Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Theorem 24. Let \( N \) be a Poisson process with intensity \( \lambda \) . Then \( {N}_{t} - {\lambda t} \) and \( {\left( {N}_{t} - \lambda t\right) }^{2} - {\lambda t} \) are martingales. | Proof. Since \( {\lambda t} \) is non-random, the process \( {N}_{t} - {\lambda t} \) has mean zero and independent increments. Therefore\n\n\[ E\left\{ {{N}_{t} - {\lambda t} - \left( {{N}_{s} - {\lambda s}}\right) \mid {\mathcal{F}}_{s}}\right\} = E\left\{ {{N}_{t} - {\lambda t} - \left( {{N}_{s} - {\lambda s}}\right... | Yes |
Theorem 25. Let \( N \) be a counting process. The natural filtration of \( N \) is right continuous. | Proof. Let \( E = \left\lbrack {0,\infty }\right\rbrack \) and \( \mathcal{B} \) be the Borel sets of \( E \), and let \( \Gamma \) be the path space given by\n\n\[ \Gamma = \left( {\mathop{\prod }\limits_{{s \in \lbrack 0,\infty )}}{E}_{s},{\bigotimes }_{s \in \lbrack 0,\infty )}{\mathcal{B}}_{s}}\right) \]\n\nDefine ... | Yes |
Theorem 27. Let \( B = {\left( {B}_{t}\right) }_{0 \leq t < \infty } \) be a one dimensional standard Brownian motion with \( {B}_{0} = 0 \) . Then \( {M}_{t} = {B}_{t}^{2} - t \) is a martingale. | Proof. \( E\left\{ {M}_{t}\right\} = E\left\{ {{B}_{t}^{2} - t}\right\} = 0 \) . Also\n\n\[ E\left\{ {{M}_{t} - {M}_{s} \mid {\mathcal{F}}_{s}}\right\} = E\left\{ {{B}_{t}^{2} - {B}_{s}^{2} - \left( {t - s}\right) \mid {\mathcal{F}}_{s}}\right\} \]\n\nand\n\n\[ E\left\{ {{B}_{t}{B}_{s} \mid {\mathcal{F}}_{s}}\right\} =... | Yes |
Theorem 28. Let \( {\pi }_{n} \) be a sequence of partitions of \( \left\lbrack {a, a + t}\right\rbrack \) . Suppose \( {\pi }_{m} \subset {\pi }_{n} \) if \( m > n \) (that is, the sequence is a refining sequence). Suppose moreover that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\operatorname{mesh}\left( {\pi }... | Proof. We first show convergence in mean square. We have\n\n\[{\pi }_{n}B - t = \mathop{\sum }\limits_{{{t}_{i} \in {\pi }_{n}}}\left\{ {{\left( {B}_{{t}_{i + 1}} - {B}_{{t}_{i}}\right) }^{2} - \left( {{t}_{i + 1} - {t}_{i}}\right) }\right\}\]\n\n\[= \mathop{\sum }\limits_{i}{Y}_{i}\]\n\nwhere \( {Y}_{i} \) are indepen... | Yes |
Theorem 32. Let \( X \) be a Lévy process and let \( T \) be a stopping time. On the set \( \{ T < \infty \} \) the process \( Y = {\left( {Y}_{t}\right) }_{0 \leq t < \infty } \) defined by \( {Y}_{t} = {X}_{T + t} - {X}_{T} \) is a Lévy process adapted to \( {\mathcal{H}}_{t} = {\mathcal{F}}_{T + t}, Y \) is independ... | Proof. First assume \( T \) is bounded. Let \( A \in {\mathcal{F}}_{T} \) and let \( \left( {{u}_{1},\ldots ,{u}_{n};{t}_{0},\ldots ,{t}_{n}}\right) \) be given with \( {u}_{j} \) in a countable dense set (for example the rationals \( \mathbb{Q} \) ) and \( {t}_{j} \in {\mathbb{R}}_{ + },{t}_{j} \) increasing with \( j... | Yes |
Theorem 33 (Reflection Principle for Brownian Motion). Let \( B = {\left( {B}_{t}\right) }_{t \geq 0} \) be standard Brownian motion \( \left( {{B}_{0} = 0}\right. \) a.s.) and \( {S}_{t} = \mathop{\sup }\limits_{{0 \leq s \leq t}}{B}_{s} \) , the Brownian maximum process. For \( y \geq 0, z > 0 \) , \[ P\left( {{B}_{t... | Proof. Let \( T = \inf \left\{ {t > 0 : {B}_{t} = z}\right\} \) . Then \( T \) is a stopping time by Theorem 4, and \( P\left( {T < \infty }\right) = 1 \) . We next define a new process \( X \) by \[ {X}_{t} = {B}_{t}{1}_{\{ t < T\} } + \left( {2{B}_{T} - {B}_{t}}\right) {1}_{\{ t \geq T\} } \] The process \( X \) is t... | Yes |
Theorem 34. Let \( X \) be a Lévy process with bounded jumps. Then \( E\left\{ {\left| {X}_{t}\right| }^{n}\right\} < \) \( \infty \) for all \( n = 1,2,3,\ldots \) . | Proof. Let \( C \) be a (non-random) bound for the jumps of \( X \) . Define the stopping times\n\n\[ \n{T}_{1} = \inf \left\{ {t : \left| {X}_{t}\right| \geq C}\right\} \n\]\n\n\[ \n\vdots \n\]\n\n\[ \n{T}_{n + 1} = \inf \left\{ {t > {T}_{n} : \left| {{X}_{t} - {X}_{{T}_{n}}}\right| \geq C}\right\} \n\]\n\nSince the p... | Yes |
