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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( {\widehat{\Omega },\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 \... | 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}... | 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\[ \nE{\\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... | No |
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(... | 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 \( {\sigma }_{E}\left( \omega \right) < s \) . Then by the con... | Yes |
Proposition 3.4. Let \( \sigma ,\tau \), and \( {\sigma }_{i} \) be stopping times. Then\n\n(i) The following are also stopping times:\n\n\[ \sigma + \tau ,\mathop{\sup }\limits_{i}{\sigma }_{i},\mathop{\inf }\limits_{i}{\sigma }_{i},\mathop{\lim }\limits_{i}{\sigma }_{i},\mathop{\lim }\limits_{i}{\sigma }_{i}. \] | For proofs of the above results, see Karatzas-Shreve [3, pp. 6-10]. | No |
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 \( \tau \) an \( {\left\{ ... | 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 \( t \geq 0 \), t... | 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 |
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\{ {\mathcal{F}}_{t}... | 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). | Yes |
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 sub... | Proof. By the monotonicity of \( \\varphi \\left( \\cdot \\right) \) and Jensen’s inequality (see (1.36)), we have\n\n(4.2)\n\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... | 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 }}\right\rbrack = 0,\;... | 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\n\[ X\left( {... | 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\rig... | 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, \(... | Note that for fixed \( \omega \in \Omega \) and \( t \geq 0, X\left( {s,\omega }\right) \) is bounded on \( \left\lbrack {0, t}\right\rbrack \) . Thus, the first integral exists. The second integral is defined as in the previous subsection. Let us make an observation. Let \( \left( {\Omega ,\mathcal{F},{\left\{ {\mathc... | Yes |
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... | 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 either (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(6.7)\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}\righ... | 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... | 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\... | Yes |
Proposition 6.6. Suppose in (5.7)-(5.8) that\n\n(6.20)\n\n\[ B = {CD} \]\n\nand \( U = \left\{ {{u}_{0},{u}_{1}}\right\} \) with\n\n(6.21)\n\n\[ D\left( {{u}_{1} - {u}_{0}}\right) \neq 0 \]\n\nThen, for any \( {x}_{0} \in {\mathbb{R}}^{n} \) and \( T > 0,\overline{{\mathcal{R}}_{S}\left( T\right) } \) is nonconvex. | 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\[ \left\{ \begin{array}{l} {d\Phi }\left( t\right) = {A\Phi }\left( t\right) {dt... | 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 \( B, D \in {\mathbb{R}}^{n \times k} \). Then \[ {\mathcal{R}}_{S}\left( T\right) \neq {L}_{{\mathcal{F}}_{T}}^{2}\left( {\Omega ;{\mathbb{R}}^{n}}\right) ,\;\forall T > 0. \] | 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... | 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), \( {L}_{{\mathcal{F}}_{T}}^{2}\left( {\Omega ;\mathbb{R}}\right) = {\overline{\mathcal{R}}}_{1} \) . Thus, \( \mathop{\inf }\limit... | 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... | (2.6)\n\n\[ \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... | 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\( \left( {2.20}\right) \)\n\n\[ \n\partial \varphi \left( x\right) \triangleq \left\{ {\xi \in {\mathbb{R}}^{n}\mid \langle \xi, y\rang... | Proof. (i) By assumption, for the given \( x \in G \) and a small \( \delta > 0 \) with \( {B}_{\delta }\left( x\right) \triangleq \left\{ {z \in {\mathbb{R}}^{n}\left| \right| z - x \mid \leq \delta }\right\} \subseteq G \), there exists a constant \( L > 0 \) such that\n\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(4.1)\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 t... | 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 ... | Yes |
