Heterogeneous Fundamentalists in a Continuous Time Model with Delays

Starting from the seminal paper by Brock and Hommes [1], attention has been placed in building on models that try to mimic the behaviours of financial markets also in the absence of stochastic shocks. A mechanism that has proved to be fruitful in this direction has been the introduction of heterogeneous expectations of financial markets operators [2].This literature has essentially been developed through the study of discrete timemodels [3]. For instance, Chiarella et al. [4] deepen themathematical properties of a two-dimensional dynamic model with fundamentalists and chartists, while Tramontana et al. [5] develop a three-dimensional nonlinear dynamic model of internationally connected financial markets. By pursuing the line of research of He et al. [6], He and Zheng [7], and He and Li [8], the aim of this paper is to inquire whether complex dynamic phenomena obtained in discrete time models also hold in continuous time models with discrete timedelays. To this purpose,we use amodel similar to that adopted by He and Westerhoff [9] and Naimzada and Ricchiuti [10]. We assume two experts that have different beliefs on the fundamental of an asset and choose their allocations by using the mean-variance criterion in every moment in time [2]. Financial operators are imitators and select allocations established by the two experts depending on performances obtained. We show that this can generate nonlinear dynamics also in a continuous time framework with time delays. In particular, coexistence of attractors as well as local and global bifurcations can occur. The rest of the paper is organised as follows. Section 2 sets up themodel. Section 3 characterises local stability properties and local bifurcations of equilibria. Section 4 provides some numerical experiments to validate the theoretical results established in Section 3, while also showing the emergence of global bifurcations. Section 5 outlines the conclusions.


Introduction
Starting from the seminal paper by Brock and Hommes [1], attention has been placed in building on models that try to mimic the behaviours of financial markets also in the absence of stochastic shocks.A mechanism that has proved to be fruitful in this direction has been the introduction of heterogeneous expectations of financial markets operators [2].This literature has essentially been developed through the study of discrete time models [3].For instance, Chiarella et al. [4] deepen the mathematical properties of a two-dimensional dynamic model with fundamentalists and chartists, while Tramontana et al. [5] develop a three-dimensional nonlinear dynamic model of internationally connected financial markets.
By pursuing the line of research of He et al. [6], He and Zheng [7], and He and Li [8], the aim of this paper is to inquire whether complex dynamic phenomena obtained in discrete time models also hold in continuous time models with discrete time delays.To this purpose, we use a model similar to that adopted by He and Westerhoff [9] and Naimzada and Ricchiuti [10].We assume two experts that have different beliefs on the fundamental of an asset and choose their allocations by using the mean-variance criterion in every moment in time [2].Financial operators are imitators and select allocations established by the two experts depending on performances obtained.
We show that this can generate nonlinear dynamics also in a continuous time framework with time delays.In particular, coexistence of attractors as well as local and global bifurcations can occur.
The rest of the paper is organised as follows.Section 2 sets up the model.Section 3 characterises local stability properties and local bifurcations of equilibria.Section 4 provides some numerical experiments to validate the theoretical results established in Section 3, while also showing the emergence of global bifurcations.Section 5 outlines the conclusions.

