Suppose we have a financial market with memory, in which there are two following investment opportunities:

(i)

A risk-free asset with unit price *S*_{0} (*t*) = 1; I*t* ≥ 1,

(ii)

A risky asset, in which investments have memory effects (or long term effects) and an infinite number of rational agents are in competition, in the following sense:

An investor can decide at time *t* ∈ [0, *T*] what amount *u*(*t*) of the current wealth *X*(*t*) to invest in the risky asset. The wealth process *X*(*t*) = *X*^{u}(t) at time *t* is described by the following controlled mean-field stochastic Volterra equation:

$$X(t)=x+\underset{0}{\overset{t}{\int}}b(t,s,X(s),E[X(s)],u(s),\omega )ds+\underset{0}{\overset{t}{\int}}\sigma (t,s,X(s),E[X(s)],u(s),\omega )dB(s),$$(1)where *B*(*t*) denotes a standard Brownian motion in a probability space (Ω, 𝓕, *P*) with the natural filtration (𝓕_{t})_{t≥0}, here *P* is the probability measure. Suppose that

$$b(t,s,x,{x}^{\prime},u,\omega ):[0,T]\times [0,T]\times \mathbb{R}\times \mathbb{R}\times \mathbb{U}\times \text{\Omega}\to \mathbb{R}$$and

$$\sigma (t,s,x,{x}^{\prime},u,\omega ):[0,T]\times [0,T]\times \mathbb{R}\times \mathbb{R}\times \mathbb{U}\times \text{\Omega}\to \mathbb{R}$$be 𝓕-adapted with respect to the second variable s for all *t*, *x*, *x*′, *u* and continuously differentiable with respect to the first variable *t*, with partial derivatives in *L*^{2}([0, *T*]×[0, *T*]×ℝ×ℝ×Ω). Udenotes a given open set containing all possible admissible investment values *u*(*t*, *ω*) for (*t*, ω) ∈ ([0, *T*] × Ω, *u* ∈ 𝓤(*x*). Here suppose 𝓤 be a given family of (𝓖_{t})_{t≥0}-predictable admissible investment, where 𝓖_{t} ⊆ 𝓕_{t} for all *t* ∈ ([0, *T*]. The introduction of 𝓤_{t} illustrates that the admissible investment amount u is decided based on only partial information available to the investor, for instance, delayed information flow.

We rewrite the controlled mean-field stochastic Volterra equation above in the following differential form

$$\begin{array}{ll}dX(t)\hfill & =b(t,t,X(t),E[X(t)],u(t))dt+\left({\displaystyle \underset{0}{\overset{t}{\int}}\frac{\partial b}{\partial t}}(t,s,X(s),E[X(s)],u(s))ds\right)dt\hfill \\ \hfill & +\sigma (t,t,X(t),E[X(t)],u(t))dB(t)+\left({\displaystyle \underset{0}{\overset{t}{\int}}\frac{\partial \sigma}{\partial t}(t,s,X(s),E[X(s)],u(s))dB(s)}\right)dt.\hfill \end{array}$$(2)From this differential form, we see that this mean-field stochastic Volterra equation differs from the mean-field stochastic differential equation, because of the two integral terms on the right hand side in (2). These terms represent memory effects of the investment *u*(·).

The mean-field investment performance functional is given by

$$J(u)=E\left[{\displaystyle \underset{0}{\overset{T}{\int}}f(s,X(s),E[X(s)],u)(s))ds+g(X(T),E[X(T)])}\right];u\in \U0001d4e4(x).$$(3)In contrast to the standard investment performance functional, the performance functional in this paper involves the mean of functions of the state variable, i.e. the mean-field term *E*([*X*(·)].

The optimal investment problem is to maximize the performance functional *J*(*u*) over all admissible investments, that is to find *u* * ∈ 𝓤(*x*) such that

$$J({u}^{\ast})=\underset{u\in \mathcal{U}}{sup}J(u).$$(4)Given an initial wealth value *x*, we say that *u*^{*} ∈ 𝓤(*x*) is an optimal investment if (4) holds.

Many papers have been devoted to the optimal investment problems. The seminal papers of [2, 3] considered the financial market modeled by a classical stochastic differential equation. Since then there have been many works concerning this subject. We refer to [4-6], for the complete market situation, to [7, 8] for constrained portfolios; to [9-13] for transactions costs; and to [14-17] for general incomplete markets.

However, to our knowledge, in all of these works, the financial markets have only one rational agent and the solutions of stochastic differential equations are Markov processes. The most natural framework to model the financial markets with an infinite number of rational agents in competition is the mean-field framework, which arise naturally in many applications. The systematic study of these problems was started, in the mathematical community by [18], and independently around the same time in the engineering community by [19, 20]. The well-known framework to model such non-Markov type solutions would be the stochastic Volterra equations where its solutions are not Markov processes. But this typically non-Markov property leads to invalidation of some methods, for instant, dynamic programming principle. In view of this, it is important to find good methods to solve the optimal investment problems for this type stochastic Volterra equations.

A good method to solve this optimal investment problem is Malliavin calculus, which has been firstly introduced in [1]. We refer to [21] for Malliavin calculus applied to optimal control of stochastic partial differential equations with jumps, and to [22] for Malliavin calculus applied to optimal control of stochastic Volterra equations but without the mean-field framework.

In the present paper following the idea of [22] we study the optimal investment problem in a finance market modeled by mean-field stochastic Volterra equation, in which both the financial market with an infinite number of rational agents in competition and non-Markov type solutions are all taken into account. The method of Malliavin calculus to solve this optimal investment problem in this paper is still adopted.

We present both a sufficient condition and a necessary condition for this optimal investment problem by mean-field stochastic maximum principle. Since the performance functional in this paper involves the mean-field term, in the proving process, we mainly use a new type of backward stochastic Volterra equations (BSVE), which is firstly studied in [23], and we refer the reader to [24, 25] for more discussions about this issue. The terminal condition and generator of this type of BSVE in this paper evolve into more complex forms because of the appearance of the mean-field term in the investment performance functional.

This paper is organized as follows. Section 2 recalls some definitions and properties about Malliavin calculus for Brownian motion. Section 3 is divided into two main parts: a sufficient and necessary conditions for optimal investment problem. An example is given in Section 4.

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