1 Models and introduction
Suppose we have a financial market with memory, in which there are two following investment opportunities:
A risk-free asset with unit price S0 (t) = 1; It ≥ 1,
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) = Xu(t) at time t is described by the following controlled mean-field stochastic Volterra equation:(1)
where B(t) denotes a standard Brownian motion in a probability space (Ω, 𝓕, P) with the natural filtration (𝓕t)t≥0, here P is the probability measure. Suppose that
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 L2([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 (𝓖t)t≥0-predictable admissible investment, where 𝓖t ⊆ 𝓕t for all t ∈ ([0, T]. The introduction of 𝓤t 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(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(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(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 , 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 . We refer to  for Malliavin calculus applied to optimal control of stochastic partial differential equations with jumps, and to  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  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 , 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.
2 Malliavin calculus for Brownian motion
For reader’s convenience, we recall the basic definition and properties of Malliavin calculus for Brownian motion. For more details about Malliavin calculus see [26, 27]. A natural starting point is the Wiener-Itô chaos expansion theorem, which states that any ξ ∈ L2(𝓕 P) can be written as
for a unique sequence of symmetric deterministic functions fn ∈ L2(Xn), where λ is a Lebesgue measure on [0; T] and
(the n-times iterated integral of fn with respect to B(·)) for n = 1; 2; … and I0(f0) = f0 when f0 is a constant.
Moreover, we have the isometry
We first present the Malliavin derivative Dtξ with respect to Brownian motion B(·) at t of a given Malliavin differentiable random variable ξ(ω);ω ∈ Ω, and then we present some basic properties about Malliavin derivative related to this paper.
Let 𝔻 denote the set of all random variables which are Malliavin differentiable with respect to Brownian motion B(·), precisely, let 𝔻 be the space of all ξ ∈ L2(𝓕, P) such that its chaos expansion satisfies
For any ξ ∈𝔻, define the Malliavin derivative Dt(ξ) of ξ at t, t ∈ [0; T] with respect to Brownian motion B(·) as where the notation In–1(fn(·, t)) means that we apply the (n – 1)-times iterated integral to the first n – 1 variables t1, t2, …, tn–1 of fn(t1, t2, …, tn) and keep the last variable tn = t as a parameter.
It is easy to check that
so (t, ω) → Dtξ(ω) belongs to L2(λ × P).
Some basic properties of the Malliavin derivative Dt are the following (a) chain rule and (b) duality formula.
Suppose ξ1, …, ξm ∈𝔻 and that f : ℝm → ℝ is C1 with bounded partial derivatives. Then f(ξ1, …, ξm) ∈𝔻 and
Suppose φ(t) is (𝓕t)t≥0-adapted with and let ξ ∈𝔻. Then
3 Main results
It is well-known that the investment performance functional (3) is related to a Hamiltonian functional. For this, we define the Hamiltonian functional(5)
here, both p(·) and q(·) are (𝓕t)t≥0-adapted processes and p (s) is assumed to be Malliavin differentiable for every s.
Define the adjoint mean-field BSDE for (p(t), q(t)) by(6)
From now on we suppose that the controlled mean-field stochastic Volterra equation (1) and the corresponding adjoint mean-field BSDE (6) have corresponding solutions Xu(·) and (pu(·), qu(·)), respectively, for any u ∈ 𝓤.
Based on the Hamiltonian functional, in this section we study two mean-field stochastic maximum principles. The first principle gives a sufficient condition for the optimal investment under the concavity of the Hamiltonian function. The second condition is a necessary maximum principle under the condition of the existence of derivative process of the wealth process X.
3.1 A sufficient condition for optimal investment
Assume that the function g(x, x') is concave with respect to (x, x') and H(t, x, x', u, p, q) is concave with respect to (x, x', u) for all (t, p, q)( Moreover, for all t (7)
Then u* is an optimal investment.
