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Proposition 3.2.9. For any \( {\mathcal{F}}_{T} \) -measurable \( F \in {L}^{2}\left( \Omega \right) \) we have\n\n\[ \mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{T}}\right\rbrack = 0,\;0 \leq T \leq t. \] | Proof. From from Relation (3.2.3) we have \( F = \mathbb{E}\left\lbrack {F \mid {\mathcal{F}}_{T}}\right\rbrack \) if and only if\n\n\[ {\int }_{T}^{\infty }\mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack d{M}_{t} = 0 \]\n\nwhich implies \( \mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{t}}\... | Yes |
Proposition 3.2.11. Let \( T > 0 \) . Under the stability Assumption 3.2.10, for any \( {\mathcal{F}}_{T} \) -measurable random variable \( F \in {L}^{2}\left( \Omega \right) \) we have \( F \in {\mathbb{D}}_{\lbrack T,\infty )} \) and\n\n\[ \n{D}_{t}F = 0,\;t \geq T.\n\] | Proof. Since \( F \) is \( {\mathcal{F}}_{T} \) -measurable, \( {D}_{t}F \) is \( {\mathcal{F}}_{T} \) -measurable, \( t \geq T \), by the stability Assumption 3.2.10, and from Proposition 3.2.9 we have\n\n\[ \n{D}_{t}F = \mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{T}}\right\rbrack = 0,\;0 \leq T \leq t.\n\] | Yes |
Proposition 3.3.1. Under the duality Assumption 3.1.1 and the Clark formula Assumption 3.2.1, the operator \( \delta \) applied to any square-integrable adapted process \( {\left( {u}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \in {L}_{ad}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) coincides with the stochasti... | Proof. Let \( u \in \mathcal{P} \) be a simple \( {\mathcal{F}}_{t} \) -predictable process. From the duality Assumption 3.1.1 and the fact (2.5.7) that\n\n\[ \mathbb{E}\left\lbrack {{\int }_{0}^{\infty }{u}_{t}d{M}_{t}}\right\rbrack = 0 \]\n\nwe have:\n\n\[ \mathbb{E}\left\lbrack {F{\int }_{0}^{\infty }{u}_{t}d{M}_{t}... | Yes |
Proposition 3.3.2. Assume that\n\ni) \( {\left( {M}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) has the predictable representation property, and\n\nii) the operator \( \delta \) coincides with the stochastic integral with respect to \( {\left( {M}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) on the space \( {L}_{ad}^{2}\l... | Proof. For all \( F \in \operatorname{Dom}\left( D\right) \) and square-integrable adapted process \( u \) we have:\n\n\[ \mathbb{E}\left\lbrack {\left( {F - \mathbb{E}\left\lbrack F\right\rbrack }\right) \delta \left( u\right) }\right\rbrack = \mathbb{E}\left\lbrack {{F\delta }\left( u\right) }\right\rbrack \]\n\n\[ =... | Yes |
Proposition 3.4.1. For any \( F, G \in {L}^{2}\left( \Omega \right) \) we have\n\n\[ \operatorname{Cov}\left( {F, G}\right) = \mathbb{E}\left\lbrack {{\int }_{0}^{\infty }\mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack \mathbb{E}\left\lbrack {{D}_{t}G \mid {\mathcal{F}}_{t}}\right\rbrack {dt}}\rig... | Proof. We have\n\n\[ \operatorname{Cov}\left( {F, G}\right) = \mathbb{E}\left\lbrack {\left( {F - \mathbb{E}\left\lbrack F\right\rbrack }\right) \left( {G - \mathbb{E}\left\lbrack G\right\rbrack }\right) }\right\rbrack \]\n\n\[ = \mathbb{E}\left\lbrack {{\int }_{0}^{\infty }\mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathc... | Yes |
Theorem 3.4.4. Let \( n \in \mathbb{N} \) and \( F, G \in \mathop{\bigcap }\limits_{{k = 1}}^{{n + 1}}{\mathbb{D}}_{2, k}\left( {\Delta }_{k}\right) \) . We have\n\n\[ \operatorname{Cov}\left( {F, G}\right) = \mathop{\sum }\limits_{{k = 1}}^{n}{\left( -1\right) }^{k + 1}\mathbb{E}\left\lbrack {{\int }_{{\Delta }_{k}}\l... | Proof. By polarization we may take \( F = G \) . For \( n = 0 \) ,((3.4.2)) is a consequence of the Clark formula. Let \( n \geq 1 \) . Applying Lemma 3.2.4 to \( {D}_{{t}_{n}}\cdots {D}_{{t}_{1}}F \) with \( t = {t}_{n} \) and \( {ds} = d{t}_{n + 1} \), and integrating on \( \left( {{t}_{1},\ldots ,{t}_{n}}\right) \in... | Yes |
Proposition 3.5.1. Let \( F \in \operatorname{Dom}\left( D\right) \) be lower bounded with \( F > \eta \) a.s. for some \( \eta > 0 \) . We have\n\n\[ \operatorname{Ent}\left\lbrack F\right\rbrack \leq \frac{1}{2}\mathbb{E}\left\lbrack {\frac{1}{F}{\int }_{0}^{\infty }\left( {2 - {\mathbf{1}}_{\left\{ {\phi }_{t} = 0\r... | Proof. Let us assume that \( F \) is bounded and \( {\mathcal{F}}_{T} \) -measurable, and let\n\n\[ {X}_{t} = \mathbb{E}\left\lbrack {F \mid {\mathcal{F}}_{t}}\right\rbrack = {X}_{0} + {\int }_{0}^{t}{u}_{s}d{M}_{s},\;t \in {\mathbb{R}}_{ + }, \]\n\nwith \( {u}_{s} = \mathbb{E}\left\lbrack {{D}_{s}F \mid {\mathcal{F}}_... | Yes |
Proposition 3.7.1. For any \( f \in {\mathcal{C}}_{b}^{2}\left( {\mathbb{R}}^{n}\right) \), the process \( {\left( {P}_{t, T}f\left( {X}_{t}\right) \right) }_{t \in \left\lbrack {0, T}\right\rbrack } \) is an \( {\mathcal{F}}_{t} \) -martingale. | Proof. By the tower property of conditional expectations, cf. Section 9.3, we have\n\n\[ \mathbb{E}\left\lbrack {{P}_{t, T}f\left( {X}_{t}\right) \mid {\mathcal{F}}_{s}}\right\rbrack = \mathbb{E}\left\lbrack {\mathbb{E}\left\lbrack {f\left( {X}_{T}\right) \mid {\mathcal{F}}_{t}}\right\rbrack \mid {\mathcal{F}}_{s}}\rig... | Yes |
Lemma 3.7.2. Let \( f \in {\mathcal{C}}_{b}^{2}\left( {\mathbb{R}}^{n}\right) \) . We have\n\n\[ \mathbb{E}\left\lbrack {{D}_{t}f\left( {X}_{T}\right) \mid {\mathcal{F}}_{t}}\right\rbrack = \left( {{L}_{t}\left( {{P}_{t, T}f}\right) }\right) \left( {X}_{t}\right) ,\;t \in \left\lbrack {0, T}\right\rbrack . \]\n\n(3.7.2... | Proof. We apply the change of variable formula (3.7.1) to \( t \mapsto {P}_{t, T}f\left( {X}_{t}\right) = \) \( \mathbb{E}\left\lbrack {f\left( {X}_{T}\right) \mid {\mathcal{F}}_{t}}\right\rbrack \), since \( {P}_{t, T}f \) is \( {\mathcal{C}}^{2} \) . Using the fact that the finite variation term\n\nvanishes since \( ... | Yes |
Proposition 3.7.3. We have for \( f \in {\mathcal{C}}_{b}\left( \mathbb{R}\right) \)\n\n\[ \n{P}_{t, T}f\left( x\right) = \frac{1}{\sqrt{2\pi }}\mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{\mathrm{e}}^{-{\Gamma }_{t}\left( T\right) }}{k!}{\int }_{-\infty }^{\infty }{\mathrm{e}}^{-{t}_{0}^{2}/2}{\int }_{{\left\lbrack... | Proof. We have \( {P}_{t, T}f\left( x\right) = \mathbb{E}\left\lbrack {f\left( {S}_{T}\right) \mid {S}_{t} = x}\right\rbrack = \mathbb{E}\left\lbrack {f\left( {S}_{t, T}^{x}\right) }\right\rbrack \), and\n\n\[ \n{P}_{t, T}f\left( x\right) = \exp \left( {-{\Gamma }_{t}\left( T\right) }\right) \mathop{\sum }\limits_{{k =... | Yes |
Proposition 4.1.3. The operators \( D \) and \( \delta \) satisfy the duality relation\n\n\[ \mathbb{E}\left\lbrack {{F\delta }\left( u\right) }\right\rbrack = \mathbb{E}\left\lbrack {\langle {DF}, u{\rangle }_{{L}^{2}\left( {\mathbb{R}}_{ + }\right) }}\right\rbrack ,\;F \in \mathcal{S}, u \in \mathcal{U}. \] | Proof. As in Proposition 1.8.2, we consider \( F = {I}_{n}\left( {f}_{n}\right) \) and \( {u}_{t} = \) \( {I}_{m}\left( {{g}_{m + 1}\left( {*, t}\right) }\right), t \in {\mathbb{R}}_{ + },{f}_{n} \in {L}^{2}{\left( {\mathbb{R}}_{ + }\right) }^{\circ n},{g}_{m + 1} \in {L}^{2}{\left( {\mathbb{R}}_{ + }\right) }^{\circ m... | Yes |
For any \( u \in \widetilde{\mathcal{U}} \) we have\n\n\[ \n{D}_{t}\delta \left( u\right) = {u}_{t} + \delta \left( {{D}_{t}u}\right) ,\;t \in {\mathbb{R}}_{ + }.\n\] | Proof. Letting \( {u}_{t} = f\left( t\right) {I}_{n}\left( {g}_{n}\right), t \in {\mathbb{R}}_{ + }, f \in {L}^{2}\left( {\mathbb{R}}_{ + }\right) ,{g}_{n} \in {L}^{2}{\left( {\mathbb{R}}_{ + }\right) }^{\circ n} \), from Proposition 4.1.3 we have, by (4.1.3),\n\n\[ \n{D}_{t}\delta \left( u\right) = {D}_{t}\delta \left... | Yes |