Theorem 35. The set function \( \Lambda \mapsto {N}_{t}^{A}\left( \omega \right) \) defines a \( \sigma \) -finite measure on \( \mathbb{R} \smallsetminus \{ 0\} \) for each fixed \( \left( {t,\omega }\right) \) . The set function \( \nu \left( \Lambda \right) = E\left\{ {N}_{1}^{\Lambda }\right\} \) also defines a \( ... | Proof. The set function \( \Lambda \mapsto {N}_{t}^{A}\left( \omega \right) \) is simply a counting measure: \( \mu \left( \Lambda \right) = \) \{number of \( s \leq t : \Delta {X}_{s}\left( \omega \right) \in \Lambda \} \) . It is then clear that \( \nu \) is also a measure. | No |
Theorem 39. Let \( {\Lambda }_{1},{\Lambda }_{2} \) be two disjoint Borel sets with \( 0 \notin {\bar{\Lambda }}_{1},0 \notin {\bar{\Lambda }}_{2} \) . Then the two processes\n\n\[ \n{J}_{t}^{1} = \mathop{\sum }\limits_{{0 < s \leq t}}\Delta {X}_{s}{1}_{{\Lambda }_{1}}\left( {\Delta {X}_{s}}\right) \n\]\n\n\[ \n{J}_{t}... | Proof. By Theorem 36 and its corollary we have that \( {J}^{1} \) and \( {J}^{2} \) are Lévy processes. To show they are independent, we begin by forming for \( u, v \) in \( \mathbb{R} \) ,\n\n\[ \n{C}_{t}^{u} = \frac{{e}^{{iu}{J}_{t}^{1}}}{E\left\{ {e}^{{iu}{J}_{t}^{1}}\right\} } - 1 \n\]\n\n\[ \n{D}_{t}^{v} = \frac{... | Yes |
Theorem 40. Let \( X \) be a Lévy process. Then \( {X}_{t} = {Y}_{t} + {Z}_{t} \), where \( Y, Z \) are Lévy processes, \( Y \) is a martingale with bounded jumps, \( {Y}_{t} \in {L}^{p} \) for all \( p \geq 1 \) and \( Z \) has paths of finite variation on compacts. | Proof. Let \( {J}_{t} = \mathop{\sum }\limits_{{0 < s \leq t}}\Delta {X}_{s}{1}_{\left\{ \left| \Delta {X}_{s}\right| \geq 1\right\} } \) . Since \( X \) has càdlàg paths, for each fixed \( \omega \) the function \( s \mapsto {X}_{s}\left( \omega \right) \) has only finitely many jumps bigger than one on \( \left\lbrac... | Yes |
Theorem 41. Let \( X \) be a Lévy process with jumps bounded by a. That is, \( \mathop{\sup }\limits_{s}\left| {\Delta {X}_{s}}\right| \leq a \) a.s. Let \( {Z}_{t} = {X}_{t} - E\left\{ {X}_{t}\right\} \) . Then \( Z \) is a martingale and \( {Z}_{t} = \) \( {Z}_{t}^{c} + {Z}_{t}^{d} \) where \( {Z}^{c} \) is a marting... | Proof. \( Z \) has mean zero and independent increments so it is a martingale, as well as a Lévy process. For a given set \( \Lambda \) we define \[ {M}_{t}^{A} = {\int }_{\Lambda }x{N}_{t}\left( {\cdot ,{dx}}\right) - t{\int }_{\Lambda }{x\nu }\left( {dx}\right) \] \[ = \mathop{\sum }\limits_{{0 < s \leq t}}\Delta {X}... | Yes |
Theorem 44. Let \( X \) be a Lévy process in \( {\mathbb{R}}^{n} \), with \( {X}_{0} = 0 \) . Then there exists a convolution semigroup of probability measures on \( {\mathbb{R}}^{n} \) such that \( \mathcal{L}\left( {X}_{t}\right) = {\mu }_{t} \) . The characteristic function of \( {\mu }_{t} \) is \( {e}^{-{t\psi }\l... | A nice proof of the above theorem can be found in Bertoin [12, pages 13-15]. | No |
Theorem 45. Definitions 1 and 2 are equivalent. Moreover \( X \) is Markov with respect to \( \mathbb{F} \) if and only if \( E\left\{ {f\left( {X}_{u}\right) \mid {\mathcal{F}}_{t}}\right\} = E\left\{ {f\left( {X}_{u}\right) \mid {X}_{t}}\right\} \) a.s. for all \( u \geq t \geq 0 \) . | Let \( X \) be a càdlàg process and let \( X \) be Markov with respect to \( \mathbb{F} \) . Since \( X \) is Markov, we can associate with it a Markov transition function on \( \left( {{\mathbb{R}}^{d},\mathcal{B}}\right) \) , where \( \mathcal{B} \) denotes the Borel sets of \( {\mathbb{R}}^{d} \) . This is a family ... | Yes |
Theorem 46. Let \( X \) be a càdlàg Feller process for a filtration \( \mathbb{F} \). Then it is also Markov with respect to the filtration \( {\mathbb{F}}_{ + } = {\left( {\mathcal{F}}_{t + }\right) }_{0 \leq t \leq \infty } \). | Proof. Let \( t > 0 \) be fixed, and assume \( f \in {\mathcal{C}}_{0} \). Let \( {Y}_{t} = f\left( {X}_{t}\right) \). For \( 0 \leq s \leq t \), let \[ {Y}_{s} = E\left\{ {f\left( {X}_{t}\right) \mid {\mathcal{F}}_{s}}\right\} = E\left\{ {f\left( {X}_{t}\right) \mid {X}_{s}}\right\} = {P}_{t - s}f\left( {X}_{s}\right)... | Yes |
Theorem 47. If \( X \) is Markov with respect to \( {P}^{\mu } \), or alternatively Markov with respect to \( {P}^{\mu } \) for all \( \mu \) with \( \parallel \mu \parallel = 1 \), then \( {\mathbb{F}}^{\mu } \) and \( \mathbb{F} \) are, respectively, right continuous. | Proof. We prove the result for \( {\mathbb{F}}^{\mu } \), from which the result for \( \mathbb{F} \) follows. Let \( \mathcal{H} \) denote all \( h \in \mathbf{b}{\mathcal{F}}^{\mu } \) such that\n\n\[ E\left\{ {h \mid {\mathcal{F}}_{t}^{\mu }}\right\} = E\left\{ {h \mid {\mathcal{F}}_{t + }^{\mu }}\right\} ,\;{P}^{\mu... | Yes |
