Algorithmic Trading: Optimal Control PIMS Summer School
|
|
- Dora Porter
- 5 years ago
- Views:
Transcription
1 Algorihmic Trading: Opimal Conrol PIMS Summer School Sebasian Jaimungal, U. Torono Álvaro Carea,U. Oxford many hanks o José Penalva,(U. Carlos III) Luhui Gan (U. Torono) Ryan Donnelly (Swiss Finance Insiue, EPFL) Damir Kinzebulaov (U. Laval) Jason Ricci (Morgan Sanley) July, 2016 (c) S. Jaimungal, 2016 Algo Trading July, / 21
2 is concerned wih maximizing / minimizing a performance crieria where he crieria is affeced by fuure unknown noise in he sysem, as well as he acions of he conroller / agen (c) S. Jaimungal, 2016 Algo Trading July, / 21
3 We aim o solve he problem [ H(x) = sup E u A G(X x,u T ) }{{} erminal reward T + 0 F (s, Xs x,u, u s ) ds } {{ } running reward/penaly where u = (u ) 0 is he conrol process and he agen chooses i A is he admissible se of conrols e.g., exclude doubling sraegies X u = (X u ) 0 is he conrolled process, which he agen parially conrols, and saisfies he SDE dx u = µ(, X u, u ) d + σ(, X u, u ) dw, X 0 = x (c) S. Jaimungal, 2016 Algo Trading July, / 21
4 Trick is o inroduce a larger class of problems indexed by ime The value funcion is defined as [ T H(, x) = sup E,x G(XT u ) + F (s, Xs u, u s ) ds u A where E,x [ means expecaion condiional on X = x. Prove a dynamic programming principle and he corresponding dynamic programming equaion (c) S. Jaimungal, 2016 Algo Trading July, / 21
5 We will flow an arbirary admissible conrol and re-wrie he value funcion recursively [ E,X u G(XT u)+ T F(s,Xu s,u s )ds {}}{ X u 4 X u Time (c) S. Jaimungal, 2016 Algo Trading July, / 21
6 Take an arbirary admissible conrol u A H u (, x) [ T = E,x G(XT u ) + F (s, Xs u, u s) ds [ T = E,x G(XT u ) + F (s, Xs u, u s) ds + F (s, Xs u, u s) ds [ [ T = E,x E,X u G(XT u ) + F (s, Xs u, us) ds + F (s, Xs u, us) ds (by ieraed expecaion) [ = E,x H u (, X u ) + F (s, Xs u, us) ds (by defn) (c) S. Jaimungal, 2016 Algo Trading July, / 21
7 However, H(, x) H u (, x), wih equaliy if u = u, hence H u (, x) E,x [H(, X u ) + F (s, Xs u, u s) ds sup E,x [H(, X u ) + F (s, Xs u, u s) ds u A and so H(, x) sup E,x [H(, X u ) + F (s, Xs u, u s) ds u A (c) S. Jaimungal, 2016 Algo Trading July, / 21
8 Take an ε-opimal conrol υ ε a conrol ha performs beer han H(, x) ε, bu of course no as good as H(, x), i.e., such ha H(, x) H υε (, x) H(, x) ε Modify he ε-opimal conrol beween and by an arbirary conrol u, i.e., define υ ε by υ ε := u 1 + υ ε 1 > (c) S. Jaimungal, 2016 Algo Trading July, / 21
9 Then, H(, x) H υε (, x) [ = E,x H υε (, X υε ) + F (s, Xs υε, υ s ε ) ds = E,x [H υε (, X u ) + F (s, Xs u, u s) ds E,x [H(, X u ) ε + F (s, Xs u, u s) ds Hence, aking ε 0, and since u is arbirary by ieraed expecaions by he modified sraegy ε-opimal conrol H(, x) sup E,x [H(, X u ) + F (s, Xs u, u s) ds u A (c) S. Jaimungal, 2016 Algo Trading July, / 21
