Mean-field FBSDE and optimal control

TL;DR

This paper develops a stochastic maximum principle for optimal control problems involving mean-field forward-backward stochastic differential equations, and demonstrates its application in a mean-field risk minimization portfolio problem.

Contribution

It introduces a new stochastic maximum principle for mean-field FBSDEs and applies it to solve a portfolio optimization problem with mean-field risk.

Findings

Derived necessary and sufficient optimality conditions.

Solved a portfolio risk minimization problem using the maximum principle.

Demonstrated the applicability of the theoretical results.

Abstract

We study optimal control for mean-field forward backward stochastic differential equations with payoff functionals of mean-field type. Sufficient and necessary optimality conditions in terms of a stochastic maximum principle are derived. As an illustration, we solve an optimal portfolio with mean-field risk minimization problem.

Equations

$$\left\{\begin{array}[c]{ll}dX(t)&=b(t,X(t),\mathcal{L}(X(t)))dt+\sigma(t,X(t),\mathcal{L}(X(t)))dB(t),\\ X(0)&=x_{0}.\end{array}\right.$$

$$\left\{\begin{array}[c]{ll}dX^{i,n}(t)&=b(t,X^{i,n}(t),\par\frac{1}{n}{\textstyle\sum_{j=1}^{n}}\delta_{X^{j,n}(t)}\par)dt+\sigma(t,X^{j,n}(t),\par\frac{1}{n}{\textstyle\sum_{j=1}^{n}}\delta_{X^{j,n}(t)}\par)dB^{i}(t),\\ X(0)&=x_{0},\end{array}\right.$$

$$\begin{array}[c]{ll}J(u)&=E[h(X(T),M(T))+\phi(Y(0),N(0))\\ &+{{\textstyle\int_{0}^{T}}}f(t,X(t),Y(t),Z(t),M(t),N(t),u(t))dt],\end{array}$$

$$\left\{\begin{array}[c]{lll}dX(t)&=&b(t,X(t),M(t),u(t))dt+\sigma(t,X(t),M(t),u(t))dB(t),\text{ \ }t\in\left[0,T\right],\\ X(0)&=&x_{0},\text{ \ }\end{array}\right.$$

$$\left\{\begin{array}[c]{lll}dY(t)&=&-g(t,X(t),Y(t),Z(t)),M(t),N(t),u(t))dt+Z(t)dB(t),\text{ \ }t\in\left[0,T\right],\\ Y(T)&=&X(T),\end{array}\right.$$

$$\begin{array}[c]{lll}\left\|\mu\right\|_{\mathcal{M}}^{2}&:=&E[{\textstyle\int_{\mathbb{R}}}|\hat{\mu}(y)|^{2}e^{-y^{2}}dy]\text{,}\end{array}$$

$$\begin{array}[c]{lll}\hat{\mu}(y)&:=&{{\textstyle\int_{\mathbb{R}}}}e^{ixy}d\mu(x);\quad y\in\mathbb{R}.\end{array}$$

$$\begin{array}[c]{lll}\hat{\mu}(y)&={{\textstyle\int_{\mathbb{R}}}}e^{ixy}d\mu(x)&=e^{ix_{0}y},\end{array}$$

$$\begin{array}[c]{lll}\left\|\mu\right\|_{\mathcal{M}_{0}}^{2}&={\textstyle\int_{\mathbb{R}}}|e^{ix_{0}y}|^{2}e^{-y^{2}}dy&<\infty\text{.}\end{array}$$

$$\begin{array}[c]{lll}\left\|\mu\right\|_{\mathcal{M}_{0}}^{2}&={\textstyle\int_{\mathbb{R}}}|\hat{\mu}(y)|^{2}e^{-y^{2}}dy&<\infty\text{.}\end{array}$$

$$\begin{array}[c]{lll}|\hat{\mu}(y)|&\leq{\textstyle\int_{\mathbb{R}}}d\mu(y)=\mu(\mathbb{R})&<\infty\text{, for all }y\text{,}\end{array}$$

