A Maximum Principle for Mean-Field SDEs with Time Change

Abstract

Time change is a powerful technique for generating noises and providing flexible models. In the framework of time changed Brownian and Poisson random measures we study the existence and uniqueness of a solution to a general mean-field stochastic differential equation. We consider a mean-field stochastic control problem for mean-field controlled dynamics and we present a necessary and a sufficient maximum principle. For this we study existence and uniqueness of solutions to mean-field backward stochastic differential equations in the context of time change. An example of a centralised control in an economy with specialised sectors is provided.

Appendix

Proof that the mapping $\Psi$ in (4.5) is a contraction. Fix $\beta >0$. We define the norm $||\cdot ||_{\beta }$ on $L^2_{ad}(\mathbb {G}) \times \mathcal {I}$ by

$$\begin{aligned} ||(Y,Z)||_{\beta }:=\left( E\left[ \int \limits _0^Te^{\beta s}(|Y_s|^2+||Z_s||^2_{\lambda _s})ds\right] \right) ^{\frac{1}{2}} \end{aligned}$$

which is equivalent to the canonical one.

Let $(y^{(1)},z^{(1)}),\,(y^{(2)},z^{(2)})\in L^2_{ad}(\mathbb {G}) \times \mathcal {I}$ be two given inputs and define $(Y^{(1)},Z^{(1)}):=\Psi (y^{(1)},z^{(1)})$, $(Y^{(2)},Z^{(2)}):=\Psi (y^{(2)},z^{(2)})$, which are indeed the corresponding solutions of (4.4). Furthermore, define

$$\begin{aligned}&\hat{Y}:=Y^{(1)}-Y^{(2)},\,\hat{y}:=y^{(1)}-y^{(2)},\,\\&\hat{Z}:=Z^{(1)}-Z^{(2)},\,\hat{z}:=z^{(1)}-z^{(2)}. \end{aligned}$$

Then $(\hat{Y},\hat{Z})$ satisfies the BSDE

$$\begin{aligned} {\left\{ \begin{array}{ll} d\hat{Y}_t&{}=E'\left[ h\left( t,\lambda _t,\lambda '_t,Y^{(1)}_t,(y^{(1)}_t)', Z^{(1)}_t,(z^{(1)}_t)'\right) \right. \\ &{}\qquad \left. -h\left( t,\lambda _t,\lambda '_t,Y^{(2)}_t, (y^{(2)}_t)',Z^{(2)}_t,(z^{(2)}_t)'\right) \right] dt\\ &{}\quad +\int _\mathbb {R}Z^{(1)}_t(z)-Z^{(2)}_t(z)\mu (dt,dz)\\ \hat{Y}_T&{}=F-F=0. \end{array}\right. } \end{aligned}$$

The application of Ito’s formula on $e^{\beta s}|\hat{Y}_s|^2$ yields

$$\begin{aligned} 0&\ge E\left[ e^{\beta \cdot T}|\hat{Y}_T|^2-e^{\beta \cdot 0}|\hat{Y}_0|^2\right] =E\Bigg [\int \limits _0^T\beta e^{\beta s}|\hat{Y}_{s-}|^2ds+\int \limits _0^T\int \limits _\mathbb {R}2e^{\beta s}\hat{Y}_{s-}\hat{Z}_s(\xi )\mu (ds,d\xi ) \\&\quad +\int \limits _0^T2e^{\beta s}\hat{Y}_{s-}E'\Big [h\Big (s,\lambda _s,\lambda '_s,Y^{(1)}_s,(y^{(1)}_s)',Z^{(1)}_s, (z^{(1)}_s)'\Big )\nonumber \\&\quad -h\Big (s,\lambda _s,\lambda '_s,Y^{(2)}_s,(y^{(2)}_s)', Z^{(2)}_s,(z^{(2)}_s)'\Big )\Big ]ds\\&\quad +\frac{1}{2}\int \limits _0^T2e^{\beta s}|\hat{Z}_s(0)|^2\lambda ^B_sds\quad \\&\quad +\int \limits _0^T\int \limits _{\mathbb {R}_0} \{ e^{\beta s}(|\hat{Y}_{s-}+\hat{Z}_s(\xi )|^2-|\hat{Y}_{s-}|^2)-2e^{\beta s}\hat{Y}_{s-}\hat{Z}_s(\xi ) \} \lambda ^B_s\nu (d\xi )ds\Bigg ] \end{aligned}$$

Since $Z^{(1)},\,Z^{(2)}\in \mathcal I$, then the process $M_t:=\int _0^t\int _\mathbb {R}\hat{Z}_s(z)\mu (ds,dz)$ is a martingale. Since the filtration $\mathbb {G}$ is right continuous (see [9, Lemma 2.4]), Doob’s Regularization Theorem (see, e.g. [14, Theorem 6.27]) implies that M has a càdlàg version and, being the integral w.r.t. ds continuous, we conclude that Y has a càdlàg version. Hence the càdlàg version of Y has only countably many discontinuities, we can replace the $\hat{Y}_{s-}$ by $\hat{Y}_{s}$ in the integrals w.r.t. ds. Rearranging terms and the Lipschitzianity of h, given by (C3) yields