10 Puing boh inequaliies ogeher we arrive a he dynamic programming principle (DPP) H(, x) = sup E,x [H(, X u ) + F (s, Xs u, u s ) ds u A The dynamic programming equaion (DPE) is he infiniesimal version of his principle (c) S. Jaimungal, 2016 Algo Trading July, / 21
11 Take in he DPP o be he following = T inf {s > : (s, X u s x ) / [0, h) [0, ɛ)} X u Time ha is, eiher 1. an amoun of ime h passes, and he processes has deviaed by less han ɛ, or 2. he process deviaes by ɛ and we sop (c) S. Jaimungal, 2016 Algo Trading July, / 21
12 From he DPP H(, x) sup E,x [H(, X u ) + u A F (s, X u s, u s) ds Hence, for a consan sraegy υ on he inerval [, H(, x) E,x [H(, X υ ) + F (s, Xs υ, υ) ds Applying Iô s lemma o he value funcion H(, X υ ) = H(, X ) + where he generaor is L υ + ( + L υ s ) H(s, X υ s ) ds xh(s, X υ s ) σ(s, X υ s, υ) dw s, = µ(, x, υ) x σ2 (, x, υ) xx (c) S. Jaimungal, 2016 Algo Trading July, / 21
13 We hen have, H(, x) E,x [H(, X ) + ( + Ls υ ) H(s, Xs υ ) ds + xh(s, Xs υ ) σ(s, Xs υ, υ) dw s + F (s, X υ s, υ) ds Since X υ x < ɛ on [,, he sochasic inegral is indeed a maringale and we have { } H(, x) E,x [H(, x) + ( + Ls υ ) H(s, Xs υ ) + F (s, Xs υ, υ) ds (c) S. Jaimungal, 2016 Algo Trading July, / 21
14 Then, [ 1 { } 0 lim E,x ( + Ls υ ) H(s, Xs υ ) + F (s, Xs υ, υ) ds h 0 h = ( + L υ ) H(, x) + F (, x, υ) which follows b/c (i) as h 0, = + h a.s. since he process will no hi he barrier of ɛ in exremely shor periods of ime, (ii) he condiion ha X u x ɛ, which implies ha if he process does hi he barrier i is bounded, (iii) he Mean-Value Theorem allows us o wrie ω s ds = ω, and lim h 0 1 h +h (iv) he process sars a X υ = x. (c) S. Jaimungal, 2016 Algo Trading July, / 21
15 This inequaliy holds for every consan υ, and herefore H(, x) + sup {L υ H(, x) + F (, x, υ)} 0 υ (c) S. Jaimungal, 2016 Algo Trading July, / 21
16 Nex, we show he opposie inequaliy, i.e., ha H(, x) + sup {L υ H(, x) + F (, x, υ)} 0 υ We do his by conradicion... assume ( 0, x 0 ) s.., H( 0, x 0 ) + sup {L υ H( 0, x 0 ) + F ( 0, x 0, υ)} < 0 (1) υ Then, define a modificaion ϕ of he value funcion H via ϕ(, x) = H(, x) + ɛ ( ( 0 ) 2 + (x x 0 ) 4) his lies above he value funcion, bu equals i a ( 0, x 0 ), and is differeniable enough o apply Iô s lemma (c) S. Jaimungal, 2016 Algo Trading July, / 21
17 Noe also ha ϕ( 0, x 0) = H( 0, x 0), xϕ( 0, x 0) = xh( 0, x 0), xxϕ( 0, x 0) = xxh( 0, x 0) Therefore, from (1), we have ha ϕ( 0, x 0) + sup {L υ ϕ( 0, x 0) + F ( 0, x 0, υ)} < 0 υ and if he Hamilonian sup υ {L υ H(, x) + F (, x, υ)} is coninuous, hen a neighbourhood N r = ( 0 r, 0 + r) (x 0 r, x 0 + r) s.. ϕ(, x) + sup {L υ ϕ(, x) + F (, x, υ)} < 0 (2) υ for all (, x) N r (c) S. Jaimungal, 2016 Algo Trading July, / 21
18 Define η = max (ϕ H)(, x) > 0 (,x) N r Take an arbirary conrol u A and define he sopping ime Since X is coninuous = inf{s > 0 : X u s / N r } X u N r and herefore ϕ(, X u ) η + H(, X u ) (3) (c) S. Jaimungal, 2016 Algo Trading July, / 21
