A random time-dependent noncooperative equilibrium problem

Abstract

The paper deals with the random time-dependent oligopolistic market equilibrium problem. For such a problem the firms’ point of view has been analyzed in Barbagallo and Guarino Lo Bianco (Optim. Lett. 14: 2479–2493, 2020) while here the policymaker’s point of view is studied. The random dynamic optimal control equilibrium conditions are expressed by means of an inverse stochastic time-dependent variational inequality which is proved to be equivalent to a stochastic time-dependent variational inequality. Some existence and well-posedness results for optimal regulatory taxes are obtained. Moreover a numerical scheme to compute the solution to the stochastic time-dependent variational inequality is presented. Finally an example is discussed.

1 Introduction

In the recent years stochastic dynamic optimization models received a lot of attention and have applications in many different areas. In particular stochastic variational inequalities arise from problems with conditions of randomness where the data are affected by a certain degree of uncertainty (see [17]). Such topics involved a lot of authors: see for example [14] or the recent paper [24] where stochastic variational inequalities with anticipativity in a dynamic multistage setting have been studied. Moreover, it is worth to highlight that inverse and control problems associated with stochastic PDEs also lead to stochastic variational inequalities (see [18]).

The purpose of this note is to analyze an effective oligopolistic market equilibrium model using the theory of stochastic and time-dependent variational inequalities combined with the Nash equilibrium theory. Similar analysis has been obtained in [7] where the authors studied a random time-dependent oligopolistic market equilibrium problem in presence of both production and demand excesses from the firms’ point of view. Here we focus our attention on the policymaker’s point of view. More precisely, control policies are implemented by imposing higher taxes or subsidies in order to restrict or encourage exportations. We prove that the equilibrium conditions can be formulated by means of an inverse stochastic time-dependent variational inequality. We investigate also the well-posedness of the inverse stochastic time-dependent variational inequality and the connection between the well-posedness of a stochastic time-dependent variational inequality and its inverse. Moreover we present a numerical method to compute the random dynamic oligopolitistic market equilibrium distribution. In the literature numerical methods to solve stochastic variational inequalities are available. A first numerical method was proposed in [21] but the convergence was ensured under very strong hypothesis on the function of the variational inequality. Later many other stochastic approximation methods for stochastic variational inequalities have been developed (see for instance [16, 20]). In our case we deal with a stochastic time-dependent variational inequality problem. Then we introduce a first instance of an iterative procedure for such inequalities which is based on the stochastic continuity result obtained in [7]. Thanks to that, a discretization of the time interval can be performed and, then, a projection method to solve the stochastic variational inequalities can be applied. Some reference for the numerical resolution of dynamic variational inequalities can be found, for instance, in [2, 3].

The time-dependent oligopolistic market equilibrium problem was intensively studied in the deterministic setting. It was introduced and deep analyzed starting by [4]. After that several variations of the model have been considered. More precisely in [11, 12] the presence of production and demand excesses, occurring during an economic crisis period or when the physical transportation of commodity between a firm and a demand market is evidently limited, has been introduced. Recently, the oligopolistic market model has been extended by allowing the possibility to a company to produce more than one good: the theoretical tools is based, in this case, on the tensor variational inequality theory (see [8,9,10] and the reference therein). The attention on the policymaker’s point of view (whose aim is to control the commodity exportations by means of the imposition of taxes or incentives) was studied in [13] where the authors formulate the resulting optimization problem as an inverse variational inequality. Inverse variational inequalities can be considered as a special case of general variational inequalities and can be used to model various control problems (see [15] for details). Note that only recently the strict connection between classical variational inequalities and inverse variational inequalities has been studied (see, e.g., [25]).

The introduction of uncertainty in the oligopolistic market equilibrium problem arises because the constraints or the data vary in a non-regular and unpredictable manner (think about unpredictable events). Thus a suitable model has to permit to handle random constraints. The models in presence of uncertainty have been analyzed in [6] while in [5] the authors add production and demand excesses.

The mathematical setting is the one of Hilbert spaces, which allows us to obtain existence and regularity results. In this paper we study inverse stochastic time-dependent variational inequalities: results are obtained concerning existence and well-posedness.

The paper is organized as follows. In Sect. 2, the random oligopolistic market equilibrium problem is presented. We describe the firms’ point of view obtaining the equivalence between the random Cournot-Nash equilibrium condition and an appropriate stochastic time-dependent variational inequality. Then we introduce the policymaker’s point of view, proving that the random dynamic optimal regulatory tax is a solution to an inverse stochastic time-dependent variational inequality. In Sect. 3 some existence results are obtained. In Sect. 4 we study the well-posedness of an inverse stochastic time-dependent variational inequality, which is, under suitable conditions, equivalent to the existence and uniqueness of its solution. In Sect. 5 a numerical scheme to compute the random dynamic oligopolistic market equilibrium distribution, based on a combination between a discretization procedure and a projection method, is presented. Finally an example is provided.

2 The random dynamic oligopolistic market equilibrium model

The aim of the section is to present the random dynamic oligopolistic market equilibrium problem. This is the problem of finding a trade equilibrium in a supply-demand market between a finite number of spatially separated firms which produce a homogeneous commodity and ship the commodity to some demand markets. We suppose that the data are affected by a certain degree of uncertainty and depending on the time. We analyze first the firms’ point of view of the problem and then we introduce the policymaker’s point of view, namely the random dynamic optimal control equilibrium problem.

2.1 The firms’ point of view

Let us start to consider the firms’ point of view of the random dynamic oligopolistic market equilibrium problem. Let $T>0$, let $\Omega$ be an open subset of $\mathbb {R}$ and let $(\Omega , \mathcal {F}, \mathbb {P})$ be a probability space. Let $([0,T] \times \Omega , \mathcal {B}([0,T])\otimes \mathcal {F}, \mathcal {L}^1\otimes \mathbb {P})$ be the product measure space, where $\mathcal {B}([0,T])$ is the Borel $\sigma$-field of [0, T] and $\mathcal {L}^1$ is the 1-dimensional Lebesgue measure on [0, T]. A point in $[0,T]\times \Omega$ will be denoted by the couple $(t,\omega )$. If $Y=Y(t,\omega )$ is a measurable function on $[0,T]\times \Omega$, then the function defined as

$$\begin{aligned} X(\omega )=\int _0^TY(\xi ,\omega )\,d\xi \end{aligned}$$

is a random variable on $\Omega$ and its expectation is given by

$$\begin{aligned} \mathbb {E}(X)=\int _\Omega X(\omega )\,d\mathbb {P}. \end{aligned}$$

A stochastic process is a family of random variables $X_t(\omega )=X(t,\omega )$ on $\Omega$ indexed by the time variable $t\in [0,T]$. Correspondingly, the random function $t \longmapsto X(t,\omega )$, $\omega \in \Omega$ is called sample path at $\omega$. Let us denote by $L^2([0,T] \times \Omega , \mathbb {R}^{k}, \mathbb {P})$ the Hilbert space of stochastic processes X from $[0,T] \times \Omega$ to $\mathbb {R}^k$ such that the expectation $\mathbb {E}(X) < \infty$. Moreover, we define in $L^2([0,T] \times \Omega , \mathbb {R}^{k}, \mathbb {P})$ the following bilinear form on $(L^2([0,T] \times \Omega , \mathbb {R}^{k}, \mathbb {P}))^*\times L^2([0,T] \times \Omega , \mathbb {R}^{k}, \mathbb {P}),$ through the expectation, by

$$\begin{aligned} \ll \Xi ,X \gg _{\mathbb {E}}= \int _0^T \int _\Omega \left\langle \Xi (t,\omega ),X(t,\omega )\right\rangle dt \, d\mathbb {P}, \end{aligned}$$

where $\Xi \in (L^2([0,T] \times \Omega ,\mathbb {R}^k, \mathbb {P}))^*=L^2([0,T] \times \Omega ,\mathbb {R}^k, \mathbb {P}),$ $X \in L^2([0,T] \times \Omega ,\mathbb {R}^k, \mathbb {P})$ and

$$\begin{aligned} \left\langle \Xi (t,\omega ), X(t,\omega )\right\rangle =\sum _{l=1}^k\Xi _l(t,\omega ) X_l(t,\omega ), \quad \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.} \end{aligned}$$

Let us consider m firms $P_i$, $i=1,\ldots ,m$, which produce a homogeneous commodity and n demand markets $Q_j$, $j=1,\ldots ,n,$ which are generally spatially separated. Assume that the homogeneous commodity, produced by the m firms and consumed by the n markets, depends on random variables. Let $p_i$ be the random time-dependent variable expressing the nonnegative commodity output produced by firm $P_i$ and suppose that $p_i=p_i(t,\omega )$, $(t,\omega )\in [0,T] \times \Omega$, $i=1,\ldots ,m$. Let $q_j$ be the random time-dependent variable expressing the nonnegative demand for the commodity of demand market $Q_j,$ namely $q_j=q_j(t,\omega ),$ $(t,\omega ) \in [0,T] \times \Omega$, $j=1,\ldots ,n$. Let $x_{ij}$ be the random time-dependent variable expressing the nonnegative commodity shipment between the supply producer $P_i$ and the demand market $Q_j$, namely $x_{ij}=x_{ij}(t,\omega )$, $(t,\omega )\in [0,T] \times \Omega$, $i=1,\ldots ,m$, $j=1,\ldots ,n$. Finally let $x_i$ be the strategy vector for the firm $P_i$, namely $x_i(t,\omega )=(x_{i1}(t,\omega ),\ldots ,x_{in}(t,\omega ))$, $(t,\omega ) \in [0,T] \times \Omega$, $i=1,\ldots ,m$. For technical reasons, we analyze the model in the Hilbert space $L^2([0,T] \times \Omega ,\mathbb {R}^{mn}_+, \mathbb {P})$.

