Multiple-Choice Test Parabolic Partial Differential Equations
Advanced Mathematical Economics...Chapter 8 Scalar parabolic partial differential equations 8.1...
Transcript of Advanced Mathematical Economics...Chapter 8 Scalar parabolic partial differential equations 8.1...
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Advanced Mathematical Economics
Paulo B. BritoPhD in Economics: 2020-2021
ISEGUniversidade de Lisboa
Lecture 72.12.2020
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Contents
8 Scalar parabolic partial differential equations 28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2 The simplest linear forward equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
8.2.1 The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.2.2 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.2.3 The forward heat equation in the infinite domain . . . . . . . . . . . . . . . . 68.2.4 The forward linear equation in the semi-infinite domain . . . . . . . . . . . . 108.2.5 The backward heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
8.3 The homogeneous linear PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138.3.1 Equation without transport term . . . . . . . . . . . . . . . . . . . . . . . . 138.3.2 The general homogeneous diffusion equation . . . . . . . . . . . . . . . . . . . 14
8.4 Non-autonomous linear equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158.5 Fokker-Planck-Kolmogorov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.5.1 The simplest problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.5.2 The distribution associated to the Ornstein Uhlenbeck equation . . . . . . . . 18
8.6 Economic applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.6.1 The distributional Solow model . . . . . . . . . . . . . . . . . . . . . . . . . . 198.6.2 The option pricing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
8.7 Bibiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.A Appendix: Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
9 Optimal control of parabolic partial differential equations 299.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.2 A simple optimal control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
9.2.1 Average optimal control problem . . . . . . . . . . . . . . . . . . . . . . . . . 319.2.2 Application: the distributional ๐ด๐พ model . . . . . . . . . . . . . . . . . . . . 32
9.3 Optimal control of the Fokker-Planck-Kolmogorov equation . . . . . . . . . . . . . . 359.3.1 Application: optimal distribution of capital with stochastic redistribution . . 36
9.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.A Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
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Chapter 8
Scalar parabolic partial differentialequations
8.1 Introduction
Parabolic partial differential equations involve a known function ๐น depending on two independentvariables (๐ก, ๐ฅ), an unknown function of them ๐ข(๐ก, ๐ฅ), the first partial derivative as regards ๐ก andfirst and second partial derivatives as regards the โspatialโ variable ๐ฅ:
๐น(๐ก, ๐ฅ, ๐ข, ๐ข๐ก, ๐ข๐ฅ, ๐ข๐ฅ๐ฅ) = 0
where ๐ข โถ ๐ฏ ร X โ โ, where ๐ฏ โ โ+ and X โ โ.In its simplest form, ๐น(๐ข๐ก, ๐ข๐ฅ๐ฅ) = 0, the equation models a distribution featuring dispersion
through time, for a cross section variable, generated by spatial contact (think about the time dis-tribution of a pollutant spreading within a lake in which the water is completely still). Equation๐น(๐ข๐ก, ๐ข๐ฅ, ๐ข๐ฅ๐ฅ) = 0 features both dispersion and advection behaviors (think about the time dis-tribution of a pollutant spreading within a river). Equation ๐น(๐ข๐ก, ๐ข, ๐ข๐ฅ, ๐ข๐ฅ๐ฅ) = 0 jointly displaysdispersion, advection and growth or decay behaviors (think about a time distribution of a pollutantspreading within a river, in which there is a permanent flow of new pollutants being dumped intothe river). The independent terms appear in function ๐น(.) if there are some time or spatial specificcomponents.
We will also see in the next chapter that there is a close connection between partial differentialequations and stochastic differential equations. This implied that continuous-time finance has beenusing parabolic PDEโs since the beginning of the 1970โs.
In economics and finance applications it is important to distinguish between forward (FPDE)and backward (BPDE) parabolic PDEโs. While the first are complemented with an initial distri-bution and generate a flow of distributions forward in time, the latter are complemented with aterminal distribution and its solution generate a flow of distributions consistent with that terminal
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constraint. While for FPDE the terminal distribution is unknown, for BPDE the distribution attime ๐ก = 0 is unknown. For planar systems, we may have forward, backward or forward-backward(FBPDE) parabolic PDEโs. The last case can be seen as a generalization of the saddle-path dy-namics for ODEโs.
In mathematical finance most applications, such as the Black and Scholes (1973) model, mostPDEโs are of the backward type. In economics there is recent interest in PDEโs related to thetopical importance of distribution issues, and, in particular spatial dynamics modelled by BPDE.Optimal control of PDEโs and the mean-field games usually lead to FBPDEโs.
Again, the body of theory and application of parabolic PDEโs is huge. We only present nextsome very introductory results and applications.
Let ๐ข(๐ก, ๐ฅ) where (๐ก, ๐ฅ) โ ๐ฏ ร X โ โ ร โ+ is an at least ๐ถ2,1(โ+, โ) function1. We can define
โข linear parabolic PDE
๐ข๐ก = ๐(๐ก, ๐ฅ)๐ข๐ฅ๐ฅ + ๐(๐ก, ๐ฅ)๐ข๐ฅ + ๐(๐ก, ๐ฅ)๐ข + ๐(๐ก, ๐ฅ)
if ๐น(.) is linear in ๐ข and all its derivatives, and the coefficients are independent from ๐ข
โข a semi-linear parabolic PDE
๐ข๐ก = ๐(๐ก, ๐ฅ)๐ข๐ฅ๐ฅ + ๐(๐ก, ๐ฅ)๐ข๐ฅ + ๐(๐ฅ, ๐ก, ๐ข)
it ๐น(.) is linear in the derivatives of ๐ข, and the coefficients are independent from ๐ข
โข a quasi-linear parabolic PDE
๐ข๐ก = ๐(๐ฅ, ๐ก, ๐ข)๐ข๐ฅ๐ฅ + ๐(๐ฅ, ๐ก, ๐ข)๐ข๐ฅ + ๐(๐ฅ, ๐ก, ๐ข)
if ๐น(.) is linear in the derivatives of ๐ข, but the coefficients can be functions of ๐ข.
Consider the simplest linear parabolic equation with constant coefficients, sometimes called thediffusion equation with advection and growth,
๐ข๐ก = ๐๐ข๐ฅ๐ฅ + ๐๐ข๐ฅ + ๐๐ข + ๐.
The time-behavior of ๐ข depends on three terms: a diffusion term, ๐๐ข๐ฅ๐ฅ, a transport term, ๐๐ข๐ฅ,and a growth term ๐๐ข + ๐. If ๐ > 0 (๐ < 0) the equation is sometimes called a forward FPDE(backward BPDE) equation, because the diffusion operator works forward (backward) in time.The second term introduces a behavior similar to the first-order PDE: it involves a translation of thesolution along the direction ๐ฅ. The third term generates a time behavior of the whole distribution๐ข(๐ฅ, .) in a way similar to a solution of a ordinary differential equation, that is, it involves stabilityor instability properties.
1It is, at least, differentiable to the second order as regards ๐ฅ and to the first order as regards ๐ก.
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In the case of a parabolic PDE the stability or instability properties are related to the wholedistribution: we have stability in a distributional sense if there is a solution ๐ข(๐ก, ๐ฅ) = ๐ข(๐ฅ)such that
lim๐กโโ
๐ข(๐ก, ๐ฅ) = ๐ข(๐ฅ)
where ๐ข(๐ฅ) is a stationary distribution.An important element regarding the existence and characterization of the solution of PDEโs
is related to the characteristics of the support of the distribution X. We can distinguish betweenthree main cases:
โข unbounded or infinite case X = (โโ, โ)
โข the semi-bounded of semi-infinite case X = [0, โ) or X = (โโ, 0], where 0 can be substitutedby any finite number
โข the bounded case X = (๐ฅ, ๐ฅ) where both limits are finite.
In order to define problems involving parabolic PDEโs we have to supplement it with adistribution referred to a point in time (an initial distribution for the forward PDE or terminaldistribution for a backward PDE), and possibly conditions involving known values for the valuesof ๐ข(๐ก, ๐ฅ) at the boundaries of X) (so called boundary conditions), i.e, ๐ฅ โ ๐X.
A problem is said to be well-posed if there is a solution to the PDE that satisfies jointly theinitial (or terminal) and the boundary conditions and it is continuous at those points. In this casewe say we have a classic solution. If a problem is not well-posed it is ill-posed. In this casethere are no solutions or classic solutions do not exist (but generalized solutions can exist).
A necessary condition for a problem involving a FPDE to be well posed is that it is supplementedwith an initial condition in time, and a necessary condition for a problem BPDE to be well-posedis that it involves a terminal condition in time.
Next we will present the solutions for some simple equations and problems.
8.2 The simplest linear forward equation
8.2.1 The heat equation
The simplest linear parabolic PDE is the heat equation, where ๐ข(๐ก, ๐ฅ) and is formalized by thelinear forward parabolic PDE
๐ข๐ก โ ๐ข๐ฅ๐ฅ = 0 (8.1)
2. It describes the dynamics of the temperature distribution when spatial differences in tempera-ture drive the change in spatial distribution of temperature across time. Consider a homogeneousrod with infinite width and let ๐ข(๐ก, ๐ฅ) be the temperature at point ๐ฅ โ (โโ, โ) at time ๐ก โฅ 0.
2The first formulation of the heat equation is attributed to Fourier in a presentation to the Institut de France,and in a book with title Theorie de la Propagation de la Chaleur dans les Solides both in 1807.
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Consider a small segment of the rod between points ๐ฅ and ๐ฅ + ฮ๐ฅ, where ฮ๐ฅ > 0. The differencein the temperature between the two boundaries of the segment
๐ข(๐ก, ๐ฅ + ฮ๐ฅ) โ ๐ข(๐ก, ๐ฅ) = โซ๐ฅ+โ๐ฅ
๐ฅ๐ข๐ฅ(๐ก, ๐ง) ๐๐ง
is a measure of the average temperature in the segment at time ๐ก. The instantaneous change inaverage temperature in the segment is
๐๐๐ก( โซ
๐ฅ+โ๐ฅ
๐ฅ๐ข(๐ก, ๐ง) ๐๐ง) = โซ
๐ฅ+โ๐ฅ
๐ฅ๐ข๐ก(๐ก, ๐ง) ๐๐ง
If there is a hotter spot located outside the segment, for instance in a leftward region, and becausethe heat flows from hot to colder regions, then temperature in the segment ฮ๐ฅ is lower then inthe leftward region, implying ๐ข๐ฅ(๐ก, ๐ฅ) < 0, and it is higher than in the rightward region, implying๐ข๐ฅ(๐ก, ๐ฅ + ฮ๐ฅ) < 0, and the gradient in the leftward boundary is higher in absolute terms that therightward ๐ข๐ฅ(๐ก, ๐ฅ) โ ๐ข๐ฅ(๐ก, ๐ฅ + ฮ๐ฅ) < 0. Therefore, the temperature flow is
๐ข๐ฅ(๐ก, ๐ฅ + ฮ๐ฅ) โ ๐ข๐ฅ(๐ก, ๐ฅ) = โซ๐ฅ+โ๐ฅ
๐ฅ๐ข๐ฅ๐ฅ (๐ก, ๐ง) ๐๐ง.
If is assumed that the instantaneous change in the segmentโs temperature is equal to the heat thatflows through the segment, then
โซ๐ฅ+โ๐ฅ
๐ฅ๐ข๐ก(๐ก, ๐ง) ๐๐ง = โซ
๐ฅ+โ๐ฅ
๐ฅ๐ข๐ฅ๐ฅ (๐ก, ๐ง) ๐๐ง.
which is equivalent to
โซ๐ฅ+โ๐ฅ
๐ฅ๐ข๐ก(๐ก, ๐ง) โ ๐ข๐ฅ๐ฅ (๐ก, ๐ง) ๐๐ง = 0,
which is holds if and only if equation (8.1) is satisfied.Next we define and solve the simplest linear scalar parabolic partial differential equation
๐ข๐ก(๐ก, ๐ฅ) = ๐ ๐ข๐ฅ๐ฅ(๐ก, ๐ฅ) to address the differences in the solution when we consider the domain of๐ฅ, X, the existence of side conditions and the sign of ๐.
We start with the forward equation, where ๐ > 0, in subsection 8.2.3 and deal next with thebackward equation 8.2.5
8.2.2 Fourier transforms
There are several methods to solve linear parabolic PDEโs. When the domain of the independent
variable ๐ฅ is (โโ, โ), the most direct method to find a solution is by using Fourier and inverseFourier transforms (see Appendix 8.7).
The method of obtaining a solution follows three steps: first, we transform function ๐ข(๐ก, ๐ฅ) suchthat the PDE is transformed into a parameterized ordinary differential equation; second we solve
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this ODE; and finally we transform back to the original function. When the domain of ๐ฅ is not thedouble-infinite we may have to adapt this method.
There are several possible transformations: sine, cosine, Laplace, Mellin or Fourier transforms.Next we use the Fourier transform approach.
The Fourier transform of ๐ข(๐ก, ๐ฅ), taking ๐ก as a parameter, is 3
๐(๐ก, ๐) = โฑ[๐ข(๐ก, ๐ฅ)](๐) โก โซโ
โโ๐ข(๐ก, ๐ฅ)๐โ2๐๐๐๐ฅ๐๐ฅ (8.2)
where ๐2 = โ1 and the inverse Fourier transform is
๐ข(๐ก, ๐ฅ) = โฑโ1[๐(๐ก, ๐)](๐ฅ) โก โซโ
โโ๐(๐ก, ๐)๐2๐๐๐๐ฅ๐๐. (8.3)
Time derivatives can also have Fourier transform representations: first derivative representationsare
๐ข๐ก(๐ก, ๐ฅ) =๐๐๐ก โซ
โ
โโ๐(๐ก, ๐)๐2๐๐๐๐ฅ๐๐ = โซ
โ
โโ๐๐ก(๐ก, ๐)๐2๐๐๐๐ฅ๐๐,
and๐ข๐ฅ(๐ก, ๐ฅ) =
๐๐๐ฅ โซ
โ
โโ๐(๐ก, ๐)๐2๐๐๐๐ฅ๐๐ = โซ
โ
โโ2๐๐๐ ๐(๐ก, ๐)๐2๐๐๐๐ฅ๐๐,
and the second derivative is
๐ข๐ฅ๐ฅ(๐ก, ๐ฅ) = โซโ
โโ(2๐๐๐)2๐(๐ก, ๐)๐2๐๐๐๐ฅ๐๐ = โ โซ
โ
โโ(2๐๐)2๐(๐ก, ๐)๐2๐๐๐๐ฅ๐๐.
