage time is difficult to derive directly. To address this problem, a weak convergence theorem is first adopted to approximately transform a fractional Brownian motion-based degradation process into a Brownian motion-based one with a time-varying coefficient. Then, with a space-time transformation, the first passage time of the degradation process with long-range dependence can be obtained in a closed form. Unknown parameters in the degradation model can be identified using discrete dyadic wavelet transform and maximum likelihood estimation. Numerical simulations and a practical example of a blast furnace wall are given to verify the effectiveness of the proposed method.
- Published
- 2017
20. Comment on: Limit of Random Measures Associated with the Increments of a Brownian Semimartingale*
- Author
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Dacheng Xiu and Jia Li
- Subjects
010104 statistics & probability ,Economics and Econometrics ,Semimartingale ,0502 economics and business ,05 social sciences ,Mathematical analysis ,Limit (mathematics) ,0101 mathematics ,01 natural sciences ,Finance ,Brownian motion ,050205 econometrics ,Mathematics - Published
- 2017
21. Itô's rule and Lévy's theorem in vector lattices
- Author
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Coenraad C.A. Labuschagne and Jacobus J. Grobler
- Subjects
Discrete mathematics ,Pure mathematics ,021103 operations research ,Applied Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Hölder condition ,02 engineering and technology ,01 natural sciences ,Functional calculus ,Semimartingale ,Local martingale ,Dedekind cut ,0101 mathematics ,Martingale (probability theory) ,Vector-valued function ,Analysis ,Brownian motion ,Mathematics - Abstract
The change of variable formula, or Ito's rule, is studied in a Dedekind complete vector lattice E with weak order unit E. Using the functional calculus we prove that for a Holder continuous semimartingale X t = X a + M t + B t , t ∈ J , and a twice continuously differentiable function f, the formula (0.1) f ( X t ) = f ( X a ) + ∫ 0 t f ′ ( X s ) d M s + ∫ 0 t f ′ ( X s ) d B s + 1 2 ∫ 0 t f ″ ( X s ) d 〈 M 〉 s , 0 ≤ s ≤ t ∈ J holds. The first integral in the formula is an Ito integral with reference to the local martingale M and the second and third integrals are Dobrakov-type integrals of a vector valued function with reference to a vector valued measure. Using the formula, we prove Levy's characterization of Brownian motion as being a continuous martingale with compensator tE. The proof of this result yields a concrete description of abstract Brownian motion defined in vector lattices.
- Published
- 2017
22. Asymptotic behavior analysis of Markovian switching neutral-type stochastic time-delay systems
- Author
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Zhao-Yan Li, Feiqi Deng, and Jiamin Liu
- Subjects
Lyapunov stability ,0209 industrial biotechnology ,Applied Mathematics ,020206 networking & telecommunications ,02 engineering and technology ,Type (model theory) ,Stability (probability) ,Noise (electronics) ,Moment (mathematics) ,Computational Mathematics ,020901 industrial engineering & automation ,Semimartingale ,Convergence (routing) ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,Brownian motion ,Mathematics - Abstract
A new integral inequality method is put forward to analyze the general decay stability for Markovian switching neutral stochastic functional differential systems. At first, in order to get around the dynamic analyses difficulty induced by the coinstantaneous presence of neutral term, Markovian switching and Brownian motion noise, an new integral inequality as a powerful tool is gained. Then, based on the integral inequality, general decay stability in the sense of p th( p > 0 ) moment and the almost sure can be taken out by utilizing the nonnegative semimartingale convergence theorem and Lyapunov stability theory. The obtained results can be especially applied to two special types of neutral stochastic differential systems that have been studied in the literature. Finally, an example has been performed to verify the obtained analytical results.
- Published
- 2021
23. Limit of Random Measures Associated with the Increments of a Brownian Semimartingale
- Author
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Jean Jacod
- Subjects
Path (topology) ,Economics and Econometrics ,Sequence ,Lebesgue measure ,05 social sciences ,Function (mathematics) ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Random measure ,Semimartingale ,0502 economics and business ,Limit (mathematics) ,0101 mathematics ,Finance ,Brownian motion ,050205 econometrics ,Mathematics - Abstract
We consider a Brownian semimartingale X (the sum of a stochastic integral w.r.t. a Brownian motion and an integral w.r.t. Lebesgue measure), and for each n an increasing sequence T(n, i) of stopping times and a sequence of positive ℱT(n,i)-measurable variables Δ(n,i) such that S(n,i):=T(n,i)+Δ(n,i)≤T(n,i+1). We are interested in the limiting behavior of processes of the form Utn(g)=δn∑i:S(n,i)≤t[g(T(n,i),ξin)−αin(g)], where δn is a normalizing sequence tending to 0 and ξin=Δ(n,i)−1/2(XS(n,i)−XT(n,i)) and αin(g) are suitable centering terms and g is some predictable function of (ω,t,x). Under rather weak assumptions on the sequences T(n, i) as n goes to infinity, we prove that these processes converge (stably) in law to the stochastic integral of g w.r.t. a random measure B which is, conditionally on the path of X, a Gaussian random measure. We give some applications to rates of convergence in discrete approximations for the p-variation processes and local times.
