On the Controller-Stopper Problems with Controlled Jumps

Abstract

We analyze the continuous time zero-sum and cooperative controller-stopper games of Karatzas and Sudderth (Ann Probab 29(3):1111–1127, 2001), Karatzas and Zamfirescu (Ann Probab 36(4):1495–1527, 2008) and Karatzas and Zamfirescu (Appl Math Optim 53(2):163–184, 2006) when the volatility of the state process is controlled as in Bayraktar and Huang (SIAM J Control Optim 51(2):1263–1297, 2013) but additionally when the state process has controlled jumps. We perform this analysis by first resolving the stochastic target problems [of Soner and Touzi (SIAM J Control Optim 41(2):404–424, 2002; J Eur Math Soc 4(3):201–236, 2002)] with a cooperative or a non-cooperative stopper and then embedding the original problem into the latter set-up. Unlike in Bayraktar and Huang (SIAM J Control Optim 51(2):1263–1297, 2013) our analysis relies crucially on the Stochastic Perron method of Bayraktar and Sîrbu (SIAM J Control Optim 51(6):4274–4294, 2013) but not the dynamic programming principle, which is difficult to prove directly for games.

Appendices

Appendix A

We provide sufficient conditions for the nonemptiness of $\mathbb {U}^+_{\text {unco}}$, $\mathbb {U}^-_{\text {unco}}$, $\mathbb {U}^{+}_{\text {co}}$ and $\mathbb {U}^{-}_{\text {co}}$.

Assumption A.1

g is bounded.

Assumption A.2

There exists $u_0 \in U$ such that $\sigma _Y(t,x,y,u_0)=0$ and $b(t,x,y,u_0(e),e)=0$ for all $(t,x,y,e)\in \mathbb {D}\times \mathbb {R}\times E$.

Proposition A.1

Under Assumptions 2.1, 2.2, A.1 and A.2, $\mathbb {U}^{+}_{\text {unco}}$ and $\mathbb {U}^{-}_{\text {co}}$ are not empty.

Proof

We will only show $\mathbb {U}^{+}_{\text {unco}}$ is not empty. A very similar proof applies to $\mathbb {U}^{-}_{\text {co}}$.

Step 1 In this step we assume that $\mu _{Y}$ is non-decreasing in its y-variable. We will show that $w(t,x)=\gamma -e^{kt}$ is a stochastic super-solution for some choice of k and $\gamma$.

By the linear growth condition on $\mu _Y$ in Assumption 2.2, there exists $L>0$ such that

$$\begin{aligned} |\mu _Y(t,x,y,u_0)|\le L(1+|y|), \end{aligned}$$

where $u_0$ is the element in U in Assumption A.2. Choose $k\ge 2L$ and $\gamma$ such that $-\,e^{k T}+\gamma \ge \Vert g\Vert _{\infty }$. Then $w(t,x)\ge w(T,x)\ge g(x)$ for all $(t,x)\in \mathbb {D}$. It suffices to show that for any $(t,x,y)\in \mathbb {D}\times \mathbb {R}$, $\tau \in \mathcal {T}_t$, $\nu \in \mathcal {U}^t_{\text {unco}}$ and $\rho \in \mathcal {T}_{\tau }$,

$$\begin{aligned} Y(\rho )\ge w(\rho , X(\rho )) \;\; \mathbb {P}\text {-a.s.}\;\; \text {on}\;\;\{Y(\tau )\ge w(\tau , X(\tau ))\}, \end{aligned}$$

(A.4)

where $X:= X_{t,x}^{\nu \otimes _{\tau }u_0}, Y:=Y_{t,x,y}^{\nu \otimes _{\tau }u_0}$.

