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Albert N. Shiryaev - One of the best experts on this subject based on the ideXlab platform.
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Kolmogorov’s Equations for jump Markov processes with unbounded jump rates
Annals of Operations Research, 2017Co-Authors: Eugene A. Feinberg, Manasa Mandava, Albert N. ShiryaevAbstract:As is well-known, transition probabilities of jump Markov processes satisfy Kolmogorov’s backward and Forward Equations. In the seminal 1940 paper, William Feller investigated solutions of Kolmogorov’s Equations for jump Markov processes. Recently the authors solved the problem studied by Feller and showed that the minimal solution of Kolmogorov’s backward and Forward Equations is the transition probability of the corresponding jump Markov process if the transition rate at each state is bounded. This paper presents more general results. For Kolmogorov’s backward Equation, the sufficient condition for the described property of the minimal solution is that the transition rate at each state is locally integrable, and for Kolmogorov’s Forward Equation the corresponding sufficient condition is that the transition rate at each state is locally bounded.
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Kolmogorov's Equations for Jump Markov Processes with Unbounded Jump Rates
arXiv: Probability, 2016Co-Authors: Eugene A. Feinberg, Manasa Mandava, Albert N. ShiryaevAbstract:As well-known, transition probabilities of jump Markov processes satisfy Kolmogorov's backward and Forward Equations. In the seminal 1940 paper, William Feller investigated solutions of Kolmogorov's Equations for jump Markov processes. Recently the authors solved the problem studied by Feller and showed that the minimal solution of Kolmogorov's backward and Forward Equations is the transition probability of the corresponding jump Markov process if the transition rate at each state is bounded. This paper presents more general results. For Kolmogorov's backward Equation, the sufficient condition for the described property of the minimal solution is that the transition rate at each state is locally integrable, and for Kolmogorov's Forward Equation the corresponding sufficient condition is that the transition rate at each state is locally bounded.
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sufficiency of markov policies for continuous time markov decision processes and solutions to kolmogorov s Forward Equation for jump markov processes
Conference on Decision and Control, 2013Co-Authors: Eugene A. Feinberg, Manasa Mandava, Albert N. ShiryaevAbstract:In continuous-time Markov decision processes (CTMDPs) with Borel state and action spaces, unbounded transition rates, for an arbitrary policy, we construct a relaxed Markov policy such that the marginal distribution on the state-action pairs at any time instant is the same for both the policies. This result implies the existence of a relaxed Markov policy that performs equally to an arbitrary policy with respect to expected discounted and non-discounted total costs as well as average costs per unit time. The proof consists of two steps. The first step describes the properties of solutions to Kolmogorov's Forward Equation for jump Markov Processes. The second step applies these results to CTMDPs.
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CDC - Sufficiency of Markov policies for continuous-time Markov decision processes and solutions to Kolmogorov's Forward Equation for jump Markov processes
52nd IEEE Conference on Decision and Control, 2013Co-Authors: Eugene A. Feinberg, Manasa Mandava, Albert N. ShiryaevAbstract:In continuous-time Markov decision processes (CTMDPs) with Borel state and action spaces, unbounded transition rates, for an arbitrary policy, we construct a relaxed Markov policy such that the marginal distribution on the state-action pairs at any time instant is the same for both the policies. This result implies the existence of a relaxed Markov policy that performs equally to an arbitrary policy with respect to expected discounted and non-discounted total costs as well as average costs per unit time. The proof consists of two steps. The first step describes the properties of solutions to Kolmogorov's Forward Equation for jump Markov Processes. The second step applies these results to CTMDPs.
Yury A. Kutoyants - One of the best experts on this subject based on the ideXlab platform.
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Approximation of BSDE with Hidden Forward Equation and Unknown Volatility.
arXiv: Statistics Theory, 2020Co-Authors: Oleg V. Chernoyarov, Yury A. KutoyantsAbstract:In the present paper the problem of approximating the solution of BSDE is considered in the case where the solution of Forward Equation is observed in the presence of small Gaussian noise. We suppose that the volatility of the Forward Equation depends on an unknown parameter. This approximation is made in several steps. First we obtain a preliminary estimator of the unknown parameter, then using Kalman-Bucy filtration Equations and Fisher-score device we construct an one-step MLE-process of this parameter. The solution of BSDE is approximated by means of the solution of PDE and the One-step MLE-process. The error of approximation is described in different metrics.
