stochastic optimal control with applications in financial engineering

... Trang 2Optimal Control with Engineering ApplicationsTrang 3Optimal Control with Engineering ApplicationsWith 12 Figures 123 Trang 4Professor of Automatic Control and MechatronicsMeasurement and Control ... “Pontryagin’s MinimumPrinciple” [29] Exploiting Pontryagin’s Minimum Principle, several optimalcontrol problems are solved completely Solving an optimal control problem using Pontryagin’s Minimum Principletypically ... optimized) 2.1 Optimal Control Problems with a Fixed Final State In this section, Pontryagin’s Minimum Principle is derived for optimal controlproblems with a fixed final state (and no state constraints)

Ngày tải lên: 05/03/2014, 18:20

... glob- ally minimized with respect to the control u. In the corresponding Nash- Pontryagin Minimax Principle, the Hamiltonian function must simultane- ously be globally minimized with respect ... maximizing control of the resulting optimal control problem of Type C.1. 4.1.3 Proof Proving the theorem proceeds in complete analogy to the proofs of Theorem C in Chapter 2.3.3 and Theorem A in ... This introduction into differential games is very short. Its raison d’ˆetre here lies in the interesting connections between differential games and the H ∞ theory of robust linear control. In most

Ngày tải lên: 12/08/2014, 16:21

... resulting in the control vector U (t) = Unom(t) + u(t) If indeed the errors x(t) and the control corrections can be kept small, the stabilizing controller can be designed by linearizing the nonlinear ... constraints are linear, au-tomatically a linear solution results, i.e., the result will be a linear state feedback controller of the form u(t) = −G(t)x(t) with the optimal time-varying controller ... equality or inequality side-constraints h(.), g(.) switching function for the control and offset function in a singular optimal control problem H(x, u, λ, λ0, t) Hamiltonian function J (x, t) optimal

Ngày tải lên: 12/08/2014, 16:21

... Fig 1.2 Optimal flying maneuver described in body-fixed coordinates. This leads to the following alternative formulation of the optimal control problem: Find a piecewise continuous turning rate ... helpful Instead of minimizing the function f with respect to the independent vari-ables x1, , x n over a constrained set (defined by the functions g i), minimize the augmented function F with ... constraints uniquely de-termine the admissible vector x o Thus, the function f to be minimized is not relevant at all Minimizing f is not the issue in this case! Nev-ertheless, minimizing the

Ngày tải lên: 12/08/2014, 16:21

... “Pontryagin’s Minimum Principle” [29] Exploiting Pontryagin’s Minimum Principle, several optimal control problems are solved completely Solving an optimal control problem using Pontryagin’s Minimum ... 2.1 Optimal Control Problems with a Fixed Final State In this section, Pontryagin’s Minimum Principle is derived for optimal control problems with a fixed final state (and no state constraints) ... 1 Introduction1.4 Exercises 1 In all of the optimal control problems stated in this chapter, the control constraint Ω is required to be a time-invariant set in the control space R m For the control

Ngày tải lên: 12/08/2014, 16:21

... Constrained Final State 43 2.4 Optimal Control Problems with a Partially Constrained Final State In this section, Pontryagin’s Minimum Principle is derived for optimal control problems with a ... 2 Optimal Control 2) If the optimal control u o (t)attimet lies in the interior of the control constraint set Ω, then the factor ∂H/∂u must vanish (and H must have a local minimum). If the optimal ... −at . 2.5 Optimal Control Problems with State Constraints In this section, Pontryagin’s Minimum Principle is derived for optimal con- trol problems of the general form of Type B, but with the

Ngày tải lên: 12/08/2014, 16:21

... superior solution and minimum or maximum for a non-inferior solution In this section, we are only interested in finding a superior solution or infimum of an optimal control problem with a non-scalar-valued ... The following optimal control problem has a globally optimal solution: Find an optimal control u : [t a , t b]→ Ω ⊂ R m, such that the dynamic system ˙ x(t) = f (x(t), u(t)) with the continuously ... up into superiority and non-inferiority [31] The latter is often called Pareto optimality Correspondingly, depending on whether we are “minimizing” or “maximizing”, an extremum is called infimum

Ngày tải lên: 12/08/2014, 16:21

... Hamiltonian to zero in order to find the known result Trang 43 Optimal State Feedback ControlChapter 2 has shown how optimal control problems can be used by exploiting Pontryagin’s Minimum Principle Once ... problem into an antecedent optimal control problem over the time interval [t a , τ ] and a succedent optimal problem over the interval [τ, t b] The antecedent optimal control problem is: Find an ... of the optimal solution of the original problem 2) The optimal solution of the antecedent optimal control problem coincides with the antecedent part of the optimal solution of the original problem

Ngày tải lên: 12/08/2014, 16:21

... following optimal control problem of Type B.1 where there is an additional state constraint x(t 1 )∈ S 1 ⊂R n at the fixed time t 1 within thetimeinterval[t a ,t b ]: Find a piecewise continuous control ... maximize using an optimal state feedback control law. Here, α>0 is a parameter by which we influence the compromise between being rich in the end and consuming a lot in the time interval [0,t ... Optimal State Feedback Control 3.3.3 Controller with a Progressive Characteristic For a linear time-invariant dynamic system of first order, we want to design a time-invariant state feedback control