Next we prove the relationship between a convolution of functions and the multiplication ofFourier transforms. The function ๐ข(๐ก, ๐ฅ) is a convolution if it can bewritten as
๐ข(๐ก, ๐ฅ) = ๐ฃ(๐ก, ๐ฅ) โ ๐ฆ(๐ก, ๐ฅ) โก โซโ
โโ๐ฃ(๐ก, ๐)๐ฆ(๐ก, ๐ฅ โ ๐)๐๐,
where ๐ฃ(๐ก, ๐ฅ) and ๐ฆ(๐ก, ๐ฅ) are integrable functions in the domain โ+ ร โ. Let Fourier transform of๐ข(๐ก, ๐ฅ) be written as a product of two Fourier transforms,
๐(๐ก, ๐) = โฑ[๐ข(๐ก, ๐ฅ)](๐) = ๐ (๐ก, ๐)๐ (๐ก, ๐)
where ๐ (๐ก, ๐) = โฑ[๐ฃ(๐ก, ๐ฅ)](๐) and ๐ (๐ก, ๐) = โฑ[๐ฆ(๐ก, ๐ฅ)](๐). Then ๐ข(๐ก, ๐ฅ) is the inverse Fouriertransform of ๐(๐ก, ๐) if and only if ๐ข(๐ก, ๐ฅ) is the convolution
๐ข(๐ก, ๐ฅ) = โฑโ1[๐(๐ก, ๐)](๐ฅ) = โฑโ1[๐ (๐ก, ๐)๐ (๐ก, ๐)](๐ฅ) = ๐ฃ(๐ก, ๐ฅ) โ ๐ฆ(๐ก, ๐ฅ).
8.2.3 The forward heat equation in the infinite domain
In this subsection we solve the slightly more general version of equation (8.1) in the infinite domainfor an arbitrary bounded initial condition and for a given initial conditions. The last two areversions of Cauchy problems in which the side conditions refer to ๐ก = 0.
3There are different definitions of Fourier transforms, we use the definition by, v.g., Kammler (2000).
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Free but bounded initial condition
The simplest linear PDE for an infinite domain ๐ = โ
๐ข๐ก โ ๐๐ข๐ฅ๐ฅ = 0, (๐ก, ๐ฅ) โ โ+ ร โ (8.4)
where ๐ > 0.
Proposition 1. Let ๐(๐ฅ) be an arbitrary but bounded function, i.e. satisfying โซโโโ |๐(๐ฅ)|๐๐ฅ < โ.Then the solution to PDE (8.4) is
๐ข(๐ก, ๐ฅ) =โง{โจ{โฉ
๐(๐ฅ), (๐ก, ๐ฅ) โ {๐ก = 0} ร โ1
2โ
๐๐๐ก โซโ
โโ๐(๐)๐โ (๐ฅโ๐)
24๐๐ก ๐๐, (๐ก, ๐ฅ) โ โ++ ร โ
(8.5)
Proof. Applying the previous definition of Fourier transform, equation (8.4) becomes
๐ข๐ก โ ๐๐ข๐ฅ๐ฅ = โซโ
โโ(๐๐ก(๐ก, ๐) + ๐(2๐๐)2๐(๐ก, ๐)) ๐2๐๐๐๐ฅ๐๐ = 0.
This equation is satisfied if ๐(๐ก, ๐) is solves
๐๐ก(๐ก, ๐) = ๐(๐) ๐(๐ก, ๐).
where ๐(๐) = โ(2๐๐)2๐. The solution for this ODE is
๐(๐ก, ๐) = ๐พ(๐) ๐บ(๐ก, ๐)
where ๐บ(.) is called the Gaussian kernel
๐บ(๐, ๐ก) โกโง{โจ{โฉ
1, ๐ก = 0๐๐(๐) ๐ก, ๐ก > 0
and the function ๐พ(๐) is arbitrary. To obtain the solution in terms of the original function,๐ข(๐ก, ๐ฅ), we perform an inverse Fourier transform
๐ข(๐ก, ๐ฅ) = โฑโ1(๐(๐ก, ๐)) = โฑโ1(๐พ(๐) ๐บ(๐ก, ๐)) = ๐(๐ฅ) โ ๐(๐ก, ๐ฅ)
where ๐(๐ฅ) โ ๐(๐ก, ๐ฅ) is a convolution, that is
๐(๐ฅ) โ ๐(๐ก, ๐ฅ) = โซโ
โโ๐(๐)๐(๐ก, ๐ฅ โ ๐)๐๐.
Using the tables in the Appendix, for ๐(๐ฅ, ๐ก) = โฑโ1[๐บ(๐ก, ๐)](๐ฅ) the Gaussian kernel in the initialvariable is
๐(๐ก, ๐ฅ) =โง{โจ{โฉ
๐ฟ(๐ฅ), ๐ก = 0๐โ ๐ฅ
24๐๐ก
2โ
๐ ๐ ๐ก, ๐ก > 0
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where ๐ฟ(.) is the Diracโs delta function.Therefore, because and ๐(๐ฅ) = โฑโ1[๐พ(๐ก๐)](๐ฅ),
๐ข(๐ก, ๐ฅ) =โง{โจ{โฉ
๐(๐ฅ), ๐ก = 01
2โ
๐๐๐ก โซโ
โโ๐(๐)๐โ (๐ฅโ๐)
24๐๐ก ๐๐, ๐ก > 0
(8.6)
where ๐(๐ฅ) is an arbitrary but bounded function, i.e. satisfying โซโโโ |๐(๐ฅ)|๐๐ฅ < โ, andbecause โซโโโ ๐(๐)๐ฟ(๐ฅ โ ๐)๐๐ = ๐(๐ฅ).
Two observations can be made concerning the solution of this PDE.First, applying the Fourier transform, we change from a distribution in the original variables ๐ฅ
to a frequency distribution ๐.The transformed PDE becomes a linear ODE the coefficient is eigenfunction
๐(๐) = โ(2๐๐)2๐
which is real and non-positive for any ๐ โ โ: ๐(0) = 0 and ๐(๐) < 0 for ๐ โ 0 and,lim๐โยฑโ ๐(๐) = โโ. This means that lim๐โยฑโ ๐(๐ก, ๐) = 0 for any ๐ก if ๐พ(๐) is bounded.
Second, associated to the previous property is the solution of ๐ข(๐ก, ๐ฅ) is an expected value of thearbitrary function where the density function is a Gaussian density function with average 0 andvariance 2 ๐ ๐ก.
Initial value problem Now we consider a well-posed linear FPDE. Assume we know the distri-bution at time ๐ก = 0, then we have an initial value problem
โง{โจ{โฉ
๐ข๐ก = ๐๐ข๐ฅ๐ฅ, (๐ก, ๐ฅ) โ (0, โ) ร (โโ, โ)๐ข(0, ๐ฅ) = ๐(๐ฅ) (๐ก, ๐ฅ) โ {๐ก = 0} ร (โโ, โ)
(8.7)
where ๐ > 0 and ๐(๐ฅ) is a known bounded function. Applying (8.6), the solution is
๐ข(๐ก, ๐ฅ) =โง{{โจ{{โฉ
๐(๐ฅ), (๐ก, ๐ฅ) โ {๐ก = 0} ร โ
โซโ
โโ
๐(๐)2โ
๐ ๐ ๐ก ๐โ
(๐ฅ โ ๐)24๐๐ก ๐๐, (๐ก, ๐ฅ) โ โ++ ร โ
because โซโโโ ๐(๐)๐ฟ(๐ฅ โ ๐)๐๐ = ๐(๐ฅ).
Example Figure 8.1 illustrates the behavior of the solution for ๐ = 1 and ๐(๐ฅ) = ๐โ๐ฅ2โ๐ , which
is simplified to
๐ข(๐ก, ๐ฅ) = 1โ๐(1 + 4 ๐ก)๐
โ๐ฅ2
1 + 4 ๐ก .
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0.4
0.5
Figure 8.1: Solution for the initial value problem for the heat equation with ๐ = 1 and ๐(๐ฅ) = ๐โ๐ฅ2โ๐ .
As can be seen, the solution decays through time and converges to a homogeneous distribution
lim๐กโโ
๐ข(๐ก, ๐ฅ) = 0, โ๐ฅ โ (โโ, โ)
However, a conservation law holds,
โซโ
โโ๐ข(๐ก, ๐ฅ) ๐๐ฅ = 1, for each ๐ก โฅ 0
Piecewise-constant initial condition We consider the heat equation with the initial condition
๐(๐ฅ) =โง{โจ{โฉ
๐0, if ๐ฅ โ [๐ฅ, ๐ฅ] 0 if ๐ฅ โ [๐ฅ, ๐ฅ)
where ๐ฅ < ๐ฅ are both finite. In this case, the solution to the problem is
๐ข(๐ก, ๐ฅ) = ๐0 [ ฮฆ (๐ฅ โ ๐ฅโ
2๐๐ก ) โ ฮฆ (๐ฅ โ ๐ฅโ
2๐๐ก )] (8.8)
where ฮฆ(๐ง) is the standard normal distribution function
ฮฆ(๐ฆ) = 1โ2๐ โซ๐ฆ
โโ๐โ ๐ง
22 ๐๐ง โ (0, 1).
Observe that โซโโโ ๐โ ๐ง22 ๐๐ง = (2๐) 12 .
The solution of equation (8.8) is illustrated in Figure 8.2.In order to prove this result, applying the general solution in equation (8.6) yields the solution
of the initial-value problem
๐ข(๐ก, ๐ฅ) = ๐02โ
๐๐๐ก โซ๐ฅ
๐ฅ๐โ (๐ฅโ๐)
24๐๐ก ๐๐.
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Paulo Brito Advanced Mathematical Economics 2020/2021 10
Figure 8.2: Solution for the initial value problem for the heat equation with ๐ = 1 and piecewiseinitial condition.
To simplify the expression, we make the transformation of variables ๐ง โก (๐ฅ โ ๐)/โ
2๐๐ก, and denote๐ง โก (๐ฅ โ ๐)/
โ2๐๐ก and ๐ง โก (๐ฅ โ ๐)/
โ2๐๐ก. Then, because ๐๐ง = โ1/
โ2๐๐ก๐๐ the solution simplifies 4
1โ4๐๐๐ก
โซ๐ฅ
๐ฅ๐โ(๐ฅโ๐)2/4๐๐ก๐๐ = โ
โ2๐๐กโ
4๐๐๐กโซ
(๐ฅโ๐ฅ)/โ
2๐๐ก
(๐ฅโ๐ฅ)/โ
2๐๐ก๐โ๐ง2/2๐๐ง
= 1โ2๐ (โซ(๐ฅโ๐ฅ)/
โ2๐๐ก
โโ๐โ๐ง2/2๐๐ง โ โซ
(๐ฅโ๐ฅ)/โ
2๐๐ก
โโ๐โ๐ง2/2๐๐ง) =
= ฮฆ ( ๐ฅ โ ๐ฅโ2๐๐ก ) โ ฮฆ (๐ฅ โ ๐ฅโ
2๐๐ก ) .
8.2.4 The forward linear equation in the semi-infinite domain
Now consider the equation defined on the semi-infinite domain for ๐ฅ, that is X = โ+. This case ismore interesting for economic applications in which the independent variable can only take non-negative values, for instance when ๐ฅ refers to a stock.
The FPDE we consider is๐ข๐ก โ ๐๐ข๐ฅ๐ฅ = 0, (๐ก, ๐ฅ) โ โ2+ (8.9)
where ๐ > 0.
Proposition 2. The solution to equation (8.11) is
๐ข(๐ก, ๐ฅ) = 12โ
๐ ๐ ๐ก โซโ
0๐ (๐) (๐โ (๐ฅโ๐)
24 ๐ ๐ก โ ๐โ (๐ฅ+๐)
24 ๐ ๐ก ) ๐๐, ๐ก > 0. (8.10)
where ๐(๐ฅ) โถ X โ โ is an arbitrary bounded function.4Recalling the formula for integration by substitution of variables, if we set ๐ง = ๐(๐) and ๐ โ (๐, ๐) then
โซ๐(๐)
๐(๐)๐(๐ง)๐๐ง = โซ
๐
๐๐(๐(๐)) ๐โฒ (๐) ๐๐.
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Paulo Brito Advanced Mathematical Economics 2020/2021 11
Proof. We solve this equation by using the method of images. It consists in introducing thefollowing extension to the arbitrary function ๐(๐ฅ)
๏ฟฝฬ๏ฟฝ(๐ฅ) =โง{โจ{โฉ
๐(๐ฅ), if ๐ฅ โฅ 0โ๐(โ๐ฅ) if ๐ฅ < 0
where ๐(.) is an odd function satisfying ๐(โ๐ฅ) = โ๐(๐ฅ). Using the solution (8.6) for ๐ก > 0 wehave
๐ข(๐ก, ๐ฅ) = 12โ
๐๐๐ก โซโ
โโ๏ฟฝฬ๏ฟฝ (๐)๐โ (๐ฅโ๐)
24๐๐ก ๐๐ =
= 12โ
๐๐๐ก (โซ0
โโ๏ฟฝฬ๏ฟฝ (๐)๐โ (๐ฅโ๐)
24๐๐ก ๐๐ + โซ
โ
0๏ฟฝฬ๏ฟฝ (๐)๐โ (๐ฅโ๐)
24๐๐ก ๐๐) =
= 12โ
๐๐๐ก (โ โซโ
0๐ (๐)๐โ (๐ฅ+๐)
24๐๐ก ๐๐ + โซ
โ
0๐ (๐)๐โ (๐ฅโ๐)
24๐๐ก ๐๐)
where the last step involves integration by substitution: i.e., if we define ๐ข = โ๐ฅ for ๐ฅ โ [0, โ)then โซ0โโ ๐(๐ข)๐๐ข = โ โซ
0โ ๐(โ๐ฅ)๐๐ฅ = โซ
โ0 ๐(โ๐ฅ)๐๐ฅ. Then the solution of equation (8.11) is equation
(8.11).