The Model
We consider a continuous time version of the model developed by Naimzada and Ricchiuti [10] augmented with discrete time delays.Specifically, we set up a model with a riskfree asset, characterised by a perfectly elastic supply and an instantaneous rate of return  > 0, and a risky asset, with a price per share () and a (stochastic) dividend process ().There are two types of market operators (fundamentalists and imitators) and a market maker that behaves as a Walrasian auctioneer.We assume there exist two types (Type 1 and Type 2) of fundamentalists with different beliefs on the fundamental of the traded assets.They are myopic and behave on the basis of the mean-variance criterion at each moment in time.By considering (1) in Naimzada and Ricchiuti [10, page 173], we get the following continuous time version of wealth dynamics of the Type  = 1, 2 fundamentalist: where   (), ẇ  (), and   () are the wealth, its time derivative, and the share of risky asset of fundamentalist  at time , respectively.
The objective of Type  fundamentalist is therefore the following: where  > 0 is a parameter that measures the degree of risk aversion of both agents.By assuming that the variance is constant and is given by  2 , the maximisation programme (2) gives the following market demands of the risky asset: Similarly with Naimzada and Ricchiuti [10], we assume that both fundamentalists have the same correct expectations on dividend dynamics, that is,   [ ẏ ()] = ẏ (), but heterogeneous expectations on price per share dynamics, that is,   [ ẋ ()+()] =   > 0. This means that every fundamentalist expects that the value of the risky asset tends to a level believed as being its fundamental.From (3), it follows that the share of risky asset of fundamentalist  is given by where () = ( + 1)() − ẏ () − () and  = 1/( 2 ).In addition, we set  1 <  2 without loss of generality.
In this model, fundamentalists play the role of experts in the market and other agents follow fundamentalists' choices depending on a mechanism that rewards the portfolio allocations based on the fundamental closer to the realised market price.Specifically, we assume the following adjustment rule: where () is the share of agents that follow Type 1 fundamentalist, L () is the time derivative of (), and (  − ()) 2 is the squared error related to expert .Then, this rule stimulates agents to adopt portfolio choices of the fundamentalist whose fundamental value deviates less from the price actually realised on the market.
In addition, we assume the existence of a Walrasian auctioneer that fixes the price of the risky asset according to the following mechanism: where  > 0 is the speed of adjustment of prices over time.The dynamics of price defined in (6) determines the variation of the market price of the risky asset depending on the price and allocation choices of imitators (followers) made at  − .This is a quite realistic assumption with regard to the adjustment rule of prices, given that the time gap  can be referred to small time intervals (e.g., minutes or hours) as those observed in actual financial markets.
In order to simplify notation, in what follows we omit the time dependence for variables and derivatives referred at time  and use V  to indicate the state of a generic variable V at time  − .Now, by using ( 4) and ( 5) to substitute into (6) to eliminate  , and   + L , respectively, we get where  :=  > 0.

Existence of Equilibria and Local Bifurcations
In this section we perform the analysis of the delay differential equation defined in (7) (see, e.g., [11]).Equilibria of ( 7) are obtained by setting Ṗ = 0 and   =  =  * for all .By doing this, one finds that the nontrivial equilibria  * of (7) must satisfy the following equation: Consequently, (7) has three positive equilibria:  * =  1 ,  * =  2 , and  * = ( 1 +  2 )/2.Let ℎ =  −  * .Then (7) can be transformed into The characteristic equation of the linear part of ( 9) is given by where When there is no delay, that is,  = 0 in (9), the characteristic equation becomes (, 0) =  −  = 0.Then,  * =  1 and  * =  2 are locally asymptotically stable ( = − < 0), while  * = ( 1 +  2 )/2 is unstable ( =  > 0).Now, assume that  > 0 in (9).We will investigate the location of the roots of the transcendental equation.First, it is immediate that (10) Proof.For  > 0,  =  is a root of ( 10 From (12), we note that there exists an inverse relationship between the value of  (and thus of the speed of adjustment of prices over time ) and the bifurcation value of .This implies that the speed of adjustment of prices over time  plays a destabilising role in the model (i.e., for a high value of  the Hopf bifurcations occur for a low value of ), while the variance  2 and the degree of risk aversion  play a stabilising role (i.e., for high values of  2 and  the Hopf bifurcations occur for a high value of ).Lemma 2.  =  0 is a simple purely imaginary root of (10), and all other roots  ̸ =  0 satisfy  ̸ =  0 for any integer .Proof.If  =  0 is not simple, that is, ( 0 ,   ) =   ( 0 ,   ) = 0, then one has 1 +  0   = 0, which is a contradiction.Let us assume that there exists a root   such that (  ,  0 ) = 0 and   =  0 for some  ̸ = 0, ±1.From (14), we get  2 =  2 / 2 0 = 1.Therefore, the statement follows immediately.
Let   = ]  () +   () be the roots of (10) close to  =   that satisfy ]  (  ) = 0 and   (  ) =  0 , where  0 and   are defined by (12).The next result indicates that each crossing of the real part of characteristic roots at   must be from left to right; that is, stability is lost at the smallest stability switch and it cannot be regarded later.
Proof.Differentiating both sides of (10) with respect to  gives Then, we have which implies concluding the proof.
Proof.The first part follows noting that the equilibrium  * is locally asymptotically stable when  = 0 and so its stability can only be lost if eigenvalues cross the imaginary axis from left to right.Let ( 0 ) = ]+ be a root of ( 10) with ] > 0. Then ] = − −] 0 cos  0 ,  =  −] 0 sin  0 , and thus, we derive (1/)tan −1 (−/]) =  0 > 0, which is a contradiction the left-hand side term of this equation being a negative number.Since the rate of change of the real part of an eigenvalue with respect to  when  =   is positive, then the number of roots with positive real parts is increasing.Due to the fact above, the number of roots of the characteristic equation with positive real part will be constant for 0 ≤  <  0 and equal to the number of eigenvalues with positive real parts when  = 0 (i.e., zero being  * stable).For each subsequent interval   <  <  +1 , the number can be determined from the number in the previous interval  −1 <  <   and the number of roots with zero real part at   .
Spectral properties in the previous lemma immediately lead to stability properties of the zero solution of (9) and, equivalently, of the positive equilibrium  * of (7).Theorem 5. Let   be defined as in (12).
Remark 6.Since when  = 0 (7) collapses into a onedimensional autonomous differential equation, then in the continuous time version of the model by Naimzada and Ricchiuti [10] no persistent oscillations can occur, and the dynamics are therefore monotonic and convergent towards one of the equilibria.This result stresses the importance of

