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Stochastic Perron for Stochastic Target Problems

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Abstract

In this paper, we adapt stochastic Perron’s method to analyze stochastic target problems in a jump diffusion setup, where the controls are unbounded. Since classical control problems can be analyzed under the framework of stochastic target problems (with unbounded controls), we use our results to generalize the results of Bayraktar and Sîrbu (SIAM J Control Optim 51(6):4274–4294, 2013) to problems with controlled jumps.

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Notes

  1. The bound may depend on the process.

  2. The convergence \(\psi \overset{\text {u.c.}}{\longrightarrow } \varphi \) is understood in the sense that \(\psi \) converges uniformly on compact subsets to \(\varphi \).

  3. This can be easily checked.

  4. C and N may depend on w and T. This also applies to Definition 2.3

  5. If this does not hold, the integral is less than \(-K\) with positive probability. Noticing this, we can carry out the proof in a similar manner when this assumption does not hold.

  6. Since we will fix \(n_{0}\) and \(\iota \) later, we still use the notation \({\tilde{\varphi }}\) when without ambiguity despite the fact that the function depends on \(n_{0}\) and \(\iota \).

  7. Here we choose \((t,x)\in \mathbb {D}_{i}\) since the case \((t,x)\in \mathbb {D}_{T}\) is trivial.

  8. The control \(\nu _0\) can be fixed outside the time interval \([\tau ,\theta )\), since we are only interested in the restriction of it to \([\tau ,\theta )\).

  9. The existence of \(n_{0}\) follows as in Step1.

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Acknowledgments

We would like to thank Bruno Bouchard who encouraged us to write this paper and for his constructive comments on its first version. We also thank the referees and the anonymous associate editor for their helpful comments, which helped us to improve our paper. This research is supported by the National Science Foundation under grant DMS-1613170.

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Correspondence to Erhan Bayraktar.

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Communicated by Lars Grüne.

Appendices

Appendix A

We provide sufficient conditions for the non-emptiness of \(\mathbb {U}^+\) and \(\mathbb {U}^-\).

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 \((t,x,y,e)\in \mathbb {D}\times {\mathbb {R}}\times E\).

Remark A.1

In the context of super-hedging in mathematical finance, the assumption above is equivalent to restricting trading to the riskless assets.

Proposition A.1

Under Assumptions 2.1, 2.2, A.1 and A.2, \(\mathbb {U}^{+}\) is not empty.

Proof

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 \(|\mu _Y(t,x,y,u_0)|\le L(1+|y|)\), 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 \(w(T,x)\ge g(x)\). It suffices to show that for any \((t,x,y)\in \mathbb {D}\times {\mathbb {R}}\), \(\tau \in {\mathcal {T}}_t\), \(\nu \in \mathcal {U}^t\) and \(\rho \in {\mathcal {T}}_{\tau }\),

$$\begin{aligned}&Y(\rho )\ge w(\rho , X(\rho )) \;\; {\mathbb {P}}\text {-a.s.}\;\; \text {on}\;\;\{Y(\tau )\ge w(\tau , X(\tau ))\},\nonumber \\&\qquad \text {where } X:= X_{t,x}^{\nu \otimes _{\tau }u_0}, Y:=Y_{t,x,y}^{\nu \otimes _{\tau }u_0}. \end{aligned}$$
(35)

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

$$\begin{aligned} dY(s)= & {} \mu _{Y}\left( s,X(s),Y(s), u_0\right) ds, \;dV(s)= -ke^{ks}ds,\;\nonumber \\ \varGamma (s)= & {} {\mathbbm {1}}_{A}\int _{\tau }^{s} (\xi (q)+ \varDelta (q)) dq, \text {where} \\ \varDelta (s):= & {} -ke^{ks}-\mu _Y(s,X(s),Y(s),u_0)\le -ke^{ks}-\mu _Y(s,X(s),\nonumber \\- & {} e^{ks},u_0)\le -ke^{ks}+L(1+e^{ks})\le 0, \nonumber \\ \xi (s):= & {} \mu _{Y}(s,X(s),V(s),u_0) -\mu _{Y}(s,X(s),Y(s),u_0).\nonumber \end{aligned}$$
(36)

Therefore, from (36) it holds that

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

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

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

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 \(\varGamma ^+(\tau )=0\), an application of Grönwall’s Inequality implies that \(\varGamma ^+(\rho )\le 0\), which further implies that (35) holds.

Step 2 We get rid of our assumption on \(\mu _{Y}\) from Step 1 by following a proof similar to those in [12] and [15]. For \(c>0\), define \(\widetilde{Y}_{t,x,y}^{\nu }\) as the strong solution of

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

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,

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

Let \( {\tilde{u}}(t,x)= \inf \{y\in \mathbb {R}: \exists \; \nu \in {\mathcal {U}}^t, \text{ s.t. }\; \widetilde{Y}^{\nu }_{t,x,y}(T)\ge {\tilde{g}}(X^{\nu }_{t,x}(T))\;\text{-a.s. }\}, \) where \({\tilde{g}}(x)=e^{c T} g(x)\). Therefore, \({\tilde{u}}(t,x)=e^{ct}u(t,x).\) Since \(\mu _{Y}\) is Lipschitz in y, we can choose \(c>0\) so that

$$\begin{aligned} \widetilde{\mu }_{Y}: (t,x,y,u) \mapsto cy + e^{c t}\mu _{Y}(t,x,e^{-c t}y,u) \end{aligned}$$

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 \(\widetilde{\mathbb {U}}^+\) be the set of stochastic super-solutions of

$$\begin{aligned} -\partial _t\varphi (t,x)+\widetilde{H}^*\varphi (t,x)\ge 0\;\;\text {on}\;\;\mathbb {D}_{i}. \end{aligned}$$

It is easy to see that \(w\in \mathbb {U}^+\) if and only if \(\widetilde{w}(t,x):=e^{ct}w(t,x)\in \widetilde{\mathbb {U}}^+\). From Step 1, \(\widetilde{\mathbb {U}}^+\) is not empty. Thus, \(\mathbb {U}^+\) 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, \({\mathbb {U}}^{-}\) is not empty.