The solution to the initial-value problem
โง{โจ{โฉ
๐ข๐ก โ ๐๐ข๐ฅ๐ฅ = 0, for ๐ > 0, (๐ก, ๐ฅ) โ โ2+๐ข(0, ๐ก) = ๐ข0(๐ฅ), for (๐ก, ๐ฅ) โ {๐ก = 0} ร โ+
(8.11)
is๐ข(๐ก, ๐ฅ) = 12
โ๐ ๐ ๐ก โซ
โ
0๐ข0(๐) (๐โ
(๐ฅโ๐)24 ๐ ๐ก โ ๐โ (๐ฅ+๐)
24 ๐ ๐ก ) ๐๐, ๐ก > 0. (8.12)
We obtain this result by direct application of equation (8.12).Example Consider the initial-value problem in which the initial distribution is log-normal
๐ข0(๐ฅ) =๐โ
(ln ๐ฅ โ ๐)22๐2
2โ
๐ ๐ฅ2 ๐2
if we substitute in equation (8.12) we have the solution depicted in Figure 8.4 for several momentsin time.
We observe that the solution is not conservative, i.e. the integral ๐(๐ก) = โซโ0 ๐ข(๐ก, ๐ฅ) ๐๐ฅ decaysin time such that lim๐กโโ ๐(๐ก) = 0.
8.2.5 The backward heat equation
In finance applications and associated to Euler equation in optimal control problems, we sometimesneed to solve backward parabolic PDE.
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Paulo Brito Advanced Mathematical Economics 2020/2021 12
5 10 15 20
0.05
0.10
0.15
0.20
0.25
0.30
Figure 8.3: Solution for the initial value problem for the heat equation in the semi-infinite line with๐ = 1 and an initial log-normal density.
The simplest parabolic BPDE equation in the infinite domain for ๐ฅ and for the semi-infinitedomain for ๐ก is
๐ข๐ก + ๐๐ข๐ฅ๐ฅ = 0, (๐ก, ๐ฅ) โ [0, ๐ ] ร (โโ, โ) (8.13)
where ๐ > 0.
Proposition 3. Consider the BPDE equation (8.13). The solution is
๐ข(๐ก, ๐ฅ) =โง{โจ{โฉ
๐(๐ฅ), ๐ก = ๐1
โ4๐๐(๐ โ๐ก) โซโ
โโ๐(๐)๐โ
(๐ฅโ๐)24๐(๐โ๐ก) ๐๐, ๐ก โ (0, ๐ )
Proof. In order to solve it we introduce a change in variables ๐ = ๐ โ ๐ก and consider a change inthe variable ๐ฃ(๐, ๐ฅ) = ๐ข(๐ก(๐), ๐ฅ) where ๐ก(๐) = ๐ โ ๐ . As
๐ฃ๐(๐, ๐ฅ) = โ๐ข๐ก(๐ก(๐), ๐ฅ), and ๐ฃ๐ฅ๐ฅ(๐, ๐ฅ) = ๐ข๐ฅ๐ฅ(๐ก(๐), ๐ฅ)
Then ๐ข๐ก(๐ก, ๐ฅ) = โ๐๐ข๐ฅ๐ฅ(๐ก, ๐ฅ) is equivalent to
๐ฃ๐(๐, ๐ฅ) = ๐๐ฃ๐ฅ๐ฅ(๐, ๐ฅ).
Using the solution already found in equation (8.6) we get
๐ฃ(๐, ๐ฅ) =โง{โจ{โฉ
๐(๐ฅ), ๐ = 0
โซโ
โโ๐(๐) (4๐๐๐)โ1/2 ๐โ (๐ฅโ๐)
24๐๐ ๐๐, ๐ โ (0, ๐ ).
Transforming back to ๐ข(๐ก, ๐ฅ) we have solution.
A problem involving a backward PDE is only well posed if together with the PDE we have aterminal condition, for example ๐ข(๐ , ๐ฅ) = ๐๐ (๐ฅ). In this case the value of the variable at time๐ก = 0 becomes endogenous.
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Consider the terminal-value problem
โง{โจ{โฉ
๐ข๐ก = โ๐๐ข๐ฅ๐ฅ, (๐ก, ๐ฅ) โ (โโ, โ) ร (0, ๐ ]๐ข(๐ , ๐ฅ) = ๐๐ (๐ฅ) (๐ก, ๐ฅ) โ (โโ, โ) ร { ๐ก = ๐ }.
The solution is
๐ข(๐ก, ๐ฅ) =โง{โจ{โฉ
๐๐ (๐ฅ), (๐ก, ๐ฅ) โ {๐ก = ๐ } ร โ 1
โ4๐๐(๐ โ ๐ก)โซ
โ
โโ๐๐ (๐)๐โ
(๐ฅโ๐)24๐(๐โ๐ก) ๐๐, (๐ก, ๐ฅ) โ (0, ๐ ) ร โ
The initial distribution can be obtained by setting ๐ก = 0
๐ข(0, ๐ ) = 1โ4๐๐(๐ โ ๐ก)โซ
โ
โโ๐๐ (๐)๐โ
(๐ฅโ๐)24๐๐ ๐๐.
8.3 The homogeneous linear PDE
The general forward linear parabolic PDE in the infinite domain is
๐ข๐ก = ๐ ๐ข๐ฅ๐ฅ + ๐ ๐ข๐ฅ + ๐ ๐ข, (๐ก, ๐ฅ) โ โ+ ร โ
where ๐ > 0. The dynamics of ๐ข(๐ก, ๐ฅ) contains three terms: a diffusion term, if ๐ โ 0, a transportterm, if ๐ โ 0, and a growth or decay term if ๐ > 0 or ๐ < 0.
In order to solve the equation, we can follow one of two alternative methods:
1. transform the equation into a heat equation, solve the heat equation and transform back tothe initial variables.
2. apply the Fourier transform method to transform the PDE into a parameterized ODE, solveit, and apply inverse Fourier transforms.
8.3.1 Equation without transport term
If the linear forward PDE does not contain a transport term, we have
๐ข๐ก = ๐ ๐ข๐ฅ๐ฅ + ๐ ๐ข, (๐ก, ๐ฅ) โ (0, โ) ร (โโ, โ) (8.14)
where ๐ > 0 and ๐ โ 0, which has solution, for an arbitrary bounded function ๐(๐ฅ)
๐ข(๐ก, ๐ฅ) =โง{{โจ{{โฉ
โซโ
โโ๐(๐)๐ฟ(๐ฅ โ ๐)๐๐ = ๐(๐ฅ), (๐ก, ๐ฅ) โ {๐ก = 0} ร โ
๐๐ ๐ก โซโ
โโ๐(๐) 1โ
4 ๐ ๐ ๐ก๐โ (๐ฅโ๐)
24 ๐ ๐ก ๐๐, (๐ก, ๐ฅ) โ โ+ ร โ.
To find the solution, we use the first method. First, define ๐ฃ(๐ก, ๐ฅ) = ๐โ๐ ๐ก๐ข(๐ก, ๐ฅ), which hasderivatives ๐ฃ๐ก = โ๐๐โ๐ ๐ก๐ข + ๐โ๐ ๐ก๐ข๐ก and ๐ฃ๐ฅ๐ฅ = ๐โ๐๐ก๐ข๐ฅ๐ฅ. Second, equation (8.14) is equivalent to the
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Paulo Brito Advanced Mathematical Economics 2020/2021 14
simplest linear equation ๐ฃ๐ก = ๐๐ฃ๐ฅ๐ฅ which has solution (8.6). Third, as ๐ข(๐ก, ๐ฅ) = ๐๐ ๐ก ๐ฃ(๐ก, ๐ฅ) we obtainthe solution
The dynamics of the solution depends crucially on the sign of ๐:
lim๐กโโ
๐ข(๐ก, ๐ฅ) =โง{โจ{โฉ
0 if ๐ < 0โ if ๐ > 0
Figure 8.4 illustrates the cases in which ๐ < 0 and ๐ > 0. In both cases we see that the long-time
behavior of the solution is commanded by ๐๐ ๐ก: if ๐ < 0 then lim๐กโโ ๐ข(๐ก, ๐ฅ) = 0, for any ๐ฅ โ โ, andif ๐ > 0 then lim๐กโโ ๐ข(๐ก, ๐ฅ) โ lim๐กโโ ๐๐ ๐ก = โ, for any ๐ฅ โ โ.
This means that the diffusion equation display asymptotic stability if ๐ < 0 and instability if๐ > 0, both in a distributional sense. In the first case the solution ๐ข(๐ก, ๐ฅ) is bounded and in thesecond case it is unbounded.
Figure 8.4: Solution for the initial value problem for the heat equation with ๐ = 1 and ๐(๐ฅ) = ๐โ๐ฅ2โ๐
and ๐ = โ0.5 and ๐ = 0.5.
8.3.2 The general homogeneous diffusion equation
The initial value problem for a a general linear homogeneous (forward) diffusion equation is
โง{โจ{โฉ
๐ข๐ก(๐ก, ๐ฅ) = ๐ ๐ข๐ฅ๐ฅ(๐ก, ๐ฅ) + ๐ ๐ข๐ฅ(๐ก, ๐ฅ) + ๐ ๐ข(๐ก, ๐ฅ), (๐ก, ๐ฅ) โ (0, โ) ร (โโ, โ)๐ข(0, ๐ฅ) = ๐(๐ฅ), (๐ก, ๐ฅ) โ {๐ก = 0} ร (โโ, โ),
(8.15)
where ๐ > 0, ๐ โ 0 and ๐ โ 0 and ๐(๐ฅ) is a bounded function over ๐ = โ.
Proposition 4. The solution to problem (8.15) is
๐ข(๐ก, ๐ฅ) = โซโ
โโ๐(๐ ) 1โ
4๐๐๐กexp (โ(๐ฅ โ ๐ )
2 + 2 ๐ (๐ฅ โ ๐ ) ๐ก + (๐2 โ 4๐๐) ๐ก24 ๐ ๐ก ) ๐๐ (8.16)
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Proof. We will solve this problem using the Fourier transform representation of equation ๐ข๐ก โ(๐๐ข๐ฅ๐ฅ + ๐๐ข๐ฅ + ๐๐ข) = 0. Using inverse Fourier transforms yields
๐ข๐ก(๐ก, ๐ฅ) โ ๐๐ข๐ฅ๐ฅ(๐ก, ๐ฅ) โ ๐๐ข๐ฅ โ ๐๐ข(๐ก, ๐ฅ) = โซโ
โโ๐2๐๐๐๐ฅ [ ๐๐ก(๐ก, ๐) + ๐(๐)๐(๐ก, ๐)] ๐๐ = 0.
where the coefficient is a complex-valued function of ๐ 5
๐(๐) โก ๐(2๐๐)2 โ ๐ โ 2๐ ๐ ๐ ๐, ๐ โกโ
โ1
Therefore, the PDE (8.15) is equivalent to the linear ODE parameterized by ๐
๐๐ก(๐ก, ๐) = โ๐(๐)๐(๐ก, ๐), (๐ก, ๐) โ โ+ ร โ,
which has the explicit solution
๐(๐ก, ๐) = ฮฆ(๐) ๐บ(๐ก, ๐), for๐ก โ [0, โ)
where ฮฆ(๐) = โฑ[ ๐(๐ฅ)] (๐) is the Fourier transform of the initial distribution, and ๐บ(๐ก, ๐) is theGaussian kernel
๐บ(๐ก, ๐) = ๐โ๐(๐)๐ก, for ๐ก > 0.
We obtain the solution of problem (8.15) by applying the inverse Fourier transform
๐ข(๐ก, ๐ฅ) = โฑโ1 [๐(๐ก, ๐)] (๐ฅ) = โฑโ1 [ฮฆ(๐) ๐บ(๐ก, ๐)] (๐ฅ) = โซโ
โโ๐(๐ )๐(๐ก, ๐ฅ โ ๐ )๐๐
where (see the Appendix 8.7 )
๐(๐ก, ๐ฆ) = โฑโ1 [ ๐โ๐(๐)๐ก] = 1โ4๐๐๐ก
exp (โ๐ฆ2 + 2๐๐ก๐ฆ + (๐2 โ 4๐๐)๐ก2
4๐๐ก ), (8.17)
because ๐๐ก > 0.
Figure 8.5 illustrates the solution (8.16) for negative (left figures) and positive (right figures)values for ๐ and negative (upper figures) and positive (lower figures) values of ๐. It is clear thatwhile ๐ introduces a transportation in the positive direction, if ๐ < 0, or in the negative direction,if ๐ > 0, ๐ is associated to the time stability of the whole distribution.
8.4 Non-autonomous linear equation
Next we consider two non-autonomous equations in which there is one term depending on theindependent variables (๐ก, ๐ฅ)
5The advection term, involving the first derivative has a complex-valued the Fourier transform representation
๐ข๐ฅ(๐ก, ๐ฅ) =๐
๐๐ฅ (โซโ
โโ๐(๐ก, ๐)๐2๐๐๐ฅ๐๐๐) = โซ
โ
โโ2๐๐๐ ๐(๐ก, ๐)๐2๐๐๐ฅ๐๐๐.
.
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Paulo Brito Advanced Mathematical Economics 2020/2021 16
Figure 8.5: Solution for the initial value problem linear PDE for ๐ = 1, ๐ = 1 and ๐ = โ1 and๐ = โ0.5 and ๐ = 1 and ๐ = โ1 and ๐ = 0.5.
