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Theorem 3.4. \( \sigma \) (r.c.l.l.) \( \subset \mathcal{O} \) . | Proof. We show that any r.c.l.l. adapted process \( X \) is a pointwise limit on \( {\mathbb{R}}_{ + } \times \Omega \) of a sequence of optional processes \( \left\{ {X}^{n}\right\} \) . For each \( n \), the definition of \( {X}^{n} \) involves optional times \( {\tau }_{k}^{n}, k \in {\mathbb{N}}_{0} \), defined ind... | Yes |
Theorem 3.6. (Optional Projection Theorem.) For any bounded \( \mathcal{B} \times \mathcal{F} \) - measurable process \( Z \), there is an optional process \( Y \) such that for each optional time \( \tau \) , \[ E\left\lbrack {{Z}_{\tau }{1}_{\{ \tau < \infty \} } \mid {\mathcal{F}}_{\tau }}\right\rbrack = {Y}_{\tau }... | Proof. For a proof via a monotone class argument, let \( \mathcal{Z} \) denote the class of all bounded \( \mathcal{B} \times \mathcal{F} \) -measurable \( Z \) for which there is such a \( Y \) . The collection of sets of the form \( \lbrack 0, s) \times F \), where \( s \in {\mathbb{R}}_{ + } \) and \( F \in \mathcal... | Yes |
Theorem 3.7. Any \( \mathcal{B} \times \mathcal{F} \) -measurable adapted process is \( {\mathcal{P}}^{ * } \) -measurable. | Proof. First we prove \( \mathcal{O} \subset {\mathcal{P}}^{ * } \), from which it follows that any optional process is \( {\mathcal{P}}^{ * } \) -measurable. By Lemma 2.2, \( \mathcal{O} \) is generated by the stochastic intervals of the form \( \lbrack \tau ,\infty ) \), and by Lemma 2.1, the stochastic interval \( \... | Yes |
Theorem 4.1. Let \( t \in {\mathbb{R}}_{ + } \) and \( \left\{ {{\pi }_{t}^{n}, n \in \mathbb{N}}\right\} \) be a sequence of partitions of \( \left\lbrack {0, t}\right\rbrack \) such that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\delta {\pi }_{t}^{n} = 0 \) . Suppose \( M \) is a continuous local martingale a... | Proof of (i). Suppose \( M \) is bounded. Then \( M \) is a martingale by Proposition 1.8. We have\n\n\[ \n{S}_{t}^{n} = \mathop{\sum }\limits_{j}\left\{ {{\left( {M}_{{t}_{j + 1}}\right) }^{2} - {\left( {M}_{{t}_{j}}\right) }^{2} - 2{M}_{{t}_{j}}\left( {{M}_{{t}_{j + 1}} - {M}_{{t}_{j}}}\right) }\right\}\n\]\n\n(4.3)\... | Yes |
Theorem 4.2. Let \( M \) be a continuous \( {L}^{2} \) -martingale. Then the following hold.\n\n(i) \( \;\left\lbrack M\right\rbrack = \left\{ {{\left\lbrack M\right\rbrack }_{t}, t \in {\mathbb{R}}_{ + }}\right\} \) is a continuous integrable increasing process with \( {\left\lbrack M\right\rbrack }_{0} = 0 \) . | Proof. For part (i), the continuity and adaptedness of \( \left\lbrack M\right\rbrack \) and the fact that \( {\left\lbrack M\right\rbrack }_{0} = 0 \) follow from (4.1). To prove integrability, for fixed \( t \) let \( {S}_{t}^{n} \) be defined as in Theorem 4.1. By the \( {L}^{2} \) -integrability of \( M \), the ort... | Yes |
Lemma 4.4. Let \( V \) be a continuous \( {L}^{2} \) -martingale that is locally of bounded variation. Then\n\n\[ P\left( {{V}_{t} = {V}_{0}\;\text{ for all }t \in {\mathbb{R}}_{ + }}\right) = 1. \] | Proof. Let \( \widehat{V}\left( t\right) = V\left( t\right) - V\left( 0\right) \) . By the path continuity of \( V \), it suffices to prove \( P\left( {\widehat{V}\left( t\right) = 0}\right) = 1 \) for each \( t \) . Since \( \widehat{V} \) is continuous and locally of bounded variation, by ordinary calculus applied pa... | Yes |
Corollary 4.5. Let \( V \) be a continuous local martingale that is locally of bounded variation. Then\n\n\[ P\left( {{V}_{t} = {V}_{0}\;}\right. \text{for all}\left. {t \in {\mathbb{R}}_{ + }}\right) = 1\text{.} \] | Proof. Apply Lemma 4.4 to appropriate localizations \( {\widehat{V}}^{k} \) of \( V - {V}_{0} \) and then let \( k \rightarrow \infty \) . | No |
Theorem 4.6. Let \( M \) be a continuous \( {L}^{2} \) -martingale. Then there is a a unique decomposition of \( {\left( M\right) }^{2} \) as the sum of a continuous martingale and a continuous integrable increasing process with initial value zero. This decomposition is given by | \[ {\left( {M}_{t}\right) }^{2} = \left( {{\left( {M}_{0}\right) }^{2} + 2{\int }_{0}^{t}{MdM}}\right) + {\left\lbrack M\right\rbrack }_{t}\;\text{ for all }\;t \geq 0. \] | Yes |
Theorem 4.7. Let \( M \) be a continuous local martingale. Then there is a unique decomposition of \( {\left( M\right) }^{2} \) as the sum of a continuous local martingale and a continuous increasing process with initial value zero. | This decomposition is given by (4.14). | No |
Theorem 4.8. Let \( M \) be a continuous local martingale and let \( Y \) be a bounded continuous adapted process. Let \( t \in {\mathbb{R}}_{ + } \) and \( \left\{ {{\pi }_{t}^{n}, n \in \mathbb{N}}\right\} \) be a sequence of partitions of \( \left\lbrack {0, t}\right\rbrack \) such that \( \mathop{\lim }\limits_{{n ... | Proof. For each \( n \in \mathbb{N} \), define\n\n\[ \n{W}_{n} = \mathop{\sum }\limits_{j}{Y}_{{t}_{j}}\left( {{\left\lbrack M\right\rbrack }_{{t}_{j + 1}} - {\left\lbrack M\right\rbrack }_{{t}_{j}}}\right) \n\]\n\nwhere the sum is over all \( j \) such that \( {t}_{j},{t}_{j + 1} \in {\pi }_{t}^{n} \) . The sequence \... | Yes |
Theorem 5.1. Let \( M \) be a continuous local martingale and \( V \) be a continuous process which is locally of bounded variation. Let \( f \) be a continuous real-valued function defined on \( {\mathbb{R}}^{2} \) such that the partial derivatives \( \frac{\partial f}{\partial x}\left( {x, y}\right) ,\frac{{\partial ... | Proof of Theorem 5.1. Since both sides of the equality in (5.2) are continuous processes, it suffices to prove for each \( t \) that (5.2) holds a.s. Let \( \left\{ {{\pi }_{t}^{n}, n \in \mathbb{N}}\right\} \) be a sequence of partitions of \( \left\lbrack {0, t}\right\rbrack \) such that \( \mathop{\lim }\limits_{{n ... | Yes |
Theorem 5.2. Let \( M \) and \( N \) be continuous local martingales. Then there is a unique decomposition of \( {MN} \) as the sum of a continuous local martingale and a continuous process which is locally of bounded variation and has initial value zero. This decomposition is given for each \( t \) by:\n\n\[{\left( MN... | Proof. The existence of the decomposition follows from (5.7), since the right member of that equation is a local martingale and \( \left\lbrack {M, N}\right\rbrack \) has the properties stated above and its initial value is zero. Indeed, by substituting the expressions for \( {\left( M \pm N\right) }^{2} - \left\lbrack... | Yes |
Theorem 5.3. Let \( M \) and \( N \) be continuous \( {L}^{2} \) -martingales. Then there is a unique decomposition of \( {MN} \) as the sum of a continuous martingale and a continuous integrable process which is locally of bounded variation and has initial value zero. Moreover, for \( 0 \leq s < t \) we have\n\n\[ E\l... | Proof. The existence of the decomposition was explained above and the uniqueness follows from Theorem 5.2. The first equality in (5.8) follows on taking conditional expectations relative to \( {\mathcal{F}}_{s} \) of the identity\n\n\[ \left( {{M}_{t} - {M}_{s}}\right) \left( {{N}_{t} - {N}_{s}}\right) = {M}_{t}{N}_{t}... | Yes |