Let $A= \{Y(\tau )> w(\tau , X(\tau ))\}$, $V(s)=w(s,X(s))$ and $\Gamma (s)=\left( V(s)-Y(s)\right) \mathbb {1}_{A}.$ Therefore, for $s\ge \tau$,

$$\begin{aligned} dY(s)&= \mu _{Y}\left( s,X(s),Y(s), u_0\right) ds, \;dV(s) \nonumber \\&= -ke^{ks}ds, \; \Gamma (s)=\mathbb {1}_{A}\int _{\tau }^{s} ( \xi (q)+ \Delta (q) ) dq + \mathbb {1}_{A}\Gamma (\tau ) , \text {where} \nonumber \\ \Delta (s)&:=-ke^{ks}-\mu _Y(s,X(s),V(s),u_0)\le -ke^{ks}-\mu _Y(s,X(s),-e^{ks},u_0) \nonumber \\&\le -ke^{ks}+L(1+e^{ks})\le 0, \nonumber \\ \xi (s)&:=\mu _{Y}(s,X(s),V(s),u_0) -\mu _{Y}(s,X(s),Y(s),u_0). \end{aligned}$$

(A.5)

Therefore, from (A.5) and the definitions of $\Gamma$ and A, it holds that

$$\begin{aligned} \Gamma (s)\le \mathbb {1}_A\int _{\tau }^{s} \xi (q) dq \;\; \text {and}\;\; \Gamma ^{+}(s)\le \mathbb {1}_A\int _{\tau }^{s} \xi ^{+}(q) dq \;\; \text {for} \;\; s\ge \tau . \end{aligned}$$

From the Lipschitz continuity of $\mu _Y$ in y-variable in Assumption 2.2,

$$\begin{aligned} \Gamma ^{+}(s)\le \mathbb {1}_A \int _{\tau }^{s} \xi ^{+}(q) dq \le \int _{\tau }^{s} L_0 \Gamma ^{+}(q) dq \;\; \text {for} \;\; s\ge \tau , \end{aligned}$$

where $L_0$ is the Lipschitz constant of $\mu _Y$ with respect to y. Note that we use the assumption that $\mu _Y$ is non-decreasing in its y-variable to obtain the second inequality. Since $\Gamma ^+(\tau )=0$, an application of Grönwall’s Inequality implies that $\Gamma ^+(\rho )\le 0$, which further implies that (A.4) holds.

Step 2 We get rid of our assumption on $\mu _{Y}$ from Step 1 by following a proof similar to those in [4] and [19]. For $c>0$, define $\widetilde{Y}_{t,x,y}^{\nu }$ as the strong solution of

$$\begin{aligned} \begin{aligned} d\widetilde{Y}(s)&=\tilde{\mu }_{Y}(s,X_{t,x}^{\nu }(s),\widetilde{Y}(s),\nu (s)) ds +\tilde{\sigma }_{Y}^{\top }(s,X_{t,x}^{\nu }(s),\widetilde{Y}(s),\nu (s))dW_{s} \\&\quad + \int _{E} \widetilde{b}^{\top }(s,X_{t,x}^{\nu }(s-),\widetilde{Y}(s-), \nu _1(s),\nu _2(s,e), e)\lambda (ds,de) \end{aligned} \end{aligned}$$

with initial data $\widetilde{Y}(t)=y$, where

$$\begin{aligned} \widetilde{\mu }_{Y}(t,x,y,u):= & {} c y+e^{ct} \mu _{Y}(t,x,e^{-c t}y,u), \; \widetilde{\sigma }_{Y}(t,x,y,u)\\:= & {} e^{c t} \sigma _{Y}(t,x,e^{-c t} y,u), \\ \widetilde{b}(t,x,y,u(e),e):= & {} e^{c t} b(t,x,e^{-c t} y,u(e),e). \end{aligned}$$

Therefore,

$$\begin{aligned} \widetilde{Y}_{t,x,y}^{\nu }(s)e^{-cs}=Y_{t,x,ye^{-ct}}^{\nu }(s), \;t\le s\le T. \end{aligned}$$

(A.6)

Let

$$\begin{aligned} \tilde{u}_{\text {unco}}(t,x)= \inf \{y\in \mathbb {R}: \exists \; \nu \in \mathcal {U}^t_{\text {unco}}, \text{ s.t. }\; \widetilde{Y}^{\nu }_{t,x,y}(\rho )\ge \tilde{g}(\rho , X^{\nu }_{t,x}(\rho ))\;\text{-a.s. }\}, \end{aligned}$$