$$\begin{aligned}&E\left[ \int \limits _0^T\beta e^{\beta s}|\hat{Y}_{s}|^2ds+\int \limits _0^Te^{\beta s}||\hat{Z}_{s}||_{\lambda _s}^2ds\right] \\&\quad \le -E\Bigg [\int \limits _0^T2e^{\beta s}\hat{Y}_{s}E'\Bigg [h\Bigg (s,\lambda _s,\lambda '_s,Y^{(1)}_s,(y^{(1)}_s)', Z^{(1)}_s,(z^{(1)}_s)'\Bigg )\\&\qquad -h\Bigg (s,\lambda _s,\lambda '_s,Y^{(2)}_s, (y^{(2)}_s)',Z^{(2)}_s,(z^{(2)}_s)'\Bigg )\Bigg ]ds\Bigg ]\\&\quad \le E\Bigg [\int \limits _0^T2e^{\beta s}|\hat{Y}_{s}|E'\Bigg [\Bigg |h\Bigg (s,\lambda _s,\lambda '_s,Y^{(1)}_s,(y^{(1)}_s)', Z^{(1)}_s,(z^{(1)}_s)'\Bigg )\\&\qquad -h\Bigg (s,\lambda _s,\lambda '_s,Y^{(2)}_s, (y^{(2)}_s)',Z^{(2)}_s,(z^{(2)}_s)'\Bigg )\Bigg |\Bigg ]ds\Bigg ]\\&\quad \le E\Bigg [\int \limits _0^T2Ke^{\beta s}|\hat{Y}_{s}|E'\Bigg [|Y^{(1)}_s-Y^{(2)}_s|+|(y^{(1)}_s)'-(y^{(2)}_s)'|\\&\qquad +||Z^{(1)}_s-Z^{(2)}_s||_{\lambda _s}+||(z^{(1)}_s)'-(z^{(2)}_s)'||_{\lambda '_s}\Bigg ]ds\Bigg ] \end{aligned}$$

By the definition of the operator $E'$, we have

$$\begin{aligned}&E'[|Y^{(1)}_s-Y^{(2)}_s|]=|Y^{(1)}_s-Y^{(2)}_s|=|\hat{Y}_s|\\&E'[|(y^{(1)}_s)'-(y^{(2)}_s)'|]=E[|y^{(1)}_s-y^{(2)}_s|]=E[|\hat{y}_s|]\\&E'[||Z^{(1)}_s-Z^{(2)}_s||_{\lambda _s}]=||Z^{(1)}_s-Z^{(2)}_s||_{\lambda _s}=||\hat{Z}_s||_{\lambda _s}\\&E'[||(z^{(1)}_s)'-(z^{(2)}_s)'||_{\lambda _s}]=E[||z^{(1)}_s-z^{(2)}_s||_{\lambda _s}]=E[||\hat{z}_s||_{\lambda _s}]. \end{aligned}$$

Making use of the fact that $2ab\le ka^2+\frac{1}{k}b^2$ for all $a,b\in \mathbb {R}$ and all $k>0$, and choosing $k:=16K$, $a:=|\hat{Y}_s|$, $b=(|\hat{Y}_s|+E[|\hat{y}_s|]+||\hat{Z}_s||_{\lambda _s}+E[||\hat{z}_s||_{\lambda _s}])$, we get

$$\begin{aligned}&E\left[ \int \limits _0^T\beta e^{\beta s}|\hat{Y}_{s}|^2ds+\int \limits _0^Te^{\beta s}||\hat{Z}_{s}||_{\lambda _s}^2ds\right] \\&\quad \le 16K^2E\left[ \int \limits _0^Te^{\beta s}|\hat{Y}_{s}|^2ds\right] +\frac{1}{16}E\left[ \int \limits _0^Te^{\beta s}(|\hat{Y}_s|+E[|\hat{y}_s|]+||\hat{Z}_s||_{\lambda _s}+E[||\hat{z}_s||_{\lambda _s}])^2ds\right] \\&\quad \le 16K^2E\left[ \int \limits _0^Te^{\beta s}|\hat{Y}_{s}|^2ds\right] +\frac{1}{4}E\left[ \int \limits _0^Te^{\beta s}|\hat{Y}_s|^2ds\right] +\frac{1}{4}E\left[ \int \limits _0^Te^{\beta s}|\hat{y}_s|^2ds\right] \\&\qquad +\frac{1}{4}E\left[ \int \limits _0^Te^{\beta s}||\hat{Z}_s||_{\lambda _s}^2ds\right] +\frac{1}{4}E\left[ \int \limits _0^Te^{\beta s}||\hat{z}_s||_{\lambda _s}^2ds\right] , \end{aligned}$$

where we also used that $(\sum _{i=1}^na_i)^2\le n\sum _{i=1}^na_i^2$ and $E[X]^2\le E[X^2]$. This yields

$$\begin{aligned}&\left( \beta -16K^2-\frac{1}{4}\right) E\left[ \int \limits _0^Te^{\beta s}|\hat{Y}_{s}|^2ds\right] +\frac{3}{4}E\left[ \int \limits _0^Te^{\beta s}||\hat{Z}_{s}||_{\lambda _s}^2ds\right] \\&\quad \le \frac{1}{4}E\left[ \int \limits _0^Te^{\beta s}(|\hat{y}_s|^2+||\hat{z}_s||_{\lambda _s}^2)ds\right] \end{aligned}$$

Choosing $\beta =16K^2+1>0$, we finally get

$$\begin{aligned}&||(\hat{Y},\hat{Z})||_{\beta }=E\left[ \int \limits _0^Te^{\beta s}(|\hat{Y}_{s}|^2ds+||\hat{Z}_{s}||_{\lambda _s}^2)ds\right] \\&\quad \le \frac{1}{3}E\left[ \int \limits _0^Te^{\beta s}(|\hat{y}_s|^2+||\hat{z}_s||_{\lambda _s}^2)ds\right] =\frac{1}{3}||(\hat{y},\hat{z})||_{\beta }. \end{aligned}$$

By this we see that $\Psi$ is a contraction.