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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\[ \lbrack \mathcal{H}\left( {t,\bar{x}\left( t\right) ,\bar{u}\left( t\right) }\right) ,\infty ) \times \{ - p\left( t\right) \} \times \lbrack - P\left( t\right) ,\infty ) \subseteq {D}_{t+, x}^{1,2, + }V\left( {t,\bar{x}\left( t\right) }\right) ,\]\n\n\[ ... | 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 |
Proposition 4.10. Under the assumptions of Theorem 4.4, we have\n\n(4.86)\n\n\[ \n\\operatorname{tr}\\left( {\\sigma {\\left( t,\\overline{x}\\left( t\\right) ,\\overline{u}\\left( t\\right) \\right) }^{\\top }\\left( {q\\left( t\\right) - P\\left( t\\right) \\sigma \\left( {t,\\overline{x}\\left( t\\right) ,\\overline... | Proof. By (4.84) and the fact that \( V \) is a viscosity solution of the HJB equation (4.6), we have\n\n\[ \n0 \\geq - \\mathcal{H}\\left( {t,\\bar{x}\\left( t\\right) ,\\bar{u}\\left( t\\right) }\\right) + \\mathop{\\sup }\\limits_{{u \\in U}}G\\left( {t,\\bar{x}\\left( t\\right), u, p\\left( t\\right), P\\left( t\\r... | Yes |
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\[ \bar{q}\left( t\right) = G\left( {t,\bar{x}\left( t\right) ,\bar{u}\left( t\right) , - \bar{p}\left( t\right) , - \bar{P}\left( t\right) }\right) \]\n\n\( \left( {5.14}\right) \)\n\n\[ = \mathop{\max }\limits_{{u \in U}}G\left( {t,\ba... | 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\[ - \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) \leq... | 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\l... | 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). Next, 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\n\n\[ \n{N}_{r} \geq 0,\;\forall r \in \left\lbrack {s, T}\right\rbrack \n\] | 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\n\[ \n\left\langle {{N}_{r}{u}_{0}\left( \cdot \right) ,{u}_{0}\left( \cdot \right) }\right\rangle < 0. \n\]\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\\( \\left( {2.27}\\right) \\)\n\n\\[ \nJ\\left( {s, y;u\\left( \\cdot \\right) }\\right) = - \\frac{1}{2}x{\\left( T\\right) }^{2}.\n\\]\n\nTake \\( {u}_{\\ell }\\left( t\\right) \\equiv \\ell \\) . Then | \\[ \nJ\\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) .\n\\]\n\nThus, the corresponding LQ problem is not finite. | 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 (2.29) admits a (unique) solution \( \left( {\bar{x}\left( \cdot \right) ,\ba... | 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), 0 = {N}_{s}\bar{u}\left( \cdot \right) + {H}_{s}\left( y\right) = \left( {R + {L}_{s}^{ * }Q{L}_{s} + S{L}_{s} + {L}_{s}^{ * }{S}^{\top } + {\widehat{L}}_{s}^{ * }G{... | 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,\n\n\[ R + {D}^{\top }{GD} = 0. \]\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))\n\n\[{N}_{s}u\left( \cdot \right) + {H}_{s}\left( y\right)\]\n\n\[= \left( {R + {L}_{s}^{ * }Q{L}_{s} + S{L}_{s} + {L}_{s}^{ * }{S... | Yes |
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 |
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 \right) ,\bar{q}\... | In this case, the following is an (the) optimal control:\n\n(5.18)\n\n\[ \bar{u}\left( t\right) = - R{\left( t\right) }^{-1}\left\lbrack {S\bar{x}\left( t\right) - {B}^{\top }\bar{p}\left( t\right) - {D}^{\top }\bar{q}\left( t\right) }\right\rbrack ,\;\forall t \in \left\lbrack {s, T}\right\rbrack . \] | No |
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 |
