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
The bound may depend on the process.
The convergence \(\psi \overset{\text {u.c.}}{\longrightarrow } \varphi \) is understood in the sense that \(\psi \) converges uniformly on compact subsets to \(\varphi \).
This can be easily checked.
C and N may depend on w and T. This also applies to Definition 2.3
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.
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 \(\iota \).
Here we choose \((t,x)\in \mathbb {D}_{i}\) since the case \((t,x)\in \mathbb {D}_{T}\) is trivial.
The control \(\nu _0\) can be fixed outside the time interval \([\tau ,\theta )\), since we are only interested in the restriction of it to \([\tau ,\theta )\).
The existence of \(n_{0}\) follows as in Step1.
References
Soner, H.M., Touzi, N.: Superreplication under gamma constraints. SIAM J. Control Optim. 39(1), 73–96 (2000)
Soner, H.M., Touzi, N.: Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. (JEMS) 4(3), 201–236 (2002)
Soner, H.M., Touzi, N.: Stochastic target problems, dynamic programming, and viscosity solutions. SIAM J. Control Optim. 41(2), 404–424 (2002)
Bouchard, B.: Stochastic targets with mixed diffusion processes and viscosity solutions. Stoch. Process. Their Appl. 101(2), 273–302 (2002)
Moreau, L.: Stochastic target problems with controlled loss in jump diffusion models. SIAM J. Control Optim. 49(6), 2577–2607 (2011)
Bouchard, B., Elie, R., Touzi, N.: Stochastic target problems with controlled loss. SIAM J. Control Optim. 48(5), 3123–3150 (2009/10)
Bayraktar, E., Sîrbu, M.: Stochastic Perron’s method and verification without smoothness using viscosity comparison: the linear case. Proc. Am. Math. Soc. 140(10), 3645–3654 (2012)
Bayraktar, E., Sîrbu, M.: Stochastic Perron’s method and verification without smoothness using viscosity comparison: obstacle problems and Dynkin games. Proc. Am. Math. Soc. 142(4), 1399–1412 (2014)
Bayraktar, E., Sîrbu, M.: Stochastic Perron’s method for Hamilton–Jacobi–Bellman equations. SIAM J. Control Optim. 51(6), 4274–4294 (2013)
Claisse, J., Talay, D., Tan, X.: A pseudo-Markov property for controlled diffusion processes. SIAM J. Control Optim. 54(2), 1017–1029 (2016)
Bouchard, B., Dang, N.M.: Optimal control versus stochastic target problems: an equivalence result. Syst. Control Lett. 61(2), 343–346 (2012)
Bayraktar, E., Li, J.: Stochastic Perron for stochastic target games. Ann. Appl. Probab. 26(2), 1082–1110 (2016)
Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(3), 567–585 (2008)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991)
Bouchard, B., Nutz, M.: Stochastic target games and dynamic programming via regularized viscosity solutions. Math. Oper. Res. 41(1), 109–124 (2016)
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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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 \(\rho \in {\mathcal {T}}_{\tau }\),
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 \(s\ge \tau \),
Therefore, from (36) it holds that
From the Lipschitz continuity of \(\mu _Y\) in y-variable in Assumption 2.2,
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
with initial data \(\widetilde{Y}(t)=y\), where
Therefore,
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
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
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 \((t,x,y,u,e)\in \mathbb {D}\times \mathbb {R}\times U\times E\),
Proposition A.2
Under Assumptions 2.1, 2.2, A.1 and A.3, \({\mathbb {U}}^{-}\) is not empty.
Proof
Assume that
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
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
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,
Therefore, it follows from (37) and the inequality above that
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
Therefore,
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
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 \(\iota >0\),
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 (t, x, y) 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
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,
For \(\kappa \in \;]0,\varepsilon \wedge \alpha [\;\), define
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 (X, Y) 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
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
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,
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
(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
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
Since \((\theta _1, X(\theta _1))\notin B_{\varepsilon /2}(t_0, x_0)\), we know
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,
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 \(\widetilde{\nu }^{\theta }\),
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
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
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 \(\iota >0\),
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
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
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,
For \(\kappa \in \;]0,\alpha \wedge \varepsilon [\;\), define
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
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
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
Since X and Y are càdlàg processes, we know that \(\theta \in {\mathcal {T}}_{\tau }\). The following also hold:
Let
We then set
where \({\mathcal {E}}(\cdot )\) denotes the Doléans-Dade exponential and
If \(s\in A_2\), then it follows from Assumption 2.1 and definitions of \(A_2\) and \(A_{3,i}\) that
Obviously, L is a nonnegative local martingale on [t, T]. Therefore, it is a super-martingale. Let \(\varGamma (s):=Y(s)-{\tilde{\varphi }}(s,X(s))-\kappa \). Applying Itô’s formula, we get
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,
By the definition of \(A_{3,i}\) and (59), it holds that
Therefore, (60) and (61) imply that \(\varGamma L\) is a local super-martingale on \([\tau ,\theta ]\). Note that
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,
Since \((\theta _1,X(\theta _1))\notin B_{\varepsilon /2}(t_0,x_0)\), it follows from the first two inequalities in (56) that
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
From (63) and (64), we get that \(Y(\theta )<w(\theta ,X(\theta ))\) on \(F\cap \{\theta < \rho \}.\) Therefore, from the definition of \(\mathbb {U}^{-}\),
From (62), it holds that
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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DOI: https://doi.org/10.1007/s10957-016-0958-2
Keywords
- Stochastic target problems
- Stochastic Perron’s method
- Jump diffusion processes
- Viscosity solutions
- Unbounded controls