- Published
- 2017
24. Support characterization for regular path-dependent stochastic Volterra integral equations
- Author
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Alexander Kalinin
- Subjects
Statistics and Probability ,Probability (math.PR) ,Mathematical analysis ,60H20, 28C20, 60G17, 45D05, 45J05 ,Hölder condition ,Absolute continuity ,Volterra integral equation ,symbols.namesake ,Semimartingale ,Mathematics::Probability ,Flow (mathematics) ,Integrator ,FOS: Mathematics ,symbols ,Almost surely ,Statistics, Probability and Uncertainty ,Brownian motion ,Mathematics - Probability ,Mathematics - Abstract
We consider a stochastic Volterra integral equation with regular path-dependent coefficients and a Brownian motion as integrator in a multidimensional setting. Under an imposed absolute continuity condition, the unique solution is a semimartingale that admits almost surely H\"older continuous paths. Based on functional It\^o calculus, we prove that the support of its law in the H\"older norm can be described by a flow of mild solutions to ordinary integro-differential equations that are constructed by means of the vertical derivative of the diffusion coefficient.
- Published
- 2019
25. ON THE RUIN PROBLEM WITH INVESTMENT WHEN THE RISKY ASSET IS A SEMIMARTINGALE
- Author
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Jérôme Spielmann, Lioudmila Vostrikova, Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), and PANORisk
- Subjects
Statistics and Probability ,[QFIN.RM]Quantitative Finance [q-fin]/Risk Management [q-fin.RM] ,01 natural sciences ,Lévy process ,[QFIN.CP]Quantitative Finance [q-fin]/Computational Finance [q-fin.CP] ,010104 statistics & probability ,Quantitative Finance - Computational Finance ,Mathematics::Probability ,Simple (abstract algebra) ,semimartingales ,Return on investment ,Applied mathematics ,ruin probability ,0101 mathematics ,Power function ,Brownian motion ,Mathematics ,010102 general mathematics ,Investment (macroeconomics) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Asymptotically optimal algorithm ,Semimartingale ,Lévy processes ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Quantitative Finance - Risk Management - Abstract
In this paper, we study the ruin problem with investment in a general framework where the business part X is a L{\'e}vy process and the return on investment R is a semimartingale. We obtain upper bounds on the finite and infinite time ruin probabilities that decrease as a power function when the initial capital increases. When R is a L{\'e}vy process, we retrieve the well-known results. Then, we show that these bounds are asymptotically optimal in the finite time case, under some simple conditions on the characteristics of X. Finally, we obtain a condition for ruin with probability one when X is a Brownian motion with negative drift and express it explicitly using the characteristics of R.
- Published
- 2019
26. BSDEs and Enlargement of Filtration
- Author
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Dongli Wu and Monique Jeanblanc
- Subjects
Mathematical analysis ,Mathematics::Optimization and Control ,Poisson random measure ,law.invention ,Mathematics::Algebraic Geometry ,Semimartingale ,Mathematics::Probability ,Mathematics::K-Theory and Homology ,law ,Projection (set theory) ,Computer Science::Databases ,Brownian motion ,Filtration ,Mathematics - Abstract
In this paper we study the solution of a BSDE in a large filtration, and we show that the projection (on a smaller filtration) of its semimartingale part has coefficients that can be explicitely given in terms of the coefficients in the large filtration.
- Published
- 2019
27. Local martingales with two reflecting barriers
- Author
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Mats Pihlsgård
- Subjects
local martingale ,Statistics and Probability ,Skorokhod problem ,General Mathematics ,Mathematical analysis ,semimartingale ,Quadratic variation ,Doob's martingale inequality ,symbols.namesake ,Semimartingale ,Wiener process ,60H05 ,60G17 ,symbols ,Local martingale ,Martingale difference sequence ,stochastic integration ,60G44 ,Brownian motion ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Martingale representation theorem ,60G51 ,Mathematics ,reflection - Abstract
We give an account of the characteristics that result from reflecting a drifting local martingale (i.e. the sum of a local martingale and a multiple of its quadratic variation process) in 0 and b > 0. We present conditions which guarantee the existence of finite moments of what is required to keep the reflected process within its boundaries. Also, we derive an associated law of large numbers and a central limit theorem which apply when the input is continuous. Similar results for integrals of the paths of the reflected process are also presented. These results are in close agreement to what has previously been shown for Brownian motion.