$$\begin{array}[c]{lll}\left\|\mu\right\|_{\mathcal{M}_{0}}^{2}&={\textstyle\int_{\mathbb{R}}}|\hat{\mu}(y)|^{2}e^{-y^{2}}dy&<\infty\text{.}\end{array}$$

$${\Vert X\Vert }_{{\mathcal{S}}^2}^2:=E[\underset{t\in [0,T]}{\sup }|X(t){|}^2]<\infty ,$$

$$\Vert Q{\Vert }_{{\mathbb{L}}^2}^2:=E[{\int }_0^T|Q(t){|}^{2}dt]<\infty .$$

$$||K|{|}_{\mathbb{K}}:=\underset{m\in {\mathcal{M}}_0,||m|{|}_{{\mathcal{M}}_0}\le 1}{\sup }|K(m)|.$$

$$||p|{|}_{{\mathcal{S}}_{\mathbb{K}}}^2:=E[\underset{t\in [0,T]}{\sup }||p(t)|{|}_{\mathbb{K}}^2]<\infty .$$

$$||q|{|}_{{\mathbb{L}}_{\mathbb{K}}^2}^2:=E[{\int }_0^T||q(t)|{|}_{\mathbb{K}}^{2}dt]<\infty .$$

$$D_{w}F(v):=\underset{\epsilon \to 0}{\lim }\frac{1}{\epsilon }(F(v+\epsilon w)-F(v))$$

$$\underset{\begin{array}{c}h\to 0\\ h\in \mathcal{X}\end{array}}{\lim }\frac{1}{\Vert h{\Vert }_{\mathcal{X}}}\Vert F(v+h)-F(v)-A(h){\Vert }_{\mathcal{Y}}=0,$$

$$A={\nabla }_{v}F.$$

$$D_{w}F(v)={\nabla }_{v}F(w)=⟨{\nabla }_{v}F,w⟩.$$

$$\left\{\begin{array}[c]{lll}dX(t)&=&b(t,X(t),M(t),u(t))dt+\sigma(t,X(t),M(t),u(t))dB(t),\text{ \ }t\in\left[0,T\right],\\ X(0)&=&x_{0},\text{ \ }\end{array}\right.$$

$$\left\{\begin{array}[c]{lll}dY(t)&=&-g(t,X(t),Y(t),Z(t)),M(t),N(t),u(t))dt+Z(t)dB(t),\text{ \ }t\in\left[0,T\right],\\ Y(T)&=&X(T),\end{array}\right.$$

$$\begin{array}[c]{lll}||\mathcal{L}(X^{(1)})-\mathcal{L}(X^{(2)})||_{\mathcal{M}_{0}}^{2}&\leq&\sqrt{\pi}\mathbb{E}[(X^{(1)}-X^{(2)})^{2}]\text{,}\end{array}$$

$$\left|\sigma \left(t,x,m,u\right)-\sigma \left(t,x^{\prime },m^{\prime },u\right)\right|+\left|b\left(t,x,m,u\right)-b\left(t,x^{\prime },m^{\prime },u\right)\right|$$

$$\le C\left(\left|x-x^{\prime }\right|+||m-m^{\prime }|{|}_{{\mathcal{M}}_0}\right).$$

$$\left|\sigma \left(t,0,{\delta }_0,u\right)\right|+\left|b\left(t,0,{\delta }_0,u\right)\right|\le C\text{,}$$

$$\left|g\left(t,x,y,z,m,n,u\right)-g\left(t,x,y^{\prime },z^{\prime },m,n^{\prime },u\right)\right|$$

$$\le C\left(\left|y-y^{\prime }\right|+\left|z-z^{\prime }\right|+||n-n^{\prime }|{|}_{{\mathcal{M}}_0}\right).$$

$$\left|g\left(t,x,0,0,m,{\delta }_0,u\right)\right|\le C\text{.}$$

$$\begin{array}[c]{ll}J(u)&=E[h(X(T),M(T))+\phi(Y(0),N(0))\\ &+{{\textstyle\int_{0}^{T}}}f(t,X(t),Y(t),Z(t),M(t),N(t),u(t))dt],\end{array}$$

Full text

This research was carried out with support of the Norwegian Research Council, within the research project Challenges in Stochastic Control, Information and Applications (STOCONINF), project number 250768/F20.