Let us assume that the following feasibility conditions hold:

$$\begin{aligned} p_i(t,\omega )=\sum _{j=1}^nx_{ij}(t,\omega ),\quad i=1,\ldots ,m, \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.}, \end{aligned}$$

(1)

$$\begin{aligned} q_j(t,\omega )=\sum _{i=1}^mx_{ij}(t,\omega ),\quad j=1,\ldots ,n, \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.} \end{aligned}$$

(2)

The condition (1) expesses that the random time-dependent quantity produced by each firm $P_i$ has to be equal to the random commodity shipments from that firm to all the demand markets. Instead, condition (2) expresses that the random time-dependent quantity demanded by each demand market $Q_j$ has to be equal to the random commodity shipments from all the firms to that demand market.

Furthermore let us assume that the nonnegative random time-dependent commodity shipment between the producer $P_i$ and the demand market $Q_j$ has to satisfy two capacity constraints, namely there exist two nonnegative random time-dependent variables $\underline{x},\overline{x} \in L^2([0,T] \times \Omega ,\mathbb {R}^{mn}_+, \mathbb {P})$ such that

$$\begin{aligned} \begin{aligned} 0 \le \underline{x}_{ij}(t,\omega )\le&x_{ij}(t,\omega )\le \overline{x}_{ij}(t,\omega ),\\ {}&\forall i=1,\ldots ,m, \ \forall j=1,\ldots ,n, \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.} \end{aligned} \end{aligned}$$

(3)

Therefore, the set of feasible distributions $x\in L^2([0,T] \times \Omega ,\mathbb {R}_+^{mn}, \mathbb {P})$ is

$$\begin{aligned} \begin{aligned} \mathbb {K}=\Bigg \{x\in L^2([0,T] \times&\Omega ,\mathbb {R}^{mn}_+, \mathbb {P}): 0\leq\underline{x}_{ij}(t,\omega )\le x_{ij}(t,\omega )\le \overline{x}_{ij}(t,\omega ),\quad \\&\forall i=1,\ldots ,m, \ \forall j=1,\ldots ,n, \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.}\Bigg \}. \end{aligned} \end{aligned}$$

It worth to underline that $\mathbb {K}$ is a convex closed bounded subset of $L^2([0,T] \times \Omega ,\mathbb {R}_+^{mn}, \mathbb {P})$.

At last we introduce the costs. More precisely, let $f_i$ be a random time-dependent variable denoting the production cost of firm $P_i$ such that $f_i=f_i(t,\omega, x(t,\omega ))$, $(t,\omega ) \in [0,T] \times \Omega$, $i=1, \ldots , m$. Similarly, let $d_j$ be a random time-dependent variable denoting the demand price for unity of the commodity for the demand market $Q_j$ such that $d_i=d_i(t, \omega, x(t,\omega ))$, $(t,\omega ) \in [0,T] \times \Omega$, $j=1, \ldots , n$. Finally, let $c_{ij}$ be the random variable expressing the transaction cost, which includes the transportation cost associated with trading the commodity between firm $P_i$ and demand market $Q_j$ such that $c_{ij}=c_{ij}(t, \omega, x(t,\omega ))$, $(t,\omega ) \in [0,T] \times \Omega$, $i=1, \ldots , m$, $j=1, \ldots , n$. Let $\eta _{ij}$ be the random variable expressing the supply or resource tax imposed on the supply market $P_i$ for the transaction with the demand market $Q_j$, namely $\eta _{ij}= \eta _{ij}(t, \omega )$, $(t,\omega ) \in [0,T] \times \Omega$, $i=1, \ldots , m$, $j=1, \ldots , n$. Let $\lambda _{ij}$ be the random variable expressing the incentive pay imposed on the supply market $P_i$ for the transaction with the demand market $Q_j$ namely $\lambda _{ij}= \lambda _{ij}(t, \omega )$, $(t,\omega ) \in [0,T] \times \Omega$, $i=1, \ldots , m$, $j=1, \ldots , n$. Moreover, let $h_{ij}$ be the random variable expressing the difference between the supply tax and the incentive pay imposed on the supply market $P_i$ for the transaction with the demand market $Q_j$, namely $h_{ij}(t,\omega )=\eta _{ij}(t,\omega )-\lambda _{ij}(t,\omega )$, $(t,\omega ) \in [0,T] \times \Omega$, $i=1, \ldots , m$, $j=1, \ldots , n$. As a consequence, the profit $v_i(t,\omega, x(t,\omega ))$ of the firm $P_i$ is

$$\begin{aligned} \begin{aligned} v_i(t,\omega, x(t,\omega )) =\sum _{j=1}^nd_j(t, \omega,x(t,\omega ))x_{ij}(t,\omega )-f_i(t, \omega,x(t,\omega ))\\-\sum _{j=1}^nc_{ij}(t, \omega,x(t,\omega ))x_{ij}(t,\omega ) - \sum _{j=1}^{n} h_{ij}(t,\omega ) x_{ij}(t,\omega ), \\ i=1,\ldots ,m, \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.}, \end{aligned} \end{aligned}$$

namely, it is equal to the price which the demand markets are disposed to pay minus the production costs, the transportation costs and the taxes.

Let us assume the following:

• (i) $v_i(\cdot )$ is continuously differentiable for each $i=1,\ldots ,m,$

• (ii) $\nabla _D v(\cdot )$ is a Carathéodory function such that

  $$\begin{aligned} \begin{aligned} \exists h \in L^2(\Omega ,\mathbb {P}):&\ \left\| \nabla _D v(t, \omega ,x(t, \omega ))\right\| \le h(t,\omega )\left\| x(t,\omega )\right\| , \\&\forall x \in L^2([0, T]\times\Omega , \mathbb {R}^{mn}_+,\mathbb {P}), \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\text{ a.s. }, \end{aligned} \end{aligned}$$

  (4)

• (iii) $v_i(\cdot )$ is pseudoconcave¹ with respect to the variables $x_i,$ $i=1,\ldots ,m$.

Let us set $\nabla _Dv=\left( \dfrac{\partial v_i}{\partial x_{ij}} \right) _{{\begin{array}{c} i=1,\ldots ,m\\ j=1,\ldots ,n\end{array}}} \in L^2([0,T]\times \Omega , \mathbb {R}^{mn}_+,\mathbb {P})$.

The model is based on the fact that the m firms supply the commodity in a noncooperative fashion, each one tries to maximize its own profit function considered the optimal distribution pattern for the other firms. The aim is to find a nonnegative commodity distribution for which the m firms and the n demand markets will be in a state of equilibrium as defined below.

Definition 1

A feasible distribution $x^*\in \mathbb {K}$ is a random time-dependent oligopolistic market equilibrium distribution if and only if, for each $i=1,\ldots ,m$, a.e. in [0, T], $\mathbb {P}$-a.s., we have

$$\begin{aligned} v_i(t,\omega,x^*(t,\omega ))\ge v_i(t,\omega, x_i(t,\omega ),x^*_{-i}(t,\omega )), \end{aligned}$$

(5)

where

$$\begin{aligned} x^*_{-i}(t,\omega )=(x^*_1(t,\omega ),\ldots ,x^*_{i-1}(t,\omega ),x^*_{i+1}(t,\omega ),\ldots ,x^*_m(t,\omega )). \end{aligned}$$

The previous definition generalizes the random Cournot-Nash principle in the random time-dependent case. Moreover, under assumptions (i), (ii) and (iii), it is characterized by means of the stochastic variational inequality (see [7])

$$\begin{aligned} \ll - \nabla _D v(x^*), x-x^* \gg _{\mathbb {E}} \ge 0,\quad \forall x\in \mathbb {K}, \end{aligned}$$

(6)

namely

$$\begin{aligned} \int _0^T \int _{\Omega } - \sum _{i=1}^m \sum _{j=1}^n \frac{\partial v_i(t, \omega, x^*(t,\omega) )}{\partial x_{ij}} \left( x_{ij}(t, \omega ) - x_{ij}^*(t, \omega ) \right) dt \, d\mathbb {P}\ge 0, \quad \forall x\in \mathbb {K}. \end{aligned}$$

2.2 The policymaker’s point of view

Let us left the producers’ point of view in the analysis of the problem and let us introduce the random time-dependent optimal control model in which the term h, which is a fixed parameter in the firms’ point of view model, will become a variable. The random time-dependent resource exploitations $x(t, \omega , h(t,\omega ))$ can be controlled by adjusting taxes $h(t,\omega )$, a.e. in [0, T], $\mathbb {P}$-a.s. In this prospective, the random time-dependent tax adjustment has the role of regulating exportation. More precisely, if the policymaker wants to reduce exportations and, hence, the production of the commodity, then higher taxes will be enforced, otherwise if the policymaker wants to force exportations of the commodity, subventions will be provided.

Let $x(h)=x(t, \omega ,h(t, \omega ))$ be the random time-dependent function of regulatory taxes, with $h \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P})$. Let us assume that $x(t,\omega ,h(t, \omega ))$ is a Carathéodory function and there exists $\gamma \in L^2([0,T] \times \Omega , \mathbb {P})$ such that

$$\begin{aligned} \begin{aligned} \left\| x(t,\omega , h(t,\omega ))\right\| \le&\gamma (t,\omega )+\left\| h(t,\omega )\right\| , \\&\forall h \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}_+, \mathbb {P}), \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.} \end{aligned} \end{aligned}$$

(7)

Hence the set of feasible states is given by

$$\begin{aligned} \begin{aligned} W=\Bigg \{w \in&L^2([0,T] \times \Omega ,\mathbb {R}^{mn}, \mathbb {P}): \underline{x}_{ij}(t, \omega )\le w_{ij}(t, \omega )\le \overline{x}_{ij}(t, \omega ),\\&\forall i=1,\ldots ,m,\;\forall j=1,\ldots ,n,\ \mathrm{a.e. \ in} \ {[}0,T], \ \mathbb {P}-\mathrm{a.s.} \Bigg \}. \end{aligned} \end{aligned}$$

We notice immediately that W is a convex closed bounded subset of $L^2([0,T] \times \Omega ,\mathbb {R}_+^{mn}, \mathbb {P})$.