Theorem 48. Let \( M, N \) be local martingales and let \( S \) and \( T \) be stopping times.\n\n(a) If \( T \) reduces \( M \) and \( S \leq T \) a.s., then \( S \) reduces \( M \) . | Proof. (a) follows from the Optional Sampling Theorem (Theorem 16) and Theorem 13. | No |
Theorem 51. Let \( X \) be a local martingale such that \( E\left\{ {X}_{t}^{ * }\right\} < \infty \) for every \( t \geq 0 \) . Then \( X \) is a martingale. If \( E\left\{ {X}^{ * }\right\} < \infty \), then \( X \) is a uniformly integrable martingale. | Proof. Let \( {\left( {T}_{n}\right) }_{n \geq 1} \) be a fundamental sequence of stopping times for \( X \) . If \( s \leq t \), then \( E\left\{ {{X}_{t \land {T}_{n}} \mid {\mathcal{F}}_{s}}\right\} = {X}_{s \land {T}_{n}} \) . The Dominated Convergence Theorem yields \( E\left\{ {{X}_{t} \mid {\mathcal{F}}_{s}}\rig... | Yes |
Theorem 52. Let \( A, C \) be adapted, strictly increasing processes such that \( C - A \) is also an increasing process. Then there exists a jointly measurable, adapted process \( H \) (defined on \( \left( {0,\infty }\right) \) ) such that \( 0 \leq H \leq 1 \) and\n\n\[ A = H \cdot C \]\n\nor equivalently\n\n\[ {A}_... | Proof. If \( \mu \) and \( \nu \) are two Borel measures on \( {\mathbb{R}}_{ + } \) with \( \mu \ll \nu \), then we can set\n\n\[ \alpha \left( {s, t}\right) = \left\{ \begin{array}{ll} \frac{\mu (\left( {s, t\rbrack }\right) }{\nu (\left( {s, t\rbrack }\right) }, & \text{ if }\nu \left( {(s, t\rbrack }\right) > 0, \\... | Yes |
Theorem 54 (Change of Variables). Let \( A \) be an FV process with continuous paths, and let \( f \) be such that its derivative \( {f}^{\prime } \) exists and is continuous. Then \( {\left( f\left( {A}_{t}\right) \right) }_{t \geq 0} \) is an FV process and\n\n\[ f\left( {A}_{t}\right) - f\left( {A}_{0}\right) = {\in... | Proof. For fixed \( \omega \), the function \( s \mapsto {f}^{\prime }\left( {{A}_{s}\left( \omega \right) }\right) \) is continuous on \( \left\lbrack {0, t}\right\rbrack \) and hence bounded. Therefore the integral \( {\int }_{0}^{t}{f}^{\prime }\left( {A}_{s}\right) d{A}_{s} \) exists. Fix \( t \) and let \( {\pi }_... | Yes |
Theorem 56. If the sums \( {S}_{n} \) of \( \left( *\right) \) converge to a limit for every continuous function \( h \) then \( x \) is of finite variation. | Proof. Let \( X \) be the Banach space of continuous functions equipped with the supremum norm. Let \( Y \) be \( \mathbb{R} \), equipped with absolute value as the norm. For \( h \in X \), let\n\n\[ {T}_{n}\left( h\right) = \mathop{\sum }\limits_{{{t}_{k},{t}_{k + 1} \in {\pi }_{n}}}h\left( {t}_{k}\right) \left( {x\le... | Yes |
Theorem 1. The set of (total) semimartingales is a vector space. | Proof. This is immediate from the definition. | No |
Theorem 2. If \( Q \) is a probability and absolutely continuous with respect to \( P \), then every (total) \( P \) semimartingale \( X \) is a (total) \( Q \) semimartingale. | Proof. Convergence in \( P \) -probability implies convergence in \( Q \) -probability. Thus the theorem follows from the definition of \( X \) . | No |
Theorem 3. Let \( {\left( {P}_{k}\right) }_{k \geq 1} \) 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 =... | Proof. Suppose \( {H}^{n} \in \mathbf{S} \) converges uniformly to \( H \in \mathbf{S} \) . Since \( X \) is a \( {P}_{k} \) semimartingale for all \( {P}_{k},{I}_{X}\left( {H}^{n}\right) \) converges to \( {I}_{X}\left( H\right) \) in probability for every \( {P}_{k} \) . This then implies \( {I}_{X}\left( {H}^{n}\rig... | Yes |
Theorem 4 (Stricker’s Theorem). Let \( X \) be a semimartingale for the filtration \( \mathbb{F} \). Let \( \mathbb{G} \) be a subfiltration of \( \mathbb{F} \), such that \( X \) is adapted to the \( \mathbb{G} \) filtration. Then \( X \) is a \( \mathbb{G} \) semimartingale. | Proof. For a filtration \( \mathbb{H} \), let \( \mathbf{S}\left( \mathbb{H}\right) \) denote the simple predictable processes for the filtration \( \mathbb{H} = {\left( {\mathcal{H}}_{t}\right) }_{t \geq 0} \). In this case we have \( \mathbf{S}\left( \mathbb{G}\right) \) is contained in \( \mathbf{S}\left( \mathbb{F}... | No |