1. 1 Introduction

2. 2 Generalities

3. 3 Optimal control problem

4. 3.1 Sufficient optimality conditions

5. 3.2 Necessary optimality conditions

6. 4 Mean-field discounted risk measure

7. 4.1 Optimal portfolio with mean-field risk minimization

1. Introduction

Stochastic differential equation (SDE) of mean-field type (a.k.a McKean-Vlasov equation) is an SDE whose coefficients depend on the marginal law of the solution (state) as well as the solution itself, i.e.

[TABLE]

where $\mathcal{L}(X(t))$ is the law of $X(t)$, which is obtained as a limit of a sequence of empirical distribution functions representing the states.

The SDE (1.1) can be viewed as the limit of a system of particles with mean-field interaction

[TABLE]

when the size of the system $n$ tends to infinity.

Optimal control of mean-field SDEs was first studied by Andersson and Djehiche [andersson2011], and Buckdahn, Djehiche and Li [buckdahngeneral], where the mean-field coupling is represented by an expected value of the state. The authors established a suitably modified stochastic maximum principle which involves mean-field backward SDEs.

Extension to the case where the marginal law of the state process is the mean-field coupling was studied by Carmona and Delarue [carmona1]. The authors use the Wassertein metric space for measures and the lifting technique introduced by Lions [Lions] to differentiate a function of a measure. In [AO1], Agram and Øksendal have introduced a Sobolev space of random measures in which, the Fréchet derivative with respect to the measure can be used directly. This approach is used in the present paper.

The purpose of our work is to derive necessary and sufficient optimality conditions in terms of a stochastic maximum principle for a set $\hat{u}$ of admissible controls which minimize a cost functional of the form

[TABLE]

with respect to admissible controls $u$, for some functions $f,\ h,\ \phi$, under dynamics governed by mean-field forward backward stochastic differential equations (MF-FBSDE). More specifically, we consider the coupled system

[TABLE]

[TABLE]

for some functions $b,\ \sigma$ and a Brownian motion $B(t).$ $M(t)$ and $N(t)$ denote the marginal laws of X and Y respectively.

Existence of a fully-coupled MF-FBSDE is studied by Carmona and Delarue under Lipschitz assumption on the coefficients but no uniqueness result was proven. Bensoussan et al [bens] prove existence and uniqueness of a fully coupled MF-FBSDE by assuming Lipschitz and monotonicity conditions. Recently, Djehiche and Hamadene in [Boualem2019] prove the same results but under weak monotonicity assumptions and without the non-degeneracy condition on the forward equation.

In the next section, we give some mathematical background. Next, we study stochastic optimal control of MF-FBSDE where sufficient and necessary optimality conditions are derived. In the last section, we construct a discounted dynamic risk measure by means of MF-BSDE and then we solve an associated risk minimization problem.

2. Generalities

Let $B=B(t),t\in[0,T]$ be a one-dimensional Brownian motion defined in a complete filtered probability space $(\Omega,\mathcal{F},\mathbb{F},P).$ The filtration $\mathbb{F=}\left\{\mathcal{F}_{t}\right\}_{t\geq 0}$ is assumed to be the $P$-augmented filtration generated by $B.$

Definition 2.1**.**

• •

Let $\mathcal{M}$ be the space of random measures $\mu$ on $\mathbb{R}$ equipped with the norm

[TABLE]

where $\hat{\mu}$ is the Fourier transform of the measure $\mu$, i.e.,

[TABLE]

We endow $\mathcal{M}$ with the inner product $\langle\mu,\eta\rangle:={\textstyle\int_{\mathbb{R}}}|\hat{\mu}(y)-\hat{\eta}(y)|^{2}e^{-y^{2}}dy,$ $\mu,\eta,y\in\mathbb{R}$, $\hat{\mu}$ and $\hat{\eta}$ are the Fourier transform of the measures $\mu$ and $\eta$. Then $(\mathcal{M},||\cdot||)$ is a pre-Hilbert space.