Corollary 5.4. Let \( M \) and \( N \) be continuous local martingales and \( \tau \) be an optional time. For each \( t \) let \( {M}_{t}^{\tau } = {M}_{t \land \tau } \) and \( {N}_{t}^{\tau } = {N}_{t \land \tau } \) . Then a.s. for each \( t \) we have\n\n(5.9)\n\n\[{\left\lbrack {M}^{\tau },{N}^{\tau }\right\rbrac... | Proof. By the decomposition of \( {MN},{\left( {M}^{\tau }{N}^{\tau }\right) }_{t} \) is the sum of a continuous local martingale evaluated at time \( t \land \tau \), and \( {\left\lbrack M, N\right\rbrack }_{t \land \tau } \) . Then (5.9) follows by the uniqueness of the decomposition of \( {M}^{\tau }{N}^{\tau } \) ... | No |
Theorem 5.5. Let \( M \) and \( N \) be continuous local martingales. For each \( t \) let \( \left\{ {{\pi }_{t}^{n}, n \in \mathbb{N}}\right\} \) be a sequence of partitions of \( \left\lbrack {0, t}\right\rbrack \) such that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\delta {\pi }_{t}^{n} = \) 0 . Then, as \(... | Proof. From the definition of \( \left\lbrack {M, N}\right\rbrack \) and Theorem 4.1 we have\n\n\[ {\left\lbrack M, N\right\rbrack }_{t} = \frac{1}{4}\left( {{\left\lbrack M + N\right\rbrack }_{t} - {\left\lbrack M - N\right\rbrack }_{t}}\right) \]\n\n\[ = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{4}\math... | Yes |
Theorem 5.7. Let \( M \) and \( N \) be continuous local martingales, \( X \in \) \( \Lambda \left( {\mathcal{P}, M}\right) \) and \( Y \in \Lambda \left( {\mathcal{P}, N}\right) \) . Then a.s. we have for all \( t \) :\n\n(5.14)\n\n\[{\left\lbrack X \cdot M, Y \cdot N\right\rbrack }_{t} = {\int }_{0}^{t}{X}_{s}{Y}_{s}... | Proof. By replacing \( M \) and \( N \) by \( M - {M}_{0} \) and \( N - {N}_{0} \), respectively, and using a localizing sequence, we may suppose that \( M \) and \( N \) are continuous \( {L}^{2} \) -martingales and \( X \in {\Lambda }^{2}\left( {\mathcal{P}, M}\right) \), and \( Y \in {\Lambda }^{2}\left( {\mathcal{P... | Yes |
Theorem 5.10. Let \( m \in \mathbb{N} \) and \( n \in \mathbb{N} \) . Let \( {M}^{i} \) be a continuous local martingale for \( 1 \leq i \leq m \), and \( {V}^{k} \) be a continuous process which is locally of bounded variation for \( 1 \leq k \leq n \) . Suppose that \( D \) is a domain in \( {\mathbb{R}}^{m + n} \) s... | Sketch of proof. The method of proof is similar to that for the one-dimensional formula. The main differences are that a theorem analogous to Theorem 4.8 must be proved for mutual variation using (5.10); and the functions \( {g}_{n} \) with bounded partial derivatives which are used to approximate \( f \) must be chose... | Yes |
A process \( M = \left\{ {{M}_{t}, t \in {\mathbb{R}}_{ + }}\right\} \) is a Brownian motion in \( \mathbb{R} \) if and only if there is a standard filtration \( \left\{ {\mathcal{F}}_{t}\right\} \) such that \( \left\{ {{M}_{t},{\mathcal{F}}_{t}, t \in {\mathbb{R}}_{ + }}\right\} \) is a continuous local martingale wi... | Proof. For the \ | No |
Theorem 6.2. Let \( M \) and \( A \) be continuous adapted processes such that \( A \) is increasing and \( {A}_{0} = 0 \) . For each \( \alpha \in \mathbb{R} \), let \( {Z}^{\alpha } \) be the process defined by\n\n\[ \n{Z}_{t}^{\alpha } = \exp \left( {\alpha {M}_{t} - \frac{1}{2}{\alpha }^{2}{A}_{t}}\right) \n\]\n\nT... | Proof. Suppose (i) holds. To prove that (ii) follows, we apply the Itô formula to \( f\left( {x, y}\right) = \exp \left( {{\alpha x} - \frac{1}{2}{\alpha }^{2}y}\right) \), to obtain a.s.:\n\n\[ \nf\left( {{M}_{t},{A}_{t}}\right) - f\left( {{M}_{0},{A}_{0}}\right) = {\int }_{0}^{t}{\alpha f}\left( {{M}_{s},{A}_{s}}\rig... | Yes |
Lemma 6.3. Suppose conditions (a) and (b) hold. Then \( M \) is an \( {L}^{2} \) - martingale with \( \left\lbrack M\right\rbrack = A \) . | Proof. For \( 0 \leq s < t, F \in {\mathcal{F}}_{s} \), and \( \left| \alpha \right| < \frac{1}{2}{\alpha }_{0} \), since \( {Z}^{\alpha } \) is a martingale we have\n\n(6.8)\n\n\[ \n{\int }_{F}\exp \left( {\alpha {M}_{s} - \frac{1}{2}{\alpha }^{2}{A}_{s}}\right) {dP} = {\int }_{F}\exp \left( {\alpha {M}_{t} - \frac{1}... | Yes |
Theorem 6.4. Let \( M \) and \( A \) be continuous adapted processes such that \( A \) is increasing and \( {A}_{0} = 0 \) . Suppose conditions (a) and (b) of Theorem 6.2 are satisfied. Then for each \( n \in {I}_{0},{H}_{n}\left( {M, A}\right) \) is an \( {L}^{2} \) -martingale. | Proof. For \( n = 0,{H}_{0}\left( {M, A}\right) \equiv 1 \) is clearly an \( {L}^{2} \) -martingale. Let \( n \in \mathbb{N} \) . Then there is a constant \( {K}_{n} > 0 \) such that for all \( x \in \mathbb{R} \) ,\n\n\[{\left| x\right| }^{m} \leq {K}_{n}\exp \left( {{\alpha }_{0}\left| x\right| /2}\right) \;\text{ fo... | Yes |
Theorem 6.8. Let \( \psi \) be a solution of the Schrödinger equation in \( D \) and suppose that \( \psi \in C\left( \bar{D}\right) \) . Further suppose that \( {u}_{D}\left( x\right) \equiv {E}^{x}\left\{ {e\left( {\tau }_{D}\right) }\right\} < \infty \) for some \( x \in D \) . Then (6.26) holds for all \( x \) in \... | Proof. Suppose \( x \in D \) and let \( \left\{ {{E}_{n}, n \in \mathbb{N}}\right\} \) be a sequence of domains such that \( x \in {E}_{n} \subset {E}_{n + 1} \subset \subset D \) for all \( n \) and \( \mathop{\bigcup }\limits_{n}{E}_{n} = D \) . By Proposition 6.7, \( {u}_{D}\left( x\right) < \infty \) and\n\n\[ {m}_... | Yes |
Corollary 6.9. Let \( \psi \) be a solution of the Schrödinger equation in \( D \) . Let \( E \) be a domain such that \( E \subset \subset D \) . Suppose that\n\n\[ \n{u}_{E}\left( x\right) = {E}^{x}\left\{ {e\left( {\tau }_{E}\right) }\right\} < \infty \;\text{ for some }x \in E.\n\]\n\nThen (6.25) holds for all \( x... | Proof. Since \( \psi \) is twice continuously differentiable in a domain containing \( \bar{E} \), it is continuous on \( \bar{E} \) and so for \( x \in \bar{E},\left( {6.25}\right) \) follows immediately from Theorem 6.8 applied to \( \bar{E} \) . For \( x \in D \smallsetminus \bar{E},{\tau }_{E} = 0{P}^{x} \) -a.s. a... | Yes |
Theorem 6.10. Suppose there is a solution \( \phi \in C\left( \bar{D}\right) \) of the Schrödinger equation in \( D \) such that \( \phi > 0 \) in \( \bar{D} \) . Then \( {u}_{D}\left( x\right) < \infty \) for all \( x \in D \) . | Proof. By Lemma 6.6, for any domain \( E \subset \subset D, x \in D \), and \( t \in {\mathbb{R}}_{ + } \) ,\n\n\[ \phi \left( x\right) = {E}^{x}\left\{ {e\left( {t \land {\tau }_{E}}\right) \phi \left( {\mathbf{B}\left( {t \land {\tau }_{E}}\right) }\right) }\right\} \geq {E}^{x}\left\{ {e\left( {t \land {\tau }_{E}}\... | Yes |