- Published
- 2015
28. Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs
- Author
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Junjian Yang, Christoph Czichowsky, Walter Schachermayer, Rémi Peyre, London School of Economics and Political Science (LSE), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Fakultät für Mathematik [Wien], Universität Wien, Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich] (ETH Zürich), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Rémi Peyre is partially supported by the Austrian Science Fund (FWF) under grant P25815., Walter Schachermayer is partially supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation and also partially supported by the Austrian Science Fund (FWF) under grants P25815 and P28661 and by the Vienna Science and Technology Fund (WWTF) under grant MA14-008., and Junjian Yang is partially supported by the Austrian Science Fund (FWF) under grant P25815
- Subjects
Statistics and Probability ,Mathematical optimization ,MSC 2010: 91G10, 93E20, 60G48 ,Shadow price ,[QFIN.PM]Quantitative Finance [q-fin]/Portfolio Management [q-fin.PM] ,fractional Brownian motion ,[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] ,01 natural sciences ,91G10, 93E20, 60G48 ,FOS: Economics and business ,010104 statistics & probability ,Mathematics::Probability ,JEL: C - Mathematical and Quantitative Methods/C.C6 - Mathematical Methods • Programming Models • Mathematical and Simulation Modeling/C.C6.C61 - Optimization Techniques • Programming Models • Dynamic Analysis ,0502 economics and business ,Economics ,JEL: G - Financial Economics/G.G1 - General Financial Markets/G.G1.G11 - Portfolio Choice • Investment Decisions ,QA Mathematics ,0101 mathematics ,proportional transaction costs ,Brownian motion ,050208 finance ,Fractional Brownian motion ,Mathematical finance ,05 social sciences ,Financial market ,Mathematical Finance (q-fin.MF) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Semimartingale ,shadow prices ,two-way crossing ,Quantitative Finance - Mathematical Finance ,Portfolio ,Arbitrage ,logarithmic utility ,Statistics, Probability and Uncertainty ,Mathematical economics ,Finance - Abstract
We continue the analysis of our previous paper (Czichowsky/Schachermayer/Yang 2014) pertaining to the existence of a shadow price process for portfolio optimisation under proportional transaction costs. There, we established a positive answer for a continuous price process $S=(S_t)_{0\leq t\leq T}$ satisfying the condition $(NUPBR)$ of "no unbounded profit with bounded risk". This condition requires that $S$ is a semimartingale and therefore is too restrictive for applications to models driven by fractional Brownian motion. In the present paper, we derive the same conclusion under the weaker condition $(TWC)$ of "two way crossing", which does not require $S$ to be a semimartingale. Using a recent result of R.~Peyre, this allows us to show the existence of a shadow price for exponential fractional Brownian motion and $all$ utility functions defined on the positive half-line having reasonable asymptotic elasticity. Prime examples of such utilities are logarithmic or power utility.
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- 2018
29. BSDEs with Default Jump
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Marie Claire Quenez, Roxana Dumitrescu, Miryana Grigorova, and Agnès Sulem
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Comparison theorem ,Stochastic differential equation ,Probability of default ,Semimartingale ,Special case ,Conditional expectation ,Martingale (probability theory) ,Mathematical economics ,Brownian motion ,Mathematics - Abstract
We study (nonlinear) Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale attached to a default jump with intensity process λ = (λ t). The driver of the BSDEs can be of a generalized form involving a singular optional finite variation process. In particular, we provide a comparison theorem and a strict comparison theorem. In the special case of a generalized λ-linear driver, we show an explicit representation of the solution, involving conditional expectation and an adjoint exponential semimartingale; for this representation, we distinguish the case where the singular component of the driver is predictable and the case where it is only optional. We apply our results to the problem of (nonlinear) pricing of European contingent claims in an imperfect market with default. We also study the case of claims generating intermediate cashflows, in particular at the default time, which are modeled by a singular optional process. We give an illustrating example when the seller of the European option is a large investor whose portfolio strategy can influence the probability of default.
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- 2018
30. Stochastic Calculus and Semimartingale Model
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Jia-An Yan
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Dominated convergence theorem ,Class (set theory) ,Semimartingale ,Mathematics::Probability ,Probability theory ,Stochastic process ,Mathematical finance ,Calculus ,Stochastic calculus ,Brownian motion - Abstract
K. Ito invented his famous stochastic calculus on Brownian motion in the 1940s. In the 1960s and 1970s, the “Strasbourg school,” headed by P.A. Meyer, developed a modern theory of martingales, the general theory of stochastic processes, and stochastic calculus on semimartingales. It turned out soon that semimartingales constitute the largest class of right continuous adapted integrators with respect to which stochastic integrals of simple predictable integrands satisfy the theorem of dominated convergence in probability. Stochastic calculus on semimartingales not only became an important tool for modern probability theory and stochastic processes but also has broad applications to many branches of mathematics, physics, engineering, and mathematical finance.
