Mean-Field Singular Stochastic Control with Regime Switching: Maximum Principles and Application

Abstract

In this paper, we study a class of mean-field singular stochastic optimal control problems for systems governed by regime-switching mean-field stochastic differential equations. The state dynamics depend on both regular and singular controls, and the coefficient of the singular component is allowed to depend explicitly on the state variable. We establish both necessary and sufficient stochastic maximum principles for this class of problems under the assumption that the control domain is convex. The presence of the state variable in the singular term leads to an adjoint process characterised by a generalised backward stochastic differential equation. As an application, we consider a one-dimensional regime-switching mean-field portfolio optimization problem with transaction costs, where the investor controls consumption and cumulative investment in a risky asset.

1. Introduction

Mean-field stochastic optimal control has grown into a vibrant area of research due to its natural ability to model systems consisting of a large number of interacting agents whose collective behaviour influences the environment faced by each individual. The origins of the theory can be traced to many-body problems in statistical physics, where the complexity arising from numerous interacting particles is approximated through aggregate or average effects. Since then, mean-field methods have found applications across a wide range of disciplines, including engineering, neuroscience, economics, finance, and insurance (Achdou et al. 2012; Bensoussan et al. 2013; Cardaliaguet 2010; Carmona and Delarue 2013, 2018, 2019; Carmona et al. 2013; Guéant 2009; Hocquet and Vogler 2021; Huang et al. 2006, 2019), with each area motivating new theoretical developments and practical challenges.

In many applications, particularly in finance and economics, the underlying environment is subject to abrupt changes caused by external shocks, policy interventions, or economic crises. Regime-switching models, in which such structural changes are represented by finite-state Markov chains, provide a natural and tractable framework for incorporating this source of uncertainty (see, e.g., Zhang et al. (2012) and the references therein). These models are sufficiently rich to capture important market features while remaining amenable to rigorous mathematical analysis.

Within this framework, singular controls arise naturally in problems where interventions occur only at discrete or irregular times rather than continuously. In contrast to classical controls, singular controls are adapted, non-decreasing càdlàg processes of finite variation whose induced measures are singular with respect to the Lebesgue measure (Øksendal and Sulem 2007). This formulation is particularly suitable for modeling irreversible or intermittent actions such as dividend payments, consumption and withdrawal decisions, portfolio adjustments under transaction costs, or harvesting policies in renewable resource management.

In this paper, we investigate a stochastic optimal control problem that simultaneously incorporates mean-field interactions, regime switching, and singular controls. The mean-field dependence is introduced through the marginal distribution of a function of the state process, providing a flexible formulation that encompasses several existing models as special cases through appropriate choices of this function. The state dynamics depend on the current state, the mean-field term, a regular control, and the regime-switching Markov chain, while the coefficient multiplying the singular control is itself state dependent. Although this general formulation significantly broadens the modeling framework, it also introduces substantial analytical challenges.

To analyze the problem, we adopt the stochastic maximum principle (SMP) rather than the dynamic programming approach. This choice is motivated by the intrinsic structure of the problem, which belongs to the class of time-inconsistent control problems because the objective functional depends on expectations involving functions of the state, rather than solely on the current state itself, making the standard Bellman principle difficult to apply. Originally developed by Kushner (1965, 1972) and subsequently extended in many directions for diffusion-type systems (see, for example, Bahlali et al. 1996; Bensoussan 1981; Cadenillas and Karatzas 1995; Elliott 1990; Haussmann 1986), the stochastic maximum principle provides a natural and powerful framework for deriving optimality conditions in such settings.

Following the pioneering work of Lasry and Lions (2007), a substantial literature has developed on mean-field control and mean-field games, including stochastic maximum principles for systems with distributional dependence (Andersson and Djehiche 2011; Buckdahn et al. 2009; Hafayed 2014; Meyer-Brandis et al. 2012). Parallel developments have established maximum principles for regime-switching systems (Donnelly 2011; Menoukeu-Pamen 2017; Menoukeu-Pamen and Momeya 2017; Nguyen et al. 2020; Tao and Wu 2012), while singular mean-field control problems admitting both necessary and sufficient optimality conditions have attracted increasing attention (Dahl and Øksendal 2017; Hu et al. 2017). In these settings, the stochastic maximum principle typically leads to backward stochastic differential equations with mean-field terms and reflection, whose well-posedness requires separate and careful analysis.

The principal contribution of this paper is the derivation of necessary and sufficient stochastic maximum principles for a class of singular mean-field control problems with Markov regime switching under the assumption that the control domain is convex. Our framework also admits a natural extension to state dynamics with jump components. It differs from that of (Wu and Zhang 2024), where impulse controls are studied for forward-backward regime-switching systems with conditional mean-field interactions, and from (Ganet Somé and Korveh 2026), where the control domain is non-convex and the coefficient associated with the singular control is independent of the state process.

A key novelty of the present work is that the coefficient of the singular control is allowed to depend explicitly on both the state variable and the current regime. This modelling feature has significant mathematical consequences. Unlike the classical setting with state-independent singular coefficients, perturbations of the state process generate additional first-order terms involving derivatives of the singular coefficient. Consequently, the variational equations, the Hamiltonian, and the associated optimality conditions acquire a different structure. These difficulties are further compounded by the simultaneous presence of mean-field interactions and regime-switching dynamics, which introduce expectation-dependent terms and Markov-chain martingales into the analysis. As a result, the adjoint process is no longer governed by the standard regime-switching BSDE encountered in earlier singular control problems. Instead, it is characterised by a generalised mean-field regime-switching BSDE coupled with an additional finite-variation component arising from the state dependence of the singular term. Establishing the existence and uniqueness of solutions to this generalised adjoint equation also constitutes one of the contributions of the paper and is essential for deriving the stochastic maximum principles. The resulting framework extends several existing formulations in the literature by simultaneously incorporating singular controls, mean-field interactions, regime-switching dynamics, and state-dependent singular coefficients.

The remainder of the paper is organised as follows. Section 2 introduces the mathematical model and formulates the control problem, together with assumptions guaranteeing the existence and uniqueness of solutions to the state equation. Section 3 presents the necessary and sufficient stochastic maximum principles and their proofs. Section 4 illustrates the theoretical results through a one-dimensional regime-switching mean-field portfolio optimization problem with transaction costs. Finally, Section 5 concludes the paper.

2. A Mean-Field Singular Markov Regime Switching Model and the Control Problem

Let $T>0$ be a fixed finite time horizon, and $(\mathsf{\Omega },\mathcal{F},{\{{\mathcal{F}}_t\}}_{t\in [0,T]},\mathbb{P})$ be a complete filtered probability space. The filtration ${\{{\mathcal{F}}_t\}}_{t\in [0,T]}$ is right-continuous and $\mathbb{P}$-completed, to which all processes defined below, including the Markov chain $\alpha :={\{\alpha (t)\}}_{t\in [0,T]}$, the N-dimensional standard Brownian motion $B:={\left\{B(t)\right\}}_{t\in [0,T]}$, and the non-decreasing processes $\xi$, are adapted. $\alpha$ is a continuous-time homogeneous and irreducible Markov chain with the finite state space $\mathcal{S}:=\{e_1,e_2,\dots ,e_D\}$, with $D\in \mathbb{N},e_i\in {\mathbb{R}}^D$ and for $i,j=1,2,\dots ,D$, the j-th component of $e_i$ is the Kronecker delta ${\delta }_{ij}$. $\mathbb{N}$ denote the set of natural numbers. $\mathcal{G}:={[{\zeta }_{ij}]}_{i,j=1,2,\dots ,D}$ denotes the generator of the Markov chain $\alpha$ under $\mathbb{P}$, also known as the rate or the Q-matrix. For $i,j=1,2,\dots ,D$, each entry ${\zeta }_{ij}$ of the rate matrix represents the constant transition intensity of the chain from state $e_i$ to state $e_j$ at time t. Without loss of generality, we suppose ${\zeta }_{ij}(t)>0,i\ne j$ and ${\displaystyle \sum _{j=1}^D}{\zeta }_{ij}=0$, that is ${\zeta }_{ii}(t)<0$. The following semi-martingale dynamics for the Markov chain hold (see Elliott et al. 2008; Zhang et al. 2012)

$$\mathrm{d}\alpha (t)=\mathcal{G}(t)\alpha (t)\mathrm{d}t+\mathrm{d}\mathbf{M}(t),\alpha (0)\in {\mathbb{R}}^D,t\in [0,T],$$

where $\{\mathbf{M}(t)|t\in [0,T]\}$ is an ${\mathbb{R}}^D$-valued, $({\{{\mathcal{F}}_t\}}_{t\in [0,T]},\mathbb{P})$-martingale. For each $i,j=1,2,\dots ,D$, with $i\ne j$ and $t\in [0,T]$, denote by ${\mathcal{J}}^{ij}(t)$ the number of jumps from state $e_i$ to state $e_j$ up to time t. Then, using the martingale dynamics (see Elliott 1994)

$$\begin{array}{cc}\hfill {\mathcal{J}}^{ij}& :=\sum _{0<s<t}〈\alpha (s-),e_i〉〈\alpha (s),e_j〉={\zeta }_{ij}{\int }_0^t〈\alpha (s-),e_i〉\mathrm{d}s+m_{ij}(t),\hfill \end{array}$$

where $m_{ij}(t):=\{m_{ij}(t)|t\in [0,T]\}$ with $m_{ij}(t):={\int }_0^t〈\alpha (s-),e_i〉〈\mathrm{d}\mathbf{M}(s),e_j〉$ is an $({\{{\mathcal{F}}_t\}}_{t\in [0,T]},\mathbb{P})$-martingale. The $m_{ij}$’s are also known as basic martingales associated with $\alpha$. For each fixed $j=1,2,\dots ,D$, let ${\mathsf{\Phi }}_j(t)$ denote the number of jumps into state $e_j$ up to time t. Then

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_j(t)& ={\displaystyle \sum _{i=1,i\ne j}^D}{\mathcal{J}}^{ij}(t)={\zeta }_j(t)+{\tilde{\mathsf{\Phi }}}_j(t),\hfill \end{array}$$

where ${\zeta }_j(t):={\displaystyle \sum _{i=1,i\ne j}^D}{\zeta }_{ij}{\int }_0^t〈\alpha (s),e_i〉\mathrm{d}s$, ${\tilde{\mathsf{\Phi }}}_j(t):={\displaystyle \sum _{i=1,i\ne j}^D}{m}_{ij}(t)$, and for each $j=1,2,\dots ,D$, ${\tilde{\mathsf{\Phi }}}_j:=\{{\tilde{\mathsf{\Phi }}}_j(t)|t\in [0,T]\}$ is an $({\{{\mathcal{F}}_t\}}_{t\in [0,T]},\mathbb{P})$-martingale. Next, we define

$$\begin{array}{cc}\hfill & {\mathcal{M}}^2({\mathbb{R}}_{+};{\mathbb{R}}^D):=\{f(·):{\mathbb{R}}_{+}\to {\mathbb{R}}^D\mathrm{s}.\mathrm{t}\parallel f(t){\parallel }_{{\mathcal{M}}^2}^2:={\displaystyle \sum _{j=1}^D}|f_j(t){|}^2{\zeta }_j(t)<\infty \}\hfill \\ \hfill & L^2({\mathcal{F}}_T;{\mathbb{R}}^d):=\{X\in {\mathbb{R}}^d,{\mathcal{F}}_T\text{-}\mathrm{measurable}\mathrm{random}\mathrm{variables},\mathrm{s}.\mathrm{t}.\mathbb{E}\left[{|X|}^2\right]<\infty \};\hfill \\ \hfill & S^2([0,T];{\mathbb{R}}^d):=\{f\in {\mathbb{R}}^d,{\{{\mathcal{F}}_t\}}_{t\ge 0}\text{-}\mathrm{adapted}\mathrm{c}\mathrm{àdlàg},\mathrm{s}.\mathrm{t}.\mathbb{E}[\underset{0\le t\le T}{sup}|f(t){|}^2]<\infty \};\hfill \\ \hfill & L_{\mathcal{F},p}^2([0,T];{\mathbb{R}}^N):=\{f\in {\mathbb{R}}^N,{\{{\mathcal{F}}_t\}}_{t\ge 0}\text{-}\mathrm{predictable}\mathrm{processes},\mathrm{s}.\mathrm{t}.\mathbb{E}[{\int }_0^T\parallel f(t){\parallel }^2\mathrm{d}t]<\infty \};\hfill \\ \hfill & M_p^2([0,T];{\mathbb{R}}^D):=\{f\in {\mathbb{R}}^D,{\{{\mathcal{F}}_t\}}_{t\ge 0}\text{-}\mathrm{predictable}\mathrm{processes},\mathrm{s}.\mathrm{t}.\mathbb{E}[{\int }_0^T\parallel f(t,·){\parallel }_{{\mathcal{M}}^2}^2\mathrm{d}t]<\infty \}.\hfill \end{array}$$

Let $A_1\subset \mathbb{R}$ be nonempty and $A_2\subset [0,\infty )$. Denote by $U_1$ (resp. $U_2$) the class of measurable, adapted processes $u(t)=u(t,\omega ):[0,T]\times \mathsf{\Omega }\to A_1$ (resp. $\xi (t)=\xi (t,\omega ):[0,T]\times \mathsf{\Omega }\to A_2$ such that $\xi$ is non-decreasing, continuous and $\xi (0)=0$). We suppose the controlled state process $X={\{X(t)\}}_{t\in [0,T]}$ satisfies

$$\begin{array}{cc}\hfill \mathrm{d}{X}^{u,\xi }(t)=& b(t,X^{u,\xi }(t),\mathbb{E}[\phi (X^{u,\xi }(t))],u(t),\alpha (t))\mathrm{d}t\hfill \\ \hfill & +\sigma (t,X^{u,\xi }(t),\mathbb{E}[\phi (X^{u,\xi }(t))],u(t),\alpha (t))\mathrm{d}B(t)+G(t,X^{u,\xi }(t))\mathrm{d}\xi (t)\hfill \\ \hfill & +\gamma (t,X^{u,\xi }(t-),\mathbb{E}[\phi (X^{u,\xi }(t-))],u(t-),\alpha (t-))\mathrm{d}\tilde{\mathsf{\Phi }}(t),\hfill \\ \hfill X^{u,\xi }(0)=& x_0\in {\mathbb{R}}^d,\hfill \end{array}$$

(1)

where $b:[0,T]\times \mathbb{R}\times \mathbb{R}\times U\times \mathcal{S}\to \mathbb{R}$, $\sigma :[0,T]\times \mathbb{R}\times \mathbb{R}\times U\times \mathcal{S}\to \mathbb{R}$, $\gamma :[0,T]\times \mathbb{R}\times \mathbb{R}\times U\times \mathcal{S}\to {\mathbb{R}}^D$, and $G:[0,T]\times \mathbb{R}\to \mathbb{R}$ are given continuous functions, and $\tilde{\mathsf{\Phi }}:=\left({\tilde{\mathsf{\Phi }}}_1(t),{\tilde{\mathsf{\Phi }}}_2(t),\dots ,{\tilde{\mathsf{\Phi }}}_D(t)\right)$ with ${\tilde{\mathsf{\Phi }}}_j(t)$ as given above. We write $X(t):=X^{u,\xi }(t)$. Furthermore, $\mathbb{E}$ denotes the expectation with respect to the probability measure $\mathbb{P}$, and $\phi :\mathbb{R}\to \mathbb{R}$ is a Lipschitz continuous function with spatial linear growth.

Definition 1.

An admissible control is an ${\{{\mathcal{F}}_t\}}_{t\ge 0}$-adapted process $(u,\xi )\in U_1\times U_2$ such that (1) has a unique strong solution and

$$\mathbb{E}[\underset{t\in [0,T]}{sup}{|u(t)|}^8+e^{\mu |\xi (T)|}]<\infty ,\mathrm{for}\mathrm{all}\mu >0.$$

(2)

We call $\mathcal{U}={\mathcal{U}}_1\times {\mathcal{U}}_2$ the set of admissible controls.

Let us consider a performance criterion defined for each $x_0\in {\mathbb{R}}^d,e_i\in \mathcal{S}$ as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J(x_0,e_i;u,\xi )& :={\mathbb{E}}_{x_0,e_i}[{\int }_0^{T}f(t,X(t),\mathbb{E}[\phi (X(t))],u(t),\alpha (t))\mathrm{d}t+{\int }_0^T\kappa (t)\mathrm{d}\xi (t)\hfill \\ \hfill & +h(X(T),\mathbb{E}[\phi (X(T))],\alpha (T))],\hfill \end{array}\end{array}$$

(3)

where ${\mathbb{E}}_{x_0,e_i}$ denotes the conditional expectation given $X(0)=x_0$, and $\alpha (0)=e_i$ under the probability measure $\mathbb{P}$. Furthermore, the functions $f:[0,T]\times \mathbb{R}\times \mathbb{R}\times U_1\times \mathcal{S}\to \mathbb{R}$, $h:{\mathbb{R}}^d\times \mathbb{R}\times \mathcal{S}\to \mathbb{R}$, and $\kappa :[0,T]\to \mathbb{R}$ are given continuous functions.