Definition 2

A random dynamic regulatory tax $h^*\in L^2([0,T] \times \Omega ,\mathbb {R}_+^{mn}, \mathbb {P})$ is a random dynamic optimal regulatory tax if $x(h^*) \in W$ and, for $i=1,\ldots ,m$, $j=1,\ldots ,n$, a.e. in [0, T] and $\mathbb {P}$-a.s. , the following conditions hold:

$$\begin{aligned} x_{ij}(t,\omega ,h^*(t,\omega ))=\underline{x}_{ij}(t,\omega )\Rightarrow & {} h^*_{ij}(t,\omega )\le 0, \end{aligned}$$

(8)

$$\begin{aligned} \underline{x}_{ij}(t,\omega )<x_{ij}(t,\omega ,h^*(t,\omega ))<\overline{x}_{ij}(t,\omega )\Rightarrow & {} h^*_{ij}(t,\omega )=0, \end{aligned}$$

(9)

$$\begin{aligned} x_{ij}(t,\omega ,h^*(t,\omega ))=\overline{x}_{ij}(t,\omega )\Rightarrow & {} h^*_{ij}(t,\omega )\ge 0. \end{aligned}$$

(10)

Definition 2 must be interpreted as follows: the random time-dependent optimal regulatory tax $h^*$ is such that the correspondig state $x(t,\omega,h^*(t,\omega))$ has to satisfy capacity constraints, namely $\underline{x}(t,\omega ) \le x(t,\omega,h^*(t,\omega)) \le \overline{x}(t,\omega )$, a.e. in [0, T], $\mathbb {P}$-a.s. Moreover, if $x_{ij}(t,\omega ,h^*(t,\omega ))=\underline{x}_{ij}(t,\omega )$, then the random time-dependent exportations have to be encouraged, namely taxes must be less than or equal to the random dynamic incentive pays. If $x_{ij}(t,\omega ,h^*(t,\omega ))=\overline{x}_{ij}(t,\omega ),$ then the random time-dependent exportations have to be reduced, hence random dynamic taxes must be greater than or equal to the random dynamic incentive pays. Finally, if $\underline{x}_{ij}(t,\omega )<x_{ij}(t,\omega , h^*(t,\omega ))<\overline{x}_{ij}(t,\omega )$ is satisfied, random dynamic taxes must be equal to random dynamic incentive pays.

Let us establish the inverse stochastic time-dependent variational formulation of the random dynamic optimal equilibrium control problem.

Theorem 1

A random dynamic regulatory tax $h^*\in L^2([0,T] \times \Omega ,\mathbb {R}_+^{mn}, \mathbb {P})$ is a random dynamic optimal regulatory tax if and only if it solves the inverse stochastic time-dependent variational inequality

$$\begin{aligned} \int _0^T \int _{\Omega } \sum _{i=1}^{m}\sum _{j=1}^n h_{ij}^*(t,\omega ) \left( w_{ij}(t,\omega )-x_{ij}(t,\omega , h^*(t,\omega ))\right) dt \, d\mathbb {P}\le 0,\quad \forall w \in W. \end{aligned}$$

(11)

Proof

For the reader’s convinience, we present the details of the proof. Let $h^*$ be a random dynamic optimal regulatory tax and let $w \in W$. For $i\in \{1,\ldots ,m\}$ and $j\in \{1,\ldots ,n\}$ fixed, it results that $\underline{x}_{ij}(t,\omega )\le w_{ij}(t,\omega )\le \overline{x}_{ij}(t,\omega )$, a.e. in [0, T], $\mathbb {P}$-a.s. One has

• 1. If $x_{ij}(t,\omega ,h^*(t,\omega ))=\underline{x}_{ij}(t,\omega )$, a.e. in [0, T], $\mathbb {P}$-a.s., by (8) it follows that $h^*_{ij}(t,\omega )\le 0$, a.e. in [0, T], $\mathbb {P}$-a.s., and, as a consequence, $h_{ij}^*(t,\omega ) \left( w _{ij}(t,\omega )-x_{ij}(t,\omega ,h^*(t,\omega ))\right) \le 0,$ a.e. in [0, T], $\mathbb {P}$-a.s.;

• 2. If $\underline{x}_{ij}(t,\omega )<x_{ij}(t,\omega ,h^*(t,\omega ))<\overline{x}_{ij}(t,\omega )$, a.e. in [0, T], $\mathbb {P}$-a.s., by using (9) we get $h^*_{ij}(t,\omega )= 0$, a.e. in [0, T], $\mathbb {P}$-a.s., and, then, $h_{ij}^*(t,\omega ) \left( w_{ij}(t,\omega )-x_{ij}(t,h^*(\omega ))\right) =0$, a.e. in [0, T], $\mathbb {P}$-a.s.;

• 3. If $x_{ij}(t,\omega ,h^*(t,\omega ))=\overline{x}_{ij}(t,\omega )$, a.e. in [0, T], $\mathbb {P}$-a.s., by (10) we deduce that $h^*_{ij}(t,\omega )\ge 0$, a.e. in [0, T], $\mathbb {P}$-a.s., and, hence, $h_{ij}^*(t,\omega ) \left( w_{ij}(t,\omega )-x_{ij}(t,\omega ,h^*(t,\omega ))\right) \le 0,$ a.e. in [0, T], $\mathbb {P}$-a.s.

We have shown that for every $i=1,\ldots ,m,$ $j=1,\ldots ,n$ and $w \in W,$ we have

$$\begin{aligned} h_{ij}^*(t,\omega ) \left( w_{ij}(t,\omega )-x_{ij}(t,\omega ,h^*(t,\omega ))\right) \le 0, \quad \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.} \end{aligned}$$

By summing over $i= 1,\ldots ,m$, $j= 1,\ldots ,n$ and integrating both on [0, T] and on $\Omega ,$ we obtain the inverse stochastic time-dependent variational inequality (11).

Vice versa, let $h^*$ be satisfied the inverse stochastic time-dependent variational inequality (11). We fix $i\in \{1,\ldots ,m\},$ $j\in \{1,\ldots ,n\}$ and set $w_{hk}(t,\omega )=x_{ij}(t,\omega ,h^*(\omega )),$ a.e. in [0, T], $\mathbb {P}$-a.s., for every $h\ne i,\,k\ne j$. Making use of (11), it results

$$\begin{aligned} \begin{aligned} \int _0^T \int _{\Omega } h_{ij}^*(t,\omega )&\left( w_{ij}(t,\omega )-x_{ij}(t,\omega ,h^*(t,\omega ))\right) dt \, d\mathbb {P}\le 0, \\&\forall w_{ij}(t,\omega )\in L^2([0,T] \times \Omega ,\mathbb {P}): \ \underline{x}_{ij}(t,\omega )\le w_{ij}(t,\omega )\le \overline{x}_{ij}(t,\omega ). \end{aligned} \end{aligned}$$

(12)

We claim that if $x_{ij}(t,\omega ,h^*(t,\omega ))=\underline{x}_{ij}(t,\omega )$, a.e. in [0, T], $\mathbb {P}$-a.s., then, $h^*_{ij}(t,\omega )\le 0$, a.e. in [0, T], $\mathbb {P}$-a.s. Indeed, by contradiction, we suppose that there exists or a set $I \subset [0,T]$ either a set $\Xi \subseteq \Omega$, with $\mathbb {P}(\Xi ) >0$, such that $h^*_{ij}(t,\omega )>0$, or a.e. in I, either $\mathbb {P}$-a.s. in $\Xi$. Let us suppose that we are in the case in which there exists a set $I \subset [0,T]$ such that $h^*_{ij}(t,\omega )>0$, a.e. in I, $\mathbb {P}$-a.s. Then we choose

$$\begin{aligned} w_{ij}(t,\omega ) = \left\{ \begin{array}{ll} \overline{x}_{ij}(t,\omega ), &{} \mathrm{a.e. \ in} \ I, \ \mathbb {P}-\mathrm{a.s. } , \\ x_{ij}(t,\omega ,h^*(t,\omega )), &{} \mathrm{a.e. \ in} \ [0,T] \setminus I, \ \mathbb {P}-\mathrm{a.s. } \end{array} \right. \end{aligned}$$

Therefore we obtain

$$\begin{aligned} \begin{aligned} \int _0^T \int _{\Omega }&h_{ij}^*(t,\omega ) \left( w_{ij}(t,\omega )-x_{ij}(t,\omega ,h^*(t,\omega ))\right) \,dt \, d\mathbb {P}\\&= \int _I \int _{\Omega } h^*_{ij}(t,\omega ) (\overline{x}_{ij}(t,\omega )-\underline{x}_{ij}(t,\omega )) \, dt \, d\mathbb {P}>0, \end{aligned} \end{aligned}$$

(13)

which is in contradiction with (12). The case in which there exists a set $\Xi \subseteq \Omega$, with $\mathbb {P}(\Xi ) >0$, such that $h^*_{ij}(t,\omega )>0$, a.e. in [0, T], $\mathbb {P}$-a.s. in $\Xi$, is analogous.

Similarly we can proceed in the other cases deducing:

• if $x_{ij}(t,\omega ,h^*(t,\omega ))=\overline{x}_{ij}(t,\omega )$, a.e. in [0, T], $\mathbb {P}$-a.s., then $h^*_{ij}(t,\omega )\ge 0$, a.e. in [0, T], $\mathbb {P}$-a.s.,

• if $\underline{x}_{ij}(t,\omega )<x_{ij}(t,\omega ,h^*(t,\omega ))<\overline{x}_{ij}(t,\omega )$, a.e. in [0, T], $\mathbb {P}$-a.s., then $h^*(\omega )=0$, a.e. in [0, T], $\mathbb {P}$-a.s.

