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Corollary 7.5.1. Let \( n \in \mathbb{N} \) and \( F, G \in \mathop{\bigcap }\limits_{{k = 1}}^{{n + 1}}\mathbb{D}\left( {\Delta }_{k}\right) \) . We have | \[ \operatorname{Cov}\left( {F, G}\right) = \mathop{\sum }\limits_{{k = 1}}^{n}{\left( -1\right) }^{k + 1}\mathbb{E}\left\lbrack {{\int }_{{\Delta }_{k}}\left( {{\widetilde{D}}_{{t}_{k}}\cdots {\widetilde{D}}_{{t}_{1}}F}\right) \left( {{\widetilde{D}}_{{t}_{k}}\cdots {\widetilde{D}}_{{t}_{1}}G}\right) d{t}_{1}\cdots d{... | Yes |
Proposition 7.5.2. The mapping \( \Theta : {L}^{p}\left( \Omega \right) \rightarrow {L}^{p}\left( W\right) \) is an isometry. Further, it satisfies the intertwining relation\n\n\[ {2\Theta }{\left| \widetilde{D}F\right| }_{{L}^{2}\left( {\mathbb{R}}_{ + }\right) }^{2} = {\left| D\Theta F\right| }_{{L}^{2}\left( {\mathb... | Proof. The proposition follows from the fact that \( F \) and \( {\Theta F} \) have same distribution since the half sum of two independent Gaussian squares has an exponential distribution. Relation (7.5.2) follows by a direct calculation. | No |
Corollary 7.5.3. Let \( F \in \operatorname{Dom}\left( \widetilde{D}\right) \) . We have\n\n\[ \mathbb{P}\left( {F - \mathbb{E}\left\lbrack F\right\rbrack \geq x}\right) \leq \exp \left( {-\frac{{x}^{2}}{2\parallel \widetilde{D}F{\parallel }_{{L}^{2}\left( {{\mathbb{R}}_{ + },{L}^{\infty }\left( \Omega \right) }\right)... | In particular if \( F \) is \( {\mathcal{F}}_{T} \) measurable and \( \parallel \widetilde{D}F{\parallel }_{\infty } \leq K \) then\n\n\[ \mathbb{P}\left( {F - \mathbb{E}\left\lbrack F\right\rbrack \geq x}\right) \leq \exp \left( {-\frac{{x}^{2}}{2{K}^{2}T}}\right) ,\;x \geq 0. \] | Yes |
Corollary 7.5.4. Let \( F \in \operatorname{Dom}\left( \widetilde{D}\right) \) . Then\n\n\[ \mathbb{P}\left( {F - \mathbb{E}\left\lbrack F\right\rbrack \geq x}\right) \leq \exp \left( {-\frac{{x}^{2}}{4\parallel \widetilde{D}F{\parallel }_{{L}^{\infty }\left( {\Omega ,{L}^{2}\left( {\mathbb{R}}_{ + }\right) }\right) }^... | The above result can also be obtained via logarithmic Sobolev inequalities, i.e. by application of Corollary 2.5 of [76] to Theorem 0.7 in [4] (or Relation (4.4) in [76] for a formulation in terms of exponential random variables). A sufficient condition for the exponential integrability of \( F \) is \( {\begin{Vmatrix... | Yes |
Proposition 7.6.3. The Lie bracket \( \{ u, v\} \) of \( u, v \in \mathcal{U} \) satisfies\n\n\[ \{ u, v\} = {\nabla }_{u}v - {\nabla }_{v}u \]\n\n(7.6.6)\n\ni.e. the connection defined by \( \nabla \) has a vanishing torsion\n\n\[ T\left( {u, v}\right) = {\nabla }_{u}v - {\nabla }_{v}u - \{ u, v\} = 0,\;u, v \in \math... | Proof. For all \( u, v \in {\mathcal{C}}_{c}^{\infty }\left( {\mathbb{R}}_{ + }\right) \) we have\n\n\[ \left( {{\widetilde{D}}_{u}{\widetilde{D}}_{v} - {\widetilde{D}}_{v}{\widetilde{D}}_{u}}\right) {T}_{n} = - {\widetilde{D}}_{u}{\int }_{0}^{{T}_{n}}{v}_{s}{ds} + {\widetilde{D}}_{v}{\int }_{0}^{{T}_{n}}{u}_{s}{ds} \]... | Yes |
Proposition 7.6.4. The Riemannian curvature tensor \( R \) of \( \nabla \) vanishes on \( \mathcal{U} \), i.e.\n\n\[ R\left( {u, v}\right) h \mathrel{\text{:=}} \left\lbrack {{\nabla }_{u},{\nabla }_{v}}\right\rbrack h - {\nabla }_{\{ u, v\} }h = 0,\;u, v, h \in \mathcal{U}. \] | Proof. We have, letting \( \widetilde{u}\left( t\right) = - {\int }_{0}^{t}{u}_{s}{ds}, t \in {\mathbb{R}}_{ + } \) :\n\n\[ \left\lbrack {{\nabla }_{u},{\nabla }_{v}}\right\rbrack h = \widetilde{u}\dot{\widehat{{\nabla }_{v}h}} - \widetilde{v}\dot{\widehat{{\nabla }_{u}h}} = \widetilde{u}\dot{\widehat{\widetilde{v}\dot... | Yes |
Proposition 7.6.5. The bracket \( \{ \cdot , \cdot \} \) satisfies the Jacobi identity\n\n\[ \n\{ \{ u, v\}, w\} + \{ w,\{ u, v\} \} + \{ v,\{ u, w\} \} = 0,\;u, v, w \in {\mathcal{C}}_{c}^{\infty }\left( {\mathbb{R}}_{ + }\right) ,\n\]\n\nhence \( \mathcal{U} \) is a Lie algebra under \( \{ \cdot , \cdot \} \) . | Proof. The vanishing of \( R\left( {u, v}\right) \) in Proposition 7.6.4 shows that\n\n\[ \n\left\lbrack {{\nabla }_{u},{\nabla }_{v}}\right\rbrack = {\nabla }_{\{ u, v\} }h,\;u, v \in \mathcal{U},\n\]\n\nhence\n\n\[ \n{\nabla }_{\{ \{ u, v\}, w\} } + {\nabla }_{\{ w,\{ u, v\} \} } + {\nabla }_{\{ v,\{ u, w\} \} }\n\]\... | Yes |
Lemma 7.6.6. We have the commutation relation\n\n\[ \n{\widetilde{D}}_{u}\widetilde{\delta }\left( v\right) = \widetilde{\delta }\left( {{\nabla }_{u}v}\right) + \langle u, v{\rangle }_{{L}^{2}\left( {\mathbb{R}}_{ + }\right) }, \]\n\n(7.6.8)\n\n\( u, v \in {\mathcal{C}}_{c}^{\infty }\left( {\mathbb{R}}_{ + }\right) \)... | Proof. We have\n\n\[ \n{\widetilde{D}}_{u}\widetilde{\delta }\left( v\right) = - \mathop{\sum }\limits_{{k = 1}}^{\infty }\dot{v}\left( {T}_{k}\right) {\int }_{0}^{{T}_{k}}{u}_{s}{ds} \]\n\n\[ \n= - \widetilde{\delta }\left( {\dot{v} \cdot {\int }_{0}^{ \cdot }{u}_{s}{ds}}\right) - {\int }_{0}^{\infty }\dot{v}\left( t\... | Yes |
Theorem 7.6.8. We have for \( u \in \mathcal{U} \) :\n\n\[ \mathbb{E}\left\lbrack {\left| \widetilde{\delta }\left( u\right) \right| }^{2}\right\rbrack + \mathbb{E}\left\lbrack {\parallel \widetilde{D}u{\parallel }_{{L}^{2}\left( {\mathbb{R}}_{ + }\right) \land {L}^{2}\left( {\mathbb{R}}_{ + }\right) }^{2}}\right\rbrac... | Proof. Relation (7.6.10) for \( u = \mathop{\sum }\limits_{{i = 1}}^{n}{h}_{i}{F}_{i} \in \mathcal{U} \) follows from Relation (7.6.7) and Proposition 7.6.7. | No |
Lemma 8.1.2. The following statements are equivalent:\n\ni) the portfolio \( {V}_{t} \) is self-financing,\n\nii) we have\n\n\[ \n{\widetilde{V}}_{t} = {\widetilde{V}}_{0} + {\int }_{0}^{t}{\sigma }_{u}{\eta }_{u}{\widetilde{S}}_{u}d{M}_{u},\;t \in {\mathbb{R}}_{ + }, \n\]\n\n(8.1.5)\n\niii) we have\n\n\[ \n{V}_{t} = {... | Proof. First, note that (8.1.5) is clearly equivalent to (8.1.6). Next, the self-financing condition (8.1.4) shows that\n\n\[ \nd{V}_{t} = {\zeta }_{t}d{A}_{t} + {\eta }_{t}d{S}_{t} \n\]\n\n\[ \n= {\zeta }_{t}{A}_{t}{r}_{t}{dt} + {\eta }_{t}{r}_{t}{S}_{t}{dt} + {\sigma }_{t}{\eta }_{t}{S}_{t}d{M}_{t} \n\]\n\n\[ \n= {r}... | Yes |
Given \( F \in {L}^{2}\left( \Omega \right) \), let\n\n\[{\eta }_{t} = \frac{\exp \left( {-{\int }_{t}^{T}{r}_{s}{ds}}\right) }{{\sigma }_{t}{S}_{t}}\mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack\]\n\n(8.2.1)\n\n\[{\zeta }_{t} = \frac{\exp \left( {-{\int }_{t}^{T}{r}_{u}{du}}\right) \mathbb{E}\le... | Proof. Applying (8.2.2) at \( t = 0 \) we get\n\n\[ \mathbb{E}\left\lbrack F\right\rbrack \exp \left( {-{\int }_{0}^{T}{r}_{u}{du}}\right) = {V}_{0} \]\n\nhence from (8.2.2), the definition (8.2.1) of \( {\eta }_{t} \) and the Clark formula we obtain\n\n\[ {V}_{t} = {\zeta }_{t}{A}_{t} + {\eta }_{t}{S}_{t} \]\n\n\[ = \... | Yes |