Lemma 7.2. There exists a family of random variables \( \{ J\left( {t, x}\right) ,\left( {t, x}\right) \in \) \( \left. {{\mathbb{R}}_{ + } \times \mathbb{R}}\right\} \) and a set \( {\Omega }_{0} \) with \( P\left( {\Omega }_{0}\right) = 1 \) such that \( \left( {t, x}\right) \rightarrow J\left( {t, x}\right) \left( \... | By shrinking the sample space to \( {\Omega }_{0} \) we may and do assume that \( J \) is continuous for all \( \omega \) . We now redefine \( L \) by means of (7.5), using \( J \) as the version of the stochastic integral there. | No |
Corollary 7.4. Let \( f \) be Borel measurable and locally integrable on \( {IR} \) , then we have for each \( t \), almost surely:\n\n(7.14)\n\n\[{\int }_{-\infty }^{\infty }L\left( {t, x}\right) f\left( x\right) {dx} = {\int }_{0}^{t}f\left( {B}_{s}\right) {ds}\] | We observe that the integral on the left side of (7.14) is actually over a bounded set of \( x \), since for \( t \) and \( \omega \) fixed, by (7.1) the set of \( x \) for which \( L\left( {t, x}\right) \left( \omega \right) \neq 0 \) is bounded, because the range of \( B\left( {s,\omega }\right) \) for \( s \in \left... | Yes |
Theorem 7.5. For each \( \left( {t, x}\right) \), we have a.s.\n\n\[ \left| {{B}_{t} - x}\right| - \left| {{B}_{0} - x}\right| = {\int }_{0}^{t}\operatorname{sgn}\left( {{B}_{s} - x}\right) d{B}_{s} + L\left( {t, x}\right) . \] | Proof. Since \( - B \) is a Brownian motion, it has a local time at \( - x \) which will be denoted by \( {L}^{ - }\left( {t, - x}\right) \) . By applying (7.1) to it, we see at once that \( {L}^{ - }\left( {t, - x}\right) = L\left( {t, x}\right) \) a.s. By combining this with (7.5) applied to \( - B \) and \( - x \), ... | Yes |
Theorem 7.6. For each \( x \), we have a.s.\n\n\[ \left| {B - x}\right| = \widehat{B}\left( {\cdot, x}\right) + L\left( {\cdot, x}\right) \]\n\nwhere \( \widehat{B}\left( {\cdot, x}\right) \) is a Brownian motion and \( L\left( {\cdot, x}\right) \) is a continuous increasing process with initial value zero. Moreover, a... | Proof. Since both sides of (7.15) are continuous in \( t \) ,(7.17) follows immediately. By (7.16), \( \widehat{B}\left( {t, x}\right) \) is the sum of its initial r.v. and the continuous \( {L}^{2} \) -martingale \( {\int }_{0}^{t}\operatorname{sgn}\left( {{B}_{s} - x}\right) d{B}_{s} \) . The quadratic variation of t... | Yes |
Lemma 8.1. Let \( x \in \mathcal{C} \) with \( x\left( 0\right) \geq 0 \) . Then \( {PR}\left( x\right) \) has a unique solution given by \( \left( {z, y}\right) \) where\n\n(8.2)\n\n\[ z = x + y;\;y\left( t\right) = \mathop{\max }\limits_{{0 \leq s \leq t}}{x}^{ - }\left( s\right) \text{ for each }t \in {\mathbb{R}}_{... | Proof. We shall first verify that \( \left( {z, y}\right) \), defined by \( \left( {8.2}\right) \), is a solution of \( {PR}\left( x\right) \) . Clearly, \( y \) and \( z \) are continuous and (i) holds. Condition (ii) is easily verified as follows:\n\n\[ z\left( t\right) = x\left( t\right) + y\left( t\right) \geq x\le... | Yes |
Theorem 8.2. Let \( \phi \in {C}^{2}\left( \mathbb{R}\right) \) and \( \alpha \geq 0 \) . Then a.s. for all \( t \) we have\n\n\[ \n{e}^{-{\alpha t}}\phi \left( {Z}_{t}\right) - \phi \left( {Z}_{0}\right) = {\int }_{0}^{t}{e}^{-{\alpha s}}{\phi }^{\prime }\left( {Z}_{s}\right) d{\widehat{B}}_{s} + {\int }_{0}^{t}{e}^{-... | Proof. This follows by applying Theorem 5.10 with \( {M}_{t} = {\widehat{B}}_{t},{V}_{t}^{1} = {L}_{t} \) , \( {V}_{t}^{2} = {e}^{-{\alpha t}} \), domain \( D = {\mathbb{R}}^{3} \), and function \( f \) defined by \( f\left( {x,{y}_{1},{y}_{2}}\right) = \) \( {y}_{2}\phi \left( {x + {y}_{1}}\right) \) for all \( \left(... | Yes |
Corollary 8.3. Suppose \( \phi \in {C}^{2}\left( \mathbb{R}\right), h \in C\left( {\mathbb{R}}_{ + }\right) \), and \( \alpha \geq 0 \), are such that\n\n(8.9)\n\n\[ \n{\alpha \phi } - \frac{1}{2}{\phi }^{\prime \prime } = h\text{ on }{\mathbb{R}}_{ + }\;\text{ and }\;{\phi }^{\prime }\left( 0\right) = 0.\n\]\n\nThen \... | Proof. Since \( L \) can only increase when \( Z \) is zero and \( {\phi }^{\prime }\left( 0\right) = 0 \), the integral with respect to \( d{L}_{s} \) in (8.8) is zero for all \( t \) . Then it follows from (8.8) and (8.9), since \( Z \geq 0 \), that\n\n(8.11)\n\n\[ \n{e}^{-{\alpha t}}\phi \left( {Z}_{t}\right) - \phi... | Yes |
Theorem 8.4. For each \( t,{W}_{t}^{n} \) converges in distribution to \( {Z}_{t} \) as \( n \rightarrow \infty \) . | Sketch of proof. The main omission in the following is of details relating to weak convergence in the function space \( D\left\lbrack {0, m}\right\rbrack \) of all functions defined on \( \left\lbrack {0, m}\right\rbrack, m \in {IN} \), which are right continuous on \( \lbrack 0, m) \) and have finite left limits on \(... | No |
Lemma 8.5. For each \( n \in \mathbb{N}_{0} \), (8.19) \[ {W}_{n}^{1} = {X}_{n}^{1} + {Y}_{n}^{1} \;\text{ and } \] (8.20) \[ {W}_{n}^{2} = {X}_{n}^{2} - {Y}_{n}^{1} + {Y}_{n}^{2} \] | Proof. Since \( {W}_{n}^{1} \) is defined in the same recursive manner as \( {W}_{n} \) in Example 1, (8.19) follows from (8.14). We shall use (8.19) and induction to prove (8.20).\n\nWhen \( n = 0 \), both sides of (8.20) are zero. Suppose \( n > 0 \) and (8.20) holds with \( n - 1 \) in place of \( n \). Then by (8.1... | Yes |
Theorem 8.6. For each \( t \), as \( n \rightarrow \infty \):\\n\\n\[\\n{\\mathbf{W}}_{t}^{n} \\rightarrow {\\mathbf{Z}}_{t}\\;\\text{ in dist. }\\]\\n | Sketch of proof. The method of proof is similar to that for Theorem 8.4. Since the sequences \( \\left\\{ {{X}_{n}^{1}, n \\in \\mathbb{N}}\\right\\} \) and \( \\left\\{ {{X}_{n}^{2}, n \\in \\mathbb{N}}\\right\\} \) are independent, it follows by applying the usual central limit theorem to each component of \( {\\math... | Yes |
Theorem 8.7. Let \( \psi \) be a twice continuously differentiable function defined on \( {\mathbb{R}}^{2} \) and \( \alpha \geq 0 \) . Then a.s. for all \( t \) we have:\n\n\[ \n{e}^{-{\alpha t}}\psi \left( {\mathbf{Z}}_{t}\right) - \psi \left( {\mathbf{Z}}_{0}\right) = \mathop{\sum }\limits_{{j = 1}}^{2}{\int }_{0}^{... | Proof. This follows by applying Theorem 5.10 with \( {M}_{t}^{1} = {B}_{t}^{1},{M}_{t}^{2} = {B}_{t}^{2} \) , \( {V}_{t}^{1} = {L}_{t}^{1},{V}_{t}^{2} = {L}_{t}^{2},{V}_{t}^{3} = {e}^{-{\alpha t}} \), domain \( D = {\mathbb{R}}^{5} \), and function \( f \) defined by\n\n\[ \nf\left( {{x}_{1},{x}_{2},{y}_{1},{y}_{2},{y}... | Yes |
Corollary 8.8. Suppose that \( \psi \) is a twice continuously differentiable function on \( {\mathbb{R}}^{2}, h \) is a continuous function on \( {\mathbb{R}}_{ + }^{2} \), and \( \alpha \geq 0 \), such that\n\n(8.25)\n\n\[ \n{\alpha \psi } - \frac{1}{2}{\Delta \psi } = h\;\text{ on }{\mathbb{R}}_{ + }^{2} \n\]\n\nand... | Proof. Since \( {L}^{j} \) increases only when \( {Z}^{j} \) is zero, for \( j = 1,2 \), it follows from (8.26) that the second and third integrals in (8.24) are zero. The remainder of the proof is similar to that of Corollary 8.3.\n\nWe can rewrite (8.26) in the vector form\n\n\[ \n\left\{ \begin{array}{ll} \nabla \ps... | Yes |