$\square$

We are interested to express the random dynamic optimal equilibrium control problem by means of a suitable stochastic time-dependent variational inequality. For this reason we set

$$\begin{aligned} Z=L^2([0,T] \times \Omega ,\mathbb {R}^{mn},\mathbb {P})\times W,\qquad F: {[}0,T] \times \Omega \times Z \rightarrow L^2([0,T]\times \Omega ,\mathbb {R}^{2mn}, \mathbb {P}),\\ z(t,\omega ) = \left( \begin{array}{c} h(t,\omega ) \\ w(t,\omega ) \end{array} \right) \in Z, \quad \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.},\\ F(t,\omega ,z(t,\omega )) = \left( \begin{array}{c} w(t,\omega )-x(t,\omega ,h(t,\omega )) \\ -h(t,\omega ) \end{array} \right) , \quad \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.}. \end{aligned}$$

Let us highlight that Z is a closed, convex (as product of convex sets) and not bounded subset of $L^2([0,T] \times \Omega ,\mathbb {R}^{2mn}_+, \mathbb {P}).$ The following result holds.

Theorem 2

The inverse stochastic time-dependent variational inequality (11) is equivalent to the stochastic time-dependent variational inequality

$$\begin{aligned} \int _0^T \int _{\Omega } \sum _{l=1}^{2m} \sum _{j=1}^n F_{lj}(t,\omega ,z^*(t,\omega ))\left( z_{lj}(t,\omega )-z^*_{lj}(t,\omega )\right) \, dt \, d\mathbb {P}\ge 0,\quad \forall z\in Z. \end{aligned}$$

(14)

Proof

We suppose that (14) holds true. Therefore, one has $z^*=(h^*,w^*)^T\in Z$, and

$$\begin{aligned} \begin{aligned}&\int _0^T \int _{\Omega } \left( \sum _{i=1}^{m}\sum _{j=1}^n(w_{ij}^*(t, \omega )-x_{ij}(t,\omega , h^*(t,\omega )))(h_{ij}(t,\omega )-h^*_{ij}(t,\omega ))\right) \, dt \, d\mathbb {P}\\&\quad - \int _0^T \int _{\Omega } \left( \sum _{i=1}^{m}\sum _{j=1}^n h^*_{ij}(t,\omega )(w_{ij}(t,\omega )-w_{ij}^*(t,\omega ))\right) \, dt \, d\mathbb {P}\ge 0, \quad \forall z=(h,w)^T\in Z. \end{aligned} \end{aligned}$$

(15)

Let us take $h(t,\omega )=h^*(t,\omega )-w^*(t,\omega )+x(t,\omega ,h^*(t,\omega ))$, a.e. in [0, T], $\mathbb {P}$-a.s., and $w(t,\omega )=w^*(t,\omega )$, a.e. in [0, T], $\mathbb {P}$-a.s., in (15). As a consequence, we get

$$\begin{aligned} - \int _0^T \int _{\Omega } \sum _{i=1}^{m}\sum _{j=1}^n(w_{ij}^*(t,\omega )-x_{ij}(t,\omega ,h^*(t,\omega )))^2 \, dt \, d\mathbb {P}\ge 0, \end{aligned}$$

then $x(t,\omega ,h^*(t,\omega ))=w^*(t,\omega ),$ a.e. in [0, T], $\mathbb {P}$-a.s. Hence $x(h^*)\in W$ and, by using (15), we obtain (11).

Vice versa if $h^*\in L^2([0,T] \times \Omega ,\mathbb {R}^{mn}_+, \mathbb {P})$ is a solution to (11), it results

$$\begin{aligned}&\underbrace{\int _0^T \int _{\Omega } \sum _{i=1}^{m}\sum _{j=1}^{n} (x_{ij}(t,\omega ,h^*(t,\omega ))-x_{ij}(t,\omega ,h^*(t,\omega )))(h_{ij}(t,\omega )-h^*_{ij}(t,\omega )) \, dt \, d\mathbb {P}}_{=0}\\&\underbrace{- \int _0^T \int _{\Omega } \sum _{i=1}^{m}\sum _{j=1}^n h^*_{ij}(t,\omega )(w_{ij}(t,\omega )-x_{ij}(t,\omega ,h^*(t, \omega ))) \, dt \, d\mathbb {P}}_{\ge 0}\ge 0. \end{aligned}$$

Hence, $z^*=(h^*,x(h^*))^T\in Z$ is a solution to (14). $\square$

3 Existence results

Let us establish some existence results for the random time-dependent oligopolistic market equilibrium problem. For what concerns the firms’ point of view model, we refer to [7]. Here, we investigate on the existence of the random dynamic optimal regulatory tax. First of all, let us remark that thanks to the equivalent stochastic time-dependent variational formulation of (11), the existence can be obtained applying the results shown in [7]. In the sequel, we concentrate our investigation on the inverse stochastic time-dependent variational inequality (11).

In order to prove an existence result for the inverse stochastic time-dependent variational inequality (11), we assume the following:

• (a) $x(t,\omega , h): [0,T] \times \Omega \times L^2([0,T] \times \Omega ,\mathbb {R}^{mn}, \mathbb {P}) \rightarrow L^2([0,T] \times \Omega ,\mathbb {R}^{mn}, \mathbb {P})$ is a Carathéodory function such that there exists a function $\gamma \in L^2([0,T] \times \Omega ,\mathbb {R}, \mathbb {P}):$

  $$\begin{aligned} \Vert x(t,\omega , h(t,\omega ))\Vert \le \gamma (t,\omega ) + \Vert h(t,\omega )\Vert , \\ \forall h \in L^2([0,T]\times \Omega ,\mathbb {R}^{mn}, \mathbb {P}), \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.}; \end{aligned}$$

• (b) $x(t,\omega , h): [0,T] \times \Omega \times L^2([0,T] \times \Omega ,\mathbb {R}^{mn}, \mathbb {P}) \rightarrow L^2([0,T] \times \Omega ,\mathbb {R}^{mn}, \mathbb {P})$ is anti-monotone with respect to h, namely

  $$\begin{aligned} \langle h_1(t, \omega) - h_2(t,\omega), x(t, \omega , h_1(t,\omega)) - x(t, \omega , h_2(t,\omega)) \rangle \le 0, \\ \forall h_1,h_2 \in L^2([0,T]\times \Omega ,\mathbb {R}^{mn}, \mathbb {P}), \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.}; \end{aligned}$$

• (c) there exists a constant $M>0$ such that for any $h \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P})$ with $\Vert h \Vert _{L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P})} > M$, it results

  $$\begin{aligned} \int _0^T \int _{\Omega } \langle h(t,\omega ), w_0^{proj}(t,\omega ) - x(t,\omega , h(t,\omega )) \rangle \, dt \, d \mathbb {P}>0, \end{aligned}$$

  (16)

  where $w_0^{proj}$ is the projection of $w_0(t,\omega )=x(t,\omega , 0)$ onto the feasible set W, namely

  $$\begin{aligned} w_0^{proj}(t,\omega )= {\left\{ \begin{array}{ll} \underline{x}(t,\omega ), &{} \mathrm{if} \ w_0(t,\omega ) \le \underline{x}(t,\omega ), \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.}, \\ w_0(t,\omega ), &{} \mathrm{if} \ \underline{x}(t,\omega ) \le w_0(t,\omega ) \le \overline{x}(t,\omega ), \ \mathrm{a.e. \ in} \ {[}0,T], \ \mathbb {P}-\mathrm{a.s.}, \\ \overline{x}(t,\omega ), &{} \mathrm{if} \ w_0(t,\omega ) \ge \overline{x}(t,\omega ), \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.} \end{array}\right. } \end{aligned}$$

Assumption (16) means that if we adjust the price in a suitable way (so if $w_0(t,\omega )$ is enough positive for $w_0(t,\omega ) \le \overline{x}(t,\omega )$ and is enough negative for $w_0(t,\omega ) \le \underline{x}(t,\omega )$), then the resultant curve $x(t,\omega , h(t,\omega ))$ will be strictly controlled in the interior of the feasible set W. Now, we are able to prove an existence result under the above assumptions and making use of Corollary 3.7 in [23].

Theorem 3

Let us assume that conditions (a), (b) and (c) are satisfied. Then the inverse stochastic time-dependent variational inequality (11) admits a solution.

Proof

Making use of assumptions (b) and (c), we have that any $h \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P})$ with $\Vert h \Vert _{L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P})} > M$ is too large to be a solution to (11). Let us set $L^2_{M} = \{ x \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P}): \ \Vert h \Vert _{L^2([0,T]\times \Omega , \mathbb {R}^{mn}, \mathbb {P})} \le M \}$. Consequently, we consider the stochastic time-dependent variational inequality on the bounded set $Z'= L^2_{M} \times W$, namely

$$\begin{aligned} \int _0^T \int _{\Omega } \sum _{l=1}^{2m} \sum _{j=1}^n F_{lj} (t,\omega , z^*(t,\omega )) (z_{lj}(t,\omega ) - z^*_{lj}(t,\omega )) \, dt \, d \mathbb {P}\ge 0, \quad \forall z \in Z'. \end{aligned}$$

By assumption (b), it is easy to prove

$$\begin{aligned} \langle z(t,\omega) -\widehat{z}(t,\omega), F(t,\omega , z(t,\omega)) - F(t,\omega , \widehat{z}(t,\omega)) \rangle = \langle x(t,\omega , h(t,\omega)) - x(t,\omega , \widehat{h}(t,\omega)), h(t,\omega) -\widehat{h}(t,\omega) \rangle \ge 0, \end{aligned}$$

(17)

for every $z=(h,w)^T, \widehat{z}=(\widehat{h},\widehat{w})^T \in Z'$, a.e. in [0, T], $\mathbb {P}$-a.s. Let us set $C: Z' \rightarrow (L^2([0,T] \times \Omega , \mathbb {R}^{mn}_+, \mathbb {P}) \times L^2([0,T] \times \Omega , \mathbb {R}^{mn}_+, \mathbb {P}))^*$ such that

$$\begin{aligned} \ll C(z), u \gg_{\mathbb{E}} = \int _0^T \int _{\Omega } \langle F(t,\omega , z(t,\omega )), u(t,\omega )\rangle \, dt \, d\mathbb {P}, \\ \forall z \in Z', \ \forall u \in L^2([0,T] \times \Omega , \mathbb {R}_{+}^{mn}, \mathbb {P}) \times L^2([0,T] \times \Omega , \mathbb {R}_{+}^{mn}, \mathbb {P}). \end{aligned}$$

