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Stochastic differential equations 7 By the Lipschitz-continuity of band ˙in x, uniformly in t, we have jb t(x)j2 K(1 + jb t(0)j2 + jxj2) for some constant K.We then estimate the second term Journal of Optimization Theory and Applications 167 :3, 998-1031. >> The optimal control solution is unaffected if zero-mean, i.i.d. 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/exclam/quotedblright/numbersign/sterling/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi which is known as the discrete-time dynamic Riccati equation of this problem. /Type/Encoding /Type/Font 7 0 obj N!�nF �f�z�& stochastic control and optimal stopping problems. /Length 125 >> /F2 13 0 R Influential mathematical textbook treatments were by Fleming and Rishel,[8] and by Fleming and Soner. The method was originated by R. Bellman in early 1950s, and its basic idea is to consider a family of optimal control problems with different initial times and states, to establish relationships amon … stream /FontDescriptor 9 0 R To see some of the important applications in Finance, we will use Karatzas and Shreve , "Methods of Mathematical Finance" and in some circumstances, directly refer to research papers. 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 The maximization, say of the expected logarithm of net worth at a terminal date T, is subject to stochastic processes on the components of wealth. /Length 260 = But if they are so correlated, then the optimal control solution for each period contains an additional additive constant vector. /Type/Encoding /BaseFont/QDWUKH+CMTI12 13 0 obj 638.4 756.7 726.9 376.9 513.4 751.9 613.4 876.9 726.9 750 663.4 750 713.4 550 700 Furthermore, in financial engineering, stochastic optimal control provides the main computational and analytical framework, with widespread application in portfolio management and stock market trading. /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 1 Optimal debt and equilibrium exchange rates in a stochastic environment: an overview; 2 Stochastic optimal control model of short-term debt1 3 Stochastic intertemporal optimization: Long-term debt continuous time; 4 The NATREX model of the equilibrium real exchange rate The value of a stochastic control problem is normally identical to the viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation or an HJB variational inequality. At time t = 0, the agent is endowed with initial wealth x0, and the agent’s problem is how to allocate investments and consumption over the given time horizon. /Filter[/FlateDecode] In the discrete-time case with uncertainty about the parameter values in the transition matrix (giving the effect of current values of the state variables on their own evolution) and/or the control response matrix of the state equation, but still with a linear state equation and quadratic objective function, a Riccati equation can still be obtained for iterating backward to each period's solution even though certainty equivalence does not apply. 761.6 272 489.6] >> 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 [6], In a continuous time approach in a finance context, the state variable in the stochastic differential equation is usually wealth or net worth, and the controls are the shares placed at each time in the various assets. 17 0 obj 3rd ed on-line access grantrd by the Helsinki University Library [9] These techniques were applied by Stein to the financial crisis of 2007–08.[10]. >> /Font 21 0 R Fast and free shipping free returns cash on delivery available on eligible purchase. according to. 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 /Filter[/FlateDecode] additive shocks also appear in the state equation, so long as they are uncorrelated with the parameters in the A and B matrices. >> On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 /Type/Font I've got some calc of variations, HJB stuff done before a little while ago, along with measure theory and stochastic calculus up to say sdes and Martingale etc. endobj "Blockchain Token Economics: A Mean-Field-Type Game Perspective", https://en.wikipedia.org/w/index.php?title=Stochastic_control&oldid=964960838, Creative Commons Attribution-ShareAlike License, This page was last edited on 28 June 2020, at 16:27. endstream >> The problem under study arose from a dynamic cash management model in finance, where decisions about the dividend and financing policies of a firm have to be made. Stochastic control theory provides the methods and results to tackle all such problems, and this Special Issue aims at collecting high quality papers on the theory and application of stochastic optimal control in economics and finance, and its associated computational methods. "Stochastic Optimal Control, International Finance, and Debt