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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 |
Lemma 6.6. (Alexandrov’s theorem) Let \( Q \subseteq {\mathbb{R}}^{n} \) be a convex body and \( \varphi : Q \rightarrow \mathbb{R} \) a semiconvex (or semiconcave) function. Then there exists a set \( N \subseteq Q \) with \( \left| N\right| = 0 \) such that at any \( x \in Q \smallsetminus N,\varphi \) is twice diffe... | \[ \varphi \left( {x + y}\right) = \varphi \left( x\right) + \langle p, y\rangle + \frac{1}{2}\langle {Py}, y\rangle + o\left( {\left| y\right| }^{2}\right) , \] \[ \text{for all}\left| y\right| \text{small enough.} \] | 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 |
Theorem 3.4. Let (D1) 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 optimal triple of Proble... | Proof of Theorem 3.4. First of all, we note that\n\n(3.39)\n\n\[ \n\mathop{\lim }\limits_{{h \rightarrow 0}}\frac{1}{h}{\int }_{t}^{t + h}\varphi \left( r\right) {dr} = \varphi \left( t\right) ,\n\] | Yes |
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\[ V\left( {t, x}\right) = \left\{ \begin{array}{ll} - x, & \text{ if }x \leq 0 \\ - x{e}^{T - t}, & \text{ if }x > 0 \end{array}\right. \]\n\nNow consider Problem \( \left( {\mathrm{D}}_{00}\right) \) (i.e., \( s = 0 \) and \( y = 0 \) ). Clearly, \( \left( {\bar{x}\le... | 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 |
Theorem 3.9. 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 {\math... | 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 |
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 \( \left( {\bar{... | 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 3.16. Let \( \overline{\mathbf{u}} \) be an admissible feedback control, and \( \overline{\mathbf{p}} \) and \( \overline{\mathbf{q}} \) two measurable functions satisfying\n\n(3.69)\n\n\[ \left( {\overline{\mathbf{q}}\left( {t, x}\right) ,\overline{\mathbf{p}}\left( {t, x}\right) }\right) \in {D}_{t, x}^{1, + ... | Proof. For any \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \), let \( \bar{x}\left( \cdot \right) \) be the corresponding trajectory under \( \overline{\mathbf{u}} \) for Problem \( \left( {\mathrm{D}}_{sy}\right) \) . Put\n\n(3.71)\n\n\[ \bar{u}\left( t\right) \triangleq \overline{\mathbf{u}}\lef... | 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 |
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 \( \left( {S}_{sy}\right) \) . Assume further ... | Proof. This is immediate from (4.17) and (4.19). | Yes |
Consider the same manufacturing firm studied in Section 3.2. Suppose now there are random disturbances in the system, so that (3.14) is modified to\n\n\[ \left\{ \begin{array}{l} {dx}\left( t\right) = r\left( {t, x\left( t\right), u\left( t\right) }\right) {dt} + \sigma \left( {t, x\left( t\right), u\left( t\right) }\r... | The first-order adjoint equation of this problem takes the form (recall that all the variables are assumed to be one-dimensional)\n\n\[ - {dp}\left( t\right) = \left\{ {{n}_{x}\left( {t,\bar{x}\left( t\right) ,\bar{u}\left( t\right) }\right) + {r}_{x}\left( {t,\bar{x}\left( t\right) ,\bar{u}\left( t\right) }\right) p\l... | Yes |
Consider the following control system \( \left( {n = m = 1}\right) \) :\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 = \left\lbr... | 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 |
Consider the optimal control problem (4.55) (4.56) in Example 4.6. Since \( V \) is given by (4.58), which is independent of \( t \), we have\n\n\[ \n{D}_{t}^{1, + }V\left( {t,\bar{x}\left( t\right) }\right) = \{ 0\} ,\;\forall t \in \left\lbrack {0, T}\right\rbrack .\n\] | However, for the optimal pair \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) given there, we have (see (4.66))\n\n(4.83)\n\n\[ \n\mathcal{H}\left( {t,\bar{x}\left( t\right) ,\bar{u}\left( t\right) }\right) = - P\left( t\right) + {\left\lbrack \operatorname{ch}\bar{x}\left( t\right) \rig... | Yes |
Theorem 4.9. Under the assumptions of Theorem 4.4, we have\n\n(4.84)\n\n\[ \lbrack \mathcal{H}\left( {t,\bar{x}\left( t\right) ,\bar{u}\left( t\right) }\right) ,\infty ) \times \{ - p\left( t\right) \} \times \lbrack - P\left( t\right) ,\infty ) \subseteq {D}_{t+, x}^{1,2, + }V\left( {t,\bar{x}\left( t\right) }\right) ... | Proof. The first conclusion can be proved by combining the proofs of (4.37) and (4.67) and making use of (4.33). For (4.85), it can be proved in a way similar to the proof of (4.38). The details are left to the reader. | No |
Theorem 5.1. Let \( {\left( S1\right) }^{\prime } \) and \( {\left( S2\right) }^{\prime } \) hold. Let \( v \in {C}^{1,2}\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) be a solution of the HJB equation (4.6). Then\n\n(5.1)\n\n\[ v\left( {s, y}\right) \leq J\left( {s, y;u\left( \cdot \right)... | Proof. For any \( u\left( \cdot \right) \in {\mathcal{U}}^{w}\left\lbrack {s, T}\right\rbrack \) with the corresponding state trajectory \( x\left( \cdot \right) \), applying Itô’s formula to \( v\left( {t, x\left( t\right) }\right) \), we obtain\n\n\[ v\left( {s, y}\right) = \operatorname{Eh}\left( {x\left( T\right) }... | Yes |
Lemma 5.2. Let \( g \in C\left\lbrack {0, T}\right\rbrack \) . Extend \( g \) to \( \left( {-\infty , + \infty }\right) \) with \( g\left( t\right) = g\left( T\right) \) for \( t > T \), and \( g\left( t\right) = g\left( 0\right) \) for \( t < 0 \) . Suppose there is a \( \rho \in {L}^{1}\left( {0, T}\right) \) such th... | Proof. In view of (5.6), we can apply Fatou's lemma to get\n\n\[ {\int }_{\alpha }^{\beta }\mathop{\lim }\limits_{{h \rightarrow 0 + }}\frac{g\left( {r + h}\right) - g\left( r\right) }{h}{dr} \geq \mathop{\lim }\limits_{{h \rightarrow 0 + }}{\int }_{\alpha }^{\beta }\frac{g\left( {r + h}\right) - g\left( r\right) }{h}{... | Yes |
