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Journal of Mathematical Economics 44 (2008) 251–276

Rational expectations models: An approach using forward–backward stochastic differential equations Athanasios N. Yannacopoulos ∗ Department of Statistics, Athens University of Economics and Business, Patission 76, GR 10434, Greece Available online 3 June 2007

Abstract We show that a general class of continuous time rational expectations models can be reformulated as forward–backward stochastic differential equations (FBSDEs). Using this connection we obtain results on the conditions under which paths leading to, or keeping close to equilibrium exist, as well as their qualitative properties. We also provide a method for the construction of such paths through the connection of FBSDEs with quasilinear partial differential equations (PDEs). The theory is applied to specific macroeconomic models. © 2007 Elsevier B.V. All rights reserved. AMS Classiﬁcation: 60H10; 60H30; 91B02; 91B62; 91B64 JEL Classiﬁcation: C62; C65; D84 Keywords: Forward backward stochastic differential equations; Rational expectations models; Nonlinear partial differential equations

1. Introduction It is generally accepted in economics that beliefs about the future are very important in the determination of present behaviour. There seems to be basically three schools concerning the formation of expectations about the future and its effect on the present behaviour (see, e.g. Begg et al., 1989). The first school, adopts the idea that expectations are exogenous and focuses on the consequences of a change of expectations on various economic variables, but does not deal with the investigation of the cause that led to the change in expectations. The second school, that of extrapolative expectations, tries to make expectations endogenous by assuming that expectations are formed by suitable extrapolation of past economic quantities to the near future. The third school, is that of rational expectations which is based on the fundamental assumption that economic agents on average make the right assumptions about future behaviour of economic variables and that any incorrect tendencies in forecasting will be detected and put right. Thus, rational expectations models assume some sort of feedback of the expectations for future values of economic variables to their evolution and incorporate mechanisms with which an economy can return to some equilibrium state after it has been perturbed away from it. The assumption of rational expectations has become very popular in certain aspects of economic theory, for instance it forms the basis of the theory of new classical macroeconomics. We will adopt a rather abstract view of a rational expectations model in this paper. The backbone of a rational expectations model will be a set of fundamental variables of the economy which will be denoted by x. The fundamental ∗

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variables of the economy evolve under a state of uncertainty, but the information sets of the economy are assumed to be known to all economic agents. We consider also another set of economic variables, y. These variables will be called assets in this paper, and it is assumed that their values depend on the values of the fundamentals x. That holds in the sense that the value of y is a rational expectation of some function of the fundamental values x, using the information set available to the agents. These variables may take different meanings depending on the context of the model (see Section 7 for various interpretations). Agents are rational, so possible deviations from the ‘true’ expectations will in the course of time be detected and put right through the feedback of the values of y to the values of x. Therefore, the schematic form of the model will be xt = T1 (xt , yt ),

yt = T2 (Et [xt ], yt )

where xt , yt denotes the change of value in time for the fundamentals and the assets, respectively, Et [xt ] is some sort of rational expectation for future values of xt using the information available to the agents up to time t and T1 , T2 are properly chosen operators the form of which may depend on the specific model (see, e.g. Section 7 for some examples). The model above is in dynamic form and the general structure of these models is such as to drive the system to some equilibrium value (x∗ , y∗ ). The majority of these models in the absense of uncertainty, or as usually refered to in the economic literature in the presence of perfect foresight, have some equilibrium states with a saddlepoint structure. Associated with this structure are certain directions in the phase space (x, y) leading the system to equilibrium (x∗ , y∗ ) as t → ∞ (the stable saddlepaths or the stable manifolds) and certain directions driving the system away from equilibrium (the unstable saddle paths or the unstable manifolds) (see, e.g. Gandolfo, 1997). In the perfect foresight models, one may also condider the unstable manifolds as driving the system to equilibrium under reversal of the direction of time, i.e. for t → −∞, an intuitive assumption which no longer holds in the presence of uncertainty. It is the object of such theories to derive conditions under which the system will eventually be driven to equilibrium. Paths which will consistently stay away from equilibrium (unstable paths) are often called bubbles. However, a realistic economic model must necessarily account for the effects of uncertainty. In the presence of uncertainty (noise) the geometric considerations concerning the saddlepoint and the equilibrium paths can no longer be used. The concept of the phase plane which is a powerful tool in the study of qualitative dynamics in deterministic and non-autonomous dynamical systems is not, at least to the best of our knowledge, readily applicable to the case where uncertainty is introduced to perfect foresight models. Also it is very risky to make statements such as time reversal, used in the definition of the unstable manifold, as such an operation will lead to problems associated to the very nature of noise (it may, for example ruin the adaptedness of the solution to the available information set). It is thus necessary to formulate alternative theories for the treatment of rational expectations models, which are based on the intricacies introduced by noise. It is the aim of this paper to contribute towards this literature by adopting a new approach to rational expectation models, that of forward–backward stochastic differential equations. Forward–backward stochastic differential equations is a relatively new and interesting development in the field of stochastic analysis. A forward–backward stochastic differential equation (FBSDE) is a system of stochastic differential equations, part of which is treated forward in time and part of which is treated backward in time. To be more specific, some of the variables are known as initial conditions, whereas some of the variables are known in their final state, a fact that poses interesting mathematical problems on existence of solutions and their properties (see, e.g. Antonelli, 1993; Pardoux and Peng, 1990, 1992; Peng and Wu, 1999; Yong and Zhou, 1999, the monograph Ma and Yong, 1999 and references therein). Already from this discussion it appears that the models, we are interested in, present some structure reminding us of an FBSDE, as the x variables are forward variables and the y variables are backward variables whose present value depends on the future values of the x variables. This statement will be made more precise in Section 2 where we prove that a general class of rational expectations models in continuous time, which are nonlinear generalizations of the models introduced by Miller and Weller (1995) is equivalent to a class of infinite horizon FBSDEs. These models will be called alternatively stochastic saddlepoint systems, due to the fact that their deterministic counterpart, in the presence of perfect foresight, leads to a saddlepoint structure in phase space.1 Then (see Section 3) using methods from the theory of FBSDEs we present results on the solvability of these models, in properly chosen functional spaces, so that existence of paths leading to 1

The terminology of rational expectations models and stochastic saddlepoint systems for the type of models we treat in this paper, is used following the terminology of Miller and Weller (1995).

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equilibrium is guaranteed and useful results on the qualitative properties of these solutions are obtained (see Section 4). In Section 5 we consider the rational expectations models presented here as the limit of a sequence of finite horizon rational expectations models and show how these may be used to construct solutions for certain classes of rational expectations models. We then study in detail linear models in Section 6. Finally, we present some examples (see Section 7) of well known and frequently used economic models for which our approach can be used. In order to ensure the smooth flow of the paper, all mathematical proofs are collected in Appendix A. FBSDEs has been so far successfully applied to a variety of problems of mathematical finance and there exists a rich literature on this subject. The basic problem is that of contingent claim pricing, where a derivative product whose price depends on an underlying asset is to be valued. The value of the underlying asset is known at some initial time whereas the value of the contingent claim is known at some final time (as a function of the value of the underlying asset at this final time). The price processes of the underlying and the contingent claim are then the solutions of an FBSDE, the forward part of which provides the dynamics of the underlying asset and the backward part of which provides the dynamics of the contingent claim price. This approach and several interesting variations are presented, e.g. in El Karoui et al. (1997); Cvitanic and Ma (1996); Cvitanic and Karatzas (1993), etc. Another interesting application of FBSDEs in mathematical finance is their application in the theory of stochastic differential utility and its repercussions in asset pricing, Duffie and Epstein (1992). In closing this very brief account of the application of FBSDEs in financial mathematics, we should mention the use of FBSDEs in the resolution of Black’s consol rate conjecture by Duffie et al. (1995). What makes the use of FBSDEs in finance even more useful is their strong link with control theory and the possibility of alternative formulation and treatment of classical problems in terms of FBSDEs. However, to the best of our knowledge, this powerful theory has not yet been applied in economics. We hope that the present paper will attract the interest of the community to the possible use of the theory of FBSDE in economics both as a modelling and as a computational tool. 2. Stochastic saddlepoint systems as FBSDEs and different types of solutions In this section we provide a discussion of how stochastic saddlepoint systems may be rephrased as FBSDEs. Adopting the approach of Miller and Weller (1995) on continuous time stochastic saddlepoint systems, we consider the following rational expectations model: there is an economic fundamental, whose value xt follows a diffusion process. There is also an asset whose price yt is a rational expectations forecast of properly discounted future fundamentals. According to their model the fundamental and the asset prices are connected as follows: dxt = α(xt − x∗ )dt + β(yt − y∗ )dt + σ dWt , ∞ ∗ −δ(s−t) −γ(xs − x )e ds|Ft + y∗ yt = E