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... | Yes |
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\[ \n{Y}^{0}\left( t\right) = 0,\;{Z}^{0}\left( t\right) = 0 \n\]\n\nand solve the following BSDEs successively:\n\n(3.74)... | 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... | We 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 (4.21) \( v\left( {t, x}\right) = \mathop{\inf }\limits_{{u\left( \cdot \right) \in {\mathcal{U}}^{w}\left\lbrack {t, T}\right\rbrack }}E\left( {{\i... | In addition, if (4.18) admits a classical solution, then (4.21) gives that classical solution. | No |
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 \( \left( {t, x}\right) \in \lbrack 0, T) \times G \),... | Then the viscosity solution \( v\left( {t, x}\right) \) of (4.40) is given by\n\n(4.42)\n\n\[ v\left( {t, x}\right) = {EY}\left( {t;t, x}\right) \equiv Y\left( {t;t, x}\right) ,\;\left( {t, x}\right) \in \left\lbrack {0, T}\right\rbrack \times G. \] | Yes |
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\\[ \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( \\beg... | 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... | No |
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 |
The isonormal Gaussian process associated to the Brownian motion: Let \( \left( {B\left( t\right) = \left( {{B}^{1}\left( t\right) ,\ldots ,{B}^{d}\left( t\right) }\right), t \geq 0}\right) \) be a d-dimensional Brownian motion defined on its canonical probability space \( \left( {\Omega ,\mathcal{F},\mathrm{P}}\right)... | \[ W\left( h\right) = \mathop{\sum }\limits_{{i = 1}}^{d}{\int }_{{\mathbb{R}}_{ + }}{h}_{i}\left( s\right) d{B}^{i}\left( s\right) \] | Yes |
The isonormal Gaussian process associated to the fractional Brownian motion: Fix \( T > 0 \) and let \( {B}^{\gamma } = \left( {{B}^{\gamma }\left( t\right), t \in \left\lbrack {0, T}\right\rbrack }\right) \) be a fractional Brownian motion with Hurst parameter \( \gamma \in \left( {0,1}\right) \) defined on its canoni... | We denote by \( \mathcal{E} \) the step functions on \( \left\lbrack {0, T}\right\rbrack \) . Let \( H \) be the Hilbert space defined as the closure of \( \mathcal{E} \) with respect to the scalar product\n\n\[ \n{\left\langle {\mathbf{1}}_{\left\lbrack 0, t\right\rbrack },{\mathbf{1}}_{\left\lbrack 0, s\right\rbrack ... | Yes |
Lemma 1.4 Let \( X{yY} \) two centered random variables with variance 1 and Gaussian joint distribution. Then, for all \( n, m \geq 0 \) ,\n\n\[ \n\mathrm{E}\left\lbrack {{H}_{n}\left( X\right) {H}_{m}\left( Y\right) }\right\rbrack = \left\{ \begin{array}{ll} 0, & \text{ if }n \neq m, \\ \frac{1}{n!}{\left( \mathrm{E}\... | Proof. For all \( s, t \in \mathbb{R} \), we have\n\n\[ \n\mathrm{E}\left\lbrack {\exp \left( {{sX} - \frac{{s}^{2}}{2}}\right) \exp \left( {{tY} - \frac{{t}^{2}}{2}}\right) }\right\rbrack = \exp \left( {{st}\mathrm{E}\left\lbrack {XY}\right\rbrack }\right) .\n\]\n\nTaking the \( \left( {n + m}\right) \) -th partial de... | Yes |
Theorem 1.5 Let \( \mathcal{G} \) be the \( \sigma \) -algebra generated by \( W \) . Then,\n\n\[ \n{L}^{2}\left( {\Omega ,\mathcal{G},\mathrm{P}}\right) = { \oplus }_{n = 0}^{\infty }{\mathcal{H}}_{n} \n\] | Proof. Let \( X \in {L}^{2}\left( {\Omega ,\mathcal{G},\mathrm{P}}\right) \) be orthogonal to \( {\mathcal{H}}_{n} \) for all \( n \geq 0 \) . That is, \( \mathrm{E}\left\lbrack {X{H}_{n}\left( {W\left( h\right) }\right) }\right\rbrack = 0 \) for all \( h \in H \) such that \( \parallel h{\parallel }_{H} = 1 \) . We no... | Yes |