Proof

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 \(w(t,x)=e^{kx}-\gamma \). Notice that w is continuous, has polynomial growth in x and \(w(T,\cdot )\le g(\cdot )\). It suffices to show that for any \((t,x,y)\in \mathbb {D}\times {\mathbb {R}}\), \(\tau \in {\mathcal {T}}_{t}\) and \(\nu \in \mathcal {U}^t\), it holds that \({\mathbb {P}}(Y(\rho )< w(\rho , X(\rho ))|B)>0\) for all \(\rho \in {\mathcal {T}}_{\tau }\) and \(B\subset \{Y(\tau )<w(\tau ,X(\tau ))\}\) 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

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

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

$$\begin{aligned} {\mathbbm {1}}_{A}\left( Y(\cdot )-V(\cdot )+\int _{\tau }^{\cdot } ke^{ks}-K(s) ds \right) \;\; \text {is a super-martingale after}\;\;\tau . \end{aligned}$$
(37)

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 (37) and the inequality above that

$$\begin{aligned} \widetilde{M}(\cdot ):= & {} {\mathbbm {1}}_{A}\left( Y(\cdot )-V(\cdot )-\int _{\tau }^{\cdot }\xi (s)ds)\right) \;\text {is a super-martingale after }\tau , \text {where} \nonumber \\ \xi (s):= & {} K(s)-\widetilde{K}(s). \end{aligned}$$
(38)

Since \(\widetilde{M}(\tau )<0\) on B, there exists a non-null set \(F\subset B \) such that \(\widetilde{M}(\rho )<0\) on F. By the definition of \(\widetilde{M}\) in (38), we get

$$\begin{aligned} \varGamma (\rho )< {\mathbbm {1}}_{A}\int _{\tau }^{\rho }\xi (s)ds \;\;\text {on}\;\;F. \end{aligned}$$
(39)

Therefore,

$$\begin{aligned} \varGamma ^{+}(\rho )\le {\mathbbm {1}}_A\int _{\tau }^{\rho } \xi ^{+}(s) ds \le \int _{\tau }^{\rho } L_0 \varGamma ^{+}(s) ds\;\;\text {on}\;\;F. \end{aligned}$$
(40)

By Grönwall’s Inequality, \(\varGamma ^+(\tau )=0\) implies that \(\varGamma ^+(\rho )=0\) on F. More precisely, for \(\omega \in F\) (\(\mathbb {P}-\text {a.s.}\)), \(\varGamma ^{+}(s)(\omega )=0\) for \(s\in [\tau (\omega ),\rho (\omega )]\). This implies that we can replace the inequalities with equalities in (40). Therefore, by (39), \(\varGamma (\rho )<0\) on F, which yields \({\mathbb {P}}(Y(\rho )< w(\rho , X(\rho ))|B)>0.\) \(\square \)

Appendix B

Proof of Theorem 3.1

Step 1 (\(u^+\) is a viscosity sub-solution). Assume, on the contrary, that for some \((t_0,x_0)\in \mathbb {D}_{i}\) and \(\varphi \in C^{1,2}(\mathbb {D})\) satisfying \(0=(u^+-\varphi )(t_0,x_0)= \max _{ \mathbb {D}_{i}}(u^+-\varphi )\), we have

$$\begin{aligned} 4\eta :=-\partial _t\varphi (t_0,x_0)+H_*\varphi (t_0,x_0)>0. \end{aligned}$$
(41)

From Lemma 3.2, there exists a non-increasing sequence . Fix such a sequence \(\{w_{k}\}_{k=1}^{\infty }\) and an arbitrary stochastic sub-solution \(w_{-}\). Let \({\tilde{\varphi }}(t,x)=\varphi (t,x)+\iota |x-x_0|^{n_{0}}\).Footnote 6 We can choose \(n_{0}\ge 2\) such that for any \(\iota >0\),

$$\begin{aligned} \min _{0\le t\le T} ({\tilde{\varphi }}(t,x)-w_{1}(t,x))\rightarrow \infty \;\; \text {as} \;\; |x|\rightarrow \infty . \end{aligned}$$
(42)

We can do this because \(\varphi (t, x)\) is bounded from below by \(w_{-}\) (which has polynomial growth in x) and \(w_{1}\) has polynomial growth in x. Since \(({\mathcal {N}}_{\varepsilon ,\eta })_{\varepsilon \ge 0}\) is non-decreasing in \(\varepsilon \), we know

By (4) and (41), we can find \(\varepsilon >0\), \(\eta >0\) and \(\iota >0\) such that for all (txy) satisfying \((t,x)\in B_{\varepsilon }(t_0,x_0)\) and \(|y-{\tilde{\varphi }}(t,x)|\le \varepsilon \), \( \mu _Y(t,x,y,u)-{\mathcal {L}}^u{\tilde{\varphi }}(t,x)\ge 2\eta \text { for some }u\in {\mathcal {N}}_{0, \eta }(t,x,y,D{\tilde{\varphi }}(t,x),{\tilde{\varphi }}). \) Fix \(\iota \). Note that \((t_0,x_0)\) is still a strict maximizer of \(u^+-{\tilde{\varphi }}\) over \(\mathbb {D}_{i}\). For \(\varepsilon \) sufficiently small, Assumption 2.4 implies that there exists a locally Lipschitz map \({\hat{\nu }}\) such that

