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Theorem 3.1. Let \( p \) be a point process of the class (QL). Then for \( U \in {\Gamma }_{p},{\widetilde{N}}_{p}\left( {\cdot, U}\right) \in {\mathcal{M}}_{2} \) and we have\n\n(3.1)\n\n\[ \left\langle {{\widetilde{N}}_{p}\left( {\cdot ,{U}_{1}}\right) ,{\widetilde{N}}_{p}\left( {\cdot ,{U}_{2}}\right) }\right\rangle... | For the proof, we need the following lemma.\n\nLemma 3.1. If \( U \in | No |
Lemma 3.1. If \( U \in {\Gamma }_{p} \) and \( f\left( s\right) = f\left( {s,\omega }\right) \) is a bounded \( \left( {\mathcal{F}}_{t}\right) \) -predictable process, then\n\n\[ \nX\left( t\right) = {\int }_{0}^{t}f\left( s\right) d{\widetilde{N}}_{p}{\left( s, U\right) }^{ * }\left( { = \mathop{\sum }\limits_{\subst... | Proof. By Proposition I-5.1, it is sufficient to assume that \( s \mapsto f\left( s\right) \) is a bounded, left continuous adapted process. Then, for every \( s \in \lbrack 0,\infty ) \) ,\n\n\[ \n{f}_{n}\left( s\right) = f\left( 0\right) {I}_{\lfloor s = 0)}\left( s\right) + \mathop{\sum }\limits_{{k = 0}}^{\infty }f... | Yes |
Let \( X\left( t\right) \) be a \( d \) -dimensional time homogeneous Lévy process (i.e. a right continuous process with stationary independent increments) and \( {\left( {\mathcal{F}}_{t}\right) }_{t \geq 0} \) be generated by the sample paths \( X\left( t\right) \) . Let \( {\mathbf{D}}_{p} = \{ t > 0;X\left( t\right... | (4.3)\n\n\[ \n{X}^{t}\left( t\right) = {X}^{t}\left( 0\right) + \mathop{\sum }\limits_{{k = 1}}^{{d}^{\prime }}{a}_{k}^{t}{B}^{k}\left( t\right) + {b}^{t}t + {\int }_{0}^{t + }{\int }_{{\mathbb{R}}^{d}\smallsetminus \{ 0\} }{x}^{t}{I}_{\left( \left| x\right| \geq 1\right) }{N}_{p}\left( {dsdx}\right) \n\] \n\n\[ \n+ {\... | Yes |
Lemma 5.1. Let \( C > 0 \) be a constant such that \( \left| {M\left( s\right) }\right| \leq C, s \in \) \( \left\lbrack {0, t}\right\rbrack \) . Set \( {V}_{l}^{A} = \mathop{\sum }\limits_{{k = 1}}^{l}{\left\{ M\left( {t}_{k}\right) - M\left( {t}_{k - 1}\right) \right\} }^{2}, l = 1,2,\ldots, n \) . Then \( E\left\lbr... | Proof. It is easy to see that \[ {\left( {V}_{n}^{\Delta }\right) }^{2} = \mathop{\sum }\limits_{{k = 1}}^{n}{\left\{ M\left( {t}_{k}\right) - M\left( {t}_{k - 1}\right) \right\} }^{4} \] \[ + 2\mathop{\sum }\limits_{{k = 1}}^{n}\left( {{V}_{n}^{4} - {V}_{k}^{4}}\right) {\left( M\left( {t}_{k}\right) - M\left( {t}_{k -... | Yes |
Theorem 6.1. Let \( X\left( t\right) = \left( {{X}^{1}\left( t\right) ,{X}^{2}\left( t\right) ,\ldots ,{X}^{d}\left( t\right) }\right) \) be a \( d \) -dimensional \( \left( {\mathcal{F}}_{t}\right) \) -semi-martingale such that\n\n(6.1)\n\n\[ \n{M}^{i}\left( t\right) = {X}^{i}\left( t\right) - {X}^{i}\left( 0\right) \... | Proof. It is enough to prove that\n\n(6.3)\n\n\[ \nE\left\lbrack {{e}^{i\langle \xi, X\left( t\right) - X\left( s\right) \rangle } \mid {\mathcal{F}}_{s}}\right\rbrack = {e}^{-\frac{1}{2}{\left| \xi \right| }^{2}\left( {t - s}\right) } \]\n\nfor every \( \xi \in {\mathbf{R}}^{d} \) and \( t > s \geq 0 \) . Let \( F\lef... | Yes |
Example 6.1. Let \( X\\left( t\\right) = \\left( {{X}^{1}\\left( t\\right) ,{X}^{2}\\left( t\\right) ,\\ldots ,{X}^{d}\\left( t\\right) }\\right) \) be a \( d \) -dimensional \( \\left( {\\mathcal{F}}_{t}\\right) \) -Brownian motion and \( p = \\left( {{p}_{i}^{k}\\left( {t,\\omega }\\right) }\\right) \) be a process w... | Indeed, setting \( {\\widetilde{M}}^{k}\\left( t\\right) = {\\widetilde{X}}^{k}\\left( t\\right) - {\\widetilde{X}}^{k}\\left( 0\\right) \) , \[ \\left\\langle {{\\widetilde{M}}^{k},{\\widetilde{M}}^{l}}\\right\\rangle \\left( t\\right) = {\\int }_{0}^{t}\\mathop{\\sum }\\limits_{{m, n = 1}}^{d}{p}_{m}^{k}\\left( {s,\\... | Yes |
Theorem 6.3. Let \( X\left( t\right) = \left( {{X}^{1}\left( t\right) ,{X}^{2}\left( t\right) ,\ldots ,{X}^{d}\left( t\right) }\right) \) be a \( d \) -dimensional \( \left( {\mathcal{F}}_{t}\right) \) semi-martingale and \( {p}_{1},{p}_{2},\ldots ,{p}_{n} \) be point processes of class (QL) with respect to \( \left( {... | Proof. Let \( {D}_{p} = \mathop{\bigcup }\limits_{{i = 1}}^{n}{D}_{{p}_{i}} \) and set, \( p\left( t\right) = {p}_{t}\left( t\right) \) if \( t \in {D}_{{p}_{i}} \) . Then we have a point process \( p \) on the sum \( \mathop{\bigcup }\limits_{{i = 1}}^{n}{X}_{i} * \) which is clearly a point process of the class \( \l... | Yes |
Theorem 6.4. Let \( X\\left( t\\right) = \\left( {{X}^{1}\\left( t\\right) ,{X}^{2}\\left( t\\right) ,\\ldots ,{X}^{d}\\left( t\\right) }\\right) \) be a \( d \) -dimensional \( \\left( {\\mathcal{F}}_{t}\\right) \) -Brownian motion and \( \\sigma \) be an \( \\left( {\\mathcal{F}}_{t}\\right) \) -stopping time such th... | Proof. By Doob’s optional sampling theorem, \( {M}^{*t}\\left( t\\right) = {X}^{t}\\left( {t + \\sigma }\\right) - \) \( {X}^{t}\\left( \\sigma \\right) \) is a local martingale with respect to \( \\left( {{\\mathcal{F}}_{t}^{ * }{}^{ * }}\\right) \) and also \( \\left\\langle {{M}^{*t},{M}^{*j}}\\right\\rangle \\left(... | Yes |
Lemma 6.1. \( {\mathcal{F}}_{t + 0}^{X} = {\mathcal{F}}_{t}^{X} \) . | Proof. Let \( p\left( {t, x}\right) \) be given by (I-7.1) and set\n\n\[ \left( {{H}_{t}f}\right) \left( x\right) = {\int }_{{R}^{d}}p\left( {t, x - y}\right) f\left( y\right) {dy},\;f \in {C}_{0}\left( {R}^{d}\right) . * \]\n\n\( \left\{ {H}_{t}\right\} \) constitutes a strongly continuous semigroup of operators on \(... | Yes |
Lemma 6.2. For any increasing sequence \( {\sigma }_{n} \) of \( \left( {{\mathcal{F}}_{t}x}\right) \) -stopping times,\n\n\[ \mathop{\bigvee }\limits_{n}{\mathcal{F}}_{{\sigma }_{n}}^{x} = {\mathcal{F}}_{\sigma }^{x} \]\n\nwhere \( \sigma = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\sigma }_{n} \) . | Proof. By the strong Markov property,\n\n\[ E\left\lbrack {{f}_{1}\left( {X\left( {t}_{1}\right) }\right) {f}_{2}\left( {X\left( {t}_{2}\right) }\right) \cdots {f}_{n}\left( {X\left( {t}_{n}\right) }\right) \mid {\mathcal{F}}_{\tau }^{X}}\right\rbrack \]\n\n\[ = \mathop{\sum }\limits_{{k = 1}}^{n}{I}_{\left( {t}_{k - 1... | No |
Lemma 6.3. Let \( M \in {\mathcal{M}}_{2} \) be bounded and suppose that \( \langle M, N\rangle = 0 \) for every \( N \in {\mathcal{M}}_{2}^{ * } \) . Then \( M = 0 \) . | Proof. Assume \( \left| {M\left( t\right) }\right| \leq \alpha \) where \( \alpha \) is a positive constant and set \( D\left( \omega \right) = 1 + M\left( {T,\omega }\right) /{2\alpha } \) . Then \( D\left( \omega \right) \geq 1/2 \) and \( E\left\lbrack {D\left( \omega \right) }\right\rbrack = 1 \) . Define a new pro... | Yes |
Theorem 7.1. Let \( {M}^{i} \in {\mathcal{M}}_{2}^{c,{loc}}, i = 1,2,\ldots, d \) . Suppose that \( {\Phi }_{ij}\left( s\right) \in {\mathcal{L}}_{1}^{loc} * \) and \( {\Psi }_{ik}\left( s\right) \in {\mathcal{L}}_{2}^{loc}, i, j, k = 1,2,\ldots, d \), exist such that\n\n(7.1)\n\n\[ \left\langle {{M}^{l},{M}^{j}}\right... | Proof. We will consider the case where \( {M}^{t} \in {\mathcal{M}}_{2}^{c},{\Phi }_{ij} \in {\mathcal{L}}_{1} \) and \( {\Psi }_{ik} \in {\mathcal{L}}_{2} \) ; the general case is easily reduced to this case. For \( N > 0 \) we set\n\n(7.5)\n\n\[ {\theta }_{ik}^{\left( N\right) }\left( {s,\omega }\right) = \left\{ \be... | Yes |
