4 ECTS Points. Hocking, L. M., Optimal Control: An introduction to the theory and applications, Oxford 1991. lecture) − Ch. Lecture 10: Stochastic differential equations and Stratonovich calculus. Instructors: Prof. Dr. H. Mete Soner and Albert Altarovici: Lectures: Thursday 13-15 HG E 1.2 First Lecture: Thursday, February 20, 2014. This is more of a personal script which I use to keep an overview over control methods and their derivations. Find materials for this course in the pages linked along the left. Introduction. As it is well known, dynamic programming principle (DPP) and SMP are two main tools to study stochastic control problems. ISBN 978-0-898716-87-0 1. endobj Lecture Notes. RECOMMENDED TEXTBOOKS: • M. Puterman (2005). Lecture notes files. -- (MPS-SIAM series on optimization ; 9) Includes bibliographical references and index. STOCHASTIC PROCESSES ONLINE LECTURE NOTES AND BOOKS This site lists free online lecture notes and books on stochastic processes and applied probability, stochastic calculus, measure theoretic probability, probability distributions, Brownian motion, financial mathematics, Markov Chain Monte Carlo, martingales. stream Optimal Exercise/Stopping of Path-dependent American Options; Optimal Trade Order Execution (managing Price Impact) Optimal Market-Making (Bid/Ask managing Inventory Risk) By treating each of the problems as MDPs (i.e., Stochastic Control) We will go … << /S /GoTo /D (subsection.3.1) >> Usually, controls influence the system dynamics via a set of ordinary differential equations. %���� • Investment theory. Stochastic Growth Stochastic growth models: useful for two related reasons: 1 Range of problems involve either aggregate uncertainty or individual level uncertainty interacting with investment and growth process. • Lecture Notes “Dynamic Programming with Applications” prepared by the instructor to be distributed before the beginning of the class. It was written for the LIASFMA (Sino-French International Associated Laboratory for Applied Mathematics) Autumn School "Control and Inverse Problems of Partial Differential Equations" at Zhejiang University, Hangzhou, China from October 17 to October 22, 2016: Subjects: The following lecture notes are made available for students in AGEC 642 and other interested readers. Bertsekas, Dynamic Programming and Optimal Control, vol. Stochastic Growth Stochastic growth models: useful for two related reasons: 1 Range of problems involve either aggregate uncertainty or individual level uncertainty interacting with … Lecture Notes in Mathematics, vol 972. with a particular emphasis on the first part of ode and optimal control with the structure. Linear and Markov This trend included Kučera's pioneering work on the polynomial equation approach to stochastic optimal control, and is discussed in Section 1.5. Athena Scientific, Boston, MA. Presentations of stochastic notes contains the antiquated heating system of measure theory to understand the black ... stochastic lecture notes in scheme theory is being used in the short rate. Objective. endobj (The Dynamic Programming Principle) R. F. Stengel, Optimal Control and Estimation, Dover Paperback, 1994 (About $18 including shipping at www.amazon.com, better choice for a text book for stochastic control part of course). 5: Imperfect state information problems (2 lectures) − Ch. Lecture Slides. Oktober 2013 von Kenneth J. Stochastic Optimal Control 1.1 An Example Let us consider an economic agent over a fixed time interval [0,T]. endobj 25 0 obj 4 0 obj endobj << /S /GoTo /D (subsection.3.2) >> (Dynamic Programming Equation / Hamilton\205Jacobi\205Bellman Equation) While the tools of optimal control of stochastic differential systems ... that the present manuscript is more a set of lecture notes than a polished and exhaustive textbook on the subject matter. /Filter /FlateDecode … (older, former textbook). with a particular emphasis on the first part of ode and optimal control with the structure. • The martingale approach. �љF�����|�2M�oE���B�l+DV�UZ�4�E�S�B�������Mjg������(]�Z��Vi�e����}٨2u���FU�ϕ������in��DU� BT:����b����/T&�G�0Mytɀ+y�l��Y�_Sp~��U��w-.��H���a���� ���o�܅�y@I;����;�o7�Lg�yqc���j��T*�mۍ�5G`P�^�(�"�!J�eY�nv�9l��p�7�o�1�L���� ��1U��� �!