• •

We denote by $\mathcal{M}_{0}$ the set of all deterministic elements of $\mathcal{M}$.

We give some examples:

Example 2.2** (Measures).**

Let us give some examples of measures in $\mathcal{M}_{0}$ and $\mathcal{M}$:

1. (1)

Suppose that $\mu=\delta_{x_{0}}$, the unit point mass at $x_{0}\in\mathbb{R}$. Then $\delta_{x_{0}}\in\mathcal{M}_{0}$ and

[TABLE]

and hence

[TABLE] 2. (2)

Suppose $d\mu(x)=f(x)dx$, where $f\in L^{1}(\mathbb{R})$. Then $\mu\in\mathcal{M}_{0}$ and by Riemann-Lebesque lemma, $\hat{\mu}(y)\in C_{0}(\mathbb{R})$, i.e. $\hat{\mu}$ is continuous and $\hat{\mu}(y)\rightarrow 0$ when $|y|\rightarrow\infty$. In particular, $|\hat{\mu}|$ is bounded on $\mathbb{R}$ and hence

[TABLE] 3. (3)

Suppose that $\mu$ is any finite positive measure on $\mathbb{R}$. Then $\mu\in\mathcal{M}_{0}$ and

[TABLE]

and hence

[TABLE] 4. (4)

Next, suppose $x_{0}=x_{0}(\omega)$ is random. Then $\delta_{x_{0}(\omega)}$ is a random measure in $\mathcal{M}$. Similarly, if $f(x)=f(x,\omega)$ is random, then $d\mu(x,\omega)=f(x,\omega)dx$ is a random measure in $\mathcal{M}$.

We denote by $U$ a nonempty convex subset of $\mathbb{R}$ and we denote by $\mathcal{U}_{\mathbb{G}}$ the set of $U$-valued $\mathbb{G}$-progressively measurable processes where $\mathbb{G}:=\{\mathcal{G}_{t}\}_{t\geq 0}$ with $\mathcal{G}_{t}\subseteq\mathcal{F}_{t}$ for all $t\geq 0$; we consider them as the admissible control processes.

We will also use the following spaces:

• •

$\mathcal{S}^{2}$ is the set of ${\mathbb{R}}$-valued $\mathbb{F}$-adapted càdlàg processes $X=X(t),t\in[0,T],$ such that

[TABLE]

• •

$\mathbb{L}^{2}$ is the set of ${\mathbb{R}}$-valued $\mathbb{F}$-adapted processes $Q=Q(t),t\in[0,T],$ such that

[TABLE]

• •

$\mathcal{K}$ denotes the set of absolutely continuous functions $m:[0,T]\rightarrow\mathcal{M}_{0}.$

• •

$\mathbb{K}$ is the set of bounded linear functionals $K:\mathcal{M}_{0}\rightarrow\mathbb{R}$ equipped with the operator norm

[TABLE]

• •

$\mathcal{S}_{\mathbb{K}}^{2}$ is the set of $\mathbb{F}$-adapted stochastic processes $p:[0,T]\times\Omega\mapsto\mathbb{K},$ such that

[TABLE]

• •

$\mathbb{L}_{\mathbb{K}}^{2}$ is the set of $\mathbb{F}$-adapted stochastic processes $q:[0,T]\times\Omega\mapsto\mathbb{K},$ such that

[TABLE]

We recall now the notion of differentiability which will be used in the sequel.

Let $\mathcal{X},\mathcal{Y}$ be two Banach spaces with norms $\|\cdot\|_{\mathcal{X}},\|\cdot\|_{\mathcal{Y}}$, respectively, and let $F:\mathcal{X}\rightarrow\mathcal{Y}$.

• •

We say that $F$ has a directional derivative (or Gateaux derivative) at $v\in\mathcal{X}$ in the direction $w\in\mathcal{X}$ if

[TABLE]

exists in $\mathcal{Y}$.