Theorem 7.2. Let \( M \in {\mathcal{M}}_{2}^{\text{coc }} \) such that \( \mathop{\lim }\limits_{{t \uparrow \infty }}\langle M\rangle \left( t\right) = \infty \) a.s. Then, if we set\n\n(7.8)\n\n\[ \n{\tau }_{t} = \inf \{ u;\langle M\rangle \left( u\right) > t\} \n\]\n\nand \( {\mathcal{T}}_{t} = {\mathcal{F}}_{{\tau ... | Proof. First, we remark that, with probability one, \( t \mapsto B\left( t\right) = \) \( M\left( {\tau }_{t}\right) \) is continuous. It is sufficient to show that, for any fixed \( r < {r}^{\prime } \), we have, except a set of probability zero,\n\n(7.10)\n\n\[ \n\left\{ {\langle M\rangle \left( {r}^{\prime }\right) ... | Yes |
Theorem 7.4. \( {}^{*1} \) Let \( \left( {\Omega ,\mathcal{F}, P}\right) \) be a probability space with a reference family \( \left( {\mathcal{F}}_{t}\right) \) . Let \( \left( {X,{\mathcal{B}}_{X}}\right) \) be a measurable space and \( p \) be an \( \left( {\mathcal{F}}_{t}\right) \) -point process of class (QL) on \... | Proof. First we prove several lemmas.\n\nLemma 7.1. There exists | No |
Lemma 7.1. There exists a predictable probability kernel \( Q\left( {t, x,{dz},\omega }\right) \) on \( \lbrack 0,\infty ) \times X \times {\mathcal{B}}_{Z} \times \Omega \) (i.e., for a fixed \( A \in {\mathcal{B}}_{Z},\left( {t, x,\omega }\right) \mapsto \) \( Q\left( {t, x, A,\omega }\right) \) is predictable and fo... | The proof of this lemma is standard and is left to the reader (cf. Chapter I, Section 3). | No |
Lemma 7.2. On an extension \( \left( {\widetilde{\Omega },\mathcal{F},\widetilde{P}}\right) \) and \( \left( {\mathcal{F}}_{t}\right) \) of \( \left( {\Omega ,\mathcal{F}, P}\right) \) and \( \left( {\mathcal{F}}_{t}\right) \), there exists an \( \left( {\mathcal{F}}_{t}\right) \) -point process \( \widetilde{p} \) of ... | Proof. We prepare a sequence of independent identically distributed random variables \( {\xi }_{n, k}, n, k = 1,2,\ldots \) on a probability space \( \left( {{\Omega }^{\prime },{\mathcal{F}}^{\prime }}\right. \) , \( \left. {P}^{\prime }\right) \) such that \( 0 \leq {\xi }_{n, k} \leq 1 \) a.s. and are uniformly dist... | Yes |
Theorem 1.1. The space \( d\mathcal{Q} \) with the operations \( \mathcal{A},\mathcal{M} \) and \( \mathcal{P} \) is a commutative algebra over \( \mathcal{B} \), i.e., a commutative ring with the operations \( \mathcal{A} \) and \( \mathcal{P} \) satisfying the relations\n\n\[ \Phi \cdot \left( {{dX} + {dY}}\right) = ... | Proof. It follows almost immediately from the property of stochastic integrals established in Chapter II that \( d\mathcal{Q} \) is a commutative algebra over \( \mathcal{B} \) . (1.8) follows at once because \( \left\langle {{M}_{X},{M}_{Y}}\right\rangle \in \mathcal{A} \) for \( X, Y \in \mathcal{Q} \) . | Yes |
Theorem 1.2. The space \( d\mathcal{Q} \) with the operations \( \mathcal{A},\mathcal{S}.\mathcal{M} \) . and \( \mathcal{P} \) is a commutative algebra over \( \mathcal{Q} \) ; we have, for \( X, Y, Z \in \mathcal{Q} \) , \n\n\[ \nX \circ \left( {{dY} + {dZ}}\right) = X \circ {dY} + X \circ {dZ}, \n\] \n\n\[ \n\left( ... | Proof. We note that since \( d\mathcal{Q} \cdot d\mathcal{A} = 0 \) and \( d\mathcal{Q} \cdot d\mathcal{Q} \cdot d\mathcal{Q} = 0 \), we have that \n\n(1.12) \n\n\[ \nX \circ {dY} = X \cdot {dY}\;\text{ if }\;X\text{ or }\;Y \in \mathcal{A}, \n\] \n\nand \n\n(1.13) \n\n\[ \n\left( {Z \circ {dX}}\right) \cdot {dY} = Z \... | No |
Theorem 1.3. If \( {X}^{1},{X}^{2},\ldots ,{X}^{d} \in \mathcal{Q} \) and \( f \in {C}^{3}\left( {{R}^{d} \rightarrow R}\right) \), then for \( Y = f\left( {{X}^{1},{X}^{2},\ldots ,{X}^{d}}\right) \in \mathcal{Q} \) we have\n\n(1.14)\n\n\[ \n{dY} = \mathop{\sum }\limits_{{i = 1}}^{d}{\partial }_{i}f \circ d{X}^{i} \n\] | Proof. By Theorem 1.2\n\n\[ \n\mathop{\sum }\limits_{{t = 1}}^{d}{\partial }_{t}f \circ d{X}^{t} \n\]\n\n\[ \n= \mathop{\sum }\limits_{{i = 1}}^{d}\left( {{\partial }_{t}f \cdot d{X}^{i} + \frac{1}{2}d\left( {{\partial }_{t}f}\right) \cdot d{X}^{i}}\right) \n\]\n\n\[ \n= \mathop{\sum }\limits_{{i = 1}}^{d}{\partial }_{... | Yes |
Theorem 1.4. For every \( X \) and \( Y \) in \( \mathcal{Q} \) ,\n\n\[ {\int }_{0}^{t}Y \circ {dX} = 1\text{ i. }\mathrm{p}.\mathop{\sum }\limits_{{i = 1}}^{n}\frac{Y\left( {t}_{i}\right) + Y\left( {t}_{i - 1}\right) }{2}\left( {X\left( {t}_{i}\right) - X\left( {t}_{i - 1}\right) }\right) \]\n\nwhere \( \Delta \) deno... | Proof.\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{n}\frac{Y\left( {t}_{i}\right) + Y\left( {t}_{i - 1}\right) }{2}\left( {X\left( {t}_{i}\right) - X\left( {t}_{i - 1}\right) }\right) \]\n\n\[ = \mathop{\sum }\limits_{{i = 1}}^{n}Y\left( {t}_{i - 1}\right) \left( {X\left( {t}_{i}\right) - X\left( {t}_{i - 1}\right) }\right)... | Yes |
Example 2.2. Let \( {A}_{k} = \mathop{\sum }\limits_{{j = 1}}^{d}{A}_{k}^{j}\left( x\right) \frac{\partial }{\partial {x}^{j}} \) be a \( {C}^{\infty } \) -vector field*1 on \( {\mathbf{R}}^{d} \) , \( k = 1,2,\ldots, r \) . We assume that the first and second order derivatives of all coefficients are bounded. For give... | [I] *2 If the vector fields \( {A}_{1},{A}_{2},\ldots ,{A}_{r} \) are commutative, i.e., \( \left\lbrack {{A}_{p},{A}_{q}}\right\rbrack \) \( = 0, p, q = 1,2,\ldots, r \), then this implies the integrability of\n\n\[ \left\{ \begin{array}{l} \frac{\partial {u}^{i}}{\partial {z}^{j}}\left( {x, z}\right) = {A}_{j}^{i}\le... | Yes |
Lemma 4.1. (A Fubini-type theorem for stochastic integrals). Let \( \left( {\Omega ,\mathcal{F}, P}\right) \) be a probability space and \( \left( {\mathcal{F}}_{t}\right) \) be a reference family. Let \( M \in \) \( {\mathcal{M}}_{2}^{c} \) ,(i.e., a continuous square-integrable martingale such that \( {M}_{0} = 0 \) ... | Proof. It is clear that \( {\int }_{{\mathbb{R}}^{1}}\Phi \left( {s, a,\omega }\right) \mu \left( {da}\right) \) is \( \left( {\mathcal{F}}_{t}\right) \) -predictable and bounded. Hence it is obvious that\n\n\[ E\left\lbrack {{\int }_{0}^{t}{\left\{ {\int }_{{\mathbf{R}}^{1}}\Phi \left( s, a,\omega \right) \mu \left( d... | Yes |
Lemma 4.2. Given \( f \in {W}_{0}^{1} \) and \( x \in {\mathbf{R}}^{ + } \), there exist unique \( g \in {\mathbf{C}}^{ + } \) and \( h \in {\mathbf{C}}^{ + } \) such that\n\n(i) \( g\left( t\right) = x + f\left( t\right) + h\left( t\right) \), \n\n(ii) \( h\left( 0\right) = 0 \) and \( t \mapsto h\left( t\right) \) is... | Proof. Set\n\n(4.9)\n\n\[ g\left( t\right) = x + f\left( t\right) - \mathop{\min }\limits_{{0 \leq s \leq t}}\{ \left( {x + f\left( s\right) }\right) \land 0\} ,\]\n\n(4.10)\n\n\[ h\left( t\right) = - \mathop{\min }\limits_{{0 \leq s \leq t}}\{ \left( {x + f\left( s\right) }\right) \land 0\} . \]\n\nThen it is easy to ... | Yes |
Lemma 4.3. \( n \) is invariant under the mapping: \( w \rightarrow \breve{w} \) . | Proof. \( {P}^{T} \) on \( {\mathcal{W}}^{ + } \cap \{ w;\sigma \left( w\right) = T\} \) is invariant under the mapping: \( w \rightarrow \breve{w} \) (cf. Example IV-8.4). Since\n\n\[ \n{n}^{ + }\left( B\right) = {\int }_{0}^{\infty }{P}^{T}\left( {B\cap \{ \sigma \left( w\right) = T\} }\right) {\left( 2\pi {T}^{3}\ri... | No |
Lemma 4.5. Let \( a > 0 \) and set, for \( w \in C\left( {\lbrack 0,\infty }\right) \rightarrow R) \) , (4.33) \[ {l}_{a}^{0}\left( w\right) = \sup \left\{ {t;t \leq {\sigma }_{0}\left( w\right), w\left( t\right) = a}\right\} \] (4.34) \[ {l}_{a}\left( w\right) = \sup \{ t;w\left( t\right) = a\} . \] Then (4.35) \[ {P}... | Proof. It is sufficient to prove that for bounded Borel functions \( {f}_{1},{f}_{2},\ldots ,{f}_{n} \) on \( \left( {0,\infty }\right) \) and \( 0 < {t}_{1} < {t}_{2} < \ldots < {t}_{n} \) , (4.36) \[ {E}^{{P}_{a}}\left\lbrack {\mathop{\prod }\limits_{{j = 1}}^{n}{f}_{j}\left( {w\left( {t}_{j}\right) }\right) {I}_{{u}... | Yes |