Non-homogeneous heat equation The non-homogeneous (forward) heat equation
๐ข๐ก โ ๐๐ข๐ฅ๐ฅ โ ๐(๐ก, ๐ฅ) = 0, (๐ก, ๐ฅ) โ (0, โ) ร (โโ, โ) (8.18)
this equation has a component which is not affected by ๐ข, although it changes with (๐ก, ๐ฅ).In order to solve it, we again use inverse Fourier transforms to get an equivalent ODE in
transformed variables ๐(๐ก, ๐),
๐(๐ก, ๐) = โ๐(๐)๐(๐ก, ๐) + ๐ต(๐ก, ๐)
where ๐ต(๐ก, ๐) = โฑ[๐(๐ก, ๐ฅ)](๐) and ๐(๐) = (2 ๐ ๐)2 ๐. The solution to equation (8.18)is
๐(๐ก, ๐) = ๐พ(๐)๐บ(๐ก, ๐) + โซ๐ก
0๐ต(๐ , ๐) ๐บ(๐ก โ ๐ , ๐)๐๐ ,
where ๐บ(๐ก, โ ) is a Gaussian kernel. Applying inverse Fourier transforms yields
๐ข(๐ก, ๐ฅ) = ๐(๐ฅ) โ ๐(๐ก, ๐ฅ) + โซ๐ก
0๐(๐ , ๐ฅ) โ ๐(๐ก โ ๐ , ๐ฅ)๐๐ .
Therefore, the solution to the parabolic PDE (8.18) is, for ๐ก > 0,
๐ข(๐ก, ๐ฅ) = 1โ4๐๐๐ก
โซโ
โโ๐(๐)๐โ (๐ฅโ๐)
24๐๐ก ๐๐ + โซ
๐ก
0
1โ4๐๐(๐ก โ ๐ )
โซโ
โโ๐โ
(๐ฅโ๐)24๐(๐กโ๐ ) ๐(๐ , ๐)๐๐๐๐ .
The solution can converge to a spatially non-homogenous distribution.
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Paulo Brito Advanced Mathematical Economics 2020/2021 17
Non-autonomous diffusion equation
Consider the equation
๐ข๐ก = ๐๐ข๐ฅ๐ฅ + ๐๐ข + ๐(๐ฅ), (๐ก, ๐ฅ) โ (โโ, โ) ร (0, โ)
where ๐ > 0. It can be proved (see Exercise 1) that the solution for ๐ก > 0 is
๐ข(๐ก, ๐ฅ) = ๐๐๐ก
โ4๐๐๐ก
โซโ
โโ๐(๐)๐โ (๐ฅโ๐)
24๐๐ก ๐๐ + 1โ4๐๐(๐ก โ ๐)
โซ๐ก
0๐๐(๐กโ๐) โซ
โ
โโ๐(๐)๐โ
(๐ฅโ๐)24๐(๐กโ๐) ๐๐๐๐
8.5 Fokker-Planck-Kolmogorov equation
We will see in the chapter on stochastic differential equations, that the probability distributionof a diffusion process follows a particular parabolic PDE, called the Fokker-Planck-Kolmogorovequation. This equation is having an increase attention, also in economics, as a model for processessatisfying a conservation law as
โซ๐
๐(๐ก, ๐ฅ)๐๐ฅ = 1, for every ๐ก โ ๐
where ๐(๐ก, ๐ฅ) โถ ๐ ร ๐ โ (0, 1).The Fokker-Planck-Kolmogorov equation is
๐๐ก๐(๐ก, ๐ฅ) =12๐๐ฅ๐ฅ (๐(๐ก, ๐ฅ)
2 ๐(๐ก, ๐ฅ)) โ ๐๐ฅ(๐(๐ก, ๐ฅ) ๐(๐ก, ๐ฅ)) (8.19)
where we assume ๐(0, ๐ฅ) is known and satisfies
โซ๐
๐(0, ๐ฅ)๐๐ฅ = 1.
In applications resulting from stochastic differential equations, the initial state is known ๐ฅ = ๐ฅ0and the dynamics of a probability distribution is given by Kolmogorov forward equation (or Fokker-Planck equation) and the initial condition ๐(0, ๐ฅ) = ๐ฟ(๐ฅโ๐ฅ0) where ๐ฟ(โ ) is Diracโs delta generalizedfunction.
8.5.1 The simplest problem
The simplest model has constant coefficients ๐(๐ก, ๐ฅ) = ๐ and ๐(๐ก, ๐ฅ) = ๐ and a Dirac delta initialfunction:
โง{โจ{โฉ
๐๐ก๐(๐ก, ๐ฅ) =๐22 ๐๐ฅ๐ฅ๐(๐ก, ๐ฅ) โ ๐๐๐ฅ๐(๐ก, ๐ฅ), (๐ก, ๐ฅ) โ โ+ ร โ
๐(0, ๐ฅ) = ๐ฟ(๐ฅ โ ๐ฅ0) (๐ก, ๐ฅ) โ { ๐ก = 0} ร โ(8.20)
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Paulo Brito Advanced Mathematical Economics 2020/2021 18
The solution is a Gamma probability density
๐(๐ก, ๐ฅ) = 1โ2 ๐ ๐2 ๐ก
exp โ(๐ฅ โ ๐ฅ0 โ ๐ ๐ก2 ๐2 ๐ก )2
= ฮ( โ ๐๐ก; ๐2
2 , ๐ฅ โ ๐ฅ0) (๐ก, ๐ฅ) โ โ+ ร โ
Figure 8.6 presents an illustration of the solution
Figure 8.6: Solution for (8.20) for ๐ฅ0 = 5, ๐ = 1 and ๐ = 0.5.
8.5.2 The distribution associated to the Ornstein Uhlenbeck equation
The simplest model has constant coefficients ๐(๐ก, ๐ฅ) = ๐0 + ๐1 ๐ฅ and ๐(๐ก, ๐ฅ) = ๐ and a Dirac deltainitial function:
โง{โจ{โฉ
๐๐ก๐(๐ก, ๐ฅ) =๐22 ๐๐ฅ๐ฅ๐(๐ก, ๐ฅ) โ (๐0 + ๐1 ๐ฅ) ๐๐ฅ๐(๐ก, ๐ฅ) โ ๐1๐(๐ก, ๐ฅ), (๐ก, ๐ฅ) โ โ+ ร โ
๐(0, ๐ฅ) = ๐ฟ(๐ฅ โ ๐ฅ0) (๐ก, ๐ฅ) โ { ๐ก = 0} ร โ(8.21)
The solution is a Normal density function
๐(๐ก, ๐ฅ) = 1
โ2 ๐ ๐2 (1 โ ๐2 ๐1 ๐ก)exp { โ
(๐ฅ โ ๐ฅ0 ๐๐1๐ก โ ๐0(1 โ ๐๐1๐ก))2
2 ๐2 (1 โ ๐2๐1๐ก ) }
= ๐(๐ฅ0๐๐๐ก + ๐0(1 โ ๐๐1๐ก),๐22 (1 โ ๐
2๐1๐ก )) (๐ก, ๐ฅ) โ โ+ ร โ
Exercise: prove this.We see that if ๐1 < 0 then
lim๐กโโ
๐(๐ก, ๐ฅ) โผ ๐(๐0,๐22 )
this means that the distribution is ergodic: for any initial value ๐ฅ0 it tends asymptotically to anormal distribution (see Figure 8.7).
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Paulo Brito Advanced Mathematical Economics 2020/2021 19
Figure 8.7: Solution for (8.20) for ๐ฅ0 = 5, ๐0 = 1, ๐1 = โ1 and ๐ = 0.5.
8.6 Economic applications
8.6.1 The distributional Solow model
In Brito (2004) we prove that in an economy in which the capital stock is distributed in an
heterogeneous way between regions, ๐พ(๐ก, ๐ฅ), if there is an infinite support, and there are freecapital flows between regions, the budget constraint for the location ๐ฅ can be represented by theparabolic PDE.
Consider the accounting balance between savings and internal and external investment for aregion ๐ฅ at time ๐ก
๐ผ(๐ก, ๐ฅ) + ๐ (๐ก, ๐ฅ) = ๐(๐ก, ๐ฅ)
where ๐ผ(๐ก, ๐ฅ) and ๐(๐ก, ๐ฅ) is investment and domestic savings of location ๐ฅ at time ๐ก and ๐ (๐ก, ๐ฅ) isthe savings flowing to other regions.
Assume that the capital flow for a region of length ฮ๐ฅ is symmetric to the capital distributiongradient to neighboring regions:
๐ (๐ก, ๐ฅ)ฮ๐ฅ = โ (๐พ๐ฅ(๐ฅ + ฮ๐ฅ, ๐ก) โ ๐พ๐ฅ(๐ก, ๐ฅ))
that is capital flows proportionaly and in a reverse direction to the โspatial gradientโ of thecapital distribution: regions with high capital intensity will tend to โleakโ capital to other regions.If ฮ๐ฅ โ 0 leads to ๐ (๐ก, ๐ฅ) = โ๐พ๐ฅ๐ฅ(๐ก, ๐ฅ).
If there is no depreciation then ๐ผ(๐ก, ๐ฅ) = ๐พ๐ก(๐ก, ๐ฅ). If the technology is ๐ด๐พ and the savings rateis determined as in the Solow model then ๐(๐ก, ๐ฅ) = ๐ ๐ด๐พ(๐ก, ๐ฅ) where 0 < ๐ < 1 and ๐ด is assume tobe spatially homogeneous.
Therefore we obtain a distributional Solow equation for an economy composed by heterogenousregions
๐พ๐ก = ๐พ๐ฅ๐ฅ + ๐ ๐ด๐พ, (๐ก, ๐ฅ) โ (โโ, โ) ร (0, โ)
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Paulo Brito Advanced Mathematical Economics 2020/2021 20
We can define a spatially-homogenous balanced growth path (BGP) as
๐พ(๐ก) = ๐พ๐๐พ๐ก
where ๐พ = ๐ ๐ด.Then, if we define the deviations as regards the BGP as ๐(๐ก, ๐ฅ) = ๐พ(๐ก, ๐ฅ)๐โ๐พ๐ก, we observe that
the transitional dynamics is given by the solution of the equation
๐๐ก = ๐๐ฅ๐ฅ
which is the heat equation. Therefore, given the initial distribution of the capital stock ๐พ(๐ฅ, 0) =๐0(๐ฅ) the solution for this spatial ๐ด๐พ model is
๐พ(๐ก, ๐ฅ) =โง{โจ{โฉ
๐0(๐ฅ), ๐ก = 0
๐๐พ๐ก โซโ
โโ๐0(๐) (4๐๐ก)
โ1/2 ๐โ (๐ฅโ๐)2
4๐ก ๐๐, ๐ก > 0
and the solution is similar to the case depicted in Figure 8.2 when ๐ > 0.The main conclusion is that: (1) there is long run growth; (2) , if there are homogenous
technologies and preferences the asymptotic distribution will become spatially homogeneous. Thatis: the so-called ๐ฝ- and ๐- convergences can be made consistent !
8.6.2 The option pricing model
The Black and Scholes (1973) model is a case in which a research paper had an immense impact
on the operation of the economy. It is related to the onset of derivative markets and basically gavebirth to stochastic finance6.
It provides a formula (the so called Black-Scholes formula) for the value of an European calloption when there are two assets, a riskless asset with interest rate ๐ and a underlying asset whoseprice, ๐ which follows a diffusion process (in a stochastic sense): ๐๐ = ๐๐๐๐ก + ๐๐๐๐ต where ๐๐ต isthe standard Brownian motion (see next chapter). An European call option offers the right to buythe underlying asset at time ๐ for a price ๐พ fixed at time ๐ก = 0, which is conventioned to be themoment of the contract.
Under the assumption that there are no arbitrage opportunities Black and Scholes (1973) provedthat the price of the option ๐ = ๐ (๐ก, ๐) is a a function of time, ๐ก โ (0, ๐ ) and the price of anunderlying asset ๐ โ (0, โ) follows the backward parabolic PDE and has a terminal constraint
โง{โจ{โฉ
๐๐ก(๐ก, ๐) = โ๐2๐22 ๐๐๐(๐ก, ๐) โ ๐๐๐๐(๐ก, ๐) + ๐๐ (๐ก, ๐), (๐ก, ๐) โ [0, ๐ ] ร (0, โ)
๐ (๐ , ๐) = max{ ๐ โ ๐พ, 0} , (๐ก, ๐) โ {๐ก = ๐ } ร (0, โ).(8.22)
6Myron Scholes was awarded the Nobel prize in 1997, together with Robert Merton another important contributerto stochastic finance, precisely for this formula. Fisher Black was deceased at the time.
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Paulo Brito Advanced Mathematical Economics 2020/2021 21
The first equation is valid for any financial option having the same underlying asset dynamics, andthe terminal constraint is characteristic of the European call option: because the writer sells theright, but not the obligation, to purchase the underlying asset at the price ๐พ at time ๐ก = ๐ , thebuyer is only interested in that purchase if he can sell it at the prevailing market price ๐ = ๐(๐ )when that price is higher than the exercise price ๐พ. In this case the payoff will be ๐(๐ ) โ ๐พ.Otherwise he will not execute the option and the terminal payoff will be zero.
The two boundary constraints
๐ (๐ก, 0) = 0, (๐ก, ๐) โ [0, ๐ ] ร {๐ = 0} lim
๐โโ๐ (๐ก, ๐) = ๐, (๐ก, ๐) โ [0, ๐ ] ร {๐ โ โ},
are sometimes referred to, but they are redundant.The same structure occurs in the Mertonโs model (see Merton (1974)) which is a seminal paper
on the pricing of default bonds. It was the first model on the so-called structural approach tomodelling credit risk which is on the foundation of the credit risk models used by rating agencies(see Duffie and Singleton (2003)). In essence, this model assumes that the value of the firm followsa linear diffusion process and it consideres the issuance of a bond with an expiring date ๐ whoseindenture gives it absolute priority on the value of the firm at the expiry date. This means thateither if the value of the firm is smaller that the face value of the bond the creditor takes possessionof the firm and in the opposite case it recovers the face value. In this case, we can interpret theposition of the equity owner as holding an European call option over the value of the firm withstrike price equal to the face value of the debt and the creditor as having an European put optionsecurity.