Proposition 1.

We wish to find $(u^{*},{\xi }^{*})\in \mathcal{U}$ such that

$$J(x_0,e_i;u^{*},{\xi }^{*})=\underset{(u,\xi )\in \mathcal{U}}{sup}J(x_0,e_i;u,\xi ).$$

(4)

Assumption A1.

The following assumptions will be used throughout this work

$(\mathcal{C}1)$ 

The function $\mathsf{\Gamma }=b,\sigma ,\gamma$ is uniformly Lipschitz continuous with respect to $(x,y)$, and of linear growth in $(x,y,u)$, i.e., there exists a constant $K>0$ such that

$$\begin{array}{cc}\hfill •& \parallel \mathsf{\Gamma }(t,x_1,y_1,u,e_i)-\mathsf{\Gamma }(t,x_2,y_2,u,e_i)\parallel \le K\left(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel \right);\hfill \\ \hfill •& \parallel \mathsf{\Gamma }(t,x,y,u,e_i)\parallel \le K\left(1+\parallel x\parallel +\parallel y\parallel +\parallel u\parallel \right).\hfill \end{array}$$

$(\mathcal{C}2)$ 

The function G is Lipschitz continuous and has linear growth, while κ is continuous.

$(\mathcal{C}3)$ 

The functions $b,\sigma ,\gamma ,f,h$ are twice continuously differentiable with respect to $(x,y)$, and G is twice continuously differentiable with respect to x.

$(\mathcal{C}4)$ 

The derivatives with respect to $(x,y)$ of the functions $b,\sigma ,\gamma ,G$ are bounded, and the derivatives of f are bounded by $K\left(1+\parallel x\parallel +\parallel y\parallel +\parallel u\parallel \right)$ and those of h are bounded by $K\left(1+\parallel x\parallel +\parallel y\parallel \right)$ for some constant $K>0$.

$(\mathcal{C}5)$ 

The control domain $\mathcal{U}$ is convex.

$(\mathcal{C}6)$

The coefficients $b,\sigma ,\gamma ,f$ are differentiable with respect to u with bounded derivatives.

Lemma 1.

Under Assumption 1, suppose $u\in L_{\mathcal{F},p}^2([0,T];\mathbb{R})$ and ξ is an ${\{{\mathcal{F}}_t\}}_{t\ge 0}$-adapted, continuous, non-decreasing process of finite variation satisfying $\mathbb{E}\left[{|\xi (T)|}^2\right]<\infty$. Then the controlled Markov-regime switching SDE (1) has a unique strong solution $X^{u,\xi }\in S_{\mathcal{F}}^2([0,T];{\mathbb{R}}^d)$.

Proof. 

See Section 3.2. □

For a differentiable function $\mathsf{\Gamma }(t),$ we denote by ${\mathsf{\Gamma }}_x(t)$ its derivative with respect to the spatial variable x. We write

$$\begin{array}{cc}\hfill \delta b(t,u(t)):=& b(t,X^{u^{*},{\xi }^{*}}(t),\mathbb{E}[\phi (X^{u^{*},{\xi }^{*}}(t))],u,\alpha (t))\hfill \\ \hfill & -b(t,X^{u^{*},{\xi }^{*}}(t),\mathbb{E}[\phi (X^{u^{*},{\xi }^{*}}(t))],u^{*}(t),\alpha (t)),\hfill \\ \hfill b_x^{\mu }(t,u(t)):=& b_x(t,X^{u^{*},{\xi }^{*}}(t)+\mu (X^{u^{\theta },{\xi }^{*}}(t)-X^{u^{*},{\xi }^{*}}(t)),\hfill \\ \hfill & \mathbb{E}[\phi (X^{u^{*},{\xi }^{*}}(t))]+\mu (\mathbb{E}[\phi (X^{u^{\theta },{\xi }^{*}}(t))]-\mathbb{E}[\phi (X^{u^{*},{\xi }^{*}}(t))]),u^{\theta },\alpha (t)),\hfill \end{array}$$

and similarly for $\delta \sigma ,\delta \gamma ,{\sigma }_x^{\mu }$ and ${\gamma }_x^{\mu }$.

3. Main Results and Proofs

In this section, we state and prove the necessary and sufficient conditions of optimality. Before stating the stochastic maximum principle, we introduce the Hamiltonian associated with the control problem. For convenience, we separate the Hamiltonian into its absolutely continuous and singular components. Specifically, we define the Hamiltonian $H:[0,T]\times {\mathbb{R}}^d\times \mathbb{R}\times U_1\times \mathcal{S}\times {\mathbb{R}}^d\times {\mathbb{R}}^{d\times d}\times {\mathbb{R}}^{d\times D}\to \mathbb{R}$ as follows:

$$\begin{array}{cc}\hfill H(t,x,y,u,e_i,p,q,s)=& H_1(t,x,y,u,e_i,p,q,s)+H_2(t,x,p),\hfill \end{array}$$

(5)

where

$$H_1(t,x,y,u,e_i,p,q,s)=f(t,x,y,u,e_i)+b(t,u,e_i)p+\sigma (t,u,e_i)q+\gamma (t,u,e_i)s(t)\zeta (t),$$

and

$$H_2(t,x,p)=G(t,x)p+\kappa (t).$$

Here $H_1$ corresponds to the absolutely continuous part of the state dynamics, whereas $H_2$ is associated with the singular control component. The corresponding first-order adjoint process $(p(t),q(t),s(t))$ associated with the Hamiltonian satisfies the following generalised regime-switching mean-field singular BSDE:

$$\begin{array}{cc}\hfill \mathrm{d}p(t)=& -\left\{H_{1,x}(t,x,y,u,e_i,p,q,s)+\mathbb{E}\left[H_{1,y}(t,x,y,u,e_i,p,q,s)\right]{\phi }_x(X(t))\right\}\mathrm{d}t\hfill \\ \hfill & -H_{2,x}(t,x,p)\mathrm{d}\xi (t)+q(t)\mathrm{d}B(t)+s(t)\mathrm{d}\tilde{\mathsf{\Phi }}(t),\hfill \\ \hfill p(T)=& h_x(X(T),\mathbb{E}[\phi (X(T))],\alpha (T))+\mathbb{E}[h_y(X(T),\mathbb{E}[\phi (X(T))],\alpha (T))]{\phi }_x(X(T)).\hfill \end{array}$$

(6)

It is well known that the above BSDE is well posed when its coefficients are independent of both regime switching and the mean-field term, under standard integrability and monotonicity assumptions (see Elmansouri and El Otmani 2023; Pardoux and Zhang 1998). By extending the arguments of (Elmansouri and El Otmani 2023; Pardoux and Zhang 1998) and incorporating those of (Buckdahn et al. 2009; Elmansouri and El Otmani 2023), one can establish that the BSDE remains well posed in a suitable functional space.

Assumption A2.

The following assumptions holds:

($B_1$) 

The function $g:[0,T]\times \mathsf{\Omega }\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^D\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^D\times \mathcal{S}⟶\mathbb{R}$ is uniformly Lipschitz continuous with respect to $(y,z,s,m_y,m_z,m_s)$, and of linear growth in $(y,z,s,m_y,m_z,m_s)$. More precisely, there exists a constant $K>0$ and two ${\{{\mathcal{F}}_t\}}_{t\ge 0}$-adapted process ${\varphi }_1$ and ${\varphi }_2$ such that, for all admissible arguments,

$$\begin{array}{cc}\hfill & |g(t,y_1,z_1,s_1,m_{y_1},m_{z_1},m_{s_1},e_i)-g(t,y_2,z_2,s_2,m_{y_2},m_{z_2},m_{s_2},e_i)|\hfill \\ \hfill \le & K\left(|y_1-y_2|+|z_1-z_2|+\parallel s_1-s_2\parallel +\parallel m_{y_1}-m_{y_2}\parallel +\parallel m_{z_1}-m_{z_2}\parallel +\parallel m_{s_1}-m_{s_2}\parallel \right),\hfill \end{array}$$

and

$$\begin{array}{c}\hfill |g(t,y,z,s,m_y,m_z,m_s,e_i)|\le {\varphi }_1(t)+K\left(|y|+|z|+|s|+\parallel m_y\parallel +\parallel m_z\parallel +\parallel m_s\parallel \right).\end{array}$$

($B_2$) 

$g(·,0,0,0,0,0,0,\alpha (·))\in L_{\mathcal{F},p}^2([0,T];\mathbb{R})$.

($B_3$) 

The terminal condition satisfies $\mathcal{Y}\in L^2({\mathcal{F}}_T;\mathbb{R})$ and $\mathbb{E}\left[e^{\lambda \xi (T)}{|\mathcal{Y}|}^2\right]<\infty$ for all $\lambda >0$. Moreover, ξ is assumed to be a continuous, adapted, finite variation, increasing process with $\xi (0)=0$.

($B_4$) 

The function $x\mapsto \phi (x)$ is uniformly Lipschitz continuous and of linear growth.

($B_5$) 

The function $\rho :[0,T]\times \mathsf{\Omega }\times \mathbb{R}\to \mathbb{R}$ is of the following linear growth

$$|\rho (t,y)|\le {\varphi }_2(t)+K|y|.$$

In addition, there exists $\mu >0$ such that, for all $y,y^{\prime }\in \mathbb{R}$,

$$(y-y^{\prime })\left(\rho (t,y)-\rho (t,y^{\prime })\right)\le -\mu {|y-y^{\prime }|}^2,dt\otimes d\mathbb{P}\text{-}\mathrm{a}.\mathrm{e}.$$

($B_6$) 

For all $\lambda >0$

$$\mathbb{E}[{\int }_0^T{e}^{\lambda \xi (t)}|{\varphi }_1{(t)|}^2\mathrm{d}t+{\int }_0^T{e}^{\lambda \xi (t)}{|{\varphi }_2(t)|}^2\mathrm{d}\xi (t)]<\infty .$$

Consider the following regime-switching singular mean-field BSDE

$$\left\{\begin{array}{cc}\mathrm{d}Y(t)=\hfill & -g(t,Y(t),Z(t),s(t),\mathbb{E}[\phi (Y(t))],\mathbb{E}[\phi (Z(t))],\mathbb{E}[\phi (s(t))],\alpha (t))\mathrm{d}t\hfill \\ \hfill & -\rho (t,Y(t))\mathrm{d}\xi (t)+Z(t)\mathrm{d}B(t)+R(t)\mathrm{d}\tilde{\mathsf{\Phi }}(t),\hfill \\ Y(T)=\hfill & \mathcal{Y}.\hfill \end{array}\right.$$

(7)

Lemma 2.

Under Assumption 2, the regime-switching singular mean-field BSDE (7) admits a unique adapted solution $(Y,Z,R)\in S^2([0,T];\mathbb{R})\times L_{\mathcal{F},p}^2([0,T];\mathbb{R})\times M_p^2([0,T];{\mathbb{R}}^D)$.

Proof. 

See Section 3.2. □

3.1. Main Results

We now state the first main result of this section.

Theorem 1

(Necessary maximum principle). Suppose Assumption 1 holds and let $(X^{u^{*},{\xi }^{*}}(·),$ $u^{*}(·),{\xi }^{*}(·))$ be an optimal solution to the control problem (1)–(4). Then, there exists a triplet $(p(·),q(·),s(·))$ satisfying the regime-switching BSDE (6).

In addition, the following necessary optimality conditions hold:

1. 

$$\begin{array}{cc}\hfill & {\partial }_u{H}_1\left(t,X^{*}(t),\mathbb{E}[\phi (X^{*}(t))],u^{*}(t),e_i,p(t),q(t),s(t)\right)(u(t)-u^{*}(t))\ge 0,\hfill \\ \hfill & \mathbb{P}\otimes \mathrm{d}t\text{-}\mathrm{a}.\mathrm{e}\mathrm{for}\mathrm{all}u\in {\mathcal{U}}_1,\hfill \end{array}$$

(8)

2. 

For every admissible singular control $\xi \in {\mathcal{U}}_2$,

$$\mathbb{E}[{\int }_0^T{H}_2\left(t,X^{*}(t),p(t)\right)\mathrm{d}\left(\xi (t)-{\xi }^{*}(t)\right)]\le 0.$$

(9)

Consequently,

$$\begin{array}{cc}\hfill H_2\left(t,X^{*}(t),p(t)\right)& \le 0,\mathrm{a}.\mathrm{s}.\mathrm{for}\mathrm{all}t\in [0,T],\mathrm{and}{\int }_0^T{H}_2\left(t,X^{*}(t),p(t)\right)\mathrm{d}{\xi }_t^{*}=0,\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.\hfill \end{array}$$

(10)

Remark 1.

We prove Theorem 1 in two steps. In step 1, we derive the variational equation (see Lemma 3). The second step is devoted to deriving the duality relations between the adjoint processes and the variational equations (see Lemma 4).

The next additional assumptions are needed for the sufficient conditions of optimality:

Theorem 2

(Sufficient maximum principle). Suppose Assumption 1 holds and let $(u^{*},{\xi }^{*})\in \mathcal{U}$ be an arbitrary control and $X^{*}(t):=X^{u^{*},{\xi }^{*}}(t)$ be the corresponding controlled state process. Suppose there exist adapted solutions $(\hat{p}(t),\hat{q}(t),\hat{s}(t))$ to the corresponding first-order adjoint Equation (6) such that the following integrability conditions hold:

$$\begin{array}{cc}\hfill & \mathbb{E}[{\int }_0^T\left(X^{*}(t)-X(t)\right)\{\hat{q}(t)\hat{q}{(t)}^{\top }+\hat{s}(t)\mathrm{Diag}(\zeta (t))\hat{s}{(t)}^{\top }\}\left(X^{*}(t)-X(t)\right)\mathrm{d}t]<\infty ,\hfill \\ \hfill & \mathbb{E}[{\int }_0^{T}p(t)\{{(\sigma \sigma )}^{\top }(t,X(t),\mathbb{E}[\phi (X(t))],u(t),\alpha (t))\}\hat{p}(t)\mathrm{d}t]<\infty .\hfill \end{array}$$

Furthermore, assume that:

1. 

For each pair $(t,e_i)\in [0,T]\times \mathcal{S}$, H, (resp. h) is a concave function in $x,y,u$ (resp. $x,y$).

2. 

For almost all $t\in [0,T]$

$$\begin{array}{cc}\hfill & H_1(t,X^{*}(t),\mathbb{E}[\phi (X^{*}(t))],u^{*}(t),e_i(t),p(t),q(t),s(t))\hfill \\ \hfill & =\underset{u\in U_1}{sup}{H}_1(t,X^{*}(t),\mathbb{E}[\phi (X^{*}(t))],u,e_i(t),p(t),q(t),s(t)),\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.\hfill \end{array}$$

(11)

3. 

The following condition holds

$$\begin{array}{c}\hfill H_2(t,X^{*}(t),p(t))\le 0,t\in [0,T],\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.,\end{array}$$

(12)

and

$$\begin{array}{c}\hfill {\int }_0^T{H}_2(t,X^{*}(t),p(t))\mathrm{d}{\xi }^{*}(t)=0,\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.\end{array}$$

(13)

Then, $(u^{*},{\xi }^{*})$ is an optimal control and $X^{*}(·)$ is the optimal state process.

We use convex perturbation for both singular and regular control $\xi$ and u, respectively and define the perturbations as follows:

$$(u^{\theta }(t),{\xi }^{\theta }(t)):=(u^{*}(t)+\theta v(t),{\xi }^{*}(t)+\theta \eta (t)),\theta \in (0,1)$$

where $\theta >0$ is sufficiently small, $v=u-u^{*}$, where $u\in {\mathcal{U}}_1$, and $\eta =\xi -{\xi }^{*}$, where $\xi \in {\mathcal{U}}_2$. Since $(u^{*},{\xi }^{*})$ is an optimal control pair, we have

$$J(u^{\theta },{\xi }^{\theta })-J(u^{*},{\xi }^{*})=J(u^{\theta },{\xi }^{\theta })-J(u^{\theta },{\xi }^{*})+J(u^{\theta },{\xi }^{*})-J(u^{*},{\xi }^{*})=J_1+J_2\le 0.$$

Then, the variational inequality will be obtained if

$$\underset{\theta \to 0}{lim}{\displaystyle \frac{1}{\theta }}{J}_1+\underset{\theta \to 0}{lim}{\displaystyle \frac{1}{\theta }}{J}_2\le 0.$$

(14)

Let $X^{\theta }(t):=X^{u^{\theta },{\xi }^{\theta }}(t),X^{*}(t):=X^{u^{*},{\xi }^{*}}(t)$ be the trajectories associated with the control pairs $(u^{\theta },{\xi }^{\theta })$, and $(u^{*},{\xi }^{*})$, respectively. We compute the limits in (14) separately. Next, we state some auxiliary results whose proofs are given later.