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- 2018
31. Spot volatility estimation using the Laplace transform
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Maria Cristina Recchioni, Imma Valentina Curato, Maria Elvira Mancino, Curato, Imma Valentina, Mancino, MARIA ELVIRA, and Recchioni, Maria Cristina
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Statistics and Probability ,Mathematical optimization ,Economics and Econometrics ,Monte Carlo method ,Asymptotic distribution ,01 natural sciences ,Convolution ,010104 statistics & probability ,0502 economics and business ,Applied mathematics ,0101 mathematics ,Fourier series ,Brownian motion ,050205 econometrics ,Mathematics ,Settore SECS-S/06 - Metodi mat. dell'economia e Scienze Attuariali e Finanziarie ,Nonparametric estimator ,Stochastic volatility ,Laplace transform ,05 social sciences ,Laplace ,Estimator ,nonparametric estimation ,Semimartingale ,Statistics, Probability and Uncertainty ,Volatility (finance) - Abstract
A new non-parametric estimator of the instantaneous volatility is defined relying on the link between the Laplace transform of the price process and that of the volatility process for Brownian semimartingale models. The proposed estimation method is a global one, in the spirit of methods based on Fourier series decomposition, with a plus for improving the precision of the volatility estimates near the boundary of the time interval. Consistency and asymptotic normality of the proposed estimator are proved. A simulation study confirms the theoretical results and Monte Carlo evidence of the favorable performance of the proposed estimator in the presence of microstructure noise effects is presented. A new non-parametric estimator of the instantaneous volatility is defined relying on the link between the Laplace transform of the price process and that of the volatility process for Brownian semimartingale models. The proposed estimation method is a global one, in the spirit of methods based on Fourier series decomposition, with a plus for improving the precision of the volatility estimates near the boundary of the time interval. Consistency and asymptotic normality of the proposed estimator are proved. A simulation study confirms the theoretical results and Monte Carlo evidence of the favorable performance of the proposed estimator in the presence of microstructure noise effects is presented. (c) 2016 EcoSta Econometrics and Statistics. Published by Elsevier B.V. All rights reserved.
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- 2018
32. Dominating Process of a Semimartingale
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Rajeeva L. Karandikar and B. V. Rao
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Semimartingale ,Mathematics::Probability ,Process (computing) ,Applied mathematics ,Computer Science::Databases ,Brownian motion ,Stochastic integral ,Mathematics - Abstract
In Chap. 7, we saw that using random time change, any continuous semimartingale can be transformed into a amenable semimartingale, and then one can have a growth estimate on the stochastic integral similar to the one satisfied by integrals w.r.t. Brownian motion.
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- 2018
33. Dependence modeling between continuous time stochastic processes : an application to electricity markets modeling and risk management
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Deschatre, Thomas, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université Paris sciences et lettres, and Marc Hoffmann
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Electricity markets ,Spikes ,Sélection de fenêtre ,Production éolienne ,Gestion des risques ,Local polynomial estimation ,Marchés de l'électricité ,Processus de Poisson ,[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM] ,Inégalité oracle ,Non parametric estimation ,Dependence ,Statistique haute fréquence ,Finance mathématique ,Oracle inequality ,Estimateur à polynômes locaux ,Pics ,Bandwidth selection ,Poisson process ,Mathematical finance ,Dépendance ,Mouvement Brownien ,Estimation non paramétrique ,Copule ,Semimartingale ,Intensité stochastique ,Risk management ,Copula ,Wind production ,High frequency statistics ,Stochastic intensity ,Brownian motion - Abstract
In this thesis, we study some dependence modeling problems between continuous time stochastic processes. These results are applied to the modeling and risk management of electricity markets. In a first part, we propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. We show that the class of admissible copulae for the Brownian motions contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. Results are applied to the joint modeling of electricity and other energy commodity prices. In a second part, we consider a stochastic process which is a sum of a continuous semimartingale and a mean reverting compound Poisson process and which is discretely observed. An estimation procedure is proposed for the mean reversion parameter of the Poisson process in a high frequency framework with finite time horizon, assuming this parameter is large. Results are applied to the modeling of the spikes in electricity prices time series. In a third part, we consider a doubly stochastic Poisson process with stochastic intensity function of a continuous semimartingale. A local polynomial estimator is considered in order to infer the intensity function and a method is given to select the optimal bandwidth. An oracle inequality is derived. Furthermore, a test is proposed in order to determine if the intensity function belongs to some parametrical family. Using these results, we model the dependence between the intensity of electricity spikes and exogenous factors such as the wind production.; Cette thèse traite de problèmes de dépendance entre processus stochastiques en temps continu. Ces résultats sont appliqués à la modélisation et à la gestion des risques des marchés de l'électricité.Dans une première partie, de nouvelles copules sont établies pour modéliser la dépendance entre deux mouvements Browniens et contrôler la distribution de leur différence. On montre que la classe des copules admissibles pour les Browniens contient des copules asymétriques. Avec ces copules, la fonction de survie de la différence des deux Browniens est plus élevée dans sa partie positive qu'avec une dépendance gaussienne. Les résultats sont appliqués à la modélisation jointe des prix de l'électricité et d'autres commodités énergétiques. Dans une seconde partie, nous considérons un processus stochastique observé de manière discrète et défini par la somme d'une semi-martingale continue et d'un processus de Poisson composé avec retour à la moyenne. Une procédure d'estimation pour le paramètre de retour à la moyenne est proposée lorsque celui-ci est élevé dans un cadre de statistique haute fréquence en horizon fini. Ces résultats sont utilisés pour la modélisation des pics dans les prix de l'électricité.Dans une troisième partie, on considère un processus de Poisson doublement stochastique dont l'intensité stochastique est une fonction d'une semi-martingale continue. Pour estimer cette fonction, un estimateur à polynômes locaux est utilisé et une méthode de sélection de la fenêtre est proposée menant à une inégalité oracle. Un test est proposé pour déterminer si la fonction d'intensité appartient à une certaine famille paramétrique. Grâce à ces résultats, on modélise la dépendance entre l'intensité des pics de prix de l'électricité et de facteurs exogènes tels que la production éolienne.