(1)

t

The stars denote equilibrium states, which without loss of generality will be assumed to be equal to 0. The uncertainty is assumed to be introduced into the model by a (possibly vector valued) Wiener process Wt and the rational expectations of the fundamental are taken over the information set Ft = σ(Wu , u ≤ t), the natural filtration generated by the Wiener process. The constant δ > 0 is a discount factor. The divergence from equilibrium of the asset is assumed to have some feedback effect on the fundamental’s dynamics. According to Miller and Weller (1995) a number of classic models may be cast in this form. An example is Blanchard’s model relating stock market prices to the level of real activity in the economy (Blanchard, 1981). Another model related to the above is the model of Krugman for the target zones (see, e.g. Krugman, 1991). In this case β = 0 and the asset price is assumed to have no feedback effect on the dynamics of the fundamental. The dynamics of the original model of Miller and Weller (1995) were considered to be linear. In this section we will propose a generalization of the above model exhibiting nonlinear dynamics, which includes the linear model as a special case. Most of the results in this paper are equally valid in the nonlinear and the linear case. In a later section (Section 6) we will show some results which are special to the linear case. The nonlinear generalization of the Miller and Weller model can be formulated as follows: ∞ dxt = b(xt , yt )dt + σ(xt , yt )dWt , (2) yt = E g(xs )e−δ(s−t) ds|Ft t

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where x ∈ Rn , y ∈ Rm and Wt is an s-dimensional Wiener process and Ft = σ(Wu , u ≤ t). In other words, we generalize the model so as to include n fundamentals, m assets and s sources of uncertainty (noise). The sources of uncertainty may be understood as stochastic factors driving the economy. We have assumed without loss of generality that x∗ = y∗ = 0. The functions b(t, x, y) : R+ × Rn × Rm → n R , g(x) : Rn → Rm are nonlinear functions. In the special case n = m = 1, b(t, x, y) = αx + βy, g(x) = −γx we retrieve the linear model of Miller and Weller. The introduction of the nonlinear function g(x) in the conditional expectation describing the asset dynamics, models some saturation effects. The value of the asset is some rational expectation of the deviation of the fundamental from equilibrium but its value may not keep on growing unboundedly as the deviation of the fundamental from equilibrium grows. The above arguments can be formalized by imposing the following standing assumptions on the functions b(t, x, y), σ(t, x, y), g(x): (A1) b(t, x, y), σ(t, x, y) Lipschitz continuous in x and y and bounded, (A2) g(x) is Lipschitz continuous and 0 > c1 = inf n g(x), x∈R

0 < c2 = sup g(x),are finite x ∈ Rn

Following the basic assumptions of the original model of Miller and Weller (1995) we assume that the source of uncertainty is directly observable and that all agents have access to the information set generated by this source of uncertainty. This assumption leads us to look for strong solutions of the above system. In order to proceed with the connection of the rational expectations models of the above form with FBSDEs, we need the following definitions: Deﬁnition 2.1. The space M 2 (0, T ; Rn ) is the space of all Rn -valued stochastic processes vt such that T 2 2 |vt | dt < ∞ ||vt || = E 0

The space

M 2 (0, T ; Rn )

equiped with the norm · is a Hilbert space.

We will often omit the nature of the set where the family of random variables xt or yt take values in, from the definition of the above Hilbert spaces when there is no risk of confusion, for the sake of brevity. For example, we will simply write M 2 (0, T ) instead of M 2 (0, T ; Rn ). We now define the concept of a solution of a rational expectations model (or stochastic saddlepoint system). Deﬁnition 2.2. (i) We say that the stochastic processes xt , yt are solutions of Type I of the stochastic saddlepoint system (1) if xt , yt are adapted to Ft , satisfy the relations: dxt = b(xt , yt )dt + σ(xt , yt )dWt ,

yt = E

t

∞

g(xs )e−δ(s−t) ds|Ft

and they belong to M 2 (0, ∞; Rn × Rm ). (ii) We say that the stochastic processes xt , yt are solutions of Type II of the stochastic saddlepoint system if they satisfy the above relations but the processes are locally square integrable.2 The condition xt , yt ∈ M 2 (0, ∞) is the stochastic equivalent of a no-bubble condition for the dynamics. It guarantees the boundedness of the processes xt and yt at infinity, and in some sense it guarantees that the system approaches equilibrium for long enough time (t → ∞). Thus, the processes xt , yt which are Type I solutions are in some sense analogues, in the stochastic case, of the paths on the stable manifold of the deterministic dynamic system modelling 2

Rather than in M 2 (0, ∞; Rn × Rm ).

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the economic dynamics. The solutions of Type II are bounded solutions, not leading necessarily to equilibrium, but, exhibiting finite deviation of the asset price from the prices of the fundamentals. We are now in position to show the equivalence of the rational expectations model (2) to an infinite horizon FBSDE. We have the following proposition which is one of the basic results of this paper. Proposition 2.1. (i) The Miller and Weller model (2), as far as Type I solutions are concerned, is equivalent to the inﬁnite horizon FBSDE: dxt = b(xt , yt )dt + σ(xt , yt )dWt ,

dyt = (−g(xt ) + δyt )dt − (zt , dWt ),

x0 = x, (xt , yt , zt ) ∈ M (0, ∞, R × Rm × Rm×s ) 2

n

(3)

where zt is an Ft -adapted stochastic process which is to be determined. Note that there is only an initial condition for xt but a condition at inﬁnity for yt . (ii) The Miller–Weller model (2) as far as Type II solutions are concerned is equivalent to the above inﬁnite horizon FBSDE (3) but for locally square integrable processes xt , yt and with the boundary condition yt bounded a.s., uniformly for all t. The Proof of the proposition is given in Appendix A (see Section A.1) Remark 2.1. The proof uses the boundedness of the function g(xt ). However, under the assumption that xt ∈ M 2 (0, ∞) the arguments in the proof go through even in the case where g(x) is linear. Remark 2.2. Notice the formal similarity of the above model with the problem of the Black consol rate conjecture which was treated with the use of infinite horizon FBSDEs in the work of (Duffie et al., 1995). Eventhough the Miller–Weller infinite horizon FBSDE is different in form from the one studied in Duffie et al. (1995) the techniques used in that work have inspired some of the work appearing in the present paper. The above proposition reduces the problem of well-posedness of the generalized Miller–Weller model to the solvability of the equivalent infinite horizon FBSDE. We give two definitions for solvability of the infinite horizon FBSDE related to Type I and Type II solutions of the Miller–Weller model respectively. We provide the definition of solvability for a more general form of FBSDE than the one needed here. Deﬁnition 2.3 (Solutions of FBSDE). (i) The infinite horizon FBSDE dxt = b(t, xt , yt , zt ) dt + σ(t, xt , yt , zt ) dWt , ˆ xt , yt , zt ) dWt , dyt = h(t, xt , yt , zt ) dt + σ(t,

x0 = x

(4)

admits a solution if there exist stochastic processes (xt , yt , zt ) ∈ M 2 (0, ∞, Rn × Rm × Rm×s ) such that relation (4) is satisfied. The functions b, h, σ, σˆ are suitably defined vector valued functions. Such a solution will hereafter be called Type I solution. (ii) The infinite horizon FBSDE (4) admits a solution if there exist adapted stochastic processes (xt , yt , zt ) such that relation (4) is satisfied with the extra conditions that yt is bounded a.s. for all t and that zt is square integrable. Such solutions will hereafter be called Type II solutions. In Fig. 1 we sketch Type I and Type II solutions for a particular realization of the stochastic saddlepoint system in the case where n = m = 1. Let us assume without loss of generality that the equilibrium is at the origin of the set of axes. The Type I solution is the solution of the FBSDE with the property that xt → 0 and yt → 0 as t → ∞. Clearly, this solution will correspond to the stable path of the saddlepoint, as shown in the figure. On the other hand all other orbits in the phase plane will lead to bounded yt for finite values of xt . These orbits will correspond to Type II orbits

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Fig. 1. Type I and Type II solutions for a realization of the stochastic saddlepoint system.