19 Apply Iô s lemma o ϕ o find ϕ(, X u ) = ϕ( 0, x 0)+ 0 ( +L u )ϕ(s, X u s ) ds+ 0 xϕ(s, X u s ) σ(s, X u s, u) dw s Therefore, V ( 0, x 0) = ϕ( 0, x 0) = E 0,x 0 [ϕ(, X u ) 0 ( + L u )ϕ(s, Xs u ) ds From (2), for (, x) N r ψ(, x) = ϕ(, x) + sup {L υ ϕ(, x) + F (, x, υ)} < 0 υ so ha ϕ(, x) + L u ϕ(, x) + F (, x, u ) ψ(, x) < 0 (c) S. Jaimungal, 2016 Algo Trading July, / 21
20 Therefore, we have V ( 0, x 0) = ϕ( 0, x 0) = E 0,x 0 [ϕ(, X u ) E 0,x 0 [ϕ(, X u ) ( + L u )ϕ(s, Xs u ) ds (F (s, Xs u, u s) ψ(s, Xs u )) ds On boundary N r, ϕ dominaes H, i.e. from (3), we have V ( 0, x 0) E 0,x 0 [η + H(, X u ) + η + E 0,x 0 [H(, X u ) (F (s, Xs u, u s) ψ(s, Xs u )) ds F (s, Xs u, u s) ds > E 0,x 0 [H(, X u ) + F (s, Xs u, u s) ds 0 This violaes he DPP! (since ψ < 0) (c) S. Jaimungal, 2016 Algo Trading July, / 21
21 Hence we obain he Dynamic Programming Equaion (DPE), aka Hamilon-Jacobi-Bellman (HJB) equaion H(, x) + sup {L υ H(, x) + F (, x, υ)} = 0 υ This is a non-linear PDE ha he value funcion mus saisfy I is no clear ha if we solve his PDE, he soluion is he value funcion! This requires a verificaion heorem (c) S. Jaimungal, 2016 Algo Trading July, / 21
Uniqueness of solutions to quadratic BSDEs. BSDEs with convex generators and unbounded terminal conditions
Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs On he uniqueness of soluions o quadraic BSDEs wih convex generaors and unbounded erminal condiions IRMAR, Universié Rennes 1 Châeau de
More informationHamilton Jacobi equations
Hamilon Jacobi equaions Inoducion o PDE The rigorous suff from Evans, mosly. We discuss firs u + H( u = 0, (1 where H(p is convex, and superlinear a infiniy, H(p lim p p = + This by comes by inegraion
More informationAn Introduction to Backward Stochastic Differential Equations (BSDEs) PIMS Summer School 2016 in Mathematical Finance.
1 An Inroducion o Backward Sochasic Differenial Equaions (BSDEs) PIMS Summer School 2016 in Mahemaical Finance June 25, 2016 Chrisoph Frei cfrei@ualbera.ca This inroducion is based on Touzi [14], Bouchard
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationUtility maximization in incomplete markets
Uiliy maximizaion in incomplee markes Marcel Ladkau 27.1.29 Conens 1 Inroducion and general seings 2 1.1 Marke model....................................... 2 1.2 Trading sraegy.....................................
More informationEssential Microeconomics : OPTIMAL CONTROL 1. Consider the following class of optimization problems
Essenial Microeconomics -- 6.5: OPIMAL CONROL Consider he following class of opimizaion problems Max{ U( k, x) + U+ ( k+ ) k+ k F( k, x)}. { x, k+ } = In he language of conrol heory, he vecor k is he vecor
More informationf(s)dw Solution 1. Approximate f by piece-wise constant left-continuous non-random functions f n such that (f(s) f n (s)) 2 ds 0.