Proposition 8.4.1. There exists a measurable function \( \widetilde{C} \) on \( {\mathbb{R}}_{ + } \times \mathbb{R} \) such that \( \widetilde{C}\left( {t, \cdot }\right) \) is \( {\mathcal{C}}^{1} \) for all \( t \in {\mathbb{R}}_{ + } \), and\n\n\[ \n{S}_{t}\widetilde{C}\left( {t,{Y}_{t}}\right) = \mathbb{E}\left\lb... | Proof. With the above notation, the price at time \( t \) of the Asian option\n\nbecomes\n\[ \n\mathbb{E}\left\lbrack {{\mathrm{e}}^{-{\int }_{t}^{T}{r}_{s}{ds}}{S}_{T}{\left( {Y}_{T}\right) }^{ + } \mid {\mathcal{F}}_{t}}\right\rbrack \n\]\n\nFor \( 0 \leq s \leq t \leq T \), we have\n\n\[ \nd\left( {{S}_{t}{Y}_{t}}\r... | Yes |
Proposition 9.4.1. Let \( F \in {L}^{2}\left( \Omega \right) \) . Then \( {\left( \mathbb{E}\left\lbrack F \mid {\mathcal{F}}_{n}\right\rbrack \right) }_{n \in \mathbb{N}} \) converges to \( F \) a.s. | Proof. This is a consequence of the martingale convergence theorem, cf. e.g. Theorem 27.1 in [67]. | No |
Proposition 9.8.2. Assume that \( T \) is closable. If \( {\left( {F}_{n}\right) }_{n \in \mathbb{N}} \) and \( {\left( {G}_{n}\right) }_{n \in \mathbb{N}} \) converge to \( F \in \operatorname{Dom}\left( T\right) \) and \( {\left( T{F}_{n}\right) }_{n \in \mathbb{N}} \) and \( {\left( T{G}_{n}\right) }_{n \in \mathbb{... | Proof. Indeed, under the above assumptions, \( {\left( T\left( {F}_{n} - {G}_{n}\right) \right) }_{n \in \mathbb{N}} \) converges to \( U - V \), hence \( U = V \) by the closability condition. | Yes |
Lemma 1.2. Let \( \mathcal{A} \subseteq \mathcal{F} \subseteq {2}^{\Omega } \) . Suppose \( \mathcal{A} \) is a \( \pi \) -system and \( \mathcal{F} \) is a \( \lambda \) -system. Then \( \sigma \left( \mathcal{A}\right) \subseteq \mathcal{F} \) . | Proof. Let\n\n\[ \widehat{\mathcal{G}} \triangleq \bigcap \{ \mathcal{G} \supseteq \mathcal{A} \mid \mathcal{G}\text{ is a }\lambda \text{-system }\} \subseteq \mathcal{F}. \]\n\nThen \( \widehat{\mathcal{G}} \) is the smallest \( \lambda \) -system containing \( \mathcal{A} \) . Set\n\n\[ \widehat{\mathcal{F}} \triang... | Yes |
Example 1.3. Let \( \Omega = \left\lbrack {0,1}\right\rbrack ,\mathcal{F} \) the set of all Lebesgue measurable sets in \( \left\lbrack {0,1}\right\rbrack \), and \( \mathbf{P} \) the Lebesgue measure on \( \left\lbrack {0,1}\right\rbrack \) . Then one can show that \( \left( {\Omega ,\mathcal{F},\mathbf{P}}\right) \) ... | \[ \mathbf{P}\left( A\right) = {\int }_{\{ \omega \in \Omega \mid x\left( \omega \right) \geq 0\} }d\mathbf{P}\left( \omega \right) . \] | Yes |
Proposition 1.5. Let \( \Omega \) be a Polish space and \( \mathcal{F} = \mathcal{B}\left( \Omega \right) \) . Then:\n\n(i) \( \left( {\Omega ,\mathcal{F}}\right) \) is standard, and it is countably determined.\n\n(ii) For any \( {\Omega }^{\prime } \in \mathcal{F} \), let \( {\mathcal{F}}^{\prime } = {\Omega }^{\prime... | See Parthasarathy [1, pp. 133-134] and Ikeda-Watanabe [1, p. 13] for a proof and descriptions. | No |
Lemma 1.6. Let \( \mathcal{A} \) be a \( \pi \) -system on \( \Omega \) . Let \( \mathcal{H} \) be a linear space of functions from \( \Omega \) to \( \mathbb{R} \) such that\n\n\[ \left\{ \begin{array}{l} 1 \in \mathcal{H};\;{I}_{A} \in \mathcal{H},\;\forall A \in \mathcal{A}; \\ {\varphi }_{i} \in \mathcal{H},0 \leq ... | Proof. Set\n\n\[ \mathcal{F} = \left\{ {A \subseteq \Omega \mid {I}_{A} \in \mathcal{H}}\right\} \]\n\nThen \( \mathcal{F} \) is a \( \lambda \) -system containing \( \mathcal{A} \) . Thus, by Lemma 1.2, \( \sigma \left( \mathcal{A}\right) \subseteq \mathcal{F} \) .\n\nNow, for any \( \sigma \left( \mathcal{A}\right) \... | Yes |
Proposition 1.8. Let \( \mathcal{G} \) be a sub- \( \sigma \) -field of \( \mathcal{F} \) . Then\n\n(i) Map \( E\left( {\cdot \mid \mathcal{G}}\right) : {L}_{\mathcal{F}}^{1}\left( {\Omega ;{\mathbb{R}}^{m}}\right) \rightarrow {L}_{\mathcal{G}}^{1}\left( {\Omega ;{\mathbb{R}}^{m}}\right) \) is linear and bounded.\n\n(i... | Proofs of the above results are straightforward by the definitions. | No |
Proposition 1.10. Let \( \\left( {\\Omega ,\\mathcal{F},\\mathbf{P}}\\right) \) be a standard probability space and \( \\left( {S,\\mathcal{B}}\\right) \) a measurable space. Let \( \\xi : \\left( {\\Omega ,\\mathcal{F}}\\right) \\rightarrow \\left( {S,\\mathcal{B}}\\right) \) be a random variable and \( {\\mathbf{P}}_... | For proofs of Propositions 1.9 and 1.10, see Parthasarathy [1, pp. 145- \( {150}\\rbrack \) . | No |
Lemma 1.11. Let \( \xi : \left( {\Omega ,\mathcal{F}}\right) \rightarrow \left( {{\mathbb{R}}^{\ell },\mathcal{B}\left( {\mathbb{R}}^{\ell }\right) }\right) \) be a random variable, and \( X \in {L}_{\mathcal{F}}^{1}\left( {\Omega ;{\mathbb{R}}^{m}}\right) \) . If \( E\left( {g\left( \xi \right) X}\right) = 0 \) for an... | Proof. Define\n\n\[ \mathcal{H} \triangleq \{ \varphi : \left( {\Omega ,\mathcal{F}}\right) \rightarrow \left( {\mathbb{R},\mathcal{B}\left( \mathbb{R}\right) }\right) \mid \varphi \text{ is measurable,}E\left( {\varphi X}\right) = 0\} \]\n\nand\n\n\[ \mathcal{A} \triangleq \left\{ {{\xi }^{-1}\left( B\right) \mid B = ... | Yes |
Proposition 1.12. Let \( {\xi }_{i} : \left( {\Omega ,\mathcal{F}}\right) \rightarrow \left( {{\mathbb{R}}^{{m}_{i}},\mathcal{B}\left( {\mathbb{R}}^{{m}_{i}}\right) }\right) \) be a sequence of random variables \( \left( {i = 1,2,\ldots }\right) \) . Let \( \mathcal{G} \triangleq \mathop{\bigvee }\limits_{i}\sigma \lef... | Proof. The \ | No |
Proposition 1.16. Let \( \Lambda \subseteq \mathcal{P}\left( U\right) \) . Then:\n\n(i) \( \Lambda \) is relatively compact if it is tight.\n\n(ii) If \( \left( {U, d}\right) \) is complete (i.e., it is a polish space), then \( \Lambda \) is tight if it is relatively compact. | Sec Ikeda-Watanabe [1, pp. 6-8] for proofs of Propositions 1.14 and 1.16. | No |
Corollary 1.17. If \( \left( {U, d}\right) \) is compact, then any \( \Lambda \subseteq \mathcal{P}\left( U\right) \) is tight and relatively compact. In particular, \( \mathcal{P}\left( U\right) \) is compact. | The proof is straightforward by Definition 1.15 and Proposition 1.16. See also Parthasarathy [1, p. 45, Theorem 6.4]. | No |
Theorem 1.20. Let \( \\left( {U, d}\\right) \) be a Polish space and \( \\left\\{ {{\\mathbf{P}}_{i}, i = 1,2,\\ldots ,\\mathbf{P}}}\\right\\} \\subseteq \) \( \\mathcal{P}\\left( U\\right) \) be such that \( {\\mathbf{P}}_{i} \) converges to \( \\mathbf{P} \) weakly. Then on some probability space \( \\left( {\\wideha... | This theorem is due to Skorohod. See Billingsley [1] for a proof. | No |
Theorem 2.2. Let \( \mathbb{F} = \left\{ {{F}_{{t}_{1},\ldots ,{t}_{j}}\left( {{x}_{1},\ldots ,{x}_{j}}\right), j \geq 1}\right\} \) be a family of functions satisfying the symmetry and compatibility conditions. Then there exists a probability space \( \left( {\Omega ,\mathcal{F},\mathbf{P}}\right) \) and a stochastic ... | For a proof, see Parthasarathy [1, pp. 143-144]. | No |