(A.7)

where $\tilde{g}(t, x)=e^{ct} g(x)$. Therefore, from (A.6), $\tilde{u}_{\text {unco}}(t,x)=e^{ct}u_{\text {unco}}(t,x).$ Since $\mu _{Y}$ is Lipschitz in y, we can choose $c>0$ so that $\widetilde{\mu }_{Y}: (t,x,y,u) \mapsto cy + e^{c t}\mu _{Y}(t,x,e^{-c t}y,u)$ is non-decreasing in y. Moreover, all the properties of $\widetilde{\mu }_{Y}, \widetilde{\sigma }_{Y}$ and $\widetilde{b}$ in Assumption 2.2 still hold. We replace $\mu _Y$, $\sigma _Y$ and b in all of the equations and definitions in Sect. 2 with $\widetilde{\mu }_{Y}, \widetilde{\sigma }_{Y}$ and $\widetilde{b}$, we get $\widetilde{H}^*$ and $\widetilde{H}_*$. Let $\widetilde{\mathbb {U}}^+_{\text {unco}}$ be the set of stochastic super-solutions of the new target problem (A.7). It is easy to see that $w\in \mathbb {U}^+_{\text {unco}}$ if and only if $\widetilde{w}(t,x):=e^{ct}w(t,x)\in \widetilde{\mathbb {U}}^+_{\text {unco}}$. From Step 1, $\widetilde{\mathbb {U}}^+_{\text {unco}}$ is not empty. Thus, $\mathbb {U}^+_{\text {unco}}$ is not empty. $\square$

Assumption A.3

There is $C\in \mathbb {R}$ such that for all $(t,x,y,u,e)\in \mathbb {D}\times \mathbb {R}\times U\times E$,

$$\begin{aligned} \left| \mu _Y(t,x,y,u)+\int _E b^{\top }(t,x,y,u(e),e) m(de)\right| \le C(1+|y|). \end{aligned}$$

Proposition A.2

Under Assumptions 2.1, 2.2, A.1 and A.3, $\mathbb {U}_{\text {unco}}^{-}$ and $\mathbb {U}^{+}_{\text {co}}$ are not empty.

Proof

We will only show that $\mathbb {U}_{\text {unco}}^{-}$ is not empty. Assume that

$$\begin{aligned} \mu _{Y}(t,x,y,u)+\int _E b^{\top }(t,x,y,u(e),e)m(de) \end{aligned}$$

is non-decreasing in its y-variable. We could remove this assumption by using the argument from previous proposition.

Choose $k\ge 2C$ (C is the constant in Assumption A.3) and $\gamma >0$ such that $e^{k T}-\gamma <- \Vert g\Vert _{\infty }$. Let $w(t,x)=e^{kt}-\gamma$. Notice that w is continuous, has polynomial growth in x and $w(T,x)\le g(x)$ for all $x\in \mathbb {R}^{d}$. It suffices to show that for any $(t,x,y)\in \mathbb {D}\times \mathbb {R}$, $\tau \in \mathcal {T}_{t}$ and $\nu \in \mathcal {U}^t_{\text {unco}}$, there exists $\rho \in \mathcal {T}_{t}$ such that $\mathbb {P}(Y(\rho )< g( X(\rho ))|B)>0$ for $B\subset \{Y(\tau )<w(\tau ,X(\tau ))\}$ satisfying $B\in \mathcal {F}_\tau ^t$ and $\mathbb {P}(B)>0$, where $X:= X_{t,x}^{\nu }$ and $Y:=Y_{t,x,y}^{\nu }$. Define

$$\begin{aligned} \begin{aligned} M(\cdot )&=Y(\cdot )-\int _{\tau }^{\cdot }K(s)ds,\; V(s)=w(s,X(s)), \\ A&= \{Y(\tau )<w(\tau , X(\tau ))\},\; \Gamma (s)=\left( Y(s)-V(s)\right) \mathbb {1}_{A}, \\ \text {where } K(s)&:=\mu _{Y}(s,X(s),Y(s),\nu (s)) \\&\quad +\int _{E}b^{\top }(s,X(s-),Y(s-),\nu _1(s), \nu _2(s,e),e)m(de),\\ \widetilde{K}(s)&:=\mu _{Y}(s,X(s),V(s),\nu (s)) \\&\quad +\int _{E}b^{\top }(s,X(s-), V(s-),\nu _1(s),\nu _2(s,e),e)m(de). \end{aligned} \end{aligned}$$