Proposition 4.2.1. The operators \( D \) and \( \delta \) are closable in the sense of Section 9.8 on \( {L}^{2}\left( \Omega \right) \) and \( {L}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) respectively. | It also follows from the density of \( \mathcal{S} \) in \( {L}^{2}\left( \Omega \right) \) that \( \mathcal{U} \) is dense in \( {L}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \). | No |
Proposition 4.2.2. The domain \( \operatorname{Dom}\left( D\right) = \mathbb{D}\left( {\lbrack 0,\infty }\right) ) \) of \( D \) consists in the space of square-integrable random variables with chaos expansion\n\n\[ F = \mathop{\sum }\limits_{{n = 0}}^{\infty }{I}_{n}\left( {f}_{n}\right) \]\n\n(4.2.1)\n\nsuch that the... | Given \( F \in \operatorname{Dom}\left( D\right) \) with the expansion (4.2.1) we have\n\n\[ \mathbb{E}\left\lbrack {\parallel {DF}{\parallel }_{{L}^{2}\left( {\mathbb{R}}_{ + }\right) }^{2}}\right\rbrack = \mathop{\sum }\limits_{{k = 1}}^{\infty }{kk}!{\begin{Vmatrix}{f}_{k}\end{Vmatrix}}_{{L}^{2}\left( {\mathbb{R}}_{... | Yes |
Proposition 4.2.3. Every \( F \in \mathcal{S} \) can be represented as\n\n\[ F = \mathbb{E}\left\lbrack F\right\rbrack + {\int }_{0}^{\infty }\mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack d{M}_{t} \] | Proof. By linearity, in order to prove the statement for \( F \in \mathcal{S} \), it suffices to consider \( F = {I}_{n}\left( {f}_{n}\right) \). By the definitions of \( {I}_{n}\left( {f}_{n}\right) \) and \( {D}_{t}{I}_{n}\left( {f}_{n}\right) \) and using Lemma 2.7.2 we have, since \( \mathbb{E}\left\lbrack {{I}_{n}... | Yes |
Proposition 4.2.5. For all \( F \in { \cap }_{n \geq 1}\operatorname{Dom}\left( {D}^{n}\right) \) we have\n\n\[ F = \mathbb{E}\left\lbrack F\right\rbrack + \mathop{\sum }\limits_{{n = 1}}^{\infty }{I}_{n}\left( {f}_{n}\right) \]\n\nwhere\n\[ {f}_{n}\left( {{t}_{1},\ldots ,{t}_{n}}\right) = \frac{1}{n!}\mathbb{E}\left\l... | Proof. It suffices to note that\n\n\[ {D}_{{t}_{1}}\cdots {D}_{{t}_{n}}F = n!{f}_{n}\left( {{t}_{1},\ldots ,{t}_{n}}\right) + \mathop{\sum }\limits_{{k = n + 1}}^{\infty }\frac{k!}{\left( {k - n}\right) !}{I}_{k - n}\left( {{f}_{k}\left( {*,{t}_{1},\ldots ,{t}_{n}}\right) }\right) ,\]\n\nand to use the fact that\n\n\[ ... | Yes |
Proposition 4.3.1. Let \( u \in \operatorname{Dom}\left( \delta \right) \) such that \( {u}_{t} \in \operatorname{Dom}\left( D\right) \), dt-a.e., and \( {\left( {D}_{s}{u}_{t}\right) }_{s, t \in {\mathbb{R}}_{ + }} \in {L}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }^{2}}\right) \) . We have\n\n\[ \mathbb{E}\left\lbrac... | Proof. By polarization, orthogonality and density it suffices to choose \( u = \) \( g{I}_{n}\left( {f}^{\otimes n}\right), f, g \in {L}^{2}\left( {\mathbb{R}}_{ + }\right) \), and to note that by the Definition 4.1.2 of \( \delta \) we have\n\n\[ \mathbb{E}\left\lbrack {\left| \delta \left( u\right) \right| }^{2}\righ... | Yes |
Proposition 4.3.4. Let \( {\left( {u}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \in {L}_{ad}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) be a square-integrable adapted process. We have\n\n\[ \delta \left( u\right) = {\int }_{0}^{\infty }{u}_{t}d{M}_{t} \] | Proof. This result can also be recovered from the definition (4.1.2) of \( \delta \) via multiple stochastic integrals. Since the adaptedness of \( {\left( {u}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} = \) \( {\left( {I}_{n - 1}\left( {f}_{n}\left( *, t\right) \right) \right) }_{t \in {\mathbb{R}}_{ + }} \) implies\n\n\... | Yes |
Lemma 4.3.5. Let \( u, v \in {L}_{ad}^{4}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) . For all \( t > 0 \) we have\n\n\[{\int }_{0}^{t}{u}_{s}{\int }_{s}^{t}{v}_{r}d{M}_{r}\delta {M}_{s} = {\int }_{0}^{t}{\int }_{0}^{r}{u}_{s}{v}_{r}\delta {M}_{s}d{M}_{r}\]\n\nwhere the indefinite Skorohod integral is defined i... | Proof. First, note that\n\n\[{\int }_{0}^{r}{u}_{s}{v}_{r}\delta {M}_{s} = \delta \left( {u \cdot {\mathbf{1}}_{\{ \cdot < r\} }{v}_{r}}\right)\]\n\nis \( {\mathcal{F}}_{r} \) -measurable, \( r \in {\mathbb{R}}_{ + } \), hence the stochastic integral in the right hand side of (4.3.5) exists in the Itô sense by Proposit... | Yes |
Proposition 4.4.1. Let \( F, G \in \operatorname{Dom}\left( D\right) \) . We have the covariance identity\n\n\[ \operatorname{Cov}\left( {F, G}\right) = \mathbb{E}\left\lbrack {{\int }_{0}^{\infty }{\int }_{0}^{\infty }{\mathrm{e}}^{-s}{D}_{u}F{P}_{s}{D}_{u}{Gduds}}\right\rbrack . \] | Proof. It suffices to prove this identity for \( F = {I}_{n}\left( {f}_{n}\right) \) and \( G = {I}_{n}\left( {g}_{n}\right) \) as\n\n\[ \operatorname{Cov}\left( {F, G}\right) = \mathbb{E}\left\lbrack {{I}_{n}\left( {f}_{n}\right) {I}_{n}\left( {g}_{n}\right) }\right\rbrack \]\n\n\[ = n!{\left\langle {f}_{n},{g}_{n}\ri... | Yes |
Proposition 4.5.2. Assume that \( \phi \in {L}^{\infty }\left( {\mathbb{R}}_{ + }\right) \) is a bounded, deterministic function. We have\n\n\[ \n{D}_{t}\left( {FG}\right) = F{D}_{t}G + G{D}_{t}F + {\phi }_{t}{D}_{t}F{D}_{t}G, \]\n\n(4.5.4)\n\n\[ \nt \in {\mathbb{R}}_{ + }, F, G \in \mathcal{S}\text{.} \]\n | Proof. We first notice that for \( F = {I}_{1}\left( u\right) \) and \( G = {I}_{n}\left( {f}_{n}\right) \), this formula is a consequence of the multiplication formula Proposition 4.5.1 since\n\n\[ \n{D}_{t}\left( {{I}_{1}\left( u\right) {I}_{n}\left( {f}_{n}\right) }\right) \]\n\n\[ \n= {D}_{t}\left( {{I}_{n + 1}\lef... | Yes |
Corollary 4.5.3. For all \( F, G \in \mathcal{S} \) we have\n\n\[ \n{D}_{{t}_{1}}\cdots {D}_{{t}_{r}}\left( {FG}\right) = \mathop{\sum }\limits_{{p = 0}}^{r}\mathop{\sum }\limits_{{q = r - p}}^{r}\n\] | \[ \n\mathop{\sum }\limits_{{\left\{ {{k}_{1} < \cdots < {k}_{p}}\right\} \cup \left\{ {{l}_{1} < \cdots < {l}_{q}}\right\} = \left\{ {1,\ldots, r}\right\} }}{D}_{{t}_{{k}_{1}}}\cdots {D}_{{t}_{{k}_{p}}}F{D}_{{t}_{{l}_{1}}}\cdots {D}_{{t}_{{l}_{q}}}G\mathop{\prod }\limits_{{i \in \left\{ {{k}_{1},\ldots ,{k}_{p}}\right... | Yes |
Proposition 4.5.4. Let \( T \in {\mathbb{R}}_{ + } \) and assume that \( \phi \in {L}^{\infty }\left( \left\lbrack {0, T}\right\rbrack \right) \) is a locally bounded deterministic function. Then for all \( u \in \mathcal{U} \) and \( F \in \mathcal{S} \) we have\n\n\[ \delta \left( u\right) F = \delta \left( {uF}\righ... | Proof. The proof of this statement follows by duality from Proposition 4.5.2. Letting \( u = {vG} \) we have for \( F,{G}_{1},{G}_{2} \in \mathcal{S} \) :\n\n\[ \mathbb{E}\left\lbrack {F{G}_{1}\delta \left( u\right) }\right\rbrack = \mathbb{E}\left\lbrack {{G}_{2}{\left\langle v, D\left( F{G}_{1}\right) \right\rangle }... | Yes |
Proposition 4.5.5. For \( F, G \in \mathcal{S} \) we have\n\n\[ \mathbb{E}\left\lbrack {{D}_{t}\left( {FG}\right) \mid {\mathcal{F}}_{t}}\right\rbrack = \mathbb{E}\left\lbrack {F{D}_{t}G \mid {\mathcal{F}}_{t}}\right\rbrack + \mathbb{E}\left\lbrack {G{D}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack + {\phi }_{t}\mathbb{E}... | Proof. We write (4.5.6) for \( u \in \mathcal{U} \) adapted and apply the duality between \( D \) and \( \delta \) :\n\n\[ \mathbb{E}\left\lbrack {\langle u, D\left( {FG}\right) \rangle }\right\rbrack = \mathbb{E}\left\lbrack {\delta \left( u\right) {FG}}\right\rbrack \]\n\n\[ = \mathbb{E}\left\lbrack {G(\delta \left( ... | Yes |
Lemma 4.6.2. The transformation \( {T}_{t}^{\phi } \) is multiplicative, i.e.\n\n\[ \n{T}_{t}^{\phi }\left( {FG}\right) = \left( {{T}_{t}^{\phi }F}\right) \left( {{T}_{t}^{\phi }G}\right) ,\;F, G \in \mathcal{E}.\n\] | Proof. From Lemma 2.13.4 we have\n\n\[ \n{T}_{t}^{\phi }\left( {\xi \left( u\right) \xi \left( v\right) }\right) = \exp \left( {\langle u, v{\rangle }_{{L}^{2}\left( {\mathbb{R}}_{ + }\right) }}\right) {T}_{t}^{\phi }\xi \left( {u + v + {\phi uv}}\right)\n\]\n\n\[ \n= \exp \left( {\langle u, v{\rangle }_{{L}^{2}\left( ... | Yes |