Ngày tải lên: 12/08/2014, 16:21

... Trang 2Stochastic Optimal Controland the U.S Financial Debt Crisis Trang 4Jerome L SteinStochastic Optimal Control and the U.S Financial Debt Crisis Trang 5Jerome L SteinDivision of Applied ... greatknowledge of financial markets I think that his behavior may be explained ratio-nally First he understands that the function of financial markets is to channel savinginto investment in the optimal ... uncertain interestrate In the second case, insurance is sold in period t and the claims in period t + 1 areuncertain What is the optimal debt in the first case and what are the optimalinsurance

Ngày tải lên: 01/11/2014, 18:49

... stochastic integral equation In a way,either in the deterministic case or in the case of a stochastic environment, aCauchy problem is no more than an integral equation in disguise, since theintegral ... rates), in biology (population dynamics model with the study of risk of extinctionand distribution of the extinction time), in fisheries (with extinction issuesand the study of the fishing policies in ... determines what can be done by the institution with whichyou have the contract in order to avoid having a loss Basically, starting with themoney you have paid for the contract and using it in a

Ngày tải lên: 08/01/2020, 09:35

... begin with, the symbolic variables within the expression must be defined within MATLAB as symbolic variables. This is done using the syms command. In the MATLAB Applications in Chemical Engineering ... Square Diamond Downward pointing triangle Upward pointing triangle Right pointing triangle Left pointing triangle Five point star Six point star ... generally ode45 works as an initial try 6 plot (t, ... they can be easily distinguished using a non-color printer. The appendix contains a list of available markers for plotting. MATLAB Applications in Chemical Engineering James A. Carnell, August

Ngày tải lên: 24/04/2014, 20:45

... transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any infor- mation storage or retrieval system, without prior permission in ... materials engineering He is affili-ated with the Polish Society for Materi-als Engineering, the International Soci-ety for Stereology and the Polish Soci-ety for Stereology.His Ph.D thesis obtained ... imageb) initial image after sharpening filtering c) initial image after advanced sharpening filtering Fig 2.7 Examples of sharpen filters Note that the side effect of sharpening is an increase in

Ngày tải lên: 05/06/2014, 11:49

... optimization problems are given in LΓ-spaces. 1 Introduction In 1979, Robinson1 studied the following parametric variational inclusion problem: given x ∈ R n , find y∈ Rmsuch that 0∈ gx, y ... D a proper convex cone in Z A point y ∈ A is called a vector minimal point of A if for any y ∈ A, y − y /∈ − D \ {0} The set of vector minimal point of A is denoted by Min D A. Lemma 4.1 see ... semicontinuous in short, u.s.c. if for any x ∈ X and any open set V in Y with Tx ⊂ V , there exists a neighborhood U of x such that Tx  ⊂ V for all x ∈ U, iii lower semicontinuous in short,

Ngày tải lên: 21/06/2014, 11:20

... Thejoint density is Trang 6on support2 Now transform to polars by setting X1= R cos and X2= R sin Thenthe joint density of R and is given by−r 2 /2r dr d on support r with 12R2 1and R2, on using ... variable arises in the following manner Suppose Xi∼ N i 1 for i= 1     n, and these random variables are independent Let Y = n i=1Xi2 By obtaining the moment generating function of ... 12R2 1and R2, on using inversion, 12R2= − ln R1 or R=−2 ln R1, and = 2R2 Transforming back to the original Cartesian coordinates gives X1=−2 ln R1cos 2R2  X2=−2 ln R1sin 2R2  The method

Ngày tải lên: 09/08/2014, 16:21

... E YX  is highly nonlinear A control variable removes only the linear part of the variation in Y In contrast, using the stratification variable X as a control variate in the post stratification ... reduction can be implemented using multiple controls with standard regression packages Given Xk Yk  k= 1     N, thecontrol variate estimator is obtained by comparing Equations (5.35) and (5.36) ... reduction in variance, it does not compare well with the estimated variance reduction ratio of 48 given in result (5.30), obtained through fine stratification of X The reason for this lies in the

Ngày tải lên: 09/08/2014, 16:21

... 6.3 Results for basket option, using naive Monte Carlo (basket) and importancesampling with post stratification (basketimppostratv2) b 25 replications, each consisting of 400 paths over 20 equiprobable ... ∗ is determined from Equations (6.40) and (6.41), and Equation (6.42) defines the stratification variable The procedure ‘basketimppoststratv2’ in Appendix 6.7.2 implements this using poststratified ... Approximate 95 % confidence interval for the variance reduction ratio. Since this is the expectation of a function of  Z only, the ideal stratification variablefor the option with price cgis is used,

Ngày tải lên: 09/08/2014, 16:21

... a homogeneous Markov chain withp yx = x y q yx for all x y∈ S, with x = y Note that the conditional probability of remaining in state x at a step in this chain is a mass of probability ... 1Subject to certain regularity conditions q  can take any form (providing the resultingMarkov chain is ergodic), which is a mixed blessing in that it affords great flexibility indesign It follows ... space is explored within a realization A final and positive observation relates to the calculation of x y in Equation (8.6) Since f appears in both the numerator and denominator of the right-hand

Ngày tải lên: 09/08/2014, 16:21

... will sell the existing   0  shares, obtaining   0  X  T  . The total cost of writing and hedging the option in this case is   0  X  0  e rT −  0  X  T   Bringing these two results ... call with strike price K and exercise time T , on an underlying asset having an initial price S 0 which earns interest continuously at rate q with constant volatility s, and where the risk-free interest ...  j  . Let the state of machine j be S  j  = 1 2 3 4 5 according to whether it is working, in the repair queue, being repaired, in the testing queue, or being tested respectively. Let

Ngày tải lên: 09/08/2014, 16:21