Advanced Financial Models Example shee 3 - Michaelmas 217 Michael Tehranchi Problem 1. Le f : [, R be a coninuous (non-random funcion and W a Brownian moion, and le σ 2 = f(s 2 ds and assume σ 2
More informationand Applications Alexander Steinicke University of Graz Vienna Seminar in Mathematical Finance and Probability,
Backward Sochasic Differenial Equaions and Applicaions Alexander Seinicke Universiy of Graz Vienna Seminar in Mahemaical Finance and Probabiliy, 6-20-2017 1 / 31 1 Wha is a BSDE? SDEs - he differenial
More informationOn a Fractional Stochastic Landau-Ginzburg Equation
Applied Mahemaical Sciences, Vol. 4, 1, no. 7, 317-35 On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn
More informationHomogenization of random Hamilton Jacobi Bellman Equations
Probabiliy, Geomery and Inegrable Sysems MSRI Publicaions Volume 55, 28 Homogenizaion of random Hamilon Jacobi Bellman Equaions S. R. SRINIVASA VARADHAN ABSTRACT. We consider nonlinear parabolic equaions
More informationBackward stochastic dynamics on a filtered probability space
Backward sochasic dynamics on a filered probabiliy space Gechun Liang Oxford-Man Insiue, Universiy of Oxford based on join work wih Terry Lyons and Zhongmin Qian Page 1 of 15 gliang@oxford-man.ox.ac.uk
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationStochastic Modelling in Finance - Solutions to sheet 8
Sochasic Modelling in Finance - Soluions o shee 8 8.1 The price of a defaulable asse can be modeled as ds S = µ d + σ dw dn where µ, σ are consans, (W ) is a sandard Brownian moion and (N ) is a one jump
More informationExample on p. 157
Example 2.5.3. Le where BV [, 1] = Example 2.5.3. on p. 157 { g : [, 1] C g() =, g() = g( + ) [, 1), var (g) = sup g( j+1 ) g( j ) he supremum is aken over all he pariions of [, 1] (1) : = < 1 < < n =
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationt dt t SCLP Bellman (1953) CLP (Dantzig, Tyndall, Grinold, Perold, Anstreicher 60's-80's) Anderson (1978) SCLP
Coninuous Linear Programming. Separaed Coninuous Linear Programming Bellman (1953) max c () u() d H () u () + Gsusds (,) () a () u (), < < CLP (Danzig, yndall, Grinold, Perold, Ansreicher 6's-8's) Anderson
More informationDual Representation as Stochastic Differential Games of Backward Stochastic Differential Equations and Dynamic Evaluations
arxiv:mah/0602323v1 [mah.pr] 15 Feb 2006 Dual Represenaion as Sochasic Differenial Games of Backward Sochasic Differenial Equaions and Dynamic Evaluaions Shanjian Tang Absrac In his Noe, assuming ha he
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More informationOptimal Investment Strategy Insurance Company
Opimal Invesmen Sraegy for a Non-Life Insurance Company Łukasz Delong Warsaw School of Economics Insiue of Economerics Division of Probabilisic Mehods Probabiliy space Ω I P F I I I he filraion saisfies
More informationMath Final Exam Solutions
Mah 246 - Final Exam Soluions Friday, July h, 204 () Find explici soluions and give he inerval of definiion o he following iniial value problems (a) ( + 2 )y + 2y = e, y(0) = 0 Soluion: In normal form,
More informationSingular control of SPDEs and backward stochastic partial diffe. reflection
Singular conrol of SPDEs and backward sochasic parial differenial equaions wih reflecion Universiy of Mancheser Join work wih Bern Øksendal and Agnès Sulem Singular conrol of SPDEs and backward sochasic
More informationExistence and uniqueness of solution for multidimensional BSDE with local conditions on the coefficient
1/34 Exisence and uniqueness of soluion for mulidimensional BSDE wih local condiions on he coefficien EL HASSAN ESSAKY Cadi Ayyad Universiy Mulidisciplinary Faculy Safi, Morocco ITN Roscof, March 18-23,
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationGeneralized Snell envelope and BSDE With Two general Reflecting Barriers
1/22 Generalized Snell envelope and BSDE Wih Two general Reflecing Barriers EL HASSAN ESSAKY Cadi ayyad Universiy Poly-disciplinary Faculy Safi Work in progress wih : M. Hassani and Y. Ouknine Iasi, July
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationCouplage du principe des grandes déviations et de l homogénisation dans le cas des EDP paraboliques: (le cas constant)
Couplage du principe des grandes déviaions e de l homogénisaion dans le cas des EDP paraboliques: (le cas consan) Alioune COULIBALY U.F.R Sciences e Technologie Universié Assane SECK de Ziguinchor Probabilié
More informationOptimal Investment, Consumption and Retirement Decision with Disutility and Borrowing Constraints
Opimal Invesmen, Consumpion and Reiremen Decision wih Disuiliy and Borrowing Consrains Yong Hyun Shin Join Work wih Byung Hwa Lim(KAIST) June 29 July 3, 29 Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis
More informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
More informationTHE BELLMAN PRINCIPLE OF OPTIMALITY
THE BELLMAN PRINCIPLE OF OPTIMALITY IOANID ROSU As I undersand, here are wo approaches o dynamic opimizaion: he Ponrjagin Hamilonian) approach, and he Bellman approach. I saw several clear discussions
More informationCash Flow Valuation Mode Lin Discrete Time
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 35-41 Cash Flow Valuaion Mode Lin Discree Time Olayiwola. M. A. and Oni, N. O. Deparmen of Mahemaics
More informationt 2 B F x,t n dsdt t u x,t dxdt
Evoluion Equaions For 0, fixed, le U U0, where U denoes a bounded open se in R n.suppose ha U is filled wih a maerial in which a conaminan is being ranspored by various means including diffusion and convecion.