Proof. Let u ∈ 𝓤 (or u* ∈ 𝓤) with corresponding solutions X(t) := Xu(t) (or X*(t) := Xu*(t)) of the controlled mean-field stochastic Volterra equation (2). Because of the definition of mean-field investment performance functional J(u), we have(8)
Firstly, we deal with the first term in the right hand side in the above equation (8). By the definition of the Hamiltonian functional, we get(9)
Secondly, we deal with the second term in the right hand side in the above equation (8). By the concavity of the function g(x, x') with respect to (x, x'), we get
Since in the adjoint mean-field BSDE, we obtain
Now, applying Itô formula to p*(t)(X(t) – X*(t)) between 0 and T, in connection with the first identity in equation (6), we obtain(10)
Next, using Fubini’s theorem, the fact that p* (t) is Malliavin differentiable for every t in connection with the duality formula for the Malliavin derivative Dt and Fubin’s theorem again all in this order, we have(11)
Using Fubini’s theorem again, we have(12)(13)(14)
Now, introduce the notion:
Hence, by the concavity of the function H(t, x, x', u, p, q) with respect to (x, x', u) for all (t, p, q), we obtain(15)
By the assumption (7) we have Therefore, we conclude that for all u ∈ 𝓤, J(u) ≤ J(u*), and this completes the proof. □
3.2 A necessary condition for optimal investment: mean-field maximum principle
In this subsection we will assume both the coefficients b, σ of SDE and performance functions f are continuously differentiable with respect to (x, x', u) and g is continuously differentiable with respect to (x, x').
For any t ∈ [0, T], let α = αt be any bounded 𝓖t-measurable random variable and taking β(s) + αI[t,t+h] (s), s ∈ [0, T]( Assume that u + α ∈ 𝓤 for all such α and all u ∈ 𝓤.
Define the derivative process Y(t) by
and suppose this derivative process exists.
From (1), after some simple computation, we have(16)
We can rewrite the derivative process (16) in differential form, for which it is convenient for us to apply Itô formula, as following:(17)
We now establish the mean-field stochastic maximum principle for mean-field stochastic Volterra equations with corresponding adjoint mean-field BSDE.
Proof. By the definition of the performance functional J(u), we can compute the derivative(20)
where the last equality results from .
Now, using Itô formula to p(t)Y(t), we find(21)(22)(23)
Since this equality holds for all 𝓖t-measurable bounded α, we conclude that(24)
4 An example
Consider the following linear mean-field stochastic Volterra dynamic system, for 0 ≤ t ≤ T,(25)
here, for simplicity, let the coefficient terms be
Hence the linear mean-field stochastic Volterra equation (25) given by
or, in differential form,(26)
From (26) above, we can get(27)
Here, the stochastic integral term in (27) be generated from the form of Volterra equation and represents memory effects of the investment u(·). But there is no stochastic integral term in classical Black-Scholes formula. If we assume the initial wealth value x be positive strictly, it follows that, from (27),
The invest performance functional is given by
Suppose there exists an optimal investment u* with corresponding processes (X*, p*, q*), then we have
From the definition of the Hamiltonian
and X(t) > 0, for all t ∈ [0, T], we can get(28)
From this, after some computation, we find that the adjoint BSDE given by
and the terminal condition
Obviously, this BSDE has the unique solution(29)
Since X(t) > 0 and E[ξ(T)|𝓕t]is a martingale, so, we can deduce from the equation above that
Obviously, , so we get(30)
Now, we define
and from (25), we get the following mean-field backward stochastic Volterra equation(31)
From the existence and uniqueness of mean-field backward stochastic Volterra equation, see , we known that the above equation has unique solution (X*, Z*).
Taking the expectation on the above equation for t = 0, we obtain
So, we have
here, (X*, Z*) be the solution of (31).
The authors would like to thank the referee for very helpful comments, particularly for his suggestions concerning the treatment of an example in Section 4. This work is supported by the National Natural Science Foundation of China (Grant No. 11501325, 11501330, 11301303), Social science planning and research project in Shandong Province (No. 14AWTJ01-17), CPSF (No. 2015M582071), Academy of Humanities and Social Science Research Plan Project of Higher Education in Shandong Province (No. J14WF61) and was partially supported by National Social Science Foundation of China (Grant No. 15BJY007), Humanities and Social Sciences Project of the Ministry Education of China (Grant No. 14YJA630088, Grant No. 13YJC630150), Foundation of Shandong Province of China (Grant No. ZR2013AQ021, No. ZR2014GQ014).