Example 2.4. Let \( \Omega = \left\lbrack {0,1}\right\rbrack, T \geq 1 \) , \( \mathbf{P} \) the Lebesgue measure, \( X\left( {t,\omega }\right) \equiv 0 \) , and\n\n\[ \bar{X}\left( {t,\omega }\right) = \left\{ \begin{array}{ll} 0, & \omega \neq t \\ 1, & \omega = t \end{array}\right. \]\n\nThen \( X\left( t\right) \)... | But each sample path \( X\left( {\cdot ,\omega }\right) \) is continuous, and none of the sample paths \( \bar{X}\left( {\cdot ,\omega }\right) \) is continuous. In the present case, we actually have\n\n\[ \mathop{\bigcup }\limits_{{t \in \left\lbrack {0,1}\right\rbrack }}{N}_{t} = \left\lbrack {0,1}\right\rbrack \equi... | Yes |
Proposition 2.8. Let \( \left( {\Omega ,\mathcal{F},\{ \mathcal{F}{\} }_{t \geq 0},\mathbf{P}}\right) \) be a filtered probability space, and let \( X\left( t\right) \) be measurable and \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) -adapted. Then there exists an \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t > 0} ... | For a proof, see Meyer [1, p. 68]. | No |
Lemma 2.11. The \( \sigma \) -field \( \sigma \left( {\mathbf{C}}_{T}\right) \) generated by \( {\mathbf{C}}_{T} \) coincides with the Borel \( \sigma \) -field \( \mathcal{B}\left( {{\mathbf{W}}^{m}\left\lbrack {0, T}\right\rbrack }\right) \) of \( {\mathbf{W}}^{m}\left\lbrack {0, T}\right\rbrack \) . | Proof. Let \( 0 \leq {t}_{1} < {t}_{2} < \cdots < {t}_{j} \leq T \) be given. We define a map \( \mathcal{T} : {\mathbf{W}}^{m}\left\lbrack {0, T}\right\rbrack \rightarrow {\mathbb{R}}^{jm} \) as follows:\n\n\[ \mathcal{T}\left( \zeta \right) = \left( {\zeta \left( {t}_{1}\right) ,\zeta \left( {t}_{2}\right) ,\ldots ,\... | Yes |
Lemma 2.12. Let \( \left( {\Omega ,\mathcal{F},\mathbf{P}}\right) \) be a complete probability space and \( \xi \) : \( \left\lbrack {0, T}\right\rbrack \times \Omega \rightarrow {\mathbb{R}}^{m} \) a continuous process. Then there exists an \( {\Omega }_{0} \in \mathcal{F} \) with \( \mathbf{P}\left( {\Omega }_{0}\rig... | Proof. Let \( t \in \left\lbrack {0, s}\right\rbrack \) and \( E \in \mathcal{B}\left( {\mathbb{R}}^{m}\right) \) be fixed. Then \[ {B}_{t} \triangleq \left\{ {\zeta \in {\mathbf{W}}^{m}\left\lbrack {0, T}\right\rbrack \mid \zeta \left( t\right) \in E}\right\} \in {\mathbf{C}}_{s}, \] and \[ \omega \in {\xi }^{-1}\left... | Yes |
Proposition 2.13. Let \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0},\mathbf{P}}\right) \) be a filtered probability space satisfying the usual condition, and let \( \xi \) be an \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) -adapted process. Then for any \( s \in \left\lbrack {0... | Proof. By definition, for any \( t \in \left\lbrack {0, s}\right\rbrack \) , \[ \mathbf{P}\left( {\left\{ {{\omega }^{\prime } \mid \xi \left( {t,{\omega }^{\prime }}\right) = \xi \left( {t,\omega }\right) }\right\} \mid {\mathcal{F}}_{s}}\right) \left( \omega \right) \] \[ = E\left( {{I}_{\left\{ {\omega }^{\prime } \... | Yes |
Corollary 2.15. Let \( X\\left( \\cdot \\right) \) be an \( m \) -dimensional stochastic process over \( \\left\\lbrack {0, T}\\right\\rbrack \) such that\n\n\[ E{\\left| X\\left( t\\right) - X\\left( s\\right) \\right| }^{\\alpha } \\leq K{\\left| t - s\\right| }^{1 + \\beta },\\;\\forall t, s \\in \\left\\lbrack {0, ... | See Ikeda-Watanabe [1, pp. 17-20] for proofs of Theorem 2.14 and Corollary 2.15. | No |
(i) A map \( \tau : \Omega \rightarrow \left\lbrack {0,\infty }\right\rbrack \) is a stopping time if and only if\n\n(3.3)\n\n\[ \left( {\tau < t}\right) \in {\mathcal{F}}_{t},\;\forall t > 0. \] | Proof. We first prove (ii). If \( A \in {\mathcal{F}}_{\tau } \), then for any \( t > 0 \) ,\n\n(3.5)\n\n\[ A \cap \left( {\tau < t}\right) = \mathop{\bigcup }\limits_{{n \geq 1}}\left\{ {A \cap \left( {\tau \leq t - \frac{1}{n}}\right) }\right\} \in {\mathcal{F}}_{t} \]\n\nConversely, if (3.4) holds, then for any \( t... | Yes |
Let \( X\left( t\right) \) be \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) -adapted and continuous. Let \( E \subseteq {\mathbb{R}}^{m} \) be an open set. Then the first hitting time of the process \( X\left( t\right) \) to \( E \), \[ {\sigma }_{E}\left( \omega \right) \triangleq \inf \{ t \geq 0 \mid X\left(... | Let us prove these two facts. First of all, for any \( s > 0 \), we claim that \[ \left( {{\sigma }_{E} < s}\right) = \mathop{\bigcup }\limits_{{r \in \mathbf{Q}, r < s}}\left( {X\left( r\right) \in E}\right) \in {\mathcal{F}}_{s}. \] To prove the equality, let \( \omega \) be such that \( {\sigma }_{E}\left( \omega \r... | Yes |
Proposition 3.5. Let \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0},\mathbf{P}}\right) \) be a filtered probability space satisfying the usual condition. Let \( X\left( t\right) \) be an \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) -progressively measurable process and \( \tau \... | Proof. We first prove that \( X\left( {\tau \land t}\right) \) is \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) -progressively measurable. To this end, by Proposition 3.4, the process \( \tau \land t \) is \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) -progressively measurable. Thus, for each \( t \geq ... | Yes |
Proposition 3.6. Let \( \tau \) be a stopping time and \( \xi \) a random variable. Then \( \xi \) is \( {\mathcal{F}}_{\tau } \) -measurable if and only if for all \( t \geq 0,\xi {I}_{\left( \tau \leq t\right) } \) is \( {\mathcal{F}}_{t} \) -measurable. | Proof. If \( \xi \) is \( {\mathcal{F}}_{\tau } \) measurable, then there exists a sequence of \( {\mathcal{F}}_{\tau } \) - measurable simple functions\n\n(3.18)\n\n\[{\xi }_{j} \equiv \mathop{\sum }\limits_{{i \geq 1}}{\xi }_{j}^{i}{I}_{{A}_{j}^{i}} \rightarrow \xi ,\text{ as }j \rightarrow \infty ,\;\mathbf{P}\text{... | Yes |
Proposition 3.7. Let \( \sigma \) and \( \tau \) be stopping times and \( X \) an integrable random variable. Then\n\n\[ \left\{ \begin{array}{l} {I}_{\left( \sigma > \tau \right) }E\left( {X \mid {\mathcal{F}}_{\tau }}\right) = E\left( {{I}_{\left( \sigma > \tau \right) }X \mid {\mathcal{F}}_{\tau }}\right) = {I}_{\le... | Proof. By Proposition 3.4-(iii), we have the first equalities in the first two assertions. Now, since\n\n\[ {I}_{\left( \sigma > \tau \right) }E\left( {X \mid {\mathcal{F}}_{\tau }}\right) {I}_{\left( \sigma \land \tau \leq t\right) } = E\left( {X \mid {\mathcal{F}}_{\tau }}\right) {I}_{\left( \tau \leq t\right) }{I}_{... | Yes |
Proposition 4.2. Let \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) and \( {\left\{ {\mathcal{G}}_{t}\right\} }_{t \geq 0} \) be two families of sub- \( \sigma \) - fields of \( \mathcal{F} \) with \( {\mathcal{G}}_{t} \subseteq {\mathcal{F}}_{t},\forall t \geq 0 \) . If \( X\left( t\right) \) is an \( {\left\{ ... | Proof. The first assertion follows easily from Proposition 1.8-(vi) and the definition of a martingale, while the second one follows from the first along with Proposition 1.8-(iv). | No |