Lemma 3.

Under Assumption 1, we have

$$\underset{\theta \to 0}{lim}\mathbb{E}[\underset{t\in [0,T]}{sup}|{\displaystyle \frac{X^{\theta }(t)-X^{*}(t)}{\theta }}-Z(t){|}^2]=0,$$

where Z is the solution to the following regime-switching linear singular SDE:

$$\begin{array}{cc}\hfill \mathrm{d}Z(t)=& \left\{b_x^{*}(t)Z(t)+b_y^{*}(t)\mathbb{E}[{\phi }_x(X^{*}(t))Z(t)]+b_u^{*}(t)v(t)\right\}\mathrm{d}t\hfill \\ \hfill & +\left\{{\sigma }_x^{*}(t)Z(t)+{\sigma }_y^{*}(t)\mathbb{E}[{\phi }_x(X^{*}(t))Z(t)]+{\sigma }_u^{*}(t)v(t)\right\}\mathrm{d}B(t)\hfill \\ \hfill & +\left\{{\gamma }_x^{*}(t)Z(t)+{\gamma }_y(t)\mathbb{E}[{\phi }_x(X^{*}(t))Z(t)]+{\gamma }_u^{*}(t)v(t)\right\}\mathrm{d}\tilde{\mathsf{\Phi }}(t)\hfill \\ \hfill & +G_x(t,X^{*}(t))Z(t)\mathrm{d}\xi (t)+G(t,X^{*}(t))\mathrm{d}\eta (t),\hfill \\ \hfill Z(0)=& 0,\hfill \end{array}$$

where

$$b_x^{*}(t):=b_x(t,X^{*}(t),\mathbb{E}[\phi (X^{*}(t))],u^{*}(t),\alpha (t)),$$

and similarly for other coefficients.

Proof. 

See Section 3.2. □

Lemma 4.

$$\begin{array}{cc}\hfill \mathbb{E}[p^{*}(T)Z(T)]=& \mathbb{E}[{\int }_0^T-Z(t)(f_x^{*}(t)+\mathbb{E}[f_y^{*}(t)]{\phi }_x(X^{*}(t)))\mathrm{d}t]\hfill \\ \hfill & +\mathbb{E}[{\int }_0^T{p}^{*}(t)b_u^{*}(t)v(t)\mathrm{d}t]+\mathbb{E}[{\int }_0^T{q}^{*}(t){\sigma }_u^{*}(t)v(t)\mathrm{d}t]\hfill \\ \hfill & +\mathbb{E}[{\int }_0^T{s}^{*}(t){\gamma }_u^{*}(t)\zeta (t)v(t)\mathrm{d}t]+\mathbb{E}[{\int }_0^T{p}^{*}(t)G(t,X^{*}(t))\mathrm{d}{\eta }_t.\hfill \end{array}$$

Proof. 

The proof uses Itô’s formula and is similar to (Zhang et al. 2012, Theorem 4.1). □

3.2. Proofs of Auxiliary and Main Results

Here, we provide the proofs for the auxiliary and main results.

Proof of Lemma 1

The proof follows by combining arguments from (Doléans-Dade 1976, Theorem 1); (Protter 2005, Theorem 6, chap. V), and (Shen and Siu 2013, Theorem 3.1). For completeness, we give a sketch. The proof can be done in several steps.

Step 1: We first consider the SDE without the regime switching and the mean-field term, that is, we consider the following SDE:

$$\left\{\begin{array}{cc}\hfill & \mathrm{d}{X}^{u,\xi }(t)=b(t,X^{u,\xi }(t),u(t))\mathrm{d}t+\sigma (t,X^{u,\xi }(t),u(t))\mathrm{d}B(t)+G(t,X^{u,\xi }(t))\mathrm{d}\xi (t),\hfill \\ \hfill & X^{u,\xi }(0)=x_0\in {\mathbb{R}}^d.\hfill \end{array}\right.$$

(15)

By (Doléans-Dade 1976, Theorem 1) (see also Protter 2005, chap. V), Equation (15) admits a unique strong solution in $S^2([0,T];{\mathbb{R}}^d)$.

Step 2: We first consider the SDE with regime switching and without the mean-field term, that is, we consider the following SDE:

$$\left\{\begin{array}{cc}\mathrm{d}{X}^{u,\xi }(t)=\hfill & b(t,X^{u,\xi }(t),u(t),\alpha (t))\mathrm{d}t+\sigma (t,X^{u,\xi }(t),u(t),\alpha (t))\mathrm{d}B(t)\hfill \\ & +G(t,X^{u,\xi }(t))\mathrm{d}\xi (t)+\gamma (t,X^{u,\xi }(t-),u(t-),\alpha (t-))\mathrm{d}\tilde{\mathsf{\Phi }}(t),\hfill \\ X^{u,\xi }(0)=\hfill & x_0\in {\mathbb{R}}^d.\hfill \end{array}\right.$$

(16)

It is known (see, for example, Mao and Yuan 2006) that there is a sequence ${\{{\tau }_i\}}_{i\ge 0}$ of stopping times such that $0={\tau }_0<{\tau }_1<\dots <{\tau }_i\to \infty$ and $\alpha (t)$ is constant on every interval $[{\tau }_i\wedge T;{\tau }_{i+1}\wedge T)$, i.e., for every $i\ge 0$ since $\alpha$ is right continuous

$$\alpha (t)=\alpha ({\tau }_i)=e_i,,{\tau }_i\le t<{\tau }_{i+1}.$$

Since the admissible control may depend on the current regime, we may also write

$$u(t)=u_i(t),{\tau }_i\le t<{\tau }_{i+1}.$$

Between $[0,{\tau }_1\wedge T)$ the equation becomes

$$\left\{\begin{array}{cc}\mathrm{d}{X}^{u,\xi }(t)=\hfill & \left(b(t,X^{u,\xi }(t),u_0(t),e_0)-\sum _{j\ne 0}{\zeta }_{0j}{\gamma }_j(t,X^{u,\xi }(t),u_0(t),e_0)\right)\mathrm{d}t\hfill \\ & +\sigma (t,X^{u,\xi }(t),u_0(t),e_0)\mathrm{d}B(t)+G(t,X^{u,\xi }(t))\mathrm{d}\xi (t),\hfill \\ X^{u,\xi }(0)=\hfill & x_0\in {\mathbb{R}}^d,\hfill \end{array}\right.$$

(17)

and by Step 1, the above equation has a unique strong solution in $[0,{\tau }_1)$. In addition, at time ${\tau }_1$, we have

$$X^{u,\xi }({\tau }_1)=X^{u,\xi }({\tau }_1^{-})+{\gamma }_j({\tau }_1,X^{u,\xi }({\tau }_1^{-}),u({\tau }_1^{-}),e_0).$$

In the above, ${\gamma }_j(t,x,u,e_i)$ denotes the jump size associated with a transition $e_i\to e_j$. Suppose that there is a unique solution on the interval $[{\tau }_{i-1}\wedge T,{\tau }_i\wedge T)$. Then, on the interval $[{\tau }_i\wedge T,{\tau }_{i+1}\wedge T)$, the regime is fixed and the SDE (16) is reduced to

$$\left\{\begin{array}{cc}\mathrm{d}{X}^{u,\xi }(t)=\hfill & \left(b(t,X^{u,\xi }(t),u_i(t),e_i)-\sum _{j\ne 0}{\zeta }_{ij}{\gamma }_j(t,X^{u,\xi }(t),u_i(t),e_i)\right)\mathrm{d}t\hfill \\ & +\sigma (t,X^{u,\xi }(t),u_i(t),e_i)\mathrm{d}B(t)+G(t,X^{u,\xi }(t))\mathrm{d}\xi (t),\hfill \\ X^{u,\xi }({\tau }_i)\hfill & =X^{u,\xi }({\tau }_i^{-})+{\gamma }_j({\tau }_i,X^{u,\xi }({\tau }_i^{-}),u({\tau }_i^{-}),e_{i-1}).\hfill \end{array}\right.$$

(18)

Hence, using once more (Doléans-Dade 1976, Theorems 1 and 2) (see also Protter 2005, Theorem 6, chap. V), the SDE (18) has a unique strong solution. Thus, one constructs the solution successively on $[0,{\tau }_1\wedge T),[{\tau }_1\wedge T,{\tau }_2\wedge T),\dots [{\tau }_i\wedge T,{\tau }_{i+1}\wedge T),\dots$. Since the finite-state Markov chain has only finitely many jumps on every bounded interval $[0,T]$, this gives a unique solution of (16) on $[0,T]$. The proof of step 2 is completed.

Step 3: For $x={\{x(t)\}}_{t\in [0,T]}\in S^2([0,T];{\mathbb{R}}^d)$, consider the following SDE:

$$\left\{\begin{array}{cc}\mathrm{d}{X}^{u,\xi }(t)=\hfill & b^{\mathbb{E}}(t,X^{u,\xi }(t),u(t),\alpha (t))\mathrm{d}t+{\sigma }^{\mathbb{E}}(t,X^{u,\xi }(t),u(t),\alpha (t))\mathrm{d}B(t)\hfill \\ & +G(t,X^{u,\xi }(t))\mathrm{d}\xi (t)+{\gamma }^{\mathbb{E}}(t,X^{u,\xi }(t-),u(t-),\alpha (t-))\mathrm{d}\tilde{\mathsf{\Phi }}(t),\hfill \\ X^{u,\xi }(0)=\hfill & x_0\in {\mathbb{R}}^d,\hfill \end{array}\right.$$

(19)

where we denote $b^{\mathbb{E}}(t,z,u,e_i)=b(t,z,m_x(t),u,e_i)$, where $m_x(t):=\mathbb{E}[\phi (x(t))]$. Then from Assumption 1(${\mathcal{C}}_1$), we deduce that The function ${\mathsf{\Gamma }}^{\mathbb{E}}=b^{\mathbb{E}},{\sigma }^{\mathbb{E}},{\gamma }^{\mathbb{E}}$ satisfies

$$\begin{array}{c}\hfill \parallel {\mathsf{\Gamma }}^{\mathbb{E}}(t,x_1,u,e_i)-{\mathsf{\Gamma }}^{\mathbb{E}}(t,x_2,u,e_i)\parallel \le K\left(\parallel x_1-x_2\parallel \right);\mathrm{and}\parallel {\mathsf{\Gamma }}^{\mathbb{E}}(t,x,u,e_i)\parallel \le K\left(1+\parallel x\parallel +\parallel u\parallel \right).\end{array}$$

Hence, Lipschitz and linear growth conditions are satisfied and, therefore, the equation has a unique strong solution.

Step 4: We now construct a solution in $S^2([0,T];{\mathbb{R}}^d)$ using Picard iteration. We define by induction the following processes:

$$\begin{array}{cc}\hfill X_0^{u,\xi }(t)=& 0,\hfill \\ \hfill X_{n+1}^{u,\xi }(t)=& {\int }_0^{t}b(s,X_n^{u,\xi }(s),\mathbb{E}[\phi (X_n^{u,\xi }(s))],u(s),\alpha (s))\mathrm{d}s\hfill \\ \hfill & +{\int }_0^t{\sigma }^{\mathbb{E}}(s,X_n^{u,\xi }(s),\mathbb{E}[\phi (X_n^{u,\xi }(s))],u(s),\alpha (s))\mathrm{d}B(s)+{\int }_0^{t}G(s,X_n^{u,\xi }(s))\mathrm{d}\xi (s)\hfill \\ \hfill & +{\int }_0^t\gamma (s,X_n^{u,\xi }(s-),\mathbb{E}[\phi (X_n^{u,\xi }(s-))],u(s),\alpha (s))\mathrm{d}\tilde{\mathsf{\Phi }}(s).\hfill \end{array}$$

(20)

Thanks to the linear growth condition of $\phi$ and the growth of the coefficients, one can show using a similar step as in (Doléans-Dade 1976) to obtain that $\mathbb{E}[\underset{0\le t\le T}{sup}|X_n^{u,\xi }(t){|}^2]<\infty$ for all n.

We focus on the interval $[0,{\tau }_1\wedge T)$, since the same argument applies to each interval $[{\tau }_i\wedge T,{\tau }_{i+1}\wedge T)$.

On $[0,{\tau }_1\wedge T)$, the regime remains fixed and equal to $e_0$. Therefore, the Picard iteration (20) reduces to

$$\begin{array}{cc}\hfill X_0^{u,\xi }(t)=& 0,\hfill \\ \hfill X_{n+1}^{u,\xi }(t)=& {\int }_0^t(b(s,X_n^{u,\xi }(s),\mathbb{E}[\phi (X_n^{u,\xi }(s))],u_0(s),e_0)\hfill \\ \hfill & -\sum _{j\ne 0}{\zeta }_{0j}{\gamma }_j(s,X_n^{u,\xi }(s),\mathbb{E}[\phi (X_n^{u,\xi }(s))],u_0(s),e_0))\mathrm{d}s\hfill \\ \hfill & +{\int }_0^t{\sigma }^{\mathbb{E}}(s,X_n^{u,\xi }(s),\mathbb{E}[\phi (X_n^{u,\xi }(s))],u_0(s),e_0)\mathrm{d}B(s)+{\int }_0^{t}G(s,X_n^{u,\xi }(s))\mathrm{d}\xi (s).\hfill \end{array}$$

(21)

By induction, it follows that the sequence ${\{X_n^{u,\xi }\}}_{n\ge 0}$ consists of semi-martingales. Indeed, for each $n\ge 0$, the processes $\mathsf{\Gamma }\left(s,X_n^{u,\xi }(s),\mathbb{E}[\phi (X_n^{u,\xi }(s))],u_0(s),e_0\right),\mathsf{\Gamma }=b,\sigma ,\gamma ,$ are adapted and locally bounded. We now prove by induction that

$$\mathbb{E}[\underset{0\le t\le {\tau }_1\wedge T}{sup}{|X_n^{u,\xi }(t)|}^2]<\infty ,n\ge 0.$$

The claim is immediate for $n=0$. Suppose that it holds for all $k\le n$. We show that it also holds for $k=n+1$.

To this end, we first use a localisation argument. Assume temporarily that the increasing process $\xi$ satisfies $\xi ({\tau }_1\wedge T)\le \delta$ for some $\delta >0$ such that $K\delta <16$. Squaring both sides of (21), taking the supremum over $t\in [0,{\tau }_1\wedge T]$, then taking expectations, and applying the Burkholder–Davis–Gundy inequality together with the Cauchy–Schwarz inequality, we obtain

$$\begin{array}{cc}\hfill \mathbb{E}[\underset{0\le t\le {\tau }_1\wedge T}{sup}{|X_{n+1}^{u,\xi }(t)|}^2]\le & 4(2{\tau }_1\wedge T{\int }_0^{{\tau }_1\wedge T}(\mathbb{E}[|b(s,X_n^{u,\xi }(s),\mathbb{E}[\phi (X_n^{u,\xi }(s))],u(s),e_0){|}^2]\hfill \\ \hfill & +2\mathbb{E}|\sum _{j\ne 0}{\zeta }_{0j}{\gamma }_j(s,X_n^{u,\xi }(s),\mathbb{E}[\phi (X_n^{u,\xi }(s))],u(s),e_0){|}^2])\mathrm{d}s\hfill \\ \hfill & +{\int }_0^{{\tau }_1\wedge T}\mathbb{E}[|{\sigma }^{\mathbb{E}}(s,X_n^{u,\xi }(s),\mathbb{E}[\phi (X_n^{u,\xi }(s))],u(s),e_0){|}^2]\mathrm{d}s\hfill \\ \hfill & +\mathbb{E}[{\xi }^2({\tau }_1\wedge T)\underset{0\le s\le {\tau }_1\wedge T}{sup}{|G(s,X_n^{u,\xi }(s))|}^2])\hfill \\ \hfill \le & 4({\int }_0^{{\tau }_1}16K^2({\tau }_1\wedge T+C_D)\left(1+\mathbb{E}[{|u(s)|}^2+\underset{0\le t\le s}{sup}{|X_n^{u,\xi }(t)|}^2]\right)\mathrm{d}s\hfill \\ \hfill & +{\int }_0^{{\tau }_1\wedge T}8K^2\left(1+\mathbb{E}[{|u(s)|}^2+\underset{0\le t\le s}{sup}{|X_n^{u,\xi }(t)|}^2]\right)\mathrm{d}s\hfill \\ \hfill & +2K^2{\delta }^2\mathbb{E}[(1+\underset{0\le s\le {\tau }_1\wedge T}{sup}|X_n^{u,\xi }(s){|}^2)])<\infty .\hfill \end{array}$$

(22)

Here, we have used the linear growth assumptions on the coefficients, the linear growth of $\phi$, the square-integrability of u, and the fact that the Markov chain has a finite state space, so that the transition rates ${\zeta }_{0j}$ are bounded.