By using (17), it results

$$\begin{aligned} \int _0^T \int _{\Omega } \langle F(t,\omega , z(t,\omega )) - F(t,\omega , \widehat{z}(t,\omega )), z(t,\omega ) - \widehat{z}(t,\omega )\rangle \, dt \, d\mathbb {P}\ge 0, \quad \forall z, \widehat{z} \in Z', \end{aligned}$$

and, then, the operator C is monotone². Taking into account assumption (a) and the Lebesgue theorem, we can prove that

$$\begin{aligned} \lim _n \int _0^T \int _{\Omega } \Vert F(t,\omega , \lambda _n z(t,\omega ) + (1- \lambda _n) \widehat{z}(t,\omega )) - F(t,\omega , \lambda z(t,\omega ) + (1-\lambda ) \widehat{z}(t,\omega )) \Vert ^2 \, dt \, d\mathbb {P}=0 \end{aligned}$$

holds for every sequence $\{ \lambda _n \} \subset [0,1]$, such that $\lambda _n \rightarrow \lambda \in [0,1]$ and for every $z, \widehat{z} \in W'$. Therefore we deduce

$$\begin{aligned} \lim _n \int _0^T \int _{\Omega } \langle F(t, \omega , \lambda _n z(t,\omega ) + (1- \lambda_n) \widehat{z}(t,\omega )), z(t,\omega )-\widehat{z}(t,\omega )\rangle \, dt \, d \mathbb {P}\\ = \int _0^T \int _{\Omega } \langle F(t,\omega , \lambda z(t,\omega) + (1- \lambda) \widehat{z}(t,\omega)), z(t,\omega )-\widehat{z}(t,\omega )\rangle \, dt \, d \mathbb {P}, \end{aligned}$$

namely the operator C(z) is hemicontinuous along line segments³. Moreover, since W is convex closed bounded and for the definition of $L^2_{M'}$, $Z'$ is also a convex closed bounded set. Hence, taking into account Corollary 3.7 in [23], (14) admits a solution. As a consequence, by Theorem 2, (11) has also a solution. $\square$

4 Well-posedness conditions

Let us investigate on the well-posedness of (11) and, then, we establish its relationship with the well-posedness of (14).

Let us define that a sequence $\{ h_n \} \subset L^2([0,T]\times \Omega ,\mathbb {R}^{mn}, \mathbb {P})$ is called an approximating sequence for (11) if and only if there exists a sequence $\{ \varepsilon _n \}$, with $\varepsilon _n >0$, for every $n \in \mathbb {N}$, and $\varepsilon _n \rightarrow 0$, such that

$$\begin{aligned} \int _0^T \int _{\Omega } \langle h_n(t, \omega ), w(t, \omega ) - x(t, \omega , h_n(t, \omega )) \rangle \, dt \, d\mathbb {P}\le \varepsilon _n, \quad \forall w \in W, \ \forall n \in \mathbb {N}. \end{aligned}$$

Definition 3

We say that (11) is well-posed if and only if (11) has a unique solution and every approximating sequence converges to the unique solution.

The following well-posedness result for the inverse stochastic time-dependent variational inequality holds.

Theorem 4

Let $x : [0,T] \times \Omega \times L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P}) \rightarrow L^2([0,T] \times \Omega ,\mathbb {R}^{mn}, \mathbb {P})$ be an hemicontinuous along line segments and anti-monotone mapping. Then, (11) is well-posed if and only if it has a unique solution.

Proof

The necessity holds trivially. Let us prove that the sufficiency holds also true. Hence we assume that (11) has a unique solution $h^*$. Consequently, we have

$$\begin{aligned} \int _0^T \int _{\Omega } \langle h^*(t, \omega ), w(t, \omega ) - x(t, \omega , h^*(t, \omega )) \rangle dt \, d\mathbb {P}\le 0, \quad \forall w \in W. \end{aligned}$$

Since x is anti-monotone, we get

$$\begin{aligned} \int _0^T \int _{\Omega } \langle w(t, \omega )- x(t, \omega , h(t, \omega )), h(t, \omega ) - h^*(t,\omega ) \rangle dt \, d\mathbb {P}\nonumber \\ + \int _0^T \int _{\Omega } \langle h(t, \omega ), w(t, \omega )- x(t, \omega , h^*(t, \omega )) \rangle dt \, d\mathbb {P}\nonumber \\ \qquad \qquad \quad \le \int _0^T \int _{\Omega } \langle h^*(t, \omega ), w(t, \omega ) - x(t, \omega , h^*(t, \omega )) \rangle dt \, d\mathbb {P}\le 0, \nonumber \\ \quad \forall h \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}_+, \mathbb {P}), \ \forall w \in W. \end{aligned}$$

(18)

Let $\{ h_n \} \subset L^2([0,T] \times \Omega , \mathbb {R}^{mn}_+, \mathbb {P})$ be an approximating sequence for (11). This means that there exists a sequence $\{\varepsilon _n\}$, with $\varepsilon _n > 0$, for every $n \in \mathbb {N}$, and $\varepsilon _n \rightarrow 0$ such that

$$\begin{aligned} \int _0^T \int _{\Omega } \langle h_n(t, \omega ), w(t, \omega ) - x(t, \omega , h_n(t, \omega )) \rangle dt \, d\mathbb {P}\le \varepsilon _n, \quad \forall w \in W, \ \forall n \in \mathbb {N}. \end{aligned}$$

For the anti-monotonicity of x, we deduce

$$\begin{aligned} \int _0^T \int _{\Omega } \langle w(t, \omega ) - x(t, \omega , h(t, \omega )), h(t, \omega ) - h_n(t,\omega ) \rangle dt \, d\mathbb {P}\nonumber \\ + \int _0^T \int _{\Omega } \langle h(t, \omega ), w(t, \omega )- x(t, \omega , h_n(t, \omega )) \rangle dt \, d\mathbb {P}\nonumber \\ \qquad \qquad \quad \le \int _0^T \int _{\Omega } \langle h_n(t, \omega ), w(t, \omega ) - x(t, \omega , h_n(t, \omega )) \rangle dt \, d\mathbb {P}\le \varepsilon _n, \nonumber \\ \quad \forall h \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}_+, \mathbb {P}), \ \forall w \in W. \end{aligned}$$

(19)

Let us consider

$$\begin{aligned} z^*(t,\omega )=(h^*(t,\omega ), x(t,\omega , h^*(t,\omega )))^T, \quad \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.} \end{aligned}$$

and

$$\begin{aligned} z_n(t,\omega )=(h_n(t,\omega ), x(t,\omega ,h_n(t,\omega )))^T, \quad \forall n \in \mathbb {N}, \ \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.} \end{aligned}$$

If $\{ z_n \}$ is unbounded, without loss of generality, we can assume that $\Vert z_n \Vert \rightarrow + \infty$. Let us set

$$\begin{aligned} \lambda _n = \frac{1}{\Vert z_n - z^* \Vert }, \end{aligned}$$

and

$$\begin{aligned} \zeta _n (t,\omega ) & = (k_n(t,\omega ), y_n(t,\omega )) =z^*(t,\omega )+\lambda _n(z_n(t,\omega )-z^*(t,\omega )) \\ & = (h^*(t,\omega )+\lambda _n (h_n(t,\omega )-h^*(t,\omega )), x(t,\omega ,h^*(t,\omega ))\\ & \quad + \lambda _n(x(t,\omega ,h_n(t,\omega ))-x(t,\omega ,h^*(t,\omega )))). \end{aligned}$$

Without loss of generality, we can assume that $\lambda _n \in (0,1]$ and $\zeta _n \rightarrow \zeta = (k,y) \ne z^*$. Moreover it results that $y \in W$, being W closed and convex. For any $w \in W$ and any $h \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}_+, \mathbb {P})$, we get