Proposition 4.3. Let \( X\left( t\right) \) be a submartingale and \( \varphi : \mathbb{R} \rightarrow \mathbb{R} \) a nondecreasing convex function such that \( {E\varphi }\left( {X\left( t\right) }\right) \) exists for all \( t \geq 0 \) . Then \( \varphi \left( {X\left( t\right) }\right) \) is a submartingale. In pa... | Proof. By the monotonicity of \( \varphi \left( \cdot \right) \) and Jensen’s inequality (see (1.36)), we have\n\n(4.2)\n\n\[ \varphi \left( {X\left( s\right) }\right) \leq \varphi \left( {E\left( {X\left( t\right) \mid {\mathcal{F}}_{s}}\right) }\right) \leq E\left( {\varphi \left( {X\left( t\right) }\right) \mid {\ma... | Yes |
Theorem 4.4. Let \( X\left( t\right) \) be an \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) -submartingale. Then there exists a \( \mathbf{P} \) -null set \( N \) such that for any \( \omega \in \bar{\Omega } \smallsetminus N \), the following limits exist:\n\n(4.3)\n\n\[ \mathop{\lim }\limits_{{r \in \mathbf{Q... | If we define\n\n(4.4)\n\n\[ \widehat{X}\left( t\right) = \mathop{\lim }\limits_{{r \in \mathbf{Q}, r \downarrow t}}X\left( r\right) ,\;\mathbf{P}\text{-a.s.,}\forall t \geq 0, \]\n\nand define \( \widehat{X}\left( t\right) \) as an arbitrary constant on \( N \), then \( \widehat{X}\left( t\right) \) is a submartingale ... | Yes |
Theorem 4.6. (Optional sampling theorem) Let \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0},\mathbf{P}}\right) \) be a filtered probability space satisfying the usual condition. Let \( X\left( t\right) \) be a right-continuous \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) -martin... | See Ikeda-Watanabe [1, pp. 32-34] for proofs of Theorems 4.4-4.6. | No |
Corollary 4.7. Let \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0},\mathbf{P}}\right) \) be the same as in Theorem 4.6. Let \( \sigma \leq \tau \) be two bounded stopping times. Then, for any \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) -martingale (resp. submartingale, supermart... | Proof. Let\n\n(4.11)\n\n\[ \n{\sigma }_{t} = \sigma {I}_{\left\lbrack 0,1\right\rbrack }\left( t\right) + \tau {I}_{\left( 1,\infty \right) }\left( t\right) ,\;t \geq 0.\n\]\n\nThen we have (4.9). Appealing to Theorem 4.6 by taking \( t > 1 \) and \( s \leq 1 \) , one obtains (4.10). | No |
Corollary 4.8. Let \( X\\left( t\\right) \) be an \( {\\left\{ {\\mathcal{F}}_{t}\\right\} }_{t \\geq 0} \) -martingale, and let \( \\sigma \\leq \\tau \) be two stopping times. Then\n\n(4.12)\n\n\[ E\\left\\lbrack {X\\left( {t \\land \\tau }\\right) - X\\left( {t \\land \\sigma }\\right) \\mid {\\mathcal{F}}_{\\sigma ... | Proof. In view of Propositions 3.4 and 3.5, \( t \\land \\tau \) and \( t \\land \\sigma \) are stopping times with \( t \\land \\tau \\geq t \\land \\sigma \), P-a.s., and \( X\\left( {t \\land \\tau }\\right) \) is \( {\\mathcal{F}}_{t} \) -measurable. Thus, by Corollary 4.7 and Proposition 3.7, we obtain\n\n(4.13)\n... | Yes |
Proposition 5.2. The Itô integral has the following properties:\n\n(i) For any \( f, g \in {L}_{\mathcal{F}}^{2}\left( {0, T;\mathbb{R}}\right) \) and stopping times \( \sigma \) and \( \tau \) with \( \sigma \leq \tau \) (P-a.s. ),\n\n(5.24)\n\n\[ E\left\{ {{\int }_{t \land \sigma }^{t \land \tau }f\left( r\right) {dW... | See Ikeda-Watanabe [1, pp. 49-51] for a proof. | No |
Proposition 5.3. Let \( f \in {L}_{\mathcal{F}}^{2}\left( {0, T;{\mathbb{R}}^{m}}\right) \) and \( \sigma ,\widehat{\sigma } \in {L}_{\mathcal{F}}^{2}\left( {0, T;{\mathbb{R}}^{n \times m}}\right) \) . Then, for all \( 0 \leq s \leq t \leq T \) , (5.37) \[ E\left\{ {{\int }_{s}^{t}{f}_{i}\left( r\right) d{W}^{i}\left( ... | The proof is straightforward. | No |
Theorem 5.4. Let \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0},\mathbf{P}}\right) \) be given as before and let \( W\left( t\right) \) be an \( m \) -dimensional standard Brownian motion. Let \( \sigma \in {L}_{\mathcal{F}}^{2,{loc}}\left( {0, T;{\mathbb{R}}^{n \times m}}\right) \) . Th... | See Karatzas-Shreve [3, p. 166] for a proof of Theorem 5.4. | No |
Theorem 5.5. (Itô’s formula) Let \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0},\mathbf{P}}\right) \) be a filtered probability space satisfying the usual condition, \( W\left( t\right) \) an \( m \) -dimensional \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) - Brownian motion, \(... | For a proof of the Itô formula, see Ikeda-Watanabe [1, pp. 66-73] (for a more general case). | No |
Corollary 5.6. Let \( Z \) and \( \widehat{Z} \) be \( {\mathbb{R}}^{n} \) -valued continuous processes satisfying\n\n(5.47)\n\n\[ \left\{ \begin{array}{l} {dZ}\left( t\right) = b\left( t\right) {dt} + \sigma \left( t\right) {dW}\left( t\right) \\ d\widehat{Z}\left( t\right) = \widehat{b}\left( t\right) {dt} + \widehat... | This lemma can be easily proved by Itô’s formula with \( F\left( {x, y}\right) = \langle x, y\rangle \) for \( \left( {x, y}\right) \in {\mathbb{R}}^{2n} \) . | Yes |
Theorem 5.7. Let \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0},\mathbf{P}}\right) \) be a filtered probability space satisfying the usual condition. Assume that \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) is the natural filtration generated by an \( m \) -dimensional standard ... | A proof can be found in Ikeda-Watanabe [1, pp. 80-83]. | Yes |
Theorem 5.10. Let \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0},\mathbf{P}}\right) \) be a filtered probability space satisfying the usual condition. Let \( M \in {\mathcal{M}}^{2}{\left\lbrack 0, T\right\rbrack }^{n} \) (resp. \( {\mathcal{M}}^{2,{loc}}{\left\lbrack 0, T\right\rbrack }... | See Ikeda-Watanabe [1, pp. 90-92] for a proof. | No |
Lemma 6.1. Let \( b \in {\mathcal{A}}^{n}\left( {\mathbb{R}}^{n}\right) \) . Let \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0},\mathbf{P}}\right) \) be given satisfying the usual condition and let \( X \) be a continuous \( {\mathbb{R}}^{\bar{n}} \) -valued \( {\left\{ {\mathcal{F}}_{t}... | Proof. For any \( t \in \lbrack 0,\infty ) \), let \( \Phi : \left\lbrack {0, t}\right\rbrack \times \Omega \rightarrow \left\lbrack {0, t}\right\rbrack \times {\mathbf{W}}^{n} \) be the map \( \left( {s,\omega }\right) \mapsto \left( {s, X\left( {\cdot ,\omega }\right) }\right) \) . Then\n\n(6.1)\n\n\[ b\left( {s, X\l... | Yes |
Theorem 6.8. Let \( b \in {\mathcal{A}}^{n}\left( {\mathbb{R}}^{n}\right) \) and \( \sigma \in {\mathcal{A}}^{n}\left( {\mathbb{R}}^{n \times m}\right) \) . Then (6.2) admits a unique strong solution if and only if for any probability measure \( \mu \) on \( \left( {{\mathbb{R}}^{n},\mathcal{B}\left( {\mathbb{R}}^{n}\r... | By and large, Theorem 6.8 tells that strong existence and uniqueness is equivalent to weak existence plus pathwise uniqueness. | No |
Theorem 6.9. Pathwise uniqueness implies weak uniqueness. | See Karatzas-Shreve [3, pp. 308-311] for proofs of Theorems 6.8-6.9. | No |