Theorem 5 (Jacod’s Countable Expansion). Let \( \mathcal{A} \) be a collection of events in \( \mathcal{F} \) such that if \( {A}_{\alpha },{A}_{\beta } \in \mathcal{A} \) then \( {A}_{\alpha } \cap {A}_{\beta } = \varnothing ,\left( {\alpha \neq \beta }\right) \) . Let \( {\mathcal{H}}_{t} \) be the filtration generat... | Proof. Let \( {A}_{n} \in \mathcal{A} \) . If \( P\left( {A}_{n}\right) = 0 \), then \( {A}_{n} \) and \( {A}_{n}^{c} \) are in \( {\mathcal{F}}_{0} \) by hypothesis. We assume, therefore, that \( P\left( {A}_{n}\right) > 0 \) . Note that there can be at most a countable number of \( {A}_{n} \in \mathcal{A} \) such tha... | Yes |
Theorem 6. Let \( X \) be a càdlàg, adapted process. Let \( \left( {T}_{n}\right) \) be a sequence of positive r.v. increasing to \( \infty \) a.s., and let \( \left( {X}^{n}\right) \) be a sequence of semimartin-gales such that for each \( n,{X}^{{T}_{n} - } = {\left( {X}^{n}\right) }^{{T}_{n} - } \) . Then \( X \) is... | Proof. We wish to show \( {X}^{t} \) is a total semimartingale, each \( t > 0 \) . Define \( {R}_{n} = {T}_{n}{1}_{\left\{ {T}_{n} \leq t\right\} } + \infty {1}_{\left\{ {T}_{n} > t\right\} } \) . Then\n\n\[ P\left\{ {\left| {{I}_{{X}^{t}}\left( H\right) }\right| \geq c}\right\} \leq P\left\{ {\left| {{I}_{{\left( {X}^... | Yes |
Theorem 7. Each adapted process with càdlàg paths of finite variation on compacts (of finite total variation) is a semimartingale (a total semimartingale). | Proof. It suffices to observe that \( \left| {{I}_{X}\left( H\right) }\right| \leq \parallel H{\parallel }_{u}{\int }_{0}^{\infty }\left| {d{X}_{s}}\right| \), where \( {\int }_{0}^{\infty }\left| {d{X}_{s}}\right| \) denotes the Lebesgue-Stieltjes total variation and \( \parallel H{\parallel }_{u} = \mathop{\sup }\lim... | Yes |
Theorem 8. Each \( {L}^{2} \) martingale with càdlàg paths is a semimartingale. | Proof. Let \( X \) be an \( {L}^{2} \) martingale with \( {X}_{0} = 0 \), and let \( H \in \mathbf{S} \) . Using Doob’s Optional Sampling Theorem and the \( {L}^{2} \) orthogonality of the increments of \( {L}^{2} \) martingales, it suffices to observe that\n\n\[ E\left\{ {\left( {I}_{X}\left( H\right) \right) }^{2}\ri... | Yes |
Corollary 1. Each càdlàg, locally square integrable local martingale is a semimartingale. | Proof. Apply Theorem 8 together with the corollary to Theorem 6. | No |
Corollary 2. A local martingale with continuous paths is a semimartingale. | Proof. Apply Corollary 1 together with Theorem 51 in Chap. I. | No |
Corollary 3. The Wiener process (that is, Brownian motion) is a semimartin-gale. | Proof. The Wiener process \( {B}_{t} \) is a martingale with continuous paths if \( {B}_{0} \) is integrable. It is always a continuous local martingale. | No |
Theorem 9. A decomposable process is a semimartingale. | Proof. Let \( {X}_{t} = {X}_{0} + {M}_{t} + {A}_{t} \) be a decomposition of \( X \) . Then \( M \) is a semi-martingale by Corollary 1 of Theorem 8, and \( A \) is a semimartingale by Theorem 7. Since semimartingales form a vector space (Theorem 1) we have the result. | Yes |
Theorem 10. The space \( \mathbf{S} \) is dense in \( \mathbb{L} \) under the ucp topology. | Proof. Let \( Y \in \mathbb{L} \) . Let \( {R}_{n} = \inf \left\{ {t : \left| {Y}_{t}\right| > n}\right\} \) . Then \( {R}_{n} \) is a stopping time and \( {Y}^{n} = {Y}^{{R}_{n}}{1}_{\left\{ {R}_{n} > 0\right\} } \) are in \( \mathbf{b}\mathbb{L} \) and converge to \( Y \) in ucp. Thus \( \mathbf{b}\mathbb{L} \) is de... | Yes |
Theorem 11. Let \( X \) be a semimartingale. Then the mapping \( {J}_{X} : {\mathbf{S}}_{\text{ucp }} \rightarrow \) \( {\mathbb{D}}_{\text{ucp }} \) is continuous. | Proof. Since we are only dealing with convergence on compact sets, without loss of generality we take \( X \) to be a total semimartingale. First suppose \( {H}^{k} \) in \( \mathbf{S} \) tends to 0 uniformly and is uniformly bounded. We will show \( {J}_{X}\left( {H}^{k}\right) \) tends to \( 0\mathrm{{ucp}} \) . Let ... | Yes |