Proposition 5.4. Condition (5.9) in Theorem 5.3 is equivalent to the following:\n\n\[ \n\bar{q}\left( t\right) = G\left( {t,\bar{x}\left( t\right) ,\bar{u}\left( t\right) , - \bar{p}\left( t\right) , - \bar{P}\left( t\right) }\right) \n\]\n\n(5.14)\n\n\[ \n= \mathop{\max }\limits_{{u \in U}}G\left( {t,\bar{x}\left( t\r... | Proof. It is clear that (5.14) implies (5.9). Suppose now (5.9) holds. Since \( v \) is the viscosity solution of the HJB (4.6), by definition we have\n\n\[ \n- \bar{q}\left( t\right) + \mathop{\sup }\limits_{{u \in U}}G\left( {t,\bar{x}\left( t\right), u, - \bar{p}\left( t\right) , - \bar{P}\left( t\right) }\right) \l... | Yes |
Consider the following control system \( \left( {n = m = 1}\right) \) :\n\n(5.15)\n\n\[ \left\{ \begin{array}{l} {dx}\left( t\right) = x\left( t\right) u\left( t\right) {dt} + x\left( t\right) {dW}\left( t\right) ,\;t \in \left\lbrack {s, T}\right\rbrack , \\ x\left( s\right) = y, \end{array}\right. \]\n\nwith the cont... | It is not difficult to directly verify that the following function is a viscosity solution of (5.17):\n\n(5.18)\n\n\[ V\left( {t, x}\right) = \left\{ \begin{array}{ll} - x, & \text{ if }x \leq 0 \\ - x{e}^{T - t}, & \text{ if }x > 0 \end{array}\right. \]\n\nwhich clearly satisfies (3.9)-(3.10) of Chapter 4. Thus, by th... | Yes |
Consider the problem in Example 4.6. The unique optimal feedback control has been computed to be \( \mathbf{u}\left( {t, x}\right) = \) th \( x \) . The admissible pair \( \left( {\bar{x}\left( t\right) ,\bar{u}\left( t\right) }\right) = \left( {\sqrt{2}W\left( t\right) ,0}\right) \) is therefore not optimal for Proble... | To this end, note that \( V\left( {s, y}\right) \) is given by (4.58). Hence, we can compute (noting (4.33))\n\n\[ \n{D}_{t+, x}^{1,2, - }V\left( {t,\bar{x}\left( t\right) }\right) = \left\{ {\left( {q, p, P}\right) \mid q \leq 0, p = - \operatorname{th}\left( {\sqrt{2}W\left( t\right) }\right) ,}\right. \n\]\n\n\[ \n\... | Yes |
Theorem 5.7. Let \( {\left( S1\right) }^{\prime } - {\left( S2\right) }^{\prime } \) hold and let \( V \) be the value function. Let \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) be fixed and let \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) be an optimal pair fo... | Proof. Fix a \( t \in \lbrack s, T) \) and an \( \omega \in \Omega \) such that \( \left( {\bar{q}\left( t\right) ,\bar{p}\left( t\right) ,\bar{P}\left( t\right) }\right) \in \) \( {D}_{t+, x}^{1,2, - }V\left( {t,\bar{x}\left( t\right) }\right) \) . As in the proof of Theorem 5.3, we have (using Chapter 4, Lemma 5.5-(i... | Yes |
Theorem 6.2. Let \( {\left( S1\right) }^{\prime } - {\left( S2\right) }^{\prime } \) hold and let \( v \in C\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n}}\right) \) be the viscosity solution of the HJB equation (4.6). Then for each \( \left( {t, x}\right) \in \left( {0, T}\right) \times \) \( {\mathb... | Proof. By the uniqueness of viscosity solutions to the HJB equation (4.6), \( v \) is the value function \( V \) of Problem \( \left( {\mathrm{S}}_{sy}\right) \) . Then, by Chapter 4, Theorem 6.2, we obtain (6.2).\n\nNext, for any \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \), let \( \bar{x}\left... | Yes |
Lemma 6.4. Let \( X \subseteq {\mathbb{R}}^{n} \) be a Lebesgue measurable set, \( Y \) a Polish space, and \( \Gamma : X \rightarrow {2}^{Y} \) be a multifunction. Then, \( \Gamma \) is measurable if and only if the multifunction \( x \mapsto \bar{\Gamma }\left( x\right) \triangleq \overline{\Gamma \left( x\right) } \... | Proof. Note that for any open set \( U \subseteq Y \) and \( x \in X \) ,\n\n\[ \Gamma \left( x\right) \bigcap U \neq \phi \; \Leftrightarrow \;\overline{\Gamma \left( x\right) }\bigcap U \neq \phi . \]\n\nHence, by the definition of \( {\Gamma }^{-1} \) (see (6.5))\n\n\[ {\Gamma }^{-1}\left( U\right) = {\bar{\Gamma }}... | Yes |
Proposition 6.5. Both of the multifunctions \( \left( {t, x}\right) \mapsto {D}_{t +, x}^{1,2, + }v\left( {t, x}\right) \) and \( \left( {t, x}\right) \mapsto \overline{{D}_{t+, x}^{1,2, + }v\left( {t, x}\right) } \) are convex-set-valued and are measurable. | Proof. For any \( \left( {s, y}\right) \in (0, T\rbrack \times {\mathbb{R}}^{n} \), define\n\n\[ W\left( {t, x, q, p, P;s, y}\right) \]\n\n\[ \begin{array}{l} \triangleq \left\{ \begin{array}{ll} \frac{1}{\left| {s - t}\right| + {\left| y - x\right| }^{2}}\{ v\left( {s, y}\right) - v\left( {t, x}\right) - q\left( {s - ... | Yes |
(i) If Problem (DLQ) is finite at some \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \), then \( \left( {2.10}\right) \)\n\[ {N}_{r} \geq 0,\;\forall r \in \left\lbrack {s, T}\right\rbrack \] | Proof. (i) Suppose (2.10) fails. Then, for some \( r \in \left\lbrack {s, T}\right\rbrack \) and some \( {u}_{0}\left( \cdot \right) \in \mathcal{V}\left\lbrack {r, T}\right\rbrack \), we have\n(2.14)\n\[ \left\langle {{N}_{r}{u}_{0}\left( \cdot \right) ,{u}_{0}\left( \cdot \right) }\right\rangle < 0.\]\nExtend \( {u}_... | Yes |
Theorem 2.3. Let Problem (DLQ) be solvable at some \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) with an optimal pair \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) . Then there exists a solution \( \bar{p}\left( \cdot \right) \) to the equation\n\n\( \left( {2.20... | Proof. By Chapter 3, Theorem 2.1, equation (2.20) is the adjoint equation associated with the optimal pair \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) . Thus, the maximum principle yields the following maximum condition for Problem (DLQ):\n\n\( \left( {2.23}\right) \)\n\n\[ \mathop{\... | Yes |