Lemma 9.1. A convex function \( f \) is continuous. Moreover, \( {D}^{ + }f \) and \( {D}^{ - }f \) are defined everywhere on \( \mathbb{R} \), they are both increasing functions, \( {D}^{ + }f \) is right continuous and \( {D}^{ - }f \) is left continuous. Furthermore, \( {D}^{ + }f \) equals \( {D}^{ - }f \) except o... | For a proof of this lemma, see e.g., loc. cit. pp. 3-7. | No |
Theorem 9.4. For the continuous local martingale \( \\left\\{ {{M}_{t},{\\mathcal{F}}_{t}, t \\in {\\mathbb{R}}_{ + }}\\right\\} \), we have \( P \) -a.s.\n\n(9.16)\n\n\[ \n\\left\\{ {\\mathop{\\lim }\\limits_{{t \\rightarrow \\infty }}{M}_{t}\\text{ exists and is finite }}\\right\\} = \\left\\{ {{\\left\\[ M\\right\\]... | Proof. As in the proof of Theorem 9.3, we may and do suppose that \( {M}_{0} = 0 \) . We prove (9.16) first, using an argument adapted from Sharpe [70, Theorem 1.11]. For each positive integer \( k \), define\n\n\[ \n{\\sigma }_{k} = \\inf \\left\\{ {t \\geq 0 : \\left| {M}_{t}\\right| \\geq k}\\right\\} \n\]\n\nThen f... | Yes |
Theorem 9.5. \( \left\{ {{B}_{{\tau }_{t}}, t \in {\mathbb{R}}_{ + }}\right\} \) is a continuous process equivalent in law to \( \left\{ {\left| {B}_{t}\right|, t \in {\mathbb{R}}_{ + }}\right\} \) | Proof. Set \( x = 0 \) in (7.5). Since \( {B}_{0} \geq 0 \) this gives\n\n(9.24)\n\n\[ \n{B}_{t}^{ + } = {B}_{0} + {\int }_{0}^{t}{1}_{\left\{ {B}_{s} \geq 0\right\} }d{B}_{s} + \frac{1}{2}{L}_{t} \n\] \n\nwhere \( {L}_{t} = L\left( {t,0}\right) \) is defined by (7.7). Let \( {Y}_{t} = {\int }_{0}^{t}{1}_{\left\{ {B}_{... | Yes |
Lemma 9.6. Suppose \( \tau \) is an optional time. Let \( {Q}_{\tau } \) and \( {P}_{\tau } \) denote the restrictions of \( Q \) and \( P \), respectively, to \( {\mathcal{F}}_{\tau } \) . Then\n\n\[ \n{\rho }_{\tau } = \frac{d{Q}_{\tau }}{d{P}_{\tau }} \n\]\n\nHere \( {\rho }_{\tau }\left( \omega \right) = {\rho }_{\... | Proof. By Doob’s stopping theorem, \( {\rho }_{\tau } = {E}^{P}\left\lbrack {\rho \mid {\mathcal{F}}_{\tau }}\right\rbrack \) . Thus, for \( F \in {\mathcal{F}}_{\tau } \),\n\n\[ \n{E}^{{P}_{\tau }}\left\lbrack {{\rho }_{\tau }{1}_{F}}\right\rbrack = {E}^{P}\left\lbrack {{\rho }_{\tau }{1}_{F}}\right\rbrack = {E}^{P}\l... | Yes |
Theorem 9.7. Let \( \\left\\{ {{M}_{t}, t \\in {\\mathbb{R}}_{ + }}\\right\\} \) be a right continuous stochastic process on \( \\left( {\\Omega ,\\mathcal{F}}\\right) \) with \( {M}_{t} \\in {\\mathcal{F}}_{t} \) for all \( t \\in {\\mathbb{R}}_{ + } \) . Then, \( \\left\\{ {{\\rho }_{t}{M}_{t},{\\mathcal{F}}_{t}, t \... | Proof. Without loss of generality, we may and do assume that \( {M}_{0} = 0 \), since \( \\left\\{ {{\\rho }_{t}{M}_{0},{\\mathcal{F}}_{t}, t \\in {\\mathbb{R}}_{ + }}\\right\\} \) is a \( P \) -local martingale and the random variable \( {M}_{0} \) defines a \( Q \) -local martingale. For the proof of the \ | No |
Proposition 9.9. Any consistent family \( \left\{ {{\nu }_{t}, t \in {\mathbb{R}}_{ + }}\right\} \) of probability measures defined on \( \left\{ {{\mathcal{F}}_{t}^{o}, t \in {\mathbb{R}}_{ + }}\right\} \) can be uniquely extended to a probability measure on \( {\mathcal{F}}^{o} \) . | Proof. The uniqueness is clear, since the \( {\mathcal{F}}_{t}^{o} \) ’s generate \( {\mathcal{F}}^{o} \) . For the existence, note that by Kolmogorov's extension theorem there is a unique probability measure on the product space \( {\mathbb{R}}^{\lbrack 0,\infty )} \) whose finite dimensional distributions are consist... | Yes |
Example 2. (Ornstein-Uhlenbeck process.) Assume \( b\left( x\right) = {\alpha x} \) for some \( \alpha \in \mathbb{R} \smallsetminus \{ 0\} \), and \( {X}_{0} \in {L}^{2} \) . Then \( b \) is locally bounded, it satisfies (9.42), | \[ {\rho }_{t} = \exp \left( {\alpha {\int }_{0}^{t}{X}_{s}d{X}_{s} - \frac{{\alpha }^{2}}{2}{\int }_{0}^{t}{X}_{s}^{2}{ds}}\right) \] defines a martingale under \( P \), and on \( \left( {\Omega ,{\mathcal{F}}^{o}, Q}\right) ,{X}_{t} = {B}_{t} + \alpha {\int }_{0}^{t}{X}_{s}{ds} \) , where \( B \) is a Brownian motion... | No |
For each \( T > 0 \) there exists a constant \( {C}_{T} \) such that for any \( Y \) and \( Z \) that are \( \left( {\mathcal{B} \times \mathcal{F}}\right) \) -measurable adapted \( d \) -dimensional processes satisfying \( {Y}_{0} - {Z}_{0} \in {L}^{2} \), we have for each \( t \in \left\lbrack {0, T}\right\rbrack \) ... | Proof. Without loss of generality we may suppose the last written expectation is finite. Then by the definition of \( \widetilde{Y} \) and \( \widetilde{Z} \), we have\n\n\[ {\left| {\widetilde{Y}}_{t} - {\widetilde{Z}}_{t}\right| }^{2} \leq 3\left\{ {{\left| {Y}_{0} - {Z}_{0}\right| }^{2} + {\left| {\int }_{0}^{t}\lef... | Yes |
Lemma 10.2. Suppose \( f \) and \( g \) are Lebesgue integrable in \( \left\lbrack {0, T}\right\rbrack \) for some \( T \in \left( {0,\infty }\right) \) and there is a constant \( C > 0 \) such that\n\n\[ f\left( t\right) \leq g\left( t\right) + C{\int }_{0}^{t}f\left( s\right) {ds}\text{ for all }t \in \left\lbrack {0... | In particular, if there is a constant \( A \) such that \( g\left( t\right) = A \) for all \( t \in \left\lbrack {0, T}\right\rbrack \) , then\n\n(10.14)\n\n\[ f\left( t\right) \leq A{e}^{Ct}\text{ for all }t \in \left\lbrack {0, T}\right\rbrack . \] | No |
Theorem 10.3. There is at most one \( \left( {\mathcal{B} \times \mathcal{F}}\right) \) -measurable adapted solution \( X \) of (10.4). | Proof. If \( Y \) and \( Z \) are two solutions of (10.4) then \( {Y}_{0} = {X}_{0},{Z}_{0} = {X}_{0} \) and \( {Y}_{t} = {\widetilde{Y}}_{t},{Z}_{t} = {\widetilde{Z}}_{t} \) for all \( t \) . Then for any \( T > 0 \), by (10.9) and Fubini’s theorem, we have\n\n\[ E\left\lbrack {\left| {Y}_{t} - {Z}_{t}\right| }^{2}\ri... | Yes |
Theorem 10.5. Suppose \( \sigma \) and \( b \) are bounded and satisfy the uniform Lipschitz conditions (10.6)-(10.7). Let \( B \) be a Brownian motion martingale on a filtered probability space \( \left( {\Omega ,\mathcal{F},\left\{ {\mathcal{F}}_{t}\right\}, P}\right) \) such that \( B\left( 0\right) = 0, P \) -a.s.,... | \[ E\left\lbrack {\mathop{\sup }\limits_{{0 \leq s \leq T}}{\left| X\left( s\right) - Y\left( s\right) \right| }^{2}}\right\rbrack \leq {3E}\left\lbrack {\left| X\left( 0\right) - Y\left( 0\right) \right| }^{2}\right\rbrack {e}^{{C}_{T}T}. \] | Yes |
Lemma 10.7. Fix \( n \in \mathbb{N} \) . Let \( {\sigma }_{Y},{\sigma }_{Z} : {\mathbb{R}}^{d} \rightarrow {\mathbb{R}}^{d} \otimes {\mathbb{R}}^{r} \) and \( {b}_{Y},{b}_{Z} \) : \( {\mathbb{R}}^{d} \rightarrow {\mathbb{R}}^{d} \) be continuous functions such that \( {\sigma }_{Y} = {\sigma }_{Z} = \sigma ,{b}_{Y} = {... | Proof. By subtracting (10.30) from (10.29) and stopping at the time \( \tau \), we obtain for all \( t \geq 0 \) ,\n\n\[ Y\left( {t \land \tau }\right) - Z\left( {t \land \tau }\right) \]\n\n(10.31)\n\n\[ = Y\left( 0\right) - Z\left( 0\right) + {\int }_{0}^{t \land \tau }\left( {{\sigma }_{Y}\left( {Y\left( s\right) }\... | Yes |