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On approximation of BSDE and multi-step MLE-processes
Probability Uncertainty and Quantitative Risk, 2016Co-Authors: Yury A. KutoyantsAbstract:We consider the problem of approximation of the solution of the backward stochastic differential Equations in Markovian case. We suppose that the Forward Equation depends on some unknown finite-dimensional parameter. This approximation is based on the solution of the partial differential Equations and multi-step estimator-processes of the unknown parameter. As the model of observations of the Forward Equation we take a diffusion process with small volatility. First we establish a lower bound on the errors of all approximations and then we propose an approximation which is asymptotically efficient in the sense of this bound. The obtained results are illustrated on the example of the Black and Scholes model.
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On Approximation of the BSDE with Unknown Volatility in Forward Equation
2015Co-Authors: Samvel B. Gasparyan, Yury A. KutoyantsAbstract:We consider the problem of the construction of the backward stochastic differential Equation in the Markovian case. We suppose that the Forward Equation has a diffusion coefficient depending on some unknown parameter. We propose an estimator of this parameter constructed by the discrete time observations of the Forward Equation and then we use this estimator for approximation of the solution of the backward Equation. The question of asymptotic optimality of this approximation is also discussed.
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Approximation of the solution of the backward stochastic differential Equation. Small noise, large sample and high frequency cases
Proceedings of the Steklov Institute of Mathematics, 2014Co-Authors: Yury A. KutoyantsAbstract:We present a review of some recently obtained results on estimation of the solution of a backward stochastic differential Equation (BSDE) in the Markovian case. We suppose that the Forward Equation depends on some finite-dimensional unknown parameter. We consider the problem of estimating this parameter and then use the proposed estimator to estimate the solution of the BSDE. This last estimator is constructed with the help of the solution of the corresponding partial differential Equation. We are interested in three observation models admitting a consistent estimation of the unknown parameter: small noise, large samples and unknown volatility. In the first two cases we have a continuous time observation, and the unknown parameter is in the drift coefficient. In the third case the volatility of the Forward Equation depends on the unknown parameter, and we have discrete time observations. The presented estimators of the solution of the BSDE in the three casesmentioned are asymptotically efficient.
Amel Bentata - One of the best experts on this subject based on the ideXlab platform.
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Forward Equations for option prices in semimartingale models
Finance and Stochastics, 2015Co-Authors: Amel Bentata, Rama ContAbstract:We derive a Forward partial integro-differential Equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuous- semimartingale. This result generalizes Dupire's Forward Equation to a large class of non-Markovian models with jumps and allows to retrieve various Forward Equations previously obtained for option prices in a unified framework.
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Markovian Projection of Stochastic Processes
2012Co-Authors: Amel BentataAbstract:This PhD thesis studies various mathematical aspects of problems related to the Markovian projection of stochastic processes, and explores some ap- plications of the results obtained to mathematical finance, in the context of semimartingale models. Given a stochastic process ξ, modeled as a semimartingale, our aim is to build a Markov process X whose marginal laws are the same as ξ. This construction allows us to use analytical tools such as integro-differential equa- tions to explore or compute quantities involving the marginal laws of ξ, even when ξ is not Markovian. We present a systematic study of this problem from probabilistic view- point and from the analytical viewpoint. On the probabilistic side, given a discontinuous semimartingale we give an explicit construction of a Markov process X which mimics the marginal distributions of ξ, as the solution of a martingale problems for a certain integro-differential operator. This con- struction extends the approach of Gy ̈ongy to the discontinuous case and applies to a wide range of examples which arise in applications, in particu- lar in mathematical finance. On the analytical side, we show that the flow of marginal distributions of a discontinuous semimartingale is the solution of an integro-differential Equation, which extends the Kolmogorov Forward Equation to a non-Markovian setting. As an application, we derive a Forward Equation for option prices in a pricing model described by a discontinuous semimartingale. This Forward Equation generalizes the Dupire Equation, orig- inally derived in the case of diffusion models, to the case of a discontinuous semimartingale. These results give an application to the evaluation of index options allowing to reduce the problem of high dimension.