$$\begin{aligned}&{\hat{\nu }}(t,x,y,D{\tilde{\varphi }}(t,x))\in {\mathcal {N}}_{0,\eta }(t,x,y,D{\tilde{\varphi }}(t,x),{\tilde{\varphi }}) \text { and} \end{aligned}$$
(43)
$$\begin{aligned}&\mu _Y(t,x,y,{\hat{\nu }}(t,x,y,D{\tilde{\varphi }}(t,x)))-{\mathcal {L}}^{{\hat{\nu }}(t,x,y,D{\tilde{\varphi }}(t,x))}{\tilde{\varphi }}(t,x)\ge \eta \nonumber \\&\text {for all } (t,x,y)\in \mathbb {D}_{i}\times \mathbb {R}\;\text {s.t.}\;(t,x)\in B_{\varepsilon }(t_0,x_0) \text { and }|y-{\tilde{\varphi }}(t,x)|\le \varepsilon . \end{aligned}$$
(44)

In the arguments above, choose \(\varepsilon \) small enough such that \(\text {cl}(B_{\varepsilon }(t_0,x_0))\cap \mathbb {D}_{T}=\emptyset \). Since (42) holds, there exists \( R_0>\varepsilon \) such that \({\tilde{\varphi }}>w_1+\varepsilon \ge w_k+\varepsilon \) on \({\mathbb {O}}:=\mathbb {D}\setminus [0,T]\times \text {cl}(B_{R_0}(x_0))\;\text {for all } k.\) On the compact set \({\mathbb {T}}:= [0,T]\times \text {cl}(B_{R_0}(x_0))\setminus B_{\varepsilon /2}(t_0, x_0)\), we know that \({\tilde{\varphi }} >u^+\) and the minimum of \({\tilde{\varphi }} -u^+\) is attained since \(u^+\) is USC. Therefore, \({\tilde{\varphi }} >u^+ +2\alpha \) on \({\mathbb {T}}\) for some \(\alpha >0\). By a Dini-type argument, for large enough n, we have \({\tilde{\varphi }} >w_n+\alpha \) on \({\mathbb {T}}\) and \({\tilde{\varphi }}>w_n-\varepsilon \) on \(\text {cl}(B_{\varepsilon /2}(t_0, x_0))\). For simplicity, fix such an n and set \(w=w_n\). In short,

$$\begin{aligned} {\tilde{\varphi }}>w+\varepsilon \;\;\text {on}\;\; {\mathbb {O}},\; {\tilde{\varphi }}>w+\alpha \;\;\text {on}\;\;{\mathbb {T}} \;\;\text {and}\;\; {\tilde{\varphi }}>w-\varepsilon \;\;\text {on}\;\; \text {cl}(B_{\varepsilon /2}(t_0, x_0)).\qquad \end{aligned}$$
(45)

For \(\kappa \in \;]0,\varepsilon \wedge \alpha [\;\), define

$$\begin{aligned} w^{\kappa }:=\left\{ \begin{array}{ll} ({\tilde{\varphi }} -\kappa )\wedge w\ \ {\text {o}n}\ \ \text {cl}(B_{\varepsilon }(t_0, x_0)),\\ w \ \ \text {outside}\ \ \text {cl}(B_{\varepsilon }(t_0, x_0)). \end{array}\right. \end{aligned}$$

Observing that \(w^{\kappa }(t_0,x_0)={\tilde{\varphi }}(t_0,x_0)-\kappa <u^{+}(t_0,x_0)\), we could obtain a contradiction if we could show that \(w^{\kappa }\in {\mathbb {U}}^+\). Obviously, \(w^{\kappa }\) is continuous, has polynomial growth in x and \(w^{\kappa }(T,x)\ge g(x)\) for all \(x\in {\mathbb {R}}^d\). Fix \((t,x,y)\in \mathbb {D}_{i}\times {\mathbb {R}}\), \(\nu \in \mathcal {U}^t\) and \(\tau \in {\mathcal {T}}_t\).Footnote 7 Now our goal is to construct an admissible control \(\widetilde{\nu }\) such that \(w^{\kappa }\) and the processes (XY) controlled by \(\nu \otimes _{\tau }\widetilde{\nu }\) satisfy the property in the definition of stochastic super-solutions.

Let \(A=\{w^{\kappa }(\tau ,X_{t,x}^{\nu }(\tau ))= w(\tau ,X_{t,x}^{\nu }(\tau ))\}.\) On A, let \(\widetilde{\nu }\) be \(\widetilde{\nu }_1\), which is “optimal” for w starting at \(\tau \). We get the existence of \(\widetilde{\nu }_1\) since \(w\in {\mathbb {U}}^+\). On \(A^c\), by an argument similar to that in [12] (see Step 1.1 of Theorem 3.1’s proof), we can construct an admissible control \(\nu _0\in \mathcal {U}^t\) such that

$$\begin{aligned} \nu _0(s):= & {} {\hat{\nu }}\left( s,X^{\nu \otimes _{\tau }\nu _0}_{t,x}(s),Y^{\nu \otimes _{\tau }\nu _0}_{t,x,y}(s),D{\tilde{\varphi }}(s, X^{\nu \otimes _{\tau }\nu _0}_{t,x}(s)\right) \;\; \text {for}\;\; \tau \le s< \theta , \;\text {where}\;\\ \theta= & {} \theta _1\wedge \theta _2\;\; \text {and} \\ \theta _{1}:= & {} \inf \left\{ s \in [ \tau ,T]: (s, X_{t,x}^{\nu \otimes _{\tau } \nu _0 }(s)) \notin B_{\varepsilon /2}(t_0, x_0) \right\} \wedge T, \; \\ \theta _2:= & {} \inf \left\{ s \in [ \tau ,T]: \left| Y_{t,x,y}^{\nu \otimes _{\tau } \nu _0}(s)-{\tilde{\varphi }}(s,X_{t,x}^{\nu \otimes _{\tau } \nu _0}(s))\right| \ge \varepsilon \right\} \wedge T. \end{aligned}$$