It remains to remove the auxiliary smallness assumption on $\xi$. To this end, define the stopping times

$${\tilde{\tau }}_0:=0,{\tilde{\tau }}_{k+1}:=inf\{t\ge {\tilde{\tau }}_k:\xi (t)-\xi ({\tilde{\tau }}_k)>\delta \}\wedge {\tau }_1\wedge T.$$

Since $\xi$ is continuous, increasing, and of finite variation, there exists an almost surely finite integer N such that ${\tilde{\tau }}_N={\tau }_1\wedge T.$ Moreover, on each interval $[{\tilde{\tau }}_k,{\tilde{\tau }}_{k+1}]$, we have $\xi ({\tilde{\tau }}_{k+1})-\xi ({\tilde{\tau }}_k)\le \delta .$ Therefore, the preceding estimate applies to each such interval. Restarting the equation at time ${\tilde{\tau }}_k$ with initial value $X_n^{u,\xi }({\tilde{\tau }}_k)$, and then pasting the estimates over the finite number of intervals, yields

$$\mathbb{E}[\underset{0\le t\le {\tau }_1\wedge T}{sup}{|X_{n+1}^{u,\xi }(t)|}^2]<\infty .$$

The same argument applies to each interval $[{\tau }_i\wedge T,{\tau }_{i+1}\wedge T]$. Since the finite-state Markov chain has only finitely many jumps on $[0,T]$, it follows that

$$\mathbb{E}[\underset{0\le t\le T}{sup}{|X_n^{u,\xi }(t)|}^2]<\infty ,n\ge 0.$$

Similarly, using the same argument as before gives

$$\begin{array}{cc}\hfill D_{n+1}({\tau }_1\wedge T):=& \mathbb{E}[\underset{0\le t\le {\tau }_1\wedge T}{sup}{|X_{n+1}^{u,\xi }(t)-X_n^{u,\xi }(t)|}^2]\hfill \\ \hfill \le & 4({\int }_0^{{\tau }_1\wedge T}8K^2(1+{\tau }_1\wedge T+C_D)\mathbb{E}[\underset{0\le t\le s}{sup}{|X_n^{u,\xi }(s)-X_{n-1}^{u,\xi }(s)|}^2]\mathrm{d}s\hfill \\ \hfill & +K^2{\delta }^2\mathbb{E}[\underset{0\le s\le {\tau }_1\wedge T}{sup}{|X_n^{u,\xi }(s)-X_{n-1}^{u,\xi }(s)|}^2])\hfill \\ \hfill \le & 32K^2(1+C_D)(1+T){\int }_0^{{\tau }_1\wedge T}\mathbb{E}[\underset{0\le t\le s}{sup}{|X_n^{u,\xi }(t)-X_{n-1}^{u,\xi }(t)|}^2]\mathrm{d}s\hfill \\ \hfill & +4K^2{\delta }^2\mathbb{E}[\underset{0\le s\le {\tau }_1\wedge T}{sup}{|X_n^{u,\xi }(s)-X_{n-1}^{u,\xi }(s)|}^2]\hfill \\ \hfill \le & ({\tilde{K}}_T\times {\tau }_1\wedge T+4K^2{\delta }^2)\mathbb{E}[\underset{0\le s\le {\tau }_1\wedge T}{sup}{|X_n^{u,\xi }(t)-X_{n-1}^{u,\xi }(t)|}^2]\hfill \\ \hfill \le & ({\tilde{K}}_T\times {\tau }_1\wedge T+4K^2{\delta }^2)D_n({\tau }_1\wedge T),\hfill \end{array}$$

(23)

where we have used the fact that

$$D_n(s):=\mathbb{E}[\underset{0\le t\le s}{sup}{|X_n^{u,\xi }(t)-X_{n-1}^{u,\xi }(t)|}^2],$$

is non-decreasing in s and thus $D_n(s)\le D_n({\tau }_1\wedge T)$.

To make this a contraction, we localise it further in time. Define

$${\rho }_0:=0,{\rho }_{m+1}:=inf\{t\ge {\rho }_m:{\tilde{K}}_T(t-{\rho }_m)+4K^2{(\xi (t)-\xi ({\rho }_m))}^2>{\displaystyle \frac{1}{2}}\}\wedge {\tau }_1\wedge T.$$

Since $\xi$ is continuous, increasing, and of finite variation, there exists an almost surely finite integer M such that ${\rho }_M={\tau }_1\wedge T.$ Moreover, on each interval $[{\rho }_m,{\rho }_{m+1}]$, it holds ${\tilde{K}}_T({\rho }_{m+1}-{\rho }_m)+4K^2{(\xi ({\rho }_{m+1})-\xi ({\rho }_m))}^2\le {\displaystyle \frac{1}{2}}$. Hence, applying the preceding estimate on $[{\rho }_m,{\rho }_{m+1}]$, we obtain $D_{n+1}^{(m)}\le {\displaystyle \frac{1}{2}}{D}_n^{(m)},$ where

$$D_n^{(m)}:=\mathbb{E}[\underset{{\rho }_m\le t\le {\rho }_{m+1}}{sup}{|X_n^{u,\xi }(t)-X_{n-1}^{u,\xi }(t)|}^2].$$

Consequently,

$$D_{n+1}^{(m)}\le 2^{-n}{D}_1^{(m)}.$$

Therefore ${(X_n^{u,\xi })}_{n\ge 0}$ is a Cauchy sequence in $S^2([{\rho }_m,{\rho }_{m+1}];{\mathbb{R}}^d)$. Arguing as in the proof of (Doléans-Dade 1976, Theorem 1), there exists a unique process $X^{u,\xi }\in S^2([{\rho }_m,{\rho }_{m+1}];{\mathbb{R}}^d)$ such that

$$X_n^{u,\xi }\to X^{u,\xi }\mathrm{in}{S}^2([{\rho }_m,{\rho }_{m+1}];{\mathbb{R}}^d).$$

Pasting the limits over the finitely many intervals $[{\rho }_m,{\rho }_{m+1}]$ yields a unique solution on $[0,{\tau }_1\wedge T]$.

Finally, since the finite-state Markov chain $\alpha$ has only finitely many jumps on $[0,T]$, there exists an a.s. finite integer N such that $0={\tau }_0<{\tau }_1<\dots <{\tau }_N\le T<{\tau }_{N+1}$. Repeating the above construction successively on each interval $[{\tau }_i\wedge T,{\tau }_{i+1}\wedge T]$, with initial condition $X^{u,\xi }({\tau }_i)$, yields a unique càdlàg adapted process $X^{u,\xi }\in S^2([0,T];{\mathbb{R}}^d)$ satisfying (1). Uniqueness follows from the same estimate (23) applied to two arbitrary solutions. This completes the proof. □

Proof of Lemma 2

The proof proceeds in four steps. First, for fixed mean-field terms, we establish the well-posedness of a frozen generalised regime-switching BSDE and define the associated solution map. Second, we show that this map is a contraction under a suitable exponentially weighted norm. Third, Banach’s fixed-point theorem yields the existence of a solution to (7). Finally, we establish the $S^2$-estimate and prove uniqueness.

Step 1: We write the equation in integral form

$$\begin{array}{cc}\hfill Y(t)=& \mathcal{Y}+{\int }_t^{T}g\left(s,Y(s),Z(s),R(s),\mathbb{E}[\phi (Y(s))],\mathbb{E}[\phi (Z(s))],\mathbb{E}[\phi (R(s))],\alpha (s)\right)\mathrm{d}s\hfill \\ \hfill & +{\int }_t^T\rho (s,Y(s))\mathrm{d}\xi (s)-{\int }_t^{T}Z(s)\mathrm{d}B(s)-{\int }_t^{T}R(s)\mathrm{d}\tilde{\mathsf{\Phi }}(s).\hfill \end{array}$$

(24)

Let ${\mathcal{H}}^2:=S^2([0,T];\mathbb{R})\times L_{\mathcal{F},p}^2([0,T];\mathbb{R})\times M_p^2([0,T];{\mathbb{R}}^D).$ For $(y,z,r)\in {\mathcal{H}}^2$, define

$$m_y(t):=\mathbb{E}[\phi (y(t))],m_z(t):=\mathbb{E}[\phi (z(t))],m_r(t):=\mathbb{E}[\phi (r(t))].$$

Consider the frozen BSDE

$$\begin{array}{cc}\hfill Y(t)=& \mathcal{Y}+{\int }_t^{T}g\left(s,Y(s),Z(s),R(s),m_y(s),m_z(s),m_r(s),\alpha (s)\right)\mathrm{d}s\hfill \\ \hfill & +{\int }_t^T\rho (s,Y(s))\mathrm{d}\xi (s)-{\int }_t^{T}Z(s)\mathrm{d}B(s)-{\int }_t^{T}R(s)\mathrm{d}\tilde{\mathsf{\Phi }}(s).\hfill \end{array}$$

(25)

For fixed $(y,z,r)$, the mean-field terms are deterministic inputs. The resulting equation is a generalised BSDE with a finite-variation term and a martingale component. Under Assumptions $(B_1)$–$(B_6)$, the coefficients satisfy the Lipschitz and integrability conditions required in the existence theory for generalised BSDEs with jumps.

Although Elmansouri and El Otmani (2023) is formulated for generalised BSDEs driven by a Brownian motion and a compensated jump measure, the arguments of (Elmansouri and El Otmani 2023, sct. 3–4) can be adapted to the present regime-switching framework since the compensated Markov-chain martingale $\tilde{\mathsf{\Phi }}$ satisfies analogous martingale isometry and predictable quadratic variation properties, with ${\parallel R(t)\parallel }_{{\mathcal{M}}^2}^2$ replacing the jump norm ${\parallel U(t,·)\parallel }_Q^2$. Furthermore, the dependence of the generator on the Markov chain through $\alpha (t)$ does not affect the estimates, since the Lipschitz and growth conditions in $(B_1)$ hold uniformly over all regimes $e_i\in \mathcal{S}$. Hence, arguing as in (Elmansouri and El Otmani 2023, sct. 3–4), the frozen generalised BSDE admits a unique solution

$$(Y,Z,R)\in S^2([0,T];\mathbb{R})\times L_{\mathcal{F},p}^2([0,T];\mathbb{R})\times M_p^2([0,T];{\mathbb{R}}^D).$$

Consequently, the frozen generalised BSDE defines a mapping $\mathsf{\Psi }:{\mathcal{H}}^2\to {\mathcal{H}}^2$, $\mathsf{\Psi }(y,z,r):=(Y,Z,R),$ where $(Y,Z,R)$ denotes the unique solution of the frozen equation corresponding to the input $(y,z,r)$.

Step 2: We now show that $\mathsf{\Psi }$ is a contraction. Let $(Y_i,Z_i,R_i)=\mathsf{\Psi }(y_i,z_i,r_i),i=1,2,$ and denote differences by bars, for example

$$\overline{Y}=Y_1-Y_2,\overline{Z}=Z_1-Z_2,\overline{R}=R_1-R_2,\overline{y}=y_1-y_2,\overline{z}=z_1-z_2,\overline{r}=r_1-r_2.$$

Similarly, set

$${\overline{m}}_y(t)=m_{y_1}(t)-m_{y_2}(t),{\overline{m}}_z(t)=m_{z_1}(t)-m_{z_2}(t),{\overline{m}}_r(t)=m_{r_1}(t)-m_{r_2}(t).$$

Applying Itô’s formula to $e^{\beta t}{|\overline{Y}(t)|}^2$, taking expectations, and using the Lipschitz property of g together with the monotonicity of $\rho$, we obtain

$$\begin{array}{cc}& \mathbb{E}[{\int }_0^T{e}^{\beta t}\left(\beta |\overline{Y}{(t)|}^2+|\overline{Z}{(t)|}^2+{\parallel \overline{R}(t)\parallel }_{{\mathcal{M}}^2}^2\right)\mathrm{d}t+2\mu {\int }_0^T{e}^{\beta t}{|\overline{Y}(t)|}^2\mathrm{d}\xi (t)]\hfill \\ & \le 2K\mathbb{E}[{\int }_0^T{e}^{\beta t}|\overline{Y}(t)|\left(|\overline{Y}(t)|+|\overline{Z}(t)|+\parallel \overline{R}{(t)\parallel }_{{\mathcal{M}}^2}+|{\overline{m}}_y(t)|+|{\overline{m}}_z(t)|+|{\overline{m}}_r(t)|\right)\mathrm{d}t]\hfill \\ & \le 2K\mathbb{E}[{\int }_0^T{e}^{\beta t}{|\overline{Y}(t)|}^2\mathrm{d}t]+16K^2\mathbb{E}\left[{\int }_0^T{e}^{\beta t}{|\overline{Y}(t)|}^2\mathrm{d}t\right]+{\displaystyle \frac{1}{2}}\mathbb{E}\left[{\int }_0^T{e}^{\beta t}\left(|\overline{Z}{(t)|}^2+{\parallel \overline{R}(t)\parallel }_{{\mathcal{M}}^2}^2\right)\mathrm{d}t\right]\hfill \\ & +{\displaystyle \frac{\beta }{2}}\mathbb{E}\left[{\int }_0^T{e}^{\beta t}{|\overline{Y}(t)|}^2\mathrm{d}t\right]+{\displaystyle \frac{12K^2}{\beta }}\mathbb{E}\left[{\int }_0^T{e}^{\beta t}\left(|{\overline{m}}_y{(t)|}^2+|{\overline{m}}_z{(t)|}^2+{|{\overline{m}}_r(t)|}^2\right)\right\}\mathrm{d}t].\hfill \end{array}$$

where we have used Young’s inequality (for any $\epsilon >0$, $ab\le \epsilon a^2+{\displaystyle \frac{1}{4\epsilon }}{b}^2$) in the last inequality. The above yields

$$\begin{array}{cc}& \mathbb{E}\left[{\int }_0^T{e}^{\beta t}\left(({\displaystyle \frac{\beta }{2}}-2K-16K^2)|\overline{Y}(t){|}^2+{\displaystyle \frac{1}{2}}|\overline{Z}(t){|}^2+{\displaystyle \frac{1}{2}}{\parallel \overline{R}(t)\parallel }_{{\mathcal{M}}^2}^2\right)\mathrm{d}t+2\mu {\int }_0^T{e}^{\beta t}{|\overline{Y}(t)|}^2\mathrm{d}\xi (t)\right]\hfill \\ & \le {\displaystyle \frac{12C^2}{\beta }}\mathbb{E}\left[{\int }_0^T{e}^{\beta t}\left(|{\overline{m}}_y{(t)|}^2+|{\overline{m}}_z{(t)|}^2+{|{\overline{m}}_r(t)|}^2\right)\right\}\mathrm{d}t].\hfill \end{array}$$

Finally, since $\phi$ is Lipschitz continuous, there exists $L_{\phi }>0$ such that

$$|{\overline{m}}_y{(t)|}^2\le L_{\phi }\mathbb{E}|\overline{y}{(t)|}^2,|{\overline{m}}_z{(t)|}^2\le L_{\phi }\mathbb{E}|\overline{z}{(t)|}^2,|{\overline{m}}_r{(t)|}^2\le L_{\phi }\mathbb{E}{\parallel \overline{r}(t)\parallel }_{{\mathcal{M}}^2}^2.$$

Therefore

$$\begin{array}{cc}\hfill & \mathbb{E}\left[{\int }_0^T{e}^{\beta t}\left(({\displaystyle \frac{\beta }{2}}-2K-16K^2)|\overline{Y}(t){|}^2+{\displaystyle \frac{1}{2}}|\overline{Z}(t){|}^2+{\displaystyle \frac{1}{2}}{\parallel \overline{R}(t)\parallel }_{{\mathcal{M}}^2}^2\right)\mathrm{d}t+2\mu {\int }_0^T{e}^{\beta t}{|\overline{Y}(t)|}^2\mathrm{d}\xi (t)\right]\hfill \\ \hfill & \le {\displaystyle \frac{12C^2{L}_{\phi }}{\beta }}\mathbb{E}\left[{\int }_0^T{e}^{\beta t}\left(|\overline{y}{(t)|}^2+|\overline{z}{(t)|}^2+{\parallel \overline{r}(t)\parallel }_{{\mathcal{M}}^2}^2\right)\mathrm{d}t\right].\hfill \end{array}$$

(26)