Theorem 6.10. Let \( b \in {\mathcal{A}}^{n}\left( {\mathbb{R}}^{n}\right) \) and \( \sigma \in {\mathcal{A}}^{n}\left( {\mathbb{R}}^{n \times m}\right) \) be bounded and continuous. Then there exists a weak solution of (6.2). | The proof can be found in Ikeda-Watanabe [1, pp. 155-158]. | Yes |
Theorem 6.11. Let \( \sigma \in {\mathcal{A}}^{n}\left( {\mathbb{R}}^{n \times n}\right) \) be bounded and continuous, and let \( b \in {\mathcal{A}}^{n}\left( {\mathbb{R}}^{n}\right) \) be bounded. Moreover, assume that \( \sigma {\left( t, x\right) }^{-1} \) exists for all \( \left( {t, x}\right) \in \lbrack 0,\infty... | Proof. Since \( \sigma \in {\mathcal{A}}^{n}\left( {\mathbb{R}}^{n}\right) \) is bounded and continuous, by Theorem 6.10, equation (6.19) has a weak solution \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0},\mathbf{P}, W, X}\right) \) . By the assumptions, it is easily verified that (6.20)... | No |
Theorem 6.12. Under the same assumptions of Theorem 6.11, if in addition, \( \sigma \left( {t, x}\right) \) is Lipschitz in \( x \), uniformly in \( t \in \left\lbrack {0, T}\right\rbrack \), then weak uniqueness holds for solutions of (6.2). | Proof. Since \( \sigma \left( {t, x}\right) \) is Lipschitz in \( x \), uniformly in \( t \in \left\lbrack {0, T}\right\rbrack \) ,(6.19) has a unique strong solution by Theorem 6.3. This implies weak uniqueness by Theorems 6.8 and 6.9. Therefore, the conclusion follows from the above observation. | Yes |
Theorem 6.13. Let \( b : \lbrack 0,\infty ) \times {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) and \( \sigma : \lbrack 0,\infty ) \times {\mathbb{R}}^{n} \rightarrow {\mathcal{S}}^{n} \) be measurable and bounded. Moreover, assume that \( \sigma \) is uniformly positive definite, namely, there is \( \delta > 0 \) ... | For a proof see Krylov [2, pp. 87-91]. | No |
Theorem 6.14. For any \( \xi \in {L}_{{\mathcal{F}}_{0}}^{2}\left( {\Omega ;{\mathbb{R}}^{n}}\right) \), equation (6.25) admits a unique strong solution \( X\left( \cdot \right) \), which is represented by the following:\n\n(6.27)\n\n\[ X\left( t\right) = \Phi \left( t\right) \xi + \Phi \left( t\right) {\int }_{0}^{t}\... | Proof. By Theorem 6.3, we see that (6.28) admits a unique solution \( \Phi \left( \cdot \right) \). To show that \( \Phi {\left( t\right) }^{-1} \) exists, let \( \Psi \left( \cdot \right) \) be the unique strong solution of (6.29), which exists by, once again, Theorem 6.3. Applying Itô’s formula to \( \Phi \left( t\ri... | Yes |
Theorem 6.16. Let \( \left( {RC}\right) \) hold. Then, for any \( \xi \in {L}_{{\mathcal{F}}_{0}}^{\ell }\left( {\Omega ;{\mathbb{R}}^{n}}\right) \left( {\ell \geq 1}\right) \) , (6.34) admits a unique solution \( X \) such that for any \( T > 0 \) , \n\n\( \left( {6.40}\right) \)\n\n\[ \nE\mathop{\max }\limits_{{0 \le... | The proof of this theorem is the same as that of Theorem 6.3. | No |
Theorem 6.17. Under (W), equation (6.43) has a unique strong solution. As a consequence, both pathwise uniqueness and weak existence and uniqueness hold. | Proof. Given any filtered probability space \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0},\mathbf{P}}\right) \) along with an \( m \) -dimensional \( {\left\{ {\mathcal{F}}_{t}\right\} }_{t \geq 0} \) -Brownian motion \( W\left( \cdot \right), Y\left( t\right) \equiv W\left( t\right) \)... | Yes |
Theorem 5.2. Under cither (H1) or (H2), if Problem (SL) is finite, then it admits an optimal control. | Proof. First suppose (H1) holds. Let \( \left( {{x}_{j}\left( \cdot \right) ,{u}_{j}\left( \cdot \right) }\right) \) be a minimizing sequence. By (5.10), we have\n\n(5.11)\n\n\[ \text{ 及 }\begin{aligned} \int \left( {{u}_{j}\left( \cdot \right) }\right) \rightarrow & \overline{\min }{\int }_{0}^{T}{\left| {u}_{j}\left(... | Yes |
Corollary 6.2. Let \( \underline{U \subseteq {\mathbb{R}}^{k}} \) be compact and suppose that (6.5) holds with M closed. Then Problem (DT) admits an optimal control. | Proof. Using the same argument as in the proof of Theorem 5.1, one can show that Problem (DT) with \( U \) replaced by \( \overline{\mathrm{{co}}}U \) admits a time optimal control, say, \( \widehat{u}\left( \cdot \right) \in \widehat{\mathcal{V}}\lbrack 0,\infty ) \equiv \{ u : \lbrack 0,\infty ) \rightarrow \overline... | Yes |
Theorem 6.3. (Lyapunov’s theorem) Suppose \( f \in {L}^{1}\left( {0, T;{\mathbb{R}}^{n}}\right) \) . Then the set\n\n\[ \mathcal{R} \triangleq \left\{ {{\int }_{S}f\left( t\right) {dt} \mid S \in \mathcal{B}\left\lbrack {0, T}\right\rbrack }\right\} \]\n\nis convex, where \( \mathcal{B}\left\lbrack {0, T}\right\rbrack ... | We refer the reader to Hermes-LaSalle [1] for proofs of Proposition 6.1 and Theorem 6.3 (see Diestel-Uhl [1] for a more detailed discussion related to Theorem 6.3). | No |
Consider the following one-dimensional control system:\n\n\[ \left\{ \begin{array}{l} {dx}\left( t\right) = u\left( t\right) {dW}\left( t\right) \\ x\left( 0\right) = 0 \end{array}\right. \]\n\nThe control set \( U \) is \( \{ - 1,1\} \) . Thus, we have a linear controlled system with a nonconvex control set (which is,... | The essence behind the above example is the nonconvexity of the range of vector measures defined through the Itô integral. Let us make this more precise. Let\n\n\[ {SN}\left( t\right) \times 2 = 1\left( t\right) \times B, B \in {Ft} \]\n\n(6.10)\n\n\[ \mathcal{S}\left\lbrack {0, T}\right\rbrack \triangleq \left\{ {S \s... | Yes |
Theorem 6.5. Suppose \( f \in {L}_{\mathcal{F}}^{2}\left( {0, T;{\mathbb{R}}^{n}}\right) \) such that\n\n(6.12)\n\n\[ E{\int }_{0}^{T}{\left| f\left( t\right) \right| }^{2}{dt} > 0 \]\n\nThen there exist \( {S}_{1},{S}_{2} \in \mathcal{S}\left\lbrack {0, T}\right\rbrack \) with\n\n(6.13)\n\n\[ \underbrace{\left| {S}_{i... | Proof. For any \( \varepsilon > 0 \), define\n\n(6.15)\n\n\[ {S}^{\varepsilon } = \{ \left( {t,\omega }\right) \in \left\lbrack {0, T}\right\rbrack \mid \left| {f\left( {t,\omega }\right) }\right| \geq \varepsilon \} \in \mathcal{S}\left\lbrack {0, T}\right\rbrack .\n\]\nLet\n\n(6.16)\n\n\[ {h}^{\varepsilon }\left( t\r... | Yes |
Proposition 6.6. Suppose in (5.7)-(5.8) that\n\n(6.20)\n\n\[ \nB = {CD} \n\]\n\nand \( U = \left\{ {{u}_{0},{u}_{1}}\right\} \) with\n\n(6.21)\n\n\[ \nD\left( {{u}_{1} - {u}_{0}}\right) \neq 0 \n\]\n\nThen, for any \( {x}_{0} \in {\mathbb{R}}^{n} \) and \( T > 0,\overline{{\mathcal{R}}_{S}\left( T\right) } \) is noncon... | Proof. Without loss of generality, we may assume \( U = \{ 0,\bar{u}\} \) with \( D\bar{u} \neq 0 \) and \( {x}_{0} = 0 \) . Let \( \Phi \left( \cdot \right) \) be an \( {\mathbb{R}}^{n \times n} \) -valued process satisfying\n\n(6.22)\n\n\[ \n\left\{ \begin{array}{l} {d\Phi }\left( t\right) = {A\Phi }\left( t\right) {... | Yes |
