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Theorem 7. Let \( T \) be a stopping time. Then \( {\left( H \cdot X\right) }^{T} = H{1}_{\left\lbrack 0, T\right\rbrack } \cdot X = \) \( H \cdot \left( {X}^{T}\right) \) . | Proof. Note that \( {1}_{\left\lbrack 0, T\right\rbrack } \in \mathbf{b}\mathbb{L} \), so \( H{1}_{\left\lbrack 0, T\right\rbrack } \in \mathbf{b}\mathcal{P} \) . Also, \( {X}^{T} \) is clearly still in \( {\mathcal{H}}^{2} \) . Since we know this result is true for \( H \in \mathbf{b}\mathbb{L} \) (Theorem 12 of Chap.... | No |
Theorem 12. Let \( X, Y \in {\mathcal{H}}^{2} \) and \( H, K \in \mathrm{b}\mathcal{P} \). Then\n\n\[ \n{\left\lbrack H \cdot X, K \cdot Y\right\rbrack }_{t} = {\int }_{0}^{t}{H}_{s}{K}_{s}d{\left\lbrack X, Y\right\rbrack }_{s},\;\left( {t \geq 0}\right) ,\n\]\n\nand in particular\n\n\[ \n{\left\lbrack H \cdot X, H \cd... | Proof. As in the proof of Theorem 29 of Chap. II, it suffices to show\n\n\[ \n{\left\lbrack H \cdot X, Y\right\rbrack }_{t} = {\int }_{0}^{t}{H}_{s}d{\left\lbrack X, Y\right\rbrack }_{s}\n\]\n\nLet \( \left( {H}^{n}\right) \in \mathbf{b}\mathbb{L} \) such that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{d}_{X}\l... | Yes |
Theorem 13. Let \( X \) be a semimartingale, \( {X}_{0} = 0 \) . Then \( X \) is prelocally in \( {\mathcal{H}}^{2} \) . That is, there exists a non-decreasing sequence of stopping times \( \left( {T}^{n}\right) \) , \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{T}^{n} = \infty \) a.s., such that \( {X}^{{T}^{n} -... | Proof. Recall that \( {X}^{{T}^{n} - } = {X}_{t}{1}_{\left\lbrack 0,{T}^{n}\right) } + {X}_{{T}^{n} - }{1}_{\left\lbrack {T}^{n},\infty \right) } \) . By the Bichteler-Dellacherie Theorem (Theorem 43 of Chap. III) we can write \( X = M + A \) , where \( M \) is a local martingale and \( A \) is an \( {FV} \) process. B... | Yes |
Theorem 14. Let \( X \) be a semimartingale and let \( H \in \mathcal{P} \) be \( \left( {{\mathcal{H}}^{2}, X}\right) \) integrable. Let \( {H}^{n} = H{1}_{\{ \left| H\right| \leq n\} } \in \mathbf{b}\mathcal{P} \) . Then \( {H}^{n} \cdot X \) is a Cauchy sequence in \( {\mathcal{H}}^{2} \) . | Proof. Since \( {H}^{n} \in \mathbf{b}\mathcal{P} \), each \( n \), the stochastic integrals \( {H}^{n} \cdot X \) are defined. Note also that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{H}^{n} = H \) and that \( \left| {H}^{n}\right| \leq \left| H\right| \), each \( n \) . Then\n\n\[ \parallel {H}^{n} \cdot X -... | Yes |
Theorem 15. Let \( X \) be a semimartingale and let \( H \in \mathcal{P} \) be locally bounded. Then \( H \in L\left( X\right) \) . That is, the stochastic integral \( H \cdot X \) exists. | Proof. Let \( {\left( {S}^{m}\right) }_{m \geq 1},{\left( {T}^{n}\right) }_{n \geq 1} \) be two sequence of stopping times, each increasing to \( \infty \) a.s., such that \( {H}^{{S}^{m}}{1}_{\left\{ {S}^{m} > 0\right\} } \) is bounded for each \( m \), and \( {X}^{{T}^{n} - } \in {\mathcal{H}}^{2} \) for each \( n \)... | Yes |
Theorem 16. Let \( X \) be a semimartingale and let \( H, J \in L\left( X\right) \) . Then \( {\alpha H} + \) \( {\beta J} \in L\left( X\right) \) and \( \left( {{\alpha H} + {\beta J}}\right) \cdot X = {\alpha H} \cdot X + {\beta J} \cdot X \) . That is, \( L\left( X\right) \) is a linear space. | Proof. Let \( \left( {R}^{m}\right) \) and \( \left( {T}^{n}\right) \) be sequences of stopping times such that \( H \) is \( \left( {{\mathcal{H}}^{2},{X}^{{R}^{m} - }}\right) \) integrable, each \( m \), and \( J \) is \( \left( {{\mathcal{H}}^{2},{X}^{{T}^{n} - }}\right) \) integrable, each \( n \) . Taking \( {S}^{... | Yes |
Theorem 23. Let \( X \) be a semimartingale, let \( H \in L\left( X\right) \), and suppose \( Q \) is another probability with \( Q \ll P \) . If \( {\mathrm{H}}_{Q} \cdot X \) exists, it is \( Q \) indistinguishable from \( {H}_{P} \cdot X \) . | Proof. \( {H}_{Q} \cdot X \) denotes the stochastic integral computed under \( Q \) . By Theorem 14 of Chap. II, we know that \( {H}_{Q} \cdot X = {H}_{P} \cdot X \) for \( H \in \mathbb{L} \), and therefore if \( X \in {\mathcal{H}}^{2} \) for both \( P \) and \( Q \), they are equal for \( H \in \mathbf{b}\mathcal{P}... | Yes |
Theorem 24. For \( X \) a semimartingale in \( {\mathcal{H}}^{2} \) , \[ \parallel X{\parallel }_{{\mathcal{H}}^{2}} \leq \mathop{\sup }\limits_{{\left| H\right| \leq 1}}{\begin{Vmatrix}{\left( H \cdot X\right) }_{\infty }^{ * }\end{Vmatrix}}_{{L}^{2}} + 2{\begin{Vmatrix}{\left\lbrack X, X\right\rbrack }_{\infty }^{1/2... | Proof. By Theorem 5 for \( \left| H\right| \leq 1 \) \[ {\begin{Vmatrix}{\left( H \cdot X\right) }_{\infty }^{ * }\end{Vmatrix}}_{{L}^{2}} \leq \sqrt{8}\parallel H \cdot X{\parallel }_{{\mathcal{H}}^{2}} \leq \sqrt{8}\parallel X{\parallel }_{{\mathcal{H}}^{2}} \] Since \[ 2\parallel {\left\lbrack X, X\right\rbrack }_{\... | Yes |
Lemma 1. Let \( A \) be a non-negative increasing \( {FV} \) process and let \( Z \) be a positive uniformly integrable martingale. Let \( T \) be a stopping time such that \( A = {A}^{T - } \) (that is, \( {A}_{\infty } = {A}_{T - } \) ) and let \( k \) be a constant such that \( Z \leq k \) on \( \lbrack 0, T) \) . T... | Proof. Since \( {A}_{0 - } = {Z}_{0 - } = 0 \), by integration by parts\n\n\[ {A}_{t}{Z}_{t} = {\int }_{0}^{t}{A}_{s - }d{Z}_{s} + {\int }_{0}^{t}{Z}_{s - }d{A}_{s} + {\left\lbrack A, Z\right\rbrack }_{t} \]\n\n\[ = {\int }_{0}^{t}{A}_{s - }d{Z}_{s} + {\int }_{0}^{t}{Z}_{s}d{A}_{s} \]\nwhere the second integral in the ... | Yes |
Lemma 2. Let \( X \) be a semimartingale with \( {X}_{0} = 0 \), let \( Q \) be another probability with \( Q \ll P \), and let \( {Z}_{t} = {E}_{P}\left\{ {\left. \frac{dQ}{dP}\right| \;{\mathcal{F}}_{t}}\right\} \) . If \( T \) is a stopping time such that \( {Z}_{t} \leq k \) on \( \lbrack 0, T) \) for a constant \(... | Proof. By Theorem 24 we have\n\n\[ \n{\begin{Vmatrix}{X}^{T - }\end{Vmatrix}}_{{\mathcal{H}}^{2}\left( Q\right) } \leq \mathop{\sup }\limits_{{\left| H\right| \leq 1}}{\mathbb{E}}_{Q}\{ {\left( {\left( H \cdot {X}^{T - }\right) }^{ * }\right) }^{2}{\} }^{\frac{1}{2}} + 2{\mathbb{E}}_{Q}\{ {\left\lbrack {X}^{T - },{X}^{... | Yes |
Theorem 25. Let \( X \) be a semimartingale and \( H \in L\left( X\right) \) . If \( Q \ll P \), then \( H \in L\left( X\right) \) under \( Q \) as well, and \( {H}_{Q} \cdot X = {H}_{P} \cdot X, Q \) -a.s. | Proof. Let \( {T}^{n} \) be a sequence of stopping times increasing to \( \infty \) a.s. such that \( H \) is \( \left( {{X}^{{T}^{n} - },{\mathcal{H}}^{2}}\right) \) integrable under \( P \), each \( n \geq 1 \) . Let \( {Z}_{t} = {E}_{P}\left\{ {\left. \frac{dQ}{dP}\right| \;{\mathcal{F}}_{t}}\right\} \), the càdlàg ... | Yes |