Proposition 4.6.3. For all \( u \in {L}^{2}\left( {\mathbb{R}}_{ + }\right) ,{T}_{t}^{\phi }{\xi }_{T}\left( u\right) \) coincides \( {dt} \times d\mathbb{P} \) -a.e. with the limit as \( T \) goes to infinity of the solution \( {Z}_{T}^{t} \) to the equation\n\n\[ \n{Z}_{s}^{t} = 1 + {\int }_{0}^{s}{Z}_{{\tau }^{ - }}... | Proof. Clearly by Proposition 2.13.1 we have \( {Z}_{s}^{t} = {\xi }_{s}\left( u\right), s < t \) . Next, at time \( t \) we have\n\n\[ \n{Z}_{t}^{t} = \left( {1 + {\phi }_{t}{u}_{t}}\right) {Z}_{{t}^{ - }}^{t}\n\]\n\n\[ \n= \left( {1 + {\phi }_{t}{u}_{t}}\right) {\xi }_{{t}^{ - }}\left( u\right)\n\]\n\n\[ \n= \left( {... | Yes |
Proposition 4.6.4. We have\n\n\[ \n{D}_{t}F = {D}_{t}^{B}F + \frac{{j}_{t}}{{\phi }_{t}}\left( {{T}_{t}^{\phi }F - F}\right) ,\;t \in {\mathbb{R}}_{ + },\;F \in \mathcal{E}.\n\]\n\n(4.6.4) | Proof. When \( {\phi }_{t} = 0 \) we have \( {D}_{t}^{B}F = {i}_{t}{u}_{t}\xi \left( u\right) = {i}_{t}{D}_{t}F \), hence\n\n\[ \n{D}_{t}\xi \left( u\right) = {i}_{t}{D}_{t}\xi \left( u\right) + {j}_{t}{D}_{t}\xi \left( u\right) \n\]\n\n\[ \n= {i}_{t}{u}_{t}\xi \left( u\right) + {j}_{t}{u}_{t}\xi \left( u\right) \n\]\n... | Yes |
Lemma 4.7.1. The continuity Assumption 4.4.2 is satisfied if \( {\left( {\phi }_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) is a deterministic function. | Proof. Let \( {\left( {M}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) be defined as in (2.10.4) on the product space \( \Omega = {\Omega }_{1} \times \) \( {\Omega }_{2} \) of independent Brownian motion \( {\left( {B}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) and Poisson process \( {\left( {N}_{t}\right) }_{t \in {\ma... | Yes |
Proposition 4.7.3. Let \( K \geq 0 \) and \( F \in \operatorname{Dom}\left( D\right) \) be such that \( {\phi }_{t}{D}_{t}F \leq K \) , \( {dtd}\mathbb{P} \) -a.e. for some \( K \geq 0 \) and \( \parallel {DF}{\parallel }_{{L}^{\infty }\left( {\Omega ,{L}^{2}\left( {\mathbb{R}}_{ + }\right) }\right) } < \infty \) . The... | Proof. We first assume that \( F \in \operatorname{Dom}\left( D\right) \) is a bounded random variable. Let us assume that \( \mathbb{E}\left\lbrack F\right\rbrack = 0 \) . From Proposition 4.5.2 we have as in the proof of Proposition 1.11.1:\n\n\[ 0 \leq \frac{{\mathrm{e}}^{-{sF}}{D}_{u}{\mathrm{e}}^{sF}}{{D}_{u}F} \]... | Yes |
Corollary 4.7.4. Assume that \( {\phi }_{t} = \phi \in {\mathbb{R}}_{ + }, t \in {\mathbb{R}}_{ + } \), is constant. Let \( F \in \) \( \operatorname{Dom}\left( D\right) \) be such that \( {DF} \leq K \) for some \( K \geq 0 \) and \( \parallel {DF}{\parallel }_{{L}^{\infty }\left( {\Omega ,{L}^{2}\left( {\mathbb{R}}_{... | \[ \mathbb{P}\left( {F - \mathbb{E}\left\lbrack F\right\rbrack \geq x}\right) \leq \exp \left( {-\frac{\parallel {DF}{\parallel }_{{L}^{\infty }\left( {\Omega ,{L}^{2}\left( {\mathbb{R}}_{ + }\right) }\right) }^{2}}{{\phi }^{2}{K}^{2}}g\left( \frac{x\phi K}{\parallel {DF}{\parallel }_{{L}^{\infty }\left( {\Omega ,{L}^{... | Yes |
Proposition 4.8.3. The operators\n\n\[ \n{\\nabla }^{ \\ominus } : \\mathcal{S} \\rightarrow {L}^{2}\\left( {\\Omega \\times {\\mathbb{R}}_{ + }}\\right) \n\]\n\nand\n\n\[ \n{\\nabla }^{ \\oplus } : \\mathcal{U} \\rightarrow {L}^{2}\\left( \\Omega \\right) \n\]\n\nsatisfy the duality relation\n\n\[ \n\\mathbb{E}\\left\... | Proof. By polarization, we need to prove the following. Letting \( F = {I}_{n}\\left( {f}^{\\otimes n}\\right) \) , \( u = h{I}_{n}\\left( {g}^{\\otimes n}\\right) \) and \( f, g, h \\in {\\mathcal{C}}_{c}^{1}\\left( {\\mathbb{R}}_{ + }\\right) \), we have\n\n\( \\mathbb{E}\\left\\lbrack {\\left\\langle {\\nabla }^{ \\... | Yes |
Proposition 5.1.2. The following statements hold on the Hermite polynomials:\n\ni) Generating function:\n\n\[ \n{\psi }_{\lambda }\left( {x,\sigma }\right) = {\mathrm{e}}^{{\lambda x} - \frac{1}{2}{\lambda }^{2}{\sigma }^{2}},\;x,\lambda \in \mathbb{R}. \n\]\n\nii) Derivation rule:\n\n\[ \n\frac{\partial {H}_{n}}{\part... | Proof. The recurrence relation (5.1.1) shows that the generating function \( {\psi }_{\lambda } \) satisfies the differential equation\n\n\[ \n\left\{ \begin{array}{l} \frac{\partial {\psi }_{\lambda }}{\partial \lambda }\left( {x,\sigma }\right) = \left( {x - \lambda {\sigma }^{2}}\right) {\psi }_{\lambda }\left( {x,\... | Yes |
Proposition 5.1.3. For any orthogonal family \( \\left\\{ {{u}_{1},\\ldots ,{u}_{d}}\\right\\} \) in \( {L}^{2}\\left( {\\mathbb{R}}_{ + }\\right) \) we have \[ {I}_{n}\\left( {{u}_{1}^{\\otimes {n}_{1}} \\circ \\cdots \\circ {u}_{d}^{\\otimes {n}_{d}}}\\right) = \\mathop{\\prod }\\limits_{{k = 1}}^{d}{H}_{{n}_{k}}\\le... | Proof. We have \[ {H}_{0}\\left( {{I}_{1}\\left( u\\right) ;\\parallel u{\\parallel }_{2}^{2}}\\right) = {I}_{0}\\left( {u}^{\\otimes 0}\\right) = 1\\;\\text{ and }\\;{H}_{1}\\left( {{I}_{1}\\left( u\\right) ;\\parallel u{\\parallel }_{2}^{2}}\\right) = {I}_{1}\\left( u\\right) , \] hence the proof follows by induction... | Yes |
Proposition 5.1.4. We have\n\n\[ \xi \left( u\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{1}{n!}{I}_{n}\left( {u}^{\otimes n}\right) = \exp \left( {{I}_{1}\left( u\right) - \frac{1}{2}\parallel u{\parallel }_{{L}^{2}\left( {\mathbb{R}}_{ + }\right) }^{2}}\right) . \] | Proof. Relation (5.1.7) follows from Proposition 5.1.2-i) and Proposition 5.1.3 which reads \( {I}_{n}\left( {u}^{\otimes n}\right) = {H}_{n}\left( {{I}_{1}\left( u\right) ;\parallel u{\parallel }_{{L}^{2}\left( {\mathbb{R}}_{ + }\right) }^{2}}\right), n \geq 1 \) . | No |
Proposition 5.1.5. The Brownian motion \( {\left( {B}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) has the chaos representation property. | Proof. Theorem 4.1, p. 134 of [50], shows by a Fourier transform argument that the linear space spanned by the exponential vectors\n\n\[ \left\{ {\exp \left( {{I}_{1}\left( u\right) - \frac{1}{2}\parallel u{\parallel }_{{L}^{2}\left( {\mathbb{R}}_{ + }\right) }^{2}}\right) \; : \;u \in {L}^{2}\left( {\mathbb{R}}_{ + }\... | Yes |
Lemma 5.1.6. Assume that \( F \) has the form \( F = g\left( {{I}_{1}\left( {e}_{1}\right) ,\ldots ,{I}_{1}\left( {e}_{k}\right) }\right) \) for some \( g \in {L}^{2}\left( {{\mathbb{R}}^{k},{\left( 2\pi \right) }^{-k/2}{\mathrm{e}}^{-{\left| x\right| }^{2}/2}{dx}}\right) \), and admits the chaos expansion \( F = \math... | Proof. The polynomial \( {P}_{n} \) is given by (5.1.8) above, which is a finite sum. | No |
Lemma 5.3.1. Let \( F \) of the form\n\n\[ F = f\left( {{I}_{1}\left( {u}_{1}\right) ,\ldots ,{I}_{1}\left( {u}_{n}\right) }\right) \]\n\nwhere \( f \in {\mathcal{C}}_{b}\left( {\mathbb{R}}^{n}\right) \) and \( {u}_{1},\ldots ,{u}_{n} \in {L}^{2}\left( {\mathbb{R}}_{ + }\right) \) are mutually orthogonal. For all \( t ... | Proof. Since, by Proposition 5.1.5, the exponential vectors are total in \( {L}^{2}\left( \Omega \right) \) and \( {P}_{t} \) is continuous on \( {L}^{2}\left( \Omega \right) \), it suffices to consider\n\n\[ {f}_{u}\left( x\right) = \exp \left( {x - \frac{1}{2}\parallel u{\parallel }_{2}^{2}}\right) \]\n\nand to note ... | Yes |
Lemma 5.3.2. We have for \( u \in {L}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) : \[ {\begin{Vmatrix}{P}_{t}u\end{Vmatrix}}_{{L}^{\infty }\left( {\Omega ,{L}^{2}\left( {\mathbb{R}}_{ + }\right) }\right) } \leq \parallel u{\parallel }_{{L}^{\infty }\left( {\Omega ,{L}^{2}\left( {\mathbb{R}}_{ + }\right) }\r... | Proof. Due to Lemma 5.3.1 we have \[ {\begin{Vmatrix}{P}_{s}u\left( \omega \right) \end{Vmatrix}}_{{L}^{2}\left( {\mathbb{R}}_{ + }\right) }^{2} = {\int }_{0}^{\infty }{\left| {P}_{s}{u}_{t}\left( \omega \right) \right| }^{2}{dt} \] \[ \leq {\int }_{0}^{\infty }{P}_{s}{\left| {u}_{t}\left( \omega \right) \right| }^{2}{... | Yes |