Footnote 8In the construction of \(\nu _0\), we take advantage of Assumption 2.2 and the Lipschitz continuity of \({\hat{\nu }}\) which guarantee the existence of \(X^{\nu \otimes _{\tau }\nu _0}_{t,x}\) and \(Y^{\nu \otimes _{\tau }\nu _0}_{t,x,y}\). Since \(X^{\nu \otimes _{\tau }\nu _0}_{t,x}\) and \(Y_{t,x,y}^{\nu \otimes _{\tau }\nu _0}\) are càdlàg, it is easy to check that \(\theta \in {\mathcal {T}}_{\tau }\). We also see that

$$\begin{aligned} (\theta _1, X_{t,x}^{\nu \otimes _{\tau } \nu _0}(\theta _1))\notin B_{\varepsilon /2}(t_0, x_0),\;\; \left| Y_{t,x,y}^{\nu \otimes _{\tau } \nu _0}(\theta _2)-{\tilde{\varphi }}(\theta _2,X_{t,x}^{\nu \otimes _{\tau } \nu _0}(\theta _2))\right| \ge \varepsilon , \nonumber \end{aligned}$$
(46)
$$\begin{aligned} \\ (\theta _1, X_{t,x}^{\nu \otimes _{\tau } \nu _0}(\theta _1-))\in \text {cl}(B_{\varepsilon /2}(t_0, x_0)),\;\; \left| Y_{t,x,y}^{\nu \otimes _{\tau } \nu _0}(\theta _2-)-{\tilde{\varphi }}(\theta _2,X_{t,x}^{\nu \otimes _{\tau } \nu _0}(\theta _2-))\right| \le \varepsilon .\nonumber \\ \end{aligned}$$
(47)

Let \(\widetilde{\nu }^{\theta }\) be the “optimal” control for w starting at \(\theta \). We define \({\tilde{\nu }}\) on \(A^c\) by \(\nu _0\otimes _{\theta }\widetilde{\nu }^{\theta }\). In short,

$$\begin{aligned} \widetilde{\nu }:=\left( {\mathbbm {1}}_{A}\widetilde{\nu }_1+{\mathbbm {1}}_{A^c}(\nu _0{\mathbbm {1}}_{[t,\theta [}+{\mathbbm {1}}_{[\theta ,T]}\widetilde{\nu }^{\theta })\right) {\mathbbm {1}}_{[\tau ,T]}. \end{aligned}$$

It is not difficult to check that \(\widetilde{\nu }\in {\mathcal {U}}^t\). To prove that the above construction works, we next show that \(Y(\rho )\ge w^{\kappa }(\rho , X(\rho ))\;\; {\mathbb {P}}-\text {a.s.}\) on \(\{Y(\tau )\ge w^{\kappa }(\tau , X(\tau ))\},\) where \(X:= X_{t,x}^{\nu \otimes _{\tau }{\tilde{\nu }}}\) and \(Y:=Y_{t,x,y}^{\nu \otimes _{\tau }{\tilde{\nu }}}\). Corresponding to the construction of \(\widetilde{\nu }\) on A and \(A^{c}\), we consider the following two cases:

(i) On the set \(A\cap \{Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\}\). We have \( Y(\tau )\ge w(\tau ,X(\tau )). \) From the definition of \(\nu \) on A and the fact that \(w\in \mathbb {U}^{+}\), we know

$$\begin{aligned} Y(\rho )= & {} Y_{t,x,y}^{\nu \otimes _{\tau }\widetilde{\nu }_1}(\rho )\ge w(\rho ,X_{t,x}^{\nu \otimes _{\tau }\widetilde{\nu }_1}(\rho ))\ge w^{\kappa }(\rho ,X(\rho )) \; {\mathbb {P}}\\- & {} \text {a.s}\;\; \text {on}\;\;A\cap \{Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\}. \end{aligned}$$

(ii) On the set \(A^c \cap \{Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\}\). Letting \(\varGamma (s):=Y(s)-{\tilde{\varphi }}(s,X(s))\), we use Itô’s formula and the definition of \(\nu _0\) to obtain

$$\begin{aligned} \varGamma (\cdot \wedge \theta )= & {} \varGamma (\tau ) +\int _{\tau }^{\cdot \wedge \theta }\int _{E}\overline{J}^{\nu _0(s),e}\left( s,Z(s-),{\tilde{\varphi }}\right) ^{\top }\lambda (ds,de)\\+ & {} \int _{\tau }^{\cdot \wedge \theta }\left( \mu _Y(s, Z(s), \nu _0(s))-{\mathscr {L}}^{\nu _0(s)}{\tilde{\varphi }}(s, X(s))\right) ds \end{aligned}$$

on \(A\cap \{Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\}.\) Therefore, by (43), (44), (47) and the definition of \(\theta \), we know that \(\varGamma (\cdot \wedge \theta )\) is non-decreasing on \([\tau ,T]\). This implies that

$$\begin{aligned} Y(\theta )- & {} {\tilde{\varphi }}(\theta ,X(\theta ))+\kappa \ge Y(\tau )- {\tilde{\varphi }}(\tau ,X(\tau ))\nonumber \\+ & {} \kappa \ge 0\;\;\text {on}\;\;A^c \cap \left\{ Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\right\} . \end{aligned}$$
(48)