Theorem 26. Let \( X,\bar{X} \) be two semimartingales, and let \( H \in L\left( X\right) ,\bar{H} \in \) \( L\left( \bar{X}\right) \) . Let \( A = \left\{ {\omega : H.\left( \omega \right) = \bar{H}.\left( \omega \right) }\right. \) and \( \left. {{X}_{ \cdot }\left( \omega \right) = \bar{X}.\left( \omega \right) }\ri... | Proof. Without loss of generality assume \( P\left( A\right) > 0 \) . Define \( Q \) by \( Q\left( \Lambda \right) = \) \( P\left( {\Lambda \mid A}\right) \) . Then \( Q \ll P \) and therefore \( H \in L\left( X\right) ,\bar{H} \in L\left( \bar{X}\right) \) under \( Q \) as well as under \( P \) by Theorem 25. However ... | Yes |
Theorem 27. Let \( {P}_{k} \) be a sequence of probabilities such that \( X \) is a \( {P}_{k} \) semimartingale for each \( k \) . Let \( R = \mathop{\sum }\limits_{{k = 1}}^{\infty }{\lambda }_{k}{P}_{k} \) where \( {\lambda }_{k} \geq 0 \), each \( k \), and \( \mathop{\sum }\limits_{{k = 1}}^{\infty }{\lambda }_{k}... | Proof. If \( {\lambda }_{k} > 0 \) then \( {P}_{k} \ll R \) . Moreover since \( {P}_{k}\left( \Lambda \right) \leq \frac{1}{{\lambda }_{k}}R\left( \Lambda \right) \), it follows that \( H \in L\left( X\right) \) under \( {P}_{k} \) . The result then follows by Theorem 25. | Yes |
Theorem 28. Let \( M \) be a locally square integrable local martingale, and let \( H \in \mathcal{P} \) . The stochastic integral \( H \cdot M \) exists (i.e., \( H \in L\left( M\right) \) ) and is a locally square integrable local martingale if there exists a sequence of stopping times \( {\left( {T}^{n}\right) }_{n ... | Proof. We assume that \( M \) is a square integrable martingale stopped at the time \( {T}^{n} \) . The result follows by applying the lemma. | No |
Theorem 29. Let \( M \) be a local martingale, and let \( H \in \mathcal{P} \) be locally bounded. Then the stochastic integral \( H \cdot M \) is a local martingale. | Proof. By stopping we may, as in the proof of Theorem 29 of Chap. III, assume that \( H \) is bounded, \( M \) is uniformly integrable, and that \( M = N + A \) where \( N \) is a bounded martingale and \( A \) is of integrable variation. We know that there exists \( {R}^{k} \) increasing to \( \infty \) a.s. such that... | Yes |
Theorem 30. Let \( M \) be a continuous local martingale and let \( H \in \mathcal{P} \) be such that \( {\int }_{0}^{t}{H}_{s}^{2}d{\left\lbrack M, M\right\rbrack }_{s} < \infty \) a.s., each \( t \geq 0 \) . Then the stochastic integral \( H \cdot M \) exists (i.e., \( H \in L\left( M\right) \) ) and it is a continuo... | Proof. Since \( M \) is continuous, we can take\n\n\[
{R}^{k} = \inf \left\{ {t > 0 : \left| {M}_{t}\right| > k}\right\}
\]\n\nThen \( \left| {M}_{t \land {R}^{k}}\right| \leq k \) and therefore \( M \) is locally bounded, hence locally square integrable. Also \( M \) continuous implies \( \left\lbrack {M, M}\right\rbr... | Yes |
Theorem 31. Let \( M \) be a local martingale with jumps bounded by a constant \( \beta \) . Let \( H \in \mathcal{P} \) be such that \( {\int }_{0}^{t}{H}_{s}^{2}d{\left\lbrack M, M\right\rbrack }_{s} < \infty \) a.s., \( t \geq 0 \), and \( E\left\{ {H}_{T}^{2}\right\} < \infty \) for any bounded stopping time \( T \... | Proof. Let \( {R}^{n} = \inf \left\{ {t > 0 : {\int }_{0}^{t}{H}_{s}^{2}d{\left\lbrack M, M\right\rbrack }_{s} > n}\right\} \), and let \( {T}^{n} = \min \left( {{R}^{n}, n}\right) \) . Then \( {T}^{n} \) are bounded stopping times increasing to \( \infty \) a.s. Note that\n\n\[ E\left\{ {{\int }_{0}^{{T}^{n}}{H}_{s}^{... | Yes |
Theorem 33. Let \( \mathbb{F} = {\left( {\mathcal{F}}_{t}\right) }_{t \geq 0} \) and \( \mathbb{G} = {\left( {\mathcal{G}}_{t}\right) }_{t \geq 0} \) be two filtrations satisfying the usual hypotheses and suppose \( {\mathcal{F}}_{t} \subset {\mathcal{G}}_{t} \), each \( t \geq 0 \), and that \( X \) is a semimartingal... | Proof. It is trivial that \( H \) is locally bounded and predictable for \( {\left( {\mathcal{G}}_{t}\right) }_{t \geq 0} \) as well. By stopping, we can assume without loss of generality that \( H \) is bounded. Let\n\n\[ \mathcal{H} = \left\{ {\text{all bounded,}\mathcal{F}\text{ predictable }H\text{ such that }{H}_{... | Yes |
Theorem 34. Let \( X \) be a semimartingale on a filtered complete probability space \( \left( {\Omega ,\mathcal{F},\mathbb{F}, P}\right) \) satisfying the usual hypotheses. Then there exists a probability \( Q \) which is equivalent to \( P \) such that under \( Q, X \) is a semimartingale in \( {\mathcal{H}}^{2} \) .... | Proof of Theorem 34. Using the lemma we make a first change of measure making all of the random variables \( {\left\lbrack X, X\right\rbrack }_{n} \) integrable. Recall that \( \left\lbrack {X, X}\right\rbrack \) is invariant under a change to an equivalent probability measure. By abusing notation, we will still denote... | Yes |
Theorem 35. Let \( F \) be a closed subspace of \( {\mathbf{M}}^{2} \). Then the following are equivalent.\n\n(a) \( F \) is closed under the following operation. For \( M \in F,\left( {M - {M}^{t}}\right) {1}_{\Lambda } \in F \) for \( \Lambda \in {\mathcal{F}}_{t} \), any \( t \geq 0 \).\n\n(b) \( F \) is a stable su... | Proof. Property (d) implies (c), and it is simple that (c) implies (b). To get (b) implies (a), let \( T = {t}_{\Lambda } \), where\n\n\[ \n{t}_{\Lambda } = \left\{ \begin{array}{ll} t, & \text{ if }\omega \in \Lambda \\ \infty , & \text{ if }\omega \notin \Lambda \end{array}\right.\n\]\n\nThen \( T = {t}_{\Lambda } \)... | Yes |
Lemma 1. If \( \mathcal{A} \) is any subset of \( {\mathbf{M}}^{2} \), then \( {\mathcal{A}}^{ \times } \) is (closed and) stable. | Proof. Let \( {M}^{n} \) be a sequence of elements of \( {\mathcal{A}}^{ \times } \) converging to \( M \), and let \( N \in \mathcal{A} \) . Then \( {M}^{n}N \) is a martingale for each \( n \) and \( {\mathcal{A}}^{ \times } \) will be shown to be closed if \( {MN} \) is also one, or equivalently that \( \left\lbrack... | Yes |
Lemma 2. Let \( N, M \) be in \( {\mathbf{M}}^{2} \) . Then the following are equivalent.\n\n(a) \( M \) and \( N \) are strongly orthogonal.\n\n(b) \( \mathcal{S}\left( M\right) \) and \( N \) are strongly orthogonal.\n\n(c) \( \mathcal{S}\left( M\right) \) and \( \mathcal{S}\left( N\right) \) are strongly orthogonal.... | Proof. If \( M \) and \( N \) are strongly orthogonal, let \( \mathcal{A} = \{ N\} \) and then \( M \in {\mathcal{A}}^{ \times } \) . Since \( {\mathcal{A}}^{ \times } \) is a closed stable subspace by Lemma \( 1,\mathcal{S}\left( M\right) \subset \{ N{\} }^{ \times } \) . Therefore (b) holds and hence (a) implies (b).... | Yes |
Theorem 36. Let \( {M}^{1},\ldots ,{M}^{n} \in {\mathbf{M}}^{2} \), and suppose \( {M}^{i},{M}^{j} \) are strongly orthogonal for \( i \neq j \) . Then \( \mathcal{S}\left( {{M}^{1},\ldots ,{M}^{n}}\right) \) consists of the set of stochastic integrals \[ {H}^{1} \cdot {M}^{1} + \cdots + {H}^{n} \cdot {M}^{n} = \mathop... | Proof. Let \( \mathcal{I} \) denote the space of processes \( \mathop{\sum }\limits_{{i = 1}}^{n}{H}^{i} \cdot {M}^{i} \), where \( {H}^{i} \) satisfy the hypotheses of the theorem. By Theorem 35 any closed, stable subspace must contain \( \mathcal{I} \) . It is simple to check that \( \mathcal{I} \) is stable, so we n... | Yes |