Theorem 5.4.3. For any non-decreasing functionals \( F, G \in {L}^{2}\left( \Omega \right) \) we have\n\n\[ \operatorname{Cov}\left( {F, G}\right) \geq 0 \] | The proof of this result is a direct consequence of Lemma 3.4.2 and the next lemma. | No |
Proposition 5.4.5. For any non-decreasing functional \( F \in {L}^{2}\left( \Omega \right) \) we have\n\n\[ \mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack \geq 0,\;{dt} \times d\mathbb{P} - \text{ a.e. } \] | Proof. Assume that \( F \in {L}^{2}\left( \Omega \right) \) is non-decreasing. Then \( {P}_{1/n}F, n \geq 1 \), is non-decreasing from (5.3.2), and belongs to \( \operatorname{Dom}\left( D\right) \) from Relation (5.3.1). From Lemma 5.4.4 we have\n\n\[ {D}_{t}{P}_{1/n}F \geq 0,\;{dt} \times d\mathbb{P} - \text{ a.e. },... | Yes |
Corollary 5.5.2. Let \( n \geq 1 \) and \( u \in {\mathbb{D}}_{n + 1,2}\left( H\right) \) such that \( \langle u, u{\rangle }_{H} \) is deterministic and\n\n\[ \operatorname{trace}{\left( Du\right) }^{k + 1} + \mathop{\sum }\limits_{{i = 2}}^{k}\frac{1}{i}{\left\langle {\left( Du\right) }^{k - i}u, D\operatorname{trace... | Proof. We have\n\n\[ {D}_{t}\langle u, u\rangle = {D}_{t}{\int }_{0}^{\infty }\left\langle {{u}_{s},{u}_{s}}\right\rangle {ds} \]\n\n\[ = {\int }_{0}^{\infty }\left\langle {{u}_{s},{D}_{t}{u}_{s}}\right\rangle {ds} + {\int }_{0}^{\infty }\left\langle {{D}_{t}{u}_{s},{u}_{s}}\right\rangle {ds} \]\n\n\[ = 2{\int }_{0}^{\... | Yes |
Lemma 5.5.3. Let \( n \geq 1 \) and \( u \in {\mathbb{D}}_{n + 1,2}\left( H\right) \) . Then for all \( 1 \leq k \leq n \) we have\n\n\[ \n\mathbb{E}\left\lbrack {{\left( \delta \left( u\right) \right) }^{n - k}\left\langle {{\left( Du\right) }^{k - 1}u,{D\delta }\left( u\right) }\right\rangle }\right\rbrack - \left( {... | Proof. We have \( {\left( Du\right) }^{k - 1}u \in {\mathbb{D}}_{\left( {n + 1}\right) /k,1}\left( H\right) ,\delta \left( u\right) \in {\mathbb{D}}_{\left( {n + 1}\right) /\left( {n - k + 1}\right) ,1}\left( \mathbb{R}\right) \) , and using Relation (5.5.2) we obtain\n\n\[ \n\mathbb{E}\left\lbrack {{\left( \delta \lef... | Yes |
Lemma 5.6.2. Let \( F, G \in \mathcal{S} \) be written as\n\n\[ F = f\left( {{I}_{1}\left( {u}_{1}\right) ,\ldots ,{I}_{1}\left( {u}_{n}\right) }\right) ,\;{u}_{1},\ldots ,{u}_{n} \in \mathcal{S}\left( {{\mathbb{R}}_{ + };{\mathbb{R}}^{d}}\right) ,\;f \in {\mathcal{C}}^{1}\left( {{\mathbb{R}}^{n};\mathbb{R}}\right) ,\]... | Proof. Let \( {e}_{1},\ldots ,{e}_{k} \in \mathcal{S}\left( {{\mathbb{R}}_{ + };{\mathbb{R}}^{d}}\right) \) be orthonormal vectors that generate \( {u}_{1},\ldots ,{u}_{n},{v}_{1},\ldots ,{v}_{m} \) . Assume that \( {u}_{i} \) and \( {v}_{i} \) are written as\n\n\[ {u}_{i} = \mathop{\sum }\limits_{{j = 1}}^{k}{\alpha }... | Yes |
Proposition 5.6.4. For \( F \in \mathcal{S} \), we have in \( {L}^{2}\left( \Omega \right) \) :\n\n\[ \frac{d}{d\varepsilon }\Lambda \left( {{U}_{\varepsilon },{\varepsilon h}}\right) {F}_{\mid \varepsilon = 0} = {\int }_{0}^{\infty }\left\langle {{h}_{0}\left( t\right) ,{D}_{t}F}\right\rangle {dt} + \delta \left( {\ma... | Proof. Let \( A : \mathcal{S} \rightarrow \mathcal{S} \) be defined by\n\n\[ {AF} = \delta \left( {\mathcal{L}{DF}}\right) + \operatorname{trace}\left( {{\operatorname{Id}}_{H} \otimes \mathcal{L}}\right) {DDF} + {\int }_{0}^{\infty }\left\langle {{h}_{0}\left( t\right) ,{D}_{t}F}\right\rangle {dt},\;F \in \mathcal{S}.... | Yes |
Corollary 5.6.5. Assume that \( \mathcal{L} : {L}^{2}\left( {{\mathbb{R}}_{ + };{\mathbb{R}}^{d}}\right) \rightarrow {L}^{2}\left( {\Omega \times {\mathbb{R}}_{ + };{\mathbb{R}}^{d}}\right) \) is antisymmetric as an endomorphism of \( {L}^{2}\left( {{\mathbb{R}}_{ + };{\mathbb{R}}^{d}}\right) ,\mathbb{P} \) -a.s., we h... | Proof. Since \( \mathcal{L} \) is antisymmetric, we have for any symmetric tensor \( u \otimes u \in \) \( \mathcal{S}\left( {{\mathbb{R}}_{ + };{\mathbb{R}}^{d}}\right) \otimes \mathcal{S}\left( {{\mathbb{R}}_{ + };{\mathbb{R}}^{d}}\right) \) :\n\n\[ \operatorname{trace}\left( {{\operatorname{Id}}_{H} \otimes \mathcal... | Yes |
Corollary 5.7.2. Assume that the Ricci curvature of \( M \) is uniformly bounded, and let \( z \in \mathcal{U}\left( {\mathbf{P}\left( M\right) \times {\mathbb{R}}_{ + };{\mathbb{R}}^{d}}\right) \) be adapted. We have\n\n\[ \n{\int }_{0}^{\infty }\left\langle {{\widehat{D}}_{t}F,\dot{z}\left( t\right) }\right\rangle {d... | Proof. We let \( {V}_{\varepsilon }\left( t\right) = \exp \left( {{\varepsilon q}\left( {t, z}\right) }\right), t \in {\mathbb{R}}_{ + },\varepsilon \in \mathbb{R} \) . Then from Proposition 3.5.3 of [44] we have\n\n\[ \n{\int }_{0}^{\infty }\langle \widehat{D}F,\dot{z}\left( t\right) \rangle {dt} = \frac{d}{d\varepsil... | Yes |
Proposition 5.7.4. We have for \( z \in \mathcal{U}\left( {\mathbf{P}\left( M\right) \times {\mathbb{R}}_{ + };{\mathbb{R}}^{d}}\right) \) :\n\n\[ \n{\int }_{0}^{\infty }\left\langle {{\widetilde{D}}_{t}F,\dot{z}\left( t\right) }\right\rangle {dt} = {\int }_{0}^{\infty }\left\langle {{\widehat{D}}_{t}F,\dot{\widetilde{... | Proof. We compute\n\n\[ \n{\int }_{0}^{\infty }\left\langle {{\widetilde{D}}_{t}F,\dot{z}\left( t\right) }\right\rangle {dt} = \mathop{\sum }\limits_{{i = 1}}^{{i = n}}{\int }_{0}^{{t}_{i}}\left\langle {{Q}_{{t}_{i}, s}^{ * }{t}_{0 \leftarrow {t}_{i}}{\nabla }_{i}^{M}f\left( {\gamma \left( {t}_{1}\right) ,\ldots ,\gamm... | Yes |
Lemma 5.7.7. Let \( F \in \operatorname{Dom}\left( \widetilde{D}\right) \) . If \( \parallel \widetilde{D}F{\parallel }_{{L}^{2}\left( {{\mathbb{R}}_{ + },{L}^{\infty }\left( {\mathrm{P}\left( M\right) }\right) }\right) } \leq C \), for some \( C > 0 \), then | \[ \nu \left( {F - \mathbb{E}\left\lbrack F\right\rbrack \geq x}\right) \leq \exp \left( {-\frac{{x}^{2}}{{2C}\parallel \widetilde{D}F{\parallel }_{\mathbf{H}}}}\right) ,\;x \geq 0. \] (5.7.7) In particular, \( \mathbb{E}\left\lbrack {e}^{\lambda {F}^{2}}\right\rbrack < \infty \), for \( \lambda < {\left( 2C\parallel \... | Yes |
Proposition 5.8.3. We have for \( F \in \mathcal{S} \)\n\n\[{\int }_{0}^{\infty }h\left( t\right) \left( {{\nabla }_{t}^{ \ominus } + \frac{1}{2}{D}_{t}{D}_{t}}\right) {Fdt} = - \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}\frac{1}{\varepsilon }\left( {F \circ {\mathcal{T}}_{\varepsilon h} - F}\right) . | Proof. We first notice that as a consequence of Proposition 5.8.1, the operator\n\n\[{\nabla }_{t}^{ \ominus } + \frac{1}{2}{D}_{t}{D}_{t}\]\n\n\( t \in {\mathbb{R}}_{ + } \), has the derivation property. Indeed, by Proposition 5.8.1 we have\n\n\[{\nabla }_{t}^{ \ominus }\left( {FG}\right) + \frac{1}{2}{D}_{t}{D}_{t}\l... | Yes |
Proposition 6.1.3. Let \( {A}_{1},\ldots ,{A}_{n} \) be compact disjoint subsets of \( X \) . Under the measure \( {\pi }_{\sigma }^{X} \) on \( \left( {{\Omega }^{X},{\mathcal{F}}^{X}}\right) \), the \( {\mathbb{N}}^{n} \) -valued vector\n\n\[ \omega \mapsto \left( {\omega \left( {A}_{1}\right) ,\ldots ,\omega \left( ... | Proof. Consider a disjoint partition \( {A}_{1} \cup \cdots \cup {A}_{n} \) of \( X \) and\n\n\[ F\left( \omega \right) = {\mathbf{1}}_{\left\{ \omega \left( {A}_{1}\right) = {k}_{1}\right\} }\cdots {\mathbf{1}}_{\left\{ \omega \left( {A}_{n}\right) = {k}_{n}\right\} } \]\n\n\[ = {\mathbf{1}}_{\left\{ \omega \left( X\r... | Yes |
Proposition 6.1.4. Let \( f \in {L}^{1}\left( {X,\sigma }\right) \) . We have\n\n\[ \n{\mathbb{E}}_{{\pi }_{\sigma }}\left\lbrack {\exp \left( {i{\int }_{X}f\left( x\right) \omega \left( {dx}\right) }\right) }\right\rbrack = \exp \left( {{\int }_{X}\left( {{\mathrm{e}}^{{if}\left( x\right) } - 1}\right) \sigma \left( {... | Proof. We first assume that \( X \) is compact. We have\n\n\[ \n{\mathbb{E}}_{{\pi }_{\sigma }}\left\lbrack {\exp \left( {i{\int }_{X}f\left( x\right) \omega \left( {dx}\right) }\right) }\right\rbrack \n\]\n\n\[ \n= {\mathrm{e}}^{-\sigma \left( X\right) }\mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{1}{n!}{\int }_{X}\... | Yes |