Theorem 14. Let \( Q \ll P \) . Then \( {H}_{Q} \cdot X \) is \( Q \) indistinguishable from \( {H}_{P} \cdot X \) . | Proof. Note that by Theorem \( 2, X \) is a \( Q \) semimartingale. The theorem is clear if \( H \in \mathbf{S} \), and it follows for \( H \in \mathbb{L} \) by passage to the limit in the \( {ucp} \) topology, since convergence in \( P \) -probability implies convergence in \( Q \) -probability. | No |
Theorem 15. 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 \( {P}_{k} \ll R \), and the result follows by Theorem 14. Note that by Theorem 3 we know that \( X \) is an \( R \) semimartingale. | Yes |
Theorem 16. Let \( \mathbb{G} = {\left( {\mathcal{G}}_{t}\right) }_{t \geq 0} \) be another filtration such that \( H \) is in both \( \mathbb{L}\left( \mathbb{G}\right) \) and \( \mathbb{L}\left( \mathbb{F}\right) \), and such that \( X \) is also a \( \mathbb{G} \) semimartingale. Then \( {H}_{\mathbb{G}} \cdot X = \... | Proof. \( \mathbb{L}\left( \mathbb{G}\right) \) denotes left continuous processes adapted to the filtration \( \mathbb{G} \) . As in the proof of Theorem 10, we can construct a sequence of processes \( {H}^{n} \) converging to \( H \) where the construction of the \( {H}^{n} \) depends only on \( H \) . Thus \( {H}^{n}... | Yes |
Theorem 17. If the semimartingale \( X \) has paths of finite variation on compacts, then \( H \cdot X \) is indistinguishable from the Lebesgue-Stieltjes integral, computed path-by-path. | Proof. The result is evident for \( H \in \mathbf{S} \) . Let \( {H}^{n} \in \mathbf{S} \) converge to \( H \) in ucp. Then there exists a subsequence \( {n}_{k} \) such that \( \mathop{\lim }\limits_{{{n}_{k} \rightarrow \infty }}{\left( {H}^{{n}_{k}} - H\right) }_{t}^{ * } = 0 \) a.s., and the result follows by inter... | No |
Theorem 18. Let \( X,\bar{X} \) be two semimartingales, and let \( H,\bar{H} \in \mathbb{L} \) . Let \( A = \{ \omega : {H}_{ \cdot }\left( \omega \right) = {\bar{H}}_{ \cdot }\left( \omega \right) \; \) and \( \;{X}_{ \cdot }\left( \omega \right) = {\bar{X}}_{ \cdot }\left( \omega \right) \} ,\; \) and \( \; \) let \(... | Proof. Without loss of generality we assume \( P\left( A\right) > 0 \) . Define a new probability law \( Q \) by \( Q\left( \Lambda \right) = P\left( {\Lambda \mid A}\right) \) . Then under \( Q \) we have that \( H \) and \( \bar{H} \) as well as \( X \) and \( \bar{X} \) are indistinguishable. Thus \( {H}_{Q} \cdot X... | Yes |
Theorem 19 (Associativity). The stochastic integral process \( Y = H \cdot X \) is itself a semimartingale, and for \( G \in \mathbb{L} \) we have\n\n\[ G \cdot Y = G \cdot \left( {H \cdot X}\right) = \left( {GH}\right) \cdot X. \] | Proof. Suppose we know \( Y = H \cdot X \) is a semimartingale. Then \( G \cdot Y = {J}_{Y}\left( G\right) \) . If \( G, H \) are in \( \mathbf{S} \), then it is clear that \( {J}_{Y}\left( G\right) = {J}_{X}\left( {GH}\right) \) . The associativity then extends to \( \mathbb{L} \) by continuity.\n\nIt remains to show ... | Yes |
Theorem 20. Let \( X \) be a locally square integrable local martingale, and let \( H \in \mathbb{L} \). Then the stochastic integral \( H \cdot X \) is also a locally square integrable local martingale. | Proof. We have seen that a locally square integrable local martingale is a semimartingale (Corollary 1 of Theorem 8), so we can formulate \( H \cdot X \). Without loss of generality, assume \( {X}_{0} = 0 \). Also, if \( {T}^{k} \) increases to \( \infty \) a.s. and \( {\left( H \cdot X\right) }^{{T}_{k}} \) is a local... | Yes |
Theorem 21. Let \( X \) be a semimartingale, and let \( Y \) be a process in \( \mathbb{D} \) or in \( \mathbb{L} \) . Let \( \left( {\sigma }_{n}\right) \) be a sequence of random partitions tending to the identity. Then the processes \( {\int }_{0 + }^{t}{Y}_{s}^{{\sigma }_{n}}d{X}_{s} = \mathop{\sum }\limits_{i}{Y}_... | Proof. (The notation \( {Y}_{ - } \) means the process whose value at \( s \) is given by \( {\left( {Y}_{ - }\right) }_{s} = \mathop{\lim }\limits_{{u \rightarrow s, u < s}}{Y}_{u} \) ; also, \( {\left( {Y}_{ - }\right) }_{0} = 0 \), by convention.) We prove the theorem for the case where \( Y \) is càdlàg, the other ... | Yes |