Theorem 37. Let \( \mathcal{A} \) be a subset of \( {\mathbf{M}}^{2} \) which is stable. Then \( {\mathcal{A}}^{ \bot } \) is a stable subspace, and if \( M \in {\mathcal{A}}^{ \bot } \) then \( M \) is strongly orthogonal to \( \mathcal{A} \) . That is, \( {\mathcal{A}}^{ \bot } = \) \( {\mathcal{A}}^{ \times } \), an... | Proof. We first show that \( {\mathcal{A}}^{ \bot } = {\mathcal{A}}^{ \times } \) . Let \( M \in \mathcal{A} \) and \( N \in {\mathcal{A}}^{ \bot } \) . Since \( N \) is orthogonal to \( \mathcal{S}\left( M\right) \), by Lemma \( 2, N \) and \( M \) are strongly orthogonal. Therefore \( {\mathcal{A}}^{ \bot } \subset {... | Yes |
Corollary 1. Let \( \mathcal{A} \) be a stable subspace of \( {\mathbf{M}}^{2} \) . Then each \( M \in {\mathbf{M}}^{2} \) has a unique decomposition \( M = A + B \), with \( A \in \mathcal{A} \) and \( B \in {\mathcal{A}}^{ \times } \) . | Proof. \( \mathcal{A} \) is a closed subspace of \( {\mathbf{M}}^{2} \), so each \( M \in {\mathbf{M}}^{2} \) has a unique decomposition into \( M = A + B \) with \( A \in \mathcal{A} \) and \( B \in {\mathcal{A}}^{ \bot } \) . However \( {\mathcal{A}}^{ \bot } = {\mathcal{A}}^{ \times } \) by Theorem 37. | Yes |
Corollary 2. Let \( M, N \in {\mathbf{M}}^{2} \), and let \( L \) be the projection of \( N \) onto \( \mathcal{S}\left( M\right) \), the stable subspace generated by \( M \). Then there exists a predictable process \( H \) such that \( L = H \cdot M \). | Proof. We know that such an \( L \) exists by Corollary 1. Since \( \{ M\} \) consists of just one element we can apply Theorem 36 to obtain the result. | No |
Corollary 3. Let \( \mathcal{A} = \left\{ {{M}^{1},\ldots ,{M}^{n}}\right\} \subset {\mathbf{M}}^{2} \), and suppose \( {M}^{i},{M}^{j} \) are strongly orthogonal for \( i \neq j \) . Suppose further that if \( N \in {\mathbf{M}}^{2}, N \bot \mathcal{A} \) in the strong sense implies that \( N = 0 \) . Then \( \mathcal... | Proof. By Theorem 36 we have \( \mathcal{S}\left( \mathcal{A}\right) = \mathcal{I} \) . The hypotheses imply that \( \mathcal{S}{\left( \mathcal{A}\right) }^{ \bot } = \{ 0\} \), hence \( \mathcal{S}\left( \mathcal{A}\right) = {\mathbf{M}}^{2} \). | Yes |
Theorem 38. Let \( \mathcal{A} \subset {\mathbf{M}}^{2} \) . If \( \mathcal{S}\left( \mathcal{A}\right) = {\mathbf{M}}^{2} \) then \( P \) is an extremal point of \( {\mathcal{M}}^{2}\left( \mathcal{A}\right) \) | Proof. Suppose \( P \) is not extremal. We will show that \( \mathcal{S}\left( \mathcal{A}\right) \neq {\mathbf{M}}^{2} \) . Since \( P \) is not extremal, there exist \( Q, R \in {\mathcal{M}}^{2}\left( \mathcal{A}\right), Q \neq R \), such that \( P = {\lambda Q} + \left( {1 - \lambda }\right) R \) ,\n\n\( 0 < \lambd... | Yes |
Theorem 39. Let \( \mathcal{A} \subset {\mathbf{M}}^{2} \) . If \( P \) is an extremal point of \( {\mathcal{M}}^{2}\left( \mathcal{A}\right) \), then every bounded \( P \) martingale strongly orthogonal to \( \mathcal{A} \) is null. | Proof. Let \( L \) be a bounded nonconstant martingale strongly orthogonal to \( \mathcal{A} \) . Let \( c \) be a bound for \( \left| L\right| \), and set\n\n\[ \n{dQ} = \left( {1 - \frac{{L}_{\infty }}{2c}}\right) {dP}\;\text{ and }\;{dR} = \left( {1 + \frac{{L}_{\infty }}{2c}}\right) {dP}.\n\]\n\nWe have \( Q, R \in... | Yes |
Theorem 41. Let \( M \in {\mathbf{M}}^{2},{Y}^{n} \in {\mathbf{M}}^{2}, n \geq 1 \), and suppose \( {Y}_{\infty }^{n} \) converges to \( {Y}_{\infty } \) in \( {L}^{2} \), and that there exists a sequence \( {H}^{n} \in L\left( M\right) \) such that \( {Y}_{t}^{n} = \) \( {\int }_{0}^{t}{H}_{s}^{n}d{M}_{s}, n \geq 1 \)... | Proof. If \( {Y}_{\infty }^{n} \) converges to \( {Y}_{\infty } \) in \( {L}^{2} \), then \( {Y}^{n} \) converges to \( Y \) in \( {\mathbf{M}}^{2} \) . By Theorem 36 we have that \( \mathcal{S}\left( M\right) = \mathcal{I}\left( M\right) \), the stochastic integrals with respect to \( M \) . Moreover \( {Y}^{n} \in \m... | Yes |
Theorem 42. Let \( \mathcal{A} = \left\{ {{M}^{1},{M}^{2},\ldots ,{M}^{n},\ldots }\right\} \), with \( {M}^{i} \in {\mathbf{M}}^{2} \), and suppose there exist disjoint predictable sets \( {\Lambda }^{i} \) such that \( {1}_{{\Lambda }^{i}}d\left\lbrack {{M}^{i},{M}^{i}}\right\rbrack = d\left\lbrack {{M}^{i},{M}^{i}}\r... | Proof. Let \( {N}^{n} = \mathop{\sum }\limits_{{i = 1}}^{n}{M}^{i} \) . Then \( {\left\lbrack {N}^{n},{N}^{n}\right\rbrack }_{t} = \mathop{\sum }\limits_{{i = 1}}^{n}{\int }_{0}^{t}{1}_{{A}_{i}}\left( s\right) d{\left\lbrack {M}^{i},{M}^{i}\right\rbrack }_{s} \) , hence \( E\left\{ {\left( {N}_{\infty }^{n}\right) }^{2... | Yes |
Theorem 43. Let \( X = \left( {{X}^{1},\ldots ,{X}^{n}}\right) \) be an \( n \) -dimensional Brownian motion and let \( \mathbb{F} = {\left( {\mathcal{F}}_{t}\right) }_{0 \leq t \leq \infty } \) denote its completed natural filtration. Then every locally square integrable local martingale \( M \) for \( \mathbb{F} \) h... | Proof. Fix \( {t}_{0},0 < t < {t}_{0} \), and assume \( X \) is stopped at \( {t}_{0} \) . Then letting \( \mathcal{A} = \left\{ {{X}^{1},\ldots ,{X}^{n}}\right\} \), we have that \( \mathcal{A} \subset {\mathbf{M}}^{2} \) . Let \( {\mathcal{M}}^{2}\left( \mathcal{A}\right) \) be all probability measures \( Q \) such t... | Yes |
Corollary 1. As in Theorem 43, let \( \mathbb{F} \) be the completed natural filtration of an \( n \) -dimensional Brownian motion. Then every local martingale \( M \) for \( \mathbb{F} \) is continuous. | Proof. In the proof of Theorem 43 we saw that the underlying probability law \( P \) is extremal for \( \mathcal{A} = \left\{ {{X}^{1},\ldots ,{X}^{n}}\right\} \) . Therefore by Theorem 40(c), every uniformly integrable martingale is continuous. The corollary follows by stopping. | No |
Corollary 2. Let \( X = \left( {{X}^{1},\ldots ,{X}^{n}}\right) \) be an \( n \) -dimensional Brownian motion and let \( \mathbb{F} \) be its completed natural filtration. Then every local martingale \( M \) for \( \mathbb{F} \) has a representation\n\n\[ \n{M}_{t} = {M}_{0} + \mathop{\sum }\limits_{{i = 1}}^{n}{\int }... | Proof. By Corollary 1 any local martingale \( M \) is continuous, hence it is locally square integrable. It remains only to apply Theorem 43. | No |
Let \( X = \left( {{X}^{1},\ldots ,{X}^{n}}\right) \) be an \( n \) -dimensional Brownian motion and let \( \overline{\mathbb{F}} \) be its completed natural filtration. Let \( Z \in {\mathcal{F}}_{\infty } \) be in \( {L}^{1} \) . Then there exist \( {H}^{i} \) predictable in \( L\left( {X}^{i}\right) \) with \( {\int... | Let \( {Z}_{t} = E\left\{ {Z \mid {\mathcal{F}}_{t}}\right\} \), taking the càdlàg (and hence continuous) version. By Corollary 2 we have \[ {Z}_{t} = {Z}_{0} + \mathop{\sum }\limits_{{i = 1}}^{n}{\int }_{0}^{t}{H}_{s}^{i}d{X}_{s}^{i} \] By Theorem 42 of Chap. II we have that \( {Z}_{t} = {B}_{{\left\lbrack Z, Z\right\... | Yes |