Numerical Simulations
In this section we provide some numerical simulations to validate the theoretical results on local bifurcations stated in Section 3, and we show the occurrence of global bifurcations as well.For this purpose, we fix the following parameter set,  = 0.2,  1 = 2, and  2 = 8, and let  vary.For this parameter values the Hopf bifurcation occurs at  0 ≃ 7.854.For  <  0 (but sufficiently high), the dynamics of the model are oscillatory and convergent towards one of the equilibria depending on the initial conditions (as shown in Figure 1).Just after the Hopf bifurcation value there exist two attracting closed invariant curves (each of which with its basin of attraction), and then the dynamics of prices (and also the shares of imitators of Type 1 and For  =  0 , the equilibrium undergoes a Hopf bifurcation and the dynamics are characterised by oscillations with a maximum and a minimum for  ∈ ( 0 ,  1 ), with  1 ≃ 11.6.Just after  1 it is possible to observe the birth of another local extremum.For  =  2 , a global bifurcation occurs and the dynamics lies in a larger portion of the phase plane.We note that for  >  2 there are several windows in which many maxima and minima coexist, so that fluctuations become more complicated.
Type 2 fundamentalists) show persistent oscillations (see Figures 2(a) and 2(b)) characterised by unique maximum and minimum points.By increasing the value of , other phenomena are possible.In particular, we observe an increase in the number of local maxima and minima that resembles the sequence of period-doubling bifurcations shown by Naimzada and Ricchiuti [10] in a discrete time model (see Figures 3(a

Conclusion
The debate on whether it is better to adopt continuous time models or discrete time models to describe the behaviours of financial markets operators is still open, as pointed out by He and Li [8, page 974]: "The [discrete time] set up facilitates economic understanding of the role of heterogeneous expectations and mathematical analysis, it, however, faces a limitation when dealing with expectations formed from the lagged prices over different time horizons and a challenge to characterise the adaptive behaviour in a continuous-time." Generally speaking, in the absence of stochastic shocks the continuous time framework tends to generate regular dynamics.Nevertheless, when one assumes that some economic processes (e.g., the adjustment of prices) react to changes occurred at a certain time gap, the dynamics generated by continuous time models tend to mimic those generated by discrete time models.This paper has confirmed the existence of nonlinear asset price dynamics in a continuous time version (with discrete time delays) of the model proposed by Naimzada and Ricchiuti [10], characterised by the existence of fundamentalists with heterogeneous expectations on the value of a risky asset.

Figure 3 :
Figure 3:  = 12.(a) Two attracting closed invariant curves.(b) Corresponding time series with two local maxima and minima.

Figure 4 :
Figure 4:  = 14.3.(a) A unique attractor generated by the global bifurcation.(b) Corresponding time series with large oscillations.

Figure 5 :
Figure 5: Bifurcation diagram for  showing local maxima and minima of () for a given value of .Given the initial condition, for  <  0 the dynamics converges to the equilibrium  2 = 8.For  =  0 , the equilibrium undergoes a Hopf bifurcation and the dynamics are characterised by oscillations with a maximum and a minimum for  ∈ ( 0 ,  1 ), with  1 ≃ 11.6.Just after  1 it is possible to observe the birth of another local extremum.For  =  2 , a global bifurcation occurs and the dynamics lies in a larger portion of the phase plane.We note that for  >  2 there are several windows in which many maxima and minima coexist, so that fluctuations become more complicated.
) and 3(b)).When  increases further, a global bifurcation occurs and the two attractors merge each other.Then, a unique attractor is born (as shown in Figures 4(a) and 4(b)).At this stage, the dynamics of prices are characterised by large oscillations and possibly a high degree of unpredictability as suggested by the bifurcation diagram shown in Figure 5.