Example 1.6 We consider \( H = \mathbb{R},\left( {\Omega ,\mathcal{F},\mathrm{P}}\right) = \left( {\mathbb{R},\mathcal{B}\left( \mathbb{R}\right) ,\mu }\right) \), where \( \mu \) is the Gaussian law \( N\left( {0,1}\right) \) . For each \( h \in H \), we define \( W\left( h\right) = {hX} \), where \( X \) is a Gaussia... | Moreover, Lemma 1.4 and Theorem 1.5 imply that the Hermite polynomials \( \left\{ {\sqrt{n!}{H}_{n}\left( x\right), n \geq 0}\right\} \) form a complete orthonormal system in \( {L}^{2}\left( {\mathbb{R},\mu }\right) \) . | Yes |
Lemma 1.7 If \( f \in {\mathcal{E}}_{n} \), then \( {I}_{n}\left( f\right) = {I}_{n}\left( \widetilde{f}\right) \) . | Proof. Because \( {I}_{n} \) is linear, it suffices to prove the lemma for a function of the form\n\n\[ f = {\mathbf{1}}_{\left\lbrack {{t}_{1}^{\left( 1\right) },{t}_{1}^{\left( 2\right) }}\right) \times \cdots \times \left\lbrack {{t}_{n}^{\left( 1\right) },{t}_{n}^{\left( 2\right) }}\right) }, \]\n\nwhere the interv... | Yes |
Lemma 1.9 The space \( {\mathcal{E}}_{n} \) is dense in \( {L}^{2}\left( {T}^{n}\right) \) . That is, for all \( f \in {L}^{2}\left( {T}^{n}\right) \), there exists a sequence \( {\left\{ {f}_{k}\right\} }_{k \geq 1},{f}_{k} \in {\mathcal{E}}_{n} \), that converges towards \( f \) in \( {L}^{2}\left( {T}^{n}\right) \),... | Proof. Because the usual set of elementary functions is dense in \( {L}^{2}\left( {T}^{n}\right) \), and the set \( D \) has Lebesgue measure zero, the proof is immediate. | No |
Theorem 1.11 Let \( f \in {L}^{2}\left( {T}^{n}\right), n \geq 1 \) . Then\n\n\[ \n{I}_{n}\left( f\right) = n!{\int }_{a}^{b}\cdots {\int }_{a}^{{t}_{n - 2}}\left( {{\int }_{a}^{{t}_{n - 1}}\widetilde{f}\left( {{t}_{1},\ldots ,{t}_{n}}\right) {dB}\left( {t}_{n}\right) }\right) {dB}\left( {t}_{n - 1}\right) \cdots {dB}\... | Proof. We observe that it suffices to show the theorem in the case where \( f \) is a characteristic function on a disjoint rectangle with the set \( D \) . That is,\n\n\[ \nf\left( {{t}_{1},\ldots ,{t}_{n}}\right) = {\mathbf{1}}_{\left\lbrack {{t}_{1}^{\left( 1\right) },{t}_{1}^{\left( 2\right) }}\right) \times \cdots... | Yes |
Theorem 1.12 For all \( f \in {L}^{2}\left( T\right) \) and \( n \geq 1 \) ,\n\n\[ \n{I}_{n}\left( {f}^{\otimes n}\right) = n!{H}_{n}\left( {W\left( f\right) ,\parallel f{\parallel }_{{L}^{2}\left( T\right) }^{2}}\right) ,\n\]\n\nwhere \( {f}^{\otimes n} \) is the \( n \) variables (symmetric) function \( {f}^{\otimes ... | Proof. We prove this result by induction over \( n \) . The case \( n = 1 \) is immediate. We assume that the result holds for \( 1,\ldots, n \) . Using Theorem 1.11 we get that\n\n\[ \n{I}_{n + 1}\left( {f}^{\otimes n + 1}\right) = \left( {n + 1}\right) !{\int }_{a}^{b}f\left( {t}_{1}\right) {X}_{{t}_{1}}{dB}\left( {t... | Yes |
Example 1.14 Let \( \\left( {B\\left( t\\right), t \\in \\left\\lbrack {0,1}\\right\\rbrack }\\right) \) be a one-dimensional Brownian motion defined on its canonical probability space \( \\left( {\\Omega ,\\mathcal{F},\\mathrm{P}}\\right) \). Let \( H = {L}^{2}\\left( {\\left\\lbrack {0,1}\\right\\rbrack ,\\mathcal{B}... | Let \( F = f\\left( {W\\left( {\\mathbf{1}}_{\\left\\lbrack 0,{t}_{1}\\right\\rbrack }\\right) ,\\ldots, W\\left( {\\mathbf{1}}_{\\left\\lbrack 0,{t}_{n}\\right\\rbrack }\\right) }\\right) \\in \\mathcal{S},0 \\leq {t}_{1} < \\cdots < {t}_{n} \\leq 1 \). Then, for each \( h \\in H \), \[ \\langle {DF}, h{\\rangle }_{H}... | Yes |