$$\begin{aligned} \begin{aligned} \int _0^T \int _{\Omega }&\langle w(t, \omega )- x(t, \omega , h(t, \omega )), h(t, \omega ) - k(t,\omega ) \rangle dt \, d\mathbb {P}\\& \qquad + \int _0^T \int _{\Omega } \langle h(t, \omega ), w(t, \omega ) - y(t, \omega ) \rangle dt \, d\mathbb {P}\\&= \int _0^T \int _{\Omega } \langle w(t, \omega ) -x(t, \omega , h(t, \omega )), k_n(t, \omega ) - k(t,\omega ) \rangle dt \, d\mathbb {P}\\& \qquad + \int _0^T \int _{\Omega } \langle w(t, \omega ) - x(t, \omega , h(t, \omega )), h^*(t, \omega ) - k_n(t,\omega ) \rangle dt \, d\mathbb {P}\\& \qquad + \int _0^T \int _{\Omega } \langle w(t, \omega ) -x(t, \omega , h(t, \omega )), h(t, \omega ) - h^*(t,\omega ) \rangle dt \, d\mathbb {P}\\& \qquad + \int _0^T \int _{\Omega } \langle h(t,\omega ), w(t, \omega ) - x(t, \omega , h^*(t, \omega )) \rangle dt \, d\mathbb {P}\\& \qquad + \int _0^T \int _{\Omega } \langle h(t,\omega ), x(t, \omega , h^*(t, \omega )) - y_n(t, \omega ) \rangle dt \, d\mathbb {P}\\& \qquad + \int _0^T \int _{\Omega } \langle h(t, \omega ), y_n(t, \omega ) - y(t, \omega ) \rangle dt \, d\mathbb {P}\\ {}&= \Big \{ \int _0^T \int _{\Omega } \langle w(t, \omega ) - x(t, \omega , h(t, \omega )), k_n(t, \omega ) - k(t,\omega ) \rangle dt \, d\mathbb {P}\\ {}& \qquad + \int _0^T \int _{\Omega } \langle h(t,\omega ), y_n(t, \omega ) - y(t, \omega ) \rangle dt \, d\mathbb {P}\Big \} \\& \qquad + \Big \{ \int _0^T \int _{\Omega } \langle w(t, \omega )- x(t, \omega , h(t, \omega )), h(t, \omega ) - h^*(t,\omega ) \rangle dt \, d\mathbb {P}\\ {}& \qquad + \int _0^T \int _{\Omega } \langle h(t, \omega ), w(t, \omega ) - x(t, \omega , h^*(t, \omega )) \rangle dt \, d\mathbb {P}\\& \qquad + \lambda _n \Big \{ \int _0^T \int _{\Omega } \langle w(t, \omega )- x(t, \omega , h(t, \omega )), h^*(t, \omega ) - h_n(t,\omega ) \rangle dt \, d\mathbb {P}\\& \qquad + \int _0^T \int _{\Omega } \langle h(t,\omega ), x(t, \omega , h^*(t, \omega )) - x(t, \omega , h_n(t,\omega )) \rangle dt \, d\mathbb {P}\Big \} \\&= \Big \{ \int _0^T \int _{\Omega } \langle w(t, \omega )- x(t, \omega , h(t, \omega )), k_n(t, \omega ) - k(t,\omega ) \rangle dt \, d\mathbb {P}\\& \qquad + \int _0^T \int _{\Omega } \langle h(t,\omega ), y_n(t, \omega ) - y(t, \omega ) \rangle dt \, d\mathbb {P}\Big \} \\& \qquad + (1-\lambda _n) \Big \{ \int _0^T \int _{\Omega } \langle w(t, \omega )- x(t, \omega , h(t, \omega )), h(t, \omega ) - h^*(t,\omega ) \rangle dt \, d\mathbb {P}\\& \qquad + \int _0^T \int _{\Omega } \langle h(t, \omega ), w(t, \omega )- x(t, \omega , h^*(t, \omega )) \rangle dt \, d\mathbb {P}\\& \qquad + \lambda _n \Big \{ \int _0^T \int _{\Omega } \langle w(t, \omega ) - x(t, \omega , h(t, \omega )), h(t, \omega ) - h_n(t,\omega ) \rangle dt \, d\mathbb {P}\\& \qquad + \int _0^T \int _{\Omega } \langle h(t,\omega ), w(t, \omega )- x(t, \omega , h_n(t, \omega )) \rangle dt \, d\mathbb {P}\Big \}. \end{aligned} \end{aligned}$$

(20)

By using (18)–(20), it follows

$$\begin{aligned} \int _0^T \int _{\Omega } \langle w(t, \omega ) - x(t, \omega , h(t, \omega )), h(t, \omega ) - k(t,\omega ) \rangle dt \, d\mathbb {P}\\+ \int _0^T \int _{\Omega } \langle h(t, \omega ), w(t, \omega )- y(t, \omega ) \rangle dt \, d\mathbb {P}\\ \le \int _0^T \int _{\Omega } \langle w(t, \omega ) - x(t, \omega , h(t, \omega )), k_n(t, \omega ) - k(t,\omega ) \rangle dt \, d\mathbb {P}\\+ \int _0^T \int _{\Omega } \langle h(t,\omega ), y_n(t, \omega ) - y(t, \omega ) \rangle dt \, d\mathbb {P}+ \lambda _n \varepsilon _n, \\ \forall w \in W, \ \forall h \in L^2([0,T] \times \Omega , \mathbb {R}_+^{mn}, \mathbb {P}), \ \forall n \in \mathbb {N}. \end{aligned}$$

Passing to the limit as $n \rightarrow + \infty$ in the above inequality, we obtain

$$\begin{aligned} \int _0^T \int _{\Omega } \langle w(t, \omega )- x(t, \omega , h(t, \omega )), h(t, \omega ) - k(t,\omega ) \rangle dt \, d\mathbb {P}\nonumber \\ + \int _0^T \int _{\Omega } \langle h(t, \omega ), w(t, \omega ) - y(t, \omega ) \rangle dt \, d\mathbb {P}\le 0, \nonumber \\ \forall w \in W, \ \forall h \in L^2([0,T] \times \Omega , \mathbb {R}_+^{mn}, \mathbb {P}). \end{aligned}$$

(21)

For any $h' \in L^2([0,T] \times \Omega , \mathbb {R}_+^{mn}, \mathbb {P})$ and any $y' \in W$, define $k_{\lambda }(t, \omega )= k(t,\omega ) + \lambda (h'(t, \omega ) - k(t, \omega ))$ and $y_{\lambda }(t, \omega )=y(t, \omega )+\lambda (y'(t, \omega ) - y(t, \omega ))$, for all $\lambda \in [0,1]$, a.e. in [0, T], $\mathbb {P}$-a.s. Making use of (21), it results

$$\begin{aligned} \int _0^T \int _{\Omega } \langle y_{\lambda }(t, \omega ) - x(t, \omega , k_{\lambda }(t, \omega )), k_{\lambda }(t, \omega ) - k(t, \omega ) \rangle dt \, d\mathbb {P}\\ + \int _0^T \int _{\Omega } \langle k_{\lambda }(t, \omega ), y_{\lambda }(t, \omega ) - y(t, \omega ) \rangle dt \, d\mathbb {P}\le 0, \end{aligned}$$

which implies

$$\begin{aligned} \int _0^T \int _{\Omega } \langle y_{\lambda }(t, \omega ) - x(t, \omega , k_{\lambda }(t, \omega )), h'(t, \omega ) - k(t, \omega ) \rangle dt \, d\mathbb {P}\\ + \int _0^T \int _{\Omega } \langle k_{\lambda }(t, \omega ), y'(t, \omega ) - y(t, \omega ) \rangle dt \, d\mathbb {P}\le 0, \quad \forall h' \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}_+, \mathbb {P}), \ \forall y' \in W. \end{aligned}$$

Being x hemicontinuous along line segments, passing to the limit as $\lambda \rightarrow 0^+$ in the above inequality, we obtain

$$\begin{aligned} \int _0^T \int _{\Omega } \langle y(t, \omega )- x(t, \omega , k(t, \omega )), h'(t, \omega ) - k(t, \omega ) \rangle dt \, d\mathbb {P}\nonumber \\ + \int _0^T \int _{\Omega } \langle k(t, \omega ), y'(t, \omega ) - y(t, \omega ) \rangle dt \, d\mathbb {P}\le 0, \quad \forall h' \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}_+, \mathbb {P}), \ \forall y' \in W. \end{aligned}$$

(22)

By using (22) and being $h'$ arbitrary, it results

$$\begin{aligned} s \int _0^T \int _{\Omega } \langle y(t, \omega ) - x(t, \omega , k(t, \omega )) , v(t, \omega ) \rangle dt \, d\mathbb {P}\le \mathrm{constant}, \end{aligned}$$

(23)

for every real s and every $v \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P})$. Taking into account (23) and the arbitrary of s and v, we can deduce that $x(t, \omega , k(t, \omega )) = y(t, \omega )$, a.e. in [0, T], $\mathbb {P}$-a.s. As a consequence, we have

$$\begin{aligned} \int _0^T \int _{\Omega } \langle y'(t, \omega ) - x(t, \omega , k(t, \omega )), k(t, \omega ) \rangle dt \, d\mathbb {P}\le 0, \quad \forall y' \in W. \end{aligned}$$

(24)

By using (24), we get that k solves (11) and then $k(t, \omega )=h^*(t, \omega )$, a.e. in [0, T], $\mathbb {P}$-a.s., since $h^*$ is the unique solution to (11), which contradicts $(h^*(t, \omega ),x(t, \omega , h^*(t, \omega )))^T \ne (k(t, \omega ),x(t, \omega ,k(t, \omega )))^T$, a.e. in [0, T], $\mathbb {P}$-a.s.

Hence we can suppose that $\{ z_n \}$ is bounded. Let $\{ z_{n_r} \}$ be any subsequence of $\{ z_n \}$ such that $z_{n_r} \rightarrow ( \overline{h}, \overline{y})$, as $r \rightarrow + \infty$. By virtue of (19), it follows

$$\begin{aligned} \int _0^T \int _{\Omega } \langle w(t, \omega )- x(t, \omega , h(t, \omega )), h(t, \omega ) - h_{n_r}(t,\omega ) \rangle dt \, d\mathbb {P}\\+ \int _0^T \int _{\Omega } \langle h(t, \omega ), w(t, \omega ) - x(t, \omega , h_{n_r}(t, \omega )) \rangle dt \, d\mathbb {P}\le \varepsilon _{n_r}, \\ \quad \forall h \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}_+, \mathbb {P}), \ \forall w \in W. \end{aligned}$$

Passing to the limit as $r \rightarrow + \infty$ in the above inequality, we obtain

$$\begin{aligned} \int _0^T \int _{\Omega } \langle w(t, \omega ) - x(t, \omega , h(t, \omega )), h(t, \omega ) - \overline{h}(t,\omega ) \rangle dt \, d\mathbb {P}\\+ \int _0^T \int _{\Omega } \langle h(t, \omega ), w(t, \omega ) - \overline{y}(t, \omega ) \rangle dt \, d\mathbb {P}\le 0, \\ \quad \forall h \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}_+, \mathbb {P}), \ \forall w \in W. \end{aligned}$$

By using same arguments as in (21)–(24), we deduce

$$\begin{aligned} x(t, \omega ,\overline{h}(t, \omega ))= \overline{y}(t, \omega ), \quad \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.}, \end{aligned}$$

and

$$\begin{aligned} \int _0^T \int _{\Omega } \langle \overline{h}(t, \omega ), x'(t, \omega ) - x(t, \omega , \overline{h}(t, \omega )) \rangle dt \, d\mathbb {P}\le 0, \quad \forall x' \in W. \end{aligned}$$

This means that $\overline{h}$ solves (11). For the uniqueness of solution to (11), it results that $\overline{h}(t, \omega ) = h^*(t, \omega )$, a.e. in [0, T], $\mathbb {P}$-a.s. As a consequence, $\{ h_n \}$ converges to $h^*$ and, hence, (11) is well-posed. $\square$

The well-posedness for the stochastic time-dependent variational inequality (14) can be introduced as done in Definition 3. More precisely, a sequence $\{ z_n \} \subset L^2([0,T]\times \Omega ,\mathbb {R}^{mn}, \mathbb {P})$ is called an approximating sequence for (14) if and only if there exists a sequence $\{ \varepsilon _n \}$, with $\varepsilon _n >0$, for every $n \in \mathbb {N}$, and $\varepsilon _n \rightarrow 0$, such that

$$\begin{aligned} \int _0^T \int _{\Omega } \langle F(t, \omega , z_n(t,\omega )), z_n(t, \omega ) - z(t, \omega ) \rangle \, dt \, d\mathbb {P}\le \varepsilon _n, \quad \forall z \in Z, \ \forall n \in \mathbb {N}. \end{aligned}$$

Similarly, we say that (14) is well-posed if and only if (14) has a unique solution and every approximating sequence converges to the unique solution.