Proposition 2.4. Suppose Problem (DLQ) is finite at some \( \left( {s, y}\right) \in \) \( \lbrack 0, T) \times {\mathbb{R}}^{n} \) . Then (2.22) holds. | Proof. It suffices to show that \( R\left( t\right) \geq 0 \) at any Lebesgue point \( t \in \lbrack s, T) \) of \( R\left( \cdot \right) \) . Suppose this is not true. Then there are a Lebesgue point \( t \in \lbrack s, T) \) and a \( {u}_{0} \in {\mathbb{R}}^{k} \) such that \( \left\langle {R\left( t\right) {u}_{0},... | Yes |
Consider the one-dimensional control system\n\n\[ \left\{ \begin{array}{ll} \dot{x}\left( t\right) = u\left( t\right) , & t \in \left\lbrack {s, T}\right\rbrack \\ x\left( s\right) = y \in \mathbb{R} & \end{array}\right. \]\nwith \( s \in \lbrack 0, T) \) and the cost functional\n\n\[ J\left( {s, y;u\left( \cdot \right... | In this case, \( R = 0 \) . By taking\n\n\[ {u}_{\varepsilon }\left( t\right) = - \frac{y}{\varepsilon }{\chi }_{\left\lbrack s, s + \varepsilon \right\rbrack }\left( t\right) ,\;\forall t \in \left\lbrack {s, T}\right\rbrack ,\]\n\nwith \( 0 < \varepsilon < T - s \), we have\n\n\[ 0 \leq J\left( {s, y;{u}_{\varepsilon... | Yes |
Consider system (2.24) with cost functional\n\n\[ J\left( {s, y;u\left( \cdot \right) }\right) = - \frac{1}{2}x{\left( T\right) }^{2}. \] | Take \( {u}_{\ell }\left( t\right) \equiv \ell \) . Then\n\n\[ J\left( {s, y;{u}_{\ell }\left( \cdot \right) }\right) = - \frac{1}{2}{\left\lbrack \ell \left( T - s\right) + y\right\rbrack }^{2} \rightarrow - \infty \;\left( {\ell \rightarrow \infty }\right) . \] | Yes |
Let \( {N}_{s} \geq 0 \) for some \( s \in \lbrack 0, T) \) and (2.28) hold. Then Problem (DLQ) is (uniquely) solvable at \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) if and only if the two-point boundary value problem\n\n(2.29)\n\n\[ \left\{ \begin{array}{l} \dot{\bar{x}}\left( t\right) = \left... | Proof. (i) If Problem (DLQ) is solvable at \( \left( {s, y}\right) \) with an optimal control \( \bar{u}\left( \cdot \right) \), then by Theorem 2.2-(ii),\n\n\[ 0 = {N}_{s}\bar{u}\left( \cdot \right) + {H}_{s}\left( y\right) \]\n\n\[ = \left( {R + {L}_{s}^{ * }Q{L}_{s} + S{L}_{s} + {L}_{s}^{ * }{S}^{\top } + {\widehat{... | Yes |
For a standard LQ problem (i.c., (2.18) holds), the Ric-cati equation (2.34) admits a unique solution \( P\left( \cdot \right) \) over \( \left\lbrack {0, T}\right\rbrack \), and Problem (DLQ) is uniquely solvable with the optimal control \( \bar{u}\left( \cdot \right) \) given by (2.30). Moreover, \( P\left( t\right) ... | Proof. Under (2.18), all the conclusions of this corollary are clear from Theorems 2.2-(iii) and 2.9, except the nonnegativity of \( P\left( \cdot \right) \), which we now prove. By Theorem 2.2-(iv), we know that in the current case, Problem (DLQ) with \( b\left( \cdot \right) = 0 \) is also uniquely solvable, for whic... | Yes |
Consider a control system (with both the state and control being one-dimensional)\n\n\[ \left\{ \begin{array}{l} \dot{x}\left( t\right) = u\left( t\right) ,\;t \in \left\lbrack {s, T}\right\rbrack , \\ x\left( s\right) = y, \end{array}\right. \]\n\nwith the cost functional\n\n\[ J\left( {s, y;u\left( \cdot \right) }\ri... | This is not a standard LQ problem, as the weight on the square of the terminal state is negative. Let \( T > 1 \) and \( s \in (T - 1, T\rbrack \) . For any \( u\left( \cdot \right) \in \) \( {L}^{2}\left( {s, T;\mathbb{R}}\right) \), let \( x\left( \cdot \right) \) be the corresponding state trajectory. Applying the N... | Yes |
Proposition 2.12. The operator \( \Gamma \) is continuous and monotonically increasing. Moreover, there is a \( \widehat{K} \in {L}^{\infty }\left( {0, T;{\mathcal{S}}_{ + }^{n}}\right) \) such that\n\n\[ \mathop{\sup }\limits_{{R \in \mathcal{R}}}\langle \Gamma \left( R\right) y, y\rangle \leq \langle \widehat{K}y, y\... | Proof. Let \( {R}_{i} \in \widehat{\mathcal{R}} \) and \( {P}_{i} = \Gamma \left( {R}_{i}\right) \left( {i = 1,2}\right) \) . Define \( \widehat{P} = {P}_{1} - {P}_{2} \) . Then \( \widehat{P} \) satisfies\n\n\[ \left\{ \begin{array}{l} \dot{\widehat{P}} = - \widehat{P}\widehat{A} - {\widehat{A}}^{\top }\widehat{P} - \... | Yes |
Consider a control system (where both the state and control are one-dimensional)\n\n\[ \left\{ \begin{array}{l} \dot{x}\left( t\right) = u\left( t\right) ,\;t \in \left\lbrack {s, T}\right\rbrack , \\ x\left( s\right) = y, \end{array}\right. \]\n\nwith the cost functional (which is the same as (2.25))\n\n\[ {J}_{1}\lef... | In Example 2.5, we have shown that this problem is finite at any \( s \in \left\lbrack {0, T}\right\rbrack \) , but not solvable at any \( \left( {s, y}\right) \) with \( y \neq 0 \) . | No |
Consider the control system (3.8) with the cost functional\n\n\[ \n{J}_{2}\left( {s, y;u\left( \cdot \right) }\right) = - \frac{1}{2}{\int }_{s}^{T}u{\left( t\right) }^{2}{dt} + \frac{1}{2}x{\left( T\right) }^{2}. \n\]\n\nWe see that (3.15) has a negative weight \( R = - 1 \) on the control term. By Proposition 2.4, th... | As a matter of fact, by taking\n\n\[ \n{u}_{\varepsilon }\left( t\right) = \frac{1}{\varepsilon }{\chi }_{\left\lbrack T - \varepsilon, T\right\rbrack }\left( t\right) ,\;t \in \left\lbrack {s, T}\right\rbrack \n\]\n\nwith \( 0 < \varepsilon < T - s \), one has\n\n(3.16)\n\n\[ \n{J}_{2}\left( {s, y;{u}_{\varepsilon }\l... | Yes |
(i) For any \( \xi \left( \cdot \right) \in \mathcal{X}\left\lbrack {s, T}\right\rbrack \), let \( \left( {{p}_{0}\left( \cdot \right) ,{q}_{0}\left( \cdot \right) }\right) \in \mathcal{X}\left\lbrack {s, T}\right\rbrack \times \mathcal{X}\left\lbrack {s, T}\right\rbrack \) be the adapted solution of (4.11) with \( \et... | Proof. For any \( \eta \in {\mathcal{X}}_{T} \) and \( \xi \left( \cdot \right) \in \mathcal{X}\left\lbrack {s, T}\right\rbrack \), by Chapter 7, Theorem 2.2, there exists a unique adapted solution \( \left( {p\left( \cdot \right), q\left( \cdot \right) }\right) \in \mathcal{X}\left\lbrack {s, T}\right\rbrack \times \m... | Yes |