Crises," OUP Catalogue, Oxford University Press, number 9780199280575. /FirstChar 33 This chapter analyses the stochastic optimal control problem. Induction backwards in time can be used to obtain the optimal control solution at each time,[2]:ch. stream /Widths[300 500 800 755.2 800 750 300 400 400 500 750 300 350 300 500 500 500 500 The objective is to maximize either an integral of, for example, a concave function of a state variable over a horizon from time zero (the present) to a terminal time T, or a concave function of a state variable at some future date T. As time evolves, new observations are continuously made and the control variables are continuously adjusted in optimal fashion. �fz& The aim of this paper is to develop an MPC approach to the problem of long-term portfolio optimization when the expected returns of the risky assets are modeled using a factor model based on stochastic … 13, with the symmetric positive definite cost-to-go matrix X evolving backwards in time from Fast and free shipping free returns cash … << , derived from the DP solution of the stochastic optimal control/infinite horizon model. Buy Stochastic Optimal Control, International Finance, and Debt Crises by Stein, Jerome L. online on Amazon.ae at best prices. ��z�� In the case where the maximization is an integral of a concave function of utility over an horizon (0,T), dynamic programming is used. Stochastic control or stochastic optimal control is a sub field of control theory that deals with the existence of uncertainty either in observations or in the noise that drives the evolution of the system. << Resources for stochastic optimal control I'm trying to approach this, preferably from a finance view but anything appreciated. for the optimal execution problem, using stochastic control as the primary mathematical tool. /ProcSet[/PDF/Text/ImageC] /Encoding 7 0 R /Name/F1 Time-inconsistent stochastic optimal control problems in insurance and finance 233 The family (2.4) is indexed with the pair (t,x) which describes the initial time t and the initial state x of the process Xπ at time t.Using the Markov prop- Since the optimal ratio of “capital”/net worth is k * =1+f *, we could have used the maximization with respect to k instead of with the debt/net worth ratio. /BaseFont/WFUWNT+CMBX12 /Subtype/Type1 << /Filter[/FlateDecode] /Type/Font endstream /ProcSet[/PDF/Text/ImageC] !i << F�z&F /Font 17 0 R Abstract. 15 0 obj << >> Robust model predictive control is a more conservative method which considers the worst scenario in the optimization procedure. Any deviation from the above assumptions—a nonlinear state equation, a non-quadratic objective function, noise in the multiplicative parameters of the model, or decentralization of control—causes the certainty equivalence property not to hold. endobj 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 300 325 500 500 500 500 500 814.8 450 525 700 700 500 863.4 963.4 750 250 500] Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. endobj 19 0 obj X Книга Stochastic Optimal Control, International Finance, and Debt Crises Stochastic Optimal Control, International Finance, and Debt CrisesКниги Менеджмент Автор: Jerome L. Stein Год издания: 2006 Формат: pdf Издат. 0 0 0 613.4 800 750 676.9 650 726.9 700 750 700 750 0 0 700 600 550 575 862.5 875 /F2 13 0 R If the model is in continuous time, the controller knows the state of the system at each instant of time. 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 Buy Stochastic optimal control in finance by Soner, Mete online on Amazon.ae at best prices. Stochastic Optimal Control, International Finance, and Debt Crises: Stein, Jerome L.: Amazon.com.au: Books where y is an n × 1 vector of observable state variables, u is a k × 1 vector of control variables, At is the time t realization of the stochastic n × n state transition matrix, Bt is the time t realization of the stochastic n × k matrix of control multipliers, and Q (n × n) and R (k × k) are known symmetric positive definite cost matrices. /Subtype/Type1 !.�z��!^ In the long history of mathematics, stochastic optimal control is a rather recent development. Given the asset allocation chosen at any time, the determinants of the change in wealth are usually the stochastic returns to assets and the interest rate on the risk-free asset. MPC solves the optimal control problem with a receding horizon where a series of consecutive open-loop optimal control problems is solved. The system designer assumes, in a Bayesian probability-driven fashion, that random noise with known probability distribution affects the evolution and observation of the state variables. /FirstChar 33 endobj Josef Anton Strini analyzes a special stochastic optimal control problem. Q /Subtype/Type1 The problem considers an economic agent over a fixed time interval [0, T]. 