Theorem 5

Let $x : [0,T] \times \Omega \times L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P}) \rightarrow L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P})$ be a continuous mapping. Then, (11) is well-posed if and only if (14) is well-posed.

Proof

Let us start supposing that (11) is well-posed. Hence (11) has a unique solution $h^* \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P})$. Making use of Theorem 2, it follows that $z^* = (h^*,x(h^*))^T \in Z$ is the unique solution to (14). Let us consider an approximating sequence $\{ z_n \} = \{ (h_n,w_n)^T \} \subset Z$ for (14). Then there exists a sequence $\{ \varepsilon _n \}$, with $\varepsilon _n > 0$, for every $n \in \mathbb {N}$, and $\varepsilon _n \rightarrow 0$, such that

$$\begin{aligned} \int _0^T \int _{\Omega } \langle F(t,\omega ,z_n(t,\omega )), z_n(t, \omega ) - z(t,\omega )\rangle \, dt \, d\mathbb {P}\le \varepsilon _n, \quad \forall z=(h,x(h))^T \in Z, \ \forall n \in \mathbb {N}. \end{aligned}$$

As a consequence, we have

$$\begin{aligned} \int _0^T \int _{\Omega } \langle w_n(t,\omega )-x(t,\omega , h_n(t,\omega )), h_n(t,\omega ) - h(t, \omega ) \rangle dt \, d\mathbb {P}\nonumber \\ - \int _0^T \int _{\Omega } \langle h_n(t,\omega ), w_n(t,\omega )-w(t,\omega ) \rangle dt \, d\mathbb {P}\le \varepsilon _n,\nonumber \\ \forall z=(h,w)^T \in Z, \ \forall n \in \mathbb {N}. \end{aligned}$$

(25)

Then, we deduce

$$\begin{aligned} \int _0^T \int _{\Omega } \langle w_n(t,\omega ) -x(t,\omega , h_n(t,\omega )) , h_n(t,\omega ) - h(t, \omega ) \rangle dt \, d\mathbb {P}\\ \le \varepsilon _n + \int _0^T \int _{\Omega } \langle h_n(t,\omega ), w_n(t,\omega ) - w(t,\omega ) \rangle dt \, d\mathbb {P}, \\ \quad z=(h,w)^T \in Z, \ \forall n \in \mathbb {N}. \end{aligned}$$

Fix $w \in W$, $v \in L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P})$ and consider $h(t, \omega )=h_n(t,\omega ) -s v(t, \omega )$, a.e. in [0, T], $\mathbb {P}$-a.s., then

$$\begin{aligned} s \int _0^T \int _{\Omega } \langle w_n(t,\omega )- x(t, \omega , h_n(t,\omega )), v(t,\omega ) \rangle dt \, d\mathbb {P}\le \mathrm{constant}, \end{aligned}$$

where s is arbitrary. Consequently $x(t, \omega , h_n(t,\omega )) = w_n(t,\omega )$, and then by (25) we get

$$\begin{aligned} - \int _0^T \int _{\Omega } \langle h_n(t,\omega ), w_n(t,\omega ) - w(t,\omega ) \rangle dt \, d\mathbb {P}\\ = - \int _0^T \int _{\Omega } \langle h_n(t,\omega ), x(t, \omega , h_n(t,\omega )) - w(t,\omega ) \rangle dt \, d\mathbb {P}\le \varepsilon _n,\\ \quad \forall w \in W, \ \forall n \in \mathbb {N}. \end{aligned}$$

We have that $\{ h_n \} \subset L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P})$ is an approximating sequence for (11). Since (11) is well-posedness, it results $h_n \rightarrow h^*$. Therefore, $z_n = (h_n, x(h_n)) \rightarrow (h^*, x(h^*))$ and, hence, (14) is well-posed.

Vice versa, let us suppose that (14) is well-posed. Hence it has a unique solution $z^*=(h^*, x(h^*))^T \in Z$. By Theorem 2, it results that $h^*$ is the unique solution to (11). Let $\{ h_n \} \subset L^2([0,T] \times \Omega , \mathbb {R}^{mn}, \mathbb {P})$ be an approximating sequence for (11). Then there exists a sequence $\{ \varepsilon _n \}$, with $\varepsilon _n > 0$, for every $n \in \mathbb {N}$, and $\varepsilon _n \rightarrow 0$, such that

$$\begin{aligned} - \int _0^T \int _{\Omega } \langle h_n(t,\omega ), x(t,\omega , h_n(t,\omega ))-w(t, \omega ) \rangle \, dt \, d\mathbb {P}\le \varepsilon _n, \quad \forall w \in W, \ \forall n \in \mathbb {N}. \end{aligned}$$

(26)

Let us consider

$$\begin{aligned} w_n(t, \omega ) = x(t, \omega ,h_n(t, \omega )), \quad \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.}, \end{aligned}$$

and

$$\begin{aligned} z_n(t, \omega )= (h_n(t, \omega ), w_n(t, \omega ))^T, \quad \mathrm{a.e. \ in} \ [0,T], \ \mathbb {P}-\mathrm{a.s.} \end{aligned}$$

By using (26), one has

$$\begin{aligned} \int _0^T \int _{\Omega } \langle F(t, \omega ,z_n(t,\omega )), z_n(t,\omega ) - z(t,\omega ) \rangle \, dt \, d\mathbb {P}\le \varepsilon _n, \quad \forall z=(h,x(h))^T \in Z, \ \forall n \in \mathbb {N}. \end{aligned}$$

This means that $\{z_n\}$ is an approximating sequence for (14). Making use of the well-posedness of (14), we obtain that $z_n \rightarrow z^*$. As a consequence, the sequence $\{ h_n \}$ converges to $h^*$. Then the claim is completely achieved. $\square$

5 Numerical method

In this section we present an algorithm for solving stochastic time-dependent variational inequalities as (6) based on a combination between a discretization procedure and a projection method. The algorithm we present is a generalization of the one in [19] extended to the evolutionary case.

In the deterministic setting, the classical projection method for the following variational inequality

$$\begin{aligned} \langle F(x^*), x-x^* \rangle \ge 0, \quad \forall x \in K, \end{aligned}$$

where K is a nonempty closed convex subset of $\mathbb {R}^m$ and $F: K \rightarrow \mathbb {R}^m$ is a mapping, akin to the projection method for convex optimization, is

$$\begin{aligned} x_{k+1} = P_K (x_k - \alpha F(x_k )), \end{aligned}$$

where $P_K$ is the projection operator onto K and $\alpha$ is a suitable real number. The convergence of this method is guaranteed assuming F is strongly monotone, Lipschitz continuous and the real number $\alpha$ satisfies a strong condition depending on the modulus of strong monotonicity and the Lipschitz constant. Nevertheless the requirement of strong monotonicity is heavy and often fails to be satisfied in applications. More refined extragradient methods are available in the literature.

In the stochastic setting, the first method for stochastic variational inequalities, proposed in [21], updates iteratively $x_k$ according to the formula

$$\begin{aligned} x_{k+1} = P_K(x_k - \alpha _k F (v_k , x_k )), \end{aligned}$$

where $\{v_k\}$ is a sample of v and $\{\alpha _k\}$ is a sequence of positive stepsizes. Again the convergence is ensured under strong hypotesis on monotonicity of F, Lipschitz constant and tight conditions on the stepsize. After this first instance, the recent research on stochastic approximation methods for stochastic variational inequalities had remarkable developments.

We introduce here an extragradient method for stochastic time-dependent variational inequalities as (6) based on [19]. In [7, Theorem 4] it is proved that under the assumptions (i), (ii), (iii), and assuming the continuity of the data, the solution to (6) is stochastic continuous on [0, T]. The stochastic continuity allows us to carry out a discretization procedure in order to reduce the time-dependent problem to some static problems. That is, firstly we consider a partition of [0, T] such that:

$$\begin{aligned} 0=t_0<t_1< \dots< t_r< \dots < t_n=T. \end{aligned}$$

For each value $t_r$, $r = 0, 1,...,n$, we apply an extragradient method to solve the point-to-point stochastic variational inequality

$$\begin{aligned} \int _\Omega \langle F(t,\omega , x^*(t, \omega )), x(t, \omega ) - x^*(t, \omega ) \rangle d\mathbb {P}\ge 0, \quad \forall x(t,\omega ) \in \mathbb {K}(t), \ \text {in} \ [0,T], \end{aligned}$$

(27)

where

$$\begin{aligned} \mathbb {K}(t)=\Bigg \{x(t, \omega )\in \mathbb {R}^{mn}: \ \underline{x}_{ij}(t,\omega )\le x_{ij}(t,\omega )\le \overline{x}_{ij}(t,\omega ),\quad \\ \forall i=1,\ldots ,m, \ \forall j=1,\ldots ,n, \ \mathbb {P}-\mathrm{a.s.}\Bigg \}, \end{aligned}$$

with $t=t_r$, $r = 0, 1,...,n$. More precisely we have the following:

Algorithm

• Discretize the interval [0, T]: $0=t_0<t_1< \dots< t_r< \dots < t_n=T$.