Corollary 10.8. Suppose all of the hypotheses of Theorem 10.6 hold and in addition that \( X\\left( 0\\right) \\in {L}^{2} \) . Then \( X\\left( t\\right) \\in {L}^{2} \) for all \( t \\geq 0 \) . | Proof. This follows immediately on letting \( n \\rightarrow \\infty \) in (10.34) and invoking Fatou’s lemma. ∎ | Yes |
Lemma 10.10. Let \( X \) be a solution of (10.4). Then for any \( f \in {C}_{b}\left( {\mathbb{R}}^{d}\right) \) and \( t \geq 0 \) we have\n\n\[ E\left\lbrack {f\left( {X}_{t}\right) \mid {X}_{0}}\right\rbrack = {E}^{{X}_{0}}\left\lbrack {f\left( {w}_{t}\right) }\right\rbrack \] | Proof. Observe that by Lemma 10.9, the right member above is measurable with respect to the \( \sigma \) -field generated by \( {X}_{0} \) . | No |
Theorem 10.12. For each \( x \in {\mathbb{R}}^{d}, f \in {C}_{b}\left( {\mathbb{R}}^{d}\right) \), and each optional time \( \tau \) (relative to \( \left\{ {\mathcal{M}}_{t}\right\} \) ), we have for \( t \geq 0 \) :\n\n(10.45)\n\n\[ \n{E}^{x}\left\lbrack {f\left( {w}_{\tau + t}\right) \mid {M}_{\tau + }}\right\rbrack... | Proof. This follows from Lemma 10.9 and Theorem 10.12 in the same manner as Theorem 1 of \( \$ {2.3} \) of Chung [12]. Note that only the property in Lemma 10.9 is needed for that proof. | No |
In low dimensions, one can picture a simplex easily. A 0-simplex is a point, of course. The 1-simplex spanned by \( {a}_{0} \) and \( {a}_{1} \) consists of all points of the form\n\n\[ x = t{a}_{0} + \left( {1 - t}\right) {a}_{1} \]\n\nwith \( 0 \leq t \leq 1 \) ; this just the line segment joining \( {a}_{0} \) and \... | A similar proof shows that a 3-simplex is a tetrahedron. | No |
Lemma 1.1. Let \( U \) be a bounded, convex, open set in \( {\mathbf{R}}^{n} \) ; let \( w \in U \) .\n\n(a) Each ray emanating from \( w \) intersects \( \operatorname{Bd}U = \bar{U} - U \) in precisely one point.\n\n(b) There is a homeomorphism of \( \bar{U} \) with \( {B}^{n} \) carrying Bd \( U \) onto \( {S}^{n - ... | Proof. (a) Given a ray \( \mathcal{R} \) emanating from \( w \), its intersection with \( U \) is convex, bounded, and open in \( \mathcal{R} \) . Hence it consists of all points of the form \( w + {tp} \), where \( t \) ranges over a half-open interval \( \lbrack 0, a) \) . Then \( \mathcal{R} \) intersects \( \bar{U}... | Yes |
Lemma 2.1. A collection \( K \) of simplices is a simplicial complex if and only if the following hold:\n\n(1) Every face of a simplex of \( K \) is in \( K \) .\n\n\( \left( {2}^{\prime }\right) \) Every pair of distinct simplices of \( K \) have disjoint interiors. | Proof. First, assume \( K \) is a simplicial complex. Given two simplices \( \sigma \) and \( \tau \) of \( K \), we show that if their interiors have a point \( x \) in common, then \( \sigma = \tau \) . Let \( s = \sigma \cap \tau \) . If \( s \) were a proper face of \( \sigma \), then \( x \) would belong to \( \op... | Yes |
Let \( K \) be the collection of all 1-simplices in \( \mathbf{R} \) of the form \( \left\lbrack {m, m + 1}\right\rbrack \) , where \( m \) is an integer different from 0, along with all simplices of the form \( \left\lbrack {1/\left( {n + 1}\right) ,1/n}\right\rbrack \) for \( n \) a positive integer, along with all f... | For instance, the set of points of the form \( 1/n \) is closed in \( \left| K\right| \) but not in \( \mathbf{R} \). | Yes |
Lemma 2.2. If \( L \) is a subcomplex of \( K \), then \( \left| L\right| \) is a closed subspace of \( \left| K\right| \) . In particular, if \( \sigma \in K \), then \( \sigma \) is a closed subspace of \( \left| K\right| \) . | Proof. Suppose \( A \) is closed in \( \left| L\right| \) . If \( \sigma \) is a simplex of \( K \), then \( \sigma \cap \left| L\right| \) is the union of those faces \( {s}_{i} \) of \( \sigma \) that belong to \( L \) . Since \( A \) is closed in \( \left| L\right| \), the set \( A \cap {s}_{i} \) is closed in \( {s... | Yes |
Lemma 2.3. A map \( f : \left| K\right| \rightarrow X \) is continuous if and only if \( f \mid \sigma \) is continuous for each \( \sigma \in K \) . | Proof. If \( f \) is continuous, so is \( f \mid \sigma \) since \( \sigma \) is a subspace of \( K \) . Conversely, suppose each map \( f \mid \sigma \) is continuous. If \( C \) is a closed set of \( X \), then \( {f}^{-1}\left( C\right) \cap \) \( \sigma = {\left( f \mid \sigma \right) }^{-1}\left( C\right) \), whic... | Yes |
Lemma 2.5. If \( K \) is finite, then \( \left| K\right| \) is compact. Conversely, if a subset \( A \) of \( \left| K\right| \) is compact, then \( A \subset \left| {K}_{0}\right| \) for some finite subcomplex \( {K}_{0} \) of \( K \) . | Proof. If \( K \) is finite, then \( \left| K\right| \) is a finite union of compact subspaces \( \sigma \), and hence is compact. Now suppose \( A \) is compact and \( A \) does not lie in the polytope of any finite subcomplex of \( K \) . Choose a point \( {x}_{s} \in A \cap \) Int \( s \) whenever this set is nonemp... | Yes |
Lemma 2.6. The complex \( K \) is locally finite if and only if the space \( \left| K\right| \) is locally compact. | Proof. Suppose \( K \) is locally finite. Given \( x \in \left| K\right| \), it lies in St \( v \) for some vertex \( v \) of \( K \) . Since \( \overline{\mathrm{{St}}}v \) is a compact set, \( \left| K\right| \) is locally compact. We leave the converse as an exercise. | No |
Lemma 2.7. Let \( K \) and \( L \) be complexes, and let \( f : {K}^{\left( 0\right) } \rightarrow {L}^{\left( 0\right) } \) be a map. Suppose that whenever the vertices \( {v}_{0},\ldots ,{v}_{n} \) of \( K \) span a simplex of \( K \), the points \( f\left( {v}_{0}\right) ,\ldots, f\left( {v}_{n}\right) \) are vertic... | Proof. Note that although the vertices \( f\left( {v}_{0}\right) ,\ldots, f\left( {v}_{n}\right) \) of \( L \) are not necessarily distinct, still they span a simplex \( \tau \) of \( L \), by hypothesis. When we \ | No |
Lemma 2.8. Suppose \( f : {K}^{\left( 0\right) } \rightarrow {L}^{\left( 0\right) } \) is a bijective correspondence such that the vertices \( {v}_{0},\ldots ,{v}_{n} \) of \( K \) span a simplex of \( K \) if and only if \( f\left( {v}_{0}\right) ,\ldots, f\left( {v}_{n}\right) \) span a simplex of \( L \) . Then the ... | Proof. Each simplex \( \sigma \) of \( K \) is mapped by \( g \) onto a simplex \( \tau \) of \( L \) of the same dimension as \( \sigma \) . We need only show that the linear map \( h : \tau \rightarrow \sigma \) induced by the vertex correspondence \( {f}^{-1} \) is the inverse of the map \( g : \sigma \rightarrow \t... | Yes |