Corollary 1. The bracket process \( \left\lbrack {X, Y}\right\rbrack \) of two semimartingales has paths of finite variation on compacts, and it is also a semimartingale. | Proof. By the polarization identity \( \left\lbrack {X, Y}\right\rbrack \) is the difference of two increasing processes, hence its paths are of finite variation. Moreover, the paths are clearly càdlàg, and the process is adapted. Hence by Theorem 7 it is a semi-martingale. | Yes |
Corollary 2 (Integration by Parts). Let \( X, Y \) be two semimartingales. Then \( {XY} \) is a semimartingale and\n\n\[{XY} = \int {X}_{ - }{dY} + \int {Y}_{ - }{dX} + \left\lbrack {X, Y}\right\rbrack\] | Proof. The formula follows trivially from the definition of \( \left\lbrack {X, Y}\right\rbrack \) . That \( {XY} \) is a semimartingale follows from the formula, Theorem 19, and Corollary 1 above. | Yes |
Corollary 3. All semimartingales on a given filtered probability space form an algebra. | Proof. Since semimartingales form a vector space, Corollary 2 shows they form an algebra. | No |
Theorem 26. If \( X \) is adapted, càdlàg, with paths of finite variation on compacts, then \( X \) is a quadratic pure jump semimartingale. | Proof. Without loss of generality we assume \( {X}_{0} = 0 \) . We have already seen that such an \( X \) is a semimartingale (Theorem 7), and that the stochastic integral with respect to \( X \) is nothing more than a path-by-path Lebesgue-Stieltjes integral (Theorem 17). The integration by parts formula for Lebesgue-... | Yes |
Theorem 27. Let \( X \) be a local martingale with continuous paths that are not everywhere constant. Then \( \left\lbrack {X, X}\right\rbrack \) is not the constant process \( {X}_{0}^{2} \), and \( {X}^{2} - \left\lbrack {X, X}\right\rbrack \) is a continuous local martingale. Moreover if \( {\left\lbrack X, X\right\... | Proof. Note that a continuous local martingale is a semimartingale (Corollary 2 of Theorem 8). We have \( {X}^{2} - \left\lbrack {X, X}\right\rbrack = 2\int {X}_{ - }{dX} \), and by the martingale preservation property (Theorem 20) we have that \( 2\int {X}_{ - }{dX} \) is a local martingale. Moreover \( ⏊\int {X}_{ - ... | Yes |
Corollary 1. Let \( X \) be a continuous local martingale, and \( S \leq T \leq \infty \) be stopping times. If \( X \) has paths of finite variation on the stochastic interval \( \left( {S, T}\right) \), then \( X \) is constant on \( \left\lbrack {S, T}\right\rbrack \) . Moreover if \( \left\lbrack {X, X}\right\rbrac... | Proof. \( M = {X}^{T} - {X}^{S} \) is also a continuous local martingale, and \( M \) has finite variation on compacts. Moreover \( \left\lbrack {M, M}\right\rbrack = \left\lbrack {{X}^{T} - {X}^{S},{X}^{T} - {X}^{S}}\right\rbrack = \) \( {\left\lbrack X, X\right\rbrack }^{T} - {\left\lbrack X, X\right\rbrack }^{S} \),... | Yes |
Corollary 2. Let \( X \) and \( Y \) be two locally square integrable local martingales. Then \( \left\lbrack {X, Y}\right\rbrack \) is the unique adapted càdlàg process \( A \) with paths of finite variation on compacts satisfying the two properties:\n\n(i) \( {XY} - A \) is a local martingale; and\n\n(ii) \( {\Delta ... | Proof. Integration by parts yields\n\n\[ {XY} = \int {X}_{ - }{dY} + \int {Y}_{ - }{dX} + \left\lbrack {X, Y}\right\rbrack \]\n\nbut the martingale preservation property tells us that both stochastic integrals are local martingales. Thus \( {XY} - \left\lbrack {X, Y}\right\rbrack \) is a local martingale. Property (ii)... | Yes |
Corollary 5. Let \( X \) be a continuous local martingale. Then \( X \) and \( \left\lbrack {X, X}\right\rbrack \) have the same intervals of constancy a.s. | Proof. Let \( r \) be a positive rational, and define\n\n\[ \n{T}_{r} = \inf \left\{ {t \geq r : {X}_{t} \neq {X}_{r}}\right\} \n\]\n\nThen \( M = {X}^{{T}_{r}} - {X}^{r} \) is a local martingale which is constant. Hence \( \left\lbrack {M, M}\right\rbrack = \) \( {\left\lbrack X, X\right\rbrack }^{{T}_{r}} - {\left\lb... | Yes |
Theorem 28. Let \( X \) be a quadratic pure jump semimartingale. Then for any semimartingale \( Y \) we have\n\n\[{\left\lbrack X, Y\right\rbrack }_{t} = {X}_{0}{Y}_{0} + \mathop{\sum }\limits_{{0 < s \leq t}}\Delta {X}_{s}\Delta {Y}_{s}\] | Proof. The Kunita-Watanabe inequality (Theorem 25) tells us \( d{\left\lbrack X, Y\right\rbrack }_{s} \) is a.s. absolutely continuous with respect to \( d\left\lbrack {X, X}\right\rbrack \) (path-by-path). Thus \( {\left\lbrack X, X\right\rbrack }^{c} = \) 0 implies \( {\left\lbrack X, Y\right\rbrack }^{c} = 0 \), and... | Yes |