Theorem 5.1. Let (L1) hold. Let Problem (SLQ) be solvable at \( \left( {s, y}\right) \in \) \( \lbrack 0, T) \times {\mathbb{R}}^{n}\; \) with \( \left( {\overline{x}\left( \cdot \right) ,\overline{u}\left( \cdot \right) }\right) \) being an optimal pair. \( \; \) Then there exist adapted processes \( \left( {\bar{p}\l... | Proof. It is clear that the first and second adjoint equations corresponding to the given optimal pair \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) are (5.1) and (5.2), respectively, for the present case. Moreover, the \( \mathcal{H} \) -function \( \mathcal{H}\left( {t,\bar{x}\left( ... | Yes |
Corollary 5.2. Let (L1) hold with \( T \) being a Lebesgue point of \( R\left( \cdot \right) \) and \( D\left( \cdot \right) \) . Suppose Problem (SLQ) is finite at some \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) . Then\n\n(5.4)\n\n\[ R\left( T\right) + D{\left( T\right) }^{\top }{GD}\left( T\... | Proof. It follows from Theorem 4.2 that (4.19) holds under our assumption. Now, for any \( \varepsilon > 0 \), we consider an LQ problem that is the same as Problem (SLQ) except that \( R\left( t\right) \) is replaced by \( R\left( t\right) + {\varepsilon I} \) . Then the corresponding operator \( {N}_{s} \) is replace... | Yes |
Consider the one-dimensional control system\n\n\[ \left\{ \begin{array}{l} {dx}\left( t\right) = \frac{1}{2}x\left( t\right) {dt} + u\left( t\right) {dW}\left( t\right) ,\;t \in \left\lbrack {s, T}\right\rbrack , \\ x\left( s\right) = y, \end{array}\right. \] \n\nwith the cost functional\n\n\[ J\left( {s, y;u\left( \cd... | In this case, \( A = \frac{1}{2}, B = C = Q = 0, D = R = 1 \), and \( G = - 1 \) . Thus, \[ R + {D}^{\top }{GD} = 0. \] \n\nApplying Itô’s formula to \( - {e}^{T - t}x{\left( t\right) }^{2} \), we have\n\n\[ J\left( {s, y;u\left( \cdot \right) }\right) = E\left\{ {{\int }_{s}^{T}u{\left( t\right) }^{2}\left( {1 - {e}^{... | Yes |
For any \( \left( {y, u\left( \cdot \right) }\right) \in {\mathbb{R}}^{n} \times {\mathcal{U}}^{w}\left\lbrack {s, T}\right\rbrack \), let \( \left( {x\left( \cdot \right), p\left( \cdot \right), q\left( \cdot \right) }\right) \) be the adapted solution of (5.9). Then | Proof. Let \( \left( {x\left( \cdot \right), p\left( \cdot \right), q\left( \cdot \right) }\right) \) be the adapted solution of (5.9). By Proposition 4.1 and (4.18), we have (noting (4.6)) | No |
Proposition 5.5. Let (L1) hold. Then Problem (SLQ) is (pathwise uniquely) solvable at \( \left( {s, y}\right) \in \left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n} \) with an (the) optimal pair \( \left( {\overline{x}\left( \cdot \right) ,\overline{u}\left( \cdot \right) }\right) \) if and only if there (uniquel... | Proof. Necessity. The condition (5.14) follows from Theorem 5.1, while (5.15) follows from Theorem 4.2-(i) along with (5.11).\n\nSufficiency. From Lemma 5.4, condition (5.15) is equivalent to \( {N}_{s} \geq 0 \) . Now let \( \left( {\overline{x}\left( \cdot \right) ,\overline{p}\left( \cdot \right) ,\overline{q}\left(... | Yes |
Corollary 5.6. Let (L1) and (5.16) hold. Let \( {N}_{s} \geq 0 \) . Then Problem (SLQ) is (pathwise uniquely) solvable at \( \left( {s, y}\right) \in \lbrack 0, T) \times {\mathbb{R}}^{n} \) if and only if the FBSDE (5.17) admits a (unique) adapted solution \( \left( {\bar{x}\left( \cdot \right) ,\bar{p}\left( \cdot \r... | In this case, the following is an (the) optimal control:\n\n(5.18)\n\n\[ \bar{u}\left( t\right) = - R{\left( t\right) }^{-1}\left\lbrack {S\bar{x}\left( t\right) - {B}^{\top }\bar{p}\left( t\right) - {D}^{\top }\bar{q}\left( t\right) }\right\rbrack ,\;\forall t \in \left\lbrack {s, T}\right\rbrack . \] | Yes |
Corollary 5.7. Let (L1) and (4.23) hold. Then FBSDE (5.17) admits a unique adapted solution \( \left( {\bar{x}\left( \cdot \right) ,\bar{p}\left( \cdot \right) ,\bar{q}\left( \cdot \right) }\right) \), and Problem (SLQ) is pathwise uniquely solvable at \( s \) with the optimal control having the representation (5.18). | It is clear that if (4.23) holds, (4.19) or equivalently (5.15) automatically holds. Thus, for a standard stochastic LQ problem, Corollary 5.7 implies that (5.13)-(5.14) completely characterize the optimal pair \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) . | No |
Theorem 6.1. Let (L1) hold. Let \( P\left( \cdot \right) \in C\left( {\left\lbrack {s, T}\right\rbrack ;{\mathcal{S}}^{n}}\right) \) and \( \varphi \left( \cdot \right) \in C(\left\lbrack {s, T}\right\rbrack \) ; \( \left. {\mathbb{R}}^{n}\right) \) be the solutions of (6.6) and (6.7), respectively, for some \( s \in \... | Proof. By (L1) and (6.9)-(6.10), the following SDE admits a unique strong solution \( \bar{x}\left( \cdot \right) \) :\n\n(6.13)\n\n\[ \begin{cases} d\bar{x}\left( t\right) = & \{ \left\lbrack {A\left( t\right) - B\left( t\right) \Psi \left( t\right) }\right\rbrack \bar{x}\left( t\right) - B\left( t\right) \psi \left( ... | Yes |
For the LQ problem with the state equation (3.10) and the cost functionals (3.15), the Riccati equation reads (noting \( \left| \delta \right| > 1 \) ) | \[ \left\{ \begin{array}{l} \dot{P}\left( t\right) = \frac{P{\left( t\right) }^{2}}{{\delta }^{2}P\left( t\right) - 1},\;t \leq T, \\ P\left( T\right) = 1, \\ - 1 + {\delta }^{2}P\left( t\right) > 1. \end{array}\right. \] A direct computation shows that the above is equivalent to \[ \left\{ {\begin{array}{l} {\delta }^... | Yes |
Proposition 7.1. Let (L1) and (L2) hold. If \( P \in C\left( {\left\lbrack {0, T}\right\rbrack ;{\mathbb{R}}^{n \times n}}\right) \) is a solution to the stochastic Riccati equation (6.6), then \( P \) is the only solution. | Proof. Suppose \( \widetilde{P} \in C\left( {\left\lbrack {0, T}\right\rbrack ;{\mathbb{R}}^{n \times n}}\right) \) is another solution of (6.6). Set \( \widehat{P} \triangleq P - \widetilde{P} \) . Then \( \widehat{P} \) satisfies\n\n\[ \begin{cases} \dot{\widehat{P}} + & \widehat{P}A + {A}^{\top }\widehat{P} + {C}^{\... | Yes |