Lemma 1.15 Let \( F \in \mathcal{S} \) and \( h \in H \) . Then,\n\n\[ \mathrm{E}\left\lbrack {\langle {DF}, h{\rangle }_{H}}\right\rbrack = \mathrm{E}\left\lbrack {{FW}\left( h\right) }\right\rbrack \] | Proof. It is sufficient to consider the case where \( \parallel h{\parallel }_{H} = 1 \) . There exists and orthonormal family of \( H,\left\{ {{e}_{1},\ldots {e}_{n}}\right\} \) such that \( h = {e}_{1} \) y \( F = f\left( {W\left( {e}_{1}\right) ,\ldots, W\left( {e}_{n}\right) }\right) \), where \( f \in {\mathcal{C}... | Yes |
Proposition 1.17 For any \( p \geq 1 \), the operator \( D \) is closable from \( {L}^{p}\left( \Omega \right) \) to \( {L}^{p}\left( {\Omega ;H}\right) \) . | Proof. Let \( \left\{ {{F}_{N}, N \geq 1}\right\} \) be a sequence of random variables in \( \mathcal{S} \) such that \( {F}_{N} \) converges to zero in \( {L}^{p}\left( \Omega \right) \) and the sequence of derivatives \( D{F}_{N} \) converges to \( \eta \) in \( {L}^{p}\left( {\Omega ;H}\right) \) .\n\nThen, for any ... | Yes |
Lemma 1.19 For any \( F \in \mathcal{S},1 \leq p \leq q \) and \( 0 \leq k \leq j, k, j \in \mathbb{N},\parallel F{\parallel }_{k, p} \leq \parallel F{\parallel }_{j, q} \) . In particular, \( {\mathbb{D}}^{k + 1, p} \subset {\mathbb{D}}^{k, q} \) for all \( k \geq 0 \) y \( p > q \) . | Proof. The case \( p = q \) is trivial. If \( k = j \), we use Hölder’s inequality. | No |
Proposition 1.21 Let \( F \in {\mathbb{D}}^{1,2} \) be a square integrable random variable with the Wiener chaos decomposition of Theorem 1.13, that is, \( F = \mathop{\sum }\limits_{{n = 0}}^{\infty }{I}_{n}\left( {f}_{n}\right) \), where \( {f}_{n} \in {L}^{2}\left( {T}^{n}\right) \) are symmetric. Then, \( F \) belo... | Proof. Assume first that \( F = {I}_{n}\left( {f}_{n}\right) \), with \( {f}_{n} \in {\mathcal{E}}_{n} \) symmetric. Then\n\n\[ {D}_{t}F = \mathop{\sum }\limits_{{j = 1}}^{n}\mathop{\sum }\limits_{{{i}_{1},\ldots ,{i}_{n} = 1}}^{k}{a}_{{i}_{1}\cdots {i}_{n}}{\xi }_{{i}_{1}}\cdots {1}_{\left\lbrack {\tau }_{{i}_{j} - 1}... | Yes |
Lemma 1.26 Let \( F \) be a random variable such that \( F \in {\mathbb{D}}^{1,2} \cap {L}^{2}\left( {\Omega ,{\mathcal{F}}_{{\left\lbrack c, d\right\rbrack }^{c}},\mathrm{P}}\right) \). Then \( {D}_{t}F = 0,\left( {\lambda \times \mathrm{P}}\right) \) -for almost every \( \left( {t,\omega }\right) \in \left\lbrack {c,... | Proof. Let \( F \in \mathcal{S}, F = f\left( {W\left( {\left| {1}_{\left\lbrack {a}_{1},{b}_{1}\right\rbrack }\right) ,\ldots, W\left( {\mathbf{1}}_{\left\lbrack {a}_{n},{b}_{n}\right\rbrack }\right) }\right) \text{, where}f \in {\mathcal{C}}_{p}^{\infty }\left( {\mathbb{R}}^{n}\right) ,\left\lbrack {{a}_{i},{b}_{i}}\r... | Yes |
Lemma 1.27 With probability one the Brownian motion attains it maximum on \( \\left\\lbrack {0,1}\\right\\rbrack \) on a unique point. | Proof. We want to show that the set\n\n\[ G = \\left\\{ {\\omega : \\mathop{\\sup }\\limits_{{t \\in \\left\\lbrack {0,1}\\right\\rbrack }}B\\left( t\\right) = B\\left( {t}_{1}\\right) = B\\left( {t}_{2}\\right) \\text{ for some }{t}_{1} \\neq {t}_{2}}\\right\\} \]\n\nhas probability zero. For each \( n \\geq 0 \), we ... | Yes |