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 \[ E\left\langle {p\left( T... | 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 |
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 |
Theorem 2.5. Let \( \left( {D1}\right) \) and \( {\left( D2\right) }^{\prime } \) hold. Then the value function \( V\left( {\cdot , \cdot }\right) \) satisfies \[ \left| {V\left( {s, y}\right) - V\left( {\bar{s},\bar{y}}\right) }\right| \leq K\left\{ {\left| {y - \bar{y}}\right| + \left( {1 + \left| y\right| \vee \left... | Proof. We split the proof into several steps.\n\nStep 1. \( V\left( {\cdot , \cdot }\right) \) satisfies (2.36).\n\nBy Gronwall’s inequality, taking into account \( {\left( \mathrm{D}2\right) }^{\prime } \), we easily get \[ \left\{ \begin{array}{l} \left| {x\left( {t;s, y, u\left( \cdot \right) }\right) }\right| \leq ... | Yes |
Lemma 2.8. 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... | (ii) \( \left( {q, p}\right) \in {D}_{t+, x}^{1, - }v\left( {{t}_{0},{x}_{0}}\right) \) if and only if there exists a function \( \varphi \in {C}^{1}(\mathbb{R} \times \n\n\( \left. {\mathbb{R}}^{n}\right) \) such that\n\n\( \left( {2.77}\right) \n\n\[ \left\{ \begin{array}{l} \left( {\varphi \left( {{t}_{0},{x}_{0}}\r... | 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 |
Proposition 4.4. A function \( \varphi : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) is semiconcave if and only if for some constant \( K \geq 0 \), \[ {\lambda \varphi }\left( x\right) + \left( {1 - \lambda }\right) \varphi \left( y\right) - \varphi \left( {{\lambda x} + \left( {1 - \lambda }\right) y}\right) \] \[ \le... | 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 |
Let \( v\left( {t, x}\right) = - {t}^{ + } - {x}^{ + },\;\left( {t, x}\right) \in {\mathbb{R}}^{2} \). | A direct computation shows that \[ \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, + }v... | 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}_... | No |
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,\;\forall \left( {... | 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 |
Proposition 5.8. Let \( v \in {C}^{1,2}\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) . Then \( v \) is a viscosity solution of (3.24) if and only if it is a classical solution of (3.24). | Proof. The result is clear from Proposition 5.6, (5.13), and the fact that the generalized Hamiltonian \( G \), defined by (3.25), is nondecreasing in its last argument \( P \in {\mathcal{S}}^{n} \) . | No |
Proposition 5.9. Let \( {v}^{\varepsilon } \) be a viscosity solution of the following:\n\n(5.35)\n\n\[ \left\{ \begin{array}{l} - {v}_{t}^{\varepsilon } + {G}^{\varepsilon }\left( {t, x, - {v}_{x}^{\varepsilon }, - {v}_{xx}^{\varepsilon }}\right) = 0,\;\left( {t, x}\right) \in \left\lbrack {0, T}\right\rbrack \times {... | Proof. First of all, \( {v}^{0} \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) . Next, let \( \varphi \in {C}^{1,2}\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) such that \( {v}^{0} - \varphi \) attains a strict local maximum at \( \left( {{t}_{0},{x}_{0}}... | Yes |
Proposition 5.10. Let \( {v}^{\varepsilon } \) be a classical solution of (5.40) and let \( {v}^{0} \) be a viscosity solution of (3.24). Then there exists a constant \( K > 0 \) such that\n\n(5.41)\n\n\[ \left| {{v}^{\varepsilon }\left( {t, x}\right) - {v}^{0}\left( {t, x}\right) }\right| \leq K\sqrt{\varepsilon },\;\... | Proof. First of all, by virtue of Theorem 5.2, the value functions \( V \) and \( {V}^{\varepsilon } \) of Problems (S) and \( \left( {\mathrm{S}}^{\varepsilon }\right) \) (see Section 4.1) are viscosity solutions of (3.24) and (5.40), respectively. Next, from Theorem 6.1, which will be proved in Section 6 below, it fo... | No |