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- 2017
34. Decomposable stationary distribution of a multidimensional SRBM
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Jim Dai, Jian Wu, and Masakiyo Miyazawa
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Statistics and Probability ,Stationary distribution ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Orthant ,010104 statistics & probability ,Semimartingale ,Modeling and Simulation ,Product (mathematics) ,FOS: Mathematics ,0101 mathematics ,Marginal distribution ,Focus (optics) ,Mathematics - Probability ,Brownian motion ,Mathematics - Abstract
We call a multidimensional distribution to be decomposable with respect to a partition of two sets of coordinates if the original distribution is the product of the marginal distributions associated with these two sets. We focus on the stationary distribution of a multidimensional semimartingale reflecting Brownian motion (SRBM) on a nonnegative orthant. An SRBM is uniquely determined (in distribution) by its data that consists of a covariance matrix, a drift vector, and a reflection matrix. Assume that the stationary distribution of an SRBM exists. We first characterize two marginal distributions under the decomposability assumption. We prove that they are the stationary distributions of some lower dimensional SRBMs. We also identify the data for these lower dimensional SRBMs. Thus, under the decomposability assumption, we can obtain the stationary distribution of the original SRBM by computing those of the lower dimensional ones. However, this characterization of the marginal distributions is not sufficient for the decomposability. So, we next consider necessary and sufficient conditions for the decomposability. We obtain those conditions for several classes of SRBMs. These classes include SRBMs arising from Brownian models of queueing networks that have two sets of stations with feed-forward routing between these two sets. This work is motivated by applications of SRBMs and geometric interpretations of the product form stationary distributions., Comment: To appear in Stochastic Processes and their Applications
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- 2015
35. A central limit theorem for the realised covariation of a bivariate Brownian semistationary process
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Andrea Granelli, Almut E. D. Veraart, and Commission of the European Communities
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multivariate setting ,Statistics and Probability ,Asymptotic analysis ,FUNCTIONALS ,Statistics & Probability ,GAUSSIAN-PROCESSES ,central limit theorem ,Mathematics - Statistics Theory ,moving average process ,Statistics Theory (math.ST) ,Bivariate analysis ,BIPOWER VARIATION ,Malliavin calculus ,01 natural sciences ,POWER VARIATION ,010104 statistics & probability ,symbols.namesake ,Mathematics::Probability ,Law of large numbers ,1403 Econometrics ,FOS: Mathematics ,MULTIPOWER VARIATION ,Applied mathematics ,fourth moment theorem ,0101 mathematics ,Gaussian process ,Brownian motion ,Central limit theorem ,Mathematics ,Science & Technology ,0104 Statistics ,Probability (math.PR) ,010102 general mathematics ,16. Peace & justice ,60F05, 60F15, 60G15 ,bivariate Brownian semistationary process ,Semimartingale ,Physical Sciences ,symbols ,VOLATILITY ,stable convergence ,Mathematics - Probability ,high frequency data - Abstract
This article presents a weak law of large numbers and a central limit theorem for the scaled realised covariation of a bivariate Brownian semistationary process. The novelty of our results lies in the fact that we derive the suitable asymptotic theory both in a multivariate setting and outside the classical semimartingale framework. The proofs rely heavily on recent developments in Malliavin calculus.