Corollary 4. Let \( X = \left( {{X}^{1},\ldots ,{X}^{n}}\right) \) be an \( n \) -dimensional Brownian motion and let \( \mathbb{F} \) be its completed natural filtration. Let \( Z \in {L}^{1}\left( {\mathcal{F}}_{\infty }\right) \) and \( Z > 0 \) a.s. Then there exist \( {J}^{i} \) predictable with \( {\int }_{0}^{\i... | Proof. By Corollary 3 there exist predictable \( {H}^{i} \) such that if \( {Z}_{t} = E\left\{ {Z \mid {\mathcal{F}}_{t}}\right\} \), then \[ {Z}_{t} = E\{ Z\} + \mathop{\sum }\limits_{{i = 1}}^{n}{\int }_{0}^{t}{H}_{s}^{i}d{X}_{s}^{i} \] Therefore \[ \log \left( {Z}_{t}\right) = \log \left( {Z}_{0}\right) + \mathop{\s... | Yes |
Corollary 5. Let \( \mathbb{F} \) be the completed natural filtration of an \( n \) -dimensional Brownian motion. If \( T \) is a totally inaccessible stopping time, then \( T = \infty \) a.s. | Proof. This is merely Theorem 40 (a). | No |
Theorem 44. Let \( \left( {\Omega ,\mathcal{F},\mathbb{F}, P}\right) \) be a filtered complete probability space satisfying the usual hypotheses. Assume also that the \( \sigma \) -algebra \( \mathcal{F} = {\mathcal{F}}_{\infty } \) and that it is separable. Then there exists a countable sequence of martingales \( \lef... | Proof. Since \( \mathcal{F} \) is separable, there exists a countable basis of \( {L}^{2} \), call it \( \left( {{M}^{0},{M}^{1},{M}^{2},\ldots }\right) \), where \( {M}^{0} \) is constant and \( E\left\{ {M}^{i}\right\} = 0 \) for \( i \neq 0 \) . One can take such a basis to be orthonormal in the usual way. That is, ... | Yes |
Theorem 45 (Lebesgue’s Change of Time Formula). Let a be a positive, finite, right continuous, increasing function on \( \lbrack 0,\infty ) \) . Let \( c \) denote its right continuous inverse (change of time). Let \( f \) be a positive Borel function on \( \lbrack 0,\infty ) \) . If \( G \) is any positive, finite, ri... | Proof. First consider \( f \) of the form \( f\left( s\right) = {1}_{\left\lbrack 0, u\right\rbrack }\left( s\right) \), for \( 0 \leq u < \infty \) . The left side of the equation then reduces to \( G\left( {a\left( u\right) }\right) \) . For the right side note that \( f\left( {c\left( {s - }\right) }\right) {1}_{\{ ... | Yes |
Theorem 46. Let \( \left( {\Omega ,\mathcal{F},\mathbb{F}, P}\right) \) be an absolutely continuous space. Then the compensators for all adapted counting processes with totally inaccessible jump times and without explosions, are absolutely continuous. | Proof. Let \( N \) be an adapted counting process and let \( \widetilde{N} \) be its compensator, so that \( X = N - \widetilde{N} \) is a locally square integrable local martingale. Since \( \widetilde{N} \) is continuous we have \( {\left\lbrack X, X\right\rbrack }_{t} = \mathop{\sum }\limits_{{s \leq t}}{\left( \Del... | Yes |
Theorem 49. \( {\mathcal{H}}^{2} \subset {\mathcal{H}}^{1} \) and local martingales of integrable variation are a subset of \( {\mathcal{H}}^{1} \) . | Proof. First note that if \( M \) is a càdlàg martingale, then \( E\left\{ \sqrt{{\left\lbrack M, M\right\rbrack }_{\infty }}\right\} \leq \) \( {\left( E\left\{ {\left\lbrack M, M\right\rbrack }_{\infty }\right\} \right) }^{1/2} \) which gives the first statement. For the second, if \( M \) has integrable variation th... | Yes |
Theorem 50. \( {\mathcal{H}}^{2} \) is dense in \( {\mathcal{H}}^{1} \) and bounded martingales are dense in \( {\mathcal{H}}^{1} \) . | Proof. Let \( M \) be a local martingale in \( {\mathcal{H}}^{1} \) . By the Fundamental Theorem of Local Martingales (Theorem 25 of Chap. III) we have \( M = N + U \) where \( N \) is a local martingale with jumps bounded by 1 and \( U \) has paths of locally integrable variation. If \( {\left( {T}_{n}\right) }_{n \ge... | Yes |
Theorem 51. Let \( M \) be a local martingale. Then \( M \) is locally in \( {\mathcal{H}}^{1} \) . | Proof. By the Fundamental Theorem of Local Martingales we know that \( M = \) \( N + U \), where \( N \) has jumps bounded by a constant \( \beta \) and \( U \) is locally of integrable variation. By stopping, we thus assume that \( N \) is bounded and \( U \) has paths of integrable variation. The result then follows ... | Yes |
Corollary 2. Let \( N \) be a local martingale in \( {BMO} \). Then \( N \) is locally bounded. | Proof. Let \( c \) be a bound for the jumps of \( N \) which we know exists by the previous corollary. Next let \( {T}_{n} = \inf \left\{ {t > 0 : \left| {N}_{t}\right| \geq n}\right\} \). Then \( \left| {N}^{{T}_{n}}\right| \leq n + c \), and \( N \) is locally bounded. | Yes |
Theorem 55 (The Dual of \( {\mathcal{H}}^{1} \) is BMO). The Banach space dual of all (bounded) linear functionals on \( {\mathcal{H}}^{1} \) can be identified with BMO. Moreover if \( {L}_{N} \) is such a functional then the norms \( \begin{Vmatrix}{L}_{N}\end{Vmatrix} \) and \( \parallel N{\parallel }_{BMO} \) are eq... | Proof. Let \( N \) be in \( {BMO} \) . By Fefferman’s inequality we have\n\n\[ \left| {{L}_{N}\left( M\right) }\right| = \left| {E\left\{ {\left\lbrack M, N\right\rbrack }_{\infty }\right\} }\right| \leq c\parallel N{\parallel }_{BMO}\parallel M{\parallel }_{{\mathcal{H}}^{1}} \]\n\n\( \left( *\right) \)\n\nfor all \( ... | Yes |
Theorem 57 (Jacod-Yor Theorem on Martingale Representation). Let \( \mathcal{A} \) be a subset of \( {\mathcal{H}}^{2} \) containing constant martingales. Then \( \mathcal{S}\left( \mathcal{A}\right) \), the stable subspace of stochastic integrals generated by \( \mathcal{A} \), equals \( {\mathcal{H}}^{2} \) if and on... | Proof. The necessity has already been proved in Theorem 38. By the Hahn-Banach Theorem, \( {\mathcal{H}}^{1} = \mathcal{S}\left( \mathcal{A}\right) \) if and only if \( L\left( {\mathcal{S}\left( \mathcal{A}\right) }\right) = 0 \) implies \( L \) is identically zero, where \( L \) is a bounded linear functional. Let \(... | Yes |
Consider the equation\n\n\[ \n{\left\lbrack X, X\right\rbrack }_{t} - t = {\int }_{0}^{t}\phi \left( {X}_{s - }\right) d{X}_{s} \]\n\n\( \left( \otimes \right) \)\n\non a filtered probability space \( \left( {\Omega ,\mathcal{F},\mathbb{F}, P}\right) \) which satisfies the usual hypotheses. Then \( X \) has martingale ... | Proof. By the Jacod-Yor Theorem (Theorem 57) we need to verify that \( P \) is extremal in the set \( {\mathcal{M}}^{2} \) of all probability measures such that \( X \) is a square integrable martingale. It is clearly true if \( P \) is extremal. Suppose then that \( P \) is not extremal, and let \( Q \) and \( R \) bo... | Yes |
Theorem 62. Let \( {Y}^{n}\left( {a, t,\omega }\right) \) be a sequence of processes that are (i) \( \mathcal{A} \otimes \) \( \mathcal{B}\left( {\mathbb{R}}_{ + }\right) \otimes \mathcal{F} \) measurable, and (ii) for each fixed a the process \( {Y}^{n}\left( {a, t,\omega }\right) \) is càdlàg. Suppose \( {Y}^{n}\left... | Proof. Let \( {S}_{u, i, j}^{a} = \mathop{\sup }\limits_{{t \leq u}}\left| {{Y}^{i}\left( {a, t, \cdot }\right) - {Y}^{j}\left( {a, t, \cdot }\right) }\right| \) . Since \( {Y}^{i} \) is càdlàg in \( t \) the function \( \left( {a,\omega }\right) \mapsto {S}_{u, i, j}^{a} \) is \( \mathcal{A} \otimes \mathcal{F} \) mea... | Yes |