The price of the European call option 7, given the former assumptions is given by
๐ (๐ก, ๐) = ๐ฮฆ(๐1) โ ๐พ๐โ๐(๐ โ๐ก)ฮฆ(๐2), ๐ก โ [0, ๐ ] (8.23)
where ฮฆ(.) is cumulative Gaussian density function such that ฮฆ(๐) = โ(๐ฅ โค ๐) where
๐1 =ln (๐/๐พ) + (๐ โ ๐ก) (๐ + ๐
2
2 )
๐โ
๐ โ ๐ก(8.24)
๐2 =ln (๐/๐พ) + (๐ โ ๐ก) (๐ โ ๐
2
2 )
๐โ
๐ โ ๐ก(8.25)
Proof. In order to solve the B-S PDE, which is a non-linear backward parabolic PDE, we transformit to to a quasi-linear parabolic forward PDE, by applying the transformations: ๐ก(๐) = ๐ โ ๐ and๐ = ๐พ๐๐ฅ and setting ๐ข(๐, ๐ฅ) = ๐ (๐ก(๐), ๐(๐ฅ)). We can transform the option-pricing problem to the
7For the credit risk model ๐ would be the value of assets of a firm, ๐พ would be the face value of loan, and ๐ theterm of the loan.
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Paulo Brito Advanced Mathematical Economics 2020/2021 22
Figure 8.8: Solution for the Black and Scholes model, for ๐ = 0.02, ๐ = 20, ๐ = 0.2, and ๐พ = 10.
equivalent initial-value problem PDE equivalent to (8.22)
โง{โจ{โฉ
๐ข๐ =๐22 ๐ข๐ฅ๐ฅ + (๐ โ
๐22 ) ๐ข๐ฅ โ ๐๐ข, (๐, ๐ฅ) โ [0, ๐ ] ร (โโ, โ)
๐ข(0, ๐ฅ) = ๐ข0(๐ฅ)(8.26)
where
๐ข0(๐ฅ) =โง{โจ{โฉ
0, if ๐ฅ โค 0๐พ (๐๐ฅ โ 1) , if ๐ฅ > 0
The PDE is a particular example of equation (8.15), which implies that the solution is
๐ข(๐, ๐ฅ) = โซ0
โโ0 ๐(๐, ๐ฅ โ ๐ )๐๐ + ๐พ โซ
โ
0(๐๐ โ 1) ๐(๐, ๐ฅ โ ๐ )๐๐
= ๐พโ2๐๐2๐
โซโ
0(๐๐ โ 1) ๐โ(๐,๐ฅโ๐ )๐๐
where (from equation (8.17))
โ(๐, ๐ฆ) โก โ๐ฆ2 + 2๐ (๐ โ ๐
2
2 ) ๐ฆ + (๐ +๐22 )
2๐2
2๐๐2 .
Then
๐ข(๐, ๐ฅ) = ๐พโ2๐๐2๐
(โซโ
0๐๐ +โ(๐,๐ฅโ๐ )๐๐ โ โซ
โ
0๐โ(๐,๐ฅโ๐ )๐๐ )
= ๐พโ2๐๐2๐
(๐ผ1 โ ๐ผ2) .
In order to simplify the integrals it is useful to remember the forms of the error function, erf(๐ฅ),and of the Gaussian cumulative distribution ฮฆ(๐ฅ),
erf(๐ฅ) = 2โ๐ โซ๐ฅ
โโ๐โ๐ง2๐๐ง, ฮฆ(๐ฅ) = 1โ2๐ โซ
๐ฅ
โโ๐โ
12 ๐ง
2
๐๐ง
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Paulo Brito Advanced Mathematical Economics 2020/2021 23
which are related asฮฆ(๐ฅ) = 12 [ 1 + erf (
๐ฅโ2
)] .
After some algebra we obtain
๐ + โ(๐, ๐ฅ โ ๐ ) = ๐ฅ โ 12(๐ฟ1(๐ ))2
โ(๐, ๐ฅ โ ๐ ) = โ๐๐ โ 12(๐ฟ2(๐ ))2
where
๐ฟ1(๐ ) โก๐ฅ โ ๐ + (๐ + ๐
2
2 )
๐โ๐ , and ๐ฟ2(๐ ) โก๐ฅ โ ๐ + (๐ โ ๐
2
2 )
๐โ๐ .
Then 8
๐ผ1 = ๐๐ฅ โซโ
0๐โ
12 (๐ฟ1(๐ ))
2
๐๐ =
= โ๐โ๐๐๐ฅ โซโโ
๐1๐โ
12 ๐ฟ
21๐๐ฟ1 =
=โ
๐2๐๐๐ฅ โซ๐1
โโ๐โ
12 ๐ฟ
21๐๐ฟ1 =
=โ
2๐๐2๐๐๐ฅฮฆ(๐1) where ๐1 = ๐ฟ1(0) as in equation (8.24) for ๐ = ๐ โ ๐ก , and also, writing that ๐2 = ๐ฟ2(0), as inequation (8.25) for ๐ = ๐ โ ๐ก,
๐ผ2 = ๐โ๐๐ โซโ
0๐โ
12 (๐ฟ2(๐ ))
2
๐๐ =
= โ๐โ๐๐โ๐๐ โซโโ
๐2๐โ
12 ๐ฟ
22๐๐ฟ2 =
=โ
๐2๐๐โ๐๐ โซ๐2
โโ๐โ
12 ๐ฟ
22๐๐ฟ2 =
=โ
2๐๐2๐๐โ๐๐ฮฆ(๐2)
,
Thus๐ข(๐, ๐ฅ) = ๐พ (๐๐ฅฮฆ(๐1) โ ๐โ๐๐ฮฆ(๐2))
and transforming back ๐ (๐ก, ๐) = ๐ข (๐ โ ๐ก, ln (๐/๐พ)) we get equation (8.23).
Observe this is a backward parabolic PDE, which implies that the terminal condition determinesthe particular solution.
8We use integration by transformation of variables: if we define ๐ง = ๐(๐ ) where ๐ โถ [๐, ๐] โ โ and ๐ โถ โ โ โ wehave that
โซ๐(๐)
๐(๐)๐(๐ง)๐๐ง = โซ
๐
๐๐ (๐(๐ )) ๐โฒ (๐ )๐๐ .
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Paulo Brito Advanced Mathematical Economics 2020/2021 24
8.7 Bibiography
โข Mathematics of PDEโs: introductory Olver (2014), Salsa (2016) and (Pinsky, 2003, ch 5).Advanced (Evans, 2010, ch 3).
โข Applications to economics (with more advanced material) : Achdou et al. (2014)
โข Applications to growth theory Brito (2004) and Brito (2011) and the references therein.
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Paulo Brito Advanced Mathematical Economics 2020/2021 25
8.A Appendix: Fourier transforms
Consider a function ๐(๐ฅ) such that ๐ฅ โ โ and โซโโโ |๐(๐ฅ)|๐๐ฅ < โ. We can define a pair of generalizedfunctions, the Fourier transform of ๐(๐ฅ), ๐น(๐ ) = โฑ[๐(๐ฅ)](๐ ) and the inverse Fourier transformโฑโ1[๐น (๐ )] (๐ฅ) = ๐(๐ฅ) (using the definition of Kammler (2000) ), where
๐น(๐ ) = โฑ[๐(๐ฅ)] โก โซโ
โโ๐(๐ฅ) ๐โ2 ๐ ๐ ๐ ๐ฅ ๐๐ฅ
where ๐2 = โ1 and๐(๐ฅ) = โฑโ1[๐น (๐ )] โก โซ
โ
โโ๐น(๐ )๐2๐๐๐ ๐ฅ๐๐ .
There are some useful properties of the Fourier transform that we use in the main text:
1. the Fourier transform preserves multiplication by a complex number ๐ โ โ:
โฑ[๐ ๐(๐ฅ)] = ๐ ๐น(๐ ), and โฑโ1[๐๐น(๐ )] = ๐๐(๐ฅ),
Proof: โฑ[๐ ๐(๐ฅ)] = โซโโโ ๐ ๐(๐ฅ) ๐โ2 ๐ ๐ ๐ ๐ฅ ๐๐ฅ = ๐ โซโโโ ๐(๐ฅ) ๐
โ2 ๐ ๐ ๐ ๐ฅ ๐๐ฅ = ๐ ๐น(๐ ), and โฑโ1[๐ ๐น(๐ )] โกโซโโโ ๐ ๐น(๐ )๐
2๐๐๐ ๐ฅ๐๐ = ๐ โซโโโ ๐น(๐ )๐2๐๐๐ ๐ฅ๐๐ = ๐ ๐(๐ฅ);
2. the Fourier transform preserves linearity:
โฑ[๐ ๐(๐ฅ) + ๐ ๐(๐ฅ)] = ๐ ๐น(๐ ) + ๐ ๐บ(๐ ), and โฑโ1[๐ ๐น(๐ ) + ๐ ๐บ(๐ )] = ๐ ๐(๐ฅ) + ๐ ๐(๐ฅ)
3. the Fourier transform does not preserve multiplication of two functions. However, there isa relationship between convolution of functions and multiplication of Fourier transforms. Aconvolution between two functions ๐(๐ฅ) and ๐(๐ฅ) is defined as
๐(๐ฅ) โ ๐(๐ฅ) = โซโ
โโ๐(๐ฆ) ๐(๐ฅ โ ๐ฆ) ๐๐ฆ.
The inverse Fourier transform of a product of two Fourier transforms is a convolution,
๐(๐ฅ) โ ๐(๐ฅ) = โฑโ1[๐น (๐ ) ๐บ(๐ )] = โซโ
โโ๐น(๐ ) ๐บ(๐ ) ๐2๐๐๐ ๐ฅ๐๐
4. โฑ[๐ฅ] = โ 12 ๐ ๐๐ฟโฒ(๐ ), where ๐ฟ(๐ฅ) is Diracโs delta. To prove this observe that
โซโ
โโ๐2๐๐๐ ๐ฅ๐ฟ(๐ )๐๐ = 1
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Paulo Brito Advanced Mathematical Economics 2020/2021 26
Therefore
๐ฅ = ๐ฅ โซโ
โโ๐2๐๐๐ ๐ฅ๐ฟ(๐ )๐๐
= โ 12 ๐ ๐ ( โซโ
โโ๐2 ๐ ๐ ๐ ๐ฅ ๐ฟ(๐ ) โ โซ
โ
โโ2 ๐ ๐ ๐ฅ ๐2 ๐ ๐ ๐ ๐ฅ๐ฟ(๐ ) ๐๐ )
= โ 12 ๐ ๐ โซโ
โโ๐2 ๐ ๐ ๐ ๐ฅ๐ฟโฒ(๐ ) ๐๐
= โ 12 ๐ ๐โฑโ1[ 12 ๐ ๐ ๐ฟ
โฒ(๐ )]
5. โฑ[๐ฅ2] = 1(2 ๐)2 ๐ฟโณ(๐ )
6. โฑ[๐ฅ ๐(๐ฅ)] = โ 12 ๐ ๐ ๐นโฒ(๐ )
Proof:
โฑ[๐ฅ ๐(๐ฅ)] = โซโ
โโ๐ฅ ๐(๐ฅ) ๐โ2 ๐ ๐ ๐ ๐ฅ ๐๐ฅ
= โ 12 ๐ ๐ โซโ
โโโ2๐ ๐ ๐ฅ ๐(๐ฅ) ๐โ2 ๐ ๐ ๐ ๐ฅ ๐๐ฅ
= โ 12 ๐ ๐ โซโ
โโ๐(๐ฅ) ๐๐๐ (๐
โ2 ๐ ๐ ๐ ๐ฅ) ๐๐ฅ
= โ 12 ๐ ๐ ๐๐๐ ๐น(๐ ) =
๐๐๐ [ โซ
โ
โโ๐(๐ฅ) ๐โ2 ๐ ๐ ๐ ๐ฅ ๐๐ฅ]
= โ 12 ๐ ๐ ๐นโฒ(๐ )
Alternative proof:
โฑ[๐ฅ ๐(๐ฅ)] = โฑ[๐ฅ] โ โฑ[๐(๐ฅ)] = โซโ
โโโ 12 ๐ ๐ ๐ฟ
โฒ(๐ฆ) ๐น(๐ โ ๐ฆ)๐๐ฆ = โ 12 ๐ ๐ ๐นโฒ(๐ )
7. if ๐ = ๐(๐ฅ, ๐ก) where ๐ก is a real variable then ๐น(๐ , ๐ก) = โฑ[๐(๐ฅ, ๐ก)] and ๐(๐ฅ, ๐ก) = โฑโ1[๐น (๐ , ๐ก)].Also ๐น๐ก(๐ , ๐ก) = โฑ[๐๐ก(๐ฅ, ๐ก)] and ๐๐ก(๐ฅ, ๐ก) = โฑโ1[๐น๐ก(๐ , ๐ก)]
8. โฑ[๐ โฒ(๐ฅ)] = 2 ๐ ๐ ๐ ๐น(๐ )
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Paulo Brito Advanced Mathematical Economics 2020/2021 27
Proof:
โฑ[๐ โฒ(๐ฅ)] = โซโ
โโ๐ โฒ(๐ฅ) ๐โ2 ๐ ๐ ๐ ๐ฅ ๐๐ฅ
integration by parts
= โซโ
โโ๐(๐ฅ) ๐โ2 ๐ ๐ ๐ ๐ฅ โ โซ
โ
โโ๐(๐ฅ) ๐๐๐ฅ(๐
โ2 ๐ ๐ ๐ ๐ฅ) ๐๐ฅ
because ๐โ2 ๐ ๐ ๐ ๐ฅ is symmetric the first integral is equal to zero
= 2 ๐ ๐ ๐ โซโ
โโ๐(๐ฅ) ๐โ2 ๐ ๐ ๐ ๐ฅ ๐๐ฅ
= 2 ๐ ๐ ๐ ๐น(๐ )
9. โฑ[๐ฅ๐ โฒ(๐ฅ)] = โ(๐น(๐ ) + ๐ ๐น โฒ(๐ )) if ๐ โ โProof:
โฑ[๐ฅ ๐ โฒ(๐ฅ)] = โฑ[๐ฅ] โ โฑ[๐ โฒ(๐ฅ)]
= โซโ
โโ( โ 12 ๐ ๐ ๐ฟ
โฒ(๐ฆ)) (2 ๐ ๐ (๐ โ ๐ฆ) ๐น(๐ โ ๐ฆ))๐๐ฆ
= โ โซโ
โโ๐ฟโฒ(๐ฆ) (๐ โ ๐ฆ) ๐น(๐ โ ๐ฆ)๐๐ฆ
= โ๐ โซโ
โโ๐ฟโฒ(๐ฆ) ๐น(๐ โ ๐ฆ)๐๐ฆ + โซ
โ
โโ๐ฟโฒ(๐ฆ) ๐ฆ ๐น(๐ โ ๐ฆ)๐๐ฆ
= โ๐ ๐น โฒ(๐ ) + โซโ
โโ๐ฟ(๐ฆ) ๐ฆ ๐น(๐ โ ๐ฆ) โ โซ
โ
โโ๐ฟ(๐ฆ) ๐น(๐ โ ๐ฆ)๐๐ฆ
= ๐ ๐น โฒ(๐ ) โ ๐น(๐ )
10. โฑ[๐โณ(๐ฅ)] = โ4 ๐2 ๐ 2 ๐น(๐ )
11. โฑ[๐ฅ ๐โณ(๐ฅ)] = 2 ๐ ๐ ๐ (2 ๐น(๐ ) + ๐ ๐นโฒ(๐ ))
12. โฑ[๐ฅ2 ๐โณ(๐ฅ)] = โ๐ 2 ๐น โณ(๐ )
Some useful results:
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Paulo Brito Advanced Mathematical Economics 2020/2021 28
Table 8.1: Fourier and inverse Fourier transforms of some functions
๐(๐ฅ) for โโ < ๐ฅ < โ ๐น(๐ ) for โโ < ๐ < โ obs๐๐ฟ(๐ฅ) ๐ ๐ constant
๐ ๐ ๐ฟ(๐ ) ๐ constant๐ฟ(๐ฅ โ ๐) ๐โ2 ๐ ๐ ๐ ๐
1โ4๐๐๐โ
๐ฅ24๐ ๐โ๐(2๐๐ )2 ๐ > 0
1โ4๐๐๐โ
(๐ฅ+๐)24๐ ๐โ๐(2๐๐ )2+๐(2๐๐๐ ) ๐ > 0, ๐ โ โ
1โ4๐๐๐๐โ
๐ฅ24๐ ๐โ๐(2๐๐ )2+๐ ๐ > 0, ๐ โ โ
1โ4๐๐๐๐โ
(๐ฅ+๐)24๐ ๐โ๐(2๐๐ )2+๐(2๐๐๐ )+๐ ๐ > 0, (๐, ๐) โ โ2
๐(๐ฅ) โ ๐(๐ฅ) ๐น(๐ ) ๐บ(๐ )
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Chapter 9
Optimal control of parabolic partialdifferential equations
9.1 Introduction
The optimal control of parabolic PDE is sometimes called optimal control of distributions.Similarly to the optimal control of ODEโs, the first order conditions include a system of forward-
backward parabolic PDEโs together with boundary conditions. The requirements for the existenceof solutions are clearly very strong, because the general solutions of the PDE system may not allowfor the boundary conditions to be satisfied. Ill-posedness is, therefore, an important issue here.