For given \( a > 0 \), let \( \{ X\left( t\right) \} \) be \( B{M}^{a} \) and \( \{ Y\left( t\right) \} \) be \( {BE}{S}^{0}\left( 3\right) \) . Then \( \left\{ {X\left( {{\sigma }_{0}^{X} - t}\right) ,0 \leq t \leq }\right. \) \( \left. {\sigma }_{0}^{X}\right\} \) is equivalent in law to \( \left\{ {Y\left( t\right) ... | Proof. From (4.30), we see that \( \{ w\left( t\right) ;0 \leq t \leq \sigma \left( w\right) \} \) is decomposed, under \( {n}^{ + }\left( {\cdot \mid {\sigma }_{a} < \infty }\right) \), into mutually independent parts \( \{ w\left( t\right) \) , \( \left. {0 \leq t \leq {\sigma }_{a}}\right\} \) and \( \left\{ {w\left... | Yes |
Lemma 4.6. (i) The point process \( {p}^{ * } \) on \( \mathbf{X} \) defined by \( {\mathbf{D}}_{{p}^{ * }} = \{ t;t + \) \( \left. {\theta \in {\mathbf{D}}_{p}}\right\} \) and \( {p}^{ * }\left( t\right) = p\left( {t + \theta }\right), t \in {\mathbf{D}}_{p} * \), is independent of \( {\mathcal{F}}_{\theta } \) . | Proof. (i) This is essentially established in Theorem II-6.5, because \( \left( {\mathcal{F}}_{t}\right) \) -stationary Poisson point process \( p \) is always independent of \( {\mathcal{F}}_{0} \) . | No |
If \( X \) is \( B{M}^{0} \) and \( Y \) is \( {BE}{S}^{0}\left( 3\right) \), then \( \left\{ {X\left( {{l}_{0}^{a, X} + t}\right) ,0 \leq t \leq {\sigma }_{a}^{X} - }\right. \) \( \left\{ {l}_{0}^{a,\mathrm{X}}\right\} \) is equivalent in law to \( \left\{ {Y\left( t\right) ,0 \leq t \leq {\sigma }_{a}^{\mathrm{Y}}}\r... | The second assertion follows from the fact that \( \{ X\left( t\right) ,0 \leq t \leq \) \( \left. {l}_{0}^{a, X}\right\} \) is \( {\mathcal{F}}_{{\theta }_{ - }} \) -measurable and \( E\left\lbrack {G\left\lbrack {p\left( \theta \right) }\right\rbrack \mid \theta = s}\right\rbrack \) is independent of \( s \) . | No |
Theorem 5.2. If \( X \in \mathcal{M} \), then the exponential quasimartingale \( {M}_{t} \) is a continuous local \( \left( {\mathcal{F}}_{t}\right) \) -martingale. Furthermore, \( {M}_{t} \) is a supermartingale, and it is a martingale if and only if\n\n\[ E\left\lbrack {M}_{t}\right\rbrack = 1\;\text{ for every }t \g... | Proof. Since \( {M}_{t} \) is the unique solution of (5.2)\n\n\[ {M}_{t} - 1 = {\int }_{0}^{t}{M}_{s}{dX}\left( s\right) \in \mathcal{M}. \]\n\nThus, it is easy to see by Fatou’s lemma that \( {M}_{t} \) is a supermartingale. | No |
Proposition 6.2. Let \( {Z}_{t} = \left( {{Z}_{t}^{1},{Z}_{t}^{2},\ldots ,{Z}_{t}^{n}}\right) \) be an \( n \) -dimensional local conformal martingale such that \( d{Z}_{t}^{\alpha } \cdot d{\bar{Z}}_{t}^{\beta } = 0,\alpha ,\beta = 1,2,\ldots \) , \( n,\alpha \neq \beta \) . Then there exists an \( n \) -dimensional c... | \[ {Z}_{t}^{\alpha } = {\zeta }^{\alpha }\left( {\left\langle {Z}^{\alpha }\right\rangle }_{t}\right) ,\;\alpha = 1,2,\ldots, n, \]\n\nwhere \( {\left\langle {Z}^{\alpha }\right\rangle }_{t} \) is defined to be the common processes \( {\left\langle {X}^{\alpha }\right\rangle }_{t} = {\left\langle {Y}^{\alpha }\right\ra... | No |
Lemma 1.1. For \( A \in {\mathcal{B}}_{t}\left( {W}^{d}\right) ,\;w \in {W}_{0}^{r} \mapsto {Q}^{w}\left( A\right) \) or \( {Q}^{\prime w}\left( A\right) \) is \( {\mathcal{B}}_{t}{\left( {W}_{0}^{r}\right) }^{PW} \) -measurable. | Proof. For fixed \( t > 0 \) and \( A \in {\mathcal{B}}_{t}\left( {W}^{d}\right) \), there exists a conditional probability \( {Q}_{t}^{w}\left( A\right) \) such that \( w \in {W}_{0}^{r} \mapsto {Q}_{t}^{w}\left( A\right) \) is \( {\overline{{\mathcal{B}}_{t}{\left( {W}_{0}\right) }^{PW}}}^{w} \) -measurable and \( {P... | Yes |
Lemma 1.2. \( {w}_{3} = \left( {{w}_{3}\left( t\right) }\right) \) is an \( r \) -dimensional \( \left( {\mathcal{F}}_{t}\right) \) -Brownian motion on \( \left( {\Omega ,\mathcal{F}, Q}\right) \) . | Proof. It is only necessary to prove the independence of \( {w}_{3}\left( t\right) - \) \( {w}_{3}\left( s\right) \) and \( {\mathcal{F}}_{s} \) for every \( t > s \) . For this, it is sufficient to prove that\n\n\[ \n{E}^{Q}\left\lbrack {{e}^{i\left\langle {\xi ,{w}_{3}\left( t\right) - {w}_{3}\left( s\right) }\right\... | Yes |
Consider the following one-dimensional stochastic differential equation of the time-homogeneous Markovian type:\n\n(1.5)\n\[ \n{dX}\left( t\right) = \sigma \left( {X\left( t\right) }\right) {dB}\left( t\right) \]\n\nwhere \( \sigma \left( x\right) = 1 \) for \( x \geq 0 \) and \( \sigma \left( x\right) = - 1 \) for \( ... | Indeed, let \( B = \left( {B\left( t\right) }\right) \) be an \( \left( {\mathcal{F}}_{t}\right) \) - Brownian motion and let \( \xi \) be an \( {\mathcal{F}}_{0} \) -measurable random variable having the distribution \( \mu \) defined on some suitable probability space with a reference family \( \left( {\mathcal{F}}_{... | Yes |
Theorem 2.4. If \( \sigma \left( x\right) = \left( {{\sigma }_{k}^{i}\left( x\right) }\right) \) and \( b\left( x\right) = \left( {{b}^{i}\left( x\right) }\right) \) are continuous and satisfy the condition\n\n(2.18)\n\n\[ \n\parallel \sigma \left( x\right) {\parallel }^{2} + \parallel b\left( x\right) {\parallel }^{2}... | Proof. Let \( {\sigma }_{n} = \inf \{ t;\left| {X\left( t\right) }\right| \geq n\} \) and \( f \in {C}_{b}^{2}\left( {R}^{d}\right) \) be chosen so that \( f\left( x\right) = {\left| x\right| }^{2} \) if \( \left| x\right| \leq n \) . Then since\n\n\[ \nf\left( {X\left( {t \land {\sigma }_{n}}\right) }\right) - f\left(... | Yes |
Theorem 3.3. Consider the equation of the time homogeneous Markovian case (3.1). If \( a\left( x\right) = \sigma \left( x\right) \sigma {\left( x\right) }^{ * } \) is uniformly positive definite, bounded and continuous and \( b\left( x\right) \) is bounded and Borel measurable, then the uniqueness of solutions holds. | Proof. We assume \( b\left( x\right) \equiv 0 \) ; the general case is obtained by a transformation of drift which will be discussed in the next section. Set\n\n\[ \n{Af}\left( x\right) = \frac{1}{2}\mathop{\sum }\limits_{{i, j = 1}}^{d}{a}^{ij}\left( x\right) \frac{{\partial }^{2}f}{\partial {x}^{i}\partial {x}^{j}}\l... | Yes |
Theorem 4.1. (i) Let \( Y \in {\mathcal{M}}_{2}^{c,{loc}} \) . If we define \( \widetilde{Y} \) by\n\n(4.4)\n\n\[ \n\widetilde{Y}\left( t\right) = Y\left( t\right) - \langle Y, X\rangle \left( t\right) \n\]\n\nthen \( \widetilde{Y} \in {\mathcal{M}}_{2}^{c,{loc}} \) . | Proof. Assume first that \( \widetilde{Y}\left( t\right) \) is bounded in the sense that, for each \( t \geq 0,\widetilde{Y}\left( t\right) \in {\mathcal{L}}^{\infty }\left( {\Omega ,\mathcal{F}}\right) \) . By Itô’s formula,\n\n\[ d\left( {M\left( t\right) \widetilde{Y}\left( t\right) }\right) \]\n\n\[ = d\{ M\left( t... | Yes |
Let \( a\left( {t, x}\right) \) be a bounded Borel measurable function on \( \lbrack 0,\infty ) \times {R}^{1} \) such that \( a\left( {t, x}\right) \geq C \) for some positive constant \( C \) . Set \( \alpha \left( {t, w}\right) = a\left( {t, w\left( t\right) }\right) \) . In this case, equation (4.19) is given as\n\... | One simple sufficient condition that (4.21)' has the unique solution is that \( a\left( {t, x}\right) \) be Lipschitz continuous in \( t \) ; in this case the stochastic differential equation\n\n\[ \n{dX}\left( t\right) = a\left( {t, X\left( t\right) }\right) {dB}\left( t\right)\n\]\n\nis solved uniquely as \( X\left( ... | Yes |
