Time consistent mean-variance asset allocation for a DC plan with regime switching under a jump-diffusion model

Abstract

In this paper, we study a time consistent solution for a defined contribution pension plan under a mean-variance criterion with regime switching in a jump-diffusion setup, during the accumulation phase. We consider a market consisting of a risk-free asset and a geometric jump-diffusion risky asset process. Our solution allows the fund manager to incorporate a clause which allows for the distribution of a member’s premiums to his surviving dependents, should the member die before retirement. Applying the extended Hamilton-Jacobi-Bellman (HJB) equation, we derive the explicit time consistent equilibrium strategy and the value function. We then provide some numerical simulations to illustrate our results.

Appendix

Proof of the Theorem 3.1.

To prove the theorem, we follow [5], Theorem 7.1. We divide the proof in two parts, first we show that for the equilibrium control $\pi ^*\in {\mathcal {A}}$, the value function $\Psi (t,x,\ell ,z,j)=\Phi (t,x,\ell ,z,j,\pi ^*)$ and $\psi$ and $\varphi$ have the following probabilistic representation:

$$\begin{aligned} \psi (t,x,\ell ,z,j,j')={\mathbb {E}}_{t,x,\ell ,z,j}[F(X^{\pi ^*}(T),j)]\ \ \ \ \mathrm{and} \ \ \ \ \varphi (t,x,\ell ,z,j)={\mathbb {E}}_{t,x,\ell ,z,j}[X^{\pi ^*}(T)]\,. \end{aligned}$$

In the second part, we prove that $\pi ^*\in {\mathcal {A}}$ is indeed the equilibrium control strategy.

Let $h(t,x,\ell ,z,j)\in {\mathcal {C}}^{1,2,2}([0,T]\times {\mathbb {R}}^2\times {\mathcal {S}})$, then by Itô’s formula ([2], Lemma A1),

$$\begin{aligned}&h(t,X^\pi (t),\ell (t),Z(t),j) \\&= {} h(0,x,\ell ,z,j)+\int _0^t{\mathcal {L}}h(s,X^\pi (s),\ell (s),Z(s),j)ds \\&+\int _0^t\ell (s)\sigma _2\frac{\partial h}{\partial \ell }(s,X^\pi (s),\ell (s),Z(s),j)dW_1(s) \\&+\int _0^t\Bigl (\pi (s,X^\pi (s),\ell (s),Z(s),j)\sigma \frac{\partial h}{\partial x}(s,X^\pi (s),\ell (s),Z(s),j) \\&\ \ \ \ +\ell (s)\sigma _1\frac{\partial h}{\partial \ell }(s,X^\pi (s),\ell (s),Z(s),j)\Bigl )dW(s) \\&+\int _0^t\sum _{\alpha \in {\mathcal {S}},\alpha \ne j}\Bigl (h(s,X^\pi (s),\ell (s),Z(s),\alpha )-h(s,X^\pi (s),\ell (s),Z(s),j)\Bigl )d{\tilde{M}}_{j\alpha } \\&+\int _0^t\int _{{\mathbb {R}}}(h(s,X^\pi (s)+\pi (s,X^\pi (s),\ell (s),Z(s),j)\gamma _S,\ell (s),j) \\&\ \ \ \ \ \ \ \ \ \ +h(s,X^\pi (s),\ell (s)(1+\sigma _1),Z(s),j)-2h(s,X^\pi (s),\ell (s),Z(s),j)){\tilde{N}}(ds,d\zeta )\,, \end{aligned}$$

where ${\mathcal {L}}h$ is given by (2.8). Since $W,{\tilde{M}}$ and ${\tilde{N}}$ are martingales, under the integrability condition $h\in L^2_T(X^{\pi ^*})$, we have that

$$\begin{aligned}h(t,X^\pi (t),\ell (t),Z(s),\alpha (t))-h(0,x,\ell ,j)-\int _0^t{\mathcal {L}}^\pi h(s,X^\pi (s),\ell (s),Z(s),\alpha (s)ds\end{aligned}$$

is a martingale. Thus, for $\Psi =h$, we have:

$$\begin{aligned}&{\mathbb {E}}_{t,x,\ell ,z,j}[\Psi (T,X(T),\ell (T),Z(T),\alpha (t)] \nonumber \\&= {} \Psi (t,x,\ell ,z,j)+{\mathbb {E}}_{t,x,\ell ,z,j}\Bigl [\int _t^T{\mathcal {L}}^{\pi ^*}\Psi (s,X^{\pi ^*}(s),\ell (s),Z(s),\alpha (t)ds\Bigl ]\,. \end{aligned}$$