Corollary 2.9. Let \( {\Delta }^{N} \) denote the complex consisting of an \( N \) -simplex and its faces. If \( K \) is a finite complex, then \( K \) is isomorphic to a subcomplex of \( {\Delta }^{N} \) for some \( N \) . | Proof. Let \( {v}_{0},\ldots ,{v}_{N} \) be the vertices of \( K \) . Choose \( {a}_{0},\ldots ,{a}_{N} \) to be geometrically independent points in \( {\mathbf{R}}^{N} \), and let \( {\Delta }^{N} \) consist of the \( N \) -simplex they span, along with its faces. The vertex map \( f\left( {v}_{i}\right) = {a}_{i} \) ... | Yes |
Theorem 3.1. (a) Every abstract complex \( \mathcal{S} \) is isomorphic to the vertex scheme of some simplicial complex \( K \) . | Proof. To prove (a), we proceed as follows: Given an index set \( J \), let \( {\Delta }^{J} \) be the collection of all simplices in \( {\mathbf{E}}^{\prime } \) spanned by finite subsets of the standard basis \( \left\{ {\epsilon }_{\alpha }\right\} \) for \( {\mathbf{E}}^{\prime } \) . It is easy to see that \( {\De... | No |
Suppose we wish to indicate a simplicial complex \( K \) whose underlying space is homeomorphic to the cylinder \( {S}^{1} \times I \) . (Here \( I \) denotes the closed unit interval \( \left\lbrack {0,1}\right\rbrack \) .) One way of doing so is to draw the picture in Figure 3.1, which specifies \( K \) as a collecti... | Let \( f : {L}^{\left( 0\right) } \rightarrow {K}^{\left( 0\right) } \) be the map that assigns to each vertex of \( L \) the correspondingly labelled vertex of \( K \) . Then \( f \) extends to a simplicial map \( g : \left| L\right| \rightarrow \left| K\right| \) . Because the spaces are compact Hausdorff, \( g \) is... | No |
Now suppose we begin with a complex \( L \) and a labelling of its vertices. Consider for instance the same complex \( L \) with a different labelling of the vertices, as in Figure 3.3. | Just as before, this diagram indicates a certain abstract complex I, whose simplices one can list. Let \( K \) be a geometric realization of \( \mathcal{S} \). As before, the vertices of \( K \) correspond to the letters \( a,\ldots, f \); we consider the linear simplicial map \( g : \left| L\right| \rightarrow \left| ... | Yes |
The torus is often defined as the quotient space obtained from a rectangle by making the identifications pictured in Figure 3.4. If we wish to construct a complex whose underlying space is homeomorphic to the torus, we can thus obtain it by using the diagram in Figure 3.5. | You can check that the resulting quotient map of \( L \) onto the geometric realization of this diagram carries out precisely the identifications needed to form the torus. | No |
Lemma 4.1. Let \( G \) be an abelian group. If \( G \) is the direct sum of the subgroups \( \left\{ {G}_{\alpha }\right\} \), then there are homomorphisms \[ {j}_{\beta } : {G}_{\beta } \rightarrow G\;\text{ and }\;{\pi }_{\beta } : G \rightarrow {G}_{\beta } \] such that \( {\pi }_{\beta } \circ {j}_{\alpha } \) is t... | Proof. Suppose \( G = \oplus {G}_{\alpha } \) . We define \( {j}_{\beta } \) to be the inclusion homomorphism. To define \( {\pi }_{\beta } \), write \( g = \sum {g}_{\alpha } \), where \( {g}_{\alpha } \in {G}_{\alpha } \) for each \( \alpha \) ; and let \( {\pi }_{\beta }\left( g\right) = \) \( {g}_{\beta } \) . Uniq... | Yes |
Theorem 4.2. Let \( F \) be a free abelian group. If \( R \) is a subgroup of \( F \), then \( R \) is also a free abelian group. If \( F \) has rank \( n \), then \( R \) has rank \( r \leq n \) ; furthermore, there is a basis \( {e}_{1},\ldots ,{e}_{n} \) for \( F \) and integers \( {t}_{1},\ldots ,{t}_{k} \) with \(... | The integers \( {t}_{1},\ldots ,{t}_{k} \) are uniquely determined by \( F \) and \( R \), although the basis \( {e}_{1},\ldots ,{e}_{n} \) is not. | Yes |
Theorem 4.3 (The fundamental theorem of finitely generated abelian groups). Let \( G \) be a finitely generated abelian group. Let \( T \) be its torsion subgroup.\n\n(a) There is a free abelian subgroup \( H \) of \( G \) having finite rank \( \beta \) such that \( G = H \oplus T \) .\n\n(b) There are finite cyclic gr... | Proof. Let \( S \) be a finite set of generators \( \left\{ {g}_{i}\right\} \) for \( G \) ; let \( F \) be the free abelian group on the set \( S \) . The map carrying each \( {g}_{i} \) to itself extends to a homomorphism carrying \( F \) onto \( G \) . Let \( R \) be the kernel of this homomorphism. Then \( F/R \con... | No |
Lemma 5.1. \( {C}_{p}\left( K\right) \) is free abelian; a basis for \( {C}_{p}\left( K\right) \) can be obtained by orienting each p-simplex and using the corresponding elementary chains as a basis. | Proof. The proof is straightforward. Once all the \( p \) -simplices of \( K \) are oriented (arbitrarily), each \( p \) -chain can be written uniquely as a finite linear combination\n\n\[ c = \sum {n}_{i}{\sigma }_{i} \]\n\nof the corresponding elementary chains \( {\sigma }_{i} \) . The chain \( c \) assigns the valu... | Yes |
Lemma 5.3. \( {\partial }_{p - 1} \circ {\partial }_{p} = 0 \) . | Proof. The proof is straightforward. We compute\n\n\[ \n{\partial }_{p - 1}{\partial }_{p}\left\lbrack {{v}_{0},\ldots ,{v}_{p}}\right\rbrack = \mathop{\sum }\limits_{{i = 0}}^{p}{\left( -1\right) }^{i}{\partial }_{p - 1}\left\lbrack {{v}_{0},\ldots ,{\widehat{v}}_{i},\ldots ,{v}_{p}}\right\rbrack \n\]\n\n\[ \n= \matho... | Yes |
Consider the complex \( K \) of Figure 5.3, whose underlying space is the boundary of a square with edges \( {e}_{1},{e}_{2},{e}_{3},{e}_{4} \) . The group \( {C}_{1}\left( K\right) \) is free abelian of rank 4; the general 1-chain \( c \) is of the form \( \sum {n}_{i}{e}_{i} \) . Computing \( {\partial }_{1}c \), we ... | One concludes that \( {Z}_{1}\left( K\right) \) is infinite cyclic, and is generated by the chain \( {e}_{1} + {e}_{2} + {e}_{3} + {e}_{4} \) . Since there are no 2-simplices in \( K,{B}_{1}\left( K\right) \) is trivial. Therefore, \[ {H}_{1}\left( K\right) = {Z}_{1}\left( K\right) \simeq \mathbf{Z}. \] | Yes |
Consider the complex \( L \) of Figure 5.4, whose underlying space is a square. The general 1-chain is of the form \( \sum {n}_{i}{e}_{t} \). One reasons as before to conclude that this 1-chain is a cycle if and only if \( {n}_{1} = {n}_{2},{n}_{3} = {n}_{4} \), and \( {n}_{5} = {n}_{3} - {n}_{2} \). One can assign val... | \[ {H}_{\mathrm{t}}\left( L\right) = {Z}_{\mathrm{t}}\left( L\right) /{B}_{\mathrm{t}}\left( L\right) = 0. \] | Yes |
Given a 1-chain \( c \), it is homologous to a chain \( {c}_{3} \) that is carried by the subcomplex of \( M \) pictured in Figure 5.6. | Given a 1-chain \( c \), let \( a \) be the value of \( c \) on \( {e}_{1} \). Then by direct computation, the chain\n\n\[ {c}_{1} = c + {\partial }_{2}\left( {a{\sigma }_{1}}\right) \]\n\nhas value 0 on the oriented simplex \( {e}_{1} \). Intuitively speaking, by modifying \( c \) by a boundary, we have \ | No |