It is easy to see that M is a martingale after $\tau .$ Due to the facts that $A\in \mathcal {F}_\tau ^t$ and $dV(s)= ke^{ks}ds$, we further know

$$\begin{aligned} \mathbb {1}_{A}\left( Y(\cdot )-V(\cdot )+\int _{\tau }^{\cdot } ke^{ks}-K(s) ds \right) \;\; \text {is a martingale after}\;\;\tau . \end{aligned}$$

(A.8)

Since Assumption A.3 holds and $\mu _{Y}(t,x,y,u)+\int _E b^{\top }(t,x,y,u(e),e)m(de)$ is non-decreasing in y,

$$\begin{aligned} \widetilde{K}(s)\le & {} \mu _Y(s,X(s),e^{ks}, \nu (s))+\int _{E}b^{\top }(s,X(s-),e^{ks},\nu _1(s),\nu _2(s,e),e)m(de)\\\le & {} 2C e^{ks}. \end{aligned}$$

Therefore, it follows from (A.8), the inequality above and the fact $k\ge 2C$ that

$$\begin{aligned} \widetilde{M}(\cdot ):=\mathbb {1}_{A}\left( Y(\cdot )-V(\cdot )-\int _{\tau } ^{\cdot }\xi (s)ds)\right) \;\;\text {is a super-martingale after }\tau , \end{aligned}$$

(A.9)

where $\xi (s):=K(s)-\widetilde{K}(s)$. Since $\widetilde{M}(\tau )<0$ on B, there exists a non-null set $F\subset B$ such that $\widetilde{M}(\rho )<0$ on F for any $\rho \in \mathcal {T}_{\tau }$. By the definition of $\widetilde{M}$ in (A.9), we get

$$\begin{aligned} \Gamma (\rho )< \mathbb {1}_{A}\int _{\tau }^{\rho }\xi (s)ds \;\;\text {on}\;\;F. \end{aligned}$$

(A.10)

Therefore,

$$\begin{aligned} \Gamma ^{+}(\rho )\le \mathbb {1}_A\int _{\tau }^{\rho } \xi ^{+}(s) ds \le \int _{\tau }^{\rho } L_0 \Gamma ^{+}(s) ds\;\;\text {on}\;\;F. \end{aligned}$$

(A.11)

By Grönwall’s Inequality, $\Gamma ^+(\tau )=0$ implies that $\Gamma ^+(\rho )=0$ on F. More precisely, for $\omega \in F$ ($\mathbb {P}-\text {a.s.}$), $\Gamma ^{+}(s)(\omega )=0$ for $s\in [\tau (\omega ),\rho (\omega )]$. This implies that we can replace the inequalities with equalities in (A.11). Therefore, by (A.10), $\Gamma (\rho )<0$ on F, which yields $\mathbb {P}(Y(\rho )< g( X(\rho ))|B)>0.$$\square$

Appendix B

Let T be a finite time horizon, given a general probability space $(\Omega , \mathcal {F},\mathbb {P})$ endowed with a filtration $\mathbb {F} = \{\mathcal {F}_t\}_{0\le t \le T}$ satisfying the usual conditions. Let $\mathcal {T}_t$ be the set of $\mathbb {F}$-stopping times valued in [t, T]. In particular, let $\mathcal {T}:=\mathcal {T}_0$. We assume that $\mathcal {F}_0$ is trivial. Let us consider an optimal control problem defined as follows. Let $\mathcal {U}$ be the collection of all $\mathbb {F}$-predictable processes valued in $U\subset \mathbb {R}^k$ and $\{G^{\nu },\nu \in \mathcal {U}\}$ be a collection of bounded, right-continuous processes valued in $\mathbb {R}$. Given $(t,\nu )\in [0,T]\times \mathcal {U}$, we consider two optimal stopping control problems:

$$\begin{aligned} V_{\text {unco}}^{\nu }(t)=\mathrm{ess}\!\inf \limits _{\mu \in \mathcal {U}(t,\nu )} \mathrm{ess}\!\sup \limits _{\tau \in \mathcal {T}_t}\mathbb {E}[G^{\mu }(\tau )|\mathcal {F}_t], \end{aligned}$$

(B.12)

and

$$\begin{aligned} V_{\text {co}}^{\nu }(t)=\mathrm{ess}\!\sup \limits _{\mu \in \mathcal {U}(t,\nu )} \mathrm{ess}\!\sup \limits _{\tau \in \mathcal {T}_t}\mathbb {E}[G^{\mu }(\tau )|\mathcal {F}_t], \end{aligned}$$

(B.13)

where $\mathcal {U}(t,\nu )=\{\mu \in \mathcal {U}, \mu = \nu \;\text {on}\;[0,t]\;\;\mathbb {P}-\text {a.s.}\}$.

Lemma B.2

Given $t\in [0,T]$ and $\nu \in \mathcal {U}_{t}$, let $\mathcal {M}$ be any family of martingales which satisfies the following:

$$\begin{aligned} \begin{array}{c} \text {For any}\;\; \mu \in \mathcal {U}(t,\nu ),\;\text {there exists an}\;\; M\in \mathcal {M} \;\text {such that}\;\; \\ \mathrm{ess}\!\sup \limits _{\tau \in \mathcal {T}_t}\mathbb {E}[G^{\mu }(\tau )|\mathcal {F}_t] + M(\rho )-M(t)\ge G^{\mu }(\rho )\;\;\text {for all }\rho \in \mathcal {T}_t. \end{array} \end{aligned}$$

(B.14)

Then $V_{\text {unco}}^{\nu }(t)=Y_{\text {unco}}^{\nu }(t),$ where

$$\begin{aligned} \begin{aligned} Y_{\text {unco}}^{\nu }(t)&=\mathrm{ess}\!\inf \limits \Big \{ Y\in L^1(\Omega , \mathcal {F}_t, \mathbb {P})\;|\; \exists (M,\mu )\in \mathcal {M}\times \mathcal {U}(t,\nu ), \text {s.t.}\; Y\\&\quad +\,M(\rho )-M(t)\ge G^{\mu }(\rho )\;\;\text {for all }\rho \in \mathcal {T}_t \Big \}. \end{aligned} \end{aligned}$$

Proof

(1) $Y_{\text {unco}}^{\nu }(t)\ge V_{\text {unco}}^{\nu }(t)$: Fix $Y\in L^1(\Omega , \mathcal {F}_t, \mathbb {P})$ and $(M,\mu )\in \mathcal {M}\times \mathcal {U}(t,\nu )$ such that

$$\begin{aligned} Y+M(\rho )-M(t)\ge G^{\mu }(\rho ) \;\;\text {for all }\rho \in \mathcal {T}_t. \end{aligned}$$

By taking the conditional expectation, we get that

$$\begin{aligned} Y \ge \mathbb {E}[G^{\mu }(\rho )|\mathcal {F}_t]\;\;\text {for all }\rho \in \mathcal {T}_t. \end{aligned}$$

which implies that $Y\ge V_{\text {unco}}^{\nu }(t)$. Therefore, $Y_{\text {unco}}^{\nu }(t)\ge V_{\text {unco}}^{\nu }(t)$.