Since \((\theta _1, X(\theta _1))\notin B_{\varepsilon /2}(t_0, x_0)\), we know

$$\begin{aligned} 0\le & {} Y(\theta _1)- {\tilde{\varphi }}(\theta _1,X(\theta _1))+\kappa \le Y(\theta _1)\nonumber \\- & {} w(\theta _1,X(\theta _1)) \;\; \text {on} \;\; \{\theta _1\le \theta _2\}\cap A^c \cap \left\{ Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\right\} \end{aligned}$$
(49)

from (45). On the other hand, it holds that \(Y(\theta _2)-{\tilde{\varphi }}(\theta _2,X(\theta _2)) \ge \varepsilon \) on \( \{\theta _1>\theta _2\}\cap A^c \cap \{Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\}\) due to (46) and (48). Therefore, since \({\tilde{\varphi }}>w-\varepsilon \) on \(\text {cl}(B_{\varepsilon /2}(t_0, x_0))\) and (47) holds,

$$\begin{aligned} Y(\theta _2)- & {} w(\theta _2,X(\theta _2))\ge \varepsilon +{\tilde{\varphi }}(\theta _2,X(\theta _2))\nonumber \\- & {} w(\theta _2,X(\theta _2))>0 \;\text {on}\; \{\theta _1>\theta _2\}\cap A^c \cap \left\{ Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\right\} . \end{aligned}$$
(50)

Combining (49) and (50), we obtain \(Y(\theta )- w(\theta ,X(\theta ))\ge 0\) on \(A^c \cap \left\{ Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\right\} .\) Therefore, from the definition of \(\widetilde{\nu }^{\theta }\),

$$\begin{aligned} Y(\rho \vee \theta )- & {} w^{\kappa }(\rho \vee \theta ,X(\rho \vee \theta )) \ge Y(\rho \vee \theta ) \nonumber \\- & {} w(\rho \vee \theta ,X(\rho \vee \theta )) \ge 0\;\;\text {on}\;\; A^c \cap \left\{ Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\right\} . \end{aligned}$$
(51)

Also, the monotonicity of \( \varGamma (\cdot \wedge \theta )\) implies that \(Y(\rho \wedge \theta )-{\tilde{\varphi }}(\rho \wedge \theta ,X(\rho \wedge \theta ))+\kappa \ge 0\) on \(A^c \cap \left\{ Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\right\} .\) This means that

$$\begin{aligned} {\mathbbm {1}}_{\{\rho <\theta \}}\left( Y(\rho )-w^{\kappa }(\rho ,X(\rho ))\right) \ge 0\;\;\text {on}\;\; A^c \cap \left\{ Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\right\} . \end{aligned}$$
(52)

From (51) and (52), we get \(Y(\rho )-w^{\kappa }(\rho ,X(\rho ))\ge 0 \;\; \text {on} \,\, A^c \cap \left\{ Y(\tau )\ge w^{\kappa }(\tau ,X(\tau ))\right\} .\) Step 2 (\(u^-\) is a viscosity super-solution). Let \((t_0,x_0)\in \mathbb {D}_{i}\) satisfy \(0=(u^--\varphi )(t_0,x_0)= \min _{\mathbb {D}_{i}}(u^{-}-\varphi )\) for some \(\varphi \in C^{1,2}(\mathbb {D})\). For the sake of contradiction, assume that

$$\begin{aligned} -2\eta :=-\partial _t\varphi (t_0,x_0)+H^*\varphi (t_0,x_0)<0. \end{aligned}$$
(53)

Let \(\{w_{k}\}_{k=1}^{\infty }\) be a sequence in \(\mathbb {U}^{-}\) such that \( w_k\nearrow u^-\). Let \({\tilde{\varphi }}(t,x):=\varphi (t,x)-\iota |x-x_0|^{n_{0}}\), where we choose \(n_{0}\ge 2\) such that for all \(\iota >0\),

$$\begin{aligned} \max _{0\le t\le T}({\tilde{\varphi }}(t,x)-w_{1}(t,x))\rightarrow -\infty \;\; \text {and}\;\; \max _{0\le t\le T} {\tilde{\varphi }}(t,x)\rightarrow -\infty \;\;\text {as}\;\; |x|\rightarrow \infty . \nonumber \\ \end{aligned}$$
(54)

Footnote 9By (53), the upper semi-continuity of \(H^*\) and the fact that \({\tilde{\varphi }}\overset{\text {u.c.}}{\longrightarrow }\varphi \) as \(\iota \rightarrow 0\), we can find \(\varepsilon >0\), \(\eta >0\) and \(\iota >0\) such that

$$\begin{aligned} \begin{array}{c} \mu _Y(t,x,y,u)-{\mathscr {L}}^{u}{\tilde{\varphi }}(t,x)\le -\eta \;\text {for all }u\in {\mathcal {N}}_{\varepsilon , -\eta }(t,x,y,D{\tilde{\varphi }}(t,x),{\tilde{\varphi }}) \\ \text {and}\;\;(t,x,y)\in \mathbb {D}_{i}\times \mathbb {R}\;\text {s.t.}\;(t,x)\in B_{\varepsilon }(t_0,x_0)\;\text {and}\;|y-{\tilde{\varphi }}(t,x)|\le \varepsilon . \end{array} \end{aligned}$$
(55)

Fix \(\iota \). Note that \((t_0,x_0)\) is still a strict minimizer of \(u^--{\tilde{\varphi }}\). Since (54) holds, there exists \(R_0>\varepsilon \) such that

$$\begin{aligned} {\tilde{\varphi }}<w_1-\varepsilon \le w_k-\varepsilon \;\;\text {on}\;\; {\mathbb {O}}:=\mathbb {D}\setminus [0,T]\times \text {cl}(B_{R_0}(x_0)). \end{aligned}$$