Theorem 7.2. Let (L1), (L2), and (4.23) hold. Then the stochastic Riccati equation (6.6) admits a unique solution over \( \left\lbrack {0, T}\right\rbrack \) . | To prove this theorem, we need the following lemma. | No |
Lemma 7.3. The linear matrix-valued differential equation\n\n\[ \left\{ \begin{array}{l} \dot{P}\left( t\right) + P\left( t\right) \widehat{A}\left( t\right) + \widehat{A}{\left( t\right) }^{\top }P\left( t\right) + \widehat{C}{\left( t\right) }^{\top }P\left( t\right) \widehat{C}\left( t\right) + \widehat{Q}\left( t\r... | Proof. Since equation (7.2) is linear with bounded coefficients, the existence and uniqueness of its solution \( P \in C\left( {\left\lbrack {0, T}\right\rbrack ;{\mathcal{S}}^{n}}\right) \) are clear. Now let \( \Phi \left( \cdot \right) \) be the solution of the following SDE on a filtered probability space:\n\n\[ \l... | Yes |
Theorem 7.5. Let (L1) and (L2) hold. Then the following are equivalent:\n\n(i) The stochastic Riccati equation (7.19) admits a solution \( P \) .\n\n(ii) There exist \( {\widehat{R}}^{ + },{\widehat{R}}^{ - } \in C\left( {\left\lbrack {0, T}\right\rbrack ;{\widehat{\mathcal{S}}}_{ + }^{k}}\right) \) such that\n\n(7.21)... | Proof. (i) \( \Rightarrow \) (ii). When (7.19) admits a solution \( P \), by taking\n\n\[{\widehat{R}}^{ + } \equiv {\widehat{R}}^{ - } \triangleq R + {D}^{\top }{PD}\]\n\nwe obtain (7.21) (with all inequalities being equalities).\n\n(ii) \( \Rightarrow \) (iii). Let \( {\widehat{R}}^{ + },{\widehat{R}}^{ - } \) be giv... | Yes |
Let (L1), (L2), and (7.21) hold. Let the sequence \( \left\{ {P}_{i}\right\} \subseteq C\left( {\left\lbrack {0, T}\right\rbrack ;{\mathcal{S}}^{n}}\right) \) be constructed by the algorithms\n\n(7.26)\n\n\[ \n{\widehat{R}}_{0} = {\widehat{R}}^{ + },\;{P}_{i} = \Gamma \left( {\widehat{R}}_{i}\right) ,\;{\widehat{R}}_{i... | We prove only the estimate (7.28) for the algorithm (7.26). The other one is the same. By definition,\n\n\[ \n{P}_{i}\left( t\right) = G - {\int }_{t}^{T}\left\lbrack {{P}_{i}A + {A}^{\top }{P}_{i} - {P}_{i}B{\widehat{R}}_{i}^{-1}{B}^{\top }{P}_{i} + Q}\right\rbrack \left( s\right) {ds}.\n\]\n\nSet \( {\widehat{P}}_{i}... | Yes |
Theorem 7.7. Let (L1) and (L2) hold. Then the stochastic Riccati equation (7.19) admits a solution if and only if there exist an \( \widehat{R} \in C\left( {\left\lbrack {0, T}\right\rbrack ;{\widehat{\mathcal{S}}}_{ + }^{k}}\right) \) such that\n\n(7.30)\n\n\[ R + {D}^{\top }\Gamma \left( \widehat{R}\right) D \geq \wi... | This theorem says that while \( R \) can be indefinite (or negative definite) for the stochastic Riccati equation to have solutions, it cannot be too negative. Indeed, in any case, \( R \) cannot be smaller than \( \mathop{\inf }\limits_{{\widehat{R} \in C\left( {\left\lbrack {0, T}\right\rbrack ;{\widehat{\mathcal{S}}... | Yes |
Consider the control system\n\n\[ \left\{ \begin{array}{l} {dx}\left( t\right) = u\left( t\right) {dt} + u\left( t\right) {dW}\left( t\right) ,\;t \in \left\lbrack {s,1}\right\rbrack \\ x\left( s\right) = y \end{array}\right. \]\n\nwith the cost functional\n\n\[ J\left( {s, y;u\left( \cdot \right) }\right) = E\left\{ {... | By Theorem 7.7,(7.33) admits a solution on some interval \( \left\lbrack {s,1}\right\rbrack \left( {0 \leq s < 1}\right) \) if and only if there is a positive continuous function \( \widehat{R}\left( \cdot \right) \) such that\n\n\[ - r + \Gamma \left( \widehat{R}\right) \left( t\right) \geq \widehat{R}\left( t\right) ... | Yes |
Theorem 7.9. Let all the coefficients \( A, B, C, D, Q, S, R \) be time-invariant with \( n = k = 1 \), and let \( \alpha ,\beta ,\gamma, g \) be defined by (7.40). Then the following hold:\n\n(i) If \( \beta = \gamma = 0 \), then \( \theta = + \infty \) . | Proof. The central idea for all the cases is to find the first time \( \theta \) at which the solution \( y\left( \cdot \right) \) changes its sign from positive to negative, i.e., the first time \( \theta \) when \( y\left( \theta \right) = 0 \) .\n\n(i) When \( \beta = \gamma = 0 \) ,(7.39) becomes\n\n\[ \dot{y} = {\... | Yes |
Theorem 7.10. Let (L1) hold. Let \( P\left( \cdot \right) \in C\left( {\left\lbrack {s, T}\right\rbrack ;{\mathcal{S}}^{n}}\right) \) and \( \varphi \left( \cdot \right) \in \) \( C\left( {\left\lbrack {s, T}\right\rbrack ;{\mathbb{R}}^{n}}\right) \) be the solutions of (7.60) and (7.61), respectively, for some \( s \i... | Then Problem (SLQ) is solvable at \( s \) with the optimal control \( \bar{u}\left( \cdot \right) \) being of a state feedback form:\n\n(7.64)\n\n\[ \bar{u}\left( t\right) = - \Psi \left( t\right) x\left( t\right) - \psi \left( t\right) ,\;t \in \left\lbrack {s, T}\right\rbrack , \]\n\nand\n\n\[ V\left( {s, y}\right) =... | Yes |
Theorem 8.2. For any \( \mu > 0 \), one has\n\n(8.15)\n\n\[{\Pi }_{P\left( \mu \right) } \subseteq \mathop{\bigcup }\limits_{{-\infty < \lambda < + \infty }}{\Pi }_{A\left( {\mu ,\lambda }\right) }\]\n\nMoreover, if \( \bar{u}\left( \cdot \right) \in {\Pi }_{P\left( \mu \right) } \), then \( \bar{u}\left( \cdot \right)... | Proof. We need to prove only the second assertion, as the first one is a direct consequence of the second. Let \( \bar{u}\left( \cdot \right) \in {\Pi }_{P\left( \mu \right) } \) . If \( \bar{u}\left( \cdot \right) \notin {\Pi }_{A\left( {\mu ,\bar{\lambda }}\right) } \), then there exists \( u\left( \cdot \right) \) s... | Yes |
Theorem 8.3. Under the assumption (8.3), the efficient frontier of the bi-criteria optimal portfolio selection problem (8.9), if it ever exists, is given by (8.40). | The set (8.40) reveals explicitly the trade-off between the mean (return) and variance (risk). For example, if one has set an expected return level, then the above can tell the risk he/she has to take; and vice versa. In particular, if one cannot take any risk, namely, \( \operatorname{Var}\left( {\bar{x}\left( T\right... | No |