• For each $r=1, \ldots , n$, choose an initial point $x^0_r$, a positive stepsize sequence $\{\alpha _k\}$, the sample rate $\{N_k\}$ and initial samples $\{\omega _j^0\}^{N_0}_{j=1}$ and $\{\eta _j^0\}^{N_0}_{j=1}$ of the random variable $\omega$.

• Given the iterate $x^k_r$, generate the samples $\{\omega _j^k\}^{N_k}_{j=1}$ and $\{\eta _j^k\}^{N_k}_{j=1}$ of $\omega$ and define

  $$\begin{aligned} z^k_r= P_{\mathbb {K}(t_r)} \left( x^k_r- \frac{\alpha _k}{N_k} \sum _{j=1}^{N_k} F(t_r,\omega _j^k, x^k_r) \right), \\ x^{k+1}_r= P_{\mathbb {K}(t_r)} \left( x^k_r- \frac{\alpha _k}{N_k} \sum _{j=1}^{N_k} F(t_r,\eta _j^k, z^k_r) \right) . \end{aligned}$$

• Interpolate the numerical solutions for every $r=1, \ldots , n$.

For each $r=1, \ldots , n$, the convergence of the method to the solution to the corresponding stochastic variational inequality (27) is guarantees by the convergence results proved in [19]. In order to obtain the convergence of our scheme in the time interval [0, T], we consider a sequence $\{ \pi _s \}$ of (not necessarily equidistant) partitions of the time interval [0, T] such that $\pi _s = (t_s^0, t^1_s, \ldots , t^{N_s}_s)$, where $0=t_s^0< t^1_s< \ldots < t^{N_s}_s=T$ and assume that

$$\begin{aligned} k_s = \max \{ t^r_s - t^{r-1}_s: \ r=1,2, \ldots , N_s\}, \end{aligned}$$

approaches zero, as $s \rightarrow + \infty$. Now we construct the approximate solution to (27) by considering

$$\begin{aligned} x_s(t,\omega )=\sum _{r=1}^{N_s}x(t_s^r,\omega ) \chi _{[t_s^{r-1},t_s^r[}(t), \end{aligned}$$

where $x(t^r_s,\omega )$ is the solution to the stochastic variational inequality (27) at $t=t_s^r$, which is computed by means of the projection method. It is possible to show the $L^1$-convergence of such approximate solutions by making use of same arguments in [1].

5.1 Example

An example of the random dynamic oligopolistic market equilibrium problem which emphasizes the central role in the study of the model of the uncertainty is provided. To this aim, let us analyze an economic network made up of two firms and two demand markets, as in Fig. 1.

Fig. 1

Network structure of the numerical dynamic spatial oligopoly problem

Let $x_{ij} (t,\omega )$ be the commodity shipment from $P_i$ to $Q_j$, $i = 1, 2,$ $j = 1, 2$, and assume that $\underline{x}_{ij}(t,\omega )\le x_{ij}(t,\omega ) \le \overline{x}_{ij} (t,\omega )$ holds, where $\underline{x}_{ij}(t,\omega )$ and $\overline{x} _{ij}(t,\omega )$ are function on $[0,25]\times \Omega$ representing the capacity constraints. Assume that $\underline{x}_{ij}(t,\omega )= t^2 \check{x}_{ij}(\omega )$ and $\overline{x}_{ij}(t,\omega )= t^2 \hat{x}_{ij}(\omega )$, where $\hat{x}_{ij}(\omega )$ and $\check{x}_{ij}(\omega )$ are uniformly distributed random variables with probability density functions given by:

$$\begin{aligned} f_{\hat{x}_{i1}}(z) = {\left\{ \begin{array}{ll} \begin{array}{ll} \frac{1}{2}, &{} \text {if}\quad 0\le z\le 2, \\ 0, &{} \text {elsewhere}, \end{array} \end{array}\right. } \\ f_{\hat{x}_{i2}}(z) = {\left\{ \begin{array}{ll} \begin{array}{ll} \frac{1}{5}, &{} \text {if}\quad 0\le z\le 5, \\ 0, &{} \text {elsewhere}, \end{array} \end{array}\right. }\\ f_{\check{x}_{i1}}(z) = {\left\{ \begin{array}{ll} \begin{array}{ll} \frac{1}{25}, &{} \text {if}\quad 75\le z\le 100, \\ 0, &{} \text {elsewhere}, \end{array} \end{array}\right. }\\ f_{\check{x}_{i2}}(z) = {\left\{ \begin{array}{ll} \begin{array}{ll} \frac{1}{20}, &{} \text {if}\quad 80\le z\le 100, \\ 0, &{} \text {elsewhere}. \end{array} \end{array}\right. } \end{aligned}$$

Set now the maximal commodity production of $P_i$, $i = 1, 2$, and the maximal commodity demand of $Q_j$, $j = 1, 2$. Let us define $p_i(t,\omega )=t^2\overline{p}_i(\omega )$, $i = 1, 2$, and $q_j(t,\omega )=t^2\overline{q}_j(\omega )$, $j = 1, 2$, where the density function of $\overline{p}_i(\omega )$, $i = 1, 2$, and $\overline{q}_j(\omega )$, $j = 1, 2$, are defined by

$$\begin{aligned} f_{\overline{p}_{1}}(z) = {\left\{ \begin{array}{ll} \begin{array}{ll} \frac{1}{20}, &{} \text {if} \ 80\le z\le 100, \\ 0,&{} \text {elsewhere} \end{array} \end{array}\right. }\\ f_{\overline{p}_{2}}(z) = {\left\{ \begin{array}{ll} \begin{array}{ll} \frac{1}{10}, &{} \text {if} \ 90\le z\le 100, \\ 0,&{} \text {elsewhere} \end{array} \end{array}\right. }\\ f_{\overline{q}_{1}}(z) = {\left\{ \begin{array}{ll} \begin{array}{ll} \frac{1}{30}, &{} \text {if} \ 240\le z\le 270, \\ 0,&{} \text {elsewhere} \end{array} \end{array}\right. }\\ f_{\overline{q}_{2}}(z) = {\left\{ \begin{array}{ll} \begin{array}{ll} \frac{1}{40}, &{} \text {if} \ 150\le z\le 190, \\ 0,&{} \text {elsewhere} \end{array} \end{array}\right. } \end{aligned}$$

The feasible set $\mathbb {K}$ is then as in (4) with the above definitions of $\underline{x}_{ij}(t,\omega )$, $\overline{x}_{ij}(t,\omega )$, $\overline{p}_{i}(t,\omega )$, $\overline{q}_{j}(t,\omega )$, $i = 1, 2$, $j = 1, 2$.

Then we left to define the profit function $v_i(t,\omega,x(t,\omega ))$ for the firms $P_i$, $i=1,2$. We set

$$\begin{aligned} \begin{aligned} v_1(t,\omega,x(t,\omega ))=&-3x^2_{11}(t,\omega ) - x^2_{12} (t,\omega ) + t x_{12} (t,\omega )x_{21} (t,\omega ) + \\&t^2 a_1(t,\omega )x_{11}(t,\omega )+ b_1(t,\omega )x_{12}(t,\omega )\\ v_2(t,\omega,x(t,\omega ))=&-2e^t x^2_{21}(t,\omega ) -3t x^2_{22} (t,\omega ) + x_{11} (t,\omega )x_{22} (t,\omega )+ \\&a_2(t,\omega )x_{21}(t,\omega )+ b_2(t,\omega )x_{22}(t,\omega ) \end{aligned} \end{aligned}$$

where $a_i$, $b_i$, $i=1,2$ are uniformly distributed random variables with supports:

$$\begin{aligned} \text {spt} \, a_1 = [36,108],&\quad \text {spt} \, b_1 = [10,40]\\ \text {spt} \, a_2 = [40,120],&\quad \text {spt} \, b_2 = [10,40] \end{aligned}$$

Let us compute the operator $\nabla _D v$

$$\begin{aligned} - \nabla _Dv(x)= \left( \begin{matrix} 6x_{11}-t^2a_1 \quad &{} 2x_{12}-tx_{21}-b_1\\ 4e^t x_{21}-a_2 \quad &{} 6t x_{22}-x_{11}-b_2 \end{matrix}\right) , \end{aligned}$$

where here, and in the following, we omit the arguments of the variables, simply writing $x_{ij}$ instead of $x_{ij}(t,\omega )$.

The equilibrium conditions is expressed by the following variational inequality problem: find $x^*\in L^2([0,25] \times \Omega ,\mathbb {R}_+^{4},\mathbb {P})$ such that

$$\begin{aligned} \ll -\nabla _D v(x^* ), x - x^*\gg _{\mathbb {E}} \ge 0,\quad \forall \, x \in \mathbb {K}. \end{aligned}$$

At first we observe that the existence and uniqueness to the solution to problem above is guaranteed by the theoretical result of [7] and Sect. 3. Moreover all the assumptions to ensure the convergence of the above computational method are satisfied. As a consequence, we can apply the method described before to obtain the numerical solutions $x_{ij}^*$, $i=1,2$, $j=1,2$, for different random evaluation of $a_i, b_i$, $i=1,2$, evolving in time. We plot the $x_{ij}^*$, $i=1,2$, $j=1,2$, in the figures of Table 1 for different random variables. Observe that the solution $x^* =(x^*_{ij})$ verifying the following

$$\begin{aligned} \text {spt} \, x^*_{11} = [6,18],&\quad \text {spt} \, x^*_{12} = [20,60]\\ \text {spt} \, x^*_{21} = [10,30],&\quad \text {spt} \, x^*_{22}= [16,58] \end{aligned}$$

and, hence, $x^*$ belongs to $\mathbb {K}$ proving that $x^*$ is the solution of the random dynamic oligopolistic market equilibrium problem.

Table 1 Curves of equilibria