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Projection Markovienne de processus stochastiques
2012Co-Authors: Amel BentataAbstract:This PhD thesis studies various mathematical aspects of problems related to the Markovian projection of stochastic processes, and explores some applications of the results obtained to mathematical finance, in the context of semimartingale models. Given a stochastic process, modeled as a semimartingale, our aim is to build a Markov process whose marginal laws are the same as the first one. This construction allows us to use analytical tools such as integro-differential Equations to explore or compute quantities involving the marginal laws of a general stochastic process, even when it is not Markovian. We present a systematic study of this problem from probabilistic viewpoint and from the analytical viewpoint. On the probabilistic side, given a discontinuous semimartingale we give an explicit construction of a Markov process which mimics the marginal distributions of a general stochastic process, as the solution of amartingale problems for a certain integro-differential operator. This construction extends the approach of Gyongy to the discontinuous case and applies to a wide range of examples which arise in applications, in particular in mathematical finance. On the analytical side, we show that the flow of marginal distributions of a discontinuous semimartingale is the solution of an integro-differential Equation, which extends the Kolmogorov Forward Equation to a non-Markovian setting. As an application, we derive a Forward Equation for option prices in a pricing model described by a discontinuous semimartingale. This Forward Equation generalizes the Dupire Equation, originally derived in the case of diffusion models, to the case of a discontinuous semimartingale. .
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Mimicking the marginal distributions of a semimartingale
2009Co-Authors: Amel Bentata, Rama ContAbstract:We exhibit conditions under which the flow of marginal distributions of a discontinuous semimartingale $\xi$ can be matched by a Markov process, whose infinitesimal generator is expressed in terms of the local characteristics of $\xi$. Our construction applies to a large class of semimartingales, including smooth functions of a Markov process. We use this result to derive a partial integro-differential Equation for the one-dimensional distributions of a semimartingale, extending the Kolmogorov Forward Equation to a non-Markovian setting.
Eugene A. Feinberg - One of the best experts on this subject based on the ideXlab platform.
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Kolmogorov’s Equations for jump Markov processes with unbounded jump rates
Annals of Operations Research, 2017Co-Authors: Eugene A. Feinberg, Manasa Mandava, Albert N. ShiryaevAbstract:As is well-known, transition probabilities of jump Markov processes satisfy Kolmogorov’s backward and Forward Equations. In the seminal 1940 paper, William Feller investigated solutions of Kolmogorov’s Equations for jump Markov processes. Recently the authors solved the problem studied by Feller and showed that the minimal solution of Kolmogorov’s backward and Forward Equations is the transition probability of the corresponding jump Markov process if the transition rate at each state is bounded. This paper presents more general results. For Kolmogorov’s backward Equation, the sufficient condition for the described property of the minimal solution is that the transition rate at each state is locally integrable, and for Kolmogorov’s Forward Equation the corresponding sufficient condition is that the transition rate at each state is locally bounded.
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Kolmogorov's Equations for Jump Markov Processes with Unbounded Jump Rates
arXiv: Probability, 2016Co-Authors: Eugene A. Feinberg, Manasa Mandava, Albert N. ShiryaevAbstract:As well-known, transition probabilities of jump Markov processes satisfy Kolmogorov's backward and Forward Equations. In the seminal 1940 paper, William Feller investigated solutions of Kolmogorov's Equations for jump Markov processes. Recently the authors solved the problem studied by Feller and showed that the minimal solution of Kolmogorov's backward and Forward Equations is the transition probability of the corresponding jump Markov process if the transition rate at each state is bounded. This paper presents more general results. For Kolmogorov's backward Equation, the sufficient condition for the described property of the minimal solution is that the transition rate at each state is locally integrable, and for Kolmogorov's Forward Equation the corresponding sufficient condition is that the transition rate at each state is locally bounded.
