1 / 23

MA409: Continous-time Optimization

MA409: Continous-time Optimization. Adam Ostaszewski. Lecture times for 2011/12 . Lectures: Monday in TW1.U208 : 4-5 pm and Thursday in NAB.1.14 : 12-1pm Classes : Wednesday in TW1.U101: 6-7pm … starting Week 1 all in Michaelmas Term Plus: Revision in Summer Term!.

cadence
Télécharger la présentation

MA409: Continous-time Optimization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MA409: Continous-time Optimization Adam Ostaszewski

  2. Lecture times for 2011/12 • Lectures: Monday in TW1.U208 : 4-5 pm and Thursday in NAB.1.14 : 12-1pm • Classes: Wednesday in TW1.U101: 6-7pm … starting Week 1 all in Michaelmas Term • Plus: Revision in Summer Term!

  3. Q: What is this course about?A: How to … • Select a curve x(t) for 0 < t < T • … subject to constraints holding over some or over all time t • To minimize (maximize) some performance index

  4. About the course…geodesics &other performance index • E.g. … subject to x(0) being given, and also x(T) • …or x(t) must lie on some fixed surface • E.g. is a geodesic = minimizes distance along the given fixed surface between initial and terminal positions

  5. About the course … time optimality • E.g. … subject to x(0) being given and maybe x(T) only to lie on a specified surface • …or x(t) must lie on some varying surface • E.g. minimize time taken to reach the prescribed surface: T

  6. Notes: • The variable x(t) may be in Rn • The constraint may be f(x(t)) = 0 • … but could be a differential equation like this ‘canonical’ one: x′(t)=a(t,x(t)) • Where dash is defined by: x′=dx/dt • …so dx = a(t,x(t)) dt

  7. State equation in differential form (first order) • … Or even like x′(t)=a(t,x(t),u(t)) • here u(t) is to be selected and is known as a control (costly, usually!) • (Think: selecting car speed with car constrained to a particular road.)

  8. A state equation in vector differential form – first order • …indeed … despite Newton’s law … • x′′ = u (accn = force) • Rewritten, with x1=x, and with x2=x1′, • …as the linear matrix first-order system x1′ = x2, x2′ = u

  9. …or in stochastic form • … could be a ‘stochastic’ differential equation • dx(t)=a(t,x(t),u(t)) dt + b(t,x(t),u(t)) dWt • ..where for each t the term Wt represents a continuously varying standard `random variable’, modelling uncertainty

  10. … risky assets • Model of asset price evolution St ‘Instantaneous’ return on the asset over the time interval dt (from t to t+dt ) is defined to be: • (St+dt - St)/St , or in compact form = dSt/ St

  11. … Black-Scholes model for asset price St • Return on the asset modelled by dSt / St =  dt +  dWt • …anticipated ‘rate’ of return  • …PLUS ‘volatility factor’  times a standard volatility term, namely dWt … • nb dWt = (Wt+dt - Wt) the increments are independent, normal, with variance dt

  12. Format of Performance Index:Deterministic Case • Bequest/gift: g(x(T)) – a Meyer problem • Running cost format: 0T L(x(t),…)dt – a Lagrange problem • Mix of above: g(x(T)) + 0T L(x(t),…) dt – a Bolza problem

  13. …Actually they’re equivalent • …proved by fiddling with the number,n, of variables, e.g. introduce xn+1 and adjoin • … a diff. eqn.: dxn+1 = L(x(t),…)dt • … initial condition: xn+1(0) = 0 • Then g(x(T))= xn+1 (T) = 0T L(x(t),…)dt

  14. .. Stochastic caseExpected performance • Thus might be • E[g(x(T))] – a Meyer problem … • So we can solve the investor’s optimal consumption problem

  15. Expected performance • At what rate to consume wealth, at what rate to invest in a risky asset and how much to salt away in a safe bank deposit • Max Ew[0T exp(-rt)U(u(t))]dt • maximize expected utility of lifetime consumption, where u(t) =fraction of wealth x(t) consumed and x(0)=w.

  16. Note the implicit ‘stopping time’ • …in the upper limit of integration • Max Ew[0T exp(-rt)U(u(t))] dt • T denotes the first time t when x(t) = 0, so T may be a special sort of random variable

  17. Solution technique • Optimality as with real-variables where • Max f(x) • is solved using the differential condition: df(x) = 0, the ‘first order condition’ (f.o.c.) • We develop a higher level calculus, which converts df(x) = 0 to …a differential eqn.

  18. …f.o.c. leads to a …differential equation …satisfied by the optimal trajectory (curve) • And so the technique amounts to setting up the problem (modelling) and then solving the d.e.

  19. Reading List • To begin with we use A.O.’s “Advanced Mathematical Methods” • Notes will be provided throughout the course by way of Moodle (& the maths web-site)

  20. Literature – Mathematical Basics • (♥) Troutman, Variational Calculus, Springer • (♥) Pinch, Optimal Control and the Calculus of Variations, OUP • Fleming & Rishel, Deterministic & Stochastic Control, Springer • (♥♥) Burghes & Graham, Control & Optimal Control Theories with Applications, Horwood.

  21. Mathematical Literature - Advanced • Fleming & Soner, Controlled Markov Processes and Viscosity Solutions, Springer • Oksendal & Sulem, Applied Stochastic Control of Jump Diffusions,Springer • Vinter, Optimal Control, Birkhauser

  22. Literature – Very Advanced • Rigorous Mathematical Background …well beyond our course: • Rogers & Williams, Diffusions, Markov Processes and Martingales: Volumes 1& 2, Itô Calculus, (Cambridge Mathematical Library) • But you should know it’s there

  23. P.S. My secret place: http://www.maths.lse.ac.uk/Courses/MA409/ This also yields access to my course materials

More Related