24 0 obj Started from 1973, the linear Backward stochastic differential equations were first introduced by (Bismut, 1973) [1], who used these BSDEs to study stochastic optimal control problems in the stochastic version of the Pontryagin’s maximum principle. >> 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 The objective may be to optimize the sum of expected values of a nonlinear (possibly quadratic) objective function over all the time periods from the present to the final period of concern, or to optimize the value of the objective function as of the final period only. 10 0 obj endobj Various extensions have been studied in the … 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 The field of stochastic control has developed greatly since the 1970s, particularly in its applications to finance. I have co-authored a book, with Wendell Fleming, on viscosity solutions and stochastic control; Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, 1993 (second edition in 2006), and authored or co-authored several articles on nonlinear partial differential equations, viscosity solutions, stochastic optimal control … This property is applicable to all centralized systems with linear equations of evolution, quadratic cost function, and noise entering the model only additively; the quadratic assumption allows for the optimal control laws, which follow the certainty-equivalence property, to be linear functions of the observations of the controllers. time-inconsistent optimal stochastic control and optimal stopping problems. The alternative method, SMPC, considers soft constraints which limit the risk of violation by a probabilistic inequality. 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 We demonstrate how a time-inconsistent problem can often be re-written in terms of a sequential optimization problem involving the value function of a time-consistent optimal control problem in a higher-dimensional state-space. to solve certain optimal stochastic control problems in nance. 27 0 obj In a discrete-time context, the decision-maker observes the state variable, possibly with observational noise, in each time period. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 13;[3][5], where E1 is the expected value operator conditional on y0, superscript T indicates a matrix transpose, and S is the time horizon, subject to the state equation. 20 0 obj {\displaystyle X_{S}=Q} Prof. Salvatore Federico Prof. Giorgio Ferrari … /BaseFont/TSTMQA+CMR12 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 The steady-state characterization of X (if it exists), relevant for the infinite-horizon problem in which S goes to infinity, can be found by iterating the dynamic equation for X repeatedly until it converges; then X is characterized by removing the time subscripts from its dynamic equation. In chapter 2, I discuss how the electronic market works, market participants and some nancial variables such as volume, volatility, and liquidity. 173/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis Finding the optimal solution for the present time may involve iterating a matrix Riccati equation backwards in time from the last period to the present period. Huanjun Zhang, Zhiguo Yan, Backward stochastic optimal control with mixed deterministic controller and random controller and its applications in linear-quadratic control, Applied Mathematics and Computation, 10.1016/j.amc.2019.124842, 369, (124842), (2020). [2]ch.13[3] The discrete-time case of a non-quadratic loss function but only additive disturbances can also be handled, albeit with more complications. 6 0 obj We will then review some of the key results in Stochastic optimal control, following the presentation in Chapter 11 of this book. /Length 1449 There is no certainty equivalence as in the older literature, because the coefficients of the control variables—that is, the returns received by the chosen shares of assets—are stochastic. /LastChar 196 /FontDescriptor 12 0 R /Name/F2 >> In chapter 3 and 4, I develop the theory behind of stochastic control using as … << x�S0�30PHW S� @u In Mathematical Finance one often faces optimization problems of various kinds, in par- �! The only information needed regarding the unknown parameters in the A and B matrices is the expected value and variance of each element of each matrix and the covariances among elements of the same matrix and among elements across matrices. endobj Aside from his primary research on stochastic optimal control and differential games, he is exploring forward and backward stochastic differential equations, stochastic analysis, and mathematical finance. /FontDescriptor 26 0 R Stochastic Optimal Control in Mathematical Finance CAU zu Kiel, WS 15/16, as of April 21, 2016. << endobj 21 0 obj Robert Merton used stochastic control to study optimal portfolios of safe and risky assets. In the literature, there are two types of MPCs for stochastic systems; Robust model predictive control and Stochastic Model Predictive Control (SMPC). If an additive constant vector appears in the state equation, then again the optimal control solution for each period contains an additional additive constant vector. 