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- 2017
36. A limit theorem for moments in space of the increments of Brownian local time
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Simon Campese
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Statistics and Probability ,Pure mathematics ,central limit theorem ,Space (mathematics) ,01 natural sciences ,010104 statistics & probability ,60H05 ,60F05 ,FOS: Mathematics ,60G44 ,Limit (mathematics) ,0101 mathematics ,Brownian motion ,Brownian local time ,Variable (mathematics) ,Mathematics ,Central limit theorem ,Conjecture ,Probability (math.PR) ,asymptotic Ray–Knight theorem ,010102 general mathematics ,Semimartingale ,60F05, 60G44, 60H05 ,Local time ,Statistics, Probability and Uncertainty ,Kailath–Segall identity ,Mathematics - Probability - Abstract
We proof a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. (Ann. Prob. 38, 2010, no. 1) and Rosen (Stoch. Dyn. 11, 2011, no. 1), which were later reproven by Hu and Nualart (Electron. Commun. Probab. 14, 2009; Electron. Commun. Probab. 15, 2010) and Rosen (S\'eminaire de Probabilit\'es XLIII, Springer, 2011) are included. Furthermore, a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally different approach by exclusively working in the space variable of the Brownian local time, which allows to give a unified argument for arbitrary orders. The main ingredients are Perkins' semimartingale decomposition, the Kailath-Segall identity and an asymptotic Ray-Knight Theorem by Pitman and Yor., Comment: 28 pages; to appear in Annals of Probability
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- 2017
37. Optimal discretization of stochastic integrals driven by general Brownian semimartingale
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Emmanuel Gobet, Uladzislau Stazhynski, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), and Gobet, Emmanuel
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Statistics and Probability ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Discretization ,Stochastic calculus ,discretization of stochastic integrals ,01 natural sciences ,Upper and lower bounds ,Quadratic variation ,010104 statistics & probability ,Mathematics::Probability ,60H05 ,0101 mathematics ,almost sure convergence ,hitting times ,60G40 ,Brownian motion ,random ellipsoids ,Mathematics ,MSC2010: 60G40, 60F15, 60H05 ,Mathematical analysis ,010101 applied mathematics ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Asymptotically optimal algorithm ,Semimartingale ,Convergence of random variables ,60F15 ,Statistics, Probability and Uncertainty - Abstract
We study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a pathwise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretiza- tion stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales (relaxing in particular the non degeneracy conditions usually requested) and we prove that the asymptotic lower bound is attainable.
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- 2017
38. Fluid and Diffusion Limits for Bike Sharing Systems
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Rui-Na Fan, Zhi-Yong Qian, and Quan-Lin Li
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Fluid limit ,Diffusion (acoustics) ,Queueing theory ,Mathematical optimization ,021103 operations research ,Computer science ,0211 other engineering and technologies ,02 engineering and technology ,Heavy traffic approximation ,01 natural sciences ,010104 statistics & probability ,Semimartingale ,Traffic congestion ,0101 mathematics ,Scaling ,Brownian motion - Abstract
Bike sharing systems have rapidly developed around the world, and they are served as a promising strategy to improve urban traffic congestion and to decrease polluting gas emissions. So far performance analysis of bike sharing systems always exists many difficulties and challenges under some more general factors. In this paper, a more general large-scale bike sharing system is discussed by means of heavy traffic approximation of multiclass closed queueing networks with non-exponential factors. Based on this, the fluid scaled equations and the diffusion scaled equations are established by means of the numbers of bikes both at the stations and on the roads, respectively. Furthermore, the scaling processes for the numbers of bikes both at the stations and on the roads are proved to converge in distribution to a semimartingale reflecting Brownian motion (SRBM) in a \(N^{2}\)-dimensional box, and also the fluid and diffusion limit theorems are obtained. Furthermore, performance analysis of the bike sharing system is provided. Thus the results and methodology of this paper provide new highlight in the study of more general large-scale bike sharing systems.
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- 2017
39. Option pricing by using a mixed fractional Brownian motion with jumps
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Sri Haryatmi Kartiko, Gunardi, Chatarina Enny Murwaningtyas, and Herry Pribawanto Suryawan
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History ,Fractional Brownian motion ,Autocorrelation ,Computer Science Applications ,Education ,symbols.namesake ,Semimartingale ,Fourier transform ,Mathematics::Probability ,Valuation of options ,symbols ,Applied mathematics ,Linear combination ,Jump process ,Brownian motion ,Mathematics - Abstract
Option pricing is conventionally based on a Brownian motion (Bm). The Bm is a semimartingale process with stationary and independent increments. However, there are several stock returns that have a long memory or have high autocorrelation for long lags. A fractional Brownian motion (fBm) is one of the models that can solve this problem, but a model option with fBm is not arbitrage-free. A mixed fractional Brownian motion (mfBm) is a linear combination of a Bm and an independent fBm which can overcome the arbitrage problem. A jump process in time series is another problem found in stock price modeling. This paper deals with the problem of options pricing by using mfBm with jumps. Based on quasi-conditional expectation and Fourier transform method, we obtain a pricing formula for a stock option.