Example 4.4. (Nisio [130]). Let \( f\left( x\right) \) be a locally bounded Borel measurable function on \( {\mathbf{R}}^{1}, a\left( x\right) \) be a bounded Borel measurable function on \( {\mathbf{R}}^{1} \) such that \( a\left( x\right) \geq C \) for some positive constant \( C \), and \( y \in {\mathbf{R}}^{1} \) ... | and the equation (4.19) is given as\n\n(4.23)\n\n\[ \n{\phi }_{t} = {\int }_{0}^{t}a{\left( y + {\int }_{0}^{{\phi }_{s}}f\left\lbrack \left( {T}^{\phi }\xi \right) \left( u\right) \right\rbrack du\right) }^{-2}{ds}. \n\]\n\nNow\n\n\[ \n{\int }_{0}^{{\phi }_{s}}f\left\lbrack {\left( {{T}^{\phi }\xi }\right) \left( u\ri... | Yes |
Theorem 5.1. Suppose that \( \left\{ {{P}_{x}, x \in {S}^{\prime }}\right\} \) is a system of probability measures on \( \left( {\bar{W}\left( S\right) ,\mathcal{B}\left( {\bar{W}\left( S\right) }\right) }\right) \) satisfying conditions (i) and (ii) of Definition 5.3. Suppose further that \( \left\{ {P}_{x}\right\} \)... | Proof. It is only necessary to show that \( \left\{ {P}_{x}\right\} \) is a strongly Markovian system, i.e., it satisfies (5.5). Since \[ {X}_{f}\left( t\right) = f\left( {w\left( t\right) }\right) - f\left( {w\left( 0\right) }\right) - {\int }_{0}^{t}\left( {Af}\right) \left( {w\left( s\right) }\right) {ds} \] is a \(... | Yes |
Let \( S = {\mathbf{R}}^{d} \) and \( {S}^{\prime } = {\mathbf{R}}^{d} \cup \{ \Delta \} \), where \( \Delta \) is attached to \( {\mathbf{R}}^{d} \) as an isolated point. Let \( \mathcal{D}\left( A\right) = \left\{ {f \in \mathbf{C}\left( {S}^{\prime }\right) ;{\left. f\right| }_{{\mathbf{R}}^{d}} \in {C}_{b}^{2}\left... | To prove this, we first show that \( {P}_{x}, x \in {\mathbf{R}}^{d} \), is conservative. Indeed, \( {\left. f\left( x\right) \text{ defined by }f\left( x\right) \right| }_{{\mathbb{R}}^{d}} \equiv 0 \) and \( f\left( \Delta \right) = 1 \) belongs to \( \mathcal{D}\left( A\right) \) and \( {Af}\left( x\right) \equiv 0.... | Yes |
Let \( {S}^{\prime } \) and \( \mathcal{D}\left( A\right) \) be as in Example 5.1. Define \( A \) on \( \mathcal{D}\left( A\right) \) by\n\n(5.9)\n\n\[ \n{Af}\left( x\right) = \left\{ \begin{array}{ll} \frac{1}{2}{\Delta f}\left( x\right) + \mathop{\sum }\limits_{{i = 1}}^{d}{c}_{i}\frac{\partial f}{\partial {x}^{i}}\l... | This can be proved in the same way as in Example 5.1. | No |
Let \( {S}^{\prime } = {\mathbf{R}}^{d} \cup \{ \Delta \} \) be as in the previous examples. Let \( \mathcal{D}\left( A\right) = \left\{ {f \in C\left( {S}^{\prime }\right) ;{\left. f\right| }_{{\mathbb{R}}^{d}} \in {C}_{b}^{2}\left( {\mathbb{R}}^{d}\right) \text{and}f\left( \Delta \right) = 0}\right\} \) and define \(... | To prove this assertion, we first note that the system \( \left\{ {P}_{x}\right\} \) clearly satisfies the conditions (i) and (ii) of Definition 5.3. Secondly the function \( {f}_{\xi }\left( x\right) ,\xi \in {\mathbf{R}}^{d} \), defined by\n\n\[ \n{f}_{\xi }\left( x\right) = \left\{ \begin{array}{ll} {e}^{i\langle \x... | Yes |
Example 5.4. Let \( S = {R}_{ + }^{d} \mathrel{\text{:=}} \left\{ {x = \left( {{x}^{1},{x}^{2},\ldots ,{x}^{d}}\right) \in {R}^{d};{x}^{d} \geq 0}\right\} \) , \( {S}^{\prime } = {\mathbf{R}}_{ + }^{d} \cup \{ \Delta \} \) where \( \Delta \) is attached to \( {\mathbf{R}}_{ + }^{d} \) as an isolated point. Let \( \math... | To prove this, we first note that by Chapter III, Section \( {4.2}{X}_{x}^{k}\left( t\right) = {x}^{k} + \) \( {\widetilde{B}}^{k}\left( t\right) + {\delta }_{kd}\phi \left( t\right) \) for \( k = 1,2,\ldots, d \) where \( \widetilde{B}\left( t\right) \) is a \( d \) -dimensional\n\n--- \n\n* \( {E}_{x}^{\prime } \) st... | Yes |
For simplicity, we consider the one-dimensional case only. Let \( S = \lbrack 0,\infty ) \) and \( {S}^{\prime } = \lbrack 0,\infty ) \cup \{ \Delta \} \), where \( \Delta \) is attached to \( S \) as an isolated point. For a given parameter \( \gamma \left( {0 \leq \gamma \leq 1}\right) \), let \( \mathcal{D}\left( A\... | The proof follows by a similar argument as in Examples 5.3 and 5.4: we take for \( \xi \in \mathbf{R} \) \[ {f}_{\xi }\left( x\right) = \left\{ \begin{array}{ll} {\gamma \xi }\cos {\xi x} + \left( {1 - \gamma }\right) \sin {\xi x}, & x \in \lbrack 0,\infty ), \\ 0, & x = \Delta . \end{array}\right. \] | Yes |
Let \( S \) be a bounded smooth domain in \( {\mathbf{R}}^{d} \) and \( {S}^{\prime } = S \cup \{ \Delta \} \), where \( \Delta \) is attached to \( S \) as a point at infinity. Let \( \mathcal{D}\left( A\right) = {C}_{0}^{2}\left( S\right) ( \mathrel{\text{:=}} \{ f \) ; twice continuously differentiable in \( S \) an... | Suppose \( \left\{ {P}_{x}^{\prime }\right\} \) also satisfies these conditions. Let \( {G}_{\alpha }\left( {x, y}\right) ,\alpha > 0 \), be the Green function of \( S \) for the operator \( {L}_{\alpha } = \alpha - \Delta /2 \) with Dirichlet boundary conditions: i.e., if \( f \in {C}_{K}^{\infty }{\left( S\right) }^{... | Yes |
Proposition 6.2. (i) Let \( {\mathfrak{S}}^{r} \) be the set of all \( r \times r \) symmetric, nonnegative definite matrices. If \( a\left( x\right) : {\mathbf{R}}^{d} \rightarrow {\mathbb{S}}^{r} \) is in the class \( {\mathbf{C}}_{b}^{2}\left( {\mathbf{R}}^{d}\right) ,{}^{*2} \) then the square root \( \sigma \left(... | Proof. Clearly it is only necessary to prove (i). We shall prove this in the case \( d = 1 \) ; the general case follows from the fact that a function is uniformly Lipschitz continuous on \( {\mathbf{R}}^{d} \) if it is uniformly Lipschitz continuous in each variable \( {x}^{i} \in {\mathbf{R}}^{1} \) for fixed \( \wid... | Yes |
Theorem 7.2. We assume for the stochastic differential equation (7.8) that \( \sigma, b,\tau ,\beta ,\delta ,\rho \) satisfy the following: \( \sigma \) and \( b \) are bounded and Lipschitz continuous on \( D,\tau ,\beta \) and \( \delta \) are bounded and Lipschitz continuous on \( \partial D \) and \( \rho \) is bou... | Proof of Theorem 7.2. Let \( c\left( x\right) \) be a continuous function on \( \partial D \) such that \( {c}_{1} \leqq c\left( x\right) \leqq {c}_{2}, x \in \partial D \), for some positive constants \( {c}_{1} \) and \( {c}_{2} \) . Then it is easy to see that \( \mathfrak{X} = \left\lbrack {X\left( t\right), B\left... | Yes |
Consider the following stochastic differential equation\n\n(8.1)\n\n\[ d{X}_{t}^{i} = \mathop{\sum }\limits_{{k = 1}}^{r}{\sigma }_{k}^{i}d{B}_{t}^{k} + {b}^{i}\left( {X}_{t}\right) {dt},\;i = 1,2,\ldots, d, \] | Then the solution \( X\left( t\right) \) of (8.1) is given as\n\n(8.2)\n\n\[ X\left( t\right) = {e}^{\beta t}\left( {X\left( 0\right) + {\int }_{0}^{t}{e}^{-{\beta s}}{\sigma dB}\left( s\right) }\right) ,\] \n\nThe proof is easily seen from the relation\n\n\[ d\left( {{e}^{-{\beta t}}X\left( t\right) }\right) = {e}^{-{... | Yes |
Let \( a, c, d \) be real constants such that \( a > 0 \) . Consider the following one-dimensional stochastic differential equation:\n\n\[ \n{dX}\left( t\right) = {\left( 2aX\left( t\right) \vee 0\right) }^{1/2}{dB}\left( t\right) + \left( {{cX}\left( t\right) + d}\right) {dt}.\n\] | Since the coefficients \( \sigma \left( x\right) = {\left( 2ax \vee 0\right) }^{1/2} \) and \( b\left( x\right) = {cx} + d \) satisfy the condition of Theorem 3.2 and also the growth condition (2.18), a global strong solution \( X\left( t\right) \) exists uniquely for every given initial value \( X\left( 0\right) \) . ... | Yes |
For \( \alpha > 0 \), let \( {L}_{\alpha } \) be the differential operator on \( \lbrack 0,\infty ) \) defined by (8.14) \[ {L}_{\alpha }f\left( x\right) = \frac{1}{2}\left\lbrack {\frac{{d}^{2}}{d{x}^{2}}f\left( x\right) + \frac{\alpha - 1}{x}\frac{d}{dx}f\left( x\right) }\right\rbrack \] with the domain \( \mathcal{D... | Let \[ {\widetilde{L}}_{\alpha }f\left( x\right) = {2x}\frac{{d}^{2}}{d{x}^{2}}f\left( x\right) + \alpha \frac{d}{dx}f\left( x\right) \text{ and }c \] \[ \mathcal{D}\left( {\widetilde{L}}_{\alpha }\right) = \left\{ {\widetilde{f}\left( x\right) = f\left( \sqrt{x}\right) ;f \in \mathcal{D}\left( {L\alpha }\right) }\righ... | Yes |