Lemma 1.28 Let \( B = \{ B\left( t\right), t \in \left\lbrack {0,1}\right\rbrack \} \) be a Brownian motion. Consider the random variable \( M = \mathop{\sup }\limits_{{t \in \left\lbrack {0,1}\right\rbrack }}B\left( t\right) \) . Then \( M \in {\mathbb{D}}^{1,2} \) and \( {D}_{t}M = {\mathbf{1}}_{\left\lbrack 0, T\rig... | Proof. We start proving that \( M \in {\mathbb{D}}^{1,2} \) . Consider the approximation of \( M \) defined by\n\n\[ \n{M}_{n} = \max \left\{ {B\left( {t}_{1}\right) ,\ldots, B\left( {t}_{n}\right) }\right\} \n\]\n\nwhere \( \left( {{t}_{n}, n \geq 1}\right) \) is a countable and dense subset of \( \left\lbrack {0,1}\r... | Yes |
Proposition 1.30 (i) If \( u \in \operatorname{Dom}\delta \), then \( \mathrm{E}\left\lbrack {\delta \left( u\right) }\right\rbrack = 0 \) . | Proof. (i) is immediate applying (1.19) with \( F = 1 \) . | No |
Proposition 1.33 Let \( F \in {\mathbb{D}}^{1,2} \) and \( u \in \operatorname{Dom}\delta \) such that \( {Fu} \in {L}^{2}\left( {\Omega, H}\right) \) and \( {F\delta }\left( u\right) - \) \( \langle {DF}, u{\rangle }_{H} \in {L}^{2}\left( \Omega \right) \) . Then \( {Fu} \in \operatorname{Dom}\delta \) and \[ \delta \... | Proof. For any random variable \( G \in \mathcal{S} \), using (1.19), it holds that \[ \mathrm{E}\left\lbrack {\langle {Fu},{DG}{\rangle }_{H}}\right\rbrack = \mathrm{E}\left\lbrack {\langle u, D\left( {FG}\right) - {GDF}{\rangle }_{H}}\right\rbrack = \mathrm{E}\left\lbrack {\left( {{F\delta }\left( u\right) -\langle u... | Yes |
Proposition 1.35 If \( u \in {L}^{2}\left( {T \times \Omega }\right) \) is an adapted process then \( u \in \) Dom \( \delta \) . Moreover, \( \delta \left( u\right) \) coincides with the Itô integral with respect to the Brownian motion, that is,\n\n\[ \delta \left( u\right) = {\int }_{a}^{b}u\left( s\right) {dB}\left(... | Proof. Let \( u \) be an elementary adapted process of the form\n\n\[ {u}_{t} = \mathop{\sum }\limits_{{j = 1}}^{n}{F}_{j}{\mathbf{1}}_{\left( {t}_{j},{t}_{j + 1}\right\rbrack }\left( t\right) \]\n\nwhere \( {F}_{j} \in {L}^{2}\left( {\Omega ,{\mathcal{F}}_{{t}_{j}},\mathrm{P}}\right) \) and \( a \leq {t}_{1} < \cdots ... | Yes |
Proposition 1.36 Let \( u \in {\mathbb{D}}^{1,2}\left( H\right) \). Assume that for any \( t \in T \), the process \( \left( {{D}_{t}u\left( s\right) }\right. \), \( s \in T \) ) is in Dom \( \delta \) and that there exists a version of the process \( \left( {\delta \left( {{D}_{t}u\left( s\right) }\right), t \in T}\ri... | Proof. Let \[ u\left( t\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{I}_{n}\left( {{f}_{n}\left( {\cdot, t}\right) }\right) \] where for all \( n \geq 1,{f}_{n} \in {L}^{2}\left( {T}^{n + 1}\right) \) is symmetric in its first \( n \) variables. Then, using Propositions 1.21 and 1.34 we have that \[ {D}_{t}\left(... | Yes |
Theorem 1.37 Let \( F \in {L}^{2}\left( \Omega \right) \), measurable with respect to \( B \) . Then there exists a unique process \( u \in {L}_{a}^{2}\left( {{\mathbb{R}}_{ + } \times \Omega }\right) \) such that\n\n\[ F = \mathrm{E}\left\lbrack F\right\rbrack + {\int }_{0}^{\infty }u\left( t\right) {dB}\left( t\right... | Proof. It suffices to show that any zero-mean square integrable random variable \( G \) that is orthogonal to all stochastic integrals \( {\int }_{{\mathbb{R}}_{ + }}u\left( t\right) {dB}\left( t\right), u \in {L}_{a}^{2}\left( {{\mathbb{R}}_{ + } \times \Omega }\right) \) must be zero. Let \( u \in {L}_{a}^{2}\left( {... | Yes |