Proposition 6.1.5. Let \( \alpha \in \left( {0,1}\right) \) . We have \( {\pi }_{\sigma ,\alpha }^{X} = {\pi }_{\alpha \sigma }^{X} \), i.e. \( {\pi }_{\sigma ,\alpha }^{X} \) is the Poisson measure with intensity \( {\alpha \sigma }\left( {dx}\right) \) on \( {\Omega }^{X} \). | Proof. It suffices to treat the case where \( X \) is compact. We have\n\n\[ \n{\mathbb{E}}_{{\pi }_{\sigma ,\alpha }}\left\lbrack {\exp \left( {i{\int }_{X}f\left( x\right) \omega \left( {dx}\right) }\right) }\right\rbrack \n\]\n\n\[ \n= {\mathrm{e}}^{-\sigma \left( X\right) }\mathop{\sum }\limits_{{n = 0}}^{\infty }\... | Yes |
Proposition 6.1.7. There exists a measurable map\n\n\[ \n\tau : X \rightarrow {\mathbb{R}}_{ + }\n\]\n\na.e. bijective, such that \( \lambda = {\tau }_{ * }\sigma \), i.e. the Lebesgue measure is the image of \( \sigma \) by \( \tau \) . | We denote by \( {\tau }_{ * }\omega \) the image measure of \( \omega \) by \( \tau \), i.e. \( {\tau }_{ * } : {\Omega }^{X} \rightarrow \Omega \) maps\n\n\[ \n\omega = \mathop{\sum }\limits_{{i = 1}}^{\infty }{\epsilon }_{{x}_{i}}\;\text{ to }\;{\tau }_{ * }\omega = \mathop{\sum }\limits_{{i = 1}}^{\infty }{\epsilon ... | Yes |
Proposition 6.1.8. The application \( {\tau }_{ * } : {\Omega }^{X} \rightarrow \Omega \) maps the Poisson measure \( {\pi }_{\sigma } \) on \( {\Omega }^{X} \) to the Poisson measure \( {\pi }_{\lambda } \) on \( \Omega \) . | Proof. It suffices to check that for all families \( {A}_{1},\ldots ,{A}_{n} \) of disjoint Borel subsets \( X \) and \( {k}_{1},\ldots ,{k}_{n} \in \mathbb{N} \), we have\n\n\[ \n{\pi }_{\sigma }\left( \left\{ {\omega \in {\Omega }^{X} : {\tau }_{ * }\omega \left( {A}_{1}\right) = {k}_{1},\ldots ,{\tau }_{ * }\omega \... | Yes |
Corollary 6.1.9. Let \( F \in \operatorname{Dom}\left( D\right) \) be such that \( {DF} \leq K \), a.s., for some \( K \geq 0 \), and \( \parallel {DF}{\parallel }_{{L}^{\infty }\left( {\Omega ,{L}^{2}\left( X\right) }\right) } < \infty \) . Then | \[ \mathbb{P}\left( {F - \mathbb{E}\left\lbrack F\right\rbrack \geq x}\right) \leq \exp \left( {-\frac{\parallel {DF}{\parallel }_{{L}^{\infty }\left( {\Omega ,{L}^{2}\left( X\right) }\right) }^{2}}{{K}^{2}}g\left( \frac{xK}{\parallel {DF}{\parallel }_{{L}^{\infty }\left( {\Omega ,{L}^{2}\left( X\right) }\right) }^{2}}... | Yes |
Proposition 6.2.1. The Poisson process \( {\left( {N}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) of Definition 2.3.1 can be constructed as\n\n\[ \n{N}_{t}\left( \omega \right) = \omega \left( \left\lbrack {0, t}\right\rbrack \right) ,\;t \in {\mathbb{R}}_{ + }.\n\] | Proof. Clearly, the paths of \( {\left( {N}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) are piecewise continuous, càdlàg (i.e. continuous on the right with left limits), with jumps of height equal to one. Moreover, by definition of the Poisson measure on \( \Omega \), the vector \( \left( {{N}_{{t}_{1}} - {N}_{{t}_{0}},... | Yes |
Proposition 6.2.2. Let \( {f}_{n} : {\mathbb{R}}_{ + }^{n} \mapsto \mathbb{R} \) be continuous with compact support in \( {\mathbb{R}}_{ + }^{n} \) . Then we have the \( \mathbb{P}\left( {d\omega }\right) \) -almost sure equality\n\n\[ \n{I}_{n}\left( {f}_{n}\right) \left( \omega \right) = n!{\int }_{0}^{\infty }{\int ... | The above formula can also be written as\n\n\[ \n{I}_{n}\left( {f}_{n}\right) = n!{\int }_{0}^{\infty }{\int }_{0}^{{t}_{n}^{ - }}\cdots {\int }_{0}^{{t}_{2}^{ - }}{f}_{n}\left( {{t}_{1},\ldots ,{t}_{n}}\right) d\left( {{N}_{{t}_{1}} - {t}_{1}}\right) \cdots d\left( {{N}_{{t}_{n}} - {t}_{n}}\right) ,\n\]\n\nand by symm... | Yes |
Proposition 6.2.4. For all symmetric functions \( {f}_{n} \in {L}^{2}{\left( X,\sigma \right) }^{\circ n},{g}_{m} \in \) \( {L}^{2}{\left( X,\sigma \right) }^{\circ m} \), we have\n\n\[ \n{\mathbb{E}}_{{\pi }_{\sigma }}\left\lbrack {{I}_{n}^{X}\left( {f}_{n}\right) {I}_{m}^{X}\left( {g}_{m}\right) }\right\rbrack = n!{\... | Proof. Denoting \( {f}_{n}\left( {{\tau }^{-1}\left( {x}_{1}\right) ,\ldots ,{\tau }^{-1}\left( {x}_{n}\right) }\right) \) by \( {f}_{n} \circ {\tau }^{-1}\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) we have\n\n\[ \n{\mathbb{E}}_{{\pi }_{\sigma }}\left\lbrack {{I}_{n}^{X}\left( {f}_{n}\right) {I}_{m}^{X}\left( {g}_{m}\ri... | Yes |
Proposition 6.2.5. We have for \( u, v \in {L}^{2}\left( {X,\sigma }\right) \) such that \( {uv} \in {L}^{2}\left( {X,\sigma }\right) \) :\n\n\[ \n{I}_{1}^{X}\left( u\right) {I}_{n}^{X}\left( {v}^{\otimes n}\right) \n\]\n\n(6.2.5)\n\n\[ \n= {I}_{n + 1}^{X}\left( {{v}^{\otimes n} \circ u}\right) + n{I}_{n}^{X}\left( {\l... | Proof. This result can be proved by direct computation from (6.2.1). Alternatively it can be proved first for \( X = {\mathbb{R}}_{ + } \) using stochastic calculus or directly from Proposition 4.5.1 with \( {\phi }_{t} = 1, t \in {\mathbb{R}}_{ + } \) :\n\n\[ \n{I}_{1}\left( {u \circ {\tau }^{-1}}\right) {I}_{n}\left(... | Yes |
Proposition 6.2.9. The multiple Poisson stochastic integral of the function\n\n\\[ \n{\\mathbf{1}}_{{A}_{1}}^{\\otimes {k}_{1}} \\circ \\cdots \\circ {\\mathbf{1}}_{{A}_{d}}^{\\otimes {k}_{d}} \n\\]\nsatisfies\n\n\\[ \n{I}_{n}\\left( {{\\mathbf{1}}_{{A}_{1}}^{\\otimes {k}_{1}} \\circ \\cdots \\circ {\\mathbf{1}}_{{A}_{... | Proof. We have\n\n\\[ \n{I}_{0}\\left( {\\mathbf{1}}_{A}^{\\otimes 0}\\right) = 1 = {C}_{0}\\left( {\\omega \\left( A\\right) ,\\sigma \\left( A\\right) }\\right) \n\\]\n\nand\n\n\\[ \n{I}_{1}\\left( {\\mathbf{1}}_{A}^{\\otimes 0}\\right) \\left( \\omega \\right) = \\omega \\left( A\\right) - \\sigma \\left( A\\right) ... | Yes |
Lemma 6.2.10. Let \( F \) of the form\n\n\[ F\left( \omega \right) = g\left( {\omega \left( {A}_{1}\right) ,\ldots ,\omega \left( {A}_{k}\right) }\right) \]\n\nwhere\n\n\[ \mathop{\sum }\limits_{{{l}_{1},\ldots ,{l}_{k} = 0}}^{\infty }{\left| g\left( {l}_{1},\ldots ,{l}_{k}\right) \right| }^{2}{p}_{{l}_{1}}\left( {\sig... | Proof. We decompose \( g \) satisfying (6.2.18) as an orthogonal series\n\n\[ g\left( {{i}_{1},\ldots ,{i}_{k}}\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{P}_{n}\left( {{i}_{1},\ldots ,{i}_{n},\sigma \left( {A}_{1}\right) ,\ldots ,\sigma \left( {A}_{n}\right) }\right) ,\]\n\nwhere\n\n\[ {P}_{n}\left( {{i}_{1},\... | Yes |
Proposition 6.3.1. For all \( u \in {L}^{2}\left( X\right) \) we have\n\n\[ \xi \left( u\right) = \exp \left( {{\int }_{X}u\left( x\right) \left( {\omega \left( {dx}\right) - \sigma \left( {dx}\right) }\right) }\right) \mathop{\prod }\limits_{{x \in \omega }}\left( {\left( {1 + u\left( x\right) }\right) {\mathrm{e}}^{-... | Proof. The case \( X = {\mathbb{R}}_{ + } \) is treated in Proposition 2.13.1, in particular when \( {\phi }_{t} = 1, t \in {\mathbb{R}}_{ + } \), and the extension to \( X \) a metric space is obtained using the isomorphism \( \tau : X \rightarrow {\mathbb{R}}_{ + } \) of Proposition 6.1.7.\n\nIn particular, from Prop... | Yes |
Proposition 6.3.2. Every square-integrable random variable \( F \in {L}^{2}\left( {\Omega }^{X}\right. \) , \( {\pi }_{\sigma } \) ) admits the Wiener-Poisson decomposition\n\n\[ F = \mathop{\sum }\limits_{{n = 0}}^{\infty }{I}_{n}\left( {f}_{n}\right) \]\n\nin series of multiple stochastic integrals. | Proof. A modification of the proof of Theorem 4.1 in [50], cf. also Theorem 1.3 of [66], shows that the linear space spanned by\n\n\[ \left\{ {{\mathrm{e}}^{-{\int }_{X}u\left( x\right) \sigma \left( {dx}\right) }\mathop{\prod }\limits_{{x \in \omega }}\left( {1 + u\left( x\right) }\right) : u \in {\mathcal{C}}_{c}\lef... | Yes |