Theorem 3.3. Let \( h,\bar{h} : \left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{k} \times {\mathbb{R}}^{k \times m} \times \Omega \rightarrow {\mathbb{R}}^{k} \) satisfy \( \left( B\right) \) and let \( \xi ,\bar{\xi } \in {L}_{\mathcal{F}}^{2}\left( {\Omega ;{\mathbb{R}}^{k}}\right) \) . Let \( \left( {Y\left( \... | Proof. Define \[ \left\{ \begin{array}{l} \widehat{Y}\left( \cdot \right) \triangleq Y\left( \cdot \right) - \bar{Y}\left( \cdot \right) ,\;\widehat{Z}\left( \cdot \right) \triangleq Z\left( \cdot \right) - \bar{Z}\left( \cdot \right) , \\ \widehat{\xi } \triangleq \xi - \bar{\xi },\;\widehat{h}\left( \cdot \right) \tr... | Yes |
Theorem 3.4. Let assumption (B) hold and \( \xi \in {L}_{{\mathcal{F}}_{\tau }}^{2}\left( {\Omega ;{\mathbb{R}}^{k}}\right) \) . Let \( \left( {Y\left( \cdot \right), Z\left( \cdot \right) }\right) \in \mathcal{M}\left\lbrack {0, T}\right\rbrack \) be the unique adapted solution of (3.1) and let the sequence \( \left( ... | Proof. By the definition of \( \mathcal{T} \) (see the proof of Theorem 3.2), we have\n\n(3.27)\n\n\[ \n\left( {{Y}^{i + 1},{Z}^{i + 1}}\right) = \mathcal{T}\left( {{Y}^{i},{Z}^{i}}\right) ,\;i \geq 0. \n\]\n\nLet us define\n\n(3.28)\n\n\[ \n\left\{ {\begin{array}{l} {\widehat{Y}}^{i + 1}\left( t\right) \triangleq {Y}^... | Yes |
Theorem 3.6. Let \( {\left( B\right) }^{\prime } \) hold. Then, for any \( \xi \in {L}_{{\mathcal{F}}_{\tau }}^{2,\beta }\left( {\Omega ;{\mathbb{R}}^{k}}\right) \), BSDE (3.32) admits a unique adapted solution \( \left( {Y\left( \cdot \right), Z\left( \cdot \right) }\right) \in {\mathcal{M}}_{\beta }\left\lbrack {0,\t... | \[ \parallel \left( {Y\left( \cdot \right), Z\left( \cdot \right) }\right) - \left( {\bar{Y}\left( \cdot \right) ,\bar{Z}\left( \cdot \right) }\right) {\parallel }_{{\mathcal{M}}_{\beta }\left\lbrack {0,\tau }\right\rbrack } \leq {KE}\left\{ {\left| {\xi - \bar{\xi }}\right| {e}^{2\beta \tau }}\right. \] (3.46) \[ \lef... | Yes |
Lemma 3.9. Let \( {\left( B\right) }^{\prime } \) and (3.47) hold. Let \( \xi ,\bar{\xi } \in {L}_{{\mathcal{F}}_{\tau }}^{2,\beta }\left( {\Omega ;{\mathbb{R}}^{k}}\right) \) and \( {h}_{0},{\bar{h}}_{0} \in {L}_{\mathcal{F}}^{2,\beta }\left( {0,\tau ;{\mathbb{R}}^{k}}\right) \) . Let \( \left( {Y\left( \cdot \right),... | Proof. We apply Itô’s formula to \( {\left| Y\left( t\right) - \bar{Y}\left( t\right) \right| }^{2}{e}^{2\beta t} \) to get the following \[ \left( {0 \leq t < T < \infty }\right) \text{:} \] \[ {\left| Y\left( T \land \tau \right) - \bar{Y}\left( T \land \tau \right) \right| }^{2}{e}^{{2\beta }\left( {T \land \tau }\r... | Yes |
Lemma 3.10. Let \( {\left( B\right) }^{\prime } \) and (3.47) hold. Then there exists a constant \( {\varepsilon }_{0} > 0 \) having the following property: If for some \( \alpha \in \lbrack 0,1) \) ,(3.68) admits a unique adapted solution \( \left( {Y\left( \cdot \right), Z\left( \cdot \right) }\right) \in {\mathcal{M... | Proof. Suppose there is an \( \alpha \in \lbrack 0,1) \) with the property specified in the statement of the lemma. Then for any \( \xi \) and \( {h}_{0} \), define\n\n\( \left( {3.73}\right) \)\n\n\[ \n{Y}^{0}\left( t\right) = 0,\;{Z}^{0}\left( t\right) = 0 \]\n\nand solve the following BSDEs successively:\n\n(3.74)\n... | Yes |
Corollary 3.11. Let \( {\left( B\right) }^{\prime } \) hold with \( \tau \equiv \infty \) . Then, for any \( \xi \in {L}_{{\mathcal{F}}_{\infty }}^{2,\beta }\left( {\Omega ;{\mathbb{R}}^{k}}\right) \) , (3.85) admits a unique adapted solution \( \left( {Y\left( \cdot \right), Z\left( \cdot \right) }\right) \in {\mathca... | Note that for equation (3.85), the terminal condition \( Y\left( \infty \right) = \xi \) is understood in the following sense:\n\n(3.86)\n\n\[ \mathop{\lim }\limits_{{T \rightarrow \infty }}E\left\{ {{\left| Y\left( T\right) - \xi \right| }^{2}{e}^{2\beta T}}\right\} = 0. \] | Yes |
Theorem 4.1. Let \( \left( F\right) \) hold. Then (4.3) admits a unique viscosity solution \( v\left( {\cdot , \cdot }\right) \), and it has the following representation:\n\n(4.5)\n\n\[ v\left( {t, x}\right) = E\left\{ {{\int }_{t}^{T}h\left( {s, X\left( {s;t, x}\right) }\right) {e}^{-{\int }_{t}^{s}c\left( {r, X\left(... | Proof. We consider the following (trivial) optimal stochastic control problem: The state equation is (4.6) with the state being \( X\left( \cdot \right) \) and the control variable \( u \) being absent, or the control set \( U \) being a singleton. The cost functional is given by\n\n(4.7)\n\n\[ J\left( {t, x;u\left( \c... | Yes |
Theorem 4.2. Let \( \left( F\right) \) hold with all the functions defined on \( \left\lbrack {0, T}\right\rbrack \times \bar{G} \) , and let \( \Psi \left( {t, x}\right) \) defined by (4.9) be continuous on \( \left( {\{ T\} \times \bar{G}}\right) \bigcup \left( {\lbrack 0, T}\right) \times \) \( \partial G) \) . Then... | In addition, if (4.8) admits a classical solution, then (4.10) gives that classical solution.\n\nWe have not discussed the viscosity solutions in bounded domains. However, the theory is pretty much parallel. Thus, the proof of the above result can be carried out similarly to that of Theorem 4.1. | No |
Theorem 4.4. Let \( {\left( F\right) }^{\prime } \) hold. Then (4.18) admits a unique viscosity solution \( v\left( {\cdot , \cdot }\right) \), which can be represented by\n\n(4.21)\n\n\[ v\left( {t, x}\right) = \mathop{\inf }\limits_{{u\left( \cdot \right) \in {\mathcal{U}}^{w}\left\lbrack {t, T}\right\rbrack }}E\left... | In addition, if (4.18) admits a classical solution, then (4.21) gives that classical solution. | Yes |