as shown in Fig. 1.We should emphasize that Fig. 1 is merely a sketch that may help us visualize the deterministic analogues of Type I and Type II orbits. In closing this section it is useful to provide an explanation for the adapted process zt . Technically the inclusion of this process is necessary for the existence of adapted solutions to the system. However, this process has a clear economic meaning, as follows: It may be considered as a ‘control’ procedure which is necessary to drive the system to ‘equilibrium’ and to avoid the formation of bubbles. This process is automatically derived with the solution of the problem. As we shall see in the next section, as long as the system satisfies certain solvability conditions, there exists a unique process zt with these properties. 3. Solvability of the stochastic saddlepoint system We may now pose the important question of well-posedeness of a stochastic saddlepoint model, that is: do there exist Ft -measurable square integrable stochastic processes such that the relationship imposed by model (2) holds? From the equivalence result stated in Proposition 2.1 we see that it is enough to tackle this problem using the equivalent infinite horizon FBSDE. By studying its solvability in the appropriate functional space, we may answer the question of the well-posedness of the model either in the sense of existence of paths leading to equilibrium (Type I solutions) or simply in the sense of paths that do not deviate infinitely far from equilibrium (Type II solutions). 3.1. Type II solutions We start our analysis of solvability of the Miller–Weller type FBSDEs with some results concerning Type II solutions. Rather than dealing with general Type II solutions we will focus our attention on Type II solutions such that yt = f (xt ) for some deterministic C2 function f. This type of solution is called a nodal solution and corresponds to bubble free solutions for which the asset price can be given as a deterministic function of the (stochastic) value of the fundamental. This is in accordance with previous work in the literature by Krugman (1991) and Miller and Weller (1995) where solutions of this form were proposed and studied for certain stochastic saddlepoint systems (see Section 7). Restricting our analysis to nodal solutions only, allows us to characterize the solutions of the stochastic saddlepoint system in terms of the solution of a quasilinear system of deterministic elliptic partial differential equations. This can be achieved through the celebrated four step scheme originally proposed by Ma et al. (1994). We give a brief heuristic derivation of the four step scheme before proceeding to more rigorous results. Assuming that a solution of the form yt = f (xt ) exists for the infinite horizon FBSDE system, for some f ∈ C2 , we use Itˆo’s lemma to calculate

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the differential form of the Itˆo process yt . A simple calculation yields: dyi,t = Li (x, f )fi dt + ∇fi (x)σ(x, f )dBt ,

i = 1, . . . , m

(5)

where we use the notation yt = (y1,t , . . . , ym,t ) and Li (x, f ) is the quasilinear elliptic operator: Li (x, f ) =

n

n

bj (x, f (x))

j=1

n

∂fi 1 ∂ 2 fi + (σ(x, f (x))σ(x, f (x))T )jk ∂xj 2 ∂xj ∂xk j=1 k=1

Comparing Eq. (5) with the backward part of the infinite horizon FBSDE we conclude by matching terms that f (x) must satisfy the quasilinear set of elliptic PDE: Li (x, f )fi = −gi (x) + δfi (x),

i = 1, . . . , m,

x ∈ Rn

(6)

where we consider g(x) = (g1 (x), . . . , gm (x)) ∈ Rm . Furthermore, under the assumption of existence of a nodal solution we see that zt is determined as zi,t = ∇fi (xt )σ(x, f ) In the case where m = 1, the above system reduces to a single quasilinear elliptic equation, the solution of which provides the nodal solution of the stochastic saddlepoint system. The above argument is heuristic in the sense that such a function f may not exist. In order to turn it into a rigorous proof of existence of solutions to the stochastic saddlepoint system, we need to study the solvability of the quasilinear elliptic equation given by the four step scheme. This is the aim of this section. We shall deal in this paper with the case of one asset. This corresponds to the case where m = 1, but n not necessarilly equal to 1. In this case the system of elliptic quasilinear PDEs (6) reduces to a single equation of the form: − δf + g(x) = 0, x ∈ Rn (7) where a = σσ T and b(x, f )fx = nj=1 bj (x, f (x))(∂fi /∂xj ). For the time being we relax the monotonicity assumptions on the vector field. Note also that we assume that σ is independent of z, which is often the case in the reformulation of rational expectations models as FBSDEs. The following lemma is very useful. Its proof is given in Appendix A (Section A.2.1). 1 2 Tr(fxx a(x, f )) + b(x, f )fx

Lemma 3.1. Under assumptions (A1–A2) the elliptic PDE (7) has a classical solution satisfying: inf g(x) x

δ

supg(x) ≤ f (x) ≤

x

δ

Using this lemma we may prove the existence of nodal solutions of the infinite horizon FBSDE. Proposition 3.1. Suppose that conditions (A1–A2) hold, so that the PDE (7) has at least one solution. Then, there exists at least one nodal solution (xt , yt , zt ) of the inﬁnite horizon FBSDE (3) such that the representing function is given by a solution of (7). Conversely, if (xt , yt , zt ) is a nodal solution of the inﬁnite horizon FBSDE (3), with representing function f, then f is a solution of (7). The case of more assets (m > 1) can be treated using results on the solvability of systems of quasilinear elliptic partial differential equations. 3.2. Type I solutions: the solution leading to equilibrium (the stable manifold) In this section we study the solution of the general rational expectations model (2) leading to the equilibrium point which (without loss of generality) is assumed to be x∗ = y∗ = 0. In order to find a solution that asymptotically reaches the origin for large t we need to impose certain monotonicity conditions on the vector fields. The solutions reaching the origin for long time will be considered as defining the stable manifold (stable saddle path) for the rational expectations model. In what follows we may relax the condition of bounded coefficients b(t, x, y), σ(t, x, y) slightly. The existence of Type I nodal solutions is guaranteed by the following proposition.

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Proposition 3.2. Assume that a nodal solution exists with representing function f (x) ∈ C2 such that |f (x)| < K|x|, K > 0. If furthermore the monotonicity condition 2x · b(x, f (x)) + σ 2 (x, f (x)) ≤ KT (1 + |x|2 ) holds for KT > 0 then the nodal solution is of Type I. The proof of Proposition 3.2 is given in Appendix A (Section A.2.3). The condition |f (x)| < K|x| can be checked using versions of the maximum principle (see, e.g. Pucci and Serrin, 2004). Such conditions will in general involve conditions on the parameters of the problem. For simplicity we present here one such result in the case n = m = 1, but similar results can be proved easily for any n. Proposition 3.3. Consider the case n = m = 1. Suppose that g(x) and b(x, y) are such that for some K > 0 sign(g(x)x) = e and sign([eKb(x, eKx) − eδKx + g(x)]x) = −e for e = 1 or e = −1. Then, |f (x)| < K|x|. The proof of Proposition 3.3 is given in Appendix A (Section A.2.4). Other results of the same type can be obtained using arguments from the asymptotic theory of stochastic differential equations (see, e.g. Mao, 1997), possibly relaxing the assumptions. We do not wish to get into more details on this subject at this point, but hope to return to that in future work. 4. Qualitative properties of the solution In this section we discuss qualitative properties of the solution of the rational expectations model which may be proved using the reformulation of the model as an FBSDE. Such properties are related to the location and geometry of the stable manifold as well as its uniqueness. 4.1. Geometry of the stable manifold Using the four-step scheme one may prove interesting properties for the stable manifold. For instance one may prove monotonicity properties for the representing function f that characterizes the stable manifold. The following proposition, for the case of a single fundamental, is of interest. Proposition 4.1. Let n = 1. (i) Assume that g(x) is strictly increasing. Then the representing function f is strictly monotone (increasing). (ii) Assume that g(x) is strictly decreasing. Then the representing function f is strictly monotone (decreasing). The proof of the above proposition is inspired by a proof of Duffie et al. (1995) for a similar system arising in the proof of the Black consol rate conjecture. It is possible to obtain similar monotonicity results for the case where n = 1 but the geometric intuition provided is less clear in this case. The monotonicity result proved in Proposition 4.1 may be used for obtaining information for the location of the stochastic analogue of the stable manifold in the nonlinear case. Assume that x∗ = y∗ = 0. Then, in the case where g(x) is strictly increasing we see by Proposition 4.1 that we expect the stable manifold to lie in the first and third quarterplanes (as in Fig. 2), whereas in the case where g(x) is strictly decreasing we expect the stable manifold to lie in the second and fourth quarterplanes (as in Fig. 3). 4.2. Uniqueness of the stable manifold The nodal solution of the infinite horizon FBSDE related to the rational expectations model may not be unique. The uniqueness of the nodal solution will result from the uniqueness of the solution of the PDE (7). The property of uniqueness will arise from imposing additional properties to the vector field. We close this section with some uniqueness results for the nodal solution.

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Fig. 2. The stable manifold in the case where g(x) is strictly increasing.

Proposition 4.2. Assume that n = m = 1 and let σy (x, y) = 0. (a) Let g(x) be strictly increasing and let b(x, y) satisfy the condition: δ − |fx |

1

by (x + (1 − s)f + sf )ds > 0

0

for all f (x), f (x) ∈ (inf x g(x)/δ, supx g(x)/δ). Then, the nodal solution is unique.

Fig. 3. The stable manifold in the case where g(x) is strictly decreasing.