Theorem 63. Let \( X \) be a semimartingale with \( {X}_{0} = 0 \) a.s. and let \( H\left( {a, t,\omega }\right) = \) \( {H}_{t}^{a}\left( \omega \right) \) be \( \mathcal{A} \otimes \mathcal{P} \) measurable \( {}^{16} \) and bounded. Then there is a function \( Z\left( {a, t,\omega }\right) \) in \( \mathcal{A} \otim... | Proof. Let \( \mathcal{H} = \{ H \in \mathbf{b}\mathcal{A} \otimes \mathcal{P} \) such that the conclusion of the theorem holds \( \} \) . If \( K = K\left( {t,\omega }\right) \in \mathbf{b}\mathcal{P} \) and \( f = f\left( a\right) \in \mathbf{b}\mathcal{A} \), and if \( H\left( {a, t,\omega }\right) = f\left( a\right... | Yes |
Theorem 65 (Fubini’s Theorem: Second Version). Let \( X \) be a semi-martingale, let \( {H}_{t}^{a} = H\left( {a, t,\omega }\right) \) be \( \mathcal{A} \otimes \mathcal{P} \) measurable, let \( \mu \) be a finite positive measure on \( A \), and assume \[ {\left( {\int }_{A}{\left( {H}_{t}^{a}\right) }^{2}\mu \left( d... | Proof. By pre-stopping we may assume without loss of generality that \( X \in {\mathcal{H}}^{2} \) and that \( {\begin{Vmatrix}{H}^{a}\end{Vmatrix}}_{{L}^{2}\left( {d\mu }\right) } \) is \( \left( {{\mathcal{H}}^{2}, X}\right) \) integrable. Let \( X = \bar{N} + \bar{A} \) be the canonical decomposition of \( X \) . Th... | Yes |
Corollary 1. Let \( X \) be a semimartingale. Then \( \left| X\right| ,{X}^{ + },{X}^{ - } \) are all semi-martingales. | Proof. The functions \( f\left( x\right) = \left| x\right|, g\left( x\right) = {x}^{ + } \), and \( h\left( x\right) = {x}^{ - } \) are all convex, so the result then follows by Theorem 66. | Yes |
Corollary 2. Let \( X, Y \) be semimartingales. Then \( X \vee Y \) and \( X \land Y \) are semimartingales. | Proof. Since semimartingales form a vector space and \( x \vee y = \frac{1}{2}\left( {\left| {x - y}\right| + x + y}\right) \) and \( x \land y = \frac{1}{2}\left( {x + y - \left| {x - y}\right| }\right) \), the result is an immediate consequence of Corollary 1. | Yes |
Theorem 67. The space of semimartingales is a vector space, an algebra, a lattice, and is stable under \( {\mathcal{C}}^{2} \), and more generally under convex transformations. | Proof. In Chap. II we saw that semimartingales form a vector space (Theorem 1), an algebra (Corollary 2 of Theorem 22: Integration by Parts), and that they are stable under \( {\mathcal{C}}^{2} \) transformations (Theorem 32: Itô’s Formula). That they form a lattice is by Corollary 2 above, and that they are stable und... | Yes |
Theorem 68. Let \( X \) be a semimartingale and let \( {L}^{a} \) be its local time at \( a \) . Then\n\n\[ \n{\left( {X}_{t} - a\right) }^{ + } - {\left( {X}_{0} - a\right) }^{ + } = {\int }_{0 + }^{t}{1}_{\left\{ {X}_{s - } > a\right\} }d{X}_{s} + \mathop{\sum }\limits_{{0 < s \leq t}}{1}_{\left\{ {X}_{s - } > a\righ... | Proof. Applying Theorem 66 to the convex functions \( f\left( x\right) = {\left( x - a\right) }^{ + } \) and \( g\left( x\right) = {\left( x - a\right) }^{ - } \) we get\n\n\[ \nf\left( {X}_{t}\right) = f\left( {X}_{0}\right) + {\int }_{0 + }^{t}{f}^{\prime }\left( {X}_{s - }\right) d{X}_{s} + {C}_{t}^{ + } \n\]\n\n\[ ... | Yes |
Theorem 69. Let \( X \) be a semimartingale, and let \( {L}_{t}^{a} \) be its local time at the level \( a \), each \( a \in \mathbb{R} \) . For a.a. \( \omega \), the measure in \( t, d{L}_{t}^{a}\left( \omega \right) \), is carried by the set \( \left\{ {s : {X}_{s - }\left( \omega \right) = {X}_{s}\left( \omega \rig... | Proof. Since \( {L}_{t}^{a} \) has continuous paths, the measure \( d{L}_{t}^{a}\left( \omega \right) \) is diffuse, and since \( \left\{ {s : {X}_{s - }\left( \omega \right) = a}\right\} \) and \( \left\{ {s : {X}_{s - }\left( \omega \right) = {X}_{s}\left( \omega \right) = a}\right\} \) differ by at most a countable ... | Yes |
Corollary 3 (Meyer-Tanaka Formula). Let \( X \) be a semimartingale with continuous paths. Then\n\n\[ \left| {X}_{t}\right| = \left| {X}_{0}\right| + {\int }_{0 + }^{t}\operatorname{sign}\left( {X}_{s}\right) d{X}_{s} + {L}_{t}^{0} \] | Proof. This is merely Theorem 70 with \( f\left( x\right) = \left| x\right| \), which implies \( \mu \left( {da}\right) = \) \( 2{\varepsilon }_{0}\left( {da}\right) \), point mass at 0 . The formula also follows trivially from the definition of \( {L}^{0} \) . | Yes |
Theorem 71. Let \( X \) be a continuous local martingale with \( {X}_{0} = 0 \), and let \( 0 < \alpha < 1/2 \) . Then \( {Y}_{t} = {\left| {X}_{t}\right| }^{\alpha } \) is not a semimartingale unless \( X \) is identically zero. | Proof. Let us suppose that \( Y = {\left| X\right| }^{\alpha } \) is a semimartingale. Let \( \beta = 1/\alpha > 2 \) . We then have\n\n\[ \left| {X}_{t}\right| = {Y}_{t}^{\beta } = \beta {\int }_{0}^{t}{Y}_{s}^{\beta - 1}d{Y}_{s} + \frac{\beta \left( {\beta - 1}\right) }{2}{\int }_{0}^{t}{Y}_{s}^{\beta - 2}d{\left\lbr... | Yes |
Corollary 2. Let \( B \) be standard Brownian motion. Then there is a version of \( B \) with continuous paths, a.s. | Proof. Since \( {B}_{t} - {B}_{s} \) is Gaussian with mean zero and variance \( t - s \), we know that \( E\left\{ {\left| {B}_{t} - {B}_{s}\right| }^{4}\right\} \leq c{\left( t - s\right) }^{2} \) . (One can give a cute proof of this moment estimate using the scaling property of Brownian motion.) If we think of time a... | No |
Theorem 73 (Burkholder’s Inequality). Let \( X \) be a continuous local martingale with \( {X}_{0} = 0,2 \leq p < \infty \), and \( T \) a finite stopping time. Then\n\n\[ E\left\{ {\left( {X}_{T}^{ * }\right) }^{p}\right\} \leq {C}_{p}E\left\{ {\left\lbrack X, X\right\rbrack }_{T}^{p/2}\right\} \]\n\nwhere \( {C}_{p} ... | Proof. By stopping, it suffices to consider the case where \( X \) and \( \left\lbrack {X, X}\right\rbrack \) are bounded. By Itô's formula we have\n\n\[ {\left| {X}_{T}\right| }^{p} = p{\int }_{0}^{T}\operatorname{sign}\left( {X}_{s}\right) {\left| {X}_{s}\right| }^{p - 1}d{X}_{s} + \frac{p\left( {p - 1}\right) }{2}{\... | Yes |
Theorem 74. Let \( X \) be a semimartingale satisfying Hypothesis \( A \) . There exists a version of \( {\left( {\widehat{X}}^{c}\right) }_{t}^{a} \) such that \( \left( {a, t,\omega }\right) \mapsto {\left( {\widehat{X}}^{c}\right) }_{t}^{a}\left( \omega \right) \) is \( \mathcal{B}\left( \mathbb{R}\right) \otimes \m... | Proof. Without loss of generality we may assume \( X - {X}_{0} \in {\mathcal{H}}^{2} \) . If it is not, we can stop \( X - {X}_{0} \) at \( {T}^{n} - \) . The continuous local martingale part of \( {X}^{{T}^{n} - } \) is then just \( {\left( {X}^{c}\right) }^{{T}^{n}} \) . Suppose \( - \infty < a < b < \infty \), and l... | Yes |
Theorem 75. Let \( X \) be a semimartingale satisfying Hypothesis \( A \) . Then there exists a \( \mathcal{B}\left( \mathbb{R}\right) \otimes \mathcal{P} \) measurable version of \( \left( {a, t,\omega }\right) \mapsto {L}_{t}^{a}\left( \omega \right) \) which is everywhere jointly right continuous in a and continuous... | Proof. Since \( X \) satisfies Hypothesis A, the process \( {J}_{t} = \mathop{\sum }\limits_{{0 < s < t}}\Delta {X}_{s} \) is an \( {FV} \) semimartingale, and \( Y = X - J \) is a continuous semimartingale. We let \( Y = M + A \) be the (unique) decomposition of \( Y \), with \( {M}_{0} = {A}_{0} = 0 \) . Then \( X = ... | Yes |