Proposition 6.3.3. Let \( f \in {\mathcal{C}}_{b}^{1}\left( {\mathbb{R}}_{ + }\right) \) . We have\n\n\[ f\left( {T}_{n}\right) = - \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{1}{k!}{I}_{k}\left( {{\int }_{{t}_{1} \vee \cdots \vee {t}_{k}}^{\infty }{f}^{\prime }\left( s\right) {P}_{n}^{\left( k\right) }\left( s\righ... | Proof. We have\n\n\[ f\left( {T}_{n}\right) = - {\int }_{0}^{\infty }{f}^{\prime }\left( s\right) {\mathbf{1}}_{\left\lbrack {T}_{n},\infty \right) }\left( s\right) {ds} \]\n\n\[ = - {\int }_{0}^{\infty }{f}^{\prime }\left( s\right) \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{1}{k!}{P}_{n}^{\left( k\right) }\left( s... | Yes |
Proposition 6.3.4. Let \( {n}_{1},\ldots ,{n}_{d} \in \mathbb{N} \) with \( 1 \leq {n}_{1} < \cdots < {n}_{d} \), and let \( f \in {\mathcal{C}}_{c}^{d}\left( {\Delta }_{d}\right) \) . The chaos expansion of \( f\left( {{T}_{{n}_{1}},\ldots ,{T}_{{n}_{d}}}\right) \) is given as\n\n\[ f\left( {{T}_{{n}_{1}},\ldots ,{T}_... | Proof. Let \( 0 = {s}_{0} \leq {s}_{1} \leq \cdots \leq {s}_{d} \), and \( {n}_{1},\ldots ,{n}_{d} \in \mathbb{N} \) . We have from (6.3.3) and (6.2.16):\n\n\[ \mathop{\prod }\limits_{{i = 1}}^{d}{\mathbf{1}}_{\left\{ {N}_{{s}_{i}} - {N}_{{s}_{i - 1}} = {n}_{i}\right\} } = \mathop{\sum }\limits_{{n = 0}}^{\infty }\math... | Yes |
Proposition 6.4.3. The operators \( {D}^{X} \) and \( {\delta }^{X} \) satisfy the duality relation\n\n\[ \mathbb{E}\left\lbrack {\left\langle {D}^{X}F, u\right\rangle }_{{L}^{2}\left( {X,\sigma }\right) }\right\rbrack = \mathbb{E}\left\lbrack {F{\delta }^{X}\left( u\right) }\right\rbrack \]\n\n(6.4.7)\n\n\( F \in \ope... | Proof. The proof is identical to those of Propositions 1.8.2 and 4.1.3, and follows from the isometry formula (6.2.4). We consider \( F = {I}_{n}\left( {f}_{n}\right) \) and \( {u}_{x} = \)\n\n\[ {I}_{m}\left( {{g}_{m + 1}\left( {*, x}\right) }\right), x \in X,{f}_{n} \in {L}^{2}{\left( X\right) }^{\circ n},{g}_{m + 1}... | Yes |
Corollary 6.4.6. For all \( F \) is bounded and measurable \( A \in \mathcal{B}\left( X\right) ,0 < \) \( \sigma \left( A\right) < \infty \), we have\n\n\[ \mathbb{E}\left\lbrack {{\int }_{A}F\left( {\omega \cup \{ x\} }\right) \sigma \left( {dx}\right) }\right\rbrack = \mathbb{E}\left\lbrack {{F\omega }\left( A\right)... | Proof. From Proposition 6.4.3, Lemma 6.4.4, and Relation (6.4.4) we have\n\n\[ \mathbb{E}\left\lbrack {{\int }_{A}F\left( {\omega \cup \{ x\} }\right) \sigma \left( {dx}\right) }\right\rbrack = \mathbb{E}\left\lbrack {{\int }_{X}{\mathbf{1}}_{A}\left( x\right) {D}_{x}{F\sigma }\left( {dx}\right) }\right\rbrack + \sigma... | Yes |
Proposition 6.4.7. For any \( F \in \operatorname{Dom}\left( {D}^{X}\right) \) we have\n\n\[ \n{D}_{x}^{X}F\left( \omega \right) = F\left( {\omega \cup \{ x\} }\right) - F\left( \omega \right) \]\n\n\( {\pi }_{\sigma }\left( {d\omega }\right) \times \sigma \left( {dx}\right) \) -a.e. | Proof. There exists a sequence \( {\left( {F}_{n}\right) }_{n \in \mathbb{N}} \) of functionals of the form (6.4.8), such that \( {\left( {D}^{X}{F}_{n}\right) }_{n \in \mathbb{N}} \) converges everywhere to \( {D}^{X}F \) on a set \( {A}_{F} \) such that \( \left( {{\pi }_{\sigma } \otimes \sigma }\right) \left( {A}_{... | Yes |
Proposition 6.4.8. We have for \( F, G \in \mathcal{S} \) :\n\n\[ \n{D}_{x}^{X}\left( {FG}\right) = F{D}_{x}^{X}G + G{D}_{x}^{X}F + {D}_{x}^{X}F{D}_{x}^{X}G, \]\n\n(6.4.14)\n\n\( \mathbb{P}\left( {d\omega }\right) {d\sigma }\left( x\right) \) -a.e. | Proof. This formula can be proved either from Propositions 4.5.2 and 6.1.8 with \( {\phi }_{t} = 1, t \in {\mathbb{R}}_{ + } \), when \( X = {\mathbb{R}}_{ + } \), or directly from (6.4.9):\n\n\[ \n{D}_{x}^{X}\left( {FG}\right) \left( \omega \right) = F\left( {\omega \cup \{ x\} }\right) G\left( {\omega \cup \{ x\} }\r... | Yes |
Lemma 6.4.10. Let \( X = {\mathbb{R}}_{ + } \) and \( \sigma \left( {dx}\right) = {dx} \) . For any \( F \) of the form \( F = f\left( {{T}_{1},\ldots ,{T}_{n}}\right) \) we have\n\n\[ \mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack \]\n\n\[ = {\mathbf{1}}_{\left\{ {N}_{t} < n\right\} }{\int }_{t}... | Proof. By application of Proposition 2.3.6 we have\n\n\[ \mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack \]\n\n\[ = {\mathbf{1}}_{\left\{ {N}_{t} < n\right\} }\mathbb{E}\left\lbrack {f\left( {{T}_{1},\ldots ,{T}_{{N}_{t}}, t,{T}_{{N}_{t} + 1},\ldots ,{T}_{n - 1}}\right) - f\left( {{T}_{1},\ldots ,... | Yes |
Proposition 6.5.1. Let \( u : X \times {\Omega }^{X} \rightarrow \mathbb{R} \) and \( F : {\Omega }^{X} \rightarrow \mathbb{R} \) such that \( u\left( {\cdot ,\omega }\right) ,{D}_{ \cdot }^{X}F\left( \omega \right) \), and \( u\left( {\cdot ,\omega }\right) {D}_{ \cdot }^{X}F\left( \omega \right) \in {L}^{1}\left( {X,... | Proof. Relation (6.5.1) follows by duality from Proposition 6.4.8, or from Proposition 4.5.6 and Proposition 6.1.8. | No |
Proposition 6.5.2. For all \( u \in \operatorname{Dom}\left( {\delta }^{X}\right) \) we have\n\n\[{\delta }^{X}\left( u\right) = {\int }_{X}{u}_{x}\left( {\omega \smallsetminus \{ x\} }\right) \left( {\omega \left( {dx}\right) - \sigma \left( {dx}\right) }\right) . | Proof. The statement clearly holds by (6.4.3) when \( g \in {L}^{2}\left( {X,\sigma }\right) \) is deterministic. Next we show using (6.5.1) that the identity also holds for a process\nof the form \( u = g{I}_{n}\left( {f}_{n}\right), g \in {L}^{2}\left( {X,\sigma }\right) \), by induction on the order of the multiple ... | Yes |
Proposition 6.5.3. When \( X = {\mathbb{R}}_{ + } \), for any square-integrable adapted pro- \( \operatorname{cess}{\left( {u}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \in {L}_{ad}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) we have\n\n\[ \delta \left( u\right) = {\int }_{0}^{\infty }{u}_{t}d\left( {{N}_{t} -... | The following is the Skorohod isometry on the Poisson space, which follows here from Proposition 6.5.1, or from Propositions 4.3.1 and 6.1.8. | No |
Proposition 6.5.4. For \( u : {\Omega }^{X} \times X \rightarrow \mathbb{R} \) measurable and sufficiently integrable we have\n\n\[{\mathbb{E}}_{{\pi }_{\sigma }}\left\lbrack {\left| {\delta }^{X}\left( u\right) \right| }^{2}\right\rbrack = \mathbb{E}\left\lbrack {\parallel u{\parallel }_{{L}^{2}\left( {X,\sigma }\righ... | Proof. Applying Proposition 6.4.2, Proposition 6.5.1 and Relation (6.4.5) we have\n\n\[{\mathbb{E}}_{{\pi }_{\sigma }}\left\lbrack {\left| {\delta }^{X}\left( u\right) \right| }^{2}\right\rbrack\n\]\n\[= {\mathbb{E}}_{{\pi }_{\sigma }}\left\lbrack {{\delta }^{X}\left( {u{\delta }^{X}\left( u\right) }\right) +\langle u,... | Yes |
Proposition 6.5.5. We have, for \( u : {\Omega }^{X} \times X \rightarrow \mathbb{R} \) a sufficiently integrable process,\n\n\[ \mathbb{E}\left\lbrack {\left( {\delta }^{X}\left( u\right) \right) }^{n + 1}\right\rbrack = \mathop{\sum }\limits_{{k = 0}}^{{n - 1}}\left( \begin{array}{l} n \\ k \end{array}\right) \mathbb... | Proof. Using the relation\n\n\[ {D}_{t}^{X}{\left( {\delta }^{X}\left( u\right) \right) }^{n} = {\varepsilon }_{t}^{ + }{\left( {\delta }^{X}\left( u\right) \right) }^{n} - {\left( {\delta }^{X}\left( u\right) \right) }^{n} \]\n\n\[ = {\left( {\varepsilon }_{t}^{ + }{\delta }^{X}\left( u\right) \right) }^{n} - {\left( ... | Yes |
Proposition 6.6.2. Let \( \pi \) be a probability measure on \( {\Omega }^{X} \) such that \( {\delta }^{X}\left( u\right) \) is integrable, \( u \in \mathcal{U} \). Assume that\n\n\[ \n{\mathbb{E}}_{\pi }\left\lbrack {{\delta }^{X}\left( u\right) }\right\rbrack = 0,\;u \in \mathcal{U},\n\]\n\nor equivalently\n\n\[ \n{... | Proof. Clearly, (6.6.3) implies (6.6.4) as in the proof of Proposition 6.6.1. The implication (6.6.4) \( \Rightarrow \) (6.6.3) follows in this case by taking \( F = 1 \). Denoting the characteristic function of \( \omega \mapsto {\int }_{X}h\left( x\right) \omega \left( {dx}\right) \) by\n\n\[ \n\psi \left( z\right) =... | Yes |