Theorem 6.1. Let \( {\left( S1\right) }^{\prime } - {\left( S2\right) }^{\prime } \) hold. Then the HJB equation (3.24) admits at most one viscosity solution \( v\left( {\cdot , \cdot }\right) \) in the class of functions satisfying (3.9)-(3.10). | Indeed, we have the following stronger result in terms of the right super- /subdifferentials (compare with Theorem 2.10). | No |
Theorem 6.2. Let \( {\left( S1\right) }^{\prime } - {\left( S2\right) }^{\prime } \) hold. Then the value function \( V\left( {\cdot , \cdot }\right) \in \) \( C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) of Problem (S) is the only function that satisfies (3.9)-(3.10) and the following: ... | Proof. We may prove the conclusion the same way as we proved Theorem 2.10, by using Theorem 5.2, Lemma 5.4-(ii), (5.9), and Theorem 6.1. | No |
Lemma 6.5. Let \( {\left( S1\right) }^{\prime } - {\left( S2\right) }^{\prime } \) hold and \( v\left( {\cdot , \cdot }\right) \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) be a viscosity subsolution of (3.24). Then, for each \( \gamma > 0,{v}^{\gamma }\left( {\cdot , \cdot }\right) \... | Proof. Let \( \varphi \in {C}^{1,2}\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) such that \( {v}^{\gamma } - \varphi \) attains a maximum at \( \left( {t, x}\right) \) . Suppose \( \left( {\widehat{t},\widehat{x}}\right) \in \left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n} \) sati... | Yes |
Proposition 2.1. Suppose for any fixed \( \left( {t, x}\right) \), Legendre’s transformation (2.12) can be performed both from \( \left( {\dot{x}, L}\right) \) to \( \left( {p, H}\right) \) and from \( \left( {p, H}\right) \) to \( \left( {\dot{x}, L}\right) \), with \( \varphi \left( {t, x, p}\right) \) in (2.9) being... | Proof. First, let \( x\left( \cdot \right) \) be a solution of (2.7) with the Lagrangian \( L\left( {t, x,\dot{x}}\right) \) given, and define \( p\left( \cdot \right) \) as in the proposition. By (2.12), the first equation in (2.13) is already satisfied. Next, since (noting (2.12))\n\n\[ L\left( {t, x,\dot{x}}\right) ... | Yes |
Theorem 2.3. Let \( v\left( {t, x, a}\right) + {a}_{0} \) be a complete integral of (2.14) and let (2.16) determine a pair of functions \( \left( {x\left( {t;a, b}\right), p\left( {t;a, b}\right) }\right) \) defined for all \( \left( {t, a, b}\right) \in {\mathbb{R}}^{1 + {2n}} \) . Then \( \left( {x\left( {\cdot ;a, b... | Proof. First of all, by the definition of \( v\left( {t, x, a}\right) \), we have\n\n\( \left( {2.17}\right) \)\n\n\[ \n{v}_{t}\left( {t, x, a}\right) + H\left( {t, x,{v}_{x}\left( {t, x, a}\right) }\right) = 0.\n\]\n\nBy differentiating (2.17) with respect to \( x \) and \( a \), respectively, we obtain the following:... | Yes |
Theorem 3.1. Let \( \left( {D1}\right) \) and \( {\left( D2\right) }^{\prime } - {\left( D3\right) }^{\prime } \) hold and \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) be fixed. Let \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right), p\left( \cdot \right) }\right) \) be an optim... | Proof. By the optimality of \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \), we have\n\n(3.11)\n\n\[ \nV\left( {t,\bar{x}\left( t\right) }\right) = h\left( {\bar{x}\left( T\right) }\right) + {\int }_{t}^{T}f\left( {r,\bar{x}\left( r\right) ,\bar{u}\left( r\right) }\right) {dr},\;\forall ... | Yes |
