Bjarne S. Jensen & Tapio Palokangas (Editors)
Stochastic
Stochastic Economic Dynamics
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Contents in Brief
Introduction 1
Part I: Developments in Stochastic Dynamics
1. Fractional Brownian Motion in Finance 11
2. Moment Evolution of Gaussian and Geometric
Wiener Diffusions 57
3. Two-Dimensional Linear Dynamic Systems with
Small Random Terms 101
4. Dynamic Theory of Stochastic Movement of Systems 133
Part II: Stochastic Dynamics in Basic Growth Models and Time Delays
5. Stochastic One-Sector and Two-Sector
Growth Models in Continuous Time 167
6. Comparative Dynamics in a Stochastic Growth
and Trade Model with a Variable Savings Rate 217
7. Inada Conditions and Global Dynamic Analysis of
Basic Growth Models with Time Delays 229
8. Hopf Bifurcation in Growth Models with Time Delays 247
Part III: Intertemporal Optimization in Consumption, Finance, and Growth
9. Optimal Consumption and Investment Strategies
in Dynamic Stochastic Economies 271
10. Differential Systems in Finance and Life Insurance 317 11. Uncertain Technological Change and Capital Mobility 361 12. Stochastic Control, Non-Depletion of Renewable
Resources, and Intertemporal Substitution 381
13. Capital Accumulation in a Growth Model with
Creative Destruction 393
14. Employment Cycles in a Growth Model with
Table of Contents
Introduction 1
Bjarne S. Jensen and Tapio Palokangas
Part I: Developments in Stochastic Dynamics
1. Fractional Brownian Motion in Finance 11
Bernt Øksendal
1.1 Introduction 11
1.2 Framework and definitions 12
1.3 Classical white noise theory and
Hida-Malliavin calculus 16
1.4 Fractional stochastic calculus 30
1.5 Summary of results 40
1.6 Concluding remarks 53
2. Moment Evolution of Gaussian and Geometric
Wiener Diffusions 57
Bjarne S. Jensen, Chunyan Wang, and Jon Johnsen
2.1 Introduction 57
2.2 Structure of basic diffusion processes 59
2.3 Dynamics of first-order and second-order moments 64
2.4 Expectation vector functions 68
2.5 Covariance matrix functions 71
2.6 Probability density functions 84
2.7 Final comments 92
Appendices 92
3. Two-Dimensional Linear Dynamic Systems
with Small Random Terms 101
Nishioka Kunio
3.1 Introduction 101
3.2 Non-random dynamic system 102
3.3 Lyapunov index of the random system 106
3.4 One-dimensional diffusion process in an interval 109
3.6 Saddle point 117
3.7 Improper and proper node 127
4. Dynamic Theory of Stochastic Movement of Systems 133
Masao Nagasawa
4.1 Dynamic theory of stochastic processes 133
4.2 Kinematic theory 134
4.3 Sample path equation in kinematic theory 135
4.4 Mechanics and the equation of motion 137
4.5 Evolution function and kinematic equation 140
4.6 Exponent of motion and initial condition 142
4.7 Examples 143
4.8 Schr¨odinger’s wave theory and dynamic theory 146 4.9 Sample paths of motion governed by
the Schr¨odinger equation 147
4.10 Interference phenomena and entangled motion 159
Part II: Stochastic Dynamics in Basic Growth
Models and Time Delays
5. Stochastic One-Sector and Two-Sector
Growth Models in Continuous Time 167
Bjarne S. Jensen and Martin Richter
5.1 Introduction 167
5.2 Neoclassical technologies and CES forms 170
5.3 Stochastic one-sector growth models 172
5.4 Boundaries, steady state, and convergence 177 5.5 Explicit steady-state distribution with
CD technologies 188
5.6 Sample paths and asymptotic densities with
CD and CES technologies 190
5.7 General equilibria of two-sector economies 202
5.8 Dynamics of two-sector economies 206
5.9 Sample paths of two-sector models and CES 210
6. Comparative Dynamics in a Stochastic Growth
and Trade Model with a Variable Savings Rate 217
Zhu Hongliang and Huang Wenzao
6.1 Introduction 217
6.2 Stochastic dynamic systems for trading economies 218 6.3 Comparative dynamics and policy parameters 221
7. Inada Conditions and Global Dynamic Analysis
of Basic Growth Models with Time Delays 229