Theorem 29. Let \( X \) and \( Y \) be two semimartingales, and let \( H, K \in \mathbb{L} \). Then\n\n\[{\left\lbrack H \cdot X, K \cdot Y\right\rbrack }_{t} = {\int }_{0}^{t}{H}_{s}{K}_{s}d{\left\lbrack X, Y\right\rbrack }_{s}\]\n\nand, in particular,\n\n\[{\left\lbrack H \cdot X, H \cdot X\right\rbrack }_{t} = {\int... | Proof. First assume (without loss of generality) that \( {X}_{0} = {Y}_{0} = 0 \). It suffices to establish the following result\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\( \left( *\right) \)\n\nand then apply it again, by the symmetry of the... | Yes |
Theorem 30. Let \( H \) be a càdlàg, adapted process, and let \( X, Y \) be two semimartingales. Let \( {\sigma }_{n} \) be a sequence of random partitions tending to the identity. Then\n\n\[ \sum {H}_{{T}_{i}^{n}}\left( {{X}^{{T}_{i + 1}^{n}} - {X}^{{T}_{i}^{n}}}\right) \left( {{Y}^{{T}_{i + 1}^{n}} - {Y}^{{T}_{i}^{n}... | Proof. By the definition of quadratic variation, \( \left\lbrack {X, Y}\right\rbrack = {XY} - {X}_{ - } \cdot Y - {Y}_{ - } \cdot X \) , where \( {X}_{ - } \cdot Y \) denotes the process \( {\left( {\int }_{0}^{t}{X}_{s - }d{Y}_{s}\right) }_{t \geq 0} \) . By the associativity of the stochastic integral (Theorem 19)\n\... | Yes |
Theorem 31 (Change of Variables). Let \( V \) be an \( {FV} \) process with right continuous paths, and let \( f \) be such that \( {f}^{\prime } \) exists and is continuous. Then \( {\left( f\left( {V}_{t}\right) \right) }_{t \geq 0} \) is an FV process and | \[ f\left( {V}_{t}\right) - f\left( {V}_{0}\right) = {\int }_{0 + }^{t}{f}^{\prime }\left( {V}_{s - }\right) d{V}_{s} + \mathop{\sum }\limits_{{0 < s \leq t}}\left\{ {f\left( {V}_{s}\right) - f\left( {V}_{s - }\right) - {f}^{\prime }\left( {V}_{s - }\right) \Delta {V}_{s}}\right\} . \] | Yes |
Theorem 32 (Itô’s Formula). Let \( X \) be a semimartingale and let \( f \) be a \( {\mathcal{C}}^{2} \) real function. Then \( f\left( X\right) \) is again a semimartingale, and the following formula holds:\n\n\[ f\left( {X}_{t}\right) - f\left( {X}_{0}\right) = {\int }_{0 + }^{t}{f}^{\prime }\left( {X}_{s - }\right) ... | Proof. Note that the jump part of the stochastic integral \( \int {f}^{\prime \prime }\left( {X}_{s - }\right) d{\left\lbrack X, X\right\rbrack }_{s} \) is given by \( \mathop{\sum }\limits_{{s < t}}{f}^{\prime \prime }\left( {X}_{s - }\right) {\left( \Delta {X}_{s}\right) }^{2} \), and this is a convergent series. By ... | Yes |
Theorem 33. Let \( X = \left( {{X}^{1},\ldots ,{X}^{n}}\right) \) be an \( n \) -tuple of semimartingales, and let \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) have continuous second order partial derivatives. Then \( f\left( X\right) \) is a semimartingale and the following formula holds: | \[ f\left( {X}_{t}\right) - f\left( {X}_{0}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{\int }_{0 + }^{t}\frac{\partial f}{\partial {x}_{i}}\left( {X}_{s - }\right) d{X}_{s}^{i} + \frac{1}{2}\mathop{\sum }\limits_{{1 \leq i, j \leq n}}{\int }_{0 + }^{t}\frac{{\partial }^{2}f}{\partial {x}_{i}\partial {x}_{j}}\left( {X... | Yes |
Theorem 34. Let \( X \) be a semimartingale and let \( f \) be \( {\mathcal{C}}^{3} \). 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 \( {\mathcal{C}}^{2} \), so that \( {f}^{\prime }\left( X\right) \) is a semimartingale by Theorem 32 and in the domain of the F-S integral. By Theorem 32 and the definition, it suffices to establish \( \frac{1}{2}{\left\lbrack {f}^{\prime }\left( X\right), X\right\rbrack }^{c} =... | Yes |
Theorem 35. Let \( X, Y \) be continuous semimartingales, let \( {Z}_{t} = {X}_{t} + i{Y}_{t} \) , and let \( f \) be analytic. Then\n\n\[ f\left( {Z}_{t}\right) = f\left( {Z}_{0}\right) + {\int }_{0 + }^{t}{f}^{\prime }\left( {Z}_{s}\right) d{Z}_{s} + \frac{1}{2}{\int }_{0 + }^{t}{f}^{\prime \prime }\left( {Z}_{s}\rig... | Proof. Using Itô’s formula for the real and imaginary parts of \( f \) yields\n\n\[ f\left( {Z}_{t}\right) = f\left( {Z}_{0}\right) + {\int }_{0 + }^{t}\frac{\partial f}{\partial x}\left( {Z}_{s}\right) d{X}_{s} + {\int }_{0 + }^{t}\frac{\partial f}{\partial y}\left( {Z}_{s}\right) d{Y}_{s} \]\n\n\[ + \frac{1}{2}{\int ... | Yes |
