Optimal Control Under Resource and Price Uncertainty

Floyd B. Hanson

Laboratory for Advanced Computing Department of Math., Stat., and Computer Science University of Illinois at Chicago E-Mail: hanson@uic.edu Web-Page: http://www.math.uic.edu/ hanson/

and

Dennis Ryan

Division of Science and Mathematics McKendree College Lebanon, IL 62254, USA Email: dryan@a1.mckendree.edu

1This work was supported by the National Science Foundation Computational Mathematics Program under grants DMS-93-01107 and DMS-96-26692.

\underline{Overview}

1. Noninflationary, Deterministic Model.

2. Inflationary, Stochastic Control Model.

3. Numerical Approximations.

4. Numerical Results.

5. Conclusions.

Outline of Abstract

• Optimal Control of Stochastic Resource in Continuous Time.

• Model Effects of Large Random Price Fluctuations.

• Influence of Continuous growth and Jump Stochastic Noise.

• Computational Stochastic Dynamic Programming.

• Pronounced Effect of Inflationary Prices on Optimal Return.

Part 1. Noninflationary, Deterministic Model: Introduction.

1.1. Ordinary Differential Equation (ODE):

• Nonlinear (Logistic) Dynamics:

dX(s) = [r_{1}X(s)(1 − X(s)/K) − H(s)] ds,
0 < t < s < T.

• Initial Conditions: X(0) = x_{0} ; 0 < t < T

• State Variable (Resource Size): X(t) = [X_{i}(t)]_{1×1} ;

• BiLinear Control-State Dynamics Assumption

(for Resource Harvesting):

H(t) = qU(t)X(t) ;

where q = Efficiency (Catchability) Coefficient;

• Control Variable (Harvesting Effort):

U(t) = [U_{i}{X(t), t}]_{1×1} , U_{min} ≤ U(t) ≤ U_{max} < ∞ ;

• Growth Parameters:

r_{1} = Resource Intrinsic Growth Rate;

K = Environment Carrying (Saturation) Capacity.

1.2. Quadratic Performance Index:

V (X, U, t) = ∫_{t}^{T} e^{−δ(s−t)} [pqU(s)X(s) − c(U(s))] ds , where

• V (x, u, t) = Current Value of Future Resources (i.e., exp(δt) times Present Value);

• T = Time Horizon (T ≥ t);

• δ = Nominal Discount Rate (NOT adjusted for inflation);

• p = Price of Resource per Unit Harvest Rate;

• c(u) = c_{1}u + c_{2}u^{2} = Quadratic Costs (Assume Increasing, Convex Quadratic Costs: c_{1} > 0 and c_{2} > 0);

• Instantaneous Net Return:

R(x, u) = pqux − c(u) .

1.3. Deterministic Dynamic Programming:

• Optimization Goal = Maximize Total Return: v∗(x, t) = V (x, u∗, t) = max_{u} [V (x, u, t)] ;

• PDE of Deterministic Dynamic Programming: v∗_{t}(x, t) + r_{1}x(1 − x/K)v∗_{x}(x, t) − δv∗(x, t) + S∗(x, t) = 0;

• Control Switching Term:

S∗(x, t) = max_{u} [(p − v∗_{x}(x, t)) qux − c_{1}u − c_{2}u^{2}];

• Regular (Unconstrained) Control:

u_{R}(x, t) = \frac{(p − v∗_{x}(x, t))qx − c_{1}}{2 · c_{2}} , c_{2} > 0;

• Optimal (Constrained) Control:

u∗(x, t) = \{ \begin{matrix}U_{max}, & U_{max} ≤ u_{R}(x, t) \\ u_{R}(x, t), & U_{min} ≤ u_{R}(x, t) ≤ U_{max} \\ U_{min}, & u_{R}(x, t) ≤ U_{min} \end{matrix} ;

• Final Boundary Condition: v∗(x, T) = 0;

• Extinction Natural Boundary Condition:

v∗(0, t) = −\frac{(c_{1} + c_{2}U_{min})U_{min}}{δ} (1 − e^{−δ(T −t)}) , δ > 0.

Part 2. Inflationary, Stochastic Control Model

2.1. Stochastic Dynamics Equation (SDE (1)):

• Nonlinear Dynamics with Gaussian (G)

and Poisson (Z) Noise:

dX(s) = [r_{1}X(s)(1 − X(s)/K) − H(s)] ds
+ σ_{1}X(s) dW_{1}(s) + X(s) n ; ∑ ; j=1 a_{j} dZ_{j}(s, f_{j}) ,
X(t) = x ,

• Initial Conditions: X(0) = x_{0} , t_{0} < t < s < T ;

• Gaussian (Wiener) Noise

(Zero Mean and Normalized):

E[dW_{1}(t)] = 0 , V ar[dW_{1}(t)] = dt σ_{1} ≤ 0 ;

• Poisson (Jump) Noise:

E[dZ_{j}(t, f_{j})] = f_{j}dt , V ar[dZ_{j}(t, f_{j})] = f_{j}dt , 1 ≤ j ≤ n , where f_{j} = Jump Rate and a_{j} = Jump Amplitude Coefficient (−1 < a_{j});

• Independent (Uncorrelated) Processes Assumption: CoVar[dW_{1}(t), dZ_{j}(t, f_{j})] = 0 ,

CoVar[dZ_{j}(t, f_{j}), dZ_{j′}(t, f_{j′})] = δ_{j,j′}f_{j}dt ;

2.2. Inflationary Factor Model:

• Nonlinear Supply–Demand Model Relation:

P(t) = (\frac{p_{0}}{H}(t) + p_{1}) Y(t),

where

* P(t) · H(t) = Gross Return on Harvest;

* p_{0} = Supply–Demand Price Coefficient;

* p_{1} = Constant Price per Unit Harvest;

* Y(t) = Fluctuating Inflationary Factor;

• Linear Fluctuating Inflationary Factor SDE (2):

dY(s) = r_{2}Y(s) ds + σ_{2}Y(s) dW_{2}(s) + Y(s) m ; ∑ ; j=1 b_{j} dQ_{j}(s; g_{j}) ,

* Y(t) = y;

* r_{2} = Annual Rate of Inflation without Fluctuations;

* g_{j} = jth component of Inflationary Jump Rate;

* b_{j} = jth component of Jump Amplitude Coefficient;

• Inflationary Gaussian (Wiener) Noise:

E[dW_{2}(t)] = 0 , V ar[dW_{2}(t)] = dt σ_{2} ≤ 0 ;

• Inflationary Poisson (Jump) Noise:

E[dQ_{j}(t, g_{j})] = g_{j}dt ,  V ar[dQ_{j}(t, g_{j})] = g_{j}dt ,  1 ≤ j ≤ m ;

• Independent (Uncorrelated) Processes Assumption:

CoVar[dW_{2}(t), dQ_{j}(t, g_{j})] = 0 , CoVar[dQ_{j}(t, g_{j}), dQ_{j′}(t, g_{j′})] = δ_{j,j′}g_{j}dt ;