Zhu Hongliang and Huang Wenzao
7.1 Introduction 229
7.2 Neoclassical growth model with a time delay 231 7.3 Dynamics with delays in production and depreciation 236 7.4 Persistent oscillation in a growth model with delay 240
7.5 Final comments 245
8. Hopf Bifurcation in Growth Models with
Time Delays 247
Morten Brøns and Bjarne S. Jensen
8.1 Introduction 247
8.2 Dynamics of growth and cycles 249
8.3 Hopf bifurcation analysis 250
8.4 CD technologies and time delays 257
8.5 CES technologies and time delays 260
8.6 CES and delays with cycles,
square waves, and chaos 261
8.7 Final comments 265
Part III: Intertemporal Optimization in
Consumption, Finance, and Growth
9. Optimal Consumption and Investment Strategies
in Dynamic Stochastic Economies 271
Claus Munk and Carsten Sørensen
9.1 Introduction 271
9.2 Consumption and investment in complete markets 276 9.3 Results for CRRA utility in general markets 281
9.4 Examples 290
9.5 Extensions 301
9.6 Concluding remarks 309
Appendix 310
10. Differential Systems in Finance and Life Insurance 317
Mogens Steffensen
10.1 Introduction 317
10.2 The differential equations of Thiele and Black-Scholes 321
10.3 Surplus and dividends 332
10.4 Intervention 338
10.5 Quadratic optimization 344
10.6 Utility optimization 352
11. Uncertain Technological Change and
Capital Mobility 361
Paul A. de Hek
11.1 Introduction 361
11.2 Framework of the model 363
11.3 The effect of uncertainty on growth 369
11.4 Conclusion 374
Appendices 375
12. Stochastic Control, Non-Depletion of Renewable
Resources, and Intertemporal Substitution 381
Nils Chr. Framstad
12.1 Introduction 381
12.2 The preferences 382
12.3 The optimal control problem 385
12.4 Non-optimality of immediate total depletion 387
12.5 Concluding remarks 391
13. Capital Accumulation in a Growth Model
with Creative Destruction 393
Klaus W¨alde
13.1 Introduction 393
13.2 Framework of the model 395
13.3 Solving the model 399
13.4 Cycles and growth 402
13.5 Conclusions 410
Appendices 411
14. Employment Cycles in a Growth Model
with Creative Destruction 423
Tapio Palokangas
14.1 Introduction 423
14.2 Technology 425
14.3 R&D and capital accumulation 427
14.4 Capitalists 428
14.5 Wage settlement 430
14.6 Economic growth 432
14.7 Cycles 435
Introduction
Bjarne S. Jensen
University of Southern Denmark and Copenhagen Business School Tapio Palokangas
University of Helsinki and HECER
A unity ofaim and a diversity of topics shaped the contents of this volume. Although each chapter can be read independently as a self-contained presentation, the book is more than a collection of the in-dividual contributions. Difficult subjects are interrelated, juxtaposed, and examined for consistency in various disciplinary, theoretical, and empirical contexts. The major unifying theme of this joint work is the coherent and rigorous treatment of uncertainty and its implications for describing relevant stochastic processes throughbasic (prototype, core) models and differential equations ofstochastic dynamics.
Part I: Developments in Stochastic Dynamics
1. Bernt Øksendal. Fractional Brownian Motion in Finance.
Stochastic processes in continuous time are here described by a frac-tional Brownian motion (Wiener process) in which the stochastic in-crements are not necessarily independent. The increments have a
covariance function, for which the size of theHurst parameter (H) is critically important. When H = 12, the fractional Brownian motion coincides with the classical Brownian motion. IfH > 12, the stochastic increments have a positive autocorrelation (motion is persistent). If
anti-Bjarne S. Jensen, Tapio Palokangas
calculus (integrals) with fractional Brownian motion and discusses the
applications of fractional stochastic calculus to financial markets. As-set pricesare described assolutionsof stochastic differential equations that are driven by the generalized stochastic processes.