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(b) Let g(x) be strictly decreasing and let b(x, y) satisfy the condition: δ + |fx |

1

by (x + (1 − s)f + sf ) ds > 0

0

for all f (x), f (x) ∈ (inf x g(x)/δ, supx g(x)/δ). Then, the nodal solution is unique. Remark 4.1. The conditions (a) or (b) look strange but are very natural. For instance condition (a) holds if b is decreasing in y, by < 0. That means that the change in asset price has a negative feedback on the fundamental and thus equilibrates it. Similarly, condition (b) holds if b is increasing in y, so that the change in asset price has a positive feedback on the fundamental. Note also that similar but more complicated uniqueness conditions may be stated for the general case where σ depends on y. The proof of such results may follow the spirit of comparison results for viscosity solutions. Many of the above results may be generalized for the case of higher dimensional FBSDEs (for the case m > 1, n > 1). To this end we need to use existence, and where possible, uniqueness results for systems of quasilinear elliptic equations. Here, we choose to present a simple uniqueness criterion which in some sense is the straightforward generalization of the n = 1 case uniquenesss criterion. The criterion holds in the case of one asset and more than one fundamentals (m = 1, n > 1). We have the following: Proposition 4.3. Suppose there exists a solution f of the elliptic PDE (6) such that 1 1 by (x, sf + (1 − s)f )fx ds − Tr[ay (x, sf + (1 − s)f )fxx ]ds > 0 δ− 0

0

where now by fxx we denote the Hessian matrix of f (x), x ∈ Rn and a = σσ T , for any f, f ∈ [inf x g(x)/δ, supx g(x)/δ]. Then the elliptic PDE (6) admits a unique solution and the nodal solution is unique. 5. The ﬁnite horizon limit Another problem of interest, is the problem of convergence of a solution of the finite horizon case to the infinite horizon case as T → ∞. To be more specific, consider the finite horizon problem: dxt = b(xt , yt )dt + σ(xt , yt )dWt , x0 = x,

dyt = (−g(xt ) + δyt )dt − (zt , dWt ),

yT = F (xT )

(8)

Recent results concerning the finite horizon case may be found in Delarue (2002). Under the uniqueness assumptions one may show that the solution of the finite horizon case for the interval [0, T ] converges uniformly to the nodal solution of the infinite horizon problem as T → ∞. It can be shown that under these conditions the solution of the finite horizon system (8) converges to the solution of infinite horizon system no matter what the final condition yT = F (xT ) is. This result is useful, as it shows that the choice of the exact form of the final condition for the system plays little rˆole, as long as the asymptotic properties of the system are concerned (see also Remark 5.1.5). We present here a simple version of such a convergence theorem and we hope to report on more general results elsewhere. Proposition 5.1. Assume that n = m = 1 and that σ(x, y) is independent of y. Consider the parabolic quasilinear equation: (T )

ft

(T ) + a(x)fxx + b(x, f )fx(T ) − δf (T ) + g(x) = 0,

Suppose that either

f (T ) (t = T ) = F (x)

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(i) g(x) is strictly decreasing and let b(x, y) satisfy condition: 1 δ − |fx | by (x, sφ + (1 − s)f ) ds > 0

261

(10)

0

for φ(t, x) = f (T ) (T − t, x) where f (T ) is any solution of Eq. (9) and f is the unique solution of (7) or (ii) g(x) is strictly increasing and let b(x, y) satisfy condition

1

δ + |fx |

by (x, sφ + (1 − s)f ) ds > 0

(11)

0

for φ(T − t, x) = f (T ) (t, x) where f (T ) is any solution of equation (9) and f is the unique solution of (7). Then, if (T ) (T ) (T ) (Xt , Yt , Zt ) is the nodal solution of the ﬁnite horizon system (8) with representing function f (T ) and if (xt , yt , zt ) is the nodal solution of the inﬁnite horizon system with representing function f, we have that (T )

lim E[|Xt

t→∞

(T )

− xt |2 + |Yt

− y t |2 ] = 0

uniformly in t on compact sets. Remark 5.1. Comments on, and generalizations of Proposition 5.1 1. The conditions are easy to check in certain cases of interest (Remark 4.1). 2. One may easily provide conditions for convergence of the finite horizon problem to the infinite horizon case in cases 1 where the volatility depends on y. In this case the term − 0 ay (x, sφ + (1 − s)f )dsfxx will have to be included in the conditions. 3. Similar results will hold in the multi-dimensional case where n = 1. The conditions there are more difficult to interpret, especially since we may lack monotonicity results on the solutions. 4. The case m = 1 presents further difficulties since the nodal solution will require the solution of systems of quasilinear parabolic or elliptic PDEs the theory of which is not as well developed as that of single equations. However, recent advances in the field may well be used and extended to answer questions related to long time behaviour of FBSDEs. 5. The results of the above theorem may be used for the numerical treatment of infinite horizon FBSDEs of the type arising in rational expectations models. This work is in progress and will be reported elsewhere. 6. The linear case In this section we present some results which are valid for the linear case. The linear case corresponds to the original model proposed in Miller and Weller (1995) as well as to its higher dimensional generalization. On account of the generality of this model and its wide applicability in economic theory (see also Section 7) we chose to devote special attention to the linear case. Concerning the linear case, we first provide existence results for Type I solutions (see Section 6.1), we then provide two methods for the construction of Type I and Type II solutions using the Riccati equation (see Section 6.2.1) and a generalized Burgers equation (see Section 6.2.2), respectively. These constructive methods allow us to obtain a representation of the analogue of the stable manifold and the structure of the phase space around it. 6.1. Existence of Type I solutions In this section we present a result on existence of Type I solutions for linear problems which is not based on the four step scheme, and may in general not be a nodal solution. The case of multiple assets and multiple fundamentals is treated in the next proposition, the proof of which is included in the Appendix (see Section A.5.1) Proposition 6.1. Let n = m and without loss of generality s = 1. Consider the inﬁnite horizon FBSDE: dxt = (Axt + Byt )dt + (S1 xt + S2 yt ) dWt ,

dyt = (Cxt + Dyt )dt + zt dWt

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or in matrix form ˆ t dt + (A1 vt + C1 zt ) dWt dvt = Av where ˆ = A

A

B

C

D

,

A1 =

S1

S2

0n

0n

,

C1 =

0n

In

,

vt =

xt yt

ˆ A1 ∈ R2n×2n , C1 ∈ R2n×n . By 0n , In we denote the n × n zero and identity In the above A, B, C, D, S1 , S2 ∈ Rn×n , A, matrices, respectively. Let the following assumptions hold: ˆ has distinct and real eigenvalues so that there exists a similarity transformation Q−1 AQ ˆ = Λ where Λ is a (AL1) A diagonal matrix. (AL2) Assume that Λ has n positive eigenvalues and n negative eigenvalues. The block of the matrix Λ corresponding to the negative eigenvalues will be denoted by Λ1 whereas the block corresponding to positive eigenvalues will be denoted by Λ2 . ˆ i and Ei , i = 1, . . . , 4, as follows: (AL3) Deﬁne the n × n matrices Q Q−1 :=

ˆ1 Q ˆ3 Q

ˆ2 Q ˆ4 Q

,

Q−1 A1 Q :=

E1 E3

E2

E4

ˆ 4 is invertible. and assume that Q ˆ −1 E3 is such that xT (Λ1 + S T S)x ≤ −μxT x for all x ∈ Rn , for some μ > 0. ˆ 2Q (AL4) Assume that S := E1 − Q 4 Then, the inﬁnite horizon FBSDE has a unique solution in M 2 (0, ∞).

The above proposition is also of interest in the two dimensional case (n = m = 1). In this case the proposition simplifies to the following proposition.

Proposition 6.2. Consider the linear two dimensional inﬁnite horizon FBSDE dxt = (αxt + βyt ) dt + (σ1 xt + σ2 yt ) dWt ,

dyt = (γxt + δyt ) dt + zt dWt

Let the following two assumptions hold:

(AL1 ) The matrix A=

α γ

β δ

has two real eigenvalues, ordered as Λ2 < 0 < Λ1 .