Theorem 4.5. Let \( {\left( B\right) }^{\prime \prime } \) hold. Then (4.24) admits a unique viscosity solution \( v\left( {\cdot , \cdot }\right) \), and it has the following representation:\n\n\[ v\left( {t, x}\right) = {EY}\left( {t;t, x}\right) \equiv Y\left( {t;t, x}\right) ,\;\forall \left( {t, x}\right) \in \lef... | Proof. First of all, suppose (4.24) admits a classical solution \( v\left( {\cdot , \cdot }\right) \) . Note that under assumption (B) | No |
Theorem 4.6. Let \( {\left( B\right) }^{\prime \prime } \) hold and let \( \Psi \left( {t, x}\right) \) defined by (4.9) be continuous on \( \left( {\{ T\} \times \bar{G}}\right) \times \left( {\left\lbrack {0, T}\right\rbrack \times \partial G}\right) \). For any \( \left( {t, x}\right) \in \lbrack 0, T) \times G \), ... | The proof is parallel to that for Theorem 4.5. The details are left to the reader. We point out that in the above representation theorem, since \( \tau \left( {t, x}\right) \leq T \), Theorem 3.6 applies in view of the remark made after Theorem 3.6. | No |
Theorem 4.7. Let \( {\left( B\right) }^{\prime \prime \prime } \) hold. For any \( x \in G \), let \( X\left( {\cdot ;x}\right) \) be the unique strong solution of (4.16) and let \( \tau \left( x\right) \) be defined by (4.17). Further, suppose that\n\n(4.47)\n\n\[ E\left\{ {{e}^{{2\beta \tau }\left( x\right) }{\left| ... | The proof of the above theorem is similar to that of Theorem 4.5. The nontrivial part is the unique solvability of the BSDE (4.48) (note here that \( \tau \) may not be bounded by any deterministic constant), which has been shown in the previous section. | Yes |
Proposition 5.2. Let the following two-point boundary value problem for a system of linear ordinary differential equations admit no solutions:\n\n(5.4)\n\n\[ \left\{ \begin{array}{l} \left( \begin{array}{l} \dot{X}\left( t\right) \\ \dot{Y}\left( t\right) \end{array}\right) = \mathcal{A}\left( \begin{array}{l} X\left( ... | Proof. Suppose (5.5) admits an adapted solution \( \left( {X\left( \cdot \right), Y\left( \cdot \right), Z\left( \cdot \right) }\right) \) . Then \( \left( {{EX}\left( \cdot \right) ,{EY}\left( \cdot \right) }\right) \) is a solution of (5.4), leading to a contradiction. This proves the assertion. | Yes |
Theorem 5.3. Assume that (5.11) admits a unique solution \( z\left( {t, x, y, p}\right) \) that is uniformly Lipschitz continuous in \( \left( {x, y, p}\right) \) with \( z\left( {t,0,0,0}\right) \) being bounded, and that (5.12) admits a classical solution \( \theta \left( {t, x}\right) \) with bounded \( {\theta }_{x... | Proof. Under our conditions both \( \widetilde{b}\left( {t, x}\right) \) and \( \widetilde{\sigma }\left( {t, x}\right) \) defined by (5.14) are uniformly Lipschitz continuous in \( x \) . Therefore, for any \( x \in {\mathbb{R}}^{n} \) ,(5.13) has a unique strong solution. Then, by defining \( Y\left( t\right) \) and ... | Yes |
Theorem 5.5. Let (FB1)-(FB3) hold. Then (5.28) admits a unique classical solution \( \theta \left( {t, x}\right) \) with \( \theta \left( {t, x}\right) ,{\theta }_{t}\left( {t, x}\right) ,{\theta }_{x}\left( {t, x}\right) \), and \( {\theta }_{xx}\left( {t, x}\right) \) bounded. Consequently, FBSDE (5.27) admits a uniq... | Proof. We first show that all the required conditions in Lemma 5.4 are satisfied. Since \( \sigma \) is independent of \( z \), the function \( z\left( {t, x, y, p}\right) \) determined by (5.11) satisfies\n\n(5.35)\n\n\[ \left| {z\left( {t, x, y, p}\right) }\right| \leq K\left| p\right| ,\;\forall \left( {t, x, y, p}\... | Yes |
Theorem 5.6. Suppose that (FB1), (FB2)', and (FB3) hold. Then (5.36) admits a unique adapted solution \( \left( {X, Y, Z}\right) \) . | Proof. In the present case, for the function \( z\left( {t, x, y, p}\right) \) determined by (5.11), we still have (5.35). Also, conditions (5.40) and (5.41) hold, which leads to the existence and uniqueness of classical solutions of (5.38) or (5.37). Next, applying Theorem 5.3, we can show that there exists a unique a... | Yes |
Theorem 5.7. Let (FB1) and (FB2) hold with \( k = 1 \) . Then there exists a unique smooth function \( z\left( {t, x, y, p}\right) \) that solves (5.45) and satisfies (5.35). If in addition (FB3) holds, then FBSDE (5.1) admits a unique adapted solution determined by the four-step scheme. | We should note that the existence and uniqueness of solutions to (5.12) in the present case \( \left( {k = 1}\right) \) follow from Ladyzhenskaya-Solonnikov-Ural’tseva [1, Chapter V, Theorem 8.1]. Therefore, Theorem 5.7 can be proved similarly. Details are left to the reader. We see that the condition (5.48) together w... | No |
Theorem 1. The event \( \{ T < t\} \in {\mathcal{F}}_{t},0 \leq t \leq \infty \), if and only if \( T \) is a stopping time. | Proof. Since \( \{ T \leq t\} = \mathop{\bigcap }\limits_{{t + \varepsilon > u > t}}\{ T < u\} \), any \( \varepsilon > 0 \), we have \( \{ T \leq \) \( t\} \in \mathop{\bigcap }\limits_{{u > t}}{\mathcal{F}}_{u} = {\mathcal{F}}_{t} \), so \( T \) is a stopping time. For the converse, \( \{ T < t\} = \) \( \mathop{\big... | Yes |
Theorem 3. Let \( X \) be an adapted càdlàg stochastic process, and let \( \Lambda \) be an open set. Then the hitting time of \( \Lambda \) is a stopping time. | Proof. By Theorem 1 it suffices to show that \( \{ T < t\} \in {\mathcal{F}}_{t},0 \leq t < \infty \) . But\n\n\[ \{ T < t\} = \mathop{\bigcup }\limits_{{s \in \mathbb{Q}\cap \lbrack 0, t)}}\left\{ {{X}_{s} \in \Lambda }\right\} \]\n\nsince \( \Lambda \) is open and \( X \) has right continuous paths. Since \( \left\{ ... | Yes |