Example 8.4. (Brownian excursions). \( {}^{*1} \) Let \( T > 0 \) be fixed. Consider the following stochastic differential equation\n\n(8.21)\n\n\[ \left\{ \begin{array}{l} {dX}\left( t\right) = 2{\left( X\left( t\right) \vee 0\right) }^{1/2}{dB}\left( t\right) + \left( {3 - \frac{{2X}\left( t\right) }{T - t}}\right) {... | This is an equation similar to (8.18). Hence it can be shown that there exists a unique solution \( X\left( t\right) \) for \( t \in \lbrack 0, T) \) and that\n\n(8.22)\n\n\[ P\left( {X\left( t\right) > 0\text{ for all }t \in \left( {0, T}\right) }\right) = 1{.}^{*2} \] | Yes |
Let \( X\left( t\right) \) be a one-dimensional Brownian motion such that \( X\left( 0\right) = 0 \) . For fixed \( {t}_{0} > 0 \) and \( x, y \in {\mathbf{R}}^{1} \), define the process \( {X}_{x}^{{t}_{0}, y} = {\left( {X}_{x}^{{t}_{0}, y}\left( t\right) \right) }_{0 \leq t \leq {t}_{0}} \) by\n\n\[ \n{X}_{x}^{{t}_{0... | Clearly the solution \( X\left( t\right) \) exists uniquely for \( t \in \left\lbrack {0,{t}_{0}}\right) \) . By (8.35) we have\n\n\[ \n\left( {t - {t}_{0}}\right) d\left( \frac{X\left( t\right) - y}{t - {t}_{0}}\right) = {dB}\left( t\right) \n\] \n\nand hence \( X\left( t\right) \) is solved as\n\n\[ \nX\left( t\right... | Yes |
Theorem 9.1. If \( \sigma \left( x\right), b\left( x\right) \) and \( f\left( {x, u}\right) \) satisfy in addition to (9.1) the Lipschitz condition\n\n(9.3)\n\n\[ \parallel \sigma \left( x\right) - \sigma \left( y\right) {\parallel }^{2} + \parallel b\left( x\right) - b\left( y\right) {\parallel }^{2} + {\int }_{{U}_{0... | Proof. Suppose \( B = \left( {{B}^{k}\left( t\right) }\right), p \) and \( \xi \) are given as above. Let \( D = \{ s \in \) \( \left. {{D}_{p};p\left( s\right) \in U \smallsetminus {U}_{0}}\right\} \) . Since \( n\left( {U \smallsetminus {U}_{0}}\right) < \infty, D \) is a discrete set in \( \left( {0,\infty }\right) ... | Yes |
Theorem 1.1. There exists a function \( F : M \times {W}_{o}^{r} \rightarrow \widehat{W}\left( M\right) \) which is \( \cap \widehat{\mathcal{B}\left( M\right) } \times {\mathcal{B}}_{t}{\left( \widehat{{W}_{o}^{\prime }}\right) }^{\mu \times P{W}^{\prime }}/{\mathcal{B}}_{t}\left( {\widehat{W}\left( M\right) }\right) ... | Proof. Take a coordinate neighborhood \( {}^{*2}U \) and express \( {A}_{\alpha } = \) \( {\sigma }_{\alpha }^{i}\left( x\right) \frac{\partial }{\partial {x}^{i}},\alpha = 0,1,\ldots, r \), under the local coordinates \( \left( {{x}^{1},{x}^{2},\ldots ,{x}^{d}}\right) \) in \( U \) . Extend the functions \( {\sigma }_... | Yes |
Theorem 1.2. Let \( {P}_{x} \) be the probability law on \( \widehat{W}\left( M\right) \) of the solution \( X = \left( {X\left( t\right) }\right) \) of (1.1) with the initial value \( X\left( 0\right) = x \) . Then \( {\left\{ {P}_{x}\right\} }_{x \in M} \) is a diffusion generated by the second order differential ope... | Proof. Using the uniqueness of solutions we can show that \( \left\{ {P}_{x}\right\} \) has the strong Markov property. Actually, we can prove the following stronger result: for any \( \left( {\mathcal{F}}_{t}\right) \) -stopping time \( \sigma \left( w\right) \), we have \( X\left( {t + \sigma \left( w\right), x, w}\r... | Yes |
Lemma 2.1. Let \( A\left( x\right) = \left( {{A}_{\alpha }^{t}\left( x\right) }\right) \in {\mathbf{R}}^{m} \otimes {\mathbf{R}}^{r} \) and \( \beta \left( x\right) = \left( {{\beta }^{t}\left( x\right) }\right) \in {\mathbf{R}}^{m} \) be given and satisfy the following conditions;\n\n(i) there exists a positive consta... | Proof. Let \( T > 0 \) be arbitrary but fixed. First we remark that (2.3) implies\n\n(2.6)\n\n\[ E\left\{ {\mathop{\sup }\limits_{{0 \leq t \leq T}}{\left| \alpha \left( t\right) \right| }^{p + 1}}\right\} < \infty \]\n\nand\n\n(2.7)\n\n\[ E\left\{ {\mathop{\sup }\limits_{{0 \leq t \leq T}}{\left| {\alpha }_{n}\left( t... | Yes |
For each \( x \in {R}^{d},1 \leq i \leq d \), and each \( \alpha \), there exists a unique process \( {Y}_{\alpha }^{t}\left( t\right) = \left( {{Y}_{\alpha }^{t}\left( {t, x, w}\right) }\right) \) such that\n\n\[ E\left\{ {\mathop{\sup }\limits_{{0 \leq t \leq T}}{\left| {Y}_{\alpha ,\left( n\right) }^{t}\left( t\righ... | First consider the case of \( \left| \alpha \right| = 1 \) . If we set \( {Y}_{j,\left( n\right) }^{i}\left( {t, x, w}\right) = \) \( \frac{\partial }{\partial {x}^{j}}{X}_{n}^{t}\left( {t, x, w}\right) ,{Y}_{\left( n\right) }\left( t\right) = \left( {{Y}_{j,\left( n\right) }^{t}\left( {t, x, w}\right) }\right) \) is d... | Yes |
Theorem 2.3. \( {}^{*1} \) Let \( X\left( {t, x, w}\right) \) be the solution of (2.24) (or (2.1)) on the Wiener space \( \left( {{W}_{0}^{r},{P}^{W}}\right) \) . Then a modification of \( X\left( {t, x, w}\right) \) can be chosen so that the mapping \( x \mapsto X\left( {t, x, w}\right) \) is a diffeomorphism of \( {\... | Thus we have a one-parameter family of diffeomorphisms \( {X}_{t}\left( w\right) \) : \( x \mapsto X\left( {t, x, w}\right) \) for \( t \in \lbrack 0,\infty ) \) . Clearly \( {X}_{0}\left( w\right) = \) the identity and \( {X}_{s}\left( {{\theta }_{t}w}\right) \) 。 \( {X}_{t}\left( w\right) = {X}_{t + s}\left( w}\right... | Yes |
Theorem 2.4. Assume that \( M \) is a compact manifold. \( X\left( {t, x, w}\right) \) has a modification, \( {}^{*2} \) which is denoted by \( X\left( {t, x, w}\right) \) again, such that the mapping \( {X}_{t}\left( w\right) : x \mapsto X\left( {t, x, w}\right) \) is \( {C}^{\infty } \) in the sense that \( x \mapsto... | Proof. Let \( {x}_{0} \in M \) and \( t \in \lbrack 0,\infty ) \) be fixed. Then for almost all \( w \) such that \( X\left( {t,{x}_{0}, w}\right) \in M \), there exist an integer \( n > 0 \) and a sequence of coordinate neighborhoods \( {U}_{1},{U}_{2},\cdots ,{U}_{n} \) such that\n\n\[ \left\{ {X\left( {s,{x}_{0}, w}... | Yes |
The function \( u\left( {t, x}\right) \) defined by\n\n\[ u\left( {t, x}\right) = E\left\lbrack {\exp \left\{ {{\int }_{0}^{t}c\left( {X\left( {s, x, w}\right) }\right) {ds}}\right\} f\left( {X\left( {t, x, w}\right) }\right) }\right\rbrack ,\;f \in {F}_{0}\left( M\right) \]\nis a solution in \( {C}^{\infty }\left( {\l... | The proof is given in the same way as in Theorem 3.1. This time, however, we use Itô's formula as follows:\n\n\[ d\left\lbrack {\exp \left\{ {{\int }_{0}^{t}c\left( {X}_{s}\right) {ds}}\right\} f\left( {X}_{t}\right) }\right\rbrack = \exp \left\{ {{\int }_{0}^{t}c\left( {X}_{s}\right) {ds}}\right\} \left( {{A}_{\alpha ... | No |
\[ {\widetilde{L}}_{m}\left( {F}_{u{j}_{1}{j}_{2}\ldots {j}_{q}}^{{i}_{1}{i}_{2}\ldots {i}_{p}}\right) \left( r\right) = {\left( {F}_{\bar{u}u}\right) }_{{j}_{1}{j}_{2}\ldots {j}_{q};m}^{{i}_{1}{i}_{2}\ldots {i}_{p}}\left( r\right) \] for every \( {i}_{l},{i}_{2},\ldots ,{i}_{p},{j}_{1},{j}_{2},\ldots ,{j}_{q} \) and \... | The proof is left to the reader. | No |
Proposition 4.3. (i) For every vector field \( b = {b}^{t}\left( x\right) \frac{\partial }{\partial {x}^{t}} \) on a Riemannian manifold \( M \), there exists an affine connection \( \bar{V} = \left\{ {\Gamma }_{ij}^{k}\right\} \) on \( M \) compatible with the Riemannian metric \( g \) such that (4.33) holds. | Proof. (i) Define\n\n\[ {\Gamma }_{jk}^{i} = \left\{ {}_{j}^{i}\right\} + \frac{2}{d - 1}\left( {{\delta }_{j}^{i}{b}_{k} - {g}_{jk}{b}^{i}}\right) \]\n\nwhere \( {b}_{t} = {g}_{tj}{b}^{j} \). Then, since \( {\delta }_{j}^{t}{b}_{k} - {g}_{jk}{b}^{t} \) are the components of a \( \left( {1,2}\right) \) - tensor, \( \ba... | Yes |