Let \( X \) be a semimartingale satisfying Hypothesis \( A \) . Let \( X = \) \( M + A + J \) be a decomposition with \( M \) and \( A \) continuous and \( J \) the jump process of \( X \), and let \( {\left( {L}_{t}^{a}\right) }_{t \geq 0} \) be its local time at the level \( a \) . Then | \[ {L}_{t}^{a} - {L}_{t}^{a - } = 2{\int }_{0}^{t}{1}_{\left\{ {X}_{s - } = a\right\} }d{A}_{s} \] \[ = 2{\int }_{0}^{t}{1}_{\left\{ {X}_{s} = a\right\} }d{A}_{s} \] | Yes |
Corollary 2. Let \( X \) be a semimartingale satisfying Hypothesis A. Let \( A \) be as in Corollary 1. The local time \( \left( {L}_{t}^{a}\right) \) is continuous in \( t \) and is continuous at \( a = {a}_{0} \) if and only if | \[ {\int }_{0}^{\infty }{1}_{\left\{ {X}_{s} = {a}_{0}\right\} }\left| {d{A}_{s}}\right| = 0. \] | Yes |
Corollary 3. Let \( X \) be a semimartingale satisfying Hypothesis A. Then for every \( \left( {a, t}\right) \) we have\n\n\[ \n{L}_{t}^{a} = \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}\frac{1}{\varepsilon }{\int }_{0}^{t}{1}_{\left\{ a \leq {X}_{s} \leq a + \varepsilon \right\} }d{\left\lbrack X, X\right\rbrac... | The lack of symmetry in the above formula stems from the definition of local time, where we defined \( \operatorname{sign}\left( x\right) \) in an asymmetric way:\n\n\[ \n\operatorname{sign}\left( x\right) = \left\{ \begin{array}{ll} 1, & x > 0 \\ - 1, & x \leq 0 \end{array}\right.\n\]\n\nA symmetrized result follows t... | No |
Theorem 76. Let \( X \) be a semimartingale satisfying Hypothesis \( A \) , \( U \) a positive random variable, \( {L}^{a} \) the local times of \( X \) . Then the operation\n\n\[ f \mapsto \mathop{\sum }\limits_{{i = 1}}^{n}{f}_{i}\left( {{L}_{U}^{{a}_{i + 1}} - {L}_{U}^{{a}_{i}}}\right) \]\n\nwhere \( f\left( x\right... | Proof. Recall that \( {L}^{0} \) denotes finite-valued random variables. By Theorem 68 we know that\n\n\[ \frac{1}{2}{L}_{t}^{a} = {\left( {X}_{t} - a\right) }^{ - } - {\left( {X}_{0} - a\right) }^{ - } + {\int }_{0 + }^{t}{1}_{\left\{ {X}_{s - } \leq a\right\} }d{X}_{s} \]\n\n\[ - \mathop{\sum }\limits_{{0 < s \leq t}... | Yes |
Theorem 77 (Bouleau-Yor Formula). Let \( X \) be a semimartingale satisfying Hypothesis \( A, U \) a positive random variable, \( f \) a bounded, Borel function, and \( F\left( x\right) = {\int }_{0}^{x}f\left( u\right) {du} \) . Then | \[ F\left( {X}_{U}\right) - F\left( {X}_{0}\right) = {\int }_{0 + }^{U}f\left( {X}_{s - }\right) d{X}_{s} - \frac{1}{2}\int f\left( a\right) {d}_{a}{L}_{U}^{a} + \mathop{\sum }\limits_{{0 < s \leq U}}\left\{ {F\left( {X}_{s}\right) - F\left( {X}_{s - }\right) - f\left( {X}_{s - }\right) \Delta {X}_{s}}\right\} \] | Yes |
Theorem 79. The process \( {g}_{t} \) is \( \mathbb{G} \) adapted, and \( P\left( {{g}_{t} \leq s}\right) = \frac{2}{\pi }\arcsin \sqrt{\frac{s}{t}} \) , \( 0 \leq s \leq t \) . | Proof. For \( s \leq t \) we have almost surely\n\n\[ \left\{ {{g}_{t} \leq s}\right\} = \left( {\mathop{\bigcap }\limits_{\substack{{u \in \mathbb{Q}} \\ {s < u < t} }}\left\{ {{B}_{u} > 0}\right\} }\right) \cup \left( {\mathop{\bigcap }\limits_{\substack{{u \in \mathbb{Q}} \\ {s < u < t} }}\left\{ {{B}_{u} < 0}\right... | Yes |
Theorem 80. Azéma’s martingale \( M \) is given by \( {M}_{t} = \operatorname{sign}\left( {B}_{t}\right) \sqrt{\frac{\pi }{2}}\sqrt{t - {g}_{t}} \) , \( t \geq 0 \) . | Proof. By definition,\n\n\[ \n{M}_{t} = E\left\{ {{B}_{t} \mid {\mathcal{G}}_{t}}\right\} = E\left\{ {\operatorname{sign}\left( {B}_{t}\right) \left| {B}_{t}\right| \mid {\mathcal{G}}_{t}}\right\} = \operatorname{sign}\left( {B}_{t}\right) E\left\{ {\left| {B}_{t}\right| \mid {\mathcal{G}}_{t}}\right\} .\n\]\n\nHowever... | Yes |
Theorem 81. The local time at zero of Azéma’s martingale is not identically zero. | Proof. By the Meyer-Itô formula (Theorem 70) for \( f\left( x\right) = \left| x\right| \) we have\n\n\[ \left| {M}_{t}\right| = {\int }_{0}^{t}\operatorname{sign}\left( {M}_{s - }\right) d{M}_{s} + {L}_{t}^{0}\left( M\right) + \mathop{\sum }\limits_{{0 < s \leq t}}\left\{ {\left| {M}_{s}\right| - \left| {M}_{s - }\righ... | Yes |
Theorem 82. The local times at all levels except zero of Azéma's martingale are 0 . That is, \( {L}_{t}^{a}\left( M\right) = 0, t \geq 0 \), if \( a \neq 0 \) . | Proof. By Theorem 69 we know that the measure \( d{L}_{s}^{a}\left( \omega \right) \) on \( {\mathbb{R}}_{ + } \) is carried by the set \( \left\{ {s : {M}_{s - }\left( \omega \right) = {M}_{s}\left( \omega \right) = a}\right\} \), a.s., for each \( a \) . However if \( a \neq 0 \), this set is countable. Since \( s \m... | Yes |
Theorem 84. The local time \( {L}^{0}\left( B\right) \) of Brownian motion is \( \mathbb{G} \) adapted. | Theorem 84 will be proved if one can express \( {L}^{0}\left( B\right) \) as the a.s. limit of measurable functionals of the zero set of Brownian motion. Such a result is classical: see, e.g., Kingman [126, page 730]. | No |
Theorem 85. The local times at zero of Brownian motion and Azéma’s martingale are the same. That is, \( {L}^{0}\left( B\right) = {L}^{0}\left( M\right) \) . | Proof. As we saw in the proof of Theorem 81, \( \left| {M}_{t}\right| - {L}_{t}^{0}\left( M\right) \) is a martingale. Since the process \( {L}^{0}\left( M\right) \) is non-decreasing, continuous and adapted, it is natural. If \( A \) is another natural process such that \( \left| M\right| - A \) is a martingale, then ... | Yes |
Corollary 1. The process \( \left| M\right| - {L}^{0}\left( B\right) \) is a \( \mathbb{G} \) martingale. | Proof. This corollary is proved at the end of the proof of Theorem 85. | No |
Corollary 2. The process \( {1}_{\left\{ {B}_{t} > 0\right\} }\sqrt{\frac{\pi }{2}}\sqrt{t - {g}_{t}} - \frac{1}{2}{L}_{t}^{0}\left( B\right) \) is a \( \mathbb{G} \) martingale. | Proof. Since \( {1}_{\left\{ {M}_{s - } > 0\right\} }{M}_{s}^{ - } = {1}_{\left\{ {M}_{s - } \leq 0\right\} }{M}_{s}^{ + } = 0 \), it follows from Theorem 68 that \( {M}_{t}^{ + } - \frac{1}{2}{L}_{t}^{0}\left( M\right) \) is a martingale. However \( {M}_{t}^{ + } = {1}_{\left\{ {B}_{t} > 0\right\} }\sqrt{\frac{\pi }{2... | Yes |
Theorem 87. Let \( M \) be Azéma’s martingale. Then \( {\left\lbrack M, M\right\rbrack }_{t} - \frac{\pi }{4}t \) is a martingale. That is, \( \langle M, M{\rangle }_{t} = \frac{\pi }{4}t \) . | Proof. Recall that \( {M}_{t}^{2} - {\left\lbrack M, M\right\rbrack }_{t} \) is a martingale (cf., Theorem 27 of Chap. II). By Theorems 80 and 86,\n\n\[ \n{M}_{t}^{2} - {\left\lbrack M, M\right\rbrack }_{t} = \frac{\pi }{2}\left\{ {\left( {t - {g}_{t}}\right) - {g}_{t}}\right\} \n\]\n\n\[ \n= \frac{\pi }{2}t - \pi {g}_... | No |