More information1 Solutions to selected problems
1 Soluions o seleced problems 1. Le A B R n. Show ha in A in B bu in general bd A bd B. Soluion. Le x in A. Then here is ɛ > 0 such ha B ɛ (x) A B. This shows x in B. If A = [0, 1] and B = [0, 2], hen
More informationLECTURE 1: GENERALIZED RAY KNIGHT THEOREM FOR FINITE MARKOV CHAINS
LECTURE : GENERALIZED RAY KNIGHT THEOREM FOR FINITE MARKOV CHAINS We will work wih a coninuous ime reversible Markov chain X on a finie conneced sae space, wih generaor Lf(x = y q x,yf(y. (Recall ha q
More informationAn introduction to the theory of SDDP algorithm
An inroducion o he heory of SDDP algorihm V. Leclère (ENPC) Augus 1, 2014 V. Leclère Inroducion o SDDP Augus 1, 2014 1 / 21 Inroducion Large scale sochasic problem are hard o solve. Two ways of aacking
More informationBackward doubly stochastic di erential equations with quadratic growth and applications to quasilinear SPDEs
Backward doubly sochasic di erenial equaions wih quadraic growh and applicaions o quasilinear SPDEs Badreddine MANSOURI (wih K. Bahlali & B. Mezerdi) Universiy of Biskra Algeria La Londe 14 sepember 2007
More informationPROOF FOR A CASE WHERE DISCOUNTING ADVANCES THE DOOMSDAY. T. C. Koopmans
PROOF FOR A CASE WHERE DISCOUNTING ADVANCES THE DOOMSDAY T. C. Koopmans January 1974 WP-74-6 Working Papers are no inended for disribuion ouside of IIASA, and are solely for discussion and informaion purposes.
More informationTHE WAVE EQUATION. part hand-in for week 9 b. Any dilation v(x, t) = u(λx, λt) of u(x, t) is also a solution (where λ is constant).
THE WAVE EQUATION 43. (S) Le u(x, ) be a soluion of he wave equaion u u xx = 0. Show ha Q43(a) (c) is a. Any ranslaion v(x, ) = u(x + x 0, + 0 ) of u(x, ) is also a soluion (where x 0, 0 are consans).
More informationANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 6 SECTION 6.1: LIFE CYCLE CONSUMPTION AND WEALTH T 1. . Let ct. ) is a strictly concave function of c
John Riley December 00 S O EVEN NUMBERED EXERCISES IN CHAPER 6 SECION 6: LIFE CYCLE CONSUMPION AND WEALH Eercise 6-: Opimal saving wih more han one commodiy A consumer has a period uiliy funcion δ u (
More informationENGI 9420 Engineering Analysis Assignment 2 Solutions
ENGI 940 Engineering Analysis Assignmen Soluions 0 Fall [Second order ODEs, Laplace ransforms; Secions.0-.09]. Use Laplace ransforms o solve he iniial value problem [0] dy y, y( 0) 4 d + [This was Quesion
More informationarxiv: v1 [math.pr] 21 May 2010
ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES, AND PASSAGE TIME PROBLEMS arxiv:15.498v1 [mah.pr 21 May 21 GERARDO HERNÁNDEZ-DEL-VALLE Absrac. In his work we relae he densiy of he firs-passage
More informationSTABILITY OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS
Elecronic Journal of Differenial Equaions, Vol. 217 217, No. 118, pp. 1 14. ISSN: 172-6691. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu STABILITY OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS
More informationThe Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing When the Horizon Is Infinite
American Journal of Operaions Research, 08, 8, 8-9 hp://wwwscirporg/journal/ajor ISSN Online: 60-8849 ISSN Prin: 60-8830 The Opimal Sopping Time for Selling an Asse When I Is Uncerain Wheher he Price Process
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationarxiv:math/ v1 [math.nt] 3 Nov 2005