(4.1)

In order to show that $\Psi (t,x,\ell ,z,j)=\Phi (t,x,\ell ,z,j,\pi ^*)$, we use the $\Psi$-equation (3.1) to obtain:

$$\begin{aligned} {\mathcal {L}}^{\pi ^*}\Psi (t,x,\ell ,z,j)= & {} {\mathcal {L}}^{\pi ^*}\psi (t,x,\ell ,z,j,j')-{\mathcal {L}}^{\pi ^*}\psi ^{j'}(t,x,\ell ,z,j) \\&+{\mathcal {L}}^{\pi ^*}(\Gamma \circ \varphi )(t,x,\ell ,z,j) -{\mathcal {M}}^{\pi ^*}\varphi (t,x,\ell ,z,j)\,. \end{aligned}$$

Then, from (3.2)–(3.2), The relation (4.1) becomes:

$$\begin{aligned}&{\mathbb {E}}_{t,x,\ell ,z,j}[\Psi (T,X(T),\ell (T),Z(T),\alpha (t)] \\& \quad = {} \Psi (t,x,\ell ,z,j) +{\mathbb {E}}_{t,x,\ell ,z,j}\Bigl [ \int _t^T\Bigl ({\mathcal {L}}^{\pi ^*}\psi (s,X(s),\ell (s),Z(s),\alpha (s),j')\\&\quad +{\mathcal {L}}^{\pi ^*}(\Gamma \circ \varphi )(s,X(s),\ell (s),Z(s),\alpha (s))\Bigl )ds\Bigl ]. \end{aligned}$$

Similarly,

$$\begin{aligned}&{\mathbb {E}}_{t,x,\ell ,z,j}\Bigl [ \int _t^T{\mathcal {L}}^{\pi ^*}\psi ( s,X(s),\ell (s),Z(s),\alpha (s),j')ds\Bigl ] \\& \quad ={} {\mathbb {E}}_{t,x,\ell ,z,j}\Bigl [ \psi (T,X(T),\ell (T),Z(T),\alpha (T),j')\Bigl ] -\psi (t,x,\ell ,z,j,j') \end{aligned}$$

and

$$\begin{aligned}&{\mathbb {E}}_{t,x,\ell ,z,j}\Bigl [ \int _t^T{\mathcal {L}}^{\pi ^*}(\Gamma \circ \varphi )(s,X(s),\ell (s),Z(s),\alpha (s))ds\Bigl ]\\=\, & {} {\mathbb {E}}_{t,x,\ell ,z,j}\Bigl [ \Gamma (T,\varphi (T,X(T),\ell (T),Z(T),\alpha (T)),\alpha (t)\Bigl ] - \Gamma (t,\varphi (t,x,\ell ,z,j),j)\,. \end{aligned}$$

Then, using the above relations, the boundary conditions and (3.4), we can easily conclude that

$$\begin{aligned} \Psi (t,x,\ell ,z,j)= \,& {} \psi (t,x,\ell ,z,j,j') +\Gamma (t,\varphi (t,x,\ell ,z,j),j) \nonumber \\=\, & {} {\mathbb {E}}_{t,x,\ell ,z,j}[F(X^{\pi ^*}(T),\alpha (T)]+{\mathbb {E}}_{t,x,\ell ,z,j}[X^{\pi ^*}(T)] \nonumber \\= \,& {} \Phi (t,x,\ell ,z,j,\pi ^*)\,. \end{aligned}$$

(4.2)