Theorem 36. Let \( X, Y \) be càdlàg semimartingales, let \( {Z}_{t} = {X}_{t} + i{Y}_{t} \), and let \( f \) be analytic. Then\n\n\[ f\left( {Z}_{t}\right) = f\left( {Z}_{0}\right) + {\int }_{0 + }^{t}{f}^{\prime }\left( {Z}_{s - }\right) d{Z}_{s} + \frac{1}{2}{\int }_{0 + }^{t}{f}^{\prime \prime }\left( {Z}_{s - }\ri... | Observe that for a complex-valued semimartingale the process \( \left\lbrack {Z, Z}\right\rbrack \) is in general a complex-valued process. For many applications, it is more appropriate to use the non-negative increasing process \n\n\[ \left\lbrack {Z,\bar{Z}}\right\rbrack = \left\lbrack {X, X}\right\rbrack + \left\lbr... | No |
Theorem 38. Let \( X \) and \( Y \) be two semimartingales with \( {X}_{0} = {Y}_{0} = 0 \) . Then \( \mathcal{E}\left( X\right) \mathcal{E}\left( Y\right) = \mathcal{E}\left( {X + Y + \left\lbrack {X, Y}\right\rbrack }\right) \) | Proof. Let \( {U}_{t} = \mathcal{E}{\left( X\right) }_{t} \) and \( {V}_{t} = \mathcal{E}{\left( Y\right) }_{t} \) . Then the integration by parts formula gives that \( {U}_{t}{V}_{t} - 1 = {\int }_{0 + }^{t}{U}_{s - }d{V}_{s} + {\int }_{0 + }^{t}{V}_{s - }d{U}_{s} + {\left\lbrack U, V\right\rbrack }_{t} \) . Since \( ... | Yes |
Theorem 39 (Lévy’s Theorem). A stochastic process \( X = {\left( {X}_{t}\right) }_{t \geq 0} \) is a standard Brownian motion if and only if it is a continuous local martingale with \( {\left\lbrack X, X\right\rbrack }_{t} = t \) . | Proof. We have already observed that a Brownian motion \( B \) is a continuous local martingale and that \( {\left\lbrack B, B\right\rbrack }_{t} = t \) (see the remark following Theorem 22). Thus it remains to show sufficiency. Fix \( u \in \mathbb{R} \) and set \( F\left( {x, t}\right) = \exp \{ {iux} + \) \( \left. ... | Yes |
Theorem 41. Let \( \mathbf{X} = \left( {{X}^{1},\ldots ,{X}^{n}}\right) \) be an \( n \) -dimensional continuous local martingale with values in an open subset \( D \) of \( {\mathbb{R}}^{n} \) . Suppose that \( \left\lbrack {{X}^{i},{X}^{j}}\right\rbrack = 0 \) if \( i \neq j \), and \( \left\lbrack {{X}^{i},{X}^{i}}\... | Proof. By Itô's formula (Theorem 33) we have\n\n\[ u\left( {\mathbf{X}}_{t}\right) - u\left( {\mathbf{X}}_{0}\right) = {\int }_{0 + }^{t}\nabla u\left( {\mathbf{X}}_{s}\right) \cdot d{\mathbf{X}}_{s} + \frac{1}{2}{\int }_{0 + }^{t}{\Delta u}\left( {\mathbf{X}}_{s}\right) d{A}_{s} \]\n\nwhere the \ | Yes |
Theorem 3. Let \( T \) be a stopping time. There exist disjoint events \( A, B \) such that \( A \cup B = \{ T < \infty \} \) a.s., \( {T}_{A} \) is accessible and \( {T}_{B} \) is totally inaccessible, and \( T = {T}_{A} \land {T}_{B} \) a.s. Such a decomposition is a.s. unique. | Proof. If \( T \) is totally inaccessible there is nothing to show. So without loss of generality we assume it is not. We proceed with an inductive construction: Let \( {R}_{1} = T \) and take\n\n\[ \n{\alpha }_{1} = \sup \left\{ {P\left( {S = {R}_{1} < \infty }\right) : S\text{ is predictable }}\right\} .\n\]\n\nChoos... | Yes |
Theorem 6 (Doob Decomposition). A potential \( {\left( {X}_{n}\right) }_{n \in \mathbb{N}} \) has a decomposition \( {X}_{n} = {M}_{n} - {A}_{n} \), where \( {A}_{n + 1} \geq {A}_{n} \) a.s., \( {A}_{0} = 0,{A}_{n} \in {\mathcal{F}}_{n - 1} \), and \( {M}_{n} = E\left\{ {{A}_{\infty } \mid {\mathcal{F}}_{n}}\right\} \)... | Proof. Let \( {M}_{0} = {X}_{0} \) and \( {A}_{0} = 0 \). Define \( {M}_{1} = {M}_{0} + \left( {{X}_{1} - E\left\{ {{X}_{1} \mid {\mathcal{F}}_{0}}\right\} }\right) \), and \( {A}_{1} = {X}_{0} - E\left\{ {{X}_{1} \mid {\mathcal{F}}_{0}}\right\} \). Define \( {M}_{n},{A}_{n} \) inductively as follows:\n\n\[ \n{M}_{n} =... | Yes |
Theorem 7 (Doob-Meyer Decomposition: Case of Totally Inaccessible Jumps). Let \( Z \) be a càdlàg supermartingale with \( {Z}_{0} = 0 \) of Class \( D \) , and such that all jumps of \( Z \) occur at totally inaccessible stopping times. Then there exists a unique, increasing, continuous, adapted process \( A \) with \(... | Proof of uniqueness. Let \( Z = M - A \) and \( Z = N - C \) be two decompositions of \( Z \) . Subtraction yields \( M - N = A - C \), which implies that \( M - N \) is a continuous martingale with paths of finite variation. We know however by the Corollary of Theorem 27 of Chap. II that \( M