Next we present the necessary conditions for three different optimal control of parabolic PDEโs:a simple infinite horizon problem in section 9.2, an average optimal control problem in subsection9.2.1, and the optimal control of a Fokker-Planck-Kolmogorov equation in section 9.3.
9.2 A simple optimal control problem
Next we consider a simple optimal control problem for a system governed by a parabolic PDE.We have two independent variables, time ๐ก โ โ+ and another independent variable ๐ฅ โ โ and
two dependent functions, the control ๐ข = ๐ข(๐ก, ๐ฅ), mapping ๐ข โถ โ+ ร โ โ โ and a state ๐ฆ = ๐ฆ(๐ก, ๐ฅ),mapping ๐ข โถ โ+ ร โ โ โ.
The system to be controlled is given by a semi-linear parabolic partial differential equation๐ฆ๐ก = ๐ฆ๐ฅ๐ฅ + ๐(๐ก, ๐ฅ, ๐ข, ๐ฆ) where ๐(โ , ๐ข, ๐ฆ) is smooth, by an initial condition ๐ฆ(0, ๐ฅ) = ๐ฆ0(๐ฅ), wherefunction ๐ฆ0 โถ โ โ โ is and bounded, and a Neumann boundary condition lim๐ฅโยฑโ ๐ฆ๐ฅ(๐ก, ๐ฅ) = 0 isgiven for every ๐ก. The boundary condition means that the state variable should be โflatโ for verylarge absolute values of variable ๐ฅ.
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The utility functional involves both integration in time and in the other independent variable
๐ฝ[๐ข, ๐ฆ] = โซโ
0โซ
โ
โโ๐(๐ก, ๐ฅ, ๐ข(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ)) ๐๐ฅ ๐๐ก
where we assume that ๐(โ , ๐ข, ๐ฆ) is smooth and measurable, in the sense 1
โซโ+รโ
|๐(๐ก, ๐ฅ)|๐(๐ก, ๐ฅ) < โ.
Therefore our problem is to find the optimal ๐ขโ = (๐ขโ(๐ก, ๐ฅ))(๐ก,๐ฅ)โโ+รโ and ๐ฆ
โ = (๐ฆโ(๐ก, ๐ฅ))(๐ก,๐ฅ)โโ+รโthat solve the problem:
max๐ข(โ )
โซโ
0โซ
โ
โโ๐(๐ก, ๐ฅ, ๐ข(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ))๐๐ฅ๐๐ก (9.1)
subject to the constraints
โง{{โจ{{โฉ
๐๐ฆ๐๐ก =
๐2๐ฆ๐๐ฅ2 + ๐(๐ก, ๐ฅ, ๐ข(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ)), for (๐ก, ๐ฅ) โ โ+ ร โ
๐ฆ(0, ๐ฅ) = ๐ฆ0(๐ฅ), for (๐ก, ๐ฅ) โ { ๐ก = 0} ร โlim๐ฅโยฑโ
๐๐ฆ(๐ก, ๐ฅ)๐๐ฅ = 0 for (๐ก, ๐ฅ) โ โ+ ร {(๐ฅ = โโ), (๐ฅ = โ) }
(9.2)
Next we find the optimality conditions for this problem applying a distributional Pontriyaginmaximum principle.
We define the Hamiltonian
๐ป(๐ก, ๐ฅ, ๐ข, ๐ฆ, ๐) = ๐(๐ก, ๐ฅ, ๐ข, ๐ฆ) + ๐(๐ก, ๐ฅ)๐(๐ก, ๐ฅ, ๐ข, ๐ฆ)
and call ๐ = ๐(๐ก, ๐ฅ) the co-state variable.
Proposition 1 (Necessary first-order conditions ). Let (๐ขโ, ๐ฆโ) be a solution to problem (9.1)-(9.2). Then there is a co-state variable ๐ such that the following necessary conditions hold
๐๐ปโ(๐ก, ๐ฅ)๐๐ข = 0 (9.3)
๐๐(๐ก, ๐ฅ)๐๐ก = โ
๐2๐(๐ก, ๐ฅ)๐๐ฅ2 โ
๐๐ปโ(๐ก, ๐ฅ)๐๐ฆ (9.4)
lim๐กโโ
๐(๐ก, ๐ฅ) = 0 (9.5)
lim๐ฅโยฑโ
๐๐(๐ก, ๐ฅ)๐๐ฅ = 0. (9.6)
together with equations (9.2)1This condition requires that the function is bounded for every value of ๐ข(.) and ๐ฆ(.) and allow for the use of
Fubiniโs theorem, i.e, for the interchange of the integration for ๐ก and ๐ฅ. Intuitively, we should consider functions suchthat the order of integration does not matter.
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See the proof in the appendix. Equation (9.3) is a static optimality condition, that if function๐ป(๐ข, .) is sufficiently smooth, allows for the determination of the optimal control ๐ขโ as a function ofthe co-state variable, and the state variable. Equation (9.4) is a Euler-equation. In this case it is abackward parabolic PDE which encodes the incentives for changing the control variable. Equations(9.5) and (9.6) are transversality conditions, which are dual to the boundary conditions in (9.2)related to the asymptotic properties of the solution.
If functions ๐(โ ) and ๐(โ ) are sufficiently smooth in (๐ข, ๐ฆ) , we can use the implicit functiontheorem to obtain from equation (9.3)
๐ขโ = ๐(๐ก, ๐ฅ, ๐ฆ(๐ก, ๐ฅ), ๐(๐ก, ๐ฅ)),
yielding๐บ(๐ก, ๐ฅ, ๐ฆ(๐ก, ๐ฅ), ๐(๐ก, ๐ฅ)) = ๐(๐ก, ๐ฅ, ๐ขโ(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ))
and
๐ฟ(๐ก, ๐ฅ, ๐ฆ(๐ก, ๐ฅ), ๐(๐ก, ๐ฅ)) = ๐๐ฆ(๐ก, ๐ฅ, ๐ขโ(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ)) + ๐(๐ก, ๐ฅ) ๐๐ฆ(๐ก, ๐ฅ, ๐ขโ(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ))
we have a distributional MHDS system
โง{โจ{โฉ
๐๐ฆ๐๐ก =
๐2๐ฆ๐๐ฅ2 + ๐บ(๐ก, ๐ฅ, ๐ฆ, ๐), for (๐ก, ๐ฅ) โ โ+ ร โ
๐๐๐๐ก = โ
๐2๐๐๐ฅ2 โ ๐ฟ(๐ก, ๐ฅ, ๐ฆ, ๐) for (๐ก, ๐ฅ) โ โ+ ร โ.
This system has two semi-linear parabolic PDEโs: a forward parabolic PDE for the state variableand a backward parabolic PDE for the co-state variable. It is a distributional generalization of theMHDS for an optimal control problem of ODEโs.
9.2.1 Average optimal control problem
Next we consider a particular case of the previous problem in which the planner maximizes thepresent-value of an average utility function
๐ฝ(๐ฆ, ๐ข) = lim๐ฅโโ
12๐ฅ โซ
๐ฅ
โ๐ฅโซ
โ
0๐(๐ฆ(๐, ๐ก), ๐ข(๐, ๐ก))๐โ๐๐ก๐๐ก๐๐ (9.7)
where ๐ > 0 and ๐(.) is continuous and differentiable. We assume the same semi-linear parabolicconstraint.
The problem is
max๐ข
lim๐ฅโโ
12๐ฅ โซ
๐ฅ
โ๐ฅโซ
โ
0๐(๐ฆ(๐, ๐ก), ๐ข(๐, ๐ก))๐โ๐๐ก๐๐ก๐๐
subject to
๐ฆ๐ก = ๐2๐ฆ๐ฅ๐ฅ + ๐(๐ฆ, ๐ข), (๐ก, ๐ฅ) โ โ+ ร โ (9.8)๐ฆ(๐ฅ, 0) = ๐(๐ฅ), ๐ฅ โ โ (9.9)
lim๐กโโ
๐ (๐ก)๐ฆ(๐ก, ๐ฅ) โฅ 0, ๐ฅ โ โ (9.10)
lim๐ฅโยฑโ
๐ฆ(๐ก, ๐ฅ)๐ฅ = 0, ๐ก โ โ+. (9.11)
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where ๐ (๐ก) โค ๐ 0๐โ๐๐ก, where โ0 is a constant.The (current-value) Hamiltonian function is
๐ป(๐ฆ, ๐ข, ๐) โก ๐(๐ฆ, ๐ข) + ๐๐(๐ฆ, ๐ข)
where ๐(๐ก, ๐ฅ) is the current value co-state variable.The necessary first order conditions, according to the Pontryaginโs maximum principle are the
following.
Proposition 2 (Necessary conditions for the optimal average problem). Assume there are optimalprocesses for the state and the control variable, ๐ฆโ = (๐ฆโ(๐ก, ๐ฅ))(๐ก,๐ฅ)โโรโ+ and ๐ขโ = (๐ขโ(๐ก, ๐ฅ))(๐ก,๐ฅ)โโรโ+then there is a (current-value) co-state variable ๐(๐ก, ๐ฅ) such that the following conditions hold:
โข the optimality condition
๐๐ป๐๐ข (๐ฆโ(๐ก, ๐ฅ), ๐ขโ(๐ก, ๐ฅ), ๐(๐ก, ๐ฅ) ) = 0, (๐ก, ๐ฅ) โ (๐ก, ๐ฅ) โ โ++ ร โ
โข the distributional Euler equation
๐๐ก = โ๐2๐๐ฅ๐ฅ + ๐ (๐ โ๐๐ป๐๐ฆ (๐ฆ
โ(๐ก, ๐ฅ), ๐ขโ(๐ก, ๐ฅ), ๐(๐ก, ๐ฅ) ))), (๐ก, ๐ฅ) โ (๐ก, ๐ฅ) โ โ++ ร โ
โข the boundary condition, dual to equation (9.11)
lim๐ฅโยฑโ
๐โ๐๐ก ๐(๐ก, ๐ฅ)๐ฅ = 0, ๐ก โ โ++
โข the transversality condition
lim๐กโโ
๐โ๐๐ก lim๐ฅโโ
โซ๐ฅ
โ๐ฅ๐(๐, ๐ก)๐ฆ(๐, ๐ก) = 0, {๐ก = โ
See the proof in the Appendix
9.2.2 Application: the distributional ๐ด๐พ modelAs application consider a simple model in which there is a central planner in a dynastic economy whowants to maximize the average (un-weighted) utility of an economy composed with heterogeneousagents, distributed in space from a central point ๐ฅ = 0. Assume that the heterogeneity is onlygiven by their initial asset position, ๐0(๐ฅ). Each agent produces a different quantity of a good,depending only on their endowment of capital and the central planner assigns consumption which
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Paulo Brito Advanced Mathematical Economics 2020/2021 33
varies between consumers and can be different from their production (given the capital endowment).Therefore there is a distribution of savings in the economy allowing some agents to use more (less)capital than they have at the beginning of every (infinitesimal) period. This section draws uponBrito (2004) and Brito (2011), which present this model with more detail.