Corollary 6.6.3. Let \( \pi \) be a probability measure on \( {\Omega }^{X} \) such that \( {I}_{n}\left( {f}^{\otimes n}\right) \) is integrable under \( \pi, f \in {\mathcal{C}}_{c}^{\infty }\left( X\right) \) . The relation\n\n\[ \n{\mathbb{E}}_{\pi }\left\lbrack {{I}_{n}\left( {f}^{\otimes n}\right) }\right\rbrack ... | Proof. If (6.6.5) holds then by polarization and the Definition 6.4.2 we get\n\n\[ \n{\mathbb{E}}_{\pi }\left\lbrack {{\delta }^{X}\left( {g \otimes {I}_{n}\left( {{f}_{1} \otimes \cdots \otimes {f}_{n}}\right) }\right) }\right\rbrack = 0, \n\]\n\n\( g,{f}_{1},\ldots ,{f}_{n} \in {\mathcal{C}}_{c}^{\infty }\left( X\rig... | Yes |
Proposition 6.7.1. For \( F \in {L}^{2}\left( \Omega \right) \), we have\n\n\[ F = \mathbb{E}\left\lbrack F\right\rbrack + {\int }_{{\mathbb{R}}^{d} \times {\mathbb{R}}_{ + }}\mathbb{E}\left\lbrack {{D}_{t, x}^{X}F \mid {\mathcal{F}}_{t}}\right\rbrack X\left( {{dt},{dx}}\right) . \] | Proof. Let\n\n\[ {\widetilde{\Delta }}_{n} = \left\{ {\left( {\left( {{x}_{1},{t}_{1}}\right) ,\ldots ,\left( {{x}_{n},{t}_{n}}\right) }\right) \in {\left( {\mathbb{R}}^{d} \times {\mathbb{R}}_{ + }\right) }^{n} : {t}_{1} < \cdots < {t}_{n}}\right\} . \]\n\nFrom (6.7.3) we have for \( F \in \mathcal{S} \) :\n\n\[ F = \... | Yes |
Lemma 6.8.1. In case \( \sigma \) is finite on \( X \) we have\n\n\[ \n{P}_{t}F\left( \omega \right) = {\int }_{{\Omega }^{X} \times {\Omega }^{X}}F\left( {\widetilde{\omega } \cup \widehat{\omega }}\right) {q}_{t}\left( {\omega, d\widetilde{\omega }, d\widehat{\omega }}\right) ,\;\omega \in {\Omega }^{X},\n\]\n\n(6.8.... | Proof. We consider random functionals of the form\n\n\[ \nF = {\mathrm{e}}^{-{\int }_{X}u\left( x\right) \sigma \left( {dx}\right) }\mathop{\prod }\limits_{{x \in \omega }}\left( {1 + u\left( x\right) }\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{1}{n!}{I}_{n}\left( {u}^{\otimes n}\right) ,\n\]\n\ncf. Propo... | Yes |
Proposition 6.8.2. We have the covariance identity\n\n\[ \n\operatorname{Cov}\left( {F, G}\right) = \mathbb{E}\left\lbrack {{\int }_{0}^{\infty }{\int }_{X}{\mathrm{e}}^{-s}{D}_{x}^{X}F{P}_{s}{D}_{x}^{X}{G\sigma }\left( {dx}\right) {ds}}\right\rbrack , \n\]\n\n(6.8.2)\n\n\( F, G \in \operatorname{Dom}\left( {D}^{X}\rig... | Proof. By the chaos representation property Proposition 6.3.2, orthogonality of multiple integrals of different orders, and continuity of \( {P}_{s}, s \in {\mathbb{R}}_{ + } \), on \( {L}^{2}\left( {{\Omega }^{X},\mathbb{P}}\right) \), it suffices to prove the identity for \( F = {I}_{n}\left( {f}_{n}\right) \) and \(... | Yes |
Corollary 6.8.3. We have\n\n\[ \operatorname{Cov}\left( {F, G}\right) = {\int }_{0}^{1}{\int }_{X}{\int }_{{\Omega }^{X} \times {\Omega }^{X}}\mathop{\sum }\limits_{{{\omega }^{\prime } \subset \omega }}{D}_{x}^{X}F\left( \omega \right) \left( {G\left( {{\omega }^{\prime } \cup \widehat{\omega }\cup \{ x\} }\right) - G... | Proof. From (6.8.1) and (6.8.2) we have\n\n\[ \operatorname{Cov}\left( {F, G}\right) = \mathbb{E}\left\lbrack {{\int }_{0}^{\infty }{\int }_{X}{\mathrm{e}}^{-s}\left( {{D}_{x}^{X}F}}\right) \left( {{P}_{s}{D}_{x}^{X}G}\right) \sigma \left( {dx}\right) {ds}}\right\rbrack \]\n\n\[ = {\int }_{0}^{\infty }{\int }_{X}{\int ... | Yes |
Proposition 6.9.1. Let \( F \in \operatorname{Dom}\left( {D}^{X}\right) \) be such that \( {\mathrm{e}}^{sF} \in \operatorname{Dom}\left( {D}^{X}\right) \) , \( 0 \leq s \leq {t}_{0} \), for some \( {t}_{0} > 0 \) . Then\n\n\[ \n{\pi }_{\sigma }\left( {F - \mathbb{E}\left\lbrack F\right\rbrack \geq x}\right) \leq \exp ... | Proof. We start by deriving the following inequality for \( F \) a centered random variable:\n\n\[ \n\mathbb{E}\left\lbrack {F{\mathrm{e}}^{sF}}\right\rbrack \leq h\left( s\right) \mathbb{E}\left\lbrack {\mathrm{e}}^{sF}\right\rbrack ,\;0 \leq s \leq {t}_{0}. \n\] \n\n(6.9.3) \n\nThis follows from (6.8.2). Indeed, usin... | Yes |
Proposition 6.9.2. Let \( F : {\Omega }^{X} \rightarrow \mathbb{R} \) and let \( K : X \rightarrow {\mathbb{R}}_{ + } \) be a function such that\n\n\[ \n{D}_{y}^{X}F\left( \omega \right) \leq K\left( y\right) ,\;y \in X,\;\omega \in {\Omega }^{X}.\n\]\n\nThen\n\n\[ \n{\pi }_{\sigma }\left( {F - \mathbb{E}\left\lbrack F... | Proof. Let \( {F}_{n} = \max \left( {-n,\min \left( {F, n}\right) }\right), n \geq 1 \) . Since when \( K \) is \( {\mathbb{R}}_{ + } \) -valued the condition \( {D}_{y}^{X}{F}_{n}\left( \omega \right) \leq K\left( y\right) ,\omega \in {\Omega }^{X}, y \in X \), is satisfied we may apply Proposition 6.9.1 to \( {F}_{n}... | Yes |
Corollary 6.9.3. Let \( F \in {L}^{2}\left( {{\Omega }^{X},{\pi }_{\sigma }}\right) \) be such that \( {D}^{X}F \leq K,{\pi }_{\sigma } \otimes \sigma \) -a.e., for some \( K \in \mathbb{R} \), and \( {\begin{Vmatrix}{D}^{X}F\end{Vmatrix}}_{{L}^{\infty }\left( {{\Omega }^{X},{L}^{2}\left( {X,\sigma }\right) }\right) } ... | Proof. If \( K \geq 0 \), let us first assume that \( F \) is a bounded random variable. The function \( h \) in (6.9.7) is such that\n\n\[ \nh\left( t\right) \leq \frac{{\mathrm{e}}^{tK} - 1}{K}{\begin{Vmatrix}{D}^{X}F\end{Vmatrix}}_{{L}^{\infty }\left( {{\Omega }^{X},{L}^{2}\left( {X,\sigma }\right) }\right) }^{2} \n... | Yes |
Corollary 6.9.4. Let\n\n\[ \nF = \left( {{F}_{1},\ldots ,{F}_{n}}\right) = {\left( {\int }_{\left\{ {\left| y\right| }_{2} \leq 1\right\} }{y}_{k}\left( \omega \left( dy\right) - \sigma \left( dy\right) \right) + {\int }_{\left\{ {\left| y\right| }_{2} > 1\right\} }{y}_{k}\omega \left( dy\right) \right) }_{1 \leq k \le... | Proof. The representation (6.9.11) shows that\n\n\[ \left| {{D}_{x}^{X}f\left( F\right) \left( \omega \right) }\right| = \left| {f\left( {F\left( {\omega \cup \{ x\} }\right) }\right) - f\left( {F\left( \omega \right) }\right) }\right| \]\n\n\[ \leq c\parallel F\left( {\omega \cup \{ x\} }\right) - F\left( \omega \righ... | Yes |
Proposition 7.1.3. We have for \( F \in \mathcal{I} \) and \( V \in {\mathcal{C}}_{c}^{1}\left( {X,{TX}}\right) \) :\n\n\[ \mathbb{E}\left\lbrack {\left\langle {\widehat{D}}^{L}F, V\right\rangle }_{{L}^{2}\left( {X,{d\omega };{TX}}\right) }\right\rbrack = \mathbb{E}\left\lbrack {F{\int }_{X}{L}^{ * }V\left( x\right) \o... | Proof. We have\n\n\[ \mathbb{E}\left\lbrack {\left\langle {\widehat{D}}^{L}F, V\right\rangle }_{{L}^{2}\left( {X,{d\omega };{TX}}\right) }\right\rbrack = \mathbb{E}\left\lbrack {\mathop{\sum }\limits_{{x \in \omega }}{\left\langle {\widehat{D}}_{x}^{L}F, V\left( x\right) \right\rangle }_{TX}}\right\rbrack \]\n\n\[ = \m... | Yes |
Lemma 7.1.4. For \( F \in \mathcal{I} \) we have\n\n\[ \n{\widehat{D}}_{x}F\left( \omega \right) = {\varepsilon }_{x}^{ - }{\nabla }^{X}{\varepsilon }_{x}^{ + }F\left( \omega \right) \;\text{ on }\;\left\{ {\left( {\omega, x}\right) \in {\Omega }^{x} \times X : x \in \omega }\right\} .\n\] | Proof. Let\n\n\[ \nF = f\left( {{\int }_{X}{\varphi }_{1}{d\omega },\ldots ,{\int }_{X}{\varphi }_{n}{d\omega }}\right) ,\;x \in X,\;\omega \in {\Omega }^{X}, \n\]\n\nand assume that \( x \in \omega \) . We have\n\n\[ \n{\widehat{D}}_{x}F\left( \omega \right) = \mathop{\sum }\limits_{{i = 1}}^{n}{\partial }_{i}f\left( ... | Yes |
Proposition 7.1.5. For \( V \in {\mathcal{C}}_{c}^{\infty }\left( {X;{TX}}\right) \) and \( F \in \mathcal{I} \) we have\n\n\[ \langle \widehat{D}F\left( \omega \right), V{\rangle }_{{L}^{2}\left( {X,{d\omega };{TX}}\right) } \]\n\n(7.1.4)\n\n\[ = {\left\langle {\nabla }^{X}DF\left( \omega \right), V\right\rangle }_{{L... | Proof. This identity follows from the relation\n\n\[ {\widehat{D}}_{x}F\left( \omega \right) = \left( {{\nabla }_{x}^{X}{D}_{x}F}\right) \left( {\omega \smallsetminus \{ x\} }\right) ,\;x \in \omega ,\]\n\nand the application to \( u = {\left\langle {\nabla }^{X}DF, V\right\rangle }_{TX} \) of the relation\n\n\[ {\delt... | No |