Theorem 1.38 Let \( F \in {\mathbb{D}}^{1,2} \) . Then\n\n\[ F = \mathrm{E}\left\lbrack F\right\rbrack + {\int }_{0}^{\infty }\mathrm{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack {dB}\left( t\right) \] | Proof. Suppose that \( F \) has the representation (1.25) with \( u \in {L}_{a}^{2}\left( {{\mathbb{R}}_{ + } \times \Omega }\right) \) . Then for any \( v \in {L}_{a}^{2}\left( {{\mathbb{R}}_{ + } \times \Omega }\right) \), using the isometry property of the Itô integral, we can write\n\n\[ \mathrm{E}\left\lbrack {\de... | Yes |
Proposition 2.1 Let \( F, G \) two random variables such that \( F \in {\mathbb{D}}^{1,2} \) . Let \( u \) be an \( H \) -valued random variable such that \( \langle {DF}, u{\rangle }_{H} \neq 0 \) a.s. and \( {Gu}{\left( \langle DF, u{\rangle }_{H}\right) }^{-1} \in \) Dom \( \delta \) . Then for any function \( f \in... | Proof. Applying the chain rule (Proposition 1.20) we have that\n\n\[ \langle D\left( {f\left( F\right) }\right), u{\rangle }_{H} = {f}^{\prime }\left( F\right) \langle {DF}, u{\rangle }_{H} \]\n\nUsing the duality relation (1.19) we obtain that\n\n\[ \mathrm{E}\left\lbrack {{f}^{\prime }\left( F\right) G}\right\rbrack ... | Yes |
Proposition 2.2 Let \( F \) be a random variable such that \( F \in {\mathbb{D}}^{1,2} \) . Assume that \( \frac{DF}{\parallel {DF}{\parallel }_{H}^{2}} \in \) Dom \( \delta \) . Then the law of \( F \) has a continuous and bounded density function given by \[ f\left( x\right) = \mathrm{E}\left\lbrack {{\mathbf{1}}_{\{... | Proof. Let \( \psi : \mathbb{R} \mapsto {\mathbb{R}}_{ + } \) be a \( {\mathcal{C}}^{1} \) function with compact support and let \( \phi \left( y\right) = {\int }_{-\infty }^{y}\psi \left( x\right) {dx} \) . Using formula (2.1) with \( G = 1 \) and \( f = \phi \), we have that \[ \mathrm{E}\left\lbrack {\psi \left( F\r... | Yes |
Lemma 2.3 Let \( F \in {\mathbb{D}}^{2,4} \) such that \( \mathrm{E}\left\lbrack {\parallel {DF}{\parallel }_{H}^{-8}}\right\rbrack < + \infty \) . Then the density \( f\left( x\right) \) satisfies the following estimate:\n\n\[ f\left( x\right) \leq {\left( \mathrm{P}\{ \left| F\right| > \left| x\right| \} \right) }^{1... | Proof. First observe that the hypotheses of the lemma and Proposition 1.31 imply that \( \frac{DF}{\parallel {DF}{\parallel }_{H}^{2}} \in {\mathbb{D}}^{1,2}\left( H\right) \subset \) Dom \( \delta \), and hence, the hypotheses of Proposition 2.2 hold. Applying the Cauchy-Schwarz inequality to the expression (2.2), we ... | Yes |
Lemma 2.8 Let \( \gamma \) be a \( d \times d \) random matrix such that det \( \gamma > 0 \) a.s. and \( {\left( \det \gamma \right) }^{-1} \in \) \( {L}^{p}\left( \Omega \right) \), for all \( p \geq 1 \) . Suppose that the entries \( {\gamma }^{ij} \) of \( \gamma \) are in \( {\mathbb{D}}^{\infty } \) . Then \( {\l... | Proof. For any \( \epsilon > 0 \), define \[ {\gamma }_{\epsilon }^{-1} = \frac{\det \gamma }{\det \gamma + \epsilon }{\gamma }^{-1} \] Note that \( {\left( \det \gamma + \epsilon \right) }^{-1} \in {\mathbb{D}}^{\infty } \) as it can be expressed as the composition of \( \det \gamma \) with a function in \( {\mathcal{... | Yes |
Proposition 2.9 Let \( F = \left( {{F}^{1},\ldots ,{F}^{d}}\right) \) be a nondegenerate random vector. Let \( G \in {\mathbb{D}}^{\infty } \) and \( g \in {\mathcal{C}}_{\rho }^{\infty }\left( {\mathbb{R}}^{d}\right) \) . Then for any multiindex \( \alpha \in \{ 1,\ldots, d{\} }^{k}, k \geq 1 \), there exists an eleme... | Proof. By the chain rule (Proposition 1.20), we have, for all \( j = 1,\ldots, d \) \[ {\left\langle D\left( g\left( F\right) \right), D{F}^{j}\right\rangle }_{H} = \mathop{\sum }\limits_{{i = 1}}^{d}{\partial }_{i}{\left\langle D{F}^{i}, D{F}^{j}\right\rangle }_{H} = \mathop{\sum }\limits_{{i = 1}}^{d}{\partial }_{i}g... | Yes |