Now choose $\beta$ big enough such that ${\displaystyle \frac{\beta }{2}}-2K-16K^2\ge {\displaystyle \frac{1}{4}}$ and let $c_0=min({\displaystyle \frac{1}{2}},2\mu )$. It follows from (26) that

$$\begin{array}{cc}\hfill & c_0\left\{\mathbb{E}\left[{\int }_0^T{e}^{\beta t}\left(|\overline{Y}{(t)|}^2+|\overline{Z}{(t)|}^2+{\parallel \overline{R}(t)\parallel }_{{\mathcal{M}}^2}^2\right)\mathrm{d}t+{\int }_0^T{e}^{\beta t}{|\overline{Y}(t)|}^2\mathrm{d}\xi (t)\right]\right\}\hfill \\ \hfill & \le {\displaystyle \frac{12C^2{L}_{\phi }}{\beta }}\mathbb{E}\left[{\int }_0^T{e}^{\beta t}\left(|\overline{y}{(t)|}^2+|\overline{z}{(t)|}^2+{\parallel \overline{r}(t)\parallel }_{{\mathcal{M}}^2}^2\right)\mathrm{d}t\right].\hfill \end{array}$$

(27)

Define

$${\parallel (Y,Z,R)\parallel }_{\beta }^2:=\mathbb{E}\left[{\int }_0^T{e}^{\beta t}\left({|Y(t)|}^2+{|Z(t)|}^2+{\parallel R(t)\parallel }_{{\mathcal{M}}^2}^2\right)\mathrm{d}t\right]+\mathbb{E}\left[{\int }_0^T{e}^{\beta t}{|Y(t)|}^2\mathrm{d}\xi (t)\right].$$

Then

$$\parallel \mathsf{\Psi }(y_1,z_1,r_1)-\mathsf{\Psi }(y_2,z_2,r_2){\parallel }_{\beta }^2\le {\displaystyle \frac{12K^2{L}_{\phi }^2}{\beta c_0}}{\parallel (y_1-y_2,z_1-z_2,r_1-r_2)\parallel }_{\beta }^2.$$

Finally, choose $\beta$ even larger so that ${\displaystyle \frac{12K^2{L}_{\phi }^2}{\beta c_0}}<1.$

Step 3: Thus $\mathsf{\Psi }$ is a strict contraction on the complete metric space $({\mathbb{H}}^2,\parallel ·{\parallel }_{\beta })$ defined by ${\mathbb{H}}^2=L_{\mathcal{F},p}^2([0,T];\mathbb{R})\times L_{\mathcal{F},p}^2([0,T];\mathbb{R})\times M_p^2([0,T];{\mathbb{R}}^D).$ Therefore, by Banach’s fixed-point theorem, there exists a unique fixed point $(Y,Z,R)\in {\mathbb{H}}^2$ satisfying

$$(Y,Z,R)=\mathsf{\Psi }(Y,Z,R).$$

Consequently, $(Y,Z,R)$ solves (7).

Step 4: It remains to check the $S^2$-estimate for Y. Applying Itô’s formula to ${|Y(t)|}^2$ and using the growth of g and $\rho$ and Young’s inequality, we obtain

$$\begin{array}{cc}\hfill {|Y(t)|}^2=& {|\mathcal{Y}|}^2+2{\int }_t^T(Y(s)g\left(s,Y(s),Z(s),R(s),m_y(s),m_z(s),m_r(s),\alpha (s)\right)\mathrm{d}s\hfill \\ \hfill & -{\int }_t^T\left({|Z(s)|}^2+{\parallel R(s)\parallel }_{{\mathcal{M}}^2}^2\right)\mathrm{d}s+{\int }_t^{T}2Y(s)\rho (s,Y(s))\mathrm{d}\xi (s)-2{\int }_t^{T}Y(s)Z(s)\mathrm{d}B(s)\hfill \\ \hfill & -{\int }_t^T{\displaystyle \sum _{j=1}^D}\left(2Y(s-)R_j(s)+{|R_j(s)|}^2\right)\mathrm{d}{\tilde{\mathsf{\Phi }}}_j(s)\hfill \\ \hfill =& {|\mathcal{Y}|}^2+{\int }_t^T\left(2Y(s)g\left(s,Y(s),Z(s),R(s),m_y(s),m_z(s),m_r(s),\alpha (s)\right)-{|Z(s)|}^2\right)\mathrm{d}s\hfill \\ \hfill & +{\int }_t^{T}2Y(s)\rho (s,Y(s))\mathrm{d}\xi (s)-2{\int }_t^{T}Y(s)Z(s)\mathrm{d}B(s)\hfill \\ \hfill & -{\int }_t^T{\displaystyle \sum _{j=1}^D}{|R_j(s)|}^2\mathrm{d}{\mathsf{\Phi }}_j(s)-{\int }_t^T{\displaystyle \sum _{j=1}^D}2Y(s-)R_j(s)\mathrm{d}{\tilde{\mathsf{\Phi }}}_j(s)\hfill \\ \hfill \le & {|\mathcal{Y}|}^2+{\int }_t^T\left\{C_{{\epsilon }_1}{|Y(s)|}^2+{\epsilon }_1\left({|Z(s)|}^2+{\parallel R(s)\parallel }_{{\mathcal{M}}^2}^2\right)\right\}\mathrm{d}s\hfill \\ \hfill & +{\int }_t^T{C}_{{\epsilon }_1,L_{\phi }}\left(|{\varphi }_1{(s)|}^2+{|Y(s)|}^2+{|Z(s)|}^2+{\parallel R(s)\parallel }_{{\mathcal{M}}^2}^2\right)\mathrm{d}s-{\int }_t^T{|Z(s)|}^2\mathrm{d}s\hfill \\ \hfill & -2{\int }_t^{T}Y(s)Z(s)\mathrm{d}B(s)-{\int }_t^T{\displaystyle \sum _{j=1}^D}{|R_j(s)|}^2\mathrm{d}{\mathsf{\Phi }}_j(s)-2{\int }_t^T{\displaystyle \sum _{j=1}^D}Y(s-)R_j(s)\mathrm{d}{\tilde{\mathsf{\Phi }}}_j(s)\hfill \\ \hfill & -{\int }_t^T{(2\mu -\mu )|Y(s)|}^2\mathrm{d}\xi (s)+{\displaystyle \frac{1}{\mu }}{\int }_t^T{|{\varphi }_2(s)|}^2\mathrm{d}\xi (s).\hfill \end{array}$$

From this, we get

$$\begin{array}{cc}\hfill & {|Y(t)|}^2+{\int }_t^T{|Z(s)|}^2\mathrm{d}s+{\int }_t^T{\displaystyle \sum _{j=1}^D}|R_j{(s)|}^2\mathrm{d}{\mathsf{\Phi }}_j(s)+{\int }_t^T\mu {|Y(s)|}^2\mathrm{d}\xi (s)\hfill \\ \hfill \le & {|\mathcal{Y}|}^2+{\int }_t^{T}C\left(|{\varphi }_1{(s)|}^2+{|Y(s)|}^2+{|Z(s)|}^2+{\parallel R(s)\parallel }_{{\mathcal{M}}^2}^2\right)\mathrm{d}s\hfill \\ \hfill & -2{\int }_t^{T}Y(s)Z(s)\mathrm{d}B(s)-2{\int }_t^T{\displaystyle \sum _{j=1}^D}Y(s-)R_j(s)\mathrm{d}{\tilde{\mathsf{\Phi }}}_j(s)\hfill \\ \hfill & +{\displaystyle \frac{1}{\mu }}{\int }_t^T{|{\varphi }_2(s)|}^2\mathrm{d}\xi (s).\hfill \end{array}$$

Applying the Burkholder–Davis–Gundy inequality to the Brownian and Markov-chain martingale terms, together with the preceding estimate and the integrability assumptions on $\mathcal{Y},{\varphi }_1,{\varphi }_2$, gives

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}{|Y(t)|}^2\right]+\mathbb{E}{\int }_0^T{|Z(t)|}^2\mathrm{d}t+\mathbb{E}{\int }_0^T{|R(t)|}_{{\mathcal{M}}^2}^2\mathrm{d}t<\infty .$$

Therefore,

$$(Y,Z,R)\in S^2([0,T];\mathbb{R})\times L_{\mathcal{F},p}^2([0,T];\mathbb{R})\times M_p^2([0,T];{\mathbb{R}}^D).$$

Hence, the solution is unique. □

Proof of Lemma 3

The proof relies on the classical strategy initiated in (Doléans-Dade 1976, Theorem 1); see also (Cohen and Elliott 2015). For completeness, we only present the first step, which establishes the key estimate under a suitable smallness condition. The removal of this restriction is then obtained via a standard stopping-time localisation combined with a pasting argument, following the same line of reasoning as in (Cohen and Elliott 2015; Doléans-Dade 1976).

We begin by assuming that every admissible singular control process $\xi$ satisfies $\xi (T)\le \delta$ for some $\delta >0$ such that $K\delta <{\displaystyle \frac{1}{36}}$, where K denotes the constant appearing in Assumption 1.

From the controlled state process dynamics (1), we have by the Cauchy–Schwarz inequality

$$\begin{array}{cc}\hfill & |X^{\theta }(t)-X^{*}{(t)|}^2\hfill \\ \hfill =& |{\int }_0^t\{b^{\theta }(s)-b^{*}(s)\}\mathrm{d}s+{\int }_0^t\{{\sigma }^{\theta }(s)-{\sigma }^{*}(s)\}\mathrm{d}B(s)+{\int }_0^t\{{\gamma }^{\theta }(s)-{\gamma }^{*}(s)\}\mathrm{d}\tilde{\mathsf{\Phi }}(s)\hfill \\ \hfill & +{\int }_0^{t}G(t,X^{\theta }(s))\mathrm{d}{\xi }^{\theta }(s)-{\int }_0^{t}G(s,X^{*}(s))\mathrm{d}{\xi }^{*}(s){|}^2\hfill \\ \hfill \le & 5(t{\int }_0^t|b^{\theta }(s)-b^{*}(s){|}^2\mathrm{d}s+|{\int }_0^t\{{\sigma }^{\theta }(s)-{\sigma }^{*}(s)\}\mathrm{d}B(s){|}^2+|{\int }_0^t\{{\gamma }^{\theta }(s)-{\gamma }^{*}(s)\}\mathrm{d}\tilde{\mathsf{\Phi }}(s){|}^2\hfill \\ \hfill & +{\xi }^{*}(t){\int }_0^t|G(s,X^{\theta }(s))-G(s,X^{*}(s)){|}^2\mathrm{d}{\xi }^{*}(s)+{\theta }^2{|\eta (t)|}_{[0,t]}{\int }_0^t|G(s,X^{\theta }(s)){|}^2\mathrm{d}|\eta |(s)),\hfill \end{array}$$

where ${|\eta |}_{[0,t]}:={sup}_{0=t_0<t_1<\dots <t_m=t}\underset{i=0}{\sum ^{m-1}}\left|\eta (t_{i+1})-\eta (t_i)\right|$ is the total variation norm of $\eta$. Taking the supremum on both sides, the expectation, and using the Lipschitz continuity of the coefficients, and applying the Burkholder–Davis–Gundy (B-D-G) inequality on the continuous part, the Kunita inequality (see Applebaum 2009, Theorem 4.4.23) on the regime-switching jump part and the Cauchy–Schwarz inequality on the singular term, we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}[\underset{t\in [0,T]}{sup}|X^{\theta }(t)-X^{*}(t){|}^2]\hfill \\ \hfill \le & 5K^2(T{\int }_0^T\left(\mathbb{E}[\underset{s\in [0,t]}{sup}|X^{\theta }(s)-X^{*}{(s)|}^2]+{\theta }^2{\mathbb{E}[|v(t)|}^2]\right)\mathrm{d}t\hfill \\ \hfill & +{\int }_0^T\left(\mathbb{E}[\underset{s\in [0,t]}{sup}|X^{\theta }(s)-X^{*}{(s)|}^2]+{\theta }^2{\mathbb{E}[|v(t)|}^2]\right)\zeta (t)\mathrm{d}t\hfill \\ \hfill & +\mathbb{E}[{({\xi }^{*}(T))}^2\underset{t\in [0,T]}{sup}|X^{\theta }(t)-X^{*}{(t)|}^2]+{\theta }^2{\mathbb{E}[|\eta |}_{[0,T]}^2(1+\underset{t\in [0,T]}{sup}|X^{\theta }{(t)|}^2)]).\hfill \end{array}$$

Using the condition $K\delta <{\displaystyle \frac{1}{36}}$, together with Grönwall’s inequality and the estimate $\mathbb{E}\left[{sup}_{t\in [0,T]}{|X^{\theta }(t)|}^2\right]<\infty$, we obtain

$$\mathbb{E}\left[\underset{t\in [0,T]}{sup}{|X^{\theta }(t)-X^{*}(t)|}^2\right]\le C{\theta }^2.$$

(28)

To remove the smallness assumption, we introduce the sequence of stopping times

$$\begin{array}{cc}\hfill {\tau }_{n+1}& :=inf\{t\ge {\tau }_n:{\xi }^{*}(t)-{\xi }^{*}({\tau }_n)+{|\eta |}_{[{\tau }_n,t]}>\delta \}\wedge T,{\tau }_0=0,\hfill \end{array}$$

(29)

Since ${\xi }^{*}$ and $\eta$ are finite-variation processes on $[0,T]$, the sequence ${({\tau }_n)}_{n\ge 0}$ is non-decreasing and satisfies ${\tau }_n\uparrow T$ a.s. Moreover, on each interval $[{\tau }_n,{\tau }_{n+1})$,

$${\xi }^{*}(t)-{\xi }^{*}({\tau }_n)+{|\eta |}_{[{\tau }_n,t]}\le \delta ,$$

so the local smallness condition holds and thus estimate (28) applies locally. By restarting the dynamics at each time ${\tau }_n$ with initial condition $X^{\theta }({\tau }_n)$ and concatenating the resulting local estimates, we recover the desired global bound.

Therefore,

$$\underset{\theta \to 0}{lim}\mathbb{E}[\underset{t\in [0,T]}{sup}|X^{\theta }(t)-X^{*}(t){|}^2]=0.$$

(30)

Using similar arguments as above, we have $\mathbb{E}[{sup}_{t\in [0,T]}|Z(t){|}^p]<\infty .$ Let

$$Z_1^{\theta }(t)={\displaystyle \frac{X^{\theta }(t)-X^{*}(t)}{\theta }}-Z(t),\mathrm{i}.\mathrm{e}.,X^{\theta }(t)-X^{*}(t)=\theta \left(Z_1^{\theta }+Z(t)\right)$$

(31)

and set

$$\begin{array}{cc}\hfill b_x^{{\mu }^{\theta }}(t)=& b_x(X^{u^{*},{\xi }^{*}}(t)+\mu (X^{u^{\theta },{\xi }^{\theta }}(t)-X^{u^{*},{\xi }^{*}}(t)),\mathbb{E}[\phi (X^{u^{*},{\xi }^{*}})(t)]+\mu (\mathbb{E}[\phi (X^{u^{\theta },{\xi }^{\theta }})(t)]\hfill \\ \hfill & -\mathbb{E}[\phi (X^{u^{*},{\xi }^{*}})(t)]),u^{*}+\mu (u^{\theta }(t)-u^{*}(t)),\alpha (t)),\hfill \end{array}$$

and similarly for $b_y^{{\mu }^{\theta }}(t),b_u^{{\mu }^{\theta }}(t),{\sigma }_x^{{\mu }^{\theta }}(t),{\sigma }_y^{{\mu }^{\theta }}(t),{\sigma }_u^{{\mu }^{\theta }}(t),{\gamma }_x^{{\mu }^{\theta }}(t),{\gamma }_y^{{\mu }^{\theta }}(t),{\gamma }_u^{{\mu }^{\theta }}(t)$ and $G_x^{{\mu }^{\theta }}(t)$. Then, using the mean-value theorem, we have

$$\begin{array}{cc}\hfill & X^{\theta }(t)-X^{*}(t)\hfill \\ \hfill =& {\int }_0^t{\int }_0^1{b}_x^{{\mu }^{\theta }}(s)(X^{\theta }(s)-X^{*}(s))\mathrm{d}\mu \mathrm{d}s+{\int }_0^t{\int }_0^1{b}_y^{{\mu }^{\theta }}(s)(\mathbb{E}[\phi (X^{\theta })(s)]-\mathbb{E}[\phi (X^{*})(s)])\mathrm{d}\mu \mathrm{d}s\hfill \\ \hfill & +\theta {\int }_0^t{\int }_0^1{b}_u^{{\mu }^{\theta }}(s)v(s)\mathrm{d}\mu \mathrm{d}s+{\int }_0^t{\int }_0^1{\sigma }_x^{{\mu }^{\theta }}(s)(X^{\theta }(s)-X^{*}(s))\mathrm{d}\mu \mathrm{d}B(s)\hfill \\ \hfill & +{\int }_0^t{\int }_0^1{\sigma }_y^{{\mu }^{\theta }}(s)(\mathbb{E}[\phi (X^{\theta })(s)]-\mathbb{E}[\phi (X^{*})(s)])\mathrm{d}\mu \mathrm{d}B(s)\hfill \\ \hfill & +\theta {\int }_0^t{\int }_0^1{\sigma }_u^{{\mu }^{\theta }}(s)v(s)\mathrm{d}\mu \mathrm{d}B(s)+{\int }_0^t{\int }_0^1{\gamma }_x^{{\mu }^{\theta }}(s)(X^{\theta }(s)-X^{*}(s))\mathrm{d}\mu \mathrm{d}\tilde{\mathsf{\Phi }}(s)\hfill \\ \hfill & +{\int }_0^t{\int }_0^1{\gamma }_y^{{\mu }^{\theta }}(s)(\mathbb{E}[\phi (X^{\theta })(s)]-\mathbb{E}[\phi (X^{*}(s))])\mathrm{d}\mu \mathrm{d}\tilde{\mathsf{\Phi }}(s)+\theta {\int }_0^t{\int }_0^1{\gamma }_u^{{\mu }^{\theta }}(s)v(s)\mathrm{d}\mu \mathrm{d}\tilde{\mathsf{\Phi }}(s)\hfill \\ \hfill & +{\int }_0^t{\int }_0^1{G}_x^{{\mu }^{\theta }}(s)(X^{\theta }(s)-X^{*}(s))\mathrm{d}\mu \mathrm{d}{\xi }^{*}(s)+\theta {\int }_0^{t}G(s,X^{\theta }(s))\mathrm{d}\eta (s).\hfill \end{array}$$

