Two Elementary Exercices on the Stochastic Gradient Option Pricing Problem and Variance Reduction Optimal Control Under Probability Constraint Stochastic Gradient Method: Applications February 03, 2015 P. Carpentier Master MMMEF — Cours MNOS 2014-2015 114 / 267
Two Elementary Exercices on the Stochastic Gradient Option Pricing Problem and Variance Reduction Optimal Control Under Probability Constraint Lecture Outline Two Elementary Exercices on the Stochastic Gradient 1 Two-Stage Recourse Problem Trade-off Between Investment and Operation Option Pricing Problem and Variance Reduction 2 Financial Problem Modeling Computing Efficiently the Price Two Algorithms Optimal Control Under Probability Constraint 3 Satellite Model and Optimization Problem Probability and Conditional Expectation Handling Stochastic Arrow-Hurwicz Algorithm Numerical Results P. Carpentier Master MMMEF — Cours MNOS 2014-2015 115 / 267
Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Optimal Control Under Probability Constraint Two Elementary Exercices on the Stochastic Gradient 1 Two-Stage Recourse Problem Trade-off Between Investment and Operation Option Pricing Problem and Variance Reduction 2 Financial Problem Modeling Computing Efficiently the Price Two Algorithms Optimal Control Under Probability Constraint 3 Satellite Model and Optimization Problem Probability and Conditional Expectation Handling Stochastic Arrow-Hurwicz Algorithm Numerical Results P. Carpentier Master MMMEF — Cours MNOS 2014-2015 116 / 267
Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Optimal Control Under Probability Constraint Two Elementary Exercices on the Stochastic Gradient 1 Two-Stage Recourse Problem Trade-off Between Investment and Operation Option Pricing Problem and Variance Reduction 2 Financial Problem Modeling Computing Efficiently the Price Two Algorithms Optimal Control Under Probability Constraint 3 Satellite Model and Optimization Problem Probability and Conditional Expectation Handling Stochastic Arrow-Hurwicz Algorithm Numerical Results P. Carpentier Master MMMEF — Cours MNOS 2014-2015 117 / 267
Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Optimal Control Under Probability Constraint A Basic Two-Stage Recourse Problem We consider the management of a water reservoir. Water is drawn from the reservoir by way of random consumers. In order to ensure the reservoir supply, 2 decisions are taken at successive time steps. A first supply decision q 1 is taken without any knowledge of the effective consumption, the associated cost being equal to � � 2 , with c 1 > 0. 1 2 c 1 q 1 Once the consumption d has been observed (realization of a r.v. D defined over a probability space (Ω , A , P )), a second supply decision q 2 is taken in order to maintain the reservoir at its initial level, that is, q 2 = d − q 1 , the cost associated to this � � 2 , with c 2 > c 1 > 0. second decision being equal to 1 2 c 2 q 2 The problem is to minimize the expected cost of operation. P. Carpentier Master MMMEF — Cours MNOS 2014-2015 118 / 267
Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Optimal Control Under Probability Constraint Mathematical Formulation and Solution Problem Formulation q 1 is a deterministic decision variable, whereas q 2 is the realization of a random variable Q 2 . � � 2 � � � 2 + 1 � 1 min s.t. q 1 + Q 2 = D . 2 c 1 q 1 2 E c 2 Q 2 ( q 1 , Q 2) Equivalent Problem � � 2 � � � 2 + c 2 � 1 min D − q 1 2 E c 1 q 1 q 1 ∈ R � � c 2 Analytical solution : q ♯ 1 = E D . c 1 + c 2 P. Carpentier Master MMMEF — Cours MNOS 2014-2015 119 / 267
Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Optimal Control Under Probability Constraint Stochastic Gradient Algorithm � − c 2 D ( k +1) � − 1 Q ( k +1) = Q ( k ) ( c 1 + c 2 ) Q ( k ) . 1 1 1 k Algorithm (initialization) Algorithm (iterations) // // // Random generator // Algorithm // // rand(’normal’); rand(’seed’,123); qk = 0.; // for k = 1:100 // Random consumption dk = moy + (ect*rand(1)); // gk = ((c1+c2)*qk) - (c2*dk); moy = 10.; ect = 5.; ek = 1/k; // qk = qk - (ek*gk); // Criterion x = [x ; k]; y = [y ; qk]; // end c1 = 3.; c2 = 1.; // // // Trajectory plot // Initialization // // plot2d(x,y); x = [ ]; y = [ ]; xtitle(’Stochastic Gradient ’,’Iter.’,’Q1’); P. Carpentier Master MMMEF — Cours MNOS 2014-2015 120 / 267
Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Optimal Control Under Probability Constraint A Realization of the Algorithm Stochastic Gradient Algorithm 5.0 4.5 4.0 3.5 3.0 Q1 2.5 2.0 1.5 1.0 0.5 0.0 0 10 20 30 40 50 60 70 80 90 100 Iter. P. Carpentier Master MMMEF — Cours MNOS 2014-2015 121 / 267
Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Optimal Control Under Probability Constraint More Realizations. . . Stochastic Gradient Algorithm 5.0 4.5 4.0 3.5 3.0 Q1 2.5 2.0 1.5 1.0 0.5 0.0 0 50 100 150 200 250 300 350 400 450 500 Iter. P. Carpentier Master MMMEF — Cours MNOS 2014-2015 122 / 267
Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Optimal Control Under Probability Constraint Slight Modification of the problem As in the basic two-stage recourse problem, a first supply decision q 1 is taken without any knowledge of the effective consumption, the associated cost being equal to � � 2 , 1 2 c 1 q 1 a second supply decision q 2 is taken once the consumption d has been observed (realization of a r.v. D ), the cost of this � � 2 . second decision being equal to 1 2 c 2 q 2 The difference between supply and demand is penalized thanks to � � 2 . The new problem is : an additional cost term 1 q 1 + q 2 − d 2 c 3 � � 2 � � � 2 + c 2 � � 2 + c 3 � 1 q 1 + Q 2 − D min 2 E c 1 q 1 Q 2 . ( q 1 , Q 2) Question: how to solve it using a stochastic gradient algorithm? P. Carpentier Master MMMEF — Cours MNOS 2014-2015 123 / 267
Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Optimal Control Under Probability Constraint Two Elementary Exercices on the Stochastic Gradient 1 Two-Stage Recourse Problem Trade-off Between Investment and Operation Option Pricing Problem and Variance Reduction 2 Financial Problem Modeling Computing Efficiently the Price Two Algorithms Optimal Control Under Probability Constraint 3 Satellite Model and Optimization Problem Probability and Conditional Expectation Handling Stochastic Arrow-Hurwicz Algorithm Numerical Results P. Carpentier Master MMMEF — Cours MNOS 2014-2015 124 / 267
Two Elementary Exercices on the Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Optimal Control Under Probability Constraint Trade-off Between Investment and Operation (1) A company owns N production units and has to meet a given (non stochastic) demand d . For each unit i , the decision maker first takes an investment decision u i ∈ R , the associated cost being I i ( u i ). Then a discrete disturbance w i ∈ { w i , a , w i , b , w i , c } occurs. Knowing all noises, the decision maker selects for each unit i an operating point v i ∈ R , which leads to a cost c i ( v i , w i ) and a production e i ( v i , w i ). The goal is to minimize the overall expected cost, subject to the following constraints: investment limitation: Θ( u 1 , . . . , u N ) ≤ 0, operating limitation: v i ≤ ϕ i ( u i ) , i = 1 . . . , N , demand satisfaction: � N i =1 e i ( v i , w i ) − d = 0. P. Carpentier Master MMMEF — Cours MNOS 2014-2015 125 / 267
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