• •

We say that $F$ is Fréchet differentiable at $v\in\mathcal{X}$ if there exists a continuous linear map $A:\mathcal{X}\rightarrow\mathcal{Y}$ such that

[TABLE]

where $A(h)=\langle A,h\rangle$ is the action of the liner operator $A$ on $h$. In this case we call $A$ the gradient (or Fréchet derivative) of $F$ at $v$ and we write

[TABLE]

• •

If $F$ is Fréchet differentiable at $v$ with Fréchet derivative $\nabla_{v}F$, then $F$ has a directional derivative in all directions $w\in\mathcal{X}$ and

[TABLE]

In particular, note that if $F$ is a linear operator, then $\nabla_{v}F=F$ for all $v$.

3. Optimal control problem

Here we denote by $M(t):=\mathcal{L}(X(t))$ the law of $X(t)$ at time $t$ and by $N(t):=\mathcal{L}(Y(t))$ the law of $Y(t)$ at time $t$. We assume that our system is gouverned by a coupled system of MF-FBSDE as follows:

The MF-SDE $X^{u}(t)=X(t)$ is given by

[TABLE]

for functions $\sigma,b:\Omega\times\left[0,T\right]\times\mathbb{R}\times\mathcal{M}_{0}\times U\rightarrow\mathbb{R}$ which are supposed to be $\mathcal{F}_{t}$-measurable and the initial value $x_{0}\in\mathbb{R}$.

The couple MF-BSDE $(Y^{u}(t),Z^{u}(t))=(Y(t),Z(t))$ satisfies

[TABLE]

where $g:\Omega\times\left[0,T\right]\times\mathbb{R}^{3}\times\mathcal{M}_{0}^{2}\times U\rightarrow\mathbb{R}$ is $\mathbb{F}$-adapted.

It is obvious from the definition of the norm (2.1) that

[TABLE]

where $X^{(1)}$ and $X^{(2)}$ are random variables that follow the distributions $\mathcal{L}(X^{(1)})$ and $\mathcal{L}(X^{(2)})$ respectively.

Assume that ($C$ is a constant that may change from line to line)

• (A1)

there exists $C>0$, such that

• •

for all $t\in\left[0,T\right]$, for all fixed $u\in U,$ $x,x^{\prime}\in\mathbb{R},m,m^{\prime}\in\mathcal{M}_{0}$

[TABLE]

• •

for all $t\in\left[0,T\right]$, for all fixed $u\in U,$

[TABLE]

where $\delta_{0}$ is the distribution law of zero, i.e., the Dirac measure with mass at zero.

• (A2)

there exists $C>0$, such that, for all fixed $u\in U$ and all knowing $X(t)\in\mathcal{S}^{2}$ of equation (3.1) and $M(t):=\mathcal{L}(X(t))\in\mathcal{M}_{0}$, we have

• •

for all $t\in\left[0,T\right],$ $y,y^{\prime},z,z^{\prime}\in\mathbb{R},n,n^{\prime}\in\mathcal{M}_{0}$

[TABLE]

• •

for all $t\in\left[0,T\right]$,

[TABLE]

Proposition 3.1**.**

Under Assumptions (A1) and (A2), the MF-FBSDE (3.1)-(3.2) admits a unique solution $(X,Y,Z)\in\mathcal{S}^{2}\times\mathcal{S}^{2}\times\mathbb{L}^{2}.$

Since the system is partially-coupled i.e., the forward equation does not depend on the solution of the backward one, we can solve the system separately as follows: we first find a solution $X(t)$ of the MF-SDE (3.1) and then we plug it into the backward equation (3.2), then we solve it.