On the compact set \({\mathbb {T}}:= [0,T]\times \text {cl}(B_{R_0}(x_0))\setminus B_{\varepsilon /2}(t_0, x_0)\), we know that \({\tilde{\varphi }} <u^-\) and the maximum of \({\tilde{\varphi }}-u^-\) is attained since \(u^-\) is LSC. Therefore, \({\tilde{\varphi }} <u^- -2\alpha \) on \({\mathbb {T}}\) for some \(\alpha >0\). By a Dini-type argument, for large enough n, we have \({\tilde{\varphi }} <w_n-\alpha \) on \({\mathbb {T}}\) and \({\tilde{\varphi }}<w_n+\varepsilon \) on \(\text {cl}(B_{\varepsilon /2}(t_0, x_0))\). For simplicity, fix such an n and set \(w=w_n\). In short,

$$\begin{aligned} {\tilde{\varphi }}<w-\varepsilon \;\;\text {on}\;\; {\mathbb {O}},\; {\tilde{\varphi }}<w-\alpha \;\;\text {on}\;\;{\mathbb {T}} \;\;\text {and}\;\; {\tilde{\varphi }}<w+\varepsilon \;\;\text {on}\;\; \text {cl}(B_{\varepsilon /2}(t_0, x_0)). \end{aligned}$$
(56)

For \(\kappa \in \;]0,\alpha \wedge \varepsilon [\;\), define

$$\begin{aligned} w^{\kappa }:=\left\{ \begin{array}{ll} ({\tilde{\varphi }} +\kappa )\vee w\ \ {\text {o}n}\ \ \text {cl}(B_{\varepsilon }(t_0, x_0)),\\ w \ \ \text {outside}\ \ \text {cl}(B_{\varepsilon }(t_0, x_0)). \end{array}\right. \end{aligned}$$

Noticing that \(w^{\kappa }(t_0,x_0)\ge {\tilde{\varphi }}(t_0,x_0)+\kappa >u^-(t_0,x_0)\), we will obtain a contradiction if we show that \(w^{\kappa }\in {\mathbb {U}}^-\). Obviously, \(w^{\kappa }\) is continuous, has polynomial growth in x and \(w^{\kappa }(T,x)\le g(x)\) for all \(x\in {\mathbb {R}}^d\). Fix \((t,x,y)\in \mathbb {D}_{i}\times {\mathbb {R}}\), \(\nu \in \mathcal {U}^t\) and \(\tau \in {\mathcal {T}}_t\). Our goal is to show that

$$\begin{aligned} {\mathbb {P}}(Y(\rho )< w^{\kappa }(\rho , X(\rho ))|B)>0 \end{aligned}$$

for all \(\rho \in {\mathcal {T}}_{\tau }\) and \(B\subset \{Y(\tau )<w^{\kappa }(\tau ,X(\tau ))\}\) satisfying \(B\in {\mathcal {F}}_\tau ^t\) and \(\mathbb {P}(B)>0\), where \(X:= X_{t,x}^{\nu }\) and \(Y:=Y_{t,x,y}^{\nu }.\) Let \(A=\{w^{\kappa }(\tau ,X(\tau ))=w(\tau ,X(\tau ))\}\) and set

$$\begin{aligned} E= & {} \left\{ Y(\tau )< w^{\kappa }(\tau ,X(\tau ))\right\} , \;\; E_0=E\cap A, \;\; E_1=E\cap A^c, \\ G= & {} \{Y(\rho )< w^{\kappa }(\rho ,X(\rho )\}, \;\; G_0=\left\{ Y(\rho )< w(\rho ,X(\rho ))\right\} . \end{aligned}$$

Then \(E=E_0\cup E_1,\; E_0\cap E_1=\emptyset \;\;\text {and}\;\; G_0\subset G. \) To prove that \(w^{\kappa }\in \mathbb {U}^{-}\), it suffices to show that \({\mathbb {P}}(G\cap B)>0\). As in [12] and [3], we will show \(\mathbb {P}(B\cap E_{0})>0\implies \mathbb {P}(G\cap B\cap E_{0})>0\) and \(\mathbb {P}(B\cap E_{1})>0\implies \mathbb {P}(G\cap B\cap E_{1})>0\). This, together with the facts \(\mathbb {P}(B)=\mathbb {P}(B\cap E_{0})+\mathbb {P}(B\cap E_{1})>0\) and \(\mathbb {P}(G\cap B)=\mathbb {P}(G\cap B\cap E_{0})+\mathbb {P}(G\cap B\cap E_{1})\), implies that \(\mathbb {P}(G\cap B)>0\).

(i) Assume that \({\mathbb {P}}(B\cap E_0)>0\). Since \(B\cap E_0 \subset \{Y(\tau )<w(\tau ,X(\tau ))\}\) and \(B\cap E_0 \in {\mathcal {F}}_\tau ^t\), \({\mathbb {P}}(G_0|B\cap E_0)>0\) from the definition of \(\mathbb {U}^{-}\). This further implies that \({\mathbb {P}}(G\cap B\cap E_0) \ge {\mathbb {P}}(G_0\cap B\cap E_0)>0\).