Theorem 6.7. Let\n\n(6.26)\n\n\[ \left\{ \begin{array}{l} {\mathcal{R}}_{1} = \left\{ {{\int }_{0}^{T}h\left( s\right) {ds} \mid h\left( \cdot \right) \in {L}_{\mathcal{F}}^{2}\left( {0, T;\mathbb{R}}\right) }\right\} , \\ {\mathcal{R}}_{2} = \left\{ {{\int }_{0}^{T}f\left( s\right) {dW}\left( s\right) \mid f\left( \cd... | Proof. First of all, it is clear that\n\n(6.28)\n\n\[ {\overline{\mathcal{R}}}_{1} \subseteq {L}_{{\mathcal{F}}_{T}}^{2}\left( {\Omega ;\mathbb{R}}\right) ,{\overline{\mathcal{R}}}_{2} + \mathbb{R} \subseteq {L}_{{\mathcal{F}}_{T}}^{2}\left( {\Omega ;\mathbb{R}}\right) \]\n\nThus, we need only to prove the other direct... | Yes |
Corollary 6.8. Let a control system be given by (5.7) with \( A, C \in {\mathbb{R}}^{n \times n} \) and \( B, D \in {\mathbb{R}}^{n \times k} \). Let \[ \text{(5.7)}\left\{ \begin{array}{l} {dx}\left( t\right) = \left\lbrack {{Ax}\left( t\right) + {Bu}\left( t\right) }\right\rbrack {dt} + \left\lbrack {{Cx}\left( t\rig... | Proof. Suppose (6.38) holds. Then we can find some \( \eta \in {\mathbb{R}}^{n},\left| \eta \right| = 1 \) , such that \[ {\eta }^{\top }D = 0 \] Multiplying the integral version of (5.7) by \( {\eta }^{\top } \), we obtain \[ {\eta }^{\top }x\left( T\right) = {\eta }^{\top }{x}_{0} + {\int }_{0}^{T}{\eta }^{\top }\lef... | Yes |
Consider the controlled system\n\n\[ \left\{ \begin{array}{l} {dx}\left( t\right) = u\left( t\right) {dt} + \beta \left( t\right) {dW}\left( t\right) ,\;x\left( 0\right) = 0 \\ {dy}\left( t\right) = {2\beta }\left( t\right) {dW}\left( t\right) ,\;y\left( 0\right) = 0 \end{array}\right. \]\n\nwhere \( \beta \left( \cdot... | We now show that this (very simple) optimal control problem does not admit an optimal control. In fact,\n\n\[ J\left( {u\left( \cdot \right) }\right) = E{\left| {\int }_{0}^{T}u\left( s\right) ds - {\int }_{0}^{T}\beta \left( s\right) dW\left( s\right) \right| }^{2}. \]\n\nBy (6.27), \( {L}_{{\mathcal{F}}_{T}}^{2}\left... | Yes |
Theorem 2.1. (Deterministic Maximum Principle) Let (D1)-(D3) hold. Let \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) be an optimal pair of Problem (D). Then there exists a \( p\left( \cdot \right) : \left\lbrack {0, T}\right\rbrack \rightarrow {\mathbb{R}}^{n} \) satisfying the followi... | \[ \left\{ \begin{array}{l} \dot{p}\left( t\right) = - {b}_{x}{\left( t,\bar{x}\left( t\right) ,\bar{u}\left( t\right) \right) }^{\top }p\left( t\right) + {f}_{x}\left( {t,\bar{x}\left( t\right) ,\bar{u}\left( t\right) }\right) ,\;\text{ a.e. }t \in \left\lbrack {0, T}\right\rbrack , \\ p\left( T\right) = - {h}_{x}\lef... | Yes |
Lemma 2.2. Let (D1)-(D3) hold. Let \( {x}^{\varepsilon }\left( \cdot \right) \equiv x\left( {\cdot ;{u}^{\varepsilon }\left( \cdot \right) }\right) \) be the solution to (2.1) under the control \( {u}^{\varepsilon }\left( \cdot \right) \), and let \( {y}^{\varepsilon }\left( \cdot \right) \) be the solution of the foll... | Proof. Let \( {\xi }^{\varepsilon }\left( t\right) \triangleq {x}^{\varepsilon }\left( t\right) - \bar{x}\left( t\right) \) . Then, by (2.3), we have\n\n(2.15)\n\n\[ \left| {{\xi }^{\varepsilon }\left( t\right) }\right| \leq {\int }_{0}^{t}L\left| {{\xi }^{\varepsilon }\left( s\right) }\right| {ds} + {K\varepsilon },\;... | Yes |
Lemma 2.3. Let \( G \subseteq {\mathbb{R}}^{n} \) be a region and \( \varphi : G \rightarrow \mathbb{R} \) be a locally Lipschitz continuous function. For any \( x \in G \), define\n\n\[ \partial \varphi \left( x\right) \triangleq \left\{ {\xi \in {\mathbb{R}}^{n}\mid \langle \xi, y\rangle \leq \underset{t \downarrow 0... | Proof. (i) By assumption, for the given \( x \in G \) and a small \( \delta > 0 \) with \( {B}_{\delta }\left( x\right) \triangleq \left\{ {z \in {\mathbb{R}}^{n} \mid \left| {z - x}\right| \leq \delta }\right\} \subseteq G \), there exists a constant \( L > 0 \) such that\n\n\[ \left| {\varphi \left( z\right) - \varph... | Yes |
Lemma 2.4. Let \( \varphi \) be a convex or concave function on \( {\mathbb{R}}^{n} \times U \) with \( U \subseteq {\mathbb{R}}^{k} \) being a convex body. Assume that \( \varphi \left( {x, u}\right) \) is differentiable in \( x \) and \( {\varphi }_{x}\left( {x, u}\right) \) is continuous in \( \left( {x, u}\right) \... | Proof. First we assume that \( \varphi \) is convex. For any \( \xi \in {\mathbb{R}}^{n} \) and \( u \in {\mathbb{R}}^{k} \) , we choose a sequence \( \left\{ \left( {{x}_{i},{h}_{i}}\right) \right\} \subseteq {\mathbb{R}}^{n} \times \mathbb{R} \) in the following way:\n\n\[ \left( {{x}_{i},{u}^{ * }}\right) \in {\math... | Yes |
Consider the following control system \( \left( {n = m = 1}\right) \) :\n\n(3.11)\n\n\[ \left\{ \begin{array}{l} {dx}\left( t\right) = u\left( t\right) {dW}\left( t\right) ,\;t \in \left\lbrack {0,1}\right\rbrack , \\ x\left( 0\right) = 0, \end{array}\right. \]\n\nwith the control domain being \( U = \left\lbrack {-1,1... | Substituting \( x\left( t\right) = {\int }_{0}^{t}u\left( s\right) {dW}\left( s\right) \) into the cost functional, we obtain via a simple calculation\n\n(3.13)\n\n\[ J = E{\int }_{0}^{1}\left( {\frac{3}{2} - t}\right) u{\left( t\right) }^{2}{dt} \]\n\nHence, the optimal control is \( \bar{u}\left( t\right) \equiv 0 \)... | Yes |
Theorem 3.2. (Stochastic Maximum Principle) Let (S0)-(S3) hold. Let \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) be an optimal pair of Problem (S). Then there are pairs of processes\n\n(3.17)\n\n\[ \left\{ \begin{array}{l} \left( {p\left( \cdot \right), q\left( \cdot \right) }\right) ... | The inequality (3.19) is called the variational inequality, and (3.20) is called the maximum condition. Note that the third term on the left-hand side of (3.19) reflects the risk adjustment, which must be present when \( \sigma \) depends on \( u \) . | Yes |
Consider the following control system \( \left( {n = m = 1}\right) \) :\n\n\[ \left\{ \begin{array}{l} {dx}\left( t\right) = u\left( t\right) {dW}\left( t\right) ,\;t \in \left\lbrack {0, T}\right\rbrack , \\ x\left( 0\right) = 0, \end{array}\right. \] \n\nwith the control domain being \( U = \{ 0,1\} \) and the cost f... | Then the optimal pair is clearly given by \( \left( {\bar{x}\left( t\right) ,\bar{u}\left( t\right) }\right) \equiv \left( {0,0}\right) \) . Now let us try to make a perturbation similar to what we have done for the deterministic case. Let \( {E}_{\varepsilon } = \left\lbrack {s, s + \varepsilon }\right\rbrack \subsete... | Yes |
Lemma 4.2. Let \( Y\left( t\right) \in {L}_{\mathcal{F}}^{2}\left( {0, T;{\mathbb{R}}^{n}}\right) \) be the solution of the following:\n\n(4.6)\n\n\[ \left\{ \begin{array}{l} {dY}\left( t\right) = \{ A\left( t\right) Y\left( t\right) + \alpha \left( t\right) \} {dt} + \mathop{\sum }\limits_{{j = 1}}^{m}\left\{ {{B}^{j}... | Proof. For notational simplicity, we prove only the case \( m = 1 \) (i.e., the Brownian motion \( W\left( t\right) \) is one-dimensional), leaving the case \( m > 1 \) to the interested reader. Thus, the index \( j \) in \( {B}^{j}\left( \cdot \right) \) and \( {\beta }^{j}\left( \cdot \right) \) will be dropped. We f... | No |