Our aim is to maximize the performance functional of the form

[TABLE]

over all admissible controls, for functions $f:\Omega\times\left[0,T\right]\times\mathbb{R}^{3}\times\mathcal{M}_{0}^{2}\times U\rightarrow\mathbb{R}$, $h:\Omega\times\mathbb{R}\times\mathcal{M}_{0}\rightarrow\mathbb{R}$ and $\phi:\Omega\times\mathbb{R}\times\mathcal{M}_{0}\rightarrow\mathbb{R}.$

Now, we can define the Hamiltonian

[TABLE]

by

[TABLE]

Remark 3.2**.**

For ease of notation we drop the dependence of all variables except for the time $t,\forall\Phi\in\{\sigma,f,H,h,g,\phi\},$ we write $\Phi(t),\forall t.$  Moreover, we will use

[TABLE]

For $u\in\mathcal{U}$ with corresponding solution $X^{u}=X$, define, whenever solutions exist, $p^{\hat{u}}=p=(p^{0},p^{1})$ and $q^{\hat{u}}=q=(q^{0},q^{1})$ and $\lambda^{\hat{u}}=\lambda=(\lambda^{0},\lambda^{1})$ by the adjoint equations:

The BSDE for the unknown processes $(p^{0},q^{0})\in\mathcal{S}^{2}\times\mathbb{L}^{2}$

[TABLE]

The MF-BSDE for the unknown processes $(p^{1},q^{1})\in\mathcal{S}_{\mathbb{K}}^{2}\times\mathbb{L}_{\mathbb{K}}^{2}$

[TABLE]

The forward SDE

[TABLE]

and

[TABLE]

Before stating and proving sufficient and necessary conditions of optimality, we need the following result, which is Lemma 2.3 in Agram and Øksendal [AO1]:

Lemma 3.3**.**

Suppose that $X(t)$ is an Itô process of the form

[TABLE]

*where $\theta,\gamma$ are adapted processes.

Then the map $t\mapsto M(t):[0,T]\rightarrow\mathcal{M}_{0}$ is absolutely continuous.*

It follows that $t\mapsto M(t)$ is differentiable for $t$-a.e. We will in the following use the notation

[TABLE]

3.1. Sufficient optimality conditions

Theorem 3.4**.**

Suppose that $\hat{u}\in\mathcal{U}_{\mathbb{G}}$ with corresponding solutions $\hat{X}(t),(\hat{Y}(t),\hat{Z}(t)),\newline (p^{0}(t),q^{0}(t)),(p^{1}(t),q^{1}(t)),\lambda^{0}(t),\lambda^{1}(t)$ to equations (3.1), (3.2), (3.4),(3.5), (3.6) and (3.7) respectively. Suppose that

• •

$x,m\mapsto h(k,m)$* ,*

• •

$y,n\mapsto\phi(y,n)$* ,*

• •

$x,y,z,m,n,u\mapsto H(\cdot,x,y,z,m,n,u)$* ,*

are concave functions $P$-a.s for each $t\in[0,T].$  Moreover,

[TABLE]

$P$-a.s for all t $\in[0,T].$ Then $\hat{u}$ is an optimal control.

Proof We show that $J(u)-J(\hat{u})\leq 0,$ for an arbitrary $u$ and a fixed

optimal $\hat{u}\in\mathcal{U}_{\mathbb{G}}.$

We introduce first the following notation $\forall\Phi\in\{\sigma,f,H,h,g,\phi,M,N,M^{\prime},N^{\prime}\}$ and $\forall t,$

[TABLE]

and

[TABLE]

From the definition of the Hamiltonian (3.3), we have

[TABLE]

and

[TABLE]

We use the concavity of $h$ and $\phi$ as well as the boundary values of equations (3.4), (3.5), (3.6) and (3.7)

[TABLE]

Applying It$\hat{o}$ formula to $p^{0}(t)\delta X(t),p^{0}(t)\delta X(t),\lambda^{0}(t)\delta Y(t)$ and $\lambda^{1}(t)\delta Y(t),$ yields the following duality relations:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By the concavity of $H,$ we obtain

[TABLE]

Finally, by substituting the derived duality relations (3.10),(3.11), (3.12) and (3.13) in (3.8) and using the estimates (3.9), (3.14), we obtain

[TABLE]

Using the tower property and the fact that $u(t)$ is $\mathbb{G}$-adapted the desired result follows

[TABLE]

and thus, $\hat{u}$ is optimal.