Theorem 4. Let \( X \) be an adapted càdlàg stochastic process, and let \( \Lambda \) be a closed set. Then the random variable\n\n\[ T\left( \omega \right) = \inf \left\{ {t > 0 : {X}_{t}\left( \omega \right) \in \Lambda \text{ or }{X}_{t - }\left( \omega \right) \in \Lambda }\right\} \]\n\nis a stopping time. | Proof. By \( {X}_{t - }\left( \omega \right) \) we mean \( \mathop{\lim }\limits_{{s \rightarrow t, s < t}}{X}_{s}\left( \omega \right) \) . Let \( {A}_{n} = \{ x : d\left( {x,\Lambda }\right) < 1/n\} \) , where \( d\left( {x,\Lambda }\right) \) denotes the distance from a point \( x \) to \( \Lambda \) . Then \( {A}_{... | Yes |
Theorem 6. Let \( T \) be a finite stopping time. Then \( {\mathcal{F}}_{T} \) is the smallest \( \sigma \) - algebra containing all càdlàg processes sampled at \( T \) . That is,\n\n\[ \n{\mathcal{F}}_{T} = \sigma \left\{ {{X}_{T};X\text{ all adapted càdlàg processes }}\right\} .\n\] | Proof. Let \( \mathcal{G} = \sigma \left\{ {{X}_{T};X}\right. \) all adapted càdlàg processes \( \} \) . Let \( \Lambda \in {\mathcal{F}}_{T} \) . Then \( {X}_{t} = {1}_{\Lambda }{1}_{\{ t \geq T\} }{}^{1} \) is a càdlàg process, and \( {X}_{T} = {1}_{\Lambda } \) . Hence \( \Lambda \in \mathcal{G} \), and \( {\mathcal... | Yes |
Theorem 7. Let \( X \) be adapted and càdlàg. If \( \Delta {X}_{T}{1}_{\{ T < \infty \} } = 0 \) a.s. for each stopping time \( T \), then \( {\Delta X} \) is indistinguishable from the zero process. | Proof. It suffices to prove the result on \( \left\lbrack {0,{t}_{0}}\right\rbrack \) for \( 0 < {t}_{0} < \infty \) . The set \( \{ t \) : \( \left. {\left| {\Delta {X}_{t}}\right| > 0}\right\} \) is countable a.s. since \( X \) is càdlàg. Moreover\n\n\[ \left\{ {t : \left| {\Delta {X}_{t}}\right| > 0}\right\} = \math... | Yes |
Theorem 8 (Monotone Class Theorem). Let \( \mathcal{M} \) be a multiplicative class of bounded real-valued functions defined on a space \( \Omega \), and let \( \mathcal{A} = \sigma \{ \mathcal{M}\} \) . If \( \mathcal{H} \) is a monotone vector space containing \( \mathcal{M} \), then \( \mathcal{H} \) contains all bo... | Theorem 8 is proved in Dellacherie-Meyer [45, page 14] with the additional hypothesis that \( \mathcal{H} \) is closed under uniform convergence. This extra hypothesis is unnecessary, however, since every monotone vector space is closed under uniform convergence. (See Sharpe [215, page 365].) | Yes |
Theorem 9. Let \( X \) be a supermartingale. The function \( t \mapsto E\left\{ {X}_{t}\right\} \) is right continuous if and only if there exists a modification \( Y \) of \( X \) which is càdlàg. Such a modification is unique. | By uniqueness we mean up to indistinguishability. Our standing assumption that the | No |
Theorem 15. Let \( Y \in {L}^{2}\left( {\Omega ,\mathcal{F}, P}\right) \) . The process \( {X}_{t} = {}^{{\pi }_{t}}Y \) is a uniformly integrable martingale. | Proof. It suffices to show \( E\left\{ {Y \mid {\mathcal{F}}_{t}}\right\} = {}^{{\pi }_{t}}Y \) . The random variable \( E\left\{ {Y \mid {\mathcal{F}}_{t}}\right\} \) is the unique \( {\mathcal{F}}_{t} \) measurable r.v. such that \( {\int }_{A}{YdP} = {\int }_{A}E\left\{ {Y \mid {\mathcal{F}}_{t}}\right\} {dP} \), fo... | Yes |
Theorem 18. Let \( X \) be a uniformly integrable right continuous martingale, and let \( T \) be a stopping time. Then \( {X}^{T} = {\left( {X}_{t \land T}\right) }_{0 \leq t \leq \infty } \) is also a uniformly integrable right continuous martingale. | Proof. \( {X}^{T} \) is clearly right continuous. By Theorem 16\n\n\[ \n{X}_{t \land T} = E\left\{ {{X}_{T} \mid {\mathcal{F}}_{t \land T}}\right\} \n\]\n\n\[ \n= E\left\{ {{X}_{T}{1}_{\{ T < t\} } + {X}_{T}{1}_{\{ T \geq t\} } \mid {\mathcal{F}}_{t \land T}}\right\} \n\]\n\n\[ \n= {X}_{T}{1}_{\{ T < t\} } + E\left\{ {... | Yes |
Theorem 19 (Jensen’s Inequality). Let \( \varphi : \mathbb{R} \rightarrow \mathbb{R} \) be convex, and let \( X \) and \( \varphi \left( X\right) \) be integrable random variables. For any \( \sigma \) -algebra \( \mathcal{G} \) , | \[ \varphi \circ E\{ X \mid \mathcal{G}\} \leq E\{ \varphi \left( X\right) \mid \mathcal{G}\} \] | Yes |
Theorem 20. Let \( X \) be a positive submartingale. For all \( p > 1 \), with \( q \) conjugate to \( p \) (i.e., \( \frac{1}{p} + \frac{1}{q} = 1 \) ), we have\n\n\[ \n{\\begin{Vmatrix}\\mathop{\\sup }\\limits_{t}\\left| {X}_{t}\\right| \\end{Vmatrix}}_{{L}^{p}} \\leq q\\mathop{\\sup }\\limits_{t}{\\begin{Vmatrix}{X}... | We let \( {X}^{ * } \) denote \( \\mathop{\\sup }\\limits_{s}\\left| {X}_{s}\\right| \) . Note that if \( M \) is a martingale with \( {M}_{\\infty } \\in {L}^{2} \) , then \( \\left| M\\right| \) is a positive submartingale, and taking \( p = 2 \) we have\n\n\[ \nE\\left\{ {\\left( {M}^{ * }\\right) }^{2}\\right\} \\l... | No |
Theorem 21. Let \( X = {\left( {X}_{t}\right) }_{0 \leq t \leq \infty } \) be an adapted process with càdlàg paths. Suppose \( E\left\{ \left| {X}_{T}\right| \right\} < \infty \) and \( E\left\{ {\bar{X}}_{T}\right\} = 0 \) for any stopping time \( T \), finite or not. Then \( X \) is a uniformly integrable martingale. | Proof. Let \( 0 \leq s < t < \infty \), and let \( \Lambda \in {\mathcal{F}}_{s} \) . Let\n\n\[ \n{u}_{\Lambda } = \left\{ \begin{array}{ll} u, & \text{ if }\omega \in \Lambda \\ \infty , & \text{ if }\omega \notin \Lambda \end{array}\right. \n\]\n\nThen \( {u}_{\Lambda } \) are stopping times for all \( u \geq s \) . ... | Yes |
Theorem 22. A counting process \( N \) is adapted if and only if the associated random variables \( {\left( {T}_{n}\right) }_{n \geq 1} \) are stopping times. | Proof. If the \( {\left( {T}_{n}\right) }_{n \geq 0} \) are stopping times (with \( {T}_{0} = 0 \) a.s.), then the event\n\n\[ \left\{ {{N}_{t} = n}\right\} = \left\{ {\omega : {T}_{n}\left( \omega \right) \leq t < {T}_{n + 1}\left( \omega \right) }\right\} \in {\mathcal{F}}_{t}, \]\n\nfor each \( n \) . Thus \( {N}_{t... | Yes |
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