Theorem 88. Let \( X \) be a semimartingale. The following are equivalent:\n\n(i) \( X \) is a sigma martingale;\n\n(ii) \( X = H \cdot M \) where \( M \) is a local martingale;\n\n(iii) \( X = H \cdot M \) where \( M \) is a martingale;\n\n(iv) \( X = H \cdot M \) where \( M \) is a martingale in \( {\mathcal{H}}^{1} ... | Proof. It is clear that (iv) implies (iii), that (iii) implies (ii), and that (iii) obviously implies (i), so we need to prove only that (i) implies (iv). Without loss assume that \( {M}_{0} = 0 \) . Since \( X \) is a sigma martingale it has a representation of the form \( X = H \cdot M \) where \( M \) is a martingal... | Yes |
Corollary 1. A local sigma martingale is a sigma martingale. | Proof. Let \( X \) be a local sigma martingale, so that there exists a sequence of stopping times tending to \( \infty \) a.s. such that \( {X}^{{T}_{n}} \) is a sigma martingale for each \( n \) . Since \( {X}^{{T}_{n}} \) is a sigma martingale, there exists a martingale \( {M}^{n} \) in \( {\mathcal{H}}^{1} \) such t... | Yes |
Corollary 2. A local martingale is a sigma martingale. | Proof. This is simply a consequence of the fact that a local martingale is locally a martingale, and trivially a martingale is a sigma martingale. | No |
Theorem 89. If \( X \) is a local martingale and \( H \in L\left( X\right) \), then the stochastic integral \( H \cdot X \) is a sigma martingale. Moreover if \( X \) is a sigma martingale (and a fortiori a semimartingale) and \( H \in L\left( X\right) \), then \( H \cdot X \) is a sigma martingale. | Proof. Clearly it suffices to prove the second statement, since a local martingale is already a sigma martingale. But the second statement is simple. Since \( X \) is a sigma martingale we know an equivalent condition is that there exists a strictly positive predictable process \( \phi \) and a local martingale \( M \)... | Yes |
Theorem 90. A sigma martingale which is also a special semimartingale is a local martingale. | Proof. Any sigma martingale is a semimartingale by definition; here we also assume it is special. Thus it has a canonical decomposition \( X = M + A \) where \( M \) is a local martingale and \( A \) is a process of finite variation on compacts which is also predictable. We want to show \( A = 0 \) . We assume \( {A}_{... | Yes |
Theorem 1. Let \( Z \) be a semimartingale \( \left( {{Z}_{0} = 0}\right) \) . Then \( {\begin{Vmatrix}{\left\lbrack Z, Z\right\rbrack }_{\infty }^{1/2}\end{Vmatrix}}_{{L}^{p}} \leq \) \( \parallel Z{\parallel }_{{\underline{H}}^{p}},\left( {1 \leq p \leq \infty }\right) . | Proof. Let \( Z = M + A,{M}_{0} = {A}_{0} = 0 \), be a decomposition of \( Z \) . Then\n\n\[{\left\lbrack Z, Z\right\rbrack }_{\infty }^{1/2} \leq {\left\lbrack M, M\right\rbrack }_{\infty }^{1/2} + {\left\lbrack A, A\right\rbrack }_{\infty }^{1/2}\]\n\n\[= {\left\lbrack M, M\right\rbrack }_{\infty }^{1/2} + {\left( \m... | Yes |
Theorem 2. For \( 1 \leq p < \infty \) there exists a constant \( {c}_{p} \) such that for any semimartingale \( Z,{Z}_{0} = 0,\parallel Z{\parallel }_{{\underline{S}}^{p}} \leq {c}_{p}\parallel Z{\parallel }_{{\underline{H}}^{p}} \) . | Proof. A semimartingale \( Z \) is in \( \mathbb{D} \), so \( \parallel Z{\parallel }_{{\underline{S}}^{p}} \) makes sense. Let \( Z = M + A \) be a decomposition with \( {M}_{0} = {A}_{0} = 0 \) . Then\n\n\[ \parallel Z{\parallel }_{{\underline{\Xi }}^{p}}^{p} = E\left\{ {\left( {Z}_{\infty }^{ * }\right) }^{p}\right\... | Yes |
Theorem 3 (Emery’s Inequality). Let \( Z \) be a semimartingale, \( H \in \mathbb{L} \) , and \( \frac{1}{p} + \frac{1}{q} = \frac{1}{r}\left( {1 \leq p \leq \infty ,1 \leq q \leq \infty }\right) \) . Then \[ {\begin{Vmatrix}{\int }_{0}^{\infty }{H}_{s}d{Z}_{s}\end{Vmatrix}}_{{\underline{H}}^{r}} \leq \parallel H{\para... | Proof. Let \( H \cdot Z \) denote \( {\left( {\int }_{0}^{t}{H}_{s}d{Z}_{s}\right) }_{t \geq 0} \) . Recall that we always assume \( {Z}_{0} = 0 \) a.s., and let \( Z = M + A \) be a decomposition of \( Z \) with \( {M}_{0} = {A}_{0} = 0 \) a.s. Then \( H \cdot M + H \cdot A \) is a decomposition of \( H \cdot Z \) . H... | Yes |
Theorem 4. Let \( Z \) be a semimartingale \( \left( {{Z}_{0} = 0}\right) \) . Then \( Z \) is prelocally in \( {\underline{\underline{H}}}^{p},1 \leq p \leq \infty \) | Proof. By the Fundamental Theorem of Local Martingales (Theorem 25 of Chap. III) and the Bichteler-Dellacherie Theorem (Theorem 43 of Chap. III) we know that for given \( \varepsilon > 0, Z \) has a decomposition \( Z = M + A,{M}_{0} = {A}_{0} = 0 \) a.s., such that the jumps of the local martingale \( M \) are bounded... | Yes |
Theorem 5. Let \( Z \) be a semimartingale with \( {Z}_{0} = 0 \) a.s.\n\n(i) For \( \alpha > 0 \), if \( Z \in \mathcal{S}\left( \alpha \right) \) then for every stopping time \( T,{Z}^{T} \in \mathcal{S}\left( \alpha \right) \) and \( {Z}^{T - } \in \mathcal{S}\left( {2\alpha }\right) \). | Proof. Since \( {Z}^{T - } = {M}^{T} + \left( {{A}^{T - } - \Delta {M}_{T}{1}_{\lbrack T,\infty )}}\right) \), and since \( {\begin{Vmatrix}{Z}^{T}\end{Vmatrix}}_{{\underline{H}}^{\infty }} \leq \parallel Z{\parallel }_{{\underline{H}}^{\infty }} \) always, one concludes \( {\begin{Vmatrix}{Z}^{T - }\end{Vmatrix}}_{{\u... | Yes |
Let \( 1 \leq p < \infty \), let \( J \in {\underline{S}}^{p} \), let \( F \) be functional Lipschitz with \( F\left( 0\right) = 0 \), and suppose \( \mathop{\sup }\limits_{t}\left| {{K}_{t}\left( \omega \right) }\right| \leq k \) a.s. Let \( Z \) be a semimartingale in \( {\underline{\underline{H}}}^{\infty } \) such ... | Proof. Define \( \Lambda : {\underline{S}}^{p} \rightarrow {\underline{S}}^{p} \) by \( \Lambda {\left( X\right) }_{t} = {J}_{t} + {\int }_{0}^{t}F{\left( X\right) }_{s}d{Z}_{s} \) . Then by Theorems 2 and 3 the operator is \( 1/2 \) Lipschitz, and the fixed point theorem gives existence and uniqueness. Indeed\n\n\[ \n... | Yes |
Theorem 10. Let \( J,{J}^{n} \in \mathbb{D};Z \) be a semimartingale; \( F,{F}^{n} \) be functional Lipschitz with constants \( K,{K}_{n} \), respectively; and let \( {X}^{n}, X \) be the unique solutions of equations \( \left( {*n}\right) \) and \( \left( *\right) \), respectively. Assume that\n\n(i) \( {J}^{n}, J \) ... | Proof. Let \( {X}^{n} \) and \( X \) be the solutions of equations \( \left( {*n}\right) \) and \( \left( *\right) \), respectively. Then\n\n\[ X - {X}^{n} = J - {J}^{n} + {\left( F\left( X\right) - {F}^{n}\left( X\right) \right) }_{ - } \cdot Z + {\left( {F}^{n}\left( X\right) - {F}^{n}\left( {X}^{n}\right) \right) }_... | Yes |
Theorem 11. Let \( J,{J}^{n} \in \mathbb{D};Z \) be a semimartingale \( \left( {{Z}_{0} = 0}\right) \) ; and \( F,{F}^{n} \) be functional Lipschitz with Lipschitz processes \( K,{K}_{n} \), respectively. Let \( {X}^{n} \) , \( X \) be solutions respectively of \[ {X}_{t}^{n} = {J}_{t}^{n} + {\int }_{0}^{t}{F}^{n}{\lef... | Proof. By stopping at \( T - \) for an arbitrarily large stopping time \( T \) we can assume without loss of generality that \( Z \in \mathcal{S}\left( \frac{1}{2\sqrt{8}a}\right) \) by Theorem 5, and that \( {J}^{n} \) converges to \( J \) in \( {\underline{S}}^{2} \) and \( F\left( {X}^{n}\right) \) converges to \( F... | Yes |