\[ {\left( Af, h\right) }_{0} = {\left( f,{A}^{ * }h\right) }_{0},{}^{*1}\;f, h \in F\left( M\right) \], where \[ {A}^{ * }h = - \frac{1}{2}{\delta dh} + \delta \left( {h{\omega }_{b}}\right) = \frac{1}{2}{\Delta }_{M}h - \operatorname{div}\left( {hb}\right) {.}^{*2} \] | The proof is immediate from the definition. | No |
An invariant measure \( \mu \left( {dx}\right) \) of the \( A \) -diffusion exists and is unique up to a multiplicative constant; moreover, \( \mu \left( {dx}\right) \) is given as \( v\left( x\right) {dx} \) where \( v \in F\left( M\right) \) is a solution of \[ {A}^{ * }v = 0\text{. } \] | Proof. The equation (4.58) is equivalent to \( {A}^{ * }\mu = 0 \) in the sense of Schwartz distributions on \( M \) . Since \( {A}^{ * } \) is elliptic, any solution must be of the form \( \mu = {vdx}, v \in F\left( M\right) \), by Weyl’s lemma ([1]). Furthermore, \( \lambda = 0 \) is the largest eigenvalue of the eig... | Yes |
The \( A \) -diffusion is symmetrizable if and only if\n\n(4.62)\n\n\[ \n{\delta \beta } = \alpha = 0\;\text{ in }\;\left( {4.51}\right) ; * \n\]\n\nand this is equivalent to\n\n(4.63)\n\n\[ \n\delta {\beta }_{1} = {\alpha }_{1} = 0\;\text{ in }\;\left( {4.61}\right) . \n\] | Proof. Let \( {U}_{0}\left( x\right) \in F\left( M\right) \) be determined by\n\n\[ \n{\int }_{M}{e}^{-{U}_{0}\left( x\right) }{dx} = 1\;\text{ and }\;{A}^{ * }\left( {e}^{-{U}_{0}}\right) = 0. \n\]\n\nThen the measure \( {e}^{-{U}_{0}\left( x\right) }{dx} \) is the unique invariant probability measure of the \( A \) -... | Yes |
Theorem 6.1. Let \( r\left( t\right) = \left( {X\left( t\right), e\left( t\right) = \left( {{e}_{m}^{t}\left( t\right) }\right) }\right) \) be the horizontal Brownian motion with reflecting boundary constructed above from the solution of (6.12).\n\n(i) For any smooth function \( F\left( {t, r}\right) \) on \( \lbrack 0... | Proof. (i) is immediately obtained from Itô's formula. (ii) is easily proved once we notice\n\n\[ \n\mathop{\sum }\limits_{\alpha }{e}_{\alpha }^{i}\left( t\right) {e}_{\alpha }^{j}\left( t\right) = {g}^{ij}\left( {X\left( t\right) }\right) \n\] | Yes |
Lemma 6.1. \( \\left\\{ {P}_{r}\\right\\} \) is invariant under the action \( {T}_{a} \) of \( a, a \\in O\\left( d\\right) \) , from the right; that is, if \( w \\cdot a \\in W\\left( {O\\left( M\\right) }\\right) \) is defined for \( w \\in \) \( W\\left( {O\\left( M\\right) }\\right) \) by \( \\left( {w \\cdot a}\\r... | Proof. Let \( r\\left( t\\right) \) be a solution of (6.12) with \( B\\left( t\\right) \) and \( \\phi \\left( t\\right) \) such that \( r\\left( 0\\right) = r \) . Then for \( a \\in O\\left( d\\right) ,\\widetilde{r}\\left( t\\right) = r\\left( t\\right) \\cdot a \) is a solution of (6.12) with \( \\widetilde{B}\\lef... | Yes |
An \( {R}^{d} \otimes {R}^{d} \) -valued process \( K\left( t\right) \) adapted to \( \left( {\mathcal{F}}_{t}\right) \) is a solution of the above stochastic differential equation (6.20) with the initial condition (6.21) if and only if\n\n(6.22)\n\n\[ \begin{cases} {K}^{1}\left( t\right) \mathrel{\text{:=}} & K\left( ... | The proof is easy and omitted. | No |
Theorem 6.2. The stochastic differential equation (6.20) with the initial condition (6.21) has one and only one solution \( K\left( t\right) \in \Xi \) . | Proof. Let \( {\xi }_{n} \in \Xi, n = 0,1,\ldots \) be defined by \( {\xi }_{0} \equiv 0 \) and \( {\xi }_{n} = \) \( \Phi \left( {\xi }_{n - 1}\right), n = 1,2,\ldots \) . Using (6.30) we can show that there exists \( \xi \in \Xi \) such that\n\n\[ \n{E}_{\mu }\left\lbrack {\mathop{\sup }\limits_{{0 \leq t \leq T}}{\b... | No |
Theorem 6.3. \( M = \{ M\left( {t, w}\right) \} \) is an \( {\mathbf{R}}^{d} \otimes {\mathbf{R}}^{d} \) -valued \( {MOF} \) of the horizontal Brownian motion on \( O\left( M\right) \) with reflecting boundary;* i.e., (i) \( M\left( {t, w}\right) \) is \( \left( {\mathcal{F}}_{t}\right) \) -adapted, (ii) for every \( t... | Proof. (i) is obvious. To prove (ii), we fix \( s \) and set \( \widetilde{K}\left( t\right) = K\left( {t + s, w}\right) \) , \( \widetilde{X}\left( t\right) = X\left( {t + s, w}\right) \) and \( \bar{e}\left( t\right) = e\left( {t + s, w}\right) \) . Then it is clear that \( \widetilde{K}\left( t\right) \) satisfies t... | Yes |
Lemma 6.3. If \( X\left( 0\right) \in \partial M \), then\n\n\[ \operatorname{Pe}{\left( 0\right) }^{-1}M\left( t\right) = 0\;\text{ for all }t \geq 0. \] | Proof. It is enough to prove that \( {Pe}{\left( 0\right) }^{-1}K\left( t\right) = 0 \) for all \( t \geq 0 \) . If \( X\left( 0\right) \in \partial M \), then\n\n\[ {Pe}{\left( 0\right) }^{-1}K\left( 0\right) = {Pe}{\left( 0\right) }^{-1}\left( {{I}_{\dot{M}}\left( {X\left( 0\right) }\right) e\left( 0\right) P + e\lef... | Yes |
Theorem 6.4. (i) \( \left\{ {H}_{t}\right\} \) defines a one-parameter semigroup of operators on \( {C}_{0}\left( {O\left( M\right) \rightarrow {R}^{d}}\right) \) . (ii) If \( F \) is \( O\left( d\right) \) -equivariant, then so is \( {H}_{t}F \) for all \( t \geq 0 \) . | Proof. If \( r \in \partial O\left( M\right) \), then by Lemma 6.3 \( P{e}^{-1}\left( {{H}_{t}F}\right) \left( r\right) = 0 \) . The continuity in \( r \) of the functions \( {H}_{t}F\left( r\right), t \geq 0 \) follows from the continuity of \( r \mapsto {P}_{r} \in \mathcal{P}\left( {W\left( {O\left( M\right) }\right... | Yes |
Theorem 6.5. Let \( F\\left( {t, r}\\right) = \\left( {{F}_{t}\\left( {t, r}\\right) }\\right) \) be a smooth function on \( \\lbrack 0,\\infty ) \\times \) \( O\\left( M\\right) \) taking values in \( {\\mathbf{R}}^{d} \) such that for each \( t \\geq 0, r \\mapsto F\\left( {t, r}\\right) \) is a function in \( {C}_{0... | Proof. As we remarked above, the diffusion \( \\left( {r\\left( t\\right) }\\right) \) is a normally reflecting diffusion on \( O\\left( M\\right) \) in the sense of Definition 6.2 given below. As we shall see, a characteristic feature of such a diffusion is that if \( f\\left( {t, r}\\right) \) is a smooth function on... | Yes |
Lemma 6.5. For any \( t \) such that \( {X}_{t} \in \dot{M} \) , (6.38) \[ {dM}\left( t\right) = \frac{1}{2}M\left( t\right) J\left( {r\left( t\right) }\right) {dt} \] i.e., if \( u \in D \), then for every \( s \leq t \) such that \( \left\lbrack {s, t}\right\rbrack \subset \left( {A\left( {u - }\right), A\left( u\rig... | Proof. First we note that \( J = {eR}{e}^{-1} \) by the convention (6.19). Then by (6.20) and Itô’s formula, if \( X\left( t\right) \in \mathring{M} \) , \[ {dM}\left( t\right) = K\left( t\right) e{\left( t\right) }^{-1}\left\{ {{de}\left( t\right) e{\left( t\right) }^{-1} + e\left( t\right) d\left( {e{\left( t\right) ... | Yes |
Theorem 6.6. (i).\n\n(6.48)\n\n\[ E\left( {\mathop{\sum }\limits_{{u \in D}}{\left\lbrack {\int }_{A\left( {u - }\right) \land t}^{A\left( u\right) \land t}g\left( s\right) d{B}^{k}\left( s\right) \right\rbrack }^{2}}\right) = E\left\lbrack {{\int }_{0}^{t}g{\left( s\right) }^{2}{ds}}\right\rbrack \]\n\n\[ k = 1,2,\ldo... | Proof. We shall first prove the special case of the reflecting Brownian motion and reduce the general case to this special case.\n\n(a) The case of the reflecting Brownian motion: i.e., the case \( {\sigma }_{k}^{f}\left( x\right) = \) \( {\delta }_{k}^{i} \) with \( r = d, b\left( x\right) \equiv 0,\tau \left( x\right... | No |