Lemma 6.1. Let \( L \) be the complex of Figure 6.1, whose underlying space is a rectangle. Let \( \mathrm{Bd}L \) denote the complex whose space is the boundary of the rectangle. Orient each 2-simplex \( {\sigma }_{i} \) of \( L \) by a counterclockwise arrow. Orient the 1-simplices arbitrarily. Then:\n\n(1) Every 1-c... | To prove (1), we proceed as in Example 6 of the preceding section. Given a 1-chain \( c \) of \( L \), one \ | No |
Theorem 6.2. Let \( T \) denote the complex represented by the labelled rectangle \( L \) of Figure 6.4; its underlying space is the torus. Then:\n\n\[ \n{H}_{1}\left( T\right) \simeq \mathbf{Z} \oplus \mathbf{Z}\;\text{ and }\;{H}_{2}\left( T\right) \simeq \mathbf{Z}.\n\]\n\nOrient each 2-simplex of \( L \) counterclo... | Proof. Let \( g : \left| L\right| \rightarrow \left| T\right| \) be the pasting map; let \( A = g\left( \left| {\operatorname{Bd}L}\right| \right) \) . Then \( A \) is homeomorphic to a space that is the union of two circles with a point in common. (Such a space is called a wedge of two circles.) Orient the 1-simplices... | Yes |
Theorem 6.3. Let \( S \) denote the complex represented by the labelled rectangle of Figure 6.6; its underlying space is the Klein bottle. Then\n\n\[ \n{H}_{1}\left( S\right) \simeq \mathbf{Z} \oplus \mathbf{Z}/2\;\text{ and }\;{H}_{2}\left( S\right) = 0.\n\] | Proof. Let \( g : \left| L\right| \rightarrow \left| S\right| \) be the pasting map. Let \( A = g\left( \left| {\operatorname{Bd}L}\right| \right) \) ; as before, it is the wedge of two circles. Orient the 2-simplices of \( S \) as before; let \( \gamma \) be their sum. Orient the 1-simplices of \( S \) arbitrarily. No... | Yes |
Theorem 6.4. Let \( {P}^{2} \) be the complex indicated by the labelled rectangle of Figure 6.7; its underlying space is called the projective plane. Then\n\n\[ \n{H}_{1}\left( {P}^{2}\right) \simeq \mathbf{Z}/2\;\text{ and }\;{H}_{2}\left( {P}^{2}\right) = 0.\n\] | Proof. Let \( g : \left| L\right| \rightarrow \left| {P}^{2}\right| \) be the pasting map. Let \( A = g\left( \left| {\operatorname{Bd}L}\right| \right) \) ; it. is homeomorphic to a circle. Let \( \gamma \) be as before; let\n\n\[ \n{z}_{1} = \left\lbrack {a, b}\right\rbrack + \left\lbrack {b, c}\right\rbrack + \left\... | Yes |
Theorem 6.5. Let \( {P}^{2}\# {P}^{2} \) be the connected sum of two projective planes. Then\n\n\[ \n{H}_{1}\left( {{P}^{2}\# {P}^{2}}\right) \simeq \mathbf{Z} \oplus \mathbf{Z}/2\;\text{ and }\;{H}_{2}\left( {{P}^{2}\# {P}^{2}}\right) = 0.\n\] | Proof. We represent \( {P}^{2}\# {P}^{2} \) by the same rectangle \( L \) as before, with an appropriate vertex-labelling. In this case, the complex \( A = g\left( \left| {\mathrm{{Bd}}L}\right| \right) \) is again the wedge of two circles. Let \( {w}_{1} \) be the 1 -cycle \ | Yes |
Theorem 7.2. The group \( {\widetilde{H}}_{0}\left( K\right) \) is free abelian, and\n\n\[ \n{\widetilde{H}}_{0}\left( K\right) \oplus \mathbf{Z} \simeq {H}_{0}\left( K\right) \n\] \n\nThus \( {\widetilde{H}}_{0}\left( K\right) \) vanishes if \( \left| K\right| \) is connected. If \( \left| K\right| \) is not connected... | Proof. Given a 0-chain \( c \), it is homologous to a 0-chain of the form \( {c}^{\prime } = \sum {n}_{\alpha }{v}_{\alpha } \) ; and the chain \( {c}^{\prime } \) bounds only if \( {n}_{\alpha } = 0 \) for all \( \alpha \) . Now if \( c \in \ker \epsilon \) , then \( \epsilon \left( c\right) = \epsilon \left( {c}^{\pr... | Yes |
If \( {K}_{\sigma } \) is the complex consisting of the simplex \( \sigma = {v}_{0}\ldots {v}_{n} \) and its faces, then \( {K}_{\sigma } = {v}_{0} * {K}_{s} \), where \( s \) is the face of \( \sigma \) opposite \( {v}_{0} \). | Thus every simplex of positive dimension is a cone. | No |
Lemma 8.1. Let \( U \) be a bounded convex open set in \( {\mathbf{R}}^{n} \) ; let \( w \in U \) . If \( K \) is a finite complex such that \( \left| K\right| = \bar{U} - U \), then \( w * K \) is a finite complex such that \( \left| {w * K}\right| = \bar{U} \) . | Proof. It follows at once from Lemma 1.1 that each ray emanating from \( w \) intersects \( \left| K\right| \) in precisely one point, and that \( \bar{U} \) is the union of all line segments joining \( w \) to points of \( \left| K\right| \) . | Yes |
Theorem 8.2. If \( w * K \) is a cone, then for all \( p \) ,\n\n\[ \n{\widetilde{H}}_{p}\left( {w * K}\right) = 0 \n\] | Proof. The reduced homology of \( w * K \) vanishes in dimension 0, because \( \left| {w * K}\right| \) is connected. Consider the case \( p > 0 \) . Let \( {z}_{p} \) be a \( p \) -cycle of \( w * K \) ; we show that \( {z}_{p} \) bounds. Let us write\n\n\[ \n{z}_{p} = {c}_{p} + \left\lbrack {w,{d}_{p - 1}}\right\rbra... | Yes |
Theorem 8.3. Let \( \sigma \) be an \( n \) -simplex. The complex \( {K}_{\sigma } \) consisting of \( \sigma \) and its faces is acyclic. If \( n > 0 \), let \( {\sum }^{n - 1} \) denote the complex whose polytope is Bd \( \sigma \) . Orient \( \sigma \) . Then \( {\widetilde{H}}_{n - 1}\left( {\sum }^{n - 1}\right) \... | Proof. Because \( {K}_{\sigma } \) is a cone, it is acyclic. Let us compare the chain groups of \( {K}_{\sigma } \) and \( {\sum }^{n - 1} \) ; they are equal except in dimension \( n \) :\n\n\[ \n{C}_{n}\left( {K}_{\sigma }\right) \overset{{\partial }_{n}}{ \rightarrow }{C}_{n - 1}\left( {K}_{\sigma }\right) \overset{... | Yes |
Let \( K \) consist of an \( n \) -simplex and its faces; let \( {K}_{0} \) be the set of proper faces of \( K \) . Then the group \( {C}_{p}\left( {K,{K}_{0}}\right) \) vanishes except when \( p = n \), in which case it is infinite cyclic. | It follows that\n\n\[ \n{H}_{i}\left( {K,{K}_{0}}\right) = 0\text{ for }i \neq n, \n\] \n\n\[ \n{H}_{n}\left( {K,{K}_{0}}\right) \simeq \mathbf{Z} \n\] | Yes |
Let \( K \) be a complex and let \( {K}_{0} \) consist of a single vertex \( v \) of \( K \) . Using the results of \( §7 \), one sees readily that \( {H}_{0}\left( {K, v}\right) \) is free abelian; one obtains a basis for \( {H}_{0}\left( {K, v}\right) \) by choosing one vertex from each component of \( \left| K\right... | It is not hard to show that \( {H}_{p}\left( {K, v}\right) \simeq {H}_{p}\left( K\right) \) for \( p > 0 \) ; see the exercises. | No |
Let \( K \) be the complex indicated in Figure 9.1, whose underlying space is a square. Let \( {K}_{0} \) be the subcomplex whose space is the boundary of the square. It is easy to see that the 2-chain \( \sum {m}_{i}{\sigma }_{i} \) represents a relative cycle of \( K \) modulo \( {K}_{0} \) if and only if \( {m}_{1} ... | \[ {H}_{2}\left( {K,{K}_{0}}\right) \simeq \mathrm{Z} \] and the chain \( \gamma = \sum {\sigma }_{i} \) represents a generator. | Yes |