Substituting the above into the expression of $Z_1^{\theta }$, we get

$$\begin{array}{cc}\hfill Z_1^{\theta }(t)& ={\int }_0^t{\int }_0^1{b}_x^{{\mu }^{\theta }}(s)Z_1^{\theta }(s)\mathrm{d}\mu \mathrm{d}s+{\int }_0^t{\int }_0^1\{b_x^{{\mu }^{\theta }}(s)-b_x^{*}(s)\}Z(s)\mathrm{d}\mu \mathrm{d}s\hfill \\ \hfill & +{\int }_0^t{\int }_0^1\{b_u^{{\mu }^{\theta }}(s)-b_u^{*}(s)\}v(s)\mathrm{d}\mu \mathrm{d}s+{\int }_0^t{\int }_0^1{b}_y^{{\mu }^{\theta }}(s)\mathbb{E}[{\int }_0^1{\phi }_x^{{\vartheta }^{\theta }}(s)\mathrm{d}\vartheta Z_1^{\theta }(s)]\mathrm{d}\mu \mathrm{d}s\hfill \\ \hfill & +{\int }_0^t{\int }_0^1\{b_y^{{\mu }^{\theta }}(s)\mathbb{E}[{\int }_0^1{\phi }_x^{{\vartheta }^{\theta }}(s)\mathrm{d}\vartheta Z(s)]-b_y^{*}(s)\mathbb{E}[{\phi }_x^{*}(X(s))Z(s)]\}\mathrm{d}\mu \mathrm{d}s\hfill \\ \hfill & +{\int }_0^t{\int }_0^1{\sigma }_x^{{\mu }^{\theta }}(s)Z_1^{\theta }(s)\mathrm{d}\mu \mathrm{d}B(s)+{\int }_0^t{\int }_0^1\{{\sigma }_x^{{\mu }^{\theta }}(s)-{\sigma }_x^{*}(s)\}Z(s)\mathrm{d}\mu \mathrm{d}B(s)\hfill \\ \hfill & +{\int }_0^t{\int }_0^1\{{\sigma }_u^{{\mu }^{\theta }}(s)-{\sigma }_u^{*}(s)\}v(s)\mathrm{d}\mu \mathrm{d}B(s)+{\int }_0^t{\int }_0^1{\sigma }_y^{{\mu }^{\theta }}(s)\mathbb{E}[{\int }_0^1{\phi }_x^{{\vartheta }^{\theta }}(s)\mathrm{d}\vartheta Z_1^{\theta }(s)]\mathrm{d}\mu \mathrm{d}B(s)\hfill \\ \hfill & +{\int }_0^t{\int }_0^1\{{\sigma }_y^{{\mu }^{\theta }}(s)\mathbb{E}[{\int }_0^1{\phi }_x^{{\vartheta }^{\theta }}(s)\mathrm{d}\vartheta Z(s)]-{\sigma }_y^{*}(s)\mathbb{E}[{\phi }_x^{*}(X(s))Z(s)]\}\mathrm{d}\mu \mathrm{d}B(s)\hfill \\ \hfill & +{\int }_0^t{\int }_0^1{\gamma }_x^{{\mu }^{\theta }}(s)Z_1^{\theta }(s)\mathrm{d}\mu \mathrm{d}\tilde{\mathsf{\Phi }}(s)+{\int }_0^t{\int }_0^1\{{\gamma }_x^{{\mu }^{\theta }}(s)-{\gamma }_x^{*}(s)\}Z(s)\mathrm{d}\mu \mathrm{d}\tilde{\mathsf{\Phi }}(s)\hfill \\ \hfill & +\theta {\int }_0^t{\int }_0^1\{{\gamma }_u^{{\mu }^{\theta }}(s)-{\gamma }_u^{*}(s)\}v(s)\mathrm{d}\mu \mathrm{d}\tilde{\mathsf{\Phi }}(s)+{\int }_0^t{\int }_0^1{\gamma }_y^{{\mu }^{\theta }}(s)\mathbb{E}[{\int }_0^1{\phi }_x^{{\vartheta }^{\theta }}(s)\mathrm{d}\vartheta Z_1^{\theta }(s)]\mathrm{d}\mu \mathrm{d}\tilde{\mathsf{\Phi }}(s)\hfill \\ \hfill & +{\int }_0^t{\int }_0^1\{{\gamma }_y^{{\mu }^{\theta }}(s)\mathbb{E}[{\int }_0^1{\phi }_x^{{\vartheta }^{\theta }}(s)\mathrm{d}\vartheta Z(s)]-{\gamma }_x^{*}(s)\mathbb{E}[{\phi }_x^{*}(X(s))Z(s)]\}\mathrm{d}\mu \mathrm{d}\tilde{\mathsf{\Phi }}(s)\hfill \\ & +{\int }_0^t{\int }_0^1\{G_x^{{\mu }^{\theta }}(s)-G_x(s,X^{*}(s))\}Z(s)\mathrm{d}\mu \mathrm{d}{\xi }^{*}(s)\hfill \\ \hfill & +{\int }_0^t{\int }_0^1{G}_x^{{\mu }^{\theta }}(s)Z_1^{\theta }(s)\mathrm{d}\mu \mathrm{d}{\xi }^{*}(s)+{\int }_0^t\{G(s,X^{\theta }(s))-G(s,X^{*}(s))\}\mathrm{d}\eta (s).\hfill \end{array}$$

Therefore, squaring both sides, taking the supremum and the expectation, and using B-D-G, Kunita and and the Cauchy–Schwarz inequality on the singular term inequalities as before gives

$$\begin{array}{cc}\hfill \mathbb{E}& [\underset{t\in [0,T]}{sup}|Z_1^{\theta }(t){|}^2]\hfill \\ \hfill \le & 18\{{\int }_0^T{\int }_0^1(\mathbb{E}[\underset{t\in [0,t]}{sup}|b_x^{{\mu }^{\theta }}(s)Z_1^{\theta }{(s)|}^2]+\mathbb{E}[\underset{s\in [0,t]}{sup}|b_y^{{\mu }^{\theta }}(s)\mathbb{E}[{\int }_0^1{\phi }_x^{{\vartheta }^{\theta }}(s)\mathrm{d}\vartheta Z_1^{\theta }(s)]{|}^2]\hfill \\ \hfill & +\mathbb{E}[\underset{t\in [0,t]}{sup}|{\sigma }_x^{{\mu }^{\theta }}(s)Z_1^{\theta }{(s)|}^2]+\mathbb{E}[\underset{s\in [0,t]}{sup}|{\sigma }_y^{{\mu }^{\theta }}(s)\mathbb{E}[{\int }_0^1{\phi }_x^{{\vartheta }^{\theta }}(s)\mathrm{d}\vartheta Z_1^{\theta }(s)]{|}^2]\hfill \\ \hfill & +\mathbb{E}[\underset{t\in [0,t]}{sup}|{\gamma }_x^{{\mu }^{\theta }}(s)Z_1^{\theta }{(s)|}^2]\zeta (s)+\mathbb{E}[\underset{s\in [0,t]}{sup}|{\gamma }_y^{{\mu }^{\theta }}(s)\mathbb{E}[{\int }_0^1{\phi }_x^{{\vartheta }^{\theta }}(s)\mathrm{d}\vartheta Z_1^{\theta }(s)]{|}^2]\zeta (s)\hfill \\ \hfill & +\mathbb{E}[\underset{s\in [0,t]}{sup}|(b_x^{{\mu }^{\theta }}(s)-b_x^{*}(s)){Z(s)|}^2]+\mathbb{E}[\underset{s\in [0,t]}{sup}|(b_y^{{\mu }^{\theta }}(s)-b_y^{*}(s))Z(s){|}^2]\hfill \\ \hfill & +\mathbb{E}[\underset{s\in [0,t]}{sup}|b_y^{*}(s)\mathbb{E}[{\int }_0^1({\phi }_x^{{\vartheta }^{\theta }}(s)-{\phi }_x(s))\mathrm{d}\vartheta Z(s)]{|}^2]\hfill \\ \hfill & +\mathbb{E}[\underset{s\in [0,t]}{sup}|({\sigma }_x^{{\mu }^{\theta }}(s)-{\sigma }_x^{*}(s)){Z(s)|}^2]+\mathbb{E}[\underset{s\in [0,t]}{sup}|({\sigma }_y^{{\mu }^{\theta }}(s)-{\sigma }_y^{*}(s))Z(s){|}^2]\hfill \\ \hfill & +\mathbb{E}[\underset{s\in [0,t]}{sup}|{\sigma }_y^{*}(s)\mathbb{E}[{\int }_0^1({\phi }_x^{{\vartheta }^{\theta }}(s)-{\phi }_x(s))\mathrm{d}\vartheta Z(s)]{|}^2]\hfill \\ \hfill & +\mathbb{E}[\underset{s\in [0,t]}{sup}|({\gamma }_x^{{\mu }^{\theta }}(s)-{\gamma }_x^{*}(s)){Z(s)|}^2]\zeta (s)+\mathbb{E}[\underset{s\in [0,t]}{sup}|({\gamma }_y^{{\mu }^{\theta }}(s)-{\gamma }_y^{*}(s))Z(s){|}^2]\zeta (s)\hfill \\ \hfill & +\mathbb{E}[\underset{s\in [0,t]}{sup}|{\gamma }_y^{*}(s)\mathbb{E}[{\int }_0^1({\phi }_x^{{\vartheta }^{\theta }}(s)-{\phi }_x(s))\mathrm{d}\vartheta Z(s)]{|}^2]\zeta (s)\hfill \\ \hfill & +\mathbb{E}[\underset{s\in [0,t]}{sup}|\{b_u^{{\mu }^{\theta }}(s)-b_u^{*}(s)\}{v(s)|}^2]++\mathbb{E}[\underset{s\in [0,t]}{sup}|\{{\sigma }_u^{{\mu }^{\theta }}(s)-{\sigma }_u^{*}(s)\}v(s){|}^2]\hfill \\ \hfill & +\mathbb{E}[\underset{s\in [0,t]}{sup}|\{{\gamma }_u^{{\mu }^{\theta }}(s)-{\gamma }_u^{*}(s)\}v(s){|}^2]\zeta (s))\mathrm{d}\mu \mathrm{d}t\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\mathbb{E}[\underset{s\in [0,T]}{sup}|Z_1^{\theta }{(t)|}^2]+{\displaystyle \frac{1}{2}}\mathbb{E}[\underset{s\in [0,T]}{sup}|{\int }_0^1\{G_x^{{\mu }^{\theta }}(s)-G_x(s,X^{u^{*},{\xi }^{*}}(s))\}Z(s)\mathrm{d}\mu {|}^2]\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\mathbb{E}[\underset{s\in [0,T]}{sup}|G(s,X^{u^{\theta },{\xi }^{\theta }}(s))-G(s,X^{u^{*},{\xi }^{*}}(s)){|}^2]\}.\hfill \end{array}$$

Using both the continuity, the boundedness of $g_i=b_x,b_y,b_u,{\sigma }_x,{\sigma }_y,{\sigma }_u,{\gamma }_x,{\gamma }_y,{\gamma }_u$, and ${\phi }_x$, and using the Lipschitz continuity of G and (28), the result follows by application of Gronwall’s lemma in $[0,{\tau }_1]$. As before, the smallness condition is removed by a pasting argument. □

Proof of Theorem 1.

Assume $(X^{*}(·),u^{*}(·),{\xi }^{*}(·))$ is an optimal solution to (1)–(4). Then,

$$\begin{array}{cc}\hfill 0\ge & {\displaystyle \frac{1}{\theta }}\left(J(x_0,e_i;u^{\theta },{\xi }^{\theta })-J(x_0,e_i;u^{*},{\xi }^{*})\right)\hfill \\ \hfill =& \mathbb{E}[p(T)Z(T)]+{\int }_0^T\mathbb{E}[(f_x(t)+\mathbb{E}[f_y(t)]{\phi }_x(X^{*}(t)))Z(t)]\mathrm{d}t\hfill \\ \hfill & +\mathbb{E}[{\int }_0^T\delta f(t,u^{\theta }(t))\mathrm{d}t]+{\int }_0^T\kappa (t)\mathrm{d}\eta (t)+o(1).\hfill \end{array}$$

Using Lemma 4, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\theta }}\left(J(x_0,e_i;u^{\theta },{\xi }^{\theta })-J(x_0,e_i;u^{*},{\xi }^{*})\right)\hfill \\ \hfill =& \mathbb{E}[{\int }_0^{T}Z(t)(f_x(t)+\mathbb{E}[f_y(t)]{\phi }_x(X^{*}(t)))\mathrm{d}t]+{\int }_0^T\kappa (t)\mathrm{d}\eta (t)\hfill \\ \hfill & +\mathbb{E}[{\int }_0^T-Z(t)(f_x(t)+\mathbb{E}[f_y(t)]{\phi }_x(X^{*}(t)))\mathrm{d}t]\hfill \\ \hfill & +\mathbb{E}[{\int }_0^{T}p(t)b_u^{*}(t,u(t))v(t)\mathrm{d}t]+\mathbb{E}[{\int }_0^{T}q(t){\sigma }_u^{*}(t,u(t))v(t)\mathrm{d}t]\hfill \\ \hfill & +\mathbb{E}\left[{\int }_0^{T}s(t){\gamma }_u^{*}(t,u(t))\zeta (t)v(t)\mathrm{d}t\right]+\mathbb{E}\left[{\int }_0^{T}p(t)G(t,X(t))\mathrm{d}{\eta }_t\right]+o(1)\hfill \\ \hfill =& \mathbb{E}[{\int }_0^T{\partial }_u{H}_1(t,X(t),\mathbb{E}[\phi (X(t))],u(t),e_i,p(t),q(t))v(t)\mathrm{d}t\hfill \\ & +\mathbb{E}\left[{\int }_0^T\left\{\kappa (t)+p(t)G(t,X(t))\right\}\mathrm{d}{\eta }_t\right]+o(1).\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\theta }}\left(J(x_0,e_i;u^{\theta },{\xi }^{\theta })-J(x_0,e_i;u^{*},{\xi }^{*})\right)\hfill \\ \hfill =& \mathbb{E}[{\int }_0^T{\partial }_u{H}_1(t,X(t),\mathbb{E}[\phi (X(t))],u(t),e_i,p(t),q(t))v(t)\mathrm{d}t\hfill \\ & +\mathbb{E}\left[{\int }_0^T{H}_2(t,X(t),p(t))\mathrm{d}{\eta }_t\right]+o(1).\hfill \end{array}$$

Taking the limit on both sides of the above, since $(u^{*},{\xi }^{*})$ is optimal,

$$\begin{array}{cc}\hfill & \mathbb{E}[{\int }_0^T{\partial }_u{H}_1(t,X(t),\mathbb{E}[\phi (X(t))],u(t),e_i,p(t),q(t))v(t)\mathrm{d}t\hfill \\ \hfill & +\mathbb{E}\left[{\int }_0^T{H}_2(t,X(t),p(t))\mathrm{d}{\eta }_t\right]\ge 0,\hfill \end{array}$$

for all admissible variations $(v,\eta )$.