Theorem 2.10 There exists a unique continuous solution \( X = \{ X\left( t\right), t \in \left\lbrack {0, T}\right\rbrack \} \) to equation (2.7). Moreover, | \[ \mathrm{E}\left\lbrack {\mathop{\sup }\limits_{{0 \leq t \leq T}}{\left| X\left( t\right) \right| }^{p}}\right\rbrack \leq C \] for any \( p \geq 2 \), where \( C > 0 \) is a positive constant depending on \( p, T, K \) . | No |
Theorem 2.11 Let \( X = \{ X\left( t\right), t \in \left\lbrack {0, T}\right\rbrack \} \) be the solution to equation (2.7) and assume that its coefficients are continuously differentiable functions. Then \( {X}_{i}\left( t\right) \) belongs to \( {\mathbb{D}}^{1,\infty } \) for any \( t \in \left\lbrack {0, T}\right\r... | Proof. Consider the Picard approximations given by \n\n\[ {X}_{i}^{0}\left( t\right) = {x}_{0}^{i} \] \n\n\[ {X}_{i}^{n + 1}\left( t\right) = {x}_{0}^{i} + \mathop{\sum }\limits_{{j = 1}}^{d}{\int }_{0}^{t}{\sigma }_{ij}\left( {{X}^{n}\left( s\right) }\right) d{B}^{j}\left( s\right) + {\int }_{0}^{t}{b}_{i}\left( {{X}^... | Yes |
Theorem 2.14 Assume that Hörmander’s condition (H) holds and that the coefficients of equation (2.7) are infinitely differentiable with bounded partial derivatives of all orders greater than or equal to one. Then for any \( t > 0, X\left( t\right) \) has an infinitely differentiable density. | The proof of this theorem is based on Norris Lemma ([N06, Lemma 2.3.2]) which essentially shows that when the quadratic variation or the bounded variation part of a continuous semimartingale is large, then the semimartingale is small with an exponentially small probability. The complete proof of Theorem 2.14 is given i... | No |
Lemma 3.1 The following call-put parity relationship follows:\n\n\[ \n{C}_{t} - {P}_{t} = {S}_{t} - K{e}^{-r\left( {T - t}\right) },\text{ for all }t < T. \n\] | Proof. Assume that \( {C}_{t} - {P}_{t} > {S}_{t} - K{e}^{-r\left( {T - t}\right) } \) . At time \( t \), if we sell a call, we buy a put and we buy an action we will obtain a net benefit of \( {Y}_{t} = {C}_{t} - {P}_{t} - {S}_{t} \) . If this quantity is positive, we can put this money at interest rate \( r > 0 \) ti... | Yes |
Consider a market with a non-risky asset with \( r = 0 \) and a risky asset with initial price \( {S}_{0} = {10} \) and such that\n\n\[ \mathrm{P}\left\{ {{S}_{1} = {20}}\right\} = p,\mathrm{P}\left\{ {{S}_{1} = 7,5}\right\} = 1 - p. \]\n\nConsider a put with \( K = {15}\mathrm{y}T = 1 \), where the time in this case i... | Under the self-financing assumption, we have that \( {V}_{0} = {10}{\beta }_{1} + {\alpha }_{1} \). On the other hand, because the portfolio needs to replicate the derivative, we have that\n\n\[ {V}_{1} = \left\{ \begin{array}{ll} 5 & \text{ if }{S}_{1} = {20} \\ 0 & \text{ if }{S}_{1} = 7,5 \end{array}\right. \]\n\nth... | Yes |
Theorem 3.4 Let \( h \) be a continuous function of at most linear growth. Assume that the following PDE admits a regular solution \( v\left( {t, y}\right) \) over \( \rbrack 0, T\rbrack \times \rbrack 0, + \infty \lbrack \) :\n\n\[ \left\{ \begin{array}{l} \frac{1}{2}{\sigma }^{2}{y}^{2}{v}_{yy}^{\prime \prime }\left(... | Proof. Applying Itô’s formula to the function \( v\left( {t,{S}_{t}}\right) \), we obtain that:\n\n\[ {dv}\left( {t,{S}_{t}}\right) = {v}_{t}^{\prime }\left( {t,{S}_{t}}\right) {dt} + {v}_{y}^{\prime }\left( {t,{S}_{t}}\right) d{S}_{t} + \frac{1}{2}{\sigma }^{2}{S}_{t}^{2}{v}_{yy}^{\prime \prime }\left( {t,{S}_{t}}\rig... | Yes |