Theorem 13. Let \( {J}^{n}, J \in \mathbb{D};Z,{Z}^{n} \) be semimartingales \( \left( {{Z}_{0}^{n} = {Z}_{0} = 0\text{a.s.}}\right) \) ; and \( F,{F}^{n} \) be functional Lipschitz with Lipschitz processes \( K,{K}_{n} \), respectively. Let \( {X}^{n}, X \) be solutions of \( \left( {*n}\right) \) and \( \left( *\righ... | Proof. By stopping at \( T \) - for an arbitrarily large stopping time \( T \) we can assume without loss that \( Z \in \mathcal{S}\left( \frac{1}{2\sqrt{8}a}\right) \) by Theorem 5, and that \( {J}^{n} \) converges to \( J \) in \( {\underline{S}}^{2},{F}^{n}\left( X\right) \) converges in \( {\underline{S}}^{2} \) to... | Yes |
Theorem 17. Let \( Z \) be a semimartingale and let \( X = \mathcal{E}\left( Z\right) \), the stochastic exponential of \( Z \) . That is, \( X \) is the solution of\n\n\[ \n{X}_{t} = 1 + {\int }_{0}^{t}{X}_{s - }d{Z}_{s}\n\]\n\nLet \( {\sigma }_{n} \) be a sequence of random partitions tending to the identity. Let\n\n... | Proof. Let \( {Y}^{n} \) be the solution of\n\n\[ \n{Y}_{t} = 1 + {\int }_{0}^{t}{Y}_{s}^{{\sigma }_{n}}d{Z}_{s}\n\]\n\nequation \( \left( {*\sigma }\right) \) of Theorem 16. By the corollary of Theorem 16 we know that \( {Y}^{n} \) converges to \( X = \mathcal{E}\left( Z\right) \) in ucp. Thus it suffices to show \( {... | Yes |
Theorem 18 (Generalized Itô’s Formula). Let \( \mathbf{X} = \left( {{X}^{1},\ldots ,{X}^{n}}\right) \) be an \( n \) -tuple of semimartingales, and let \( f : {\mathbb{R}}_{ + } \times \Omega \times {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) be such that (i) there exists an adapted FV process \( A \) and a function \( ... | Proof. We sketch the proof for \( n = 1 \) . We have, letting \( 0 = {t}_{0} \leq {t}_{1} \leq \cdots \leq \) \( {t}_{m} = t \) be a partition of \( \left\lbrack {0, t}\right\rbrack \), and assuming temporarily \( \left| X\right| \leq k \) for all \( s \leq t \) , \( k \) a constant, \[ f\left( {t,\omega ,{X}_{t}}\righ... | Yes |
Theorem 19. Let \( \mathbf{X} = \left( {{X}^{1},\ldots ,{X}^{d}}\right) \) be a vector of semimartingales and let \( f : {\mathbb{R}}_{ + } \times \Omega \times {\mathbb{R}}^{d} \rightarrow \mathbb{R} \) be such that\n\n(i) there exists an adapted FV process \( A \) and a function \( g \) such that\n\n\[ f\left( {t,\om... | Proof. First assume that for fixed \( \left( {t,\omega }\right) \) the function \( f \) and all its first partials are bounded functions of \( \mathbf{x} \) . Then by optional stopping at times \( T - \) we can assume without loss of generality that \( f \) and all its first partials (in \( \mathbf{x} \) ) are bounded ... | Yes |
Theorem 20. Let \( X \) be a semimartingale and let \( f \) be \( {\mathcal{C}}^{2} \). Then\n\n\[ f\left( {X}_{t}\right) - f\left( {X}_{0}\right) = {\int }_{0 + }^{t}{f}^{\prime }\left( {X}_{s - }\right) \circ d{X}_{s} + \mathop{\sum }\limits_{{0 < s \leq t}}\left\{ {f\left( {X}_{s}\right) - f\left( {X}_{s - }\right) ... | Proof. Note that \( {f}^{\prime } \) is \( {\mathcal{C}}^{1} \), so that \( {f}^{\prime }\left( X\right) \) is in the domain of definition of the F-S integral by Theorem 19. Also by definition we have\n\n\[ {\int }_{0 + }^{t}{f}^{\prime }\left( {X}_{s - }\right) \circ d{X}_{s} = {\int }_{0 + }^{t}{f}^{\prime }\left( {X... | Yes |
Theorem 21. Let \( \mathbf{X} = \left( {{X}^{1},\ldots ,{X}^{n}}\right) \) be an \( n \) -tuple of semimartingales, and let \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) have second order continuous partial derivatives. Then \( f\left( \mathbf{X}\right) \) is a semimartingale and the following formula holds: | \[ f\left( {\mathbf{X}}_{t}\right) - f\left( {\mathbf{X}}_{0}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{\int }_{0 + }^{t}\frac{\partial f}{\partial {x}_{i}}\left( {\mathbf{X}}_{s - }\right) \circ d{X}_{s}^{i} + \mathop{\sum }\limits_{{0 < s \leq t}}\left\{ {f\left( {\mathbf{X}}_{s}\right) - f\left( {\mathbf{X}}_{s -... | Yes |
Corollary 1 (Stratonovich Integration by Parts Theorem). Let \( X \) and \( Y \) be semimartingales. Then\n\n\[ \n{X}_{t}{Y}_{t} - {X}_{0}{Y}_{0} = {\int }_{0 + }^{t}{X}_{s - } \circ d{Y}_{s} + {\int }_{0 + }^{t}{Y}_{s - } \circ d{X}_{s} + \mathop{\sum }\limits_{{0 < s \leq t}}\Delta {X}_{s}\Delta {Y}_{s} \n\] | Proof. Let \( f\left( {x, y}\right) = {xy} \) and apply Theorem 21. | No |
Corollary 2. Let \( X \) and \( Y \) be semimartingales, with at least one of \( X \) or \( Y \) continuous. Then\n\n\[{X}_{t}{Y}_{t} - {X}_{0}{Y}_{0} = {\int }_{0 + }^{t}{X}_{s - } \circ d{Y}_{s} + {\int }_{0 + }^{t}{Y}_{s - } \circ d{X}_{s}.\] | Recall that since \( {X}_{0 - } = 0 \) by convention for a càdlàg process \( X \), we did not really need to write \( {\int }_{0 + }^{t} \) ; the formula also holds for \( {\int }_{0}^{t} \) . | No |
Given a vector of semimartingales \( Z = \left( {{Z}^{1},\ldots ,{Z}^{k}}\right) \), semi-martingales \( {J}^{i}\left( {1 \leq i \leq d}\right) \), and \( F \) - \( S \) acceptable functions \( {f}_{j}^{i}(1 \leq i \leq d,1 \leq \) \( j \leq k \) ), then the system of equations\n\n\[ \n{X}_{t}^{i} = {J}_{t}^{i} + \math... | Proof. We note that equation \( \left( {* * }\right) \) has a unique solution as a trivial consequence of Theorem 7. Since \( \mathbf{X} \) is a \( d \) -dimensional semimartingale, we know that \( {f}_{j}^{i}\left( {s-,\cdot ,{\mathbf{X}}_{s - }}\right) \) is in the domain of definition of the F-S integral by Theorem ... | Yes |
Theorem 23. Let \( Z \) be a semimartingale, \( {Z}_{0} = 0 \) . The unique solution of the equation\n\n\[ \n{X}_{t} = {X}_{0} + {\int }_{0}^{t}{X}_{s - } \circ d{Z}_{s} \n\]\n\nis given by\n\n\[ \n{X}_{t} = {X}_{0}\exp \left\{ {Z}_{t}\right\} \mathop{\prod }\limits_{{0 < s \leq t}}\left( {1 + \Delta {Z}_{s}}\right) {e... | Proof. By Theorem 22 the equation above is equivalent to\n\n\[ \n{X}_{t} = {X}_{0} + {\int }_{0}^{t}{X}_{s - }d{Z}_{s} + \frac{1}{2}{\int }_{0}^{t}{X}_{s - }d{\left\lbrack Z, Z\right\rbrack }_{s}^{c} \n\]\n\n\[ \n= {X}_{0} + {\int }_{0}^{t}{X}_{s - }d\left( {{Z}_{s} + \frac{1}{2}{\left\lbrack Z, Z\right\rbrack }_{s}^{c... | Yes |
Corollary 2. Let \( B \) be a Brownian motion with \( {B}_{0} = 0 \) . Then the unique solution of \[ {X}_{t} = {X}_{0} + {\int }_{0}^{t}{X}_{s} \circ d{B}_{s} \] | is given by \( {X}_{t} = {X}_{0}\exp \left\{ {B}_{t}\right\} \) . | Yes |
Theorem 24. The solution \( \mathbf{X} \) of equation \( \left( {* * * }\right) \) above exists, is unique, and it always stays on the sphere of center \( \mathbf{0} \) and radius \( \begin{Vmatrix}{\mathbf{x}}_{0}\end{Vmatrix} \) . | Proof. Since \( a \) is singular at the origin, we need to modify it slightly. Let \( g\left( \mathbf{x}\right) \) be a \( {\mathcal{C}}^{\infty } \) function equal to \( a\left( \mathbf{x}\right) \) outside of a ball centered at the origin, \( {N}_{0} \) , such that \( {\mathbf{x}}_{0} \notin {N}_{0} \) . Let \( \math... | Yes |
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