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\square$

3.2. Necessary optimality conditions

Given an arbitrary but fixed control $u\in\mathcal{U}_{\mathbb{G}}$, we define

[TABLE]

Note that, the convexity of $U$ and $\mathcal{U}_{\mathbb{G}}$ guarantees that $u^{\rho}\in\mathcal{U}_{\mathbb{G}},\rho\in\left[0,1\right]$. We denote by $X^{\rho}:=X^{u^{\rho}}$ and by $\hat{X}:=X^{\hat{u}},$ the solution processes corresponding to $u^{\rho}$ and $\hat{u},$ respectively.

For each $t_{0}\in\left[0,T\right]$ and all bounded $\mathcal{G}_{t_{0}}$-measurable random variables $\alpha,$ the process

[TABLE]

belongs to $\mathcal{U}_{\mathbb{G}}$.

In general, if $K^{\hat{u}}(t)$ is a process depending on $\hat{u}$, we define the operator $D$ on $K$ by

[TABLE]

whenever the derivative exists.

Define the following derivative processes

[TABLE]

such that

[TABLE]

and

[TABLE]

Moreover, we assume that all the partial derivatives of $\Phi\in\{\sigma,f,H,h,g,\phi\}$ are bounded.

Theorem 3.5**.**

*Let $\hat{u}\in\mathcal{U}_{\mathbb{G}}$ be the optimal control and $\mathcal{X}^{\rho}(t),(\mathcal{Y}^{\rho}(t),\mathcal{Z}^{\rho}(t)),\newline (p^{0}(t),q^{0}(t)),(p^{1}(t),q^{1}(t)),\lambda^{0}(t),\lambda^{1}(t)$ be the corresponding solutions to the

equations (3.17),(3.18), (3.4),(3.5), (3.6),(3.7). Then, the following statements are

equivalent*

1. (i)

$\frac{d}{d\rho}J(\hat{u}+\rho u)|_{\rho=0}=0$* for all bounded $\beta\in\mathcal{U}_{\mathbb{G}}.$* 2. (ii)

$E[\frac{\partial}{\partial u}\hat{H}(t)|\mathcal{G}_{t}]=0$* for all $t\in[0,T].$*

Proof We first prove theorem 3.5 by assuming (i) and aiming to show (ii)

[TABLE]

{ we substitute $f(t)$ from equation $\eqref{H}$ }

[TABLE]

by using the chain rule, we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We apply It$\hat{o}$ formula to $p^{0}(t)\mathcal{X}^{\rho}(t),\langle p^{1}(t),DM^{\rho}(t)\rangle,\lambda^{0}(t)\mathcal{Y}^{\rho}(t)$ and

$\langle\lambda^{1}(t),DN^{\rho}(t)\rangle$ then we take the expectation, we obtain the following important duality relations:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By substituting the derived duality relations and the partial derivatives of $f(t)$ the desired result follows. This proof can be reversed to prove $(ii)\Rightarrow(i).$ We omit the details. $\qquad\qquad\ \qquad\qquad\ \ \ \ \ \qquad\qquad\qquad\ \qquad\qquad\qquad\square$

4. Mean-field discounted risk measure

In this section we are interested in a particular class of MF-BSDE of the following form

[TABLE]

where

[TABLE]

We assume that the generator $(y,\bar{y},z)\rightarrow f(t,Y(t),E[Y(t)],Z(t)):\Omega\times[0,T]\times\mathcal{\mathbb{R}}\times\mathcal{\mathbb{R}}\times\mathcal{\mathbb{R}}\rightarrow\mathcal{\mathbb{R}}$ is $\mathbb{F}$-adapted, uniformly Lipschitz and concave, and the terminal condition $\xi\in L^{2}\left(\Omega,\mathcal{F}_{T}\right).$

Definition 4.1**.**

Define $\varphi_{t}:$ $(T;\xi)\rightarrow\varphi_{t}(T;\xi)$ by

[TABLE]

where $Y_{t}(T;\xi)$ is a component of the solution of the MF-BSDE (4.1) with terminal horizon $T$, terminal condition $\xi$ and driver $f$. Then $\varphi_{t}(T;\xi)$ is a dynamic risk measure induced by a MF-BSDE.