Lemma 6.6. Let \( g\left( s\right) \) be a step process. Then (6.65) holds. | Proof. If, for example, \( g\left( s\right) \equiv 1 \), then (6.65) is trivially true: we have\n\n\[ \n{\int }_{A\left( {s - }\right) \land t}^{A\left( s\right) \land t}d{B}^{d}\left( u\right) = {\int }_{A\left( {s - }\right) \land t}^{A\left( s\right) \land t}d{X}^{d}\left( t\right) = {X}^{d}\left( {A\left( s\right) ... | Yes |
Lemma 6.7. Let \( g\left( s\right) \) be a \( {\mathcal{B}}_{t}\left( {W\left( D\right) }\right) \) -adapted process such that \( s \rightarrow \) \( g\left( s\right) \) is right-continuous with left-hand limits. Then for every \( \varepsilon > 0 \), there exists a step process \( {g}_{\varepsilon }\left( s\right) \) s... | Proof. Let \( \left\{ {\sigma }_{n}\right\} \) be defined by \( {\sigma }_{0} = 0 \) and\n\n\[ {\sigma }_{n} = \inf \left\{ {t > {\sigma }_{n - 1};\left| {g\left( t\right) - g\left( {\sigma }_{n - 1}\right) }\right| \geq \varepsilon }\right\} \land n. \]\n\nThen \( {\sigma }_{n} \uparrow \infty \), and \( {g}_{\varepsi... | Yes |
For \( \xi \in \partial D \) and \( t > 0 \), let\n\n\[ \n{\mu }^{\xi, t}\left( B\right) = {n}^{\xi }\left( {B \mid \sigma > t}\right) = \frac{{n}^{\xi }\left( {B : \sigma > t}\right) }{{n}^{\xi }\left( {\sigma > t}\right) }\;\text{ for }\;B \in \mathcal{B}\left( {\mathcal{W}\left( D\right) }\right) .\n\]\n\nThen \( {\... | In the above,\n\n\[ \n{K}^{\xi }\left( {s, x}\right) = \frac{{K}^{ + }\left( {s, x - \xi }\right) h\left( {t - s, x}\right) }{K\left( t\right) },\;s > 0,\;x \in \dot{D}\n\]\n\n\[ \np\left( {s, x;u, y}\right) = \frac{h\left( {t - u, y}\right) }{h\left( {t - s, x}\right) }{p}^{0}\left( {u - s, x, y}\right) ,0 < s < u < t... | Yes |
Lemma 6.9. Let \( g\left( s\right) \) be a bounded \( \left\{ {{\mathcal{B}}_{t}\left( {W\left( D\right) }\right) }\right\} \) -well measurable process. Then for fixed \( s, w \in W\left( D\right) \) and for any \( \varepsilon \geq {\varepsilon }^{\prime } > 0 \) , \[ {\int }_{\mathcal{W}\left( D\right) }\left| {{\int ... | Proof. By the corollary to Lemma 6.8, \[ {\int }_{\mathcal{W}\left( D\right) }\left| {{\int }_{0}^{{\varepsilon }^{\prime }}{\Phi }_{g}^{s}\left( {u, w,{w}^{\prime }}\right) d{w}^{\prime d}\left( u\right) }\right| {I}_{\{ \sigma \left( {w}^{\prime }\right) > \varepsilon \} }{n}^{w\left( s\right) }\left( {d{w}^{\prime }... | Yes |
Lemma 6.11. Let \( g\left( s\right) \) be a \( \left\{ {{\mathcal{B}}_{t}\left( {W\left( D\right) }\right) }\right\} \) -adapted process such that \( s \mapsto g\left( s\right) \) is right-continuous with left limits and \( s \mapsto E\left\lbrack {g{\left( s\right) }^{2}}\right\rbrack \) is locally bounded. Let \( {Y}... | Proof. First assume that \( g\left( s\right) \) is a step process. Then by Lemma 6.6\n\n\[ \n{S}_{\varepsilon }\left( t\right) \rightarrow {\int }_{0}^{t}g\left( s\right) d{B}^{d}\left( u\right) + {\int }_{0}^{t}g\left( u\right) {d\phi }\left( u\right) \;\text{ a.s.,} \n\]\n\nand by Lemma 6.10,\n\n\[ \n{M}_{\varepsilon... | Yes |
Theorem 6.7. Let \( g\left( t\right) \) be an \( \left( {\mathcal{F}}_{t}^{0}\right) \) -adapted process such that \( t \mapsto \) \( g\left( t\right) \) is right-continuous with left-hand limits and \( t \mapsto E\left\lbrack {g{\left( t\right) }^{2}}\right\rbrack \) is locally bounded. Then we have\n\n\[ \mathop{\sum... | Proof. Let \( A \) and \( L \) be defined by (6.45) and (6.46). By Itô’s formula,\n\n\[ g\left( u\right) {df}\left( {u, X\left( u\right) }\right) = g\left( u\right) \frac{\partial f}{\partial t}\left( {u, X\left( u\right) }\right) {du} \]\n\n\[ + \mathop{\sum }\limits_{{i = 1}}^{d}\mathop{\sum }\limits_{{k = 1}}^{r}g\l... | Yes |
Let \( D \) be a polydisc in \( {C}^{2} \) given by\n\n\[ D = \left\{ {z = \left( {{z}^{1},{z}^{2}}\right) \in {C}^{2};\left| {z}^{i}\right| < 1, i = 1,2}\right\} \]\n\nand let \( {z}_{0} = \left( {{z}_{0}^{1},{z}_{0}^{2}}\right) \in D \) . Let \( Z\left( t\right) = \left( {{Z}^{1}\left( t\right) ,{Z}^{2}\left( t\right... | Indeed \( \phi \left( {Z\left( t\right) }\right) \) is a bounded martingale and \( \mathop{\lim }\limits_{{t \uparrow \infty }}\phi \left( {Z\left( t\right) }\right) = \) \( \phi \left( {Z}_{\infty }\right) \) . Hence\n\n\[ \phi \left( {z}_{0}\right) = E\left\lbrack {\phi \left( {Z\left( 0\right) }\right) }\right\rbrac... | Yes |
Proposition 8.1. (i) \( \left\{ {\sqrt{a!}{H}_{a}\left( w\right) ;a \in \Lambda }\right\} \) is an \( {ONB} \) in \( {L}_{2} \) . | Proof. First recall that \( {\left. \left\{ \sqrt{n!}{H}_{n}\left( x\right) \right\} \right\} }_{n = 0}^{\infty } \) is an \( {ONB} \) in \( {\mathcal{L}}_{2}\left( {R,{N}_{1}}\right) \) , where \( {N}_{1} \) is the one-dimensional standard Gaussian measure, i.e.\n\n\[ \n{N}_{1}\left( {dx}\right) = \frac{1}{\sqrt{2\pi ... | Yes |
(i) (Monotonicity) if \( s \leqq {s}^{\prime } \) and \( 1 < {p}^{\prime } \leqq p < \infty \), then,\n\n\[ \parallel F{\parallel }_{p, s} \leqq \parallel F{\parallel }_{{p}^{\prime },{s}^{\prime }},\;F \in P\left( E\right) . \] | Proof. (i) Since \( {T}_{t} \) is a contraction semigroup on \( {L}_{p}\left( E\right) ,1 < p < \infty \) , the operator\n\n\[ {\left( I - L\right) }^{-s} = \frac{1}{\Gamma \left( s\right) }{\int }_{0}^{\infty }{e}^{-t}{t}^{s - 1}{T}_{t}{dt} \]\n\nis also a contraction on \( {L}_{p}\left( E\right) \) if \( s > 0 \) :\n... | No |
Lemma 8.1. (Commutation relations involving \( D \) ). For a real sequence \( \phi = {\left( {\phi }_{n}\right) }_{n = 0}^{\infty } \), \[ D{T}_{\phi } = {T}_{{\phi }^{ + }}D\;\text{ on }\;P\left( E\right) \] where \[ {\phi }^{ + }\left( n\right) = \phi \left( {n + 1}\right) ,\;n = 0,1,\cdots . \] | Proof. For simplicity, we only show the lemma in case of \( E = \mathbf{R} \) . By Proposition 8.1, it suffices to prove \[ D{T}_{\phi }{H}_{a}\left( w\right) = {T}_{{\phi }^{ + }}D{H}_{a}\left( w\right) \;\text{ for every }\;a \in \Lambda , \] where \[ {H}_{a}\left( w\right) = \mathop{\prod }\limits_{{j = 1}}^{\infty ... | Yes |
Proposition 8.7. \( {J}_{n} : {L}_{2} \rightarrow {C}_{n} \) is a bounded operator on \( {L}_{p}, p \in \) \( \left( {1,\infty }\right) \) . | Proof. Let \( p > 2 \) and \( t \) be the positive constant such that\n\n\[ \n{e}^{2t} = p - 1 \]\n\nThen by Proposition 8.6 we have\n\n\[ \n{\begin{Vmatrix}{T}_{t}F\end{Vmatrix}}_{p} \leqq \parallel F{\parallel }_{2}\;F \in {L}_{p}. \]\n\nIn particular\n\n\[ \n{\begin{Vmatrix}{T}_{t}{J}_{n}F\end{Vmatrix}}_{p} \leqq {\... | Yes |
Proposition 8.8. Let \( {E}_{1} \) and \( {E}_{2} \) be real separable Hilbert spaces. For any \( p, q \in \left( {1,\infty }\right) \) and \( k = 0,1,2,\cdots \) such that\n\n\[ \frac{1}{p} + \frac{1}{q} = \frac{1}{r} < 1 \]\n\nthere exists a constant \( {C}_{p, q, k} > 0 \) such that\n\n(8.46)\n\n\[ \parallel F \otim... | Proof. Indeed,\n\n\[ D\left( {F \otimes G}\right) = {DF} \otimes G + F \otimes {DG} \]\n\nand hence\n\n\[ {\left| D\left( F \otimes G\right) \right| }_{H \otimes {E}_{1} \otimes {E}_{2}} \leqq {\left| DF\right| }_{H \otimes {E}_{1}}{\left| G\right| }_{{E}_{2}} + {\left| F\right| }_{{E}_{1}}{\left| DG\right| }_{H \otime... | Yes |
Theorem 9.1. Let \( F : W \rightarrow {R}^{d} \) be a \( d \) -dimensional Wiener mapping satisfying Malliavin's conditions of (A.1) and (A.2). Then, for every \( p > 1 \) and \( k = 0,1,2,\cdots \), there exists a positive constant \( C = C\left( {p, k;F}\right) \) such that the following estimate holds: For every \( ... | Proof. Define \( q \) by \( 1/p + 1/q = 1 \). Then by the duality \[ \parallel \phi \circ F{\parallel }_{p, - {2k}} = \sup \left\{ {{\int }_{W}\phi \circ F\left( w\right) \cdot g\left( w\right) P\left( {dw}\right) ;g \in {D}_{q,{2k}},}\right. \] \[ \parallel g{\parallel }_{q,{2k}} \leqq 1\} \] and, by (9.8) and (9.9), ... | Yes |