Let \( K \) be the complex indicated in Figure 9.2. Its underlying space is called an annulus. Let \( {K}_{0} \) denote the 1-dimensional complex whose space is the union of the inner and outer edges of \( K \). We compute the homology of \( K \) modulo \( {K}_{0} \). | First, \( {H}_{0}\left( {K,{K}_{0}}\right) = 0 \) because the relative chain group itself vanishes in dimension 0 . To compute \( {H}_{1} \) and \( {H}_{2} \), one first verifies three facts:\n\n(i) If \( c \) is a 1-chain of \( K \), then \( c \) is homologous to a 1-chain of \( K \) that is carried by the subcomplex ... | Yes |
Theorem 9.1 (Excision theorem). Let \( K \) be a complex; let \( {K}_{0} \) be a subcom-plex. Let \( U \) be an open set contained in \( \left| {K}_{0}\right| \), such that \( \left| K\right| - U \) is the polytope of a subcomplex \( L \) of \( K \) . Let \( {L}_{0} \) be the subcomplex of \( K \) whose polytope is \( ... | Proof. Consider the composite map \( \phi \) ,\n\n\[ \n{C}_{p}\left( L\right) \rightarrow {C}_{p}\left( K\right) \rightarrow {C}_{p}\left( K\right) /{C}_{p}\left( {K}_{0}\right) ,\n\]\n\nwhich is inclusion followed by projection. Then \( \phi \) is surjective, because \( {C}_{p}\left( K\right) / \) \( {C}_{p}\left( {K}... | Yes |
Let us calculate the homology of the torus and the Klein bottle using these coefficients. | The argument given in §6 goes through essentially unchanged to show that\n\n\[ \n{H}_{1}\left( {T;\mathbf{Z}/2}\right) \simeq \mathbf{Z}/2 \oplus \mathbf{Z}/2 \n\]\n\n\[ \n{H}_{2}\left( {T;\mathbf{Z}/2}\right) \simeq \mathbf{Z}/2 \n\]\n\nFor the Klein bottle \( S \), the argument goes through but with some changes. One... | Yes |
Let us compute the homology of the torus and Klein bottle using the rational numbers \( \mathbf{Q} \) as coefficients. | For the torus, one has\n\n\[ \n{H}_{1}\left( {T;\mathbf{Q}}\right) \simeq \mathbf{Q} \oplus \mathbf{Q},\;{H}_{2}\left( {T;\mathbf{Q}}\right) \simeq \mathbf{Q}.\n\]\n\nFor the Klein bottle, one has\n\n\[ \n{H}_{1}\left( {S;\mathrm{Q}}\right) \simeq \mathrm{Q},\;{H}_{2}\left( {S;\mathrm{Q}}\right) = 0.\n\]\n\nFor with \(... | Yes |
Theorem 11.3. Let \( G \) and \( {G}^{\prime } \) be free abelian groups of ranks \( n \) and \( m \), respectively; let \( f : G \rightarrow {G}^{\prime } \) be a homomorphism. Then there are bases for \( G \) and \( {G}^{\prime } \) such that, relative to these bases, the matrix of \( f \) has the form ![dda81674-cb4... | Proof. We begin by choosing bases in \( G \) and \( {G}^{\prime } \) arbitrarily. Let \( A \) be the matrix of \( f \) relative to these bases. We shall give shortly a procedure for modifying these bases so as to bring the matrix into the normal form described. It is called \ | No |
Theorem 11.5. The homology groups of a finite complex \( K \) are effectively computable. | Proof. By the preceding theorem, there is a decomposition\n\n\[ \n{C}_{p}\left( K\right) = {U}_{p} \oplus {V}_{p} \oplus {W}_{p} \n\]\n\nwhere \( {Z}_{p} = {V}_{p} \oplus {W}_{p} \) is the group of \( p \) -cycles and \( {W}_{p} \) is the group of weak \( p \) -boundaries. Now\n\n\[ \n{H}_{p}\left( K\right) = {Z}_{p}/{... | Yes |
Lemma 12.1. The homomorphism \( {f}_{y} \) commutes with \( \partial \) ; therefore \( {f}_{y} \) induces a homomorphism \( {f}_{ * } : {H}_{p}\left( K\right) \rightarrow {H}_{p}\left( L\right) \) . | Proof. We need to show that\n\n(*) \n\n\[ \partial {f}_{ij}\left( \left\lbrack {{v}_{0},\ldots ,{v}_{p}}\right\rbrack \right) = {f}_{ij}\left( {\partial \left\lbrack {{v}_{0},\ldots ,{v}_{p}}\right\rbrack }\right) . \]\n\nLet \( \tau \) be the simplex of \( L \) spanned by \( f\left( {v}_{0}\right) ,\ldots, f\left( {v}... | Yes |
Theorem 12.2. (a) Let \( i : K \rightarrow K \) be the identity simplicial map. Then \( {i}_{ * } : {H}_{p}\left( K\right) \rightarrow {H}_{p}\left( K\right) \) is the identity homomorphism. | Proof. It is immediate from the definition that \( {i}_{ij} \) is the identity and \( {\left( g \circ f\right) }_{\# } = {g}_{\# } \circ {f}_{\# } \), as you can check. The theorem follows. | No |
Lemma 12.3. The chain map \( {f}_{\# } \) preserves the augmentation map \( \epsilon \) ; therefore, it induces a homomorphism \( {f}_{ * } \) of reduced homology groups. | Proof. Let \( f : K \rightarrow L \) be simplicial. Then \( \epsilon {f}_{\# }\left( v\right) = 1 \) and \( \epsilon \left( v\right) = 1 \) for each vertex \( v \) of \( K \) . Thus \( \epsilon \circ {f}_{\# } = \epsilon \) . This equation implies that \( {f}_{\# } \) carries the kernel of \( {\epsilon }_{K} : {C}_{0}\... | Yes |
Consider the complexes \( K \) and \( T \) indicated in Figure 12.1. Their underlying spaces are the circle and torus, respectively. Now \( {H}_{1}\left( K\right) \simeq \mathbf{Z} \) ; let us use the cycle \( z \) indicated in the figure as a generator. Similarly, \( {H}_{1}\left( T\right) \simeq \mathbf{Z} \oplus \ma... | Now consider the simplicial maps:\n\n\[ f : a \rightarrow A\;g : a \rightarrow A\;h : a \rightarrow A \]\n\[ b \rightarrow F\;b \rightarrow B\;b \rightarrow I \]\n\[ c \rightarrow D \]\n\[ d \rightarrow D \]\n\[ e \rightarrow F\;e \rightarrow E\;e \rightarrow G \]\n\[ f \rightarrow E\;f \rightarrow D\;f \rightarrow A \... | Yes |
Theorem 12.4. If there is a chain homotopy between \( {f}_{\# } \) and \( {g}_{\# } \), then the induced homomorphisms \( {f}_{ * } \) and \( {g}_{ * } \), for both reduced and ordinary homology, are equal. | Proof. Let \( z \) be a \( p \) -cycle of \( K \) . Then\n\n\[ \n{g}_{\# }\left( z\right) - {f}_{\# }\left( z\right) = \partial {Dz} + D\partial z = \partial {Dz} + 0, \n\]\n\nso \( {g}_{\# }\left( z\right) \) and \( {f}_{\# }\left( z\right) \) are in the same homology class. Thus \( {g}_{ * }\left( {\{ z\} }\right) = ... | Yes |
Theorem 12.6. Let \( f, g : \left( {K,{K}_{0}}\right) \rightarrow \left( {L,{L}_{0}}\right) \) be contiguous as maps of pairs. Then there is for all \( p \) a homomorphism\n\n\[ D : {C}_{p}\left( {K,{K}_{0}}\right) \rightarrow {C}_{p + 1}\left( {L,{L}_{0}}\right) \]\n\nsuch that \( \partial D + D\partial = {g}_{\# } - ... | Proof. The chain homotopy \( D \) constructed in the preceding proof automatically maps \( {C}_{p}\left( {K}_{0}\right) \) into \( {C}_{p + 1}\left( {L}_{0}\right) \) . For if \( \sigma \in {K}_{0} \), the complex \( L\left( \sigma \right) \) is by definition a subcomplex of \( {L}_{0} \) . Given \( \sigma \), the chai... | Yes |
Lemma 13.1. Let \( \mathcal{C} \) and \( {\mathcal{C}}^{\prime } \) be non-negative chain complexes. Let \( \phi ,\psi : \mathcal{C} \rightarrow {\mathcal{C}}^{\prime } \) be chain maps; let \( D \) be a chain homotopy between them. Suppose \( \mathcal{O} \) and \( {\mathcal{O}}^{\prime } \) are augmented by \( \epsilo... | Proof. If \( {c}_{0} \in {C}_{0} \), we have\n\n\[ \partial D{c}_{0} = \phi \left( {c}_{0}\right) - \psi \left( {c}_{0}\right) \]\n\nbecause \( \partial {c}_{0} = 0 \) . Then\n\n\[ 0 = {\epsilon }^{\prime }\left( {\partial D{c}_{0}}\right) = {\epsilon }^{\prime }\phi \left( {c}_{0}\right) - {\epsilon }^{\prime }\psi \l... | Yes |
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