2. Bjarne S. Jensen, Chunyan Wang, and Jon Johnsen. Moment Evolution of Gaussian and Geometric Wiener Diffusions.
This chapter analyzes two basic stochastic models: The time homogeneous Gaussian and the geometric Wiener diffusion of two-dimensional vector processes. Using the theory of stochastic processes and Ito’s lemma, the probability distributions of the stochastic state vectors are described by the evolution of their moments (expectation
vector andcovariance matrix as functions of time). These moments satisfy certain systems of ordinary (deterministic) differential equa-tions. By solving these ODE, the authors present explicit solutions for the first-order and second-order moment functions. Kolmogorov’s forward equation is used to derive the results by alternative methods and to gain information on the probability distributions. The general
closed form results for these moment evolutions still unavailable -have many applications in models of lineardynamicswithuncertainty. 3. Nishioka Kunio. Two-Dimensional Linear Dynamic Systems with Small Random Terms.
Chapter 3 links up with chapter 2 and further studies theasymptotic behavior of thetime paths of two-dimensional linear dynamic systems that are perturbed by small random terms. Economic growth is tra-ditionally treated as a non-random dynamic system. If the system is linear and two-dimensional, it can be classified as one of five well-known types, according to its long-run (asymptotic) behavior. With
uncertainty involved in economic growth, the asymptotic behavior of a system with random perturbations is important to investigate, for example, when analyzingsteady state properties of economic growth. If the random perturbations are small, theasymptotic behavior (time paths) of the linear stochastic system is the same as in non-random
cases, unless the relevant dynamic system is acircle or a propernode.
Introduction
4. Masao Nagasawa. Dynamic Theory of Stochastic Movement of Systems.
The dynamic theory of stochastic movement of systems contains a gen-eral mathematical theory of random motion - consisting of two parts,
stochastic kinematics and stochastic mechanics. The stochastic kine-maticsis analytically described by Kolmogorov’s PDE equation, which - with its drift coefficient anddiffusion coefficient - uniquely charac-terizes thetransition probability distribution when aninitial distribu-tion is prescribed. The stochastic mechanics contains the mechanical
equation of motion, which, in addition to the Kolmogorov equation, includes apotential function ofexternal forces.
The potential function determines a so-called induced drift coeffi-cient. This induced drift coefficient in turn enters a new kinematic (Kolmogorov) equation that fully describes the relevant transition probability density of the observed stochastic process. However, Kol-mogorov equations are not easy to solve, except in some simple cases. Therefore, to analyze the stochastic processes, it is often better to use Ito’s stochastic differential equations (SDE), and in solving them, we have the powerful tools of sample path analysis, in particular, L´evy’s formula and Ito’s formula.
The dynamic theory of stochastic motion (mechanics) is then ap-plied toQuantum Mechanics(Schr¨odinger’s complex “wave equation”). Sample paths in one and two dimensions of simple motions governed by Schr¨odinger‘s equation are illustrated. Finally, the methodology is applied to the Schr¨odinger equation with Coulomb potential to obtain the sample path of the electron in the hydrogen atom. Here thecritical (“attractive”)radius of thesolved radial motions (sample paths) agrees with theclassic Bohrradiusexpression for the “station-ary states” of hydrogen. The conceptual existence of sample paths (stochastic trajectories) has been controversial (even denied) in quan-tum dynamics. Becausesample paths and stochastic differential equa-tions are the naturalgeneralization ofdeterministic dynamics in Eco-nomics, the chapter has devoted a keen effort of calculation to demon-strate some particular sample paths for states of hydrogen motion that are governed by the universal Schr¨odinger equation.