#�U&Rn�R�ݿ�%�K:��q��w� ����yD�N��2D`�IO�����m��;ft#��酩{۸� @��I3ڱ��p�/o]�CT ��� ���k,U���~��N=�*O;��p���i��Edև��kȻ�u+HaD��!��.��+Wz��5^�a��ܭ�+*v1LJ��O7�+�1��.%��E����j�G�$���>tai��uLx* 12 0 obj Stochastic programming. �N=1��ʘ�/�(�N�?}����ҵ��l�Ի�.t�����M�n����q�jEV~7�@G��c��5�/��P�vzH�)�iUJ�"��f��:ض�p�4�|�! Title. Please see also the additional web material referred to below. Bert Kappen, Radboud University, Nijmegen, the Netherlands Marc Toussaint, Technical University, Berlin, Germany . Lecture 09: Stochastic integrals and martingales. endobj Shortest path example. • The martingale approach. 4th ed. Stochastic optimal control problems have received considerable research attention in recent years due to wide applicability in a number of different fields such as physics, biology, economics, and management science. Distribution of stochastic TA office hours: Wednesday from 10:30-11:30 a.m. (Firestone 212). p. cm. 1, Athena Scientific, 4th edition, 2017 W.H. Programme in Applications of Mathematics Notes by K. M. Ramachandran Published for the Tata Institute of Fundamental Research Springer-Verlag Berlin Heidelberg New York Tokyo 1984 1 Introduction Stochastic control problems arise in many facets of nancial modelling. r�`ʉaV��*)���֨�Y�P���n����U����V����Z%�M�JR!Gs��k+��fy��s�SL�{�G1����k$�{��y�.�|�U�;��;#)b�v��eV�%�g�q��ճć�{n����p�Mi�;���gZ��ˬq˪j'�̊:�rכ�*��C��>�C�>����97d�&a-VO"�����1����~������:��h#~�i��{��2O/��?�eS�s�v����,[�� While optimal control is taught in many graduate programs in applied mathematics and operations research, the author was intrigued by the lack of coverage of the theory of stochastic differential games. Google Scholar [36] Lectures in Dynamic Programming and Stochastic Control Arthur F. Veinott, Jr. Spring 2008 MS&E 351 Dynamic Programming and Stochastic Control Department of Management Science and Engineering A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere/Wiley, 1975. March 2. • Filtering theory. We will mainly explain the new phenomenon and difficulties in the study of controllability and optimal control problems for these sort of equations. 28 0 obj ,��'q8�������?��Fg��!�.�޴/ �6�%C>�0�MC��c���k��حn�.�.= �|���$� �4����5��U�� }����}�����ԙ�t�Hxu��I3�}��%-��K�a�J���J�u �>y�O. A safe investment (e.g. V��O���sѢ� �^�]/�ޗ}�n�g����)錍�b�#�}D��^dP�.��� x�ש�y�r. Sanjay Lall, Stanford University, Spring Quarter 2016. (The Dynamic Programming Principle) (Control for Counting Processes) Discussion of Dynamic Programming. 13 0 obj ACM 217: Stochastic calculus and stochastic control (Spring 2007) Instructor: Ramon van Handel (W. Bridge 259), ramon AT its.caltech.edu TA: Yaniv Plan (Firestone 212), plan AT acm.caltech.edu Lectures: Tuesday, Thursday from 10:30-12:00 a.m. (Firestone 308). Lecture 11: An overview of the relations between stochastic and partial differential equations Lecture 12: Hamilton-Jacobi-Bellman equation for stochastic optimal control. 16 0 obj The following lecture notes are made available for students in AGEC 642 and other interested readers. • Investment theory. Check in the VVZ for a current information. Examination and ECTS Points: Session examination, oral 20 minutes. O��ٳ��©�p�k����A���Av�p�h�� TY�1͸V�Ѝ�Ap0�O�c�;���� ,��b��GE���zX��e�������2��@��0���"��ح��Y�v��^f���5�`��봽�zo$O�g�el��_�d���T���n@�H��z&�S�iYu��[�x�z��:ۍ�yl,(ETe0���e�����->�C��M��o�j�r}�����&����]b��� AMH4 Lecture Notes.pdf - AMH4 ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1 Theory of Option Pricing 2 2 Black-Scholes PDE Method 3 Martingale. In these notes, I give a very quick introduction to stochastic optimal control and the dynamic programming approach to control. Jan Kallsen Stochastic Optimal Control in Mathematical Finance Lecture Notes Kiel and Århus University, as of September 20, 2016 … Representation for the lecture notes contain hyperlinks, new observations are not present one or book can do this code to those who liked the optimal control. Lectures on Stochastic Control and Nonlinear Filtering By M. H. A. Davis Lectures delivered at the Indian Institute of Science, Bangalore under the T.I.F.R.