We may remark that the driver $f$ depends linearly on $Y$ and its expected value $E[Y]$, and nonlinear with respect to $Z$. This is interpreted as a market with interest rates $(r(t),r^{\prime}(t))$. We can reformulated this as a problem with a driver independent of $Y$ and $E[Y]$ by discounting the financial position $\xi$. We assume that the instantaneous interest rates $r(t)$ and $r^{\prime}(t)$ are deterministic. We denote by $\varphi_{\cdot}$, the corresponding discounted risk-measure.

Define the discounted process

[TABLE]

Then $Y^{r}$ with driver

[TABLE]

and terminal value $\xi^{r}:=e^{-{\textstyle\int_{0}^{t}}(r(s)+r^{\prime}(s))ds}\xi$ is a part of the solution of the associated BSDE. We obtain also a discounted risk-measure accordingly

[TABLE]

This discounted risk-measure is translation-invariant because $F^{r}$ does not depend on $Y$, we have for $\xi\in L^{2}\left(\Omega,\mathcal{F}_{T}\right)$ and $a\in\mathbb{R},$

[TABLE]

Similarly we can get for each $t\in[0,T]$, that

[TABLE]

is translation-invariant.

4.1. Optimal portfolio with mean-field risk minimization

Consider a financial market with two investment possibilities:

(i) Safe, or risk free asset with unit price

[TABLE]

(ii) Risky asset with unit price

[TABLE]

Let $\pi(t)$ be a self-financing portfolio invested in the risky asset at time $t$. We want to minimize the risk $\varphi(X^{\pi}(T))$ of the terminal value of the wealth process $X^{\pi}(t)$ corresponding to a portfolio $\pi$ which satisfies the linear SDE

[TABLE]

such that

[TABLE]

where $Y^{\pi}(t)$ satisfies a MF-BSDE

[TABLE]

Here we assume that $b_{0}(t),$ $\sigma_{0}(t),$ $r_{0}(t)$ are given deterministic functions and $F:\mathbb{R}\rightarrow\mathbb{R}$ is some given concave function. We want to find $\hat{\pi}\in\mathcal{U}_{\mathbb{G}}$ such that

[TABLE]

Define the Hamiltonian $H$ that correspondds to our problem by

[TABLE]

The couple $(p^{0},q^{0})$ solution of the following BSDE

[TABLE]

and $(p^{1},q^{1})$ satisfies

[TABLE]

$\lambda^{0}$ is given by the forward SDE

[TABLE]

and

[TABLE]

The first order necessary optimality condition gives

[TABLE]

where we denoted by $\hat{X}(t)=X^{\hat{\pi}}(t)$ and so on. Since $\hat{X}(t)>0$ for all $t$ $P$-a.s., we obtain

[TABLE]

which implies

[TABLE]

this together with equation (4.4), yields

[TABLE]

From (4.5), we get

[TABLE]

For example, if we choose

[TABLE]

That is

[TABLE]

Substituting the expression of $\hat{Z}(t)$ above into the MF-BSDE (4.3), we obtain

[TABLE]

Consequently

[TABLE]

thus

[TABLE]

Define $\Gamma(t)$ to be the solution of the linear SDE

[TABLE]

or explicitely

[TABLE]

By the Girsanov theorem of change of measures, we know that there exists an equivalent local martingale measure $Q<<P$, such that

[TABLE]

with $\Gamma(T)=\frac{dQ}{dP}$ is called the Radon-Nikodym derivative of $Q$ with respect to $P$ on $\mathcal{F}_{T}$.

Substituting (4.8)-(4.9) into (4.7) we have

[TABLE]

Taking the expectation but now with respect to the new measure $Q$, we get

[TABLE]

where $E[\Gamma(T)\ln\Gamma(T)]$ is the entropy of $Q$ with respect to $P$.

Since we obtained the optimal value of $\hat{Y}(0)$, we can get the corresponding optimal terminal wealth $\hat{X}(T).$

Summarizing, we have the following conclusion:

Theorem 4.2**.**

Suppose that (4.6) holds. Then the minimal risk of our problem is given by (4.10).