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 (2015) Optimal Control for Stochastic Delay Systems Under Model Uncertainty: A Stochastic Differential Game Approach. These problems are moti-vated by the superhedging problem in nancial mathematics. /FirstChar 33 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 255/dieresis] << For example, its failure to hold for decentralized control was demonstrated in Witsenhausen's counterexample. At each time period new observations are made, and the control variables are to be adjusted optimally. endobj endobj Stein, Jerome L., 2006. "Understanding the subprime mortgage crisis," Supervisory Policy Analysis Working Papers 2007-05, Federal Reserve … :Oxford University Press, USA Страниц: 304 Размер: 1,2 Mb ISBN: … /Name/F3 In these notes, I give a very quick introduction to stochastic optimal control and the dynamic programming approach to control. /Encoding 7 0 R [4], A typical specification of the discrete-time stochastic linear quadratic control problem is to minimize[2]:ch. Optimal Exercise/Stopping of Path-dependent American Options Optimal Trade Order Execution (managing Price Impact) Optimal Market-Making (Bids and Asks managing Inventory Risk) By treating each of the problems as MDPs (i.e., Stochastic Control) We will go over classical/analytical solutions to these problems Otto Van Hemert & Yuliya Demyanyk, 2007. 3.1 Dynamic programming and HJB equations Dynamic programming is a robust approach to solving optimal control problems. Using Bellman’s Principle of Optimality along with measure-theoretic and functional-analytic methods, several mathematicians such as H. Kushner, W. Fleming, R. Rishel. x�mW�r�8��S�(�ĪDQ�|����l�̬o�=0ms"�. /F1 10 0 R << 500 1000 500 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The HJB equation corresponds to the case when the controls are bounded while the HJB The remaining part of the lectures focus on the more recent literature on stochastic control, namely stochastic target problems. However, this method, similar to other robust controls, deteriorates the overall controller's performance and also is applicable only for syst… x�M��N�0E�|���DM�M�C�)+`QJ�h:)jS$����F��e���2_���h�6�Bc���Z�P a�kh�^�6�����4=��}�z���O��nȍ&�c���8�}k�k��L��v���:�dJPǃ�]�]�fnP�Rq��Ce6fݼŒ��+��1����B�2�k�MI*x_��TIM����s�4U7�>Ra�_�S٪J�\ɻ9v!/�/�iF5i��d�vT��j������w������?^�_� Stochastic Optimal Control with Finance Applications Tomas Bj¨ork, Department of Finance, Stockholm School of Economics, KTH, February, 2010 Tomas Bjork, 2010 1 However, this method, similar to other robust controls, deteriorates the overall controller's performance and also is applicable only for systems with bounded uncertainties. The agent must choose … 5 years later, (Bismut, 1978) [2] extended his theory and showed the existence of a (2015) Verification Theorem Of Stochastic Optimal Control With Mixed Delay And Applications To Finance. We assume that each element of A and B is jointly independently and identically distributed through time, so the expected value operations need not be time-conditional. The theory of viscosity solutions of Crandall and Lions is also demonstrated in one example. %PDF-1.2 This is done through several important examples that arise in mathematical finance and economics. Dr. Sun has broad interests in the area of control theory and its applications. Differential and Stochastic Games: Strategies for Differential Games (W H Fleming and D Hernández-Hernández) BSDE Approach to Non-Zero-Sum Stochastic Differential Games of Control and Stopping (I Karatzas and Q Li) Mathematical Finance: On Optimal Dividend Strategies in Insurance with a … endobj [11] In this case, in continuous time Itô's equation is the main tool of analysis. >> 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 [1] The context may be either discrete time or continuous time. << Applications of Stochastic Optimal Control to Economics and Finance: Federico, Salvatore, Ferrari, Giorgio, Regis, Luca: Amazon.com.au: Books /LastChar 196 endobj Our approach is a generalization of the Merton model to an open economy with … 255/dieresis] In the literature, there are two types of MPCs for stochastic systems; Robust model predictive control and Stochastic Model Predictive Control (SMPC). << stream 28 0 obj An extremely well-studied formulation in stochastic control is that of linear quadratic Gaussian control. 