By choosing $\xi ={\xi }^{*}$, we recover (8). On the other hand, setting $u=u^{*}$ and proceeding along the same lines as in the proof of (Cadenillas and Haussmann 1994, Theorem 4.2), we obtain (9). □

We now proceed to the proof of the sufficient maximum principle

Proof of Theorem 2

For any $(u(·),\xi (·))\in \mathcal{U}$, consider the difference

$$\begin{array}{cc}\hfill & J(x_0,e_i,u(·),\xi (·))-J(x_0,e_i,u^{*}(·),{\xi }^{*}(·))\hfill \\ \hfill & =\mathbb{E}[{\int }_0^T\{f(t,X(t),\mathbb{E}[\phi (X(t))],u(t),\alpha (t))\hfill \\ \hfill & -f(t,\hat{X}(t),\mathbb{E}[\phi (X^{*}(t))],u^{*}(t),\alpha (t))\}\mathrm{d}t]\hfill \\ \hfill & +\mathbb{E}[h(X(T),\mathbb{E}[\phi (X(T))],\alpha (T))-h(X^{*}(T),\mathbb{E}[\phi (X^{*}(T))],\alpha (T))]\hfill \\ \hfill & +\mathbb{E}[{\int }_0^T\kappa (t)\mathrm{d}(\xi (t)-{\xi }^{*}(t))]=I_1+I_2+I_3.\hfill \end{array}$$

Using the concavity of h, we have

$$\begin{array}{cc}\hfill I_2\le & \mathbb{E}[h_x(T)(X(T)-X^{*}(T))+h_y(T)(\mathbb{E}[{\phi }_x(X(T))(X(T)-X^{*}(T))])]\hfill \\ \hfill =& \mathbb{E}[p(T)(X(T)-X^{*}(T))].\hfill \end{array}$$

Now, applying Itô’s product rule to $p(T)(X(t)-X^{*}(T))$, we get

$$\begin{array}{cc}\hfill & \mathbb{E}[p(T)(X(T)-X^{*}(T))]\hfill \\ \hfill =& \mathbb{E}[-{\int }_0^T(X(t)-X^{*}(t))\{(f_x^{*}(t)+b_x^{*}(t)p(t)+{\sigma }_x^{*}(t)q(t)\hfill \\ \hfill & +{\gamma }_x^{*}(t)s(t)\zeta (t)+\mathbb{E}[f_y^{*}(t)]{\phi }_x(X^{*}(t))+\mathbb{E}[b_y^{*}(t)p(t)]{\phi }_x(X^{*}(t))\hfill \\ \hfill & +\mathbb{E}[{\gamma }_y^{*}(t)s(t)\zeta (t)]{\phi }_x(X^{*}(t)))\mathrm{d}t+G_x(t,X^{*}(t))p(t)\mathrm{d}{\xi }^{*}(t)\}]\hfill \\ \hfill & +\mathbb{E}[{\int }_0^{T}p(t)\{(b(t)-b^{*}(t))\mathrm{d}t+G(t,X(t))\mathrm{d}\xi (t)-G(t,X^{*}(t))\mathrm{d}{\xi }^{*}(t)\}]\hfill \\ \hfill & +\mathbb{E}\left[{\int }_0^T(\sigma (t)-{\sigma }^{*}(t))q(t)\mathrm{d}t\right]+\mathbb{E}\left[{\int }_0^T\{(\gamma (t)-{\gamma }^{*}(t))s(t)\zeta (t)\}\mathrm{d}t\right].\hfill \end{array}$$

Using the definition of the Hamiltonian, we have

$$\begin{array}{cc}\hfill I_1=& \mathbb{E}[{\int }_0^T\{f(t)-f^{*}(t)\}\mathrm{d}t]=\mathbb{E}[{\int }_0^T\{H(t)-H^{*}(t)+(b^{*}(t)-b(t))p(t)\hfill \\ \hfill & +({\sigma }^{*}(t)-\sigma (t))q(t))+({\gamma }^{*}(t)-\gamma (t))s(t)\zeta (t)\}\mathrm{d}t+G(t,X^{*}(t))\mathrm{d}\xi (t)-G(t,X(t))\mathrm{d}\xi (t)].\hfill \end{array}$$

In $I_1$ above and in the sequel, we have used the following shorthand notations:

$$\begin{array}{cc}\hfill H(t)& =H(t,X(t),\mathbb{E}[\phi (X(t))],u(t),\alpha (t),p(t),q(t),s),\hfill \\ \hfill H^{*}(t)& =H(t,X^{*}(t),\mathbb{E}[\phi (X^{*}(t))],u^{*}(t),\alpha (t),p(t),q(t),s),\hfill \\ \hfill H_1(t)& =H_1(t,X(t),\mathbb{E}[\phi (X(t))],u(t),\alpha (t),p(t),q(t),s),\hfill \\ \hfill H_1^{*}(t)& =H_1(t,X^{*}(t),\mathbb{E}[\phi (X^{*}(t))],u^{*}(t),\alpha (t),p(t),q(t),s),\hfill \\ \hfill H_2(t)& =H_2(t,X(t),p(t)),\hfill \\ \hfill b(t)& =b(t,X(t),\mathbb{E}[\phi (X(t))],u(t),\alpha (t)),\hfill \\ \hfill b^{*}(t)& =b(t,X^{*}(t),\mathbb{E}[\phi (X^{*}(t))],u^{*}(t),\alpha (t)),\hfill \end{array}$$

and similarly for $\sigma ,{\sigma }^{*},\gamma$ and ${\gamma }^{*}$. Putting everything together, we get

$$\begin{array}{cc}\hfill I_1+I_2+I_3& \le \mathbb{E}\left[{\int }_0^T\{H_1(t)-H_1^{*}(t)\}\mathrm{d}t\right]-\mathbb{E}[{\int }_0^T(X(t)-X^{*}(t))G_x(t,X^{*}(t))p(t)\mathrm{d}{\xi }^{*}(t)]\hfill \\ \hfill & -\mathbb{E}[{\int }_0^T(X(t)-X^{*}(t))\{f_x^{*}(t)+b_x^{*}(t)p(t)+{\sigma }_x^{*}(t)q(t)\hfill \\ \hfill & +{\gamma }_x^{*}(t)s(t)\zeta (t)+\mathbb{E}[f_y^{*}(t)]{\phi }_x(X^{*}(t))+\mathbb{E}[b_y^{*}(t)p(t)]{\phi }_x(X^{*}(t))\hfill \\ \hfill & +\mathbb{E}[{\sigma }_y^{*}(t)q(t)]{\phi }_x(X^{*}(t))+\mathbb{E}\left[{\gamma }_y^{*}(t)s(t)\zeta (t)]{\phi }_x(X^{*}(t))\}\mathrm{d}t\right]\hfill \\ \hfill & +\mathbb{E}\left[{\int }_0^{T}p(t)G(t,X^{*}(t))\mathrm{d}\{\xi (t)-{\xi }^{*}(t)\}\right]+\mathbb{E}\left[{\int }_0^T\kappa (t)\mathrm{d}(\xi (t)-{\xi }^{*}(t))\right]\hfill \\ \hfill =& \mathbb{E}[{\int }_0^T\{H_1(t)-H_1^{*}(t)\}\mathrm{d}t]-\mathbb{E}\left[{\int }_0^T(X(t)-X^{*}(t)){\partial }_x{H}_1^{*}(t)\mathrm{d}t\right]\hfill \\ \hfill & -\mathbb{E}[{\int }_0^T\mathbb{E}[{\partial }_y{H}_1^{*}(t)]{\phi }_x(X^{*}(t))(X(t)-X^{*}(t))\mathrm{d}t]+\mathbb{E}\left[{\int }_0^T{H}_2(t)\mathrm{d}(\xi (t)-{\xi }^{*}(t))\right].\hfill \end{array}$$

By the concavity of the Hamiltonian in u, we have

$$\begin{array}{cc}\hfill I_1+I_2+I_3\le & \mathbb{E}\left[{\int }_0^T\langle u(t)-u^{*}(t),{\displaystyle \frac{\partial H^{*}(t)}{\partial u}}\rangle \mathrm{d}t\right]+\mathbb{E}\left[{\int }_0^T{H}_2(t)\mathrm{d}(\xi (t)-{\xi }^{*}(t))\right].\hfill \end{array}$$

Since $u^{*}$ satisfies (11), we have

$$\mathbb{E}[{\int }_0^T\langle u(t)-u^{*}(t),{\partial }_u{H}_1^{*}(t,\alpha (t))\rangle \mathrm{d}t]\le 0.$$

For any admissible singular control $\xi \in U_2$ such that $\mathbb{P}\left({\int }_0^T|G(t,X^{*}(t))|\mathrm{d}\xi (t)<\infty \right)=1$,

$$\begin{array}{c}\hfill \mathbb{E}\left[{\int }_0^T{H}_2(t)\mathrm{d}\left(\xi (t)-{\xi }^{*}(t)\right)\right]=\mathbb{E}\left[{\int }_0^T{H}_2(t)\mathrm{d}\xi (t)\right]-\mathbb{E}\left[{\int }_0^T{H}_2(t)\mathrm{d}{\xi }^{*}(t)\right].\end{array}$$

Since $H_2(t)\le 0$ a.s. by (12) and $\xi$ is non-decreasing, we have $\mathbb{E}\left[{\int }_0^T{H}_2(t)\mathrm{d}\xi (t)\right]\le 0.$ Moreover, by (13), $\mathbb{E}\left[{\int }_0^T{H}_2(t)\mathrm{d}{\xi }^{*}(t)\right]=0.$ Therefore,

$$\mathbb{E}\left[{\int }_0^T{H}_2(t)\mathrm{d}\left(\xi (t)-{\xi }^{*}(t)\right)\right]\le 0.$$

□

4. Application: A One-Dimensional Regime-Switching Mean-Field Singular Portfolio Model

We consider a financial market consisting of one risk-free asset and one risky asset. The market is affected by a finite-state Markov chain $\alpha (t)$ taking values in $\mathcal{S}=\{e_1,\dots ,e_D\}$, representing the prevailing economic regime (e.g., expansion, recession, or high volatility). The risk-free asset satisfies

$$\mathrm{d}{S}_0(t)=r(\alpha (t))S_0(t)\mathrm{d}t,$$

where $r(\alpha (t))$ is a bounded regime-dependent interest rate.

In contrast to the classical transaction-cost model (see, e.g., Chen and Dai 2013, sct. 2), we allow both the appreciation rate and volatility of the risky asset to depend on the regime

$${\displaystyle \frac{\mathrm{d}{S}_1(t)}{S_1(t)}}=\mu (\alpha (t))\mathrm{d}t+\sigma (\alpha (t))\mathrm{d}B(t),$$

where B is a one-dimensional Brownian motion. Let $X_1(t)$ and $X_2(t)$ denote the wealth invested in the bank account and the risky asset, respectively. The investor chooses:

• A consumption rate $c(t)\ge 0$;
• A continuous, non-decreasing, adapted process $\xi (t)$ representing cumulative investment activity.

The controlled system is given by

$$\begin{array}{cc}\hfill \mathrm{d}{X}_1(t)& =\left[r(\alpha (t))X_1(t)-c(t)\right]\mathrm{d}t+\left(1-\lambda (\alpha (t))\right)X_1(t)\mathrm{d}\xi (t),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill \mathrm{d}{X}_2(t)& =a(\alpha (t))\left(X_2(t)-\beta \mathbb{E}[X_2(t)]\right)\mathrm{d}t+\sigma (\alpha (t))X_2(t)\mathrm{d}B(t)+\mathrm{d}\xi (t),\hfill \end{array}$$

(33)

with

$$X_1(0)=x_1,X_2(0)=x_2,\alpha (0)=e_i.$$

The regime-switching component allows the parameters r, a, $\sigma$, and $\lambda$ to vary with economic conditions. For instance, crisis regimes may exhibit higher volatility and transaction costs. The parameter $\beta >0$ measures the strength of the mean-field interaction: (1) $\beta =1$: drift depends on deviation from the population mean; (2) $0<\beta <1$: weaker interaction; and (3) $\beta >1$: stronger aggregate effect.

If $a(\alpha (t))<0$, the system exhibits regime-dependent mean reversion toward the population average. If $a(\alpha (t))>0$, the interaction becomes destabilizing.

The term $\mathrm{d}\xi (t)$ increases the risky position additively, while $\left(1-\lambda (\alpha (t))\right)X_1(t)\mathrm{d}\xi (t)$ represents a proportional effect on the bank account. Since $\xi$ is continuous, trading occurs gradually, excluding instantaneous block transactions.

We consider the quadratic mean-field cost

$$\begin{array}{cc}\hfill J(c,\xi )=& {\displaystyle \frac{1}{2}}\mathbb{E}[{\int }_0^T{\theta }_2(t)c{(t)}^2\mathrm{d}t+{\int }_0^T{\theta }_3(t){\left(X_2(t)-\mathbb{E}[X_2(t)]\right)}^2\mathrm{d}t\hfill \\ \hfill & +M_1{\left(X_1(T)-\mathbb{E}[X_1(T)]\right)}^2+M_2{\left(X_2(T)-\mathbb{E}[X_2(T)]\right)}^2+2{\int }_0^T\kappa (t)\mathrm{d}\xi (t)],\hfill \end{array}$$

(34)

where ${\theta }_2,{\theta }_3,M_1,M_2,\kappa \ge 0$. The admissible set $\mathcal{U}$ consists of all $(c,\xi )$ such that

$$\mathbb{E}\left[{\int }_0^{T}c{(t)}^2\mathrm{d}t+e^{k\xi {(T)}^2}\right]<\infty ,\mathrm{for}\mathrm{all}k>0,$$

with c progressively measurable and $\xi$ continuous, non-decreasing, and adapted.

The control problem is

$$\underset{(c,\xi )\in \mathcal{U}}{inf}J(c,\xi ).$$

The Hamiltonian becomes

$$\begin{array}{cc}\hfill H_1& ={\displaystyle \frac{{\theta }_2(t)}{2}}{c}^2+{\displaystyle \frac{{\theta }_3(t)}{2}}{(x_2-y_2)}^2-\left(r(\alpha (t))x_1-c\right)p_1+a(\alpha (t))(x_2-\beta y_2)p_2+\sigma (\alpha (t))x_2{q}_2,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill H_2& =\left(1-\lambda (\alpha (t))\right)x_1{p}_1+p_2+\kappa (t).\hfill \end{array}$$

(36)

The associated first order adjoint processes satisfy

$$\begin{array}{cc}\hfill \mathrm{d}{p}_1(t)& =-r(\alpha (t))p_1(t)\mathrm{d}t-\left(1-\lambda (\alpha (t))\right)p_1(t)\mathrm{d}\xi (t)+{\displaystyle \sum _{j=1}^D}{s}_{1,j}(t)\mathrm{d}{\tilde{\mathsf{\Phi }}}_j(t),\hfill \\ \hfill p_1(T)& =M_1\left(\mathbb{E}[X_1(T)]-X_1(T)\right),\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill \mathrm{d}{p}_2(t)& =-\left\{{\theta }_3(t)(X_2-\mathbb{E}[X_2])+a(\alpha (t))p_2(t)-\beta \mathbb{E}[a(\alpha (t))p_2(t)]+\sigma (\alpha (t))q_2(t)\right\}\mathrm{d}t\hfill \\ \hfill & +q_2(t)\mathrm{d}B(t)+{\displaystyle \sum _{j=1}^D}{s}_{2,j}(t)\mathrm{d}{\tilde{\mathsf{\Phi }}}_j(t),\hfill \\ \hfill p_2(T)& =M_2\left(\mathbb{E}[X_2(T)]-X_2(T)\right).\hfill \end{array}$$

(38)

Observe that the state process $X_1$ does not contain a Brownian diffusion term and evolves only through its finite-variation dynamics and the regime-switching process $\alpha$. Therefore, the randomness in $X_1$ is generated solely by the Markov chain. As a result, the corresponding adjoint equation does not contain a Brownian martingale component and is driven only by the compensated Markov-chain martingales.

Under the additional assumption that $\xi$ has some exponential moments, since r is uniformly bounded and $(1-\lambda (\alpha (t))$ is bounded and positive, thus by Lemma 2, the BSDE (37) has unique solution. In addition, the BSDE (38) also has a unique solution.

Proposition 2.

Let ξ be a continuous non-decreasing adapted process and let $\alpha (t)$ be a finite-state Markov chain.

1. 