arxiv:mah/0511092v1 [mah.nt] 3 Nov 2005 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION D. A. GOLDSTON AND S. M. GONEK Absrac. Le πs denoe he argumen of he Riemann zea-funcion a he poin 1 + i. Assuming
More informationVanishing Viscosity Method. There are another instructive and perhaps more natural discontinuous solutions of the conservation law
Vanishing Viscosiy Mehod. There are anoher insrucive and perhaps more naural disconinuous soluions of he conservaion law (1 u +(q(u x 0, he so called vanishing viscosiy mehod. This mehod consiss in viewing
More informationOptimal Consumption and Investment Portfolio in Jump markets. Optimal Consumption and Portfolio of Investment in a Financial Market with Jumps
Opimal Consumpion and Invesmen Porfolio in Jump markes Opimal Consumpion and Porfolio of Invesmen in a Financial Marke wih Jumps Gan Jin Lingnan (Universiy) College, China Insiue of Economic ransformaion
More informationSolutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore
Soluions of Sample Problems for Third In-Class Exam Mah 6, Spring, Professor David Levermore Compue he Laplace ransform of f e from is definiion Soluion The definiion of he Laplace ransform gives L[f]s
More informationFINM 6900 Finance Theory
FINM 6900 Finance Theory Universiy of Queensland Lecure Noe 4 The Lucas Model 1. Inroducion In his lecure we consider a simple endowmen economy in which an unspecified number of raional invesors rade asses
More informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
More informationExpert Advice for Amateurs
Exper Advice for Amaeurs Ernes K. Lai Online Appendix - Exisence of Equilibria The analysis in his secion is performed under more general payoff funcions. Wihou aking an explici form, he payoffs of he
More information(MS, ) Problem 1
MS, 7.6.4) AKTUAREKSAMEN KONTROL I FINANSIERING OG LIVSFORSIKRING ved Københavns Universie Sommer 24 Skriflig prøve den 4. juni 24 kl..-4.. All wrien aids are allowed. The wo problems of oally 3 quesions
More informationExistence Theory of Second Order Random Differential Equations
Global Journal of Mahemaical Sciences: Theory and Pracical. ISSN 974-32 Volume 4, Number 3 (22), pp. 33-3 Inernaional Research Publicaion House hp://www.irphouse.com Exisence Theory of Second Order Random
More informationExamples of Dynamic Programming Problems
M.I.T. 5.450-Fall 00 Sloan School of Managemen Professor Leonid Kogan Examples of Dynamic Programming Problems Problem A given quaniy X of a single resource is o be allocaed opimally among N producion
More information6. Stochastic calculus with jump processes
A) Trading sraegies (1/3) Marke wih d asses S = (S 1,, S d ) A rading sraegy can be modelled wih a vecor φ describing he quaniies invesed in each asse a each insan : φ = (φ 1,, φ d ) The value a of a porfolio
More informationAnn. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:
Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS
More informationResearch Article Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations
Hindawi Publishing Corporaion Boundary Value Problems Volume 11, Aricle ID 19156, 11 pages doi:1.1155/11/19156 Research Aricle Exisence and Uniqueness of Periodic Soluion for Nonlinear Second-Order Ordinary
More informationDISCRETE GRONWALL LEMMA AND APPLICATIONS
DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE MAA NORTH CENTRAL SECTION MEETING AT UND 24 OCTOBER 29 Gronwall s lemma saes an inequaliy ha is useful in he heory of differenial equaions. Here is
More informationMath 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11.