Bjarne S. Jensen, Tapio Palokangas
Part II: Stochastic Dynamics in Basic Growth Models and Time Delays
5. Bjarne S. Jensen and Martin Richter. Stochastic One-Sector and Two-Sector Growth Models in Continuous Time.
This chapter extends the basicdeterministicone-sector and two-sector growth models to astochasticcontext in continuous time, usingWiener processes for the description of various sources of uncertainty in the growth rate of the labor force, the rate of capital depreciation, and the saving rate. The drift and diffusion coefficients of the stochas-tic dynamic systems arehomogeneous of degree one in the two state variables, labor and capital - which allows a reduction to the one-dimensional stochastic dynamics of the capital-labor ratio. The cru-cial issue ofabsorbing boundaries for the stochastic growth models is rigorously examined, and simple criteria (sufficient conditions) for in-accessibleboundaries are established, a subject that the literature has not yet adequately addressed. The steady state probability distribu-tion of the capital-labor ratios is derived from Kolmogorov’s forward equation. For stochastic one-sector and two-sector growth models, the
sample paths- of thetransition to steady states orpersistent (endoge-nous) growth - of particular state variables are simulated for many parametric specifications of the CD and CES sector technologies in-volved. The impacts of technology shocks are similarly demonstrated. All relevant sample paths are exhibited on both shorter and longer time horizons.
6. Zhu Hongliang and Huang Wenzao. Comparative Dynamics in a Stochastic Growth and Trade Model with a Variable Savings Rate.
The authors consider neoclassical two-sectorgrowth models of a small country that istrading in both commodities instochastic environment
in continuous time, and they use asaving function for which the rate of saving depends on the capital-labor ratio and a policy parameter. Theglobal comparativedynamic properties of the capital accumulation process are studied with respect to changes in the policy parameter. By characterizing the entire time path of the capital accumulation process, the effect of the policy parameter can be determined. The
time path of the capital-labor ratio satisfies a monotonicity property if the saving function changes monotonically with respect to a policy parameter. In addition, the impact of the policy parameter on the
steady-state distribution of the capital-labor ratio is analyzed. 4
Introduction
7. Zhu Hongliang and Huang Wenzao. Inada Conditions and Global Dynamic Analysis of Basic Growth Models with Time Delays.
In economics, time delays are often neglected in continuous time dy-namics, no doubt due to the difficulty in solving and analyzing such models. Nevertheless, economic development depends not only on the current state, but also onpast states (history), so delay phenom-ena also influence the dynamic characteristics of economic systems. This chapter introducestime delays ina particularneoclassical growth
model. The global conditions for steady-state stability and/or persis-tent oscillation around a steady state are obtained and analyzed. It is shown that oscillations in growing economies are not rare, but com-mon.
8. Morten Brøns and Bjarne S. Jensen. Hopf Bifurcation in Growth Models with Time Delays.
Chapter 8 complements the global analysis of particular difference-differential equations in chapter 7 by performing a non-linear local analysis of the dynamics of the delay model when the size of time delay is close to a critical value. For this critical value, a Hopf bi-furcation (of a fixed-point into a closed orbit in the neighborhood of the equilibrium) occurs, that is,periodic solutions (“limit cycles”) are created when the steady state solution of the capital-labor ratio loses its local stability. Analytical criteria are derived to determine the
stability types (supercritical or subcritical) of the periodic solutions (limit cycles). Finally, it is shown that the delay model with CES production functions can exhibit dynamics with solutionswhich have been observed in other delay-differential equations: square waves and
chaos (aperiodic waves/cycles). Simulations illustrate the analytical results and theorems.
Bjarne S. Jensen, Tapio Palokangas
Part III: Intertemporal Optimization in Consumption, Finance, and Growth
9. Claus Munk and Carsten Sørensen. Optimal Consumption and Investment Strategies in Dynamic Stochastic Economies.
The authors derive optimal consumption and investmentstrategies of aninvestor with a CRRA utility of consumption andterminal wealth
and with access to trade in a complete, but otherwise very general,
financial market. Interest rates, excess expected returns, price volatil-ities, correlations, and consumer prices may all evolve stochastically over time, even with non-Markovian dynamics. The risks that indi-viduals want to hedge are shown, as well as how to finance a desired real consumption process by investing in a market of nominal securi-ties. The general results are extended to the case of a HARA utility and power-linear habit utility. The chapter also discusses how labor income and undiversifiable shocks should be included in the consumer price index. In the special case where realinterest rates are Gaussian and real market prices of risk are deterministic, the chapter shows that CRRAinvestors hedge with a single real bond, with a utility of terminal wealth (real zero-coupon bond maturing at the horizon), and with a utility of intermediate consumption (a bond with continuous coupons proportional to the expected future real consumption rate under the forward martingale measure). Theresults areillustrated by two examples: (i) non-Markovian HJM term structure dynamics, (ii) stochastic volatility and excess returns in the stock market.