(2) $V_{\text {unco}}^{\nu }(t)\ge Y_{\text {unco}}^{\nu }(t)$: we get from (B.14), for each $\mu \in \mathcal {U}(t,\nu )$, there exists an $M\in \mathcal {M}$ such that

$$\begin{aligned} \mathrm{ess}\!\sup \limits _{\tau \in \mathcal {T}_t}\mathbb {E}[G^{\mu }(\tau )|\mathcal {F}_t] +M(\rho )-M(t)\ge G^{\mu }(\rho )\text { for all }\rho \in \mathcal {T}. \end{aligned}$$

This implies that

$$\begin{aligned} \mathrm{ess}\!\sup \limits _{\tau \in \mathcal {T}_t}\mathbb {E}[G^{\mu }(\tau )|\mathcal {F}_t]\ge Y_{\text {unco}}^{\nu }(t), \end{aligned}$$

which further implies $V_{\text {unco}}^{\nu }(t)\ge Y_{\text {unco}}^{\nu }(t)$. $\square$

Lemma B.3

Let $\mathcal {M}$ be any family of martingales which satisfies the following:

$$\begin{aligned} \text {For any}\;\; \nu \in \mathcal {U} \;\text {and}\; \rho \in \mathcal {T},\;\text {there exists an}\;\; M\in \mathcal {M} \;\text {such that}\;\; G^{\nu }(\rho ) = M(\rho ). \end{aligned}$$

(B.15)

Then for each $(t,\nu )\in [0,T]\times \mathcal {U}$, $V_{\text {co}}^{\nu }(t)=Y_{\text {co}}^{\nu }(t),$ where

$$\begin{aligned} \begin{aligned} Y_{\text {co}}^{\nu }(t)&=\mathrm{ess}\!\sup \limits \Big \{ Y\in L^1(\Omega , \mathcal {F}_t, \mathbb {P})\;| \exists (M,\mu ,\rho )\in \mathcal {M}\times \mathcal {U}(t,\nu )\times \mathcal {T}_t, \text {s.t.}\;\;Y\\&\quad +\,M(\rho )-M(t)\le G^{\mu }(\rho ) \Big \}. \end{aligned} \end{aligned}$$

Proof

(1) $Y_{\text {co}}^{\nu }(t)\le V_{\text {co}}^{\nu }(t)$: Fix $Y\in L^1(\Omega , \mathcal {F}_t, \mathbb {P})$ and $(M,\mu ,\rho )\in \mathcal {M}\times \mathcal {U}(t,\nu )\times \mathcal {T}_t$ such that

$$\begin{aligned} Y+M(\rho )-M(t)\le G^{\mu }(\rho ). \end{aligned}$$

Then by taking the conditional expectation, we get that

$$\begin{aligned} Y \le \mathbb {E}[G^{\mu }(\rho )|\mathcal {F}_t]\le V_{\text {co}}^{\nu }(t), \end{aligned}$$

which implies that $Y_{\text {co}}^{\nu }(t)\le V_{\text {co}}^{\nu }(t)$.

(2) $Y_{\text {co}}^{\nu }(t)\ge V_{\text {co}}^{\nu }(t)$: we get from (B.15), for each $\mu \in \mathcal {U}(t,\nu )$ and $\rho \in \mathcal {T}_t$, there exists an $M\in \mathcal {M}$ such that

$$\begin{aligned} \mathbb {E}[G^{\mu }(\rho )|\mathcal {F}_t]+M(\rho )-M(t)=G^{\mu }(\rho ). \end{aligned}$$

In particular,

$$\begin{aligned} \mathbb {E}[G^{\mu }(\rho )|\mathcal {F}_t]+M(\rho )-M(t)\le G^{\mu }(\rho ). \end{aligned}$$

Therefore, $\mathbb {E}[G^{\mu }(\rho )|\mathcal {F}_t]\le Y_{\text {co}}^{\nu }(t)$, which implies $V_{\text {co}}^{\nu }(t)\le Y_{\text {co}}^{\nu }(t)$. $\square$

Remark B.3

It is clear that a collection of martingales which satisfies (B.15) always exists. In particular, one can take

$$\begin{aligned} \mathcal {M}_{\text {co}}=\{\{\mathbb {E}[G^{\nu }(\rho )|\mathcal {F}_t]\}_{0\le t\le T}, \nu \in \mathcal {U}, \rho \in \mathcal {T}\}. \end{aligned}$$