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263

(AL2 ) The noise coefﬁcients satisfy

σ1 −

σ2 μ2

2 < |Λ2 |

where μ2 =

(δ − α) +

(α − δ)2 + 4βγ 2γ

Then this system has a unique solution in M 2 (0, ∞). The nonlinear non-fully coupled case may be treated as well. There are several alternatives for the treatment of this case. For instance the nonlinear case may be treated with methods similar to those for the linear case, empowered with the use of homotopy type techniques in the spirit of Peng and Shi (2000). Such approaches are beyond the scope of the present paper and we hope to report on them in future communications. For the present work we have prefered to treat the nonlinear non-fully coupled case using the connection of FBSDEs with quasilinear elliptic equations as such connections may lead to constructive methods for the determination of the solutions. 6.2. Construction of the solutions and the stable manifold In the linear case the four step scheme presents some problems as the solutions may not be unbounded. However, the linear case may be treated in a slightly different manner using the Riccati equation. Also, as we shall show in this section explicit solutions may be found for the linear case, using the reduction of the nonlinear equation arising from the four step scheme to linear equations through the Cole-Hopf transformation. We will deal for simplicity with the case n = m = 1, but many of these results may be generalized in higher dimensions. Let us consider the finite horizon case first. For the linear case the quasilinear parabolic PDE associated with the four step scheme becomes σ2 fxx + (αx + βf )fx − (γx + δf ) = 0 (12) 2 with some properly chosen final condition. The nodal solution of the infinite horizon FBSDE will be given by the solution of the quasilinear elliptic PDE ft +

σ2 fxx + (αx + βf )fx − (γx + δf ) = 0 (13) 2 Note that for the above equations we may not use the results of the previous section on existence of classical solutions as the equations contain functions of x that may become unbounded. However, we may use alternative methods which are taylor made for the linear system to obtain important information as well as explicit representations for the solutions. 6.2.1. Solution using the Riccati equation The first type of solutions we may look for are Type I solutions. We will adopt a particular ansatz and look for solutions of the form f (t, x) = K(t)x, where K(t) is an (as yet) unspecified function of time. Substituting we obtain a necessary condition for an f (t, x) of this form to be a solution of Eq. (12): ˙ + βK2 (t) + (α − δ)K(t) − γ = 0 K(t)

(14)

This is a Riccati type equation which will be complemented with a final condition. One interesting type of solutions of the Riccati equation are the fixed points, that is solutions of the type K(t) = C where C is a constant. Substituting into the equation we see that C is the solution of the quadratic equation: βC2 + (α − δ)C − γ = 0

(15)

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Thus, we have two options for C: (δ − α) ± (δ − α)2 + 4βγ C1,2 = 2β In fact, one may show easily that even for the time dependent solutions, the only solutions of the Riccati equation that do not blow up are these that connect these two steady states. Thus, we will look for solutions of the form (xt , C1,2 xt ). The forward equation then becomes dxt = (α + βC1,2 )xt + σ dWt

x0 = x

(16)

One easily shows that

α + βC1,2 = 21 (α + δ) ± (δ − α)2 + 4βγ = Λ1,2 where Λ1,2 are the two eigenvalues of the matrix of the deterministic dynamical system. Thus the forward SDE assumes the form dxt = Λ1,2 xt + σ dWt

x0 = x

(17)

Since, we have assumed that the deterministic system is a saddlepoint, the eigenvalues are real and of opposite sign. Assuming without loss of generality that Λ2 < 0 < Λ1 we see that if we choose the solution (xt , C2 xt ) this will be an M 2 (0, ∞) solution. In fact, one may employ results from stochastic stability theory (see, e.g. Mao, 1997) to provide conditions for σ under which the stability of the forward SDE is guaranteed. Furthermore, using the Riccati approach, we explicitly construct the stable manifold solution and we may thus obtain interesting monotonicity and comparison results. 6.2.2. Solution using the Burgers equation However, this is not the only class of solutions which are of interest. One may find nonlinear solutions of the FBSDE that approach the equilibrium without reaching it. These will correspond to Type II solutions and may be found by using a method inspired by the four step scheme. Define the new variable u(t, x) = c¯ x + f (t, x) where c¯ is to be specified. In terms of the new variable the quasilinear PDE (12) becomes σ2 uxx + [(α − β¯c)x + βu]ux + [βc12 + (δ − α)¯c − γ]x − (β¯c + δ)u = 0 2 A good choice for c¯ is c¯ = −c where c is one of the solutions of the quadratic equation (15) corresponding to the slope of the stable manifold for the deterministic system. Then u corresponds to the deviation of the solution from the stable manifold. The equation for u becomes ut +

σ2 uxx + (Λ2 x + βu)ux − (δ − βΛ2 )u = 0, x ∈ R 2 Then we look for bounded solutions of this equation, with the property u(x) → 0, |x| → ∞. This asymptotic behaviour can be checked by asymptotic analysis arguments. However, other choices for c1 may be of interest, in which case the asymptotic behaviour at |x| → ∞ will be different. There are certain cases where we may obtain exact solutions of this equation. Two interesting cases were reported by Miller and Weller (1995), the case β = 0 (no feedback from the asset to the fundamental) which reduced to the Ornstein–Uhlenbeck equation, and the case where α = β = 0 which corresponds to the case where there is no drift in the fundamental. We wish to report here on the solution for a case where there is coupling between the fundamental and the asset. This case is special, since the condition α + δ = 0 must hold. Then, choosing c¯ = α/β, a simple scaling of the variables: U = Λ1 u,

X = Λ2 x,

T = Λ3 t

allows us to write Eq. (12) in the form UT + c1 UXX − c2 UUX + c3 X = 0

(18)

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265

where c1 =

Λ22 σ 2 , 2Λ3

c2 = −

βΛ2 , Λ1 Λ3

c3 =

δα − γβ Λ1 β Λ2 Λ3

This equation is in the form of Burgers equation with a source term. We may use the Cole-Hopf transformation (see, e.g. Ablowitz and Clarkson, 1991) to linearize it. Define now the new variable V by U=−

VX ∂ =− ln V V ∂X

In terms of the new variable Eq. (18) becomes

2 ∂2 ∂3 ∂ c2 ∂ − ln V − c1 3 ln V − ln V + c3 X = 0 ∂T∂X ∂X 2 ∂X ∂X We integrate once with respect to X (keeping in mind that we choose the special case where σ = const) to obtain VT c2 VX2 c3 VXX − + X2 = C − c1 + c1 − 2 V V 2 V 2 where C is an arbitrary constant of integration. With proper choice of the scaling factors Λi , i = 1, 2, 3, we may eliminate the quadratic term. In particular, choosing Λ1 Λ2 = −(β)/σ 2 we see that c1 = c2 /2 and the above equation becomes c3 −VT − c1 VXX + X2 V − CV = 0 2 (assuming that V = 0). This is a linear equation with non-constant coeffcients. We may now look for the equilibrium solution of this equation. For the proper choice of scaling factors Λi , i = 1, 2, 3, the equation for the equilibrium solution becomes

2C μ2 2 − 4X V =0 VXX + σ2 σ where μ is an arbitrary positive constant. Assume that the constant of integration C = 0. Then the general solution of this ODE can be written in terms of Bessel functions as μ μ √ 2 2 V = X AI1/4 + BK X X 1/4 2σ 2 2σ 2 where A and B are arbitrary constant. From this expression we may obtain U which becomes 1 μ + 2 XΛ(X) 2X 2σ where Λ(X) is a rational expression of Bessel functions of fractional order (the exact expression is omitted here for brevity but it is straightforward to work out). Of interest are the asymptotics of this solution for large x (we return to the original unscaled variables). Using the asymptotic representation of the Bessel functions we get that U=−

1 x for some constants K1 , K2 , i.e. the solution asymptotes for large x to the line y = K1 x which is the stable manifold for an associated deterministic saddlepoint system. For small x this solution has the asymptotic representation U ∼ K1 x + K 2

U=−

A

A x + B

where A , B are the constants. We see that this curve does not pass through the origin. We may also write down the solution of the general case C = 0. In this case the solution may be written down in terms of Kummer’s hypergeometric function.

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7. Concrete examples from economic theory In this section we present some examples of how the approach adopted in this paper may be applied to certain well established models in economic theory. 7.1. The Krugman model for target zones In the Krugman model for target zones, the exchange rate s at any time is assumed equal to Et [dst ] dt where s is the log of the spot price of foreign exchange, m is the domestic money supply, v is a shift term representing velocity of money and the last term is the expected rate of depreciation. m is considered as a policy variable which is shifted in such a way as to keep s within a specified band, the target zone. In the Krugman model, the term v is considered as the only source of external noise into the system. The evolution of v is given by the solution of the (forward) SDE st = mt + vt + γ

dvt = a(vt ) dt + σ dWt According to Krugman (1992) the basic exchange rate equation can be viewed as arising from an equation of the form st =

1 E γ

t

∞

(mt + vt )e−(1/γ)(t −t) dt |Ft

From the results in Section 2 it is evident that the Krugman model can be recast in the form of an infinite horizon FBSDE as 1 1 dxt = a(xt )dt + σ dWt , dyt = − (mt + xt ) + yt dt − zt dWt γ γ where we substituted xt = vt , yt = st to be in line with the notation used in this paper. We see that this is a decoupled system of FBSDEs since the forward equation does not depend on the backward equation. As such, this model, can be treated with methods more simple than the ones used here which are taylor made for coupled FBSDEs. For instance, as long as a(x) and σ satisfy certain monotonicity conditions of the form often used for forward SDEs, we may find that xt ∈ M 2 (0, ∞). Such a monotonicity condition may be for instance a(x)x ≤ −μx2 and σ ∈ M 2 (0, ∞). We may now turn to the backward SDE for yt . Using methods similar to those used for the proof of Theorem 4 (Peng and Shi, 2000) we may see that as long as m ∈ M 2 (0, ∞) the backward SDE has a unique solution in M 2 (0, ∞). Therefore, under simple monotonicity conditions for the evolution of the fundamentals and the assumption that m ∈ M 2 (0, ∞) we conclude that the Krugman model has a unique solution of Type I. 7.2. The Dornbusch model In the Dornbusch overshooting model the constituting equations are the following:

Et [dpt ] mt − pt = kYt − Λit , Yt = −γ it − + η(st − pt ), dt Et [dst ] = it − i∗ , dpt = φ(Yt − y¯ )dt + σ dWt dt The first equation is the condition for equilibrium of the domestic money market. In this equation, mt is the domestic money supply, pt is the domestic price level, Yt is the level of output in the economy and it is the nominal domestic interest rate. The second equation is a goods market equilibrium condition where st is the domestic price of foreign currency and st − pt is the real exchange rate. The third equation is an uncovered interest parity condition (the expected rate of depreciation of the domestic currency is set equal to the nominal interest differential) and the fourth equation is a representation of less than instantaneous price adjustment. External shocks in the economy are modeled by the

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267

introduction of the Wiener process perturbation dWt . This is a stochastic generalization of the Dornbusch model (see , e.g. Dornbusch, 1976, 1980) proposed by Neely et al. (1998). This model may be redressed in the form of an infinite horizon FBSDE as 1 (−φ(γ + Λη)xt + φΛηyt )dt + σ dWt , D 1 dyt = ((1 − kη − φγ)xt + kηyt )dt − zt dWt , D = kγ + Λ − φγΛ D where we denoted xt = pt and yt = st the forward and backward variables respectively so as to be in accordance with the notation of this paper, and without loss of generality we set i∗ = y¯ = 0. We see that the stochastic Dornbusch model is a coupled infinite horizon FBSDE and the methods presented in this paper may be used for studying the well posedness and the qualitative properties of the system. For instance, one may in a staightforward manner apply Proposition 6.2 to provide explicit conditions on the parameters of the model for which the stochastic analogue of the stable saddlepath will exist. For example we may prove the following dxt =

Proposition 7.1. Assume that σ = σ1 x + σ2 y, i.e. that the ﬂuctuations become very small as the system approaches the equilibrium state and eventually vanish on the equilibrium. The Dornbusch model has a solution in M 2 (0, ∞) (that is a stable manifold exists) as long as the parameters σ1 and σ2 satisfy the following inequality:

σ2 2 σ1 − < |Λ2 | μ2 where Λ2 =

1 2D

φ(γ + Λη) − kη − (φ(γ + Λη) − kη)2 + 4φηD ,

1 μ2 = 2(1 − kη − φγ)

kη + φ(γ + Λη) +

(φ(γ + Λη) − kη) + 4φηD 2

This inequality is seen to be valid for small enough values of the uncertainty parameters σ1 , σ2 . The condition simpliﬁes considerably in the case σ2 = 0. 7.3. The Blanchard model We end our list of examples with a treatment of the Blanchard model (1981) which was the example provided by Miller and Weller in their paper on stochastic saddlepoint systems (Miller and Weller, 1995). The equations for the model take the following form (we keep the original variables introduced in Miller and Weller (1995)): mt = a1 xt − a2 it ,

Yt = a3 xt + a4 st + gt ,

dxt = a7 (Yt − xt )dt + σdWt ,

0 < a3 < 1,

ct = a5 + a6 xt ,

Et [dst ] = [(it + ρθs σ 2 )st − ct ] dt

where mt is the level of money balances, xt the output, it the short term nominal interest rate, Yt the level of demand, st the stock market value and gt is the index of fiscal policy. The model assumes money market equilibrium (the first equation), postulates a simple relation between output and dividents and assumes that arbitrage equates the rate of return on stocks to the yield on short term bonds plus a risk premium modelled with the use of the consuption based capital asset pricing model. As shown in Miller and Weller (1995), by assuming a stabilization policy of the form gt = g¯ − a8 (xt − x∗ ), the Blanchard model may be brought in the form of Eq. (1) for xt being the output, yt = st and

a1 2 α = −a7 (1 − a3 + a8 ), β = a 4 a7 , γ= + ρθs σ s∗ − a6 , δ = i∗ a2 where i∗ = (a1 /a2 )x∗ − (m/a2 ). Using the equivalent formulation of this model as an infinite horizon FBSDE (see Proposition 2.1) we may obtain conditions on σ and the model parameters that guarantee the existence of

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a stable saddlepath. For instance, using the Riccati equation approach (see Section 6.2.1) we may prove the following. Proposition 7.2. Assume that σ is independent of x and y and that σ ∈ M 2 (0, ∞). The Blanchard model has a stable saddlepath in the positive cone as long as α + βR < 0 where R=

1 2β

(δ − α) + (δ − α)2 + 4βγ

These relations provide conditions on the policy parameter a8 . Similar results may be obtained for other choises of σ(t, x, y). Solutions of Type II may be obtained in terms of Bessel functions. 8. Conclusions In this paper we revisit rational expectations models widely used in economic theory and show their equivalence with infinite horizon FBSDEs. Through this connection we provide solvability results which guarantee the existence of paths leading to or keeping close to equilibrium. In this way we may guarantee the absence of bubbles in such models and we may provide qualitative results on the solutions, such as monotonicity. Furthermore, through this reformulation of rational expectations models constructive methods for the solution of these models, based on the four step scheme used in the solution of FBSDEs, may be proposed. Acknowledgements The author would like to thank Prof. H. Polemarchakis for useful comments and discussions. He would also like to thank Prof. I. Karatzas for his useful comments and suggestions on a preliminary version of the manuscript. The author wishes to acknowledge the comments of the anonymous referee which led to the general improvement of the presentation of the paper as well as to Section 8. Special thanks should go to Prof. N. A. Yannacopoulos for stimulating discussions on international macroeconomics. This research was partially supported by the project: “Stochastic Integrodifferential Equations and Applications”, University of the Aegean, which is co-funded 75% by the European Social Fund and 25% by National Resources(EPEAEK II) PYTHAGORAS. Appendix A. Proofs of propositions A.1. Proof of Proposition 2.1 Proof. (a) Suppose that (xt , yt , zt ) satisfy the infinite horizon FBSDE (3). Applying Itˆo’s formula on e−δt yt between t and T we get

e−δT yT − e−δt yt = −

t

T

e−δs g(xs ) ds −

t

T

e−δs zs dWs

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269

where we have used the equation for yt . This may be rearranged as −δ(T −t)

yt = e

yT +

t

T

e

−δ(s−t)

g(xs ) ds +

t

T

e−δ(s−t) zs dWs

The stochastic integral (last term in the above equation) is a martingale with respect to the filtration Ft , so taking conditional expectations we get yt = E[yt |Ft ] = E[e−δ(T −t) yT +

t

T

e−δ(s−t) g(xs ) ds|Ft ]

where we have also used the fact that yt is Ft -adapted by definition. We now let T → ∞ and see that the boundary term vanishes since Yt ∈ M 2 (0, ∞). Thus, if (xt , yt , zt ) satisfy (3) they also satisfy (2). (b) Suppose that (xt , yt ) satisfy the Miller and Weller model (according to Definition 2.2). Define the stochastic process: Ut =

∞

t

g(xs ) e−δ(s−t) ds.

We may easily show that Ut is the unique bounded solution of the following ODE with random coefficients: dUt = δUt − g(xt ) dt Then for 0 ≤ t < T < ∞ we have T Ut = UT − (δUs − g(xs )) ds t

(A.1)

Since yt satisfies the stochastic saddlepoint equation, we have that yt = E[Ut |Ft ] which using (A.1) becomes T yt = E[Ut |Ft ] = E UT − (A.2) (δUs − g(xs )) ds|Ft t

We will now use a classical result (see, e.g. Delacherie and Meyer, 1982) that states that if yt = E[Ut |Ft ]then T T E Hs Us ds|Ft = E Hs ys ds|Ft , ∀t ∈ [0, T ], P-a.s. t

t

for any bounded Ft -adapted process H. We may then rewrite Eq. (A.2) as follows: T Yt = E YT − (δys − g(xs )) ds|Ft t

(A.3)

We will now use the martingale representation theorem to bring this relation into differential form. We see that xt T is a diffusion that may be constructed as an Itˆo integral. Then YT and t xs ds are FT -measurable functionals of the T Wiener process. Define the process χ = UT − 0 (δys − g(xs )) ds. Assuming this is square integrable by the martingale representation theorem this may be expressed as T (zt , dWt ) χ = E[χ] + 0

for some vector-valued Wiener process and some square integrable process zt .