Theorem 4.4. Let (S1)-(S3) hold. Then, for any \( k \geq 1 \) ,\n\n(4.26)\n\n\[ \mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}E{\left| {x}^{\varepsilon }\left( t\right) - \bar{x}\left( t\right) \right| }^{2k} = O\left( {\varepsilon }^{k}\right) \] | Proof. For simplicity of presentation, we carry out the proof only for the case \( n = m = 1 \) (thus, the indices \( i \) and \( j \) will be omitted below).\n\n1. Proof of (4.26) and (4.27).\n\nLet \( {\xi }^{\varepsilon }\left( t\right) \triangleq {x}^{\varepsilon }\left( t\right) - \bar{x}\left( t\right) \) . Then ... | No |
Lemma 4.5. Let (S0)-(S3) hold. Let \( {y}^{\varepsilon }\left( \cdot \right) \) and \( {z}^{\varepsilon }\left( \cdot \right) \) be the solutions of (4.23) and (4.24), respectively. Let \( \left( {p\left( \cdot \right), q\left( \cdot \right) }\right) \) be the adapted solution of (3.8). Then\n\n\[ E\left\langle {p\left... | The proof follows immediately from Itô's formula, and we leave it to the reader (see also Chapter 1, Corollary 5.6). | No |
Lemma 4.6. Let \( Y\left( \cdot \right), P\left( \cdot \right) \in {L}_{\mathcal{F}}^{2}\left( {0, T;{\mathbb{R}}^{n \times n}}\right) \) satisfy the following:\n\n(4.62)\n\n\[ \left\{ \begin{array}{l} {dY}\left( t\right) = \Phi \left( t\right) {dt} + \mathop{\sum }\limits_{{j = 1}}^{m}{\Psi }_{j}\left( t\right) d{W}^{... | (4.63)\n\n\[ = E{\int }_{0}^{T}\left\{ {\operatorname{tr}\left\lbrack {\Theta \left( t\right) Y\left( t\right) + P\left( t\right) \Phi \left( t\right) + \mathop{\sum }\limits_{{j = 1}}^{m}{Q}_{j}\left( t\right) {\Psi }_{j}\left( t\right) }\right\rbrack }\right\} {dt}. \] | Yes |
Lemma 5.1. Let (S0)-(S4) hold. Let \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right), p\left( \cdot \right), q\left( \cdot \right), P\left( \cdot \right), Q\left( \cdot \right) }\right) \) be a given admissible 6-tuple and let \( \mathcal{H} \) be the corresponding \( \mathcal{H} \) -function. Then fo... | Proof. Fix a \( t \in \left\lbrack {0, T}\right\rbrack \) and \( \omega \in \Omega \) . Define\n\n\[ \n\left\{ \begin{array}{l} H\left( u\right) \triangleq H\left( {t,\bar{x}\left( t\right), u, p\left( t\right), q\left( t\right) }\right) , \\ \mathcal{H}\left( u\right) \triangleq \mathcal{H}\left( {t,\bar{x}\left( t\ri... | Yes |
Consider the following control system \( \left( {n = m = 1}\right) \) :\n\n(5.8)\n\n\[ \left\{ \begin{array}{l} {dx}\left( t\right) = u\left( t\right) {dW}\left( t\right) ,\;t \in \left\lbrack {0,1}\right\rbrack , \\ x\left( 0\right) = 0, \end{array}\right. \]\n\nwith the control domain being \( U = \left\lbrack {0,1}\... | Suppose \( \left( {\overline{x}\left( \cdot \right) ,\overline{u}\left( \cdot \right) }\right) \) is an optimal pair (which we are going to identify). Then the corresponding adjoint equations are\n\n(5.10)\n\n\[ \left\{ \begin{array}{l} {dp}\left( t\right) = q\left( t\right) {dW}\left( t\right) ,\;t \in \left\lbrack {0... | Yes |
Theorem 6.1. Let (S0)-(S3) and (S5) hold. Let \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) be an optimal pair of Problem (SC). Then there exist a \( \left( {{\psi }^{0},\psi }\right) \in {\mathbb{R}}^{1 + \ell } \) satisfying\n\n(6.5)\n\n\[ \n{\psi }^{0} \geq 0,\;{\left| {\psi }^{0}\r... | and adapted solutions\n\n\[ \n\left\{ \begin{array}{l} \left( {p\left( \cdot \right), q\left( \cdot \right) }\right) \in {L}^{2}\left( {0, T;{\mathbb{R}}^{n}}\right) \times {\left\lbrack {L}^{2}\left( 0, T;{\mathbb{R}}^{n}\right) \right\rbrack }^{m}, \\ \left( {P\left( \cdot \right), Q\left( \cdot \right) }\right) \in ... | Yes |
Corollary 6.3. Let the assumptions of Lemma 6.2 hold. Let \( \rho > 0 \) and \( {v}_{0} \in V \) be such that\n\n(6.26)\n\n\[ F\left( {v}_{0}\right) \leq \mathop{\inf }\limits_{{v \in V}}F\left( v\right) + \rho \]\n\nThen there exists a \( {v}_{\rho } \in V \) such that\n\n(6.27)\n\n\[ F\left( {v}_{\rho }\right) \leq F... | Proof. We take \( \lambda = \sqrt{\rho } \) . By Lemma 6.2, there exists a \( {v}_{\rho } \in V \) such that (noting (6.26))\n\n(6.29)\n\n\[ F\left( {v}_{\rho }\right) + \sqrt{\rho }d\left( {{v}_{\rho },{v}_{0}}\right) \leq F\left( {v}_{0}\right) \leq \mathop{\inf }\limits_{{v \in V}}F\left( v\right) + \rho \leq F\left... | Yes |
Lemma 6.4. Let \( \bar{d} \) be defined by the following:\n\n(6.31)\n\n\[ \bar{d}\left( {u\left( \cdot \right) ,\widehat{u}\left( \cdot \right) }\right) \triangleq \left| {\{ \left( {t,\omega }\right) \in \left\lbrack {0, T}\right\rbrack \times \Omega \mid u\left( {t,\omega }\right) \neq \widehat{u}\left( {t,\omega }\r... | Proof. Let \( \left\{ {{u}_{n}\left( \cdot \right) }\right\} \) be a Cauchy sequence in \( \mathcal{U}\left\lbrack {0, T}\right\rbrack \) under the metric \( \bar{d} \), i.e.,\n\n(6.32)\n\n\[ \bar{d}\left( {{u}_{n}\left( \cdot \right) ,{u}_{m}\left( \cdot \right) }\right) \rightarrow 0,\;n, m \rightarrow \infty . \]\n\... | Yes |
Theorem 2.1. Let \( \left( {D1}\right) \) and \( {\left( D2\right) }^{\prime } \) hold. Then for any \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \), (2.7) \[ V\left( {s, y}\right) = \mathop{\inf }\limits_{{u\left( \cdot \right) \in \mathcal{V}\left\lbrack {s, T}\right\rbrack }}\left\{ {{\int }_{s}... | Proof. Let us denote the right-hand side of (2.7) by \( \bar{V}\left( {s, y}\right) \). By (2.6), we have \[ V\left( {s, y}\right) \leq J\left( {s, y;u\left( \cdot \right) }\right) = {\int }_{s}^{\widehat{s}}f\left( {t, x\left( t\right), u\left( t\right) }\right) {dt} + J\left( {\widehat{s}, x\left( \widehat{s}\right) ... | Yes |
Proposition 2.2. Let \( \left( {D1}\right) \) and \( {\left( D2\right) }^{\prime } \) hold. Suppose \( V \in {C}^{1}\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) . Then \( V \) is a solution to the following terminal value problem of a first-order partial differential equation:\n\n(2.14)\n... | Proof. Fix a \( u \in U \) . Let \( x\left( \cdot \right) \) be the state trajectory corresponding to the control \( u\left( t\right) \equiv u \) . By (2.7) with \( \widehat{s} \downarrow s \) ,\n\n\[ 0 \geq - \frac{V\left( {\widehat{s}, x\left( \widehat{s}\right) }\right) - V\left( {s, y}\right) }{\widehat{s} - s} - \... | Yes |
Lemma 2.7. Let \( v \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) and \( \left( {{t}_{0},{x}_{0}}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) be given. Then\n\n(i) \( \left( {q, p}\right) \in {D}_{t, x}^{1, + }v\left( {{t}_{0},{x}_{0}}\right) \) if and only if there exists a f... | Proof. We prove (ii). The proof of (i) is similar. Suppose \( \left( {q, p}\right) \in \) \( {D}_{t+, x}^{1, + }v\left( {{t}_{0},{x}_{0}}\right) \) . Let\n\n\[ \Phi \left( {t, x}\right) = \left\{ \begin{array}{l} \frac{\left( {v\left( {t, x}\right) - v\left( {{t}_{0},{x}_{0}}\right) - q\left( {t - {t}_{0}}\right) -\lan... | Yes |