Lemma 9.1. Let \( {\delta }_{y} \in {\mathcal{S}}^{\prime }\left( {R}^{d}\right) \) be the Dirac \( \delta \) -function at \( y \in {R}^{d} \) and \( m \) be a positive integer. Then\n\n\[ \n{\delta }_{y} \in {\mathcal{S}}_{-{2m}}\text{ and }{D}_{\alpha }{\delta }_{y} \in {\mathcal{S}}_{-{2m} - {2k}}\;\text{ if }m > \f... | Proof. First we note that the operator \( A = : {\left( 1 + {\left| x\right| }^{2} - \Delta /2\right) }^{-1} \) is defined by a kernel \( A\left( {x, y}\right) \) : \n\n\[ \n{Af}\left( x\right) = {\int }_{{R}^{d}}A\left( {x, y}\right) f\left( y\right) {dy} \n\] \n\nand \( A\left( {x, y}\right) \) is given by \n\n\[ \nA... | Yes |
Theorem 9.2. Let \( F : W \rightarrow {R}^{d} \) satisfy the conditions (A.1) and (A.2) and let \( {F}_{ * }\left( P\right) \) be the probability law of \( F \) . Then \( {F}_{ * }\left( P\right) \) has the smooth density \( {p}_{F}\left( y\right) \) with respect to the Lebesgue measure \( {dy} \) on \( {\mathbf{R}}^{d... | Proof. Let \( k \) be a non-negative integer. By the lemma above and the continuity of the mapping: \( {\mathcal{S}}_{-{2m}} \ni T \rightarrow T \circ F \in {D}_{p, - {2m}} \), we can deduce at once that the mapping:\n\n\[ \n{R}^{d} \ni y \rightarrow {\delta }_{y}\left( F\right) \in {D}_{p, - 2{m}_{0} - {2k}} \n\]\n\ni... | Yes |
Let \( r = d \) and \( F : W\left( { = {W}_{0}^{d}}\right) \rightarrow {R}^{d} \) be defined, for fixed \( t > 0 \) and \( x \in {\mathbf{R}}^{d} \), by \( F\left( w\right) = x + w\left( t\right) \) . Then \( F \in {D}_{ - }\left( {\mathbf{R}}^{d}\right) \) (actually \( \in P\left( {R}^{d}\right) ) \) and \( {\sigma }^... | \[ \mathop{\lim }\limits_{{\left| x\right| \rightarrow \infty }}k\left( x\right) /{\left| x\right| }^{2} = \alpha < \infty . \] If \( \alpha \) is sufficiently small (actually it is sufficient to assume \( \alpha < 1/{2t} \) ), we can easily see that \[ g\left( w\right) = \exp \left\{ {{\int }_{0}^{t}k\left( {x + w\lef... | Yes |
Theorem 9.4. Let \( \{ F\left( {\varepsilon, w}\right) ,\varepsilon \in (0,1\rbrack \} \) be a family of elements in \( {D}_{\infty }\left( {R}^{d}\right) \) such that it has the asymptotic expansion:\n\n(9.31)\n\n\[ F\left( {\varepsilon, w}\right) \sim {f}_{0} + \varepsilon {f}_{1} + \cdots \;\text{ in }{\mathbf{D}}_{... | Proof. We set \( F\left( {0, w}\right) = {f}_{0},{\sigma }^{ij}\left( 0\right) = < D{f}_{0}^{i}, D{f}_{0}^{j}{ > }_{H},\gamma \left( \varepsilon \right) = \left( {{\gamma }_{ij}\left( \varepsilon \right) }\right) \) : \( = {\sigma }^{-1} | Yes |
Lemma 10.1. Let \( \eta : W \rightarrow R \) be a real Wiener functional. Suppose that \( {c}_{i}, i = 0,1,2,3 \) exist such that\n\n(10.17)\n\n\[ P\left\lbrack {\left| \eta \right| < {n}^{-{c}_{0}}}\right\rbrack \leq {c}_{1}\exp \left\lbrack {-{c}_{2}{n}^{{c}_{3}}}\right\rbrack, n = 1,2,\cdots . \]\n\nThen \( \;E\left... | The proof is obvious and is ommitted. | No |
Lemma 10.2. Let \( \eta \) and \( \widetilde{\eta } \) be \( {\mathcal{S}}^{d} \) -valued Wiener functionals such that \( \eta \geqq \widetilde{\eta } \) a.s. Suppose that \( {c}_{i}, i = 0,1,2,3,4 \) exist such that \( \parallel \widetilde{\eta }\parallel \leqq {c}_{0} \) a.a.w. and\n\n(10.18)\n\[ \mathop{\sup }\limit... | Proof. Let \( \bar{W} = \left\{ {w;\eta \geq \widetilde{\eta }}\right. \) and \( \left. {\parallel \widetilde{\eta }\parallel \leq {c}_{0}}\right\} \) . Then \( P\left( \bar{W}\right) = 1 \) . Set\n\n\[ {W}_{n}\left( l\right) = \left\{ {w \in \bar{W};\widetilde{\eta }\left( l\right) \geq {n}^{-{c}_{1}}}\right\} \;\text... | Yes |
Lemma 10.3. Suppose that there exist constants \( {c}_{i}, i = 0,1,2,3,4 \) and \( \left\{ {\mathcal{B}}_{t}\right\} \) -stopping times \( 0 \leq {\sigma }_{1} \leq {\sigma }_{2} \leq 1 \) such that \( \left| {{f}_{\alpha }\left( s\right) }\right| \leq {c}_{0} \) for \( s \in \left( {{\sigma }_{1},{\sigma }_{2}}\right\... | Proof is immediate if we notice\n\n\[ \eta \left( l\right) = \mathop{\sum }\limits_{{\alpha = 1}}^{r}{\int }_{0}^{1}{\left\lbrack l \cdot {f}_{\alpha }\left( s\right) \right\rbrack }^{2}{ds} \]\n\nand choose\n\n\[ {\widetilde{\eta }}^{ij} = \mathop{\sum }\limits_{{\alpha = 1}}^{r}{\int }_{{\sigma }_{1}}^{{\sigma }_{2}}... | Yes |
Lemma 10.4. (Key lemma). Let \( \xi \left( t\right) \) be an Itô process with characteristics \( {\xi }_{\alpha }\left( t\right) ,\alpha = 0,1,\cdots, r \) . Assume furthermore, that \( {\xi }_{0}\left( t\right) \) is also an Itô process with characteristics \( {\xi }_{0,\alpha }\left( t\right) ,\alpha = 0,1,\cdots, r ... | Here we present a proof of this lemma along the lines of [203] and [232]. It should be remarked that a much simplified different proof was given recently by Norris [222]. First we give a series of probabilistic lemmas. | No |
Lemma 10.5. Let \( K > 0 \) and \( X\left( t\right) \) be a one-dimensional continuous semimartingale\n\n\[ X\left( t\right) = X\left( 0\right) + m\left( t\right) + A\left( t\right) \]\n\nsuch that \( \langle m\rangle \left( t\right) = {\int }_{0}^{t}\alpha \left( s\right) {ds}, A\left( t\right) = {\int }_{0}^{t}\beta ... | Proof.* By Theorem II-7.2', there exists a one-dimensional Brownian motion \( b\left( t\right) \left( {b\left( 0\right) = 0}\right) \) such that\n\n\[ X\left( t\right) - X\left( 0\right) = b\left( {\langle m\rangle \left( t\right) }\right) + A\left( t\right) .\n\nSince \( \{ \;\left| {X\left( t\right) - X\left( 0\right... | Yes |
Lemma 10.6. Let \( b = \left( {b\left( t\right) }\right) \) be a one-dimensional Brownian motion. Then for every \( a > 0 \) and \( \varepsilon > 0 \) , \[ P\left( {{\sigma }_{\lbrack 0, a\rbrack }\left( b\right) < \varepsilon }\right) \leq \sqrt{2}\exp \left\lbrack {-\frac{a}{{2}^{7}{\varepsilon }^{2}}}\right\rbrack .... | Proof. Without loss of generality, we may assume that \( b\left( 0\right) = 0 \) . We use the following well known expansion formula of \( b\left( t\right) \) in Fourier series: \[ b\left( t\right) = t{\xi }_{0} + \sqrt{2}\mathop{\sum }\limits_{{k = 1}}^{\infty }\left\lbrack {{\xi }_{k}\frac{\left( \cos 2\pi kt - 1\rig... | Yes |
Lemma 10.9. Let \( f\left( t\right) \) be a continuous function on \( \left\lbrack {a, b}\right\rbrack \) and set\n\n\[ g\left( t\right) = g\left( a\right) + {\int }_{a}^{t}f\left( s\right) {ds},\;t \in \left\lbrack {a, b}\right\rbrack .\n\]\n\nIf\n\n\[ \mathop{\sup }\limits_{{a \leq s < t \leq b}}\frac{\left| f\left( ... | Proof is easily provided if we note that, first \( {t}_{0} \in \left\lbrack {a, b}\right\rbrack \) exists such\n\nthat\n\n\[ \left| {f\left( {t}_{0}\right) }\right| > \varepsilon {\left( b - a\right) }^{-1/2} \]\n\nand then an interval \( I, I \subset \left\lbrack {a, b}\right\rbrack \), exists with length \( {\varepsi... | Yes |
Theorem 10.4. Let \( x \in {R}^{d} \) be fixed. Then \( {X}^{e}\left( {1, x, w}\right) \in {D}_{\infty }\left( {R}^{d}\right) \) has the asymptotic expansion\n\n\[ \n{X}^{\varepsilon }\left( {1, x, w}\right) \sim {f}_{0} + \varepsilon {f}_{1} + {\varepsilon }^{2}{f}_{2} + \cdots \;\text{ in }{D}_{\infty }\left( {\mathb... | Proof. The proof is easily provided by successive applications of the Itô formula:\n\n\[ \n{X}^{\varepsilon }\left( {1, x, w}\right) - x \n\]\n\n\[ \n= \varepsilon \mathop{\sum }\limits_{{\alpha = 1}}^{r}{\int }_{0}^{1}{V}_{\alpha }\left( {{X}^{\varepsilon }\left( s\right) }\right) \circ d{w}_{s}^{\alpha } + {\varepsil... | Yes |
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