1 Mah 334 Tes 1 KEY Spring 21 Secion: 1 Insrucor: Sco Glasgow Daes: Ma 1 and 11. Do NOT wrie on his problem saemen bookle, excep for our indicaion of following he honor code jus below. No credi will be
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More information556: MATHEMATICAL STATISTICS I
556: MATHEMATICAL STATISTICS I INEQUALITIES 5.1 Concenraion and Tail Probabiliy Inequaliies Lemma (CHEBYCHEV S LEMMA) c > 0, If X is a random variable, hen for non-negaive funcion h, and P X [h(x) c] E
More informationApproximation Algorithms for Unique Games via Orthogonal Separators
Approximaion Algorihms for Unique Games via Orhogonal Separaors Lecure noes by Konsanin Makarychev. Lecure noes are based on he papers [CMM06a, CMM06b, LM4]. Unique Games In hese lecure noes, we define
More informationMODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE
Topics MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES 2-6 3. FUNCTION OF A RANDOM VARIABLE 3.2 PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE 3.3 EXPECTATION AND MOMENTS
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE BACKWARD STOCHASTIC LORENZ SYSTEM
Communicaions on Sochasic Analysis Vol. 1, No. 3 (27) 473-483 EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE BACKWARD STOCHASTIC LORENZ SYSTEM P. SUNDAR AND HONG YIN Absrac. The backward sochasic Lorenz
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More information4 Sequences of measurable functions
4 Sequences of measurable funcions 1. Le (Ω, A, µ) be a measure space (complee, afer a possible applicaion of he compleion heorem). In his chaper we invesigae relaions beween various (nonequivalen) convergences
More informationOn Oscillation of a Generalized Logistic Equation with Several Delays
Journal of Mahemaical Analysis and Applicaions 253, 389 45 (21) doi:1.16/jmaa.2.714, available online a hp://www.idealibrary.com on On Oscillaion of a Generalized Logisic Equaion wih Several Delays Leonid
More informationThis document was generated at 1:04 PM, 09/10/13 Copyright 2013 Richard T. Woodward. 4. End points and transversality conditions AGEC
his documen was generaed a 1:4 PM, 9/1/13 Copyrigh 213 Richard. Woodward 4. End poins and ransversaliy condiions AGEC 637-213 F z d Recall from Lecure 3 ha a ypical opimal conrol problem is o maimize (,,
More information7 The Itô/Stratonovich dilemma
7 The Iô/Sraonovich dilemma The dilemma: wha does he idealizaion of dela-funcion-correlaed noise mean? ẋ = f(x) + g(x)η() η()η( ) = κδ( ). (1) Previously, we argued by a limiing procedure: aking noise
More informationMATH 5720: Gradient Methods Hung Phan, UMass Lowell October 4, 2018
MATH 5720: Gradien Mehods Hung Phan, UMass Lowell Ocober 4, 208 Descen Direcion Mehods Consider he problem min { f(x) x R n}. The general descen direcions mehod is x k+ = x k + k d k where x k is he curren
More informationConvergence of the Neumann series in higher norms
Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More informationOptima and Equilibria for Traffic Flow on a Network
Opima and Equilibria for Traffic Flow on a Nework Albero Bressan Deparmen of Mahemaics, Penn Sae Universiy bressan@mah.psu.edu Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 1 / 1 A Traffic
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More informationMA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions
MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by
More informationLocal Strict Comparison Theorem and Converse Comparison Theorems for Reflected Backward Stochastic Differential Equations
arxiv:mah/07002v [mah.pr] 3 Dec 2006 Local Sric Comparison Theorem and Converse Comparison Theorems for Refleced Backward Sochasic Differenial Equaions Juan Li and Shanjian Tang Absrac A local sric comparison
More informationTime discretization of quadratic and superquadratic Markovian BSDEs with unbounded terminal conditions
Time discreizaion of quadraic and superquadraic Markovian BSDEs wih unbounded erminal condiions Adrien Richou Universié Bordeaux 1, INRIA équipe ALEA Oxford framework Le (Ω, F, P) be a probabiliy space,
More informationComments on Window-Constrained Scheduling
Commens on Window-Consrained Scheduling Richard Wes Member, IEEE and Yuing Zhang Absrac This shor repor clarifies he behavior of DWCS wih respec o Theorem 3 in our previously published paper [1], and describes
More informationLaplace Transform. Inverse Laplace Transform. e st f(t)dt. (2)
Laplace Tranform Maoud Malek The Laplace ranform i an inegral ranform named in honor of mahemaician and aronomer Pierre-Simon Laplace, who ued he ranform in hi work on probabiliy heory. I i a powerful