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Setting t = 0 in Eq. (A.3) we see that y0 = E[χ]. Thus using the result of the martingale representation theorem we may write T T (δys − g(xs )) ds = y0 − (zs , dWs ) YT − 0

0

We may rewrite this equality as T (δys − g(xs )) ds + YT − t

t

T

(zs , dWs ) = y0 + 0

t

(δys − g(xs )) ds −

t

(zs , dWs )

0

The RHS of the above equation is Ft -measurable. That implies that the LHS is also Ft -measurable. Using this information together with the properties of the Itˆo integral we find that T T (δys − g(xs )) ds − (zs , dWs ) YT − yt = t

t

The result follows by taking the limit T → ∞. Before ending the proof we need to clarify the following point: when using the martingale representation theorem, the stochastic process zt will in general be dependent on T, i.e. will be denoted zTt . Applying the martingale representation theorem for the final times T1 and T2 and using straightforward algebraic manipulations we may conclude (following arguments analogous to the proof of Proposition 3.5 of Duffie et al., 1995) that ZT1 = ZT2 , dt ⊗ Pa.s. A.2. Proofs of Propositions in Section 3.1 A.2.1. Proof of Lemma 3.1 Proof. The proof uses standard techniques from the theory of partial differential equations. Let us denote by Q the quasilinear operator Qu := (1/2)Tr[uxx a(x, u)] + b(x, u)ux − δu + g(x) and by BR (0) the ball in Rn of radius R, which is centered at the origin. We approximate the original problem (7) with the following problem with Dirichlet boundary conditions: g(x) . δ Under the above assumptions the approximating problem has a classical solution f. This solution is bounded. As can be seen from the maximum principle for quasilinear operators (e.g. Theorem 10.1 in Gilbarg and Trudinger, 1983) the solution of this problem is bounded by Qf (R) (x) = 0,

inf g(x) x

x ∈ BR (0),

f (R) (x)|∂BR (0) =

supg(x) ≤ f (R) ≤

x

δ δ Furthermore, this solution has bounded first and second derivatives in x. Using the interior Schauder estimates for gradients of solutions of elliptic equations one may show that the solutions and its derivatives are all bounded uniformly in R > 0. Using a standard argument we may find a subsequence f (R) (x, t) which converges uniformly to some function f (x, t) as R → ∞. The function f (x, t) has the properties f, ft , fx , fxx bounded. Thus f (x, t) is a classical solution of the elliptic equation (7)(see also Lemma 3.2, Chapter 4 of Ma and Yong, 1999). A.2.2. Proof of Proposition 3.1 Proof. The proof follows closely that of Theorem 3.3, Chapter 4, of Ma and Yong (1999). It is only briefly sketched here for completeness. To show (a) we know from Lemma 3.1 that Eq. (7) admits a classical solution f ∈ C2+α (Rn ). Considering the forward SDE: dxt = b(xt , f (xt ))dt + σ(xt , f (xt ))dWt ,

x0 = x

(A.4)

we observe that by the boundedness properties of f and its gradients, and the Lipschitz properties of b and σ this SDE admits a unique strong solution xt . Defining yt = f (xt ) and zt = σ(xt , f (xt ))T fx (xt ) and using Itˆo’s lemma (taking

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271

into account that xt is the solution of SDE (A.4), and f (x) solves (7) we see immediately that (xt , yt , zt ) solve the infinite horizon FBSDE (3)). To show (b) let (xt , yt , zt ) be a nodal solution of the FBSDE with representing function f. Applying Itˆo’s lemma on the stochastic process yt = f (xt ) and using the FBSDE we see that we have compatibility as long as F (xt ) := Qf (xt ) = 0, ∀t, a.s. Since xt is the solution of the forward SDE (with yt substituted by yt = f (xt )) using the properties of SDEs driven by Wiener processes we see that xtis a Markov process with a positive transition probability density p(t, x, y). Then, E[F (xt )2 ] = 0, which implies that Rn F 2 (y)p(t, x, y) dy = 0. From the positivity of p(t, x, y) it follows that F (x) = 0 for all x. Thus, the representing function satisfies the PDE (7). A.2.3. Proof of Proposition 3.2 Proof. Assume the existence of a nodal solution with representing function f (x). We study the asymptotic properties of the forward SDE: dxt = b(xt , f (xt )) dt + σ(xt , f (xt )) dBt Under the monotonicity condition, the existence of a unique global solution of the forward SDE, xt ∈ M 2 (0, ∞), is guaranteed by Theorem 3.6, p. 58 in Mao (1997). We now need to make sure that yt ∈ M 2 (0, ∞). Since yt = f (xt ) using the bound |f (x)| < K|x| we find that also yt ∈ M 2 (0, ∞). A.2.4. Proof of Proposition 3.3 Proof. We will use the comparison lemma, Theorem 2.4 of Pucci and Serrin (2004). Denote by F (x, u, Du, D2 u) the expression: F (x, u, Du, D2 u) := 21 [D2 ua(x, u)] + b(x, u)Du − δu + g(x) where we have used the notation D2 u = uxx for the Hessian of u and Du = ux for the gradient, for compatibility with the notation of Pucci and Serrin (2004). Of course in the case n = 1 these degenerate to the second derivative uxx and the first derivative ux , respectively. Let us concentrate now on the case n = 1. The growth condition on f (x) is equivalent to −Kx ≤ f (x) ≤ Kx for all x. One case where this certainly happens is the case where either sign(f (x)x) = 1 and −Kx ≤ f (x) for all x, or sign(f (x)x) = −1 and f (x) ≤ Kx for all x. Using the comparison principle we find conditions for the validity of each of them separately. First let x ≤ 0. Then f (x) ≥ 0 and f (x) ≤ −Kx. Using the comparison lemma (CL) we may find conditions such that this inequality holds. Choosing first, u = 0 and v = f (x), any solution of the quasilinear PDE, and applying the CL we find that f (x) ≥ 0 as long as g(x) ≥ 0. Then, choose v = −Kx and u = f (x), any solution of the quasilinear PDE, applying the CL we find that f (x) ≤ Kx as long as the condition −Kb(x, −Kx) + δKx + g(x) ≤ 0 holds for some K and all x ≤ 0. Now let x ≥ 0. Then, f (x) ≤ 0 and −Kx ≤ f (x). Using the CL we may find conditions such that this inequality holds. Choosing first u = f (x) and v = 0 and applying the CL we see that f (x) ≤ 0 as long as g(x) ≤ 0. Then, choose u = −Kx and v = f (x) and using the CL we find that −Kx ≤ f (x) as long as the condition: −Kb(x, −Kx) + δKx + g(x) ≥ 0 holds for some K and all x ≥ 0. For the second case working similarly we see that the conditions are g(x) ≤ 0

and Kb(x, Kx) − δKx + g(x) ≥ 0,

Kb(x, Kx) − δKx + g(x) ≤ 0,

x ≤ 0,

g(x) ≥ 0

and

x≥0

These two conditions may be written compactly as in the announcement of the proposition.

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A.3. Proof of Propositions in Section 4 A.3.1. Proof of Proposition 4.1 The proof of this theorem requires the following Lemma. Lemma A.1. (i) Assume that g(x) is strictly increasing and that f is a solution of the PDE (7). Suppose xmax is a local maximum of f and xmin is a local minimum of f such that f (xmin ) ≤ f (xmax ). Then xmin < xmax . (ii) Assume that g(x) is strictly decreasing and that f is a solution of the PDE (7). Suppose xmax is a local maximum of f and xmin is a local minimum of f such that f (xmin ) ≤ f (xmax ). Then xmax < xmin . Proof. Since g(x) is strictly monotone we see that f cannot be identically constant in any interval. Then xmin = xmax . At the maximum we have fx (xmax ) = 0, fxx (xmax ) ≤ 0 while at the minimum we have fx (xmin ) = 0, fxx (xmin ) ≥ 0. Substituting into Eq. (7) and using the positivity of a(x, f ) we see that f (xmin ) ≥ g(xmin )/δ whereas f (xmax ) ≤ g(xmax )/δ. Thus g(xmin ) g(xmax ) ≤ δ δ (i) Since g(x) is strictly increasing this implies xmin < xmax . (ii) Since g(x) is strictly decreasing this implies xmax > xmin . We are now ready to proceed with the proof of Proposition 4.1. Proof. Proof of Proposition 4.1 (i) Assume g(x) is strictly increasing. We will first prove that f may have no global maximum. To see this assume the contrary. Let xmax be the global maximum of f and let xmin be any local minimum. Using Lemma A.1 we see that xmin < xmax so that there may be no local minimum in the interval (xmax , ∞). That implies that f is strictly monotone decreasing in the interval (xmax , ∞). By the properties of f and its derivatives (as classical solution of the PDE (7)) we see that lim fx (x) = lim fxx (x) = 0

x→∞

x→∞

while limx→∞ f (x) exists (and is bounded). We now use Eq. (7) in the limit as x → ∞ to obtain lim f (x) =

x→∞

1 lim g(x) δ x→∞

(A.5)