Meyer-Brandis, T., Øksendal, B., Zhou, X.: A mean-field stochastic maximum principle via Malliavin calculus, Stochastics An International Journal of Probability and Stochastic Processes, 84:5-6, 643-666 (2012) Google Scholar
Merton, R. C.: Lifetime Portfolio Selection Under Uncertainty: The Continuous-Time Model, Rev. Econ. Statist. 51, 247-257 (1969) Google Scholar
Merton, R. C.: Optimum Consumption and Portfolio Rules in a Continuous-Time Model, J. Econ. Theory 3, 373-413 (1971) Google Scholar
Pliska, S. R.: A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios, Math. Oper. Res. 11, 371-382 (1986) Google Scholar
Cox, J., Huang, C. F.: Optimal Consumption and Portfolio Policies When Asset Prices Follow a Diffusion Process, J. Econ. Theory 49, 33-83. (1989)Google Scholar
Karatzas, I., Lehoczky, J. P., Shreve, S. E.: Optimal Portfolio and Consumption Decisions for a Small Investor on a Finite Horizon, SIAM J. Control Optim. 25, 1557-1586 (1987) Google Scholar
Cvitanic, J., Karatzas, I.: Convex Duality in Constrained Portfolio Optimization, Ann. Appl. Probab. 2, 767-818. (1992) Google Scholar
Zariphopoulou, T.: Consumption-Investment Models with Constraints, SIAM J. Control Optim. 32(1), 59-85 (1994) Google Scholar
Constantinides, G. M., Magill, M. J. P.: Portfolio Selection with Transaction Costs, J. Econ. Theory 13, 245-263. (1976) Google Scholar
Davis, M. H. A., Norman, A. R.: Portfolio Selection with Transaction Costs, Math. Oper. Res. 15(4), 676-713. (1990)Google Scholar
Shreve, S. E., Soner, H. M.: Optimal Investment and Consumption with Transaction Costs, Ann. Appl. Probab. 4, 609-692 (1994)Google Scholar
Duffie, D., Sun, T.: Transaction Costs and Portfolio Choice in a Discrete-Continuous-Time Setting, J. Econ. Dyn. Control 14, 35-51. (1990)Google Scholar
Akian, M., Menaldi, J.L., Sulem, A.: Multi-Asset Portfolio Selection Problem with Transaction Costs. Probabilités Numériques (Paris, 1992), Math. Comput. Simul. 38, 163-172. (1995) Google Scholar
Karatzas, I., Lehoczky, J. P., Shreve, S. E., Xu, G. : Martingale and Duality Methods for Utility Maximization in an Incomplete Market, SIAM J. Control Optim. 29, 702-730 (1991) Google Scholar
Kramkov, D., Schachermayer, W.: The Condition on the Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets, Ann. Appl. Probab. 9, 904-950 (1999)Google Scholar
Kramkov, D., Schachermayer, W.: Necessary and Sufficient Conditions in the Problem of Optimal Investment in Incomplete Markets, Ann. Appl. Probab. 13(4), 1504-1516 (2003)Google Scholar
Kramkov, D., Sirbu, M.: On the Two-Times Differentiability of the Value Functions in the Problem of Optimal Investment in Incomplete Markets, Ann. Appl. Probab. 16(3), 1352-1384 (2006) Google Scholar
Lasry, J.M, Lions P.L: Mean field games. Jpn J Math 2(1):229-260 (2007) Google Scholar
Huang, M., Malhamé, R. P., Caines, P.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6(3):221-251 (2006) Google Scholar
Huang, M., Caines, P., Malhamé, R. P.: Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized _-Nash equilibria. IEEE Trans Autom Control 52(9):1560-1571 (2007) Google Scholar
Pamen, O., Meyer-Brandis, T., Proske, F., Salleh, H.: Malliavin calculus applied to optimal control of stochastic partial differential equations with jumps, Stochastics An International Journal of Probability and Stochastic Processes, 85:3, 431-463 (2013) Google Scholar
Agram, Nacira, and Bernt Øksendal. Malliavin calculus and optimal control of stochastic Volterra equations. Journal of Optimization Theory and Applications 167, 1070-1094. (2015) Google Scholar
Yong, J.: Backward stochastic Volterra integral equations and some related problems. Stochastic Proc. Appl. 116: 779-795 (2006) Google Scholar
Shi, Y., Wang, T., Yong, J.: Mean-field backward stochastic Volterra integral equations. Discrete and Continuous Dynamical Systems Series B 18, 1929-1967 (2013) Google Scholar
Wang, T., Zhu, Q., Shi, Y.: Necessary and sufficient conditions of optimality for stochastic integral systems with partial information. Proceedings of the 30th Chinese Control Conference. July 22-24, Yantai, China (2011) Google Scholar
Di Nunno, G., Øksendal, B., Proske, F.: Malliavin Calculus for Lévy Processes with Applications to Finance, Universitext Springer, Berlin, Heidelberg. (2008) Google Scholar
Nualart, D.,: The Malliavin Calculus and Related Topics, 2nd ed., Springer, Berlin, Heidelberg, New York (2006) Google Scholar
About the article
Published Online: 2016-05-19
Published in Print: 2016-01-01