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- 2019
40. A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales
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Neil Shephard, Svend Erik Graversen, Ole E. Barndorff–Nielsen, Mark Podolskij, Jean Jacod, Dept. of Mathematical Science, Aarhus University [Aarhus], Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Dept. of Probability and Statistics, Ruhr-Universität Bochum [Bochum], Nuffield College, University of Oxford, Benassù, Serena, University of Oxford [Oxford], Kabanov, Yuri, Liptser, Robert, and Stoyanov, Jordan
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[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,05 social sciences ,central limit theorem ,Poisson random measure ,01 natural sciences ,Quadratic variation ,Combinatorics ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Semimartingale ,quadratic variation ,Bounded function ,0502 economics and business ,0101 mathematics ,Martingale (probability theory) ,Predictable process ,60F17 60G44 ,bipower variation ,Brownian motion ,050205 econometrics ,Mathematics ,Central limit theorem - Abstract
Consider a semimartingale of the form $Y_t=Y_0+\int_0^ta_sds+\int_0^t\si_{s-}~dW_s$, where $a$ is a locally bounded predictable process and $\si$ (the ``volatility'') is an adapted right--continuous process with left limits and $W$ is a Brownian motion. We define the realised bipower variation process $V(Y;r,s)^n_t=n^{{r+s\over2}-1}\sum_{i=1}^{[nt]} |Y_{i\over n}-Y_{i-1\over n}|^r|Y_{i+1\over n}-Y_{i\over n}|^s$, where $r$ and $s$ are nonnegative reals with $r+s>0$. We prove that $V(Y;r,s)^n_t$ converges locally uniformly in time, in probability, to a limiting process $V(Y;r,s)_t$ (the ''bipower variation process''). If further $\si$ is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with $W$ and by a Poisson random measure, we prove a central limit theorem, in the sense that $\rn~(V(Y;r,s)^n-V(Y;r,s))$ converges in law to a process which is the stochastic integral with respect to some other Brownian motion $W'$, which is independent of the driving terms of $Y$ and $\si$. We also provide a multivariate version of these results.
- Published
- 2016
41. Backward stochastic dynamics on a filtered probability space
- Author
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Terry Lyons, Zhongmin Qian, and Gechun Liang
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Statistics and Probability ,Pure mathematics ,Partial differential equation ,Differential equation ,Probability (math.PR) ,semimartingale ,BSDE ,Adapted process ,Omega ,SDE ,Nonlinear system ,Stochastic differential equation ,Mathematics::Probability ,60J45 ,FOS: Mathematics ,60H10 ,Brownian motion ,Statistics, Probability and Uncertainty ,60H30 ,Martingale (probability theory) ,Mathematics - Probability ,Mathematics - Abstract
We demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither It\^{o}'s integrals nor martingale representation formulate are needed. This approach provides new tools for the study of BSDE, and is particularly useful for the study of BSDE with partial information. The approach allows us to study the following type of backward stochastic differential equations: \[dY_t^j=-f_0^j(t,Y_t,L(M)_t) dt-\sum_{i=1}^df_i^j(t,Y_t), dB_t^i+dM_t^j\] with $Y_T=\xi$, on a general filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_t,P)$, where $B$ is a $d$-dimensional Brownian motion, $L$ is a prescribed (nonlinear) mapping which sends a square-integrable $M$ to an adapted process $L(M)$ and $M$, a correction term, is a square-integrable martingale to be determined. Under certain technical conditions, we prove that the system admits a unique solution $(Y,M)$. In general, the associated partial differential equations are not only nonlinear, but also may be nonlocal and involve integral operators., Comment: Published in at http://dx.doi.org/10.1214/10-AOP588 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
- Published
- 2016
42. On the semimartingale property of Brownian bridges on complete manifolds
- Author
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Batu Güneysu
- Subjects
Mathematics - Differential Geometry ,Statistics and Probability ,Pure mathematics ,Property (philosophy) ,010102 general mathematics ,Connection (principal bundle) ,Probability (math.PR) ,Riemannian manifold ,Brownian bridge ,01 natural sciences ,010104 statistics & probability ,Mathematics - Analysis of PDEs ,Semimartingale ,Differential Geometry (math.DG) ,FOS: Mathematics ,Fiber bundle ,0101 mathematics ,Mathematics - Probability ,Brownian motion ,Gradient estimate ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
I prove that every adapted Brownian bridge on a geodesically complete connected Riemannian manifold is a semimartingale including its terminal time, without any further assumptions on the geometry. In particular, it follows that every such process can be horizontally lifted to a smooth principal fiber bundle with connection, including its terminal time. The proof is based on a localized Hamilton-type gradient estimate by Arnaudon/Thalmaier.
- Published
- 2016
- Full Text
- View/download PDF
43. Change of Time Methods: Definitions and Theory
- Author
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Anatoliy Swishchuk
- Subjects
Geometric Brownian motion ,symbols.namesake ,Stochastic differential equation ,Semimartingale ,Probabilistic method ,Stochastic volatility ,Wiener process ,Computer science ,symbols ,Applied mathematics ,Martingale (probability theory) ,Brownian motion - Abstract
In this chapter, we consider the general theory of a change of time method (CTM). One of probabilistic methods which is useful in solving stochastic differential equations (SDEs) arising in finance is the “change of time method”. We give the definition of CTM and describe CTM in martingale, semimartingale, and the SDEs settings. We also point out the association of CTM with subordinators and stochastic volatilities.