(ii) Assume that \({\mathbb {P}}(B\cap E_1)>0\). Let \(\theta =\theta _1\wedge \theta _2\), where

$$\begin{aligned} \theta _{1}:= & {} \inf \left\{ s \in [ \tau ,T]: (s, X(s)) \notin B_{\varepsilon /2}(t_0, x_0) \right\} \wedge T, \;\; \theta _2 \\:= & {} \inf \left\{ s \in [ \tau ,T]: \left| Y(s)-{\tilde{\varphi }}(s,X(s))\right| \ge \varepsilon \right\} \wedge T. \end{aligned}$$

Since X and Y are càdlàg processes, we know that \(\theta \in {\mathcal {T}}_{\tau }\). The following also hold:

$$\begin{aligned}&(\theta _1, X(\theta _1))\notin B_{\varepsilon /2}(t_0, x_0),\;\; \left| Y(\theta _2)-{\tilde{\varphi }}(\theta _2,X(\theta _2))\right| \ge \varepsilon , \end{aligned}$$
(57)
$$\begin{aligned}&(\theta _1, X(\theta _1-))\in \text {cl}(B_{\varepsilon /2}(t_0, x_0)),\;\; \left| Y(\theta _2-)-{\tilde{\varphi }}(\theta _2,X(\theta _2-))\right| \le \varepsilon . \end{aligned}$$
(58)

Let

$$\begin{aligned} c^e_i(s)= & {} J_i^{u,e}(s,X(s-), Y(s-), {\tilde{\varphi }}), \;\;\\ d_i(s)= & {} \int _E c^e_i(s) m_i(de), \;\; d(s)=\sum _{i=1}^{I}d_i(s), \\ a(s)= & {} \mu _Y(s, X(s), Y(s), \nu (s))-{\mathscr {L}}^{\nu (s)}{\tilde{\varphi }}(s, X(s)),\;\;\\ \pi (s)= & {} N^{\nu (s)}(s, X(s), Y(s), D{\tilde{\varphi }}(s, X(s))), \\ A_0= & {} \left\{ s\in [\tau , \theta ]: |\pi (s)|\le \varepsilon \right\} ,\; A_{3,i}=\left\{ (s,e)\in [\tau ,\theta ]\times E: c_i^e(s)\le -\eta /2 \right\} , \\ A_1= & {} \left\{ s\in [\tau , \theta ]: c_i^e(s)\ge -\eta \text { for } {\hat{m}}-a.s.\;e\in E \text { for all } i=1,\ldots , I \right\} ,\;A_2 = (A_1)^c. \end{aligned}$$

We then set

$$\begin{aligned} L(\cdot ):={\mathcal {E}}\left( \int _t^{\cdot \wedge \theta }\int _E \sum \delta _i^e(s)\tilde{\lambda }_i(ds,de)+\int _{t}^{\cdot \wedge \theta }\alpha ^{\top }(s) dW_s\right) , \end{aligned}$$

where \({\mathcal {E}}(\cdot )\) denotes the Doléans-Dade exponential and

$$\begin{aligned} x^{+}:= & {} \max \{0,x\},\;\;x^{-}:=\max \{0,-x\},\;\; \alpha (s):=-\frac{a(s)+d(s)}{|\pi (s)|^2}\pi (s){\mathbbm {1}}_{A_0^c}(s), \;\;\\ M_i(s):= & {} \int _E{\mathbbm {1}}_{A_{3,i}}(s,e)m_i(de), \\ K_i(s,e):= & {} \left\{ \begin{array}{ll} \frac{\mathbbm {1}_{A_{3,i}}(s,e)}{M_i(s)} \;\;\text {if}\;\; M_i(s)=0 \\ 0 \quad \quad \quad \quad \text {otherwise} \end{array} \right. , \;\;\delta _i^e(s):=\left( \frac{\eta }{2(1+|d(s)|)}-1+{\mathbbm {1}}_{A_2}(s)\cdot \frac{2a(s)^++\eta }{\eta }\cdot K_i(s,e)\right) {\mathbbm {1}}_{A_0}(s). \end{aligned}$$

If \(s\in A_2\), then it follows from Assumption 2.1 and definitions of \(A_2\) and \(A_{3,i}\) that

$$\begin{aligned} M_{i_0}(s) >0 \text { for some }i_0\in \{1,2,\ldots , I\}. \end{aligned}$$
(59)

Obviously, L is a nonnegative local martingale on [tT]. Therefore, it is a super-martingale. Let \(\varGamma (s):=Y(s)-{\tilde{\varphi }}(s,X(s))-\kappa \). Applying Itô’s formula, we get

$$\begin{aligned}&\varGamma (\cdot \wedge \theta )L(\cdot \wedge \theta )= \varGamma (\tau )L(\tau )\\+ & {} \int _{\tau }^{\cdot \wedge \theta } L(s)\left\{ \left( a(s)+d(s)\right) {\mathbbm {1}}_{A_0}(s)+\int _E\sum c^e_i(s)\delta ^e_i(s)m_i(de)\right\} ds \\&\int _{\tau }^{\cdot \wedge \theta }\int _E\sum L(s)\left\{ c^e_i(s)+\varGamma (s)\delta ^e_i(s)+c^e_i(s)\delta ^e_i(s)\right\} \tilde{\lambda }(ds,de)\\+ & {} \int _{\tau }^{\cdot \wedge \theta }L(s)\left( \pi (s)+ \varGamma (s)\alpha (s)\right) ^{\top }dW_s. \end{aligned}$$

By the definition of \(\delta _i^e\) and the fact that \({\mathbbm {1}}_{A_1}+{\mathbbm {1}}_{A_2}=1\) on \([\tau , \theta ]\), the first integral in the equation above is

By (55), \(a(s)\le -\eta \) on \(A_0\cap A_1\). Then,

$$\begin{aligned} \left( a(s)+\frac{\eta d(s)}{2(|d(s)|+1)}\right) {\mathbbm {1}}_{A_0\cap A_1}(s) \le \left( -\eta +\frac{\eta }{2}\right) {\mathbbm {1}}_{A_0\cap A_1}(s)\le 0. \end{aligned}$$
(60)