Consider agents located at ๐ and having the capital stock ๐พ(๐ก, ๐) and having savings ๐(๐ก, ๐) attime ๐ก. Savings is equal to income minus consumption, where we assume that income is generatedby a linear production function
๐ (๐ก, ๐) = ๐ด ๐พ(๐ก, ๐).
Savings can be applied in the own region, ๐ผ(๐ก, ๐), or in other regions ๐ (๐ก, ๐) : therefore ๐(๐ก, ๐) =๐ผ(๐ก, ๐) + ๐ (๐ก, ๐) where is trade balance. If there is no depreciation then ๐ผ(๐ก, ๐ ) = ๐๐พ๐๐ก . We order theregions according to their capital endowment then ๐ฅ can be used as a index for the regions. If, inaddition, we consider that: first, the flow of capital runs from regions with high capital intensityto regions to low capital intensity and, second, that the flow is proportional to the gradient of thecapital intensity at the boundary of region ๐ = [๐ฅ, ๐ฅ + ฮ๐ฅ], then flow
๐ (๐ก, ๐) = ๐2 โซโ๐ฅ
๐ฅ
๐๐พ๐๐ฅ (๐ก, ๐ ) ๐๐ .
If we let ฮ๐ฅ โ 0 then we find distributional capital accumulation constraint for every location ๐ฅ๐๐พ(๐ก, ๐ฅ)
๐๐ก = ๐2 ๐2๐พ(๐ก, ๐ฅ)
๐๐ฅ2 + ๐ด๐พ(๐ก, ๐ฅ) โ ๐ถ(๐ก, ๐ฅ) โ(๐ก, ๐ฅ) โ โ+ ร โ (9.12)
The problem is to maximize the average intertemporal discounted utility of consumption, ๐ถ,
๐ฝ(๐ถ, ๐พ) โก max[๐ถ]
lim๐ฅโโ
12๐ฅ โซ
๐ฅ
โ๐ฅโซ
โ
0
๐ถ(๐, ๐ก)1โ๐1 โ ๐ ๐
โ๐๐ก๐๐ก๐๐. (9.13)
subject to the distributional capital accumulation equation (9.12) and the terminal, the boundaryand the initial conditions
lim๐กโโ
๐โ โซ๐ก
0 ๐(๐ฅ,๐ )๐๐ ๐พ(๐ก, ๐ฅ) โฅ 0, โ๐ฅ โ โ (9.14)
lim๐ฅโโโ
๐พ(๐ก, ๐ฅ)๐ฅ = 0, โ๐ก โ โ+ (9.15)
๐พ(๐ฅ, 0) = ๐(๐ฅ), โ๐ฅ โ โ given. (9.16)
According to the distributional Pontriyagin maximum principle (see the Appendix) the distri-butional MHDS is
๐๐พ๐๐ก = ๐
2 ๐2๐พ๐๐ฅ2 + ๐ด๐พ โ ๐ถ, ๐ฅ โ โ, ๐ก > 0 (9.17)
๐๐ถ๐๐ก = โ๐
2 [๐2๐ถ
๐๐ฅ2 โ1 + ๐
๐ถ (๐๐ถ๐๐ฅ )
2] + ๐พ๐ถ, ๐ฅ โ โ, ๐ก > 0 (9.18)
where the endogenous growth rate i s๐พ โก ๐ด โ ๐๐ , (9.19)
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Paulo Brito Advanced Mathematical Economics 2020/2021 34
the transversality condition
lim๐กโโ
lim๐ฅโโ
12๐ฅ โซ
๐ฅ
โ๐ฅ๐โ๐๐ก๐พ(๐, ๐ก)๐ถ(๐, ๐ก)โ๐๐๐ = 0, (9.20)
and the dual boundary conditions
lim๐ฅโยฑโ
(๐๐๐ก๐ถ(๐ก, ๐ฅ)๐๐ฅ)โ1 = 0, ๐ก โฅ 0. (9.21)
In system (9.18)-(9.17), ๐ด is the net total factor productivity and ๐พ is equal to the endogenousgrowth rate in the benchmark homogeneous ๐ด๐พ model. The initial condition ๐พ(๐ฅ, 0) = ๐(๐ฅ) andthe boundary condition (9.15) should also hold.
The coupled system (9.18)-(9.17) has the closed form solution
๐พ(๐ก, ๐ฅ) = ๐๐พ๐ก๐(๐ก, ๐ฅ), ๐ถ(๐ก, ๐ฅ) = ๐๐พ๐ก๐(๐ก, ๐ฅ), ๐ก โฅ 0, ๐ฅ โ โ (9.22)
where๐(๐ก, ๐ฅ) = 1
2๐โ
๐๐๐กโซ
โ
โโ๐(๐)๐โ( ๐ฅโ๐2๐ )
2 1๐๐ก ๐๐, ๐ก > 0, ๐ฅ โ โ
and
๐(๐ก, ๐ฅ) = 12๐
โ๐๐๐ก
โซโ
โโ๐(๐) [๐ โ ๐พ + (๐ โ 1) ( 12๐๐ก โ (
๐ฅ โ ๐2๐๐๐ก )
2)] ๐โ( ๐ฅโ๐2๐ )
2 1๐๐ก ๐๐, ๐ก > 0, ๐ฅ โ โ.
A necessary condition for the existence of a solution of the centralized problem is that ๐ด >
๐พ. This model displays convergence to a time-unbounded balanced growth path similar to thehomogeneous-agent ๐ด๐พ model: capital will be equalized among regions. Figure ?? presents agraphic depiction of the detrended-solution for ๐(๐ฅ) = ๐โ|๐ฅ|. We observe that the initial hetero-geneity is eliminated along the transition
Figure 9.1: Convergence to a homogeneous asymptotic: local dynamics for the detrended ๐(๐ก, ๐ฅ)and ๐(๐ก, ๐ฅ) for the case ๐ด๐พ.
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9.3 Optimal control of the Fokker-Planck-Kolmogorov equation
In this section we consider a problem in which the PDE constraint of the economy is representedby a Fokker-Planck-Kolmogorov equation, which, as we saw models probabilities of distributionsacross time.
This problem has two particularities: first, the transport and the diffusion terms are controledendogeneously by the. control variable, second, the state variable enters in the objective functionalas a weighting variable.
In principle, this problem has other particular properties that should be highlighted:
1. the boundary conditions (9.25) introduce an initial condition is a density function such that
โซ๐
๐ฆ0(๐ฅ) ๐๐ฅ = 1
and for every point in time the density zero in the extremes of the support. This impliesthat the a conservation law should hold for every ๐ก โ ๐ ,
โซ๐
๐ฆ(๐ก, ๐ฅ) ๐๐ฅ = 1
and that the state variable is bounded in ๐ for every point in time (it is a ๐ฟ2 function);
However, in order to have this conservative property several technical problems have to be
solved. Although first-order PDEโs satisfy a conservation law for ๐ก > 0 if the initial condition, for๐ก = 0 does satisfy it, this property does not hold generally for parabolic PDEโs. Some normalizationhas to be introduced in the solution for single equations. We are not aware of the effect of this onthe solution to optimal control problem.
Therefore, the version we present next does not necessary satisfy a conservation law.Let ๐ = [๐ก,ฬฒ ๐ก] and ๐ = [๏ฟฝฬฒฬฒฬฒฬฒ๏ฟฝ, ๐ฅ], the state variable ๐ฆ โถ ๐ ร ๐ โ โ and the control variable
๐ข โถ ๐ ร ๐ โ โ.We consider the optimal control problem of a Fokker-Planck equation
max๐ข(โ )
โซ๐
โซ๐
๐(๐ก, ๐ฅ, ๐ข(๐ก, ๐ฅ)) ๐ฆ(๐ก, ๐ฅ) ๐๐ฅ ๐๐ก (9.23)
subject to
๐๐ก๐ฆ(๐ก, ๐ฅ) + ๐๐ฅ(๐(๐ฅ, ๐ข(๐ก, ๐ฅ)) ๐ฆ(๐ก, ๐ฅ)) โ ๐๐ฅ๐ฅ(โ(๐ฅ, ๐ข(๐ก, ๐ฅ)) ๐ฆ(๐ก, ๐ฅ)) = 0, a.e. (๐ก, ๐ฅ) โ ๐ ร ๐ (9.24)
and the boundary constraints
โง{โจ{โฉ
๐ฆ(๐ก,ฬฒ ๐ฅ) = ๐ฆ0(๐ฅ), (๐ก, ๐ฅ) โ {๐ก = ๐ก}ฬฒ ร ๐๐ฆ(๐ก, ๐ฅ) = 0, (๐ก, ๐ฅ) โ ๐ ร {๐ฅ = ๏ฟฝฬฒฬฒฬฒฬฒ๏ฟฝ, ๐ฅ = ๐ฅ}
(9.25)
We assume that functions ๐(โ ), ๐(โ ) and โ(โ ) are continuous and continuously differentiable asregards the control variable ๐ข(โ ).
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Paulo Brito Advanced Mathematical Economics 2020/2021 36
Proposition 3 (Optimal control of the FPK equation). Let ๐ขโ(โ ) and ๐ฆโ(โ ) be the solution of problem (9.23)-(9.24)-(9.25). Then there is a function ๐ โถ๐ ร ๐ โ โ such that:
1. the optimality condition
๐๐ข๐(๐ฅ, ๐ขโ(๐ก, ๐ฅ))๐๐ฅ๐(๐ก, ๐ฅ) + ๐๐ขโ(๐ฅ, ๐ขโ(๐ก, ๐ฅ))๐๐ฅ๐ฅ๐(๐ก, ๐ฅ) + ๐๐ข๐(๐ก, ๐ฅ, ๐ขโ(๐ก, ๐ฅ)) = 0 a.e (๐ก, ๐ฅ) โ ๐ ร ๐(9.26)
2. the distributional Euler equation
๐๐ก๐(๐ก, ๐ฅ)+๐(๐ฅ, ๐ขโ(๐ก, ๐ฅ))๐๐ฅ๐(๐ก, ๐ฅ)+โ(๐ฅ, ๐ขโ(๐ก, ๐ฅ))๐๐ฅ๐ฅ๐(๐ก, ๐ฅ)+๐(๐ก, ๐ฅ, ๐ขโ(๐ก, ๐ฅ)) = 0 a.e (๐ก, ๐ฅ) โ ๐ ร๐(9.27)
3. the transversality condition
๐(๐ก, ๐ฅ) = 0, (๐ก, ๐ฅ) โ {๐ก = ๐ก} ร ๐
4. and the admissibility constraints (9.24) and (9.25) evaluated at the optimum.
9.3.1 Application: optimal distribution of capital with stochastic redistribution
Consider an economy with heterogeneous households and that the household with capital stock๐(๐ก), at time ๐ก has the accumulation equation
๐๐(๐ก) = (๐ด๐(๐ก) โ ๐(๐ก)) ๐๐ก + ๐๐(๐ก)๐๐(๐ก)
where ๐๐ is a Wiener process. We assume that ๐ โ [0, โ). If ๐(๐ก, ๐) is the density of householdswith capital ๐ at time ๐ก the distribution function for households satisfies
โซโ
0๐(๐ก, ๐)๐๐ก = 1, for every ๐ก โ [0, โ).
Therefore, the distribution of households satisfies the FPK equation
๐๐(๐ก, ๐) + ๐๐((๐ด๐ โ ๐) ๐(๐ก, ๐)) โ12๐๐๐ ((๐๐)
2 ๐(๐ก, ๐)) = 0
where ๐(0, ๐) = ๐(๐) is given and ๐(๐ก, 0) = lim๐โโ ๐(๐ก, ๐) = 0.We assume a central planer wants to allocate consuming among households, and through time,
in order to maximize a social welfare function. We assume the social welfare function is
โซโ
0โซ
โ
0ln (๐(๐, ๐ก)) ๐(๐, ๐ก)๐โ๐๐ก ๐๐ ๐๐ก, ๐ > 0
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Paulo Brito Advanced Mathematical Economics 2020/2021 37
Applying Proposition 3 the necessary first order conditions lead to the forward-backward parabolicPDE-system
๐๐ก๐(๐ก, ๐) + ๐ด๐ ๐๐ ๐(๐ก, ๐) + ln ((๐๐๐(๐ก, ๐))โ1) +12 ๐
2๐2๐๐๐๐(๐ก, ๐) โ ๐๐(๐ก, ๐) โ 1 = 0 (9.28)
๐๐ก๐(๐ก, ๐) + ((๐ด๐ โ (๐๐๐(๐ก, ๐))โ1) ๐(๐ก, ๐))๐ โ๐22 ๐๐๐ (๐
2๐(๐ก, ๐)) = 0 (9.29)
together with a transversality condition
lim๐กโโ
๐โ๐๐ก๐(๐ก, ๐) = 0.
where ๐(๐ก, ๐) = ๐๐๐ก๐(๐ก, ๐) is the current distributional co-state variable.Because the system is recursive, we can solve the Euler equation together with the transversality
condition for ๐(๐ก, ๐). Substituting in the constraint (9.29) we obtain a linear parabolic PDE for theoptimal dynamics of the distribution
๐๐ก๐(๐, ๐ก) + ๐๐ (๐พ๐๐(๐, ๐ก)) โ๐22 ๐๐๐ (๐
2๐(๐, ๐ก)) = 0.
Solving the Cauchy problem with ๐(0, ๐) = ๐(๐) a closed form solution can be obtained
๐โ(๐ก, ๐ฅ) = โซโ
0๐(๐) ๐ (๐ก, ln (๐๐ ))
1๐ ๐๐. (9.30)
where
๐(๐ก, ๐ฆ) = (2๐๐2๐ก)โ 12 exp [(๐พ โ ๐2)๐ก โ (๐ฆ โ ๐ก(๐พ โ32 ๐2))
2
2๐2๐ก ], ๐ฅ โ โ. . (9.31)
This solution contains both a transport mechanism, which tends to generate growth at a rate
๐พ โก ๐ด โ ๐ > 0 and a diffusion mechanism, with strength ๐2. We can show that the average capitalstock is
๐๐(๐ก) = โซโ
0๐ ๐(๐ก, ๐) ๐๐ = ๐๐(0) ๐(๐พโ๐
2)๐ก, ๐ก โ [0, โ)
meaning that there is long run growth if ๐พ โ๐2 > 0, that is if the stochastic distribution of growthis not too volatile.