- Published
- 2016
44. Continuous Time Models
- Author
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Bruno Bouchard and Jean-François Chassagneux
- Subjects
Work (thermodynamics) ,Semimartingale ,Discrete time and continuous time ,Computer science ,Continuous modelling ,Applied mathematics ,Barrier option ,Diffusion (business) ,Predictable process ,Brownian motion - Abstract
In this chapter, we extend the results obtained in discrete time markets to a continuous time setting. We work with Ito semimartingale models in which the risky assets are modeled as a diffusion driven by a Brownian motion. Note however that most of the results presented below remain true in much more general setting, see e.g. [23] and [24]. The most technical results will be stated without proofs.
- Published
- 2016
45. A Probabilistic Approach to the Zero-Mass Limit Problem for Three Magnetic Relativistic Schrodinger Heat Semigroups
- Author
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Taro Murayama
- Subjects
Subordinator ,semimartingale ,magnetic relativistic Schrödinger operator ,functional limit theorem ,Type (model theory) ,01 natural sciences ,Feynman-Kac-Itô type path integral formula ,60G51, 60f17, 60H05, 35S10, 81S40 ,60H05 ,0103 physical sciences ,81S40 ,FOS: Mathematics ,Limit (mathematics) ,0101 mathematics ,Brownian motion ,Mathematics ,Mathematical physics ,Lévy process ,010102 general mathematics ,Probability (math.PR) ,Zero (complex analysis) ,subordinator ,Exponential function ,35S10 ,Semimartingale ,60F17 ,Path integral formulation ,010307 mathematical physics ,Mathematics - Probability ,60G51 - Abstract
We consider three magnetic relativistic Schrödinger operators which correspond to the same classical symbol $\sqrt{(\xi - A(x))^2 + m^2} + V(x)$ and whose heat semigroups admit the Feynman-Kac-Itô type path integral representation $E[e^{ - S^m (x,\,t;\,X)} g(x + X(t))]$. Using these representations, we prove the convergence of these heat semigroups when the mass-parameter $m$ goes to zero. Its proof reduces to the convergence of $e^{- S^m (x,\,t;\,X)}$, which yields a limit theorem for exponentials of semimartingales as functionals of Lévy processes $X$.
- Published
- 2016
- Full Text
- View/download PDF
46. Some existence results for advanced backward stochastic differential equations with a jump time
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Monique Jeanblanc, Thomas Lim, Nacira Agram, Lim, Thomas, Laboratoire de Mathématiques et Modélisation d'Evry (LaMME), Institut National de la Recherche Agronomique (INRA)-Université d'Évry-Val-d'Essonne (UEVE)-ENSIIE-Centre National de la Recherche Scientifique (CNRS), Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE), University of Oslo (UiO), Laboratoire de Mathématiques et Modélisation d'Evry, and Institut National de la Recherche Agronomique (INRA)-Université d'Évry-Val-d'Essonne (UEVE)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Single Jump ,0209 industrial biotechnology ,Mathematical optimization ,02 engineering and technology ,01 natural sciences ,Stochastic differential equation ,020901 industrial engineering & automation ,Mathematics::Probability ,Immersion ,Immersion (mathematics) ,QA1-939 ,0101 mathematics ,Diffusion (business) ,Brownian motion ,Mathematics ,T57-57.97 ,Applied mathematics. Quantitative methods ,010102 general mathematics ,Mathematical analysis ,[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC] ,Semimartingale ,Advanced Backward Stochastic Differential Equations ,Jump ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Jump process ,Generator (mathematics) - Abstract
In this paper, we are interested by advanced backward stochastic differential equations (ABSDEs), in a probability space equipped with a Brownian motion and a single jump process, with a jump at time τ. ABSDEs are BSDEs where the driver depends on the future paths of the solution. We show, that under immersion hypothesis between the Brownian filtration and its progressive enlargement with τ, assuming that the conditional law of τ is equivalent to the unconditional law of τ, and a Lipschitz condition on the driver, the ABSDE has a solution.
- Published
- 2016
47. Markov Properties of SDEs
- Author
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Samuel N. Cohen and Robert J. Elliott
- Subjects
symbols.namesake ,Semimartingale ,Mathematics::Probability ,Markov chain ,Lebesgue measure ,symbols ,Applied mathematics ,Markov process ,Poisson random measure ,Focus (optics) ,Selection (genetic algorithm) ,Brownian motion ,Mathematics - Abstract
In the previous chapter, we have considered SDEs where the integral is with respect to a general semimartingale. In this chapter, we focus our attention on a much more specialized setting, where the integral is taken with respect to time (i.e. Lebesgue measure), a Brownian motion and a compensated Poisson random measure. Working in this setting allows the Markovian properties of the Brownian motion and the Poisson process to be inherited by the SDE solution. A full treatment of this topic would require consideration of general Markov processes. For this, see Ethier and Kurtz [77], or the more specialized treatments in Karatzas and Shreve [117] or Revuz and Yor [155]. We shall instead present only a selection of these issues.
- Published
- 2015
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