Corollary 7.1.7. The isometry relation\n\n\\[ \n{\\mathbb{E}}_{\\pi }\\left\\lbrack {\\langle \\widehat{D}F,\\widehat{D}G{\\rangle }_{{L}_{\\omega }^{2}\\left( {TX}\\right) }}\\right\\rbrack = {\\mathbb{E}}_{\\pi }\\left\\lbrack {\\left\\langle {\\nabla }^{X}DF,{\\nabla }^{X}DG\\right\\rangle }_{{L}_{\\sigma }^{2}\\lef... | Proof.\n\ni) Relations (6.4.5) and (7.1.6) show that (7.1.7) holds when \\( \\pi = {\\pi }_{\\sigma } \\) . ii) If (7.1.7) is satisfied, then taking \\( F = {I}_{n}\\left( {u}^{\\otimes n}\\right) \\) and \\( G = {I}_{1}\\left( h\\right), h, u \\in \\) \\( {\\mathcal{C}}_{c}^{\\infty }\\left( X\\right) \\), Relation (7... | Yes |
Corollary 7.1.10. Let \( F \in \mathcal{S} \) and \( h \in {\mathcal{C}}_{b}^{1}\left( {\mathbb{R}}_{ + }\right) \) with \( h\left( 0\right) = 0 \) . We have the integration by parts formula\n\n\[ \mathbb{E}\left\lbrack {\langle \widehat{D}F, h{\rangle }_{{L}^{2}\left( {{\mathbb{R}}_{ + },{d\omega }}\right) }}\right\rb... | Proof. From Lemma 7.1.9 it suffices to notice that if \( k > d \) ,\n\n\[ \mathbb{E}\left\lbrack {F{h}^{\prime }\left( {T}_{k}\right) }\right\rbrack = {\int }_{0}^{\infty }{\mathrm{e}}^{-{t}_{k}}{h}^{\prime }\left( {t}_{k}\right) {\int }_{0}^{{t}_{k}}\cdots {\int }_{0}^{{t}_{d}}\cdots {\int }_{0}^{{t}_{2}}f\left( {{t}_... | Yes |
Proposition 7.1.11. For \( F \in \mathcal{S} \) and \( h \in {\mathcal{C}}_{c}^{1}\left( {\mathbb{R}}_{ + }\right) \) we have :\n\n\[ \mathbb{E}\left\lbrack {\langle \widehat{D}F,{hG}{\rangle }_{{L}^{2}\left( {{\mathbb{R}}_{ + }, d{N}_{t}}\right) }}\right\rbrack = \mathbb{E}\left\lbrack {F\widehat{\delta }\left( {hG}\r... | Proof. We have\n\n\[ \mathbb{E}\left\lbrack {\langle \widehat{D}F,{hG}{\rangle }_{{L}^{2}\left( {{\mathbb{R}}_{ + }, d{N}_{t}}\right) }}\right\rbrack = \mathbb{E}\left\lbrack {\langle \widehat{D}\left( {FG}\right), h{\rangle }_{{L}^{2}\left( {{\mathbb{R}}_{ + }, d{N}_{t}}\right) } - F\langle \widehat{D}G, h{\rangle }_{... | Yes |
Lemma 7.2.2. For \( F \) of the form \( F = f\left( {{T}_{1},\ldots ,{T}_{n}}\right) \) we have\n\n\[ \mathbb{E}\left\lbrack {{\widetilde{D}}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack = - \mathop{\sum }\limits_{{{N}_{t} < k \leq n}}\mathbb{E}\left\lbrack {{\partial }_{k}f\left( {{T}_{1},\ldots ,{T}_{n}}\right) \mid {\m... | \[ = - \mathop{\sum }\limits_{{{N}_{t} < k \leq n}}{\int }_{t}^{\infty }{\mathrm{e}}^{-\left( {{s}_{n} - t}\right) }{\int }_{t}^{{s}_{n}}\cdots {\int }_{t}^{{s}_{{N}_{t} + 2}} \]\n\n\[ {\partial }_{k}f\left( {{T}_{1},\ldots ,{T}_{{N}_{t}},{s}_{{N}_{t} + 1},\ldots ,{s}_{n}}\right) d{s}_{{N}_{t} + 1}\cdots d{s}_{n}. \] | No |
Lemma 7.2.3. Let \( T > 0 \) . For any \( {\mathcal{F}}_{T} \) -measurable random variable \( F \in \) \( {L}^{2}\left( \Omega \right) \) we have \( F \in {\mathbb{D}}_{\lbrack T,\infty )} \) and \[ {\widetilde{D}}_{t}F = 0,\;t \geq T. \] | Proof. In case \( F = f\left( {{T}_{1},\ldots ,{T}_{n}}\right) \) with \( f \in {\mathcal{C}}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right), F \) does not depend on the future of the Poisson process after \( T \), it does not depend on the \( k \) -th jump time \( {T}_{k} \) if \( {T}_{k} > T \), i.e. \[ {\partial }_{i}f... | Yes |
Proposition 7.2.4. We have for \( F \in \mathcal{S} \) and \( u \in {\mathcal{C}}_{c}\left( {\mathbb{R}}_{ + }\right) \) :\n\n\[ \mathbb{E}\left\lbrack {\langle \widetilde{D}F, u{\rangle }_{{L}^{2}\left( {{\mathbb{R}}_{ + },{dt}}\right) }}\right\rbrack = \mathbb{E}\left\lbrack {F{\int }_{0}^{\infty }u\left( t\right) \l... | Proof. We have, using (7.2.1),\n\n\[ \mathbb{E}\left\lbrack {\langle \widetilde{D}F, u{\rangle }_{{L}^{2}\left( {{\mathbb{R}}_{ + },{dt}}\right) }}\right\rbrack = \mathbb{E}\left\lbrack {{\int }_{0}^{\infty }{\int }_{0}^{\infty }{r}^{\left( 1\right) }\left( {s, t}\right) {\widehat{D}}_{s}{Fu}\left( t\right) d{N}_{s}{dt... | Yes |
Proposition 7.2.6. The divergence operator\n\n\[ \n\widetilde{\delta } : {L}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \rightarrow {L}^{2}\left( \Omega \right) \n\]\n\nis the adjoint of the gradient operator\n\n\[ \n\widetilde{D} : {L}^{2}\left( \Omega \right) \rightarrow {L}^{2}\left( {\Omega \times {\mathbb... | Proof. It suffices to note that Proposition 7.2.4 implies\n\n\[ \n\mathbb{E}\left\lbrack {\langle \widetilde{D}F,{hG}{\rangle }_{{L}^{2}\left( {{\mathbb{R}}_{ + },{dt}}\right) }}\right\rbrack = \mathbb{E}\left\lbrack {\langle \widetilde{D}\left( {FG}\right), h{\rangle }_{{L}^{2}\left( {{\mathbb{R}}_{ + },{dt}}\right) }... | Yes |
Proposition 7.2.8. For any \( F \in {L}^{2}\left( \Omega \right) \) we have\n\n\[ F = \mathbb{E}\left\lbrack F\right\rbrack + {\int }_{0}^{\infty }\mathbb{E}\left\lbrack {{D}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack d\left( {{N}_{t} - t}\right) \] | \[ = \mathbb{E}\left\lbrack F\right\rbrack + {\int }_{0}^{\infty }\mathbb{E}\left\lbrack {{\widetilde{D}}_{t}F \mid {\mathcal{F}}_{t}}\right\rbrack d\left( {{N}_{t} - t}\right) \] | No |
Proposition 7.2.9. The adjoint of \( \widetilde{D} \) extends the compensated Poisson stochastic integral, i.e. for all adapted square-integrable process \( u \in {L}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) we have | Proof. We consider first the case where \( v \) is a cylindrical elementary predictable process \( v = F{\mathbf{1}}_{(s, T\rbrack }\left( \cdot \right) \) with \( F = f\left( {{T}_{1},\ldots ,{T}_{n}}\right), f \in {\mathcal{C}}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . Since \( v \) is predictable, \( F \) is ... | No |
Proposition 7.4.1. For any \( F \in \mathop{\bigcap }\limits_{{n = 0}}^{\infty }\operatorname{Dom}\left( {{D}^{n}\widetilde{D}}\right) \) we have the chaos expansion\n\n\[ F = \mathbb{E}\left\lbrack F\right\rbrack + \mathop{\sum }\limits_{{n \geq 1}}^{\infty }{\widetilde{I}}_{n}\left( {{1}_{{\Delta }_{n}}{f}_{n}}\right... | Proof. We apply Proposition 4.2 .5 to \( {\widetilde{D}}_{t}F, t \in {\mathbb{R}}_{ + } \) :\n\n\[ {\widetilde{D}}_{t}F = \mathbb{E}\left\lbrack {{\widetilde{D}}_{t}F}\right\rbrack + \mathop{\sum }\limits_{{n = 1}}^{\infty }{\widetilde{I}}_{n}\left( {{1}_{{\widetilde{\Delta }}_{n}}\mathbb{E}\left\lbrack {{D}^{n}{\widet... | Yes |
Lemma 7.4.2. We have for \( f \in {\mathcal{C}}_{c}^{1}\left( \mathbb{R}\right) \) and \( n \geq 1 \) \n\n\[ \n{D}_{t}{\widetilde{D}}_{s}f\left( {T}_{n}\right) = {\widetilde{D}}_{s \vee t}f\left( {T}_{n - 1}\right) - {\widetilde{D}}_{s \vee t}f\left( {T}_{n}\right) - {\mathbf{1}}_{\{ s < t\} }{\mathbf{1}}_{\left\lbrack... | Proof. From Relation (6.4.15) we have \n\n\[ \n{D}_{t}{\widetilde{D}}_{s}f\left( {T}_{n}\right) = - {\mathbf{1}}_{\left\lbrack 0,{T}_{n - 1}\right\rbrack }\left( t\right) \left( {{\mathbf{1}}_{\left\lbrack 0,{T}_{n - 1}\right\rbrack }\left( s\right) {f}^{\prime }\left( {T}_{n - 1}\right) - {\mathbf{1}}_{\left\lbrack 0,... | Yes |
Proposition 7.4.3. For \( k \geq 1 \), the chaos expansion of \( f\left( {T}_{k}\right) \) is given as\n\n\[ f\left( {T}_{k}\right) = \mathbb{E}\left\lbrack {f\left( {T}_{k}\right) }\right\rbrack + \mathop{\sum }\limits_{{n \geq 1}}\frac{1}{n!}{I}_{n}\left( {f}_{n}^{k}\right) ,\] \n\nwhere \( {f}_{n}^{k}\left( {{t}_{1}... | Proof. of Proposition 7.4.3. Let us first assume that \( f \in {\mathcal{C}}_{c}^{1}\left( {\mathbb{R}}_{ + }\right) \) . We have\n\n\[ {f}_{1}^{k}\left( t\right) = \mathbb{E}\left\lbrack {{\widetilde{D}}_{t}f\left( {T}_{k}\right) }\right\rbrack \] \n\n\[ = - \mathbb{E}\left\lbrack {{\mathbf{1}}_{\left\lbrack 0,{T}_{k}... | Yes |
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