Theorem 3.2. Let (D1) and \( {\left( D2\right) }^{\prime } - {\left( D3\right) }^{\prime } \) hold. Suppose there exists a control \( u\left( \cdot \right) \in \mathcal{V}\left\lbrack {0, T}\right\rbrack \) such that for any \( \left( {s, y}\right) \in \left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n} \) there e... | Proof. First, by a standard result in ordinary differential equation theory \( {x}^{s, y}\left( \cdot \right) \) is differentiable in \( \left( {s, y}\right) \) for any \( y \in {\mathbb{R}}^{n} \) and almost all \( s \in \) \( \lbrack 0, T) \) . Define\n\n(3.22)\n\n\[ m\left( t\right) \triangleq \frac{\partial }{\part... | Yes |
Corollary 3.3. Let \( {x}^{s, y}\left( \cdot \right) \) be the solution of\n\n(3.31)\n\n\[ \left\{ \begin{array}{l} {\dot{x}}^{s, y}\left( t\right) = b\left( {t,{x}^{s, y}\left( t\right) }\right) ,\;t \in \left\lbrack {s, T}\right\rbrack , \\ {x}^{s, y}\left( s\right) = y. \end{array}\right. \]\n\nThen the solution of ... | Proof. Note that (3.20) is valid automatically in the present case. Thus, Theorem 3.2 applies. | No |
Consider the following control system: \( \left( {n = 1}\right) \)\n\n\[ \left\{ \begin{array}{l} \dot{x}\left( t\right) = x\left( t\right) u\left( t\right) ,\;t \in \left\lbrack {s, T}\right\rbrack , \\ x\left( s\right) = y, \end{array}\right. \]\n\nwith the control domain being \( U = \left\lbrack {0,1}\right\rbrack ... | The value function can be easily calculated as\n\n\[ V\left( {t, x}\right) = \left\{ \begin{array}{ll} - x, & \text{ if }x \leq 0 \\ - x{e}^{T - t}, & \text{ if }x > 0 \end{array}\right. \] | Yes |
Corollary 3.6. Let \( \left( {D1}\right) \) and \( {\left( D2\right) }^{\prime } - {\left( D3\right) }^{\prime } \) hold. Let a state trajectory \( x\left( \cdot \right) \) be such that the set \[ \left\{ {t \in \left\lbrack {0, T}\right\rbrack \mid {D}_{t, x}^{1, - }V\left( {t, x\left( t\right) }\right) }\right. \text... | Proof. If \( x\left( \cdot \right) \) is optimal, then for almost every \( t \in \left\lbrack {0, T}\right\rbrack ,{D}_{t, x}^{1, - }V\left( {t, x\left( t\right) }\right) \) is either empty or a singleton by (3.35). This results in a contradiction. | Yes |
Theorem 3.7. Let \( \left( {D1}\right) \) and \( {\left( D2\right) }^{\prime } \) hold. Let \( v \in {C}^{1,1}\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) be a solution of the HJB equation (3.6). Then\n\n(3.52)\n\n\[ v\left( {s, y}\right) \leq J\left( {s, y;u\left( \cdot \right) }\right) ... | Proof. For any \( u\left( \cdot \right) \in \mathcal{V}\left\lbrack {s, T}\right\rbrack \) with the corresponding state trajectory \( x\left( \cdot \right) \), we have\n\n\[ \frac{d}{dt}v\left( {t, x\left( t\right) }\right) = {v}_{t}\left( {t, x\left( t\right) }\right) + \left\langle {{v}_{x}\left( {t, x\left( t\right)... | Yes |
Lemma 3.8. Let \( Q \) be an open subset of \( {\mathbb{R}}^{n} \), and \( v : \bar{Q} \rightarrow \mathbb{R} \) a continuous function. If \( {v}^{\prime }\left( {\widehat{x};\xi }\right) \) exists for given \( \widehat{x} \in Q \) and \( \xi \in {\mathbb{R}}^{n} \), then\n\n(3.57)\n\n\[ \mathop{\sup }\limits_{{p \in {... | Proof. The result is clear if \( \xi = 0 \) . So we assume \( \xi \neq 0 \) . For any \( p \in {D}_{x}^{1, + }v\left( \widehat{x}\right) \), by definition,\n\n\[ \mathop{\lim }\limits_{{h \rightarrow 0 + }}\frac{v\left( {\widehat{x} + {h\xi }}\right) - v\left( \widehat{x}\right) -\langle p,{h\xi }\rangle }{h\left| \xi ... | Yes |
Let \( \left( {D1}\right) \) and \( {\left( D2\right) }^{\prime } \) hold. Let \( v \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) be a viscosity solution of the HJB equation (3.6). Then (3.52) holds. Furthermore, let \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \)... | Proof. The first conclusion is clear, because \( v \) coincides with the value function \( V \) due to the uniqueness of the viscosity solutions to the HJB equation (3.6) (see Chapter 4, Theorem 2.5). Now we prove the second assertion. The \ | No |