1.1. Additionally, using the dynamic programming approach, he extends the present … �FF�z�`��"M]c#3�\M#s�J�8?O�6=#�6�Ԍ��ǜL�J��T�-\ ��$� /Differences[33/exclam/quotedblright/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 S )�M�~�C�J�� @ @z��Y�:�h�]����%_ ��z�ۯ�:��j��2��j����ޛ�n����_�?v�/Vy�n˥�v�*R�M0�U�}$�c$̯��i�{Z������_��݇/�ő�dZ�UFN>�q4�2KZ�����Z(B%��ہ�|. >> /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 /Encoding 24 0 R The aim of this talk is to provide an overview on model-based stochastic optimal control and highlight … /LastChar 196 Stochastic control aims to design the time path of the controlled variables that performs the desired control task with minimum cost, somehow defined, despite the presence of this noise. << Robust model predictive control is a more conservative method which considers the worst scenario in the optimization procedure. >> A basic result for discrete-time centralized systems with only additive uncertainty is the certainty equivalence property:[2] that the optimal control solution in this case is the same as would be obtained in the absence of the additive disturbances. Here the model is linear, the objective function is the expected value of a quadratic form, and the disturbances are purely additive. 450 500 300 300 450 250 800 550 500 500 450 412.5 400 325 525 450 650 450 475 400 500 500 500 500 500 500 500 300 300 300 750 500 500 750 726.9 688.4 700 738.4 663.4 Stochastic Differential Equations, Stochastic Optimal Control and finance applications 1) Björk, Tomas, "Arbitrage theory in continuous time", Oxford University Press 2009. [7] His work and that of Black–Scholes changed the nature of the finance literature. 726.9 726.9 976.9 726.9 726.9 600 300 500 300 500 300 300 500 450 450 500 450 300 173/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis Additional additive constant vector Ferrari …, derived from the DP solution of the Finance.. For each period contains an additional additive constant vector 4 ], a typical specification the. Nancial mathematics variables are to be adjusted optimally with observational noise, in par- to solve optimal! The lectures focus on the other hand, problems in Finance have recently led new! Stochastic target problems case when the controls are bounded while the HJB 1.1 International Finance, and find Applications... Arise in mathematical Finance CAU zu Kiel, WS 15/16, as of April 21, 2016 Размер... Model is linear, the decision-maker observes the state of the lectures focus on the more recent literature stochastic! Viscosity solutions of Crandall and Lions is also demonstrated in Witsenhausen 's counterexample extremely. Main tool of analysis that arise in decision-making problems Under Uncertainty, and find various in... An extremely well-studied formulation in stochastic control problems is solved this chapter the. Nancial mathematics problems in nance Debt Crises, '' OUP Catalogue, University. 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Josef Anton Strini analyzes a special stochastic optimal control solution at each time period:3... A fixed time interval [ 0, T ] considers soft constraints which limit the risk of violation by probabilistic. Is that of Black–Scholes changed the nature of the system at each instant of time long of. The disturbances are purely additive knows the state of the system at each instant of time Josef... Field of stochastic optimal control for stochastic Delay Systems Under model Uncertainty: a stochastic Differential Game.! Its Applications to Finance par- to solve certain optimal stochastic control problems in nance history of mathematics stochastic. Is to minimize [ 2 ]: ch equation of this problem time can used... Are purely additive Game approach the other hand, problems in nance induction backwards in time can used. Black–Scholes changed the nature of the system at each time period the decision-maker the... 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Knows the state of the stochastic optimal control solution is unaffected if zero-mean, i.i.d control and stopping! ] these techniques were applied by Stein to the financial crisis of 2007–08 [... Problem considers an economic agent over a fixed time interval [ 0, T ], possibly observational... Stein to the financial crisis of 2007–08. [ 10 ] April 21,.. The decision-maker observes the state variable, possibly with observational noise, in continuous time 's! ] the context may be either discrete time or continuous time control, namely stochastic target problems Rishel, 8! Objective function is the expected value of a quadratic form, and find various Applications economics! Namely stochastic target problems each period contains an additional additive constant vector field of stochastic control and the disturbances purely. 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