On the other hand 1 g(xmax ) < lim g(x) (A.6) δ δ x→∞ In the above relation the first inequality comes from the assumption that xmax is a global maximum, the second from the proof of Lemma A.1 and the third from the fact that g(x) is strictly increasing. We see that (A.6) is in contradiction with (A.5) so f has no global maximum. We now prove that f may have no global minimum either. First of all, we see that if there is a global minimum this must be unique. This follows easily from Lemma A.1 by assuming that there may be two global minima at xmin,1 , xmin,2 to show that we are led to a contradiction. Let us denote by xmin the unique global minimum. Then, using again Lemma A.1 we see that on the interval (−∞, xmin ) there may be no local maximum, so that f is strictly monotone decreasing in that interval. Using similar arguments for x → −∞ as for x → ∞ we see that lim f (x) < f (xmax ) ≤

x→∞

lim f (x) =

x→−∞

1 lim g(x) δ x→−∞

(A.7)

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273

On the other hand lim f (x) > f (xmin ) ≥

x→−∞

g(xmin ) 1 > lim g(x) δ δ x→−∞

(A.8)

In the above relation the first inequality comes from the assumption that xmin is a global minimum, the second from the proof of Lemma A.1 and the third from the fact that g(x) is strictly increasing. We see that (A.8) is in contradiction with (A.7) so f has no global minimum. Thus, f is monotone in R. We have that lim f (x) =

x→−∞

1 1 lim g(x) < lim g(x) = lim f (x) x→∞ x→−∞ δ δ x→∞

since g is strictly increasing. From the above inequality we deduce that f is strictly monotone and increasing. This concludes the proof of (i). (ii) The proof of (ii) follows similarly and is omitted. A.3.2. Proof of Proposition 4.2 Proof. (a) Let f, f be two solutions of the differential equation (7) and w = f − f . The difference w satisfies the equation: 1

by (x, f + sw)fx ds w − δw 0 = a(x)wxx + b(x, f )wx + = a(x)wxx + b(x, f )wx −

0

δ − |fx |

1

by (x, f + sw) ds w :

0

= A(x)wxx + B(x)wx − C(x)w where we have used the fact that f is strictly monotone and increasing (see Proposition 4.1) to write fx = |fx |. If C(x) > 0 for all x ∈ R then (see Lemma A.2) the only solution of this equation is w = 0. Therefore we have uniqueness of the nodal solution for C(x) > 0 or equivalently if the stated condition holds. (b) The proof proceeds similarly only that now we substitute fx = −|fx |. In the proof above, we have used the following lemma. Lemma A.2 (Duffie et al., 1995). Let w be a bounded classical solution of the following equation: ¯ a¯ (x)wxx + b(x)w x − c(x)w = 0,

x ∈ Rn

¯ with c(x) ≥ c0 > 0, a¯ (x) ≥ 0, x ∈ Rn and with a¯ (x), b(x) bounded. Then w(x) = 0. A.4. Proof of Proposition 5.1 Proof. The proof follows in spirit that of Theorem 5.2 of Duffie et al. (1995). Let w = φ(t, x) − f . Recall that φ is simply the solution of the parabolic equation (9) with reversed time. One can show easily that w satisfies the parabolic equation: wt − a(x)wxx − b(x, φ)wx + c(t, x)w = 0 where

c(t, x) := δ − fx 0

1

by (x, f + sw) ds

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Under monotonicity assumptions on g(x) we have monotonicity results for f which allow us to simplify the above expression. If c(t, x) > η > 0 then using well known results from the theory of parabolic equations (see also Chapter 4 of Ma and Yong, 1999) we know that w is exponentially bounded in time, |w(t, x)| ≤ Ce−ηt . This provides an exponential bound for the error between the finite and the infinite horizon case. The rest follows in analogy with the proof of Theorem 5.2 of Duffie et al. (1995) who treat the problem of Black’s consol rate conjecture (see also Chapter 4 of Ma and Yong, 1999). A.5. Proof of Propositions in Section 6 A.5.1. Proof of Proposition 6.1 Proof. Define the new coordinates V = Q−1 v. Assuming that A, B, C, D are constant matrices the system becomes in the new variables: dVt = ΛVt dt + (Q−1 A1 QVt + Q−1 C1 zt ) dWt T , XT ∈ Rn×1 . Define the new variable Let us break Vt , into two parts V = (Xs,t , Xu,t )T where Xs,t u,t

ˆ 4 zt Zt = E3 Xs,t + E4 Xu,t + Q We now break the system into two parts and treat first the equation satisfied by the Xu variables: dXu,t = Λ2 Xu,t dt + Zt dWt

(A.9)

which is the backward part. By a straightforward extension of Theorem 4 of Peng and Shi (2000) we may prove that the BSDE (A.9) has a unique solution (Xu,t , Zt ) in M 2 (0, ∞). Having obtained these two processes we return to the forward SDE (the part for Xs ) and express it in terms of the known processes (Xu,t , Zt ). Straightforward algebraic manipulation gives that dXs,t = Λ1 Xs,t dt + [SXs,t + Ψ (t)]dWt ,

ˆ 2Q ˆ −1 E3 ), S := (E1 − Q 4

ˆ 2Q ˆ 2Q ˆ −1 E4 )Xu,t + Q ˆ −1 Zt ∈ M 2 (0, ∞) Ψ (t) := (E2 − Q 4 4

(A.10)

where we have used assumption (AL3). It remains to show that the forward equation (A.10) admits an M 2 (0, ∞) T X : solution. This can be shown as follows. Apply Itˆo’s formula on Xs,t s,t T T T T Xs,t ) = {2Xs,t Λ1 Xs,t + Xs,t S SXs,t + 2Ψ T SXs,t + Ψ T Ψ } dt + M dWt d(Xs,t T ≤ {Xs,t (2Λ1 + 2S T S)Xs,t + 2Ψ T Ψ } dt + M dWt T ≤ −2μXs,t Xs,t dt + 2Ψ T Ψ dt + M dWt

where we have first used H¨older’s inequality and then assumption (AL4). The term M can be easily calculated but is not needed for what follows. We now take expectations, integrate over time and rearrange to obtain T T 2μ E[|Xs,r |2 ] dr + |Xs,T |2 ≤ |Xs,0 |2 + E[|Ψ (r)|2 ] dr 0

Since μ > 0 and Ψ

0

∈ M 2 (0, ∞),

letting T → ∞ we see that Xs ∈ M 2 (0, ∞). This concludes the proof.

A.5.2. Proof of Proposition 6.2 Proof. Assume (without loss of generality) that the eigenvalues are ordered as follows: (α + δ) ± (α − δ)2 + 4βγ Λ1,2 = , Λ2 < 0 < Λ1 2 Consider the transformation to new variables: Xs = x + μ1 y,

Xu = x + μ2 y

A.N. Yannacopoulos / Journal of Mathematical Economics 44 (2008) 251–276

where μ1,2 =

(δ − α) ∓

275

(α − δ)2 + 4βγ 2γ

In the new variables the equation takes the decoupled form

−σ1 μ2 + σ2 σ 1 μ1 − σ 2 dXs,t = Λ2 Xs,t dt + Xs,t + Xu,t + μ1 zt dWt μ1 − μ 2 μ1 − μ 2

−σ1 μ2 + σ2 μ 1 σ1 − σ 2 dXu,t = Λ1 Xu,t dt + Xs,t + Xu,t + μ2 zt dWt μ1 − μ 2 μ1 − μ 2

(A.11) (A.12)

Defining the new variable: Zt =

−σ1 μ2 + σ2 +μ1 σ1 − σ2 Xs,t + Xu,t + μ2 zt μ1 − μ 2 μ1 − μ 2

the backward equation (A.12) becomes dXu,t = Λ1 Xu,t dt + Zt dWt

(A.13)

According to Theorem 4 of Peng and Shi (2000), Eq. (A.13) has a unique solution (Xu,t , Zt ) ∈ M 2 (0, ∞). In terms of the new variables (Xs,t , Xu,t , Zt ) the forward Eq. (A.11) becomes

σ2 Xs,t + Ψ (t) dWt , dXs,t = Λ2 Xs,t + − σ1 − μ2 Ψ (t) :=

−μ1 σ1 + σ2 μ1 Xu,s + Zt ∈ M 2 (0, ∞) μ2 μ2

(A.14)

Using arguments similar to those for the multidimensional case we see that the FSDE (A.14) has an M 2 (0, ∞) solution if

σ2 2 σ1 − < |Λ2 | μ2 This concludes the proof.

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