Corollary 3.10. Let \( \left( {D1}\right) \) and \( {\left( D2\right) }^{\prime } \) hold. Let \( v \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) be the viscosity solution of the HJB equation (3.6). Let \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) be fixed and ... | Proof. By (3.37), we have the necessity of condition (3.65) (in fact, the conclusion would be stronger, i.e.,(3.65) holds for any point \( \left( {q, p}\right) \in \) \( \left. {{D}_{t, x}^{1, + }v\left( {t,\bar{x}\left( t\right) }\right) \cup {D}_{t, x}^{1, - }v\left( {t,\bar{x}\left( t\right) }\right) }\right) \) . W... | Yes |
Consider the same problem as in Example 3.5. Take an admissible pair \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \equiv \left( {0,0}\right) \) for Problem \( \left( {\mathrm{D}}_{00}\right) \) . Theorem 3.7 cannot tell whether this pair is optimal, since \( {V}_{x}\left( {t, x}\right) \... | On the other hand, from (3.51) we have\n\n\[ \left\{ {\begin{array}{l} {D}_{t, x}^{1, + }V\left( {t,\bar{x}\left( t\right) }\right) = {D}_{t, x}^{1, + }V\left( {t,0}\right) = \{ 0\} \times \left\lbrack {-{e}^{T - t}, - 1}\right\rbrack , \\ H\left( {t,\bar{x}\left( t\right), u, - p}\right) = 0,\;\forall p \in \mathbb{R}... | Yes |
Corollary 3.12. Let \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) be an admissible pair of Problem \( \left( {D}_{sy}\right) \) . If there exists a set \( {T}_{0} \subseteq \left\lbrack {s, T}\right\rbrack \) with positive Lebesgue measure, such that for any \( t \in {T}_{0} \), there ... | Proof. The result follows immediately from Theorem 3.4. | No |
Lemma 3.14. Let \( v \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) be the viscosity solution of the HJB equation (3.6). Then for each \( \left( {t, x}\right) \in \left( {0, T}\right) \times {\mathbb{R}}^{n} \) , (3.67) \[ \left\{ \begin{array}{l} - q + \mathop{\sup }\limits_{{u \in U}... | Proof. The first relation follows from the definition of viscosity solutions (see (3.34)). For the second one, recall that \( {D}_{t, x}^{1, - }v\left( {t, x}\right) \subseteq \partial v\left( {t, x}\right) \), the latter being Clarke's generalized gradient, which equals the convex hull of the set of all the limits \( ... | Yes |
Theorem 4.1. Let \( {\left( S1\right) }^{\prime } - {\left( S2\right) }^{\prime } \) hold and let \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) be fixed. Let \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right), p\left( \cdot \right), q\left( \cdot \right) }\right) \) be an optimal... | Proof. By Chapter 4, Theorem 3.4, we have \[ V\left( {t,\bar{x}\left( t\right) }\right) = E\left\{ {{\int }_{t}^{T}f\left( {r,\bar{x}\left( r\right) ,\bar{u}\left( r\right) }\right) {dr} + h\left( {\bar{x}\left( T\right) }\right) \mid {\mathcal{F}}_{t}^{s}}\right\} , \] \[ \forall t \in \left\lbrack {s, T}\right\rbrack... | Yes |
Corollary 4.2. Let \( {\left( S1\right) }^{\prime } - {\left( S2\right) }^{\prime } \) hold and \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) be fixed. Let \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) be an optimal pair of Problem \( \left( {S}_{sy}\right) \) . ... | Proof. This is immediate from (4.17) and (4.19). | Yes |
Consider the following control system \( \left( {n = m = 1}\right) \) :\n\n(4.55)\n\n\[ \left\{ \begin{array}{l} {dx}\left( t\right) = {2u}\left( t\right) {dt} + \sqrt{2}{dW}\left( t\right) ,\;t \in \left\lbrack {0, T}\right\rbrack , \\ x\left( s\right) = y, \end{array}\right. \]\n\nwith the control domain being \( U =... | For any fixed \( \left( {s, y}\right) \) and \( u\left( \cdot \right) \in {\mathcal{U}}^{w}\left\lbrack {s, T}\right\rbrack \), applying Itô’s formula to the process \( \log \operatorname{ch}x\left( t\right) \), then combining with (4.56), we get\n\n\[ J\left( {s, y;u\left( \cdot \right) }\right) + \log \operatorname{c... | Yes |
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