In order to show that $\pi ^*$ is indeed an equilibrium control strategy, we construct, for any $\epsilon >0$ and $\pi \in {\mathcal {A}}$, the control strategy $\pi _{\epsilon }$ defined in Definition 2.2. For any $s\in [t,t+\epsilon ]$ we have: (See Lemma 2.2, [5])

$$\begin{aligned}&\Phi (t,x,\ell ,z,j,\pi ) \nonumber \\& \quad= {} {\mathbb {E}}_{t,x,\ell ,z,j}[\Psi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,\alpha )] \nonumber \\&\quad\quad-\Bigl \{{\mathbb {E}}_{t,x,\ell ,z,j}[\psi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ,j')] -{\mathbb {E}}_{t,x,\ell ,z,j}[\psi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ,j')]\Bigl \} \nonumber \\&-\Bigl \{{\mathbb {E}}_{t,x,\ell ,z,j}[\Gamma (t+\epsilon ,\varphi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ),\alpha )] \nonumber \\&\quad\quad -\Gamma (t+\epsilon ,{\mathbb {E}}_{t,x,\ell ,z,j}[\varphi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ),\alpha ])\Bigl \}\,. \end{aligned}$$

(4.3)

Moreover, for all $\pi \in {\mathcal {A}}$ and (3.1), we have

$$\begin{aligned}&{\mathcal {L}}^{\pi }\Psi (t,x,\ell ,z,j)-{\mathcal {L}}^\pi \psi (t,x,\ell ,z,j,t,j)+{\mathcal {L}}^\pi \psi ^{t,j}(t,x,\ell ,z,j) -{\mathcal {L}}^\pi (\Gamma \circ \varphi )(t,x,\ell ,z,j) \\&\ \ \ \ \ \ \ + {\mathcal {M}}^\pi \varphi (t,x,\ell ,z,j)\,\le \,0\,. \end{aligned}$$

Discretizing the above expression, we have

$$\begin{aligned}&{\mathbb {E}}_{t,x,\ell ,z,j}[\Psi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha )]- \Psi (t,x,\ell ,z,j) -\Bigl \{{\mathbb {E}}_{t,x,\ell ,z,j}[\psi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ,j')] \\&- \psi (t,x,\ell ,z,j,j')\Bigl \} +{\mathbb {E}}_{t,x,\ell ,z,j}[\psi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ,j')] - \psi (t,x,\ell ,z,j,j') \\&-{\mathbb {E}}_{t,x,\ell ,z,j}[\Gamma (t+\epsilon ,\varphi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ),\alpha )] + \Gamma (t,\varphi (t,x,\ell ,z,j),j) \\&+\Gamma (t+\epsilon ,{\mathbb {E}}_{t,x,\ell ,z,j}[\varphi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ),\alpha ])-\Gamma (t,\varphi (t,x,\ell ,z,j),j)\,\le o(\epsilon )\,. \end{aligned}$$

Hence

$$\begin{aligned}&\Psi (t,x,\ell ,z,j) \\ &\quad\ge {} {\mathbb {E}}_{t,x,\ell ,z,j}[\Psi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha )] -{\mathbb {E}}_{t,x,\ell ,z,j}[\psi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ,j')] \\&\quad\quad+{\mathbb {E}}_{t,x,\ell ,z,j}[\psi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ,j')] -{\mathbb {E}}_{t,x,\ell ,z,j}[\Gamma (t+\epsilon ,\varphi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ),\alpha )] \\&\quad\quad+\Gamma (t+\epsilon ,{\mathbb {E}}_{t,x,\ell ,z,j}[\varphi (t+\epsilon ,X^{\pi }_{t+\epsilon },\ell ,Z,\alpha ),\alpha ]) + o(\epsilon )\,. \end{aligned}$$

Therefore, from (4.2) and (4.3), we obtain

$$\begin{aligned}\Phi (t,x,\ell ,z,j,\pi ^*)-\Phi (t,x,\ell ,z,j,\pi _{\epsilon })\ge o(\epsilon )\,,\end{aligned}$$

that is,

$$\begin{aligned}\lim _{\epsilon \rightarrow 0}\inf \frac{\Phi (t,x,\ell ,z,j,\pi ^*)-\Phi (t,x,\ell ,z,j,\pi _{\epsilon })}{\epsilon }\ge 0\,,\end{aligned}$$

which completes the proof.