Lemma 3.5\n\n\[ \n{du}\left( {t,{\widetilde{B}}_{t}}\right) = {e}^{-{rt}}\beta \left( {t,{S}_{t}}\right) {S}_{t}{\sigma d}{\widetilde{B}}_{t} \n\] | Proof. We will use the notation \( {\beta }_{t} = \beta \left( {t,{S}_{t}}\right) \) . We have that\n\n\[ \n{du}\left( {t,{\widetilde{B}}_{t}}\right) = - r{e}^{-{rt}}v\left( {t,{S}_{t}}\right) {dt} + {e}^{-{rt}}{dv}\left( {t,{S}_{t}}\right) .\n\]\n\nOn the other hand, \( v\left( {t,{S}_{t}}\right) = {\alpha }_{t}{e}^{r... | No |
Theorem 1.2 (Hörmander) Consider 1.1, as well as the vector spaces \( \mathcal{V}\left( x\right) \subset \) \( {\mathbf{R}}^{n} \) constructed as above. If the parabolic Hörmander condition is satisfied, then the transition probabilities for \( \left( \underline{1.1}\right) \) have smooth densities with respect to Lebe... | The original proof of this result goes back to [Hör67] and relied on purely analytical techniques. However, since it has a clear probabilistic interpretation, a more \ | No |
Lemma 2.2 Let \( x, y \in \mathbf{R} \) and \( a, b \) with \( {a}^{2} + {b}^{2} = 1 \) . Then, one has the identity\n\n\[ \n{H}_{n}\left( {{ax} + {by}}\right) = \mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {a}^{k}{b}^{n - k}{H}_{k}\left( x\right) {H}_{n - k}\left( y\right) .\n\] | Proof. As a consequence of (2.4) we have\n\n\[ \n{H}_{n}\left( {{ax} + {by}}\right) = \exp \left( {-\frac{{D}_{x}^{2}}{2{a}^{2}}}\right) {\left( ax + by\right) }^{n} = \exp \left( {-\frac{{D}_{x}^{2}}{2}}\right) \exp \left( {-\frac{{b}^{2}{D}_{x}^{2}}{2{a}^{2}}}\right) {\left( ax + by\right) }^{n}.\n\]\n\nNoting that \... | Yes |
Theorem 2.4 In the above context, one has\n\n\[ \n{L}^{2}\left( {\Omega ,\mathcal{F},\mathbf{P}}\right) = {\bigoplus }_{n \geq 0}{\mathcal{H}}_{n} \n\] | Proof. Denote by \( {H}_{N} \subset H \) the subspace generated by \( {\left\{ {e}_{k}\right\} }_{k \leq N} \), by \( {\mathcal{F}}_{N} \) the \( \sigma \) -algebra generated by \( \left\{ {W\left( h\right) : h \in {H}_{N}}\right\} \), and assume that one has\n\n\[ \n{L}^{2}\left( {\Omega ,{\mathcal{F}}_{N},\mathbf{P}}... | Yes |
Lemma 2.6 Let \( H = {L}^{2}\left( {{\mathbf{R}}_{ + },{\mathbf{R}}^{m}}\right) \) . For \( n \geq 1 \), the space \( {\mathcal{H}}_{n} \) consists of all random variables of the form\n\n\[ \n{\widetilde{I}}_{n}\left( \widetilde{h}\right) = \mathop{\sum }\limits_{{{j}_{1}\cdots {j}_{n}}}{\int }_{0}^{\infty }{\int }_{0}... | Proof. We identify \( {L}^{2}\left( {{\Delta }_{n},{\mathbf{R}}^{{m}^{n}}}\right) \) with a subspace of \( {H}^{\otimes n} \) and define the symmetrisation \( \Pi : {H}^{\otimes n} \rightarrow {H}^{\otimes n} \) as before. The map \( \sqrt{n!}\Pi \) is then an isometry between \( {L}^{2}\left( {{\Delta }_{n},{\mathbf{R... | Yes |
Proposition 3.1 For every \( X \in \mathcal{W} \) and \( h \in H \), one has the identity\n\n\[ \mathbf{E}\langle \mathcal{D}X, h\rangle = \mathbf{E}\left( {{XW}\left( h\right) }\right) \] | Proof. By Grahm-Schmidt, we can assume that \( X \) is of the form 3.3 with the \( {h}_{i} \) orthonormal. One then has\n\n\[ \mathbf{E}\langle \mathcal{D}X, h\rangle = \mathop{\sum }\limits_{{k = 1}}^{N}\mathbf{E}{\partial }_{k}F\left( {W\left( {h}_{1}\right) ,\ldots, W\left( {h}_{N}\right) }\right) \left\langle {{h}_... | Yes |
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