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sufficiency of markov policies for continuous time markov decision processes and solutions to kolmogorov s Forward Equation for jump markov processes
Conference on Decision and Control, 2013Co-Authors: Eugene A. Feinberg, Manasa Mandava, Albert N. ShiryaevAbstract:In continuous-time Markov decision processes (CTMDPs) with Borel state and action spaces, unbounded transition rates, for an arbitrary policy, we construct a relaxed Markov policy such that the marginal distribution on the state-action pairs at any time instant is the same for both the policies. This result implies the existence of a relaxed Markov policy that performs equally to an arbitrary policy with respect to expected discounted and non-discounted total costs as well as average costs per unit time. The proof consists of two steps. The first step describes the properties of solutions to Kolmogorov's Forward Equation for jump Markov Processes. The second step applies these results to CTMDPs.
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CDC - Sufficiency of Markov policies for continuous-time Markov decision processes and solutions to Kolmogorov's Forward Equation for jump Markov processes
52nd IEEE Conference on Decision and Control, 2013Co-Authors: Eugene A. Feinberg, Manasa Mandava, Albert N. ShiryaevAbstract:In continuous-time Markov decision processes (CTMDPs) with Borel state and action spaces, unbounded transition rates, for an arbitrary policy, we construct a relaxed Markov policy such that the marginal distribution on the state-action pairs at any time instant is the same for both the policies. This result implies the existence of a relaxed Markov policy that performs equally to an arbitrary policy with respect to expected discounted and non-discounted total costs as well as average costs per unit time. The proof consists of two steps. The first step describes the properties of solutions to Kolmogorov's Forward Equation for jump Markov Processes. The second step applies these results to CTMDPs.
M.m.r. Williams - One of the best experts on this subject based on the ideXlab platform.
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The Forward Equation of probability balance for neutrons in a medium with randomly time-varying properties: Space and angle dependence
Progress in Nuclear Energy, 2009Co-Authors: M.m.r. WilliamsAbstract:Abstract The recent contributions to combined zero-power and at-power neutron noise analysis by Pazsit and Pal [Pazsit, I., Pal, L., 2008, Neutron Fluctuations, a Treatise on the Physics of Branching Processes. Elsevier Science Ltd.] have been restricted to the point model approximation. In this paper we extend that work and develop a set of Equations which allows a full description of the space and angle variations of the neutron fluctuations to be calculated. These Equations are based upon the Forward Kolmogorov formalism as developed by Govorkov [Govorkov, A.B., 1962a. Statistical scattering of pulse amplitudes in fast neutron pulse reactors. Translated from Atomnaya Energiya 13, 152; Govorkov, A.B., 1962b. Statistical reactor kinetic Equations. Translated from Atomnaya Energiya 17, 474.]. The nature of the randomly time-varying medium is described by a dichotomic Markov process and this is incorporated into the moment Equations by the use of a stochastic ansatz for the first and second moments of the probability distribution. The result of these calculations is a pair of coupled integro-differential Equations for the second moment from which the variance of the neutron density may be found.
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On some matters concerning the backward and Forward Equations of probability balance for neutrons in a medium with randomly time-varying properties
Progress in Nuclear Energy, 2009Co-Authors: M.m.r. WilliamsAbstract:Abstract The recent contributions to combined zero-power and at-power neutron noise analysis raises a number of interesting questions which deserve further discussion [Kitamura, Y., Pal, L., Pazsit, I., Yamamoto, A., Yamane, Y., Some properties of zero power neutron noise in a time-varying medium with delayed neutrons. Annals of Nuclear Energy, in press.]. It is shown, for example, by direct calculation, that the Forward Equation of probability balance is exact when combined with a dichotomic Markov process to describe the random physical behaviour of a medium, whereas the backward Equation contains errors. This confirms and corroborates the assertions of Pal L. and Pazsit I. [2006. Neuron fluctuations in a multiplying medium randomly varying in time. Physica Scripta 74, 62.] who first pointed out this anomaly. Extensions to spatially random media are discussed, together with an application to the calculation of the extinction probability. Other methods of solution such as that of polynomial chaos are briefly touched upon. In order to carry out the calculations described we use the stochastic ansatz concept [Williams, M.M.R., 2008a. A stochastic ansatz and its relationship with the dichotomic Markov process. Physica A 387, 4997.].