Suppose that $p_1$ satisfies the linear backward Equation (37) and define

$${\mathsf{\Lambda }}_1(t)=exp\left({\int }_0^{t}r(\alpha (s))\mathrm{d}s+{\int }_0^t\left(1-\lambda (\alpha (s))\right)\mathrm{d}\xi (s)\right).$$

Then, the solution is given by

$$p_1(t)=M_1\mathbb{E}\left[{\displaystyle \frac{{\mathsf{\Lambda }}_1(T)}{{\mathsf{\Lambda }}_1(t)}}(\mathbb{E}[X_1(T)]-X_1(T))|{\mathcal{F}}_t\right],0\le t\le T.$$

(39)

Equivalently,

$$p_1(t)=M_1\mathbb{E}\left[exp\left({\int }_t^{T}r(\alpha (s))\mathrm{d}s+{\int }_t^T\left(1-\lambda (\alpha (s))\right)\mathrm{d}\xi (s)\right)(\mathbb{E}[X_1(T)]-X_1(T))|{\mathcal{F}}_t\right].$$

(40)

2. 

Similarly, suppose that $p_2$ satisfies the linear backward Equation (38) and define

$${\mathsf{\Lambda }}_2(t)=exp\left({\int }_0^t\{a(\alpha (s))-{\displaystyle \frac{1}{2}}{\sigma }^2(\alpha (s))\}\mathrm{d}s+{\int }_0^t\sigma (\alpha (s))\mathrm{d}B(s)\right).$$

Then, the solution is given by

$$\begin{array}{cc}\hfill p_2(t)=& M_2{\mathbb{E}}_t\left[(\mathbb{E}[X_2(T)]-X_2(T)){\displaystyle \frac{{\mathsf{\Lambda }}_2(T)}{{\mathsf{\Lambda }}_2(t)}}\right]+{\int }_t^T{\mathbb{E}}_t\left[{\displaystyle \frac{{\mathsf{\Lambda }}_2(s)}{{\mathsf{\Lambda }}_2(t)}}{\theta }_3(s)(X_2(s)-\mathbb{E}[X_2(s)])\right]\mathrm{d}s\hfill \\ & -\beta {\int }_t^T{\mathbb{E}}_t\left[{\displaystyle \frac{{\mathsf{\Lambda }}_2(s)}{{\mathsf{\Lambda }}_2(t)}}\right]\mathbb{E}[a(\alpha (s))p_2(s)]\mathrm{d}s,\hfill \end{array}$$

(41)

where $\mathbb{E}[a(\alpha (s))p_2(s)]$ satisfies the Volterra Equation (45).

Proof. 

(1) The proof follows the integrating-factor argument for linear singular BSDEs, as in (Dahl and Øksendal 2017, Theorem 4.1). Define

$$\mathrm{d}{\mathsf{\Lambda }}_1(t)={\mathsf{\Lambda }}_1(t)r(\alpha (t))\mathrm{d}t+{\mathsf{\Lambda }}_1(t)\left(1-\lambda (\alpha (t))\right)\mathrm{d}\xi (t),{\mathsf{\Lambda }}_1(0)=1.$$

Since $\xi$ is continuous and of finite variation, we have

$${\mathsf{\Lambda }}_1(t)=exp\left({\int }_0^{t}r(\alpha (s))\mathrm{d}s+{\int }_0^t\left(1-\lambda (\alpha (s))\right)\mathrm{d}\xi (s)\right).$$

Applying the product rule to $\mathsf{\Lambda }(t)p_1(t)$ gives

$$\begin{array}{cc}\hfill \mathrm{d}\left({\mathsf{\Lambda }}_1(t)p_1(t)\right)=& {\mathsf{\Lambda }}_1(t)\mathrm{d}{p}_1(t)+p_1(t)\mathrm{d}{\mathsf{\Lambda }}_1(t)\hfill \\ \hfill =& {\mathsf{\Lambda }}_1(t)\left[-r(\alpha (t))p_1(t)\mathrm{d}t-\left(1-\lambda (\alpha (t))\right)p_1(t)\mathrm{d}\xi (t)+{\displaystyle \sum _{j=1}^D}{s}_{1,j}(t)\mathrm{d}{\tilde{\mathsf{\Phi }}}_j(t)\right]\hfill \\ \hfill & +p_1(t)\left[{\mathsf{\Lambda }}_1(t)r(\alpha (t))\mathrm{d}t+{\mathsf{\Lambda }}_1(t)\left(1-\lambda (\alpha (t))\right)\mathrm{d}\xi (t)\right]\hfill \\ \hfill =& {\mathsf{\Lambda }}_1(t){\displaystyle \sum _{j=1}^D}{s}_{1,j}(t)\mathrm{d}{\tilde{\mathsf{\Phi }}}_j(t).\hfill \end{array}$$

Using the terminal condition $p_1(T)=M_1(\mathbb{E}[X_1(T)]-X_1(T))$, we obtain

$${\mathsf{\Lambda }}_1(t)p_1(t)=M_1\mathbb{E}\left[\left.{\mathsf{\Lambda }}_1(T)(\mathbb{E}[X_1(T)]-X_1(T))\right|{\mathcal{F}}_t\right].$$

Dividing by $\mathsf{\Lambda }(t)>0$ yields

$$p_1(t)=M_1\mathbb{E}\left[{\displaystyle \frac{{\mathsf{\Lambda }}_1(T)}{{\mathsf{\Lambda }}_1(t)}}(\mathbb{E}[X_1(T)]-X_1(T))|{\mathcal{F}}_t\right],$$

which proves (39). Since

$${\displaystyle \frac{{\mathsf{\Lambda }}_1(T)}{{\mathsf{\Lambda }}_1(t)}}=exp\left({\int }_t^{T}r(\alpha (s))\mathrm{d}s+{\int }_t^T\left(1-\lambda (\alpha (s))\right)\mathrm{d}\xi (s)\right),$$

we also obtain (40).

(2) The proof follows once more the integrating-factor argument for linear BSDEs. We first consider the following linear SDE

$$\begin{array}{cc}\hfill \mathrm{d}{\mathsf{\Lambda }}_2(t)=& {\mathsf{\Lambda }}_2(t)\left\{a(\alpha (t))\mathrm{d}t+\sigma (\alpha (t))\mathrm{d}B(t)\right\},{\mathsf{\Lambda }}_2(0)=1.\hfill \end{array}$$

(42)

Using Itô’s product rule, we have

$$\begin{array}{cc}\hfill \mathrm{d}(p_2(t){\mathsf{\Lambda }}_2(t))=& p(t){\mathsf{\Lambda }}_2(t)\left\{a(\alpha (t))\mathrm{d}t+\sigma (\alpha (t))\mathrm{d}B(t)\right\}\hfill \\ \hfill & +{\mathsf{\Lambda }}_2(t)[\{-{\theta }_3(t)(X_2(t)-\mathbb{E}[X_2(t)])-a(\alpha (t))p_2(t)+\beta \mathbb{E}[a(\alpha (t))p_2(t)]\hfill \\ \hfill & -\sigma (\alpha (t))q_2(t)\}\mathrm{d}t+q_2(t)\mathrm{d}B(t)+{\displaystyle \sum _{j=1}^D}{s}_{2,j}(t)\mathrm{d}{\tilde{\mathsf{\Phi }}}_j(t)]\hfill \\ \hfill & +\sigma (\alpha (t))q_2(t){\mathsf{\Lambda }}_2(t)\mathrm{d}t\hfill \\ \hfill =& {\mathsf{\Lambda }}_2(t)\left\{-{\theta }_3(t)(X_2(t)-\mathbb{E}[X_2(t)])+\beta \mathbb{E}[a(\alpha (t))p_2(t)]\right\}\mathrm{d}t\hfill \\ \hfill & +{\mathsf{\Lambda }}_2(t)\left\{p(t)\sigma (\alpha (t))+q_2(t)\right\}\mathrm{d}B(t)+{\mathsf{\Lambda }}_2(t){\displaystyle \sum _{j=1}^D}{s}_{2,j}(t)\mathrm{d}{\tilde{\mathsf{\Phi }}}_j(t).\hfill \end{array}$$

Integrating both sides from t to T, taking the conditional expectation and using the fact that ${\mathsf{\Lambda }}_2(0)=1$ give

$$\begin{array}{cc}\hfill p_2(t)=& \mathbb{E}\left[p_2(T){\displaystyle \frac{{\mathsf{\Lambda }}_2(T)}{{\mathsf{\Lambda }}_2(t)}}-{\int }_t^T{\displaystyle \frac{{\mathsf{\Lambda }}_2(s)}{{\mathsf{\Lambda }}_2(t)}}\left\{-{\theta }_3(s)(X_2(s)-\mathbb{E}[X_2(s)])+\beta \mathbb{E}[a(\alpha (s))p_2(s)]\right\}\mathrm{d}s|{\mathcal{F}}_t\right]\hfill \\ \hfill =& {\mathbb{E}}_t\left[p_2(T){\displaystyle \frac{{\mathsf{\Lambda }}_2(T)}{{\mathsf{\Lambda }}_2(t)}}\right]+{\int }_t^T{\mathbb{E}}_t\left[{\displaystyle \frac{{\mathsf{\Lambda }}_2(s)}{{\mathsf{\Lambda }}_2(t)}}{\theta }_3(s)(X_2(s)-\mathbb{E}[X_2(s)])\right]\mathrm{d}s\hfill \\ & -\beta {\int }_t^T{\mathbb{E}}_t\left[{\displaystyle \frac{{\mathsf{\Lambda }}_2(s)}{{\mathsf{\Lambda }}_2(t)}}\right]\mathbb{E}[a(\alpha (s))p_2(s)]\mathrm{d}s.\hfill \end{array}$$

(43)

Multiplying both sides of (43) by $a(\alpha (t))$ and using the fact that $a(\alpha (t))$ is ${\mathcal{F}}_t$-measurable

$$\begin{array}{cc}\hfill & a(\alpha (t))p_2(t)\hfill \\ \hfill =& {\mathbb{E}}_t\left[a(\alpha (t))p_2(T){\displaystyle \frac{{\mathsf{\Lambda }}_2(T)}{{\mathsf{\Lambda }}_2(t)}}\right]+{\int }_t^T{\theta }_3(s){\mathbb{E}}_t\left[a(\alpha (t)){\displaystyle \frac{{\mathsf{\Lambda }}_2(s)}{{\mathsf{\Lambda }}_2(t)}}(X_2(s)-\mathbb{E}[X_2(s)])\right]\mathrm{d}s\hfill \\ & -\beta {\int }_t^T{\mathbb{E}}_t\left[a(\alpha (t)){\displaystyle \frac{{\mathsf{\Lambda }}_2(s)}{{\mathsf{\Lambda }}_2(t)}}\right]\mathbb{E}[a(\alpha (s))p_2(s)]\mathrm{d}s.\hfill \end{array}$$

(44)

Taking the expectation on both sides of (44) yields

$$\begin{array}{cc}\hfill & \mathbb{E}\left[a(\alpha (t))p_2(t)\right]\hfill \\ \hfill =& -M_{2}Cov\left(X_2(T),{\displaystyle \frac{a(\alpha (t)){\mathsf{\Lambda }}_2(T)}{{\mathsf{\Lambda }}_2(t)}}\right)+{\int }_t^T{\theta }_3(s)Cov\left(X_2(s),{\displaystyle \frac{a(\alpha (t)){\mathsf{\Lambda }}_2(s)}{{\mathsf{\Lambda }}_2(t)}}\right)\mathrm{d}s\hfill \\ & -\beta {\int }_t^T\mathbb{E}\left[a(\alpha (t)){\displaystyle \frac{{\mathsf{\Lambda }}_2(s)}{{\mathsf{\Lambda }}_2(t)}}\right]\mathbb{E}[a(\alpha (s))p_2(s)]\mathrm{d}s.\hfill \end{array}$$

(45)

The above is a Volterra-type equation and substituting its solution into (43) gives the solution to the adjoint equation. □

Using the maximum condition for the Hamiltonian $H_1$, the optimal consumption satisfies

$$\hat{c}(t)={\displaystyle \frac{p_1(t)}{{\theta }_2(t)}}.$$

(46)

Moreover, since $\xi$ is continuous and non-decreasing, the singular control acts through continuous finite-variation adjustments. The complementary slackness condition is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(1-\lambda (\alpha (t))\right)X_1(t)p_1(t)+p_2(t)+\kappa (t)\ge 0,\hfill \\ {\displaystyle {\int }_0^T\left\{\left(1-\lambda (\alpha (t))\right)X_1(t)p_1(t)+p_2(t)+\kappa (t)\right\}\mathrm{d}\hat{\xi }(t)=0.}\hfill \end{array}\right.\end{array}$$

(47)

Thus $\hat{\xi }$ may increase only on the set where the singular Hamiltonian vanishes. We can summarise the above result as follows:

Corollary 1.

Suppose that the wealth dynamics are given by (32) and (33), and that the objective is to minimise the cost functional (34) over $\mathcal{U}$. Furthermore, assume that the integrability assumptions in the Proposition 2 hold. Then, the optimal pair $(\hat{c},\hat{\xi })$ satisfies

$$\hat{c}(t)={\displaystyle \frac{p_1(t)}{{\theta }_2(t)}},$$

where $p_1$ is given by (39), and the optimal singular control $\hat{\xi }$ is characterised by the complementary slackness condition (47), with $p_2$ given by (41).

This model provides a tractable regime-switching mean-field extension of classical transaction-cost portfolio problems and illustrates the applicability of the stochastic maximum principle developed in this paper.

The complementary slackness condition (47) provides a useful economic interpretation of the optimal singular policy. Define the regime-dependent intervention function

$${\mathsf{\Psi }}_{\alpha (t)}(t)=\left(1-{\lambda }_i\right)X_1(t)p_1(t)+p_2(t)+\kappa (t).$$

Then the optimal singular control can increase only when

$${\mathsf{\Psi }}_{\alpha (t)}(t)=0.$$

Consequently, the optimal singular policy is characterised by a regime-dependent intervention boundary. Since the coefficient $\lambda (\alpha (t))$ varies across regimes, the marginal effect of singular investment activity on the bank-account wealth also changes with the prevailing economic conditions. As a result, the intervention threshold is shifted whenever the economy switches from one regime to another. Therefore, the same portfolio configuration may trigger singular investment activity in one regime but not in another. In particular, if the transaction-cost parameter is higher in a crisis regime than in a normal regime, then singular interventions become less attractive during periods of economic stress, leading to less frequent adjustments of the risky position. This illustrates how regime switching affects not only the evolution of wealth but also the timing and intensity of the optimal singular investment strategy.

For example, in the two-state setting with a normal regime $e_1$ and a crisis regime $e_2$, suppose that ${\lambda }_1<{\lambda }_2.$ Then $1-{\lambda }_1>1-{\lambda }_2,$ so that the contribution of the bank-account component $X_1(t)p_1(t)$ to the intervention function is larger in the normal regime than in the crisis regime. Consequently, the intervention boundaries

$$(1-{\lambda }_1)X_1(t)p_1(t)+p_2(t)+\kappa (t)=0,$$

and

$$(1-{\lambda }_2)X_1(t)p_1(t)+p_2(t)+\kappa (t)=0,$$

are different. Hence, a change of regime alters the location of the intervention boundary and may therefore modify the optimal timing of singular investment decisions. In particular, when ${\lambda }_2>{\lambda }_1$, the contribution of the bank-account component $X_1(t)p_1(t)$ to the intervention function is reduced in the crisis regime relative to the normal regime, leading to a different balance between immediate intervention and postponement of investment activity.

5. Conclusions

In this paper, we investigated a class of mean-field singular stochastic optimal control problems under Markov regime switching, where the state dynamics are driven simultaneously by regular controls and state-dependent singular controls. By allowing the coefficients to depend on both the distribution of the state process and the prevailing economic regime, the proposed framework extends several existing formulations in the literature and provides a flexible setting for modeling stochastic systems with collective interactions and structural uncertainty.

Our main contribution was the derivation of both necessary and sufficient stochastic maximum principles under the assumption of a convex control domain. The resulting optimality conditions are characterised through an adjoint equation in the form of a singular regime-switching mean-field backward stochastic differential equation. The state dependence of the singular coefficient introduces additional analytical challenges that are accommodated within our framework and distinguishes the present work from related studies in which the singular term is independent of the state process.

To illustrate the applicability of the theoretical results, we studied a one-dimensional mean-field portfolio optimization problem with transaction costs in a regime-switching market. The model demonstrates how the maximum principle can be used to characterise optimal consumption and cumulative investment strategies in the presence of both aggregate effects and changing economic conditions, while also yielding explicit representations for the adjoint processes in the linear setting.

The methodology developed in this work admits several natural extensions. In particular, incorporating jump-diffusion dynamics, partial observation, common noise, or non-convex control domains would broaden the scope of the framework and present interesting theoretical challenges. Another promising direction is the numerical approximation of the associated singular mean-field backward stochastic differential equations and the implementation of computational algorithms for high-dimensional applications arising in finance, economics, and engineering.