Theorem 2.9. A function \( v \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) is a viscosity solution of (2.14) if and only if \( v\left( {T, x}\right) = h\left( x\right) \) for all \( x \in {\mathbb{R}}^{n} \), and for all \( \left( {t, x}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n... | Proof. Suppose \( v \) is a viscosity solution of (2.14). Then, for any \( \left( {q, p}\right) \in \) \( {D}_{t, x}^{1, + }v\left( {t, x}\right) \), by Lemma 2.7 we can find a \( \varphi \in {C}^{1}\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) such that \( v - \varphi \) attains a strict ... | Yes |
Theorem 2.10. Let \( \left( {D1}\right) \) and \( {\left( D2\right) }^{\prime } \) hold. Then the value function \( V \in \) \( C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) is the only function that satisfies the following: For all \( \left( {t, x}\right) \in \lbrack 0, T) \times {\mathb... | Proof. For any \( \left( {q, p}\right) \in {D}_{t +, x}^{1, \pm }V\left( {t, x}\right) \), take the function \( \varphi \) as specified by Lemma 2.7-(ii) or Lemma 2.8-(ii) (with \( v \) replaced by \( V \) ). Then we can use exactly the same argument as in Step 2 of the proof of Theorem 2.5 (note that only the right li... | Yes |
Proposition 3.1. Let \( {\left( S1\right) }^{\prime } - {\left( S2\right) }^{\prime } \) hold. Then the value function \( V\left( {s, y}\right) \) satisfies the following:\n\n(3.9)\n\n\[ \left| {V\left( {s, y}\right) }\right| \leq K\left( {1 + \left| y\right| }\right) ,\;\forall \left( {s, y}\right) \in \left\lbrack {0... | Proof. Let \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) be fixed. For any \( \left( {\Omega ,\mathcal{F},\mathbf{P}, W\left( \cdot \right), u\left( \cdot \right) }\right) \in \) \( {\mathcal{U}}^{w}\left\lbrack {s, T}\right\rbrack \), by Theorem 6.16 of Chapter 1, we have\n\n(3.11)\n\n\[ E\matho... | Yes |
Lemma 3.2. Let \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) and \( \left( {\Omega ,\mathcal{F},\mathbf{P}, W\left( \cdot \right), u\left( \cdot \right) }\right) \in {\mathcal{U}}^{w}\left\lbrack {s, T}\right\rbrack \) . Then, for any \( \widehat{s} \in \lbrack s, T) \) and \( {\mathcal{F}}_{\wid... | Proof. Since \( u\left( \cdot \right) \) is \( {\left\{ {\mathcal{F}}_{t}^{s}\right\} }_{t \geq s} \) -adapted with \( {\mathcal{F}}_{t}^{s} = \sigma \{ W\left( r\right) : s \leq r \leq t\} \) , by Theorem 2.10 in Chapter 1 there is a function \( \psi \in {\mathcal{A}}_{T}^{m}\left( U\right) \) such that \[ u\left( {t,... | Yes |
Theorem 3.4. Let \( {\left( S1\right) }^{\prime } - {\left( S2\right) }^{\prime } \) hold. If \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) is optimal for Problem \( \left( {S}_{sy}\right) \), then\n\n\[ V\left( {t,\bar{x}\left( t\right) }\right) = E\left\{ {{\int }_{t}^{T}f\left( {r,\... | Proof. By the same argument as that in the proof of Theorem 3.3, we have\n\n\[ V\left( {s, y}\right) = J\left( {s, y;\bar{u}\left( \cdot \right) }\right) \]\n\n\[ = E\left\{ {{\int }_{s}^{t}f\left( {r,\bar{x}\left( r\right) ,\bar{u}\left( r\right) }\right) {dr}}\right. \]\n\n\[ \left. {+E\left\lbrack {{\int }_{t}^{T}f\... | Yes |
Corollary 4.2. Let \( {\left( S1\right) }^{\prime } - {\left( S2\right) }^{\prime } \) hold. Then there exists a constant \( K > 0 \) such that\n\n\[ \left| {{V}^{\varepsilon }\left( {s, y}\right) - V\left( {s, y}\right) }\right| \leq K\sqrt{\varepsilon },\;\forall \left( {s, y}\right) \in \left\lbrack {0, T}\right\rbr... | Proof. In the present case, (4.4) can be improved to\n\n\[ E\left\lbrack {\mathop{\sup }\limits_{{s \leq t \leq T}}{\left| {x}^{\varepsilon }\left( t\right) - x\left( t\right) \right| }^{2}}\right\rbrack \leq {K\varepsilon } \]\n\nand (4.7) is replaced by\n\n\[ \left| {{J}^{\varepsilon }\left( {s, y;u\left( \cdot \righ... | Yes |
A function \( \varphi : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) is semiconcave if and only if for some constant \( K \geq 0 \), \[ {\lambda \varphi }\left( x\right) + \left( {1 - \lambda }\right) \varphi \left( y\right) - \varphi \left( {{\lambda x} + \left( {1 - \lambda }\right) y}\right) \] \[ \leq {K\lambda }\lef... | Proof. Note that \( \varphi \) is semiconcave in the sense of Definition 4.3 if and only if there exists a constant \( K \geq 0 \) such that for any \( \lambda \in \left\lbrack {0,1}\right\rbrack \) and \( x, y \in {\mathbb{R}}^{n}, \) \[ \varphi \left( {{\lambda x} + \left( {1 - \lambda }\right) y}\right) - K{\left| \... | Yes |
Example 5.3. Let\n\n(5.14)\n\n\[ v\left( {t, x}\right) = - {t}^{ + } - {x}^{ + },\;\left( {t, x}\right) \in {\mathbb{R}}^{2}. \] | A direct computation shows that\n\n\[ \left\{ \begin{array}{l} {D}_{t, x}^{1,2, + }v\left( {0,0}\right) = \left\lbrack {-1,0}\right\rbrack \times \{ \left\lbrack {\left( {-1,0}\right) \times \mathbb{R}}\right\rbrack \cup \left\lbrack {\{ - 1,0\} \times \lbrack 0,\infty )\rbrack \} }\right\rbrack \\ {D}_{t +, x}^{1,2, +... | Yes |
Lemma 5.4. Let \( v \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) and \( \left( {{t}_{0},{x}_{0}}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) be given. Then:\n\n(i) \( \left( {q, p, P}\right) \in {D}_{t, x}^{1,2, + }v\left( {{t}_{0},{x}_{0}}\right) \) if and only if there exis... | Proof. We prove (ii). The proof of (i) is similar. Suppose \( \left( {q, p, P}\right) \in \) \( {D}_{t+, x}^{1,2, + }v\left( {{t}_{0},{x}_{0}}\right) \) . Let\n\n\[ \Phi \left( {t, x}\right) = \left\{ \begin{matrix} \frac{1}{t - {t}_{0} + {\left| x - {x}_{0}\right| }^{2}}\left\lbrack {v\left( {t, x}\right) - v\left( {{... | Yes |
Lemma 5.5. Let \( v \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) and \( \left( {{t}_{0},{x}_{0}}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) . (i) \( \left( {q, p, P}\right) \in {D}_{t, x}^{1,2, - }v\left( {{t}_{0},{x}_{0}}\right) \) if and only if there exists a function \( ... | (ii) \( \left( {q, p, P}\right) \in {D}_{t +, x}^{1,2, - }v\left( {{t}_{0},{x}_{0}}\right) \) if and only if there exists a function \( \varphi \in \) \( {C}^{1,2}\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) such that (5.31) \[ \left\{ \begin{array}{l} \left( {\varphi \left( {{t}_{0},{x}_... | Yes |
Proposition 5.6. A function \( v \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) is a viscosity subsolution of (3.24) if and only if \( v\left( {T, x}\right) \leq h\left( x\right) \) and\n\n(5.32)\n\n\[ \n- q + \mathop{\sup }\limits_{{u \in U}}G\left( {t, x, u, - p, - P}\right) \leq 0,\... | Proof. The result is immediate in view of Lemmas 5.4 and 5.5. | No |
Corollary 5.7. Let \( {\left( S1\right) }^{\prime } - {\left( S2\right) }^{\prime } \) hold. Let \( v \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) be a viscosity solution of (3.24). Then there exists a constant \( K > 0 \) such that\n\n(5.34)\n\n\[ \left\{ \begin{matrix} q \geq - K\l... | Proof. The result follows from Proposition 5.6 and \( {\left( \mathrm{S}1\right) }^{\prime } - {\left( \mathrm{S}2\right) }^{\prime } \) immediately. | Yes |
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