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationSupplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence
Supplemen for Sochasic Convex Opimizaion: Faser Local Growh Implies Faser Global Convergence Yi Xu Qihang Lin ianbao Yang Proof of heorem heorem Suppose Assumpion holds and F (w) obeys he LGC (6) Given
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationConvex Analysis for LQG Systems with Applications to Major Minor LQG Mean Field Game Systems
arxiv:181.7551v1 [cs.sy 16 Oc 218 Convex Analysis for LQG Sysems wih Applicaions o Major Minor LQG Mean Field Game Sysems 1 Absrac Dena Firoozi, Sebasian Jaimungal, Peer E. Caines LQG mean field game sysems
More informationOnline Appendix to Optimal Regulation of Financial Intermediaries
Online Appendix o Opimal Regulaion of Financial Inermediaries Sebasian Di Tella Sanford GSB June 217 Absrac In Secion 1 I develop he conrac environmen in deail. I provide a verificaion heorem for he HJB
More informationProperties Of Solutions To A Generalized Liénard Equation With Forcing Term
Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick
More informationA NOTE ON S(t) AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION
Bull. London Mah. Soc. 39 2007 482 486 C 2007 London Mahemaical Sociey doi:10.1112/blms/bdm032 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION D. A. GOLDSTON and S. M. GONEK Absrac Le πs denoe he
More informationHeat kernel and Harnack inequality on Riemannian manifolds
Hea kernel and Harnack inequaliy on Riemannian manifolds Alexander Grigor yan UHK 11/02/2014 onens 1 Laplace operaor and hea kernel 1 2 Uniform Faber-Krahn inequaliy 3 3 Gaussian upper bounds 4 4 ean-value
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Noes for EE7C Spring 018: Convex Opimizaion and Approximaion Insrucor: Moriz Hard Email: hard+ee7c@berkeley.edu Graduae Insrucor: Max Simchowiz Email: msimchow+ee7c@berkeley.edu Ocober 15, 018 3
More informationA proof of Ito's formula using a di Title formula. Author(s) Fujita, Takahiko; Kawanishi, Yasuhi. Studia scientiarum mathematicarum H Citation
A proof of Io's formula using a di Tile formula Auhor(s) Fujia, Takahiko; Kawanishi, Yasuhi Sudia scieniarum mahemaicarum H Ciaion 15-134 Issue 8-3 Dae Type Journal Aricle Tex Version auhor URL hp://hdl.handle.ne/186/15878
More informationSOLUTIONS TO ASSIGNMENT 2 - MATH 355. with c > 3. m(n c ) < δ. f(t) t. g(x)dx =
SOLUTIONS TO ASSIGNMENT 2 - MATH 355 Problem. ecall ha, B n {ω [, ] : S n (ω) > nɛ n }, and S n (ω) N {ω [, ] : lim }, n n m(b n ) 3 n 2 ɛ 4. We wan o show ha m(n c ). Le δ >. We can pick ɛ 4 n c n wih
More informationRepresentation of Stochastic Process by Means of Stochastic Integrals
Inernaional Journal of Mahemaics Research. ISSN 0976-5840 Volume 5, Number 4 (2013), pp. 385-397 Inernaional Research Publicaion House hp://www.irphouse.com Represenaion of Sochasic Process by Means of
More informationu(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x
. 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih
More information4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
More informationCorrespondence should be addressed to Nguyen Buong,
Hindawi Publishing Corporaion Fixed Poin Theory and Applicaions Volume 011, Aricle ID 76859, 10 pages doi:101155/011/76859 Research Aricle An Implici Ieraion Mehod for Variaional Inequaliies over he Se
More informationarxiv: v1 [math.pr] 28 Nov 2016
Backward Sochasic Differenial Equaions wih Nonmarkovian Singular Terminal Values Ali Devin Sezer, Thomas Kruse, Alexandre Popier Ocober 15, 2018 arxiv:1611.09022v1 mah.pr 28 Nov 2016 Absrac We solve a
More informationTO our knowledge, most exciting results on the existence
IAENG Inernaional Journal of Applied Mahemaics, 42:, IJAM_42 2 Exisence and Uniqueness of a Periodic Soluion for hird-order Delay Differenial Equaion wih wo Deviaing Argumens A. M. A. Abou-El-Ela, A. I.
More informationLaplace Transforms. Examples. Is this equation differential? y 2 2y + 1 = 0, y 2 2y + 1 = 0, (y ) 2 2y + 1 = cos x,
Laplace Transforms Definiion. An ordinary differenial equaion is an equaion ha conains one or several derivaives of an unknown funcion which we call y and which we wan o deermine from he equaion. The equaion
More informationSolution of Integro-Differential Equations by Using ELzaki Transform
Global Journal of Mahemaical Sciences: Theory and Pracical. Volume, Number (), pp. - Inernaional Research Publicaion House hp://www.irphouse.com Soluion of Inegro-Differenial Equaions by Using ELzaki Transform
More information