9.4 References
โข Optimal control problem of partial differential equations or an optimal distributed controlproblem Butkovskiy (1969), Lions (1971), Derzko et al. (1984) or Neittaanmaki and Tiba(1994) present optimality results with varying generality. We draw mainly upon the last tworeferences. See also the textbooks: Fattorini (1999) and Trรถltzsch (2010).
โข Applications in economics Carlson et al. (1996, chap.9)
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Paulo Brito Advanced Mathematical Economics 2020/2021 38
9.A Proofs
Next we present heuristic proofs of the three versions of the distributional PMP presented in themain text.
Proof of Proposition 1. The value functional is
๐ [๐ข, ๐ฆ] = โซโ
0โซ
โ
โโ๐(๐ก, ๐ฅ, ๐ข(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ))๐๐ฅ๐๐ก,
considering the constraint we have
๐ [๐ข, ๐ฆ] = โซโ
0โซ
โ
โโ[๐(๐ก, ๐ฅ, ๐ข(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ)) + ๐(๐ก, ๐ฅ) (๐(๐ก, ๐ฅ, ๐ข(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ)) โ ๐๐ฆ๐๐ก +
๐2๐ฆ๐๐ฅ2 )] ๐๐ฅ ๐๐ก =
= โซโ
0โซ
โ
โโ[๐ป(๐ก, ๐ฅ, ๐ข(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ), ๐(๐ก, ๐ฅ)) โ ๐(๐ก, ๐ฅ) ๐๐ฆ๐๐ก + ๐(๐ก, ๐ฅ)
๐2๐ฆ๐๐ฅ2 ] ๐๐ฅ ๐๐ก
= โซโ
0โซ
โ
โโ[๐ป(๐ก, ๐ฅ, ๐ข(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ), ๐(๐ก, ๐ฅ)) + (๐๐(๐ก, ๐ฅ)๐๐ก +
๐2๐(๐ก, ๐ฅ)๐๐ฅ2 (๐ก, ๐ฅ)) ๐ฆ(๐ก, ๐ฅ)] ๐๐ฅ ๐๐ก+
โ โซโ
โโ๐(๐ก, ๐ฅ)๐ฆ(๐ก, ๐ฅ) ๐๐ฅ โฃ
โ
๐ก=0+ โซ
โ
0(๐(๐ก, ๐ฅ)๐๐ฆ(๐ก, ๐ฅ)๐๐ฅ โ
๐๐(๐ก, ๐ฅ)๐๐ฅ ๐ฆ(๐ก, ๐ฅ)) ๐๐ก โฃ
โ
๐ฅ=โโ
for any control and state variables.Let us assume we know the optimal control and state variables ๐ขโ(๐ก, ๐ฅ) and ๐ฆโ(๐ก, ๐ฅ) then
๐ [๐ขโ, ๐ฆโ] = โซโ
0โซ
โ
โโ๐(๐ก, ๐ฅ, ๐ขโ(๐ก, ๐ฅ), ๐ฆโ(๐ก, ๐ฅ))๐๐ฅ๐๐ก,
and let ๐ข(๐ก, ๐ฅ) and ๐ฆ(๐ก, ๐ฅ) be admissible perturbations over the optimal levels
๐ข(๐ก, ๐ฅ) = ๐ขโ(๐ก, ๐ฅ) + ๐โ๐ข(๐ก, ๐ฅ)๐ฆ(๐ก, ๐ฅ) = ๐ฆโ(๐ก, ๐ฅ) + ๐โ๐ฆ(๐ก, ๐ฅ)
where ๐ is a constant, and โ๐ฆ(0, ๐ฅ) = 0, for every ๐ฅ โ โ, and lim๐ฅยฑโ๐โ๐ฆ(๐ก, ๐ฅ)
๐๐ฅ = 0, for every๐ก โ โ+.The integral derivative evaluated at ๐ = 0 is
๐ฟ๐ [๐ขโ, ๐ฆโ] = โซโ
0โซ
โ
โโ[ ๐๐ป
โ(๐ก, ๐ฅ)๐๐ข โ๐ข(๐ก, ๐ฅ) + (
๐๐ปโ(๐ก, ๐ฅ)๐๐ฆ +
๐๐(๐ก, ๐ฅ)๐๐ก +
๐2๐(๐ก, ๐ฅ)๐๐ฅ2 ) โ๐ฆ(๐ก, ๐ฅ)] ๐๐ฅ ๐๐กโ
โ โซโ
โโ๐(๐ก, ๐ฅ)โ๐ฆ (๐ก, ๐ฅ) ๐๐ฅ โฃ
โ
๐ก=0+ โซ
โ
0๐(๐ก, ๐ฅ)๐โ๐ฆ (๐ก, ๐ฅ)๐๐ฅ โ
๐๐(๐ก, ๐ฅ)๐๐ฅ โ๐ฆ (๐ก, ๐ฅ)) ๐๐ก
โ
๐ฅ=โโ=
where ๐ปโ(๐ก, ๐ฅ) = ๐ป(๐ก, ๐ฅ๐ขโ(๐ก, ๐ฅ), ๐ฆโ(๐ก, ๐ฅ), ๐(๐ก, ๐ฅ)). From admissibility conditions, we have
๐ฟ๐ [๐ขโ, ๐ฆโ] = โซโ
0โซ
โ
โโ[ ๐๐ป
โ(๐ก, ๐ฅ)๐๐ข โ๐ข(๐ก, ๐ฅ) + (
๐๐ปโ(๐ก, ๐ฅ)๐๐ฆ +
๐๐(๐ก, ๐ฅ)๐๐ก +
๐2๐(๐ก, ๐ฅ)๐๐ฅ2 ) โ๐ฆ(๐ก, ๐ฅ)] ๐๐ฅ๐๐กโ
โ lim๐กโโ
โซโ
โโ๐(๐ก, ๐ฅ)โ๐ฆ (๐ก, ๐ฅ)๐๐ฅ โ โซ
โ
0(๐๐(๐ก, ๐ฅ)๐๐ฅ โ๐ฆ (๐ก, ๐ฅ)) ๐๐ก โฃ
โ
๐ฅ=โโ
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Paulo Brito Advanced Mathematical Economics 2020/2021 39
Optimality requires that ๐ [๐ขโ, ๐ฆโ] โฅ ๐ [๐ข, ๐ฆ] which holds only if ๐ฟ๐ [๐ขโ, ๐ฆโ] = 0. Then optimality
conditions are as in equation (9.2).
Proof of Proposition 2. Let us assume that there is a solution (๐ขโ, ๐ฆโ), for the problem, and definethe value function as
๐ [๐ขโ, ๐ฆโ] = lim๐ฅโโ
12๐ฅ โซ
๐ฅ
โ๐ฅโซ
โ
0๐(๐ฆโ(๐, ๐ก), ๐ขโ(๐, ๐ก))๐โ๐๐ก๐๐ก๐๐.
Consider a small continuous perturbation (๐ข(๐), ๐ฆ(๐)) = {(๐ข(๐ก, ๐ฅ), ๐ฆ(๐ก, ๐ฅ)) โถ (๐ก, ๐ฅ) โ โ ร โ+}, where๐ is any positive constant, such that ๐ข(๐ก, ๐ฅ) = ๐ขโ(๐ก, ๐ฅ) + ๐โ๐ข(๐ก, ๐ฅ) and ๐ฆ(๐ก, ๐ฅ) = ๐ฆโ(๐ก, ๐ฅ) + ๐โ๐ฆ(๐ก, ๐ฅ),for ๐ก > 0, and โ๐ข(๐ฅ, 0) = โ๐ฆ(๐ฅ, 0) = 0, for every ๐ฅ โ โ. The value of this strategy is
๐ (๐) = lim๐ฅโโ
12๐ฅ โซ
๐ฅ
โ๐ฅโซ
โ
0๐(๐ข(๐, ๐ก), ๐ฆ(๐, ๐ก))๐โ๐๐ก๐๐ก๐๐.
But,
๐ (๐) โถ= lim๐ฅโโ
12๐ฅ โซ
๐ฅ
โ๐ฅโซ
โ
0๐(๐ข(๐, ๐ก), ๐ฆ(๐, ๐ก))๐โ๐๐ก๐๐ก๐๐โ
โ lim๐ฅโโ
12๐ฅ โซ
๐ฅ
โ๐ฅโซ
โ
0๐(๐, ๐ก) [๐๐ฆ(๐, ๐ก)๐๐ก โ
๐2๐ฆ(๐, ๐ก)๐๐2 โ ๐(๐ข(๐, ๐ก), ๐ฆ(๐, ๐ก))] ๐๐ก๐๐+
+ lim๐กโโ
lim๐ฅโโ
12๐ฅ โซ
๐ฅ
โ๐ฅ๐โ๐(๐,๐ก)๐(๐, ๐ก)๐ฆ(๐, ๐ก)๐๐ (9.32)
where ๐(.) is the co-state variable and ๐(.) is a Lagrange multiplier associated with the solvabilitycondition. In the optimum, the Kuhn-Tucker condition should hold
lim๐กโโ
lim๐ฅโโ
12๐ฅ โซ
๐ฅ
โ๐ฅ๐โ๐(๐,๐ก)๐(๐, ๐ก)๐ฆ(๐, ๐ก)๐๐ = 0.
By using integration by parts we find that
โซโ
0๐(๐ก, ๐ฅ)๐๐ฆ(๐ก, ๐ฅ)๐๐ก ๐๐ก = ๐(๐ก, ๐ฅ)๐ฆ(๐ก, ๐ฅ)|
โ๐ก=0 โ โซ
โ
0
๐๐(๐ก, ๐ฅ)๐๐ก ๐ฆ(๐ก, ๐ฅ)๐๐ก
and that
โซ๐ฅ
โ๐ฅโซ
โ
0๐(๐, ๐ก)๐
2๐ฆ(๐, ๐ก)๐๐2 ๐๐ก๐๐ =
= โซโ
0๐(๐, ๐ก)๐๐ฆ(๐, ๐ก)๐๐ โฃ
๐ฅ
๐=โ๐ฅโ ๐ฆ(๐, ๐ก)๐๐(๐, ๐ก)๐๐ โฃ
๐ฅ
๐=โ๐ฅ๐๐ก + โซ
๐ฅ
โ๐ฅโซ
โ
0
๐2๐(๐, ๐ก)๐๐2 ๐ฆ(๐, ๐ก)๐๐ก๐๐, (9.33)
where the second term is canceled by the boundary conditions (9.15). Then
๐ (๐) = lim๐ฅโโ
12๐ฅ โซ
๐ฅ
โ๐ฅโซ
โ
0(๐(๐ข(๐, ๐ก), ๐ฆ(๐, ๐ก))๐โ๐๐ก+
+๐๐(๐, ๐ก)๐๐ก ๐ฆ(๐, ๐ก) +๐2๐(๐, ๐ก)
๐๐2 ๐ฆ(๐, ๐ก) + ๐(๐, ๐ก)๐(๐ข(๐, ๐ก), ๐ฆ(๐, ๐ก))) ๐๐ก๐๐ โ
โ lim๐ฅโโ
12๐ฅ (โซ
๐ฅ
โ๐ฅ๐(๐, ๐ก)๐ฆ(๐, ๐ก)|โ๐ก=0 ๐๐ + โซ
โ
0๐(๐, ๐ก)๐๐ฆ(๐, ๐ก)๐๐ โฃ
๐ฅ
๐=โ๐ฅ๐๐ก)
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Paulo Brito Advanced Mathematical Economics 2020/2021 40
If an optimal solution exists, then we may characterize it by applying the variational principle,๐๐ฝ(๐ขโ, ๐ฆโ)
๐๐ = lim๐โ0๐ฝ(๐ข(๐), ๐ฆ(๐)) โ ๐ฝ(๐ขโ, ๐ฆโ)
๐ = 0.
But, defining the Hamiltonian function as ๐ป(๐ข, ๐ฆ, ๐) = ๐(๐ข, ๐ฆ) + ๐๐(๐ข, ๐ฆ), then
๐๐ฝ๐๐ = lim๐ฅโโ
12๐ฅ {โซ
๐ฅ
โ๐ฅโซ
โ
0[ ๐ป๐ข(๐ขโ(๐, ๐ก), ๐ฆโ(๐, ๐ก), ๐(๐, ๐ก))โ๐ข(๐, ๐ก)+
+ (๐๐(๐, ๐ก)๐๐ก +๐2๐(๐, ๐ก)
๐๐2 + ๐(๐, ๐ก)๐ป๐ฅ(๐ขโ(๐, ๐ก), ๐ฆโ(๐, ๐ก), ๐(๐, ๐ก))) โ๐ฆ(๐, ๐ก)] ๐๐ก๐๐
โ โซ๐ฅ
โ๐ฅ๐(๐, ๐ก)โ๐ฆ(๐, ๐ก)โฃ
โ๐ก=0 ๐๐ + โซ
โ
0๐(๐, ๐ก)๐โ๐ฆ(๐, ๐ก)๐๐ โฃ
๐ฅ
๐=โ๐ฅ๐๐ก โ
โ lim๐กโโ
โซ๐ฅ
โ๐ฅ๐(๐, ๐ก)๐โ๐(๐,๐ก)โ๐ฆ(๐, ๐ก)๐๐} . (9.34)
The last and the third to last expressions are canceled if lim๐กโโ[๐(๐ก, ๐ฅ)๐โ๐(๐ก,๐ฅ) โ ๐(๐ก, ๐ฅ)] = 0,and by the fact that โ๐(๐ฅ, 0) = 0, for any ๐ฅ. Then, substituting in the Kuhn-Tucker conditionwe get a generalized transversality condition. We get the first order conditions by equating tozero all the remaining components of ๐๐ฝ๏ฟฝ