10. Mogens Steffensen. Differential Systems in Finance and Life Insurance.
Financial and life insurance mathematics share a common problem of valuation of future payment streams. However, thevaluation princi-ples - no arbitrage and diversification - differ because risks differ. In both financial and life insurance mathematics, the valuation problem reduces to calculating conditional expected values and the extrema of these values. If the risk process is Markovian, expected values can be characterized by solutions to systems of deterministic differential equations. Deterministic differential systems also appear in financial and life insurance decision-making. They characterize optimal ex-pected values of future utility and optimal decisions. For both valua-tion and optimizavalua-tion, we derive someclassical examples fromfinance
and life insurance and generalize to situations that are relevant in 6
Introduction
both fields. We study valuation with participation and early exercise options, applications of the linear regulator, and generalized consump-tion problems. Thecollection ofresults and proofs demonstrate both thesimilarities and the small butimportant differences in the various problems.
11. Paul A. de Hek. Uncertain Technological Change and Capital Mobility.
Unpredictable variations in economic productivity may have a posi-tive or negaposi-tive effect on theaverage growth rate of output. This the-oretical ambiguity result is not solely determined by the value of the
elasticity of intertemporal substitution. The growth-uncertainty rela-tionship depends on two factors: whetherreturns to scaleinknowledge creation are increasing or non-increasing, and whether the elasticity of intertemporalsubstitution (of profits) is higher or lower than some
critical value. Empirical studies concerning these two factors indicate that unpredictable variations in economic productivity have a negative effect on the average long-run growth rate.
12. Nils Chr. Framstad. Stochastic Control, Non-Depletion of Renewable Resources, and Intertemporal Substitution.
For a wide class of models concerning the optimal extraction of a re-newable resource, it is well known that an expected profit maximizer
with an infinite horizon does not deplete the resource completely if its relative growth rate is strictly greater than the discount rate. This principle is extended to preferences that have intertemporal substi-tution in direct utility rates and that exhibit risk aversion (or risk neutrality) sufficiently close to zero. For a CRRA utility, the effect of intertemporal substitution is seen more clearly. The model in this chapter is an Itˆo process driven bysemi-martingales.
Bjarne S. Jensen, Tapio Palokangas
13. Klaus W¨alde. Capital Accumulation in a Growth Model with Creative Destruction.
Capital accumulation and creative destruction are modeled together with risk-averse households. The novel aspect -risk-averse households
- allows the use of well-known models not only for analyzing long-run
growthas in the literature, but also short-runfluctuations. The model remains analytically tractable because of a very convenient property of household investment decisions in this stochastic setup.
14. Tapio Palokangas. Employment Cycles in a Growth Model with Creative Destruction.
This chapter constructs a model that would explain economic
growth with fluctuations in output and employment. The particular features of the model are the following. There iscreative destruction
in the sense that a new technology renders an old technology obsolete. There are efficiency wages in R&D. In production, there is union-employer bargaining over wages. The firms can increase the proba-bility of a technological change for themselves by R&D. Learning-by-investment increases the productivity of labor in the consumption-good sector in proportion to the expected accumulation of capital.
The main results are: In the long run, the economy follows a balanced-growth path that satisfiesKaldor’s stylized facts. Wages in production grow on average in proportion to the level of productivity. The economy generates cycles around this long-run equilibrium. Capi-tal stock swings up and down due to endogenous technological shocks. Because union-firm bargaining keeps real wages in proportion to the level of productivity, the labor-capital ratio is fixed and employment swings in proportion to capital stock. Thus, a stationary state equi-librium is characterized by involuntary unemployment, employment cycles and stable real wages in production.
Chapter 1
Fractional Brownian Motion in Finance
Bernt Øksendal
Center of Mathematics for Applications (CMA) Department of Mathematics, University of Oslo, and
Norwegian School of Economics and Business Administration
1.1 Introduction
How can we model (as a function of time) (i) the levels of a river?
(ii) the characters of solar activity?
(iii) the widths of consecutive annual rings of a tree? (iv) the outdoor temperature at a given point? (v) the values of the log returns hn, defined by
hn= log S(tn)
S(tn−1)
whereS(t) is the observed price at timetof a given stock? And how can we model