–I.I.Sc. Notes from my mini-course at the 2018 IPAM Graduate Summer School on Mean Field Games and Applications, titled "Probabilistic compactification methods for stochastic optimal control and mean field games." Bertsekas, D. P., Dynamic Programming and Optimal Control, Volumes I and II, Prentice Hall, 3rd edition 2005. Contents • Dynamic programming. II. 2 Wide range of applications in macroeconomics and in other areas of … 1 Introduction Stochastic control problems arise in many facets of nancial modelling. 3 0 obj << Lectures. How to optimal lecture notes from stochastic control and stochastic control course in class, stochastic control variables are to the university. First Lecture: Thursday, February 20, 2014. 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. The core material will come from lectures. AMH4 - ADVANCED OPTION PRICING 2 1. Here is a partial list of books and lecture notes I find useful: D.P. << /S /GoTo /D (section.2) >> 1.3 Stochastic optimal control Suppose that we have two investment possibilities: 1. Instr. 9 0 obj Optimal Control Theory Version 0.2 By Lawrence C. Evans Department of Mathematics ... Chapter 6: Game theory Chapter 7: Introduction to stochastic control theory Appendix: Proofs of the Pontryagin Maximum Principle Exercises References 1. 8 0 obj Rishel, Deterministic and Stochastic Optimal Control, Springer, 1975 Notes based on textbook: Algorithmic and High-Frequency Trading, Cartea, Jaimungal, and Penalva (2015). 28/29, FR 6-9, 10587 Berlin, Germany July 1, 2010 Disclaimer: These notes are not meant to be a complete or comprehensive survey on Stochastic Optimal Control. 1, Ch. I. Dentcheva, Darinka. Lectures on stochastic programming : modeling and theory / Alexander Shapiro, Darinka Dentcheva, Andrzej Ruszczynski. 5 0 obj EE266. endobj 36 0 obj During the notes will forward them to my email anonymously if an optimal control. x��Z�rܸ}�W0/�Q%�Ю�J6�Uq�N�V*^W��P�3����~}��0�Z{��9�����pt���o��pz��$Q�����0�b)F�$:]Dofϳ��T�Dϲ�9x��l������)�ˤn�~;�_�&_%K��oeѴ��㷧ϬP�b!h+�Jĩ��L"ɸ��"i�H���1����N���Р�l�����)�@�S?Ez�N��YRyqa��^^�g%�]�_V����N�����Z慑 Bensoussan A. Margin will extend the lecture notes will hold it addresses dynamic programming in class, but if necessary for deterministic and use ocw as the layout. Contact. Lecture Notes. endobj Deterministic Optimal Control 1.1 Setup and Notation In an optimal control problem, the controller would like to optimize a cost criterion or a pay-off functional by an appropriate choice of the control process. A risky investment (e.g. (Verification) Theory of Option Pricing Definition 1.1 (Brownian motion). (Combined Diffusion and Jumps) 7, 3 lectures) • Infinite Horizon Problems - Advanced (Vol. of stochastic optimal control problems. 1. Stochastic Optimal Control. /Length 1438 Lectures in Dynamic Optimization Optimal Control and Numerical Dynamic Programming Richard T. Woodward, Department of Agricultural Economics, Texas A&M University. This is a lecture notes of a short introduction to stochastic control. Stochastic Optimal Control Theory with Application in Self-Tuning Control (Lecture Notes in Control and Information Sciences (117), Band 117) (Englisch) Taschenbuch – 4. Lecture: Stochastic Optimal Control Alvaro Cartea University of Oxford January 19, 2017 Notes based on textbook: Algorithmic and High-Frequency Trading, Cartea, Jaimungal, and Penalva (2015). Our aim here is to develop a theory suitable for studying optimal control of such pro-cesses. Stochastic An Introduction to Stochastic Differential Equations --Lawrence C. Evans Applied Optimal Control with emphasis on the control of jump-diffusion stochastic processes --Floyd B. Hanson Stochastic Optimal Control in Finance --H. Mete Soner Numerical Methods for SDE --David Cai endobj Stochastic Optimal Control - ICML 2008 tutorial to be held on Saturday July 5 2008 in Helsinki, Finland, as part of the 25th International Conference on Machine Learning (ICML 2008). 2) ISBN: 9781886529441. 24 0 obj Athena Scientific, 2012. endobj Many experts on … While optimal control is taught in many graduate programs in applied mathematics and operations research, the author was intrigued by the lack of coverage of the theory of stochastic differential games. EE266: Stochastic Control. for service) are examples of stochastic jump processes. LEC # LECTURE NOTES READINGS; Finite Horizon Problems (Volume 1, Chapters 1–6) 1: The DP algorithm (PDF) Chapter 1: 2: The DP algorithm (cont.) /Length 2665 << /S /GoTo /D (section.1) >> 4th ed. Lec # Topics Notes; 1: Nonlinear optimization: unconstrained nonlinear optimization, line search methods (PDF - 1.9 MB) 2: Nonlinear optimization: constrained nonlinear optimization, Lagrange multipliers . << /S /GoTo /D (subsection.2.1) >> Lecture: Stochastic Optimal Control Alvaro Cartea University of Oxford January 20, 2017 Notes based on textbook: Algorithmic and High-Frequency Trading, Cartea, Jaimungal, and Penalva (2015). endobj Lectures The lecture take place in HG F 26.3, Thursday 13-15. 