By the definition of \(A_{3,i}\) and (59), it holds that

$$\begin{aligned}&{\mathbbm {1}}_{A_0\cap A_2}(s)\left( a(s)+\frac{\eta d(s)}{2(|d(s)|+1)}+ \frac{2a(s)^++\eta }{\eta }\int _E\sum c^e_i(s)K_i(s,e)m_i(de)\right) \nonumber \\&\le {\mathbbm {1}}_{A_0\cap A_2}(s)\left( a(s)+\frac{\eta }{2}-\frac{2a(s)^++\eta }{\eta }\cdot \frac{\eta }{2}\right) =- {\mathbbm {1}}_{A_0\cap A_2}(s)a(s)^-. \end{aligned}$$
(61)

Therefore, (60) and (61) imply that \(\varGamma L\) is a local super-martingale on \([\tau ,\theta ]\). Note that

$$\begin{aligned} \varGamma (\theta )-\varGamma (\theta -)=\int _{E} \overline{J}^{\nu (\theta ),e}\left( \theta ,X(\theta -),Y(\theta -),{\tilde{\varphi }}\right) ^{\top }\lambda (\{\theta \},de). \end{aligned}$$

Since \({\tilde{\varphi }}\in C(\mathbb {D})\) and (54) holds, \({\tilde{\varphi }}\) is locally bounded and globally bounded from above. This, together with (58) and the admissibility condition (2), implies that \( \varGamma (\theta )-\varGamma (\theta -)\ge -K\) almost surely for some \(K>0\) (K may depend on \((t_{0}, x_{0}), \varepsilon \), \(\nu \) and \({\tilde{\varphi }}\)). Since \( \varGamma (s)= Y(s)-{\tilde{\varphi }}(s,X(s))-\kappa \ge -(\varepsilon +\kappa ) \text { on }[\tau ,\theta [\; \), \(\varGamma L\) is bounded from below by a sub-martingale \(-(\varepsilon +\kappa +K)L\) on \([\tau , \theta ]\). This further implies that \(\varGamma L\) is a super-martingale by Fatou’s Lemma. Since \(\varGamma (\tau )L(\tau )<0\) on \(B\cap E_1\), the super-martingale property implies that there exists \(F \subset B \cap E_1\) such that \(F\in {\mathcal {F}}^t_{\tau }\) and \(\varGamma (\theta \wedge \rho )L(\theta \wedge \rho )<0\) on F. The non-negativity of L then yields \(\varGamma (\theta \wedge \rho )<0\). Therefore,

$$\begin{aligned}&Y(\theta _1)<{\tilde{\varphi }}(\theta _1,X(\theta _1)) +\kappa \text { on }F\cap \{\theta _1\le \theta _2, \theta<\rho \},\; Y(\theta _2)<{\tilde{\varphi }}(\theta _2,X(\theta _2))\nonumber \\&+\kappa \text { on }F\cap \{\theta _1>\theta _2, \theta<\rho \}\text { and} \nonumber \\&Y(\rho )-({\tilde{\varphi }}(\rho ,X(\rho ))+\kappa )<0 \text { on } F\cap \{\theta \ge \rho \}. \end{aligned}$$
(62)

Since \((\theta _1,X(\theta _1))\notin B_{\varepsilon /2}(t_0,x_0)\), it follows from the first two inequalities in (56) that

$$\begin{aligned} Y(\theta _1)<{\tilde{\varphi }}(\theta _1,X(\theta _1))+\kappa<w(\theta _1,X(\theta _1))\;\;\text {on}\;\; F\cap \{\theta _1\le \theta _2, \theta <\rho \}. \end{aligned}$$
(63)

On the other hand, since \(Y(\theta _2)<{\tilde{\varphi }}(\theta _2,X(\theta _2))+\kappa \) on \( F\cap \{\theta _1>\theta _2, \theta <\rho \}\) and (57) holds, \(Y(\theta _2)-{\tilde{\varphi }}(\theta _2,X(\theta _2))\le -\varepsilon \) on \(F\cap \{\theta _1>\theta _2, \theta <\rho \}\). Observing that \((\theta _2,X(\theta _2))\in B_{\varepsilon /2}(t_0,x_0)\) on \(\{\theta _{1}>\theta _{2}\}\), we get from the last inequality of (45) that

$$\begin{aligned}&Y(\theta _2)-w(\theta _2,X(\theta _2))<{\tilde{\varphi }}(\theta _2,X(\theta _2))-\varepsilon -w(\theta _2,X(\theta _2))< 0 \;\nonumber \\&\quad \text {on}\;\; F\cap \{\theta _1>\theta _2, \theta <\rho \}. \end{aligned}$$
(64)

From (63) and (64), we get that \(Y(\theta )<w(\theta ,X(\theta ))\) on \(F\cap \{\theta < \rho \}.\) Therefore, from the definition of \(\mathbb {U}^{-}\),

$$\begin{aligned} {\mathbb {P}}(G_0|F\cap \{\theta< \rho \})>0 \;\; \text { if} \;\; {\mathbb {P}}(F\cap \{\theta < \rho \})>0. \end{aligned}$$
(65)

From (62), it holds that

$$\begin{aligned} {\mathbb {P}}(G|F\cap \{\theta \ge \rho \})>0 \;\; \text { if} \;\; {\mathbb {P}}(F\cap \{\theta \ge \rho \})>0. \end{aligned}$$
(66)

Since \(G_0\subset G\), (65) and (66) imply that \({\mathbb {P}}(G\cap F)>0.\) Therefore, \({\mathbb {P}}(G\cap B\cap E_1)>0\). \(\square \)

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Bayraktar, E., Li, J. Stochastic Perron for Stochastic Target Problems. J Optim Theory Appl 170, 1026–1054 (2016). https://doi.org/10.1007/s10957-016-0958-2

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