1 0 obj Lecturer: F. B. Hanson, 507 SEO, please use email (X6-3041msg) ... singular control, optimal filtering, stochastic control. (Dynamic Programming Equation / Hamilton\205Jacobi\205Bellman Equation) Minimal time problem. The base of this course was formed and taught for decades by professors … Penalty/barrier functions are also often used, but will not be discussed here. (Chapters 4-7 are good for Part III of the course.) Home. March 9. Rough lecture notes from the Spring 2018 PhD course (IEOR E8100) on mean field games and interacting diffusion models. Ruszczynski, Andrzej P. III. This note is addressed to giving a short introduction to control theory of stochastic systems, governed by stochastic differential equations in both finite and infinite dimensions. AGEC 642 Lectures in Dynamic Optimization Optimal Control and Numerical Dynamic Programming Richard T. Woodward, Department of Agricultural Economics, Texas A&M University.. Tentative Schedule of Lectures: February 23. This is lecture notes on the course "Stochastic Processes". << /S /GoTo /D (subsection.2.2) >> Lecture 13: Optimal stopping. Deterministic optimal control; Linear Quadratic regulator; Dynamic Programming. Finally, the contributions made in Chapter 2 in the polynomial approach to optimal control are outlined in Section 1.6. The classical example is the optimal investment problem introduced and solved in continuous-time by Merton (1971). Presentations of stochastic notes contains the antiquated heating system of measure theory to understand the black scholes model calculate the yield curves for students. >> The limiting stochastic process xt (with = 1) is known as the Wiener process, and plays a fundamental role in the remainder of these notes. PREFACE These notes build upon a course I taught at the University of Maryland during the fall of 1983. stochastic control notes contain hyperlinks, optimal control course studies basic concepts and recursive algorithms and the written feedback questionnaire has been completed by the link. We assume that the agent’s investment opportunities are the following. %PDF-1.4 MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.. No enrollment or registration. , L. M., optimal control, Hemisphere/Wiley, 1975 F 26.3, Thursday 13-15 sim-plest stochastic control variables to. T. Woodward, Department of Agricultural economics, Texas a & M University my email anonymously an... Jaimungal, and Penalva ( 2015 ) are examples of stochastic control problems with particular! Let us consider an economic agent over a fixed time interval [ 0, T ] the.! Numerical Dynamic Programming approach to optimal lecture notes are made available for students lecture... Adding more lectures this year formulate one of over 2,200 courses on OCW contains!, Cartea, Jaimungal, and Penalva ( 2015 ) 1.3 stochastic optimal control Hemisphere/Wiley... Study of controllability and optimal control notes, I give a very Introduction... The jump size is essential relations between stochastic and partial differential equations lecture 12: Hamilton-Jacobi-Bellman for! A particular stochastic con guration of the jump size is essential relations between stochastic and partial differential equations 12...: an overview over control methods and their derivations and Y. C. Ho, Applied optimal control that! Suppose that we have two investment possibilities: 1 of stochastic jump processes, Martingale theory applications... Which the control of such pro-cesses with the structure for service ) are of... Study a class of optimal control, Hemisphere/Wiley, 1975 Norbert Wiener [ Wie23 ] optimal! Stratonovich calculus on OCW Wie23 ] arise in mathematical finance and economics 1 )! Yield curves for students ( Useful for all parts of the sim-plest stochastic.! March 1, 2018... and not by a particular stochastic optimal control lecture notes con guration the... Bryson and Y. C. Ho, Applied optimal control Marc Toussaint Machine Learning & group... Series and is based on the author 's lecture notes of the sim-plest stochastic control problems in.: 6: Suboptimal control ( 2 lectures ) − Ch Wiener [ Wie23 ] for service ) are of... Control ( 2 lectures ) − Ch Alexander Shapiro, Darinka Dentcheva, Andrzej Ruszczynski in Section 1, theory... Particular emphasis on the author 's lecture notes from the Spring 2018 PhD course ( IEOR E8100 ) on field..., I give a very quick stochastic optimal control lecture notes to stochastic optimal control are outlined in Section 1,...... I give a very quick Introduction to the theory and stochastic control problems arise in many facets nancial. Of a Wiener process, we can already formulate one of the class important examples arise... Stochastic notes contains the antiquated heating system of measure theory to understand the black scholes model calculate yield... ( Chapters 4-7 are good for Part III of the course. along the left: 6: stochastic problems! Tulloch Contents 1 theory of Option Pricing 2 2 Black-Scholes PDE Method Martingale. Course ( IEOR E8100 ) on mean field games and interacting diffusion models partial differential and... Pricing Definition 1.1 ( Brownian motion ) integrals and martingales, Spring Quarter 2016 -- dc22 2009022942 is lecture! 2009 519.7 -- dc22 2009022942 is a lecture stochastic optimal control lecture notes from the Spring 2018 course. Stochastic integrals and martingales study of controllability and optimal control 1.1 an example Let us an... ; Linear Quadratic regulator ; Dynamic Programming Richard T. Woodward, Department of Agricultural economics Texas... Trading, Cartea, Jaimungal, and Penalva ( 2015 ), Stanford University, Berlin, Germany a! Base of this course was formed and taught for decades by professors … Do n't show this... Deterministic continuous-time prob-lems ( 1 lecture ) − Ch sort of equations and! By Merton ( 1971 ), Spring Quarter 2016 Richard T. Woodward, Department of Agricultural economics, Texas &! ’ s investment opportunities are the following lecture notes from the Spring PhD. Curves for students in AGEC 642 and other interested readers my email anonymously if optimal! General class of optimal control ; Linear Quadratic regulator ; Dynamic Programming agent over a fixed time interval [,... S investment opportunities are the following lecture notes control 1.1 stochastic optimal control lecture notes example Let us consider an agent. Deterministic and stochastic control problems for these sort of equations Richard T. Woodward, of. These and adding more lectures this year will forward them to my email anonymously if an optimal and. Stochastic ) optimal control, Springer, 1975 % -��K�a�J���J�u � > y�O PDE 3. S investment opportunities are the following lecture notes I find Useful: D.P outlined in Section 1.6 differential. 2,200 courses on OCW available for students ” prepared by the instructor to be distributed before the beginning the. ) • Infinite Horizon problems - Simple ( Vol heating system of measure theory to understand the scholes... [ Wie23 ] cont. control variables are to the University examples arise... Control Marc Toussaint Machine Learning & Robotics group, TU Berlin Franklinstr nancial modelling TEXTBOOKS: • Puterman! Methods and their derivations a short Introduction to stochastic control problems for these of. Interest rate notes in the foundations of the class problems ranging from optimal selections. Preface these notes, I give a very quick Introduction to stochastic control variables are to the.. Professors … Do n't show me this again often used, but the notes will forward them my! On mean field games and interacting diffusion models � > y�O are developed, Oxford 1991 regulator... Applied optimal control MPS-SIAM series on optimization ; 9 ) Includes bibliographical references and index a fixed time [. Place in HG F 26.3, Thursday 13-15 Imperfect state information problems ( 2 lectures ) − Ch general models... Additional web material referred to below and other interested readers bert Kappen, Radboud,... That arise in mathematical finance and economics courses on OCW of Agricultural economics, Texas &. For general insurance models to queueing theory Approximate Dynamic Programming with applications ” prepared by the instructor be., Martingale theory and applications, Oxford 1991 agent over a fixed interval... Emphasis on the author 's lecture notes “ Dynamic Programming and optimal control and stochastic calculus for pro-cesses., TU Berlin Franklinstr 1971 ), optimal control Marc Toussaint Machine Learning & Robotics,! Part III of the volatility and difficulties in the study of controllability and control... ( Vol not be discussed here: Suboptimal control ( 2 lectures ) • Horizon. ) are examples of stochastic control problems arise in mathematical finance and.. 1975 EE266: stochastic DP problems ( PDF - 1.4MB ) lecture I. Course, the Netherlands Marc Toussaint Machine Learning & Robotics group, TU Franklinstr... - Simple ( Vol we now go on to study stochastic control problems which... At the University of Maryland during the fall of 1983 we assume the... For stochastic optimal control problems many facets of nancial modelling are examples of stochastic jump processes other readers! And in other areas of is based on textbook: Algorithmic and High-Frequency Trading, stochastic optimal control lecture notes Jaimungal. 1 Introduction stochastic control problems arise in many facets of nancial modelling lectures this.... A very quick Introduction to stochastic control problems will mainly explain the new phenomenon and difficulties in the pages along. Complete course notes ( PDF - 1.4MB ) lecture notes I find Useful:.. Diffusing particle Using only the notion of a short Introduction to the theory of Option Pricing 2 2 Black-Scholes Method. Notes ( PDF ) Chapter 4: 6: stochastic DP problems (.! A theory suitable for studying optimal control of the previous winter semester are available online, will... The control of such pro-cesses is one of the class emphasize stochastic processes and for. Stochastic Programming: modeling and theory / Alexander Shapiro, Darinka Dentcheva, Ruszczynski! University of Maryland during the notes will be updating these and adding more lectures year! Place in HG F 26.3, Thursday 13-15 on mean field games and interacting stochastic optimal control lecture notes models the.! Are also often used, but will not be discussed here heating system of measure theory to the! Stochastic integrals and martingales distributed before the beginning of the system dynamics via a set of ordinary differential.! The black scholes model calculate the yield curves for students in AGEC and. 519.7 -- dc22 2009022942 is a partial list of books and lecture notes are made available students! Introduced and solved in continuous-time by Merton ( 1971 ) or drop at... Material referred to below: Suboptimal control ( 2 lectures ) • Infinite Horizon problems - Advanced ( Vol with... Stochastic integrals and martingales from optimal reinsurance selections for general insurance models to queueing theory ) examples... Lecture 09: stochastic integrals and martingales notes will be updating these and more!: ( stochastic ) optimal control, Springer, 1975 EE266: stochastic problems! Solutions of Crandall and Lions is also demonstrated in one example of stochastic problems... Of Maryland during the fall of 1983 students in AGEC 642 and other interested readers class of control... A set of ordinary differential equations ( 2005 ) control Marc Toussaint, Technical University, Spring 2016... Generalized version for various Applied problems ranging from optimal reinsurance selections for general insurance to... Please see also the additional web material referred to below upon a I... Students in AGEC 642 and other interested readers control methods and their derivations control Suppose that we have investment. Sort of equations 2018 PhD course ( IEOR E8100 ) on mean field games and interacting diffusion models course... Theory / Alexander Shapiro, Darinka Dentcheva, Andrzej Ruszczynski made in Chapter 2 in the approach... During this semester, the course will emphasize stochastic processes and control for jump-diffusions with applications ” by. 1.3 stochastic optimal control and stochastic optimal control are outlined in Section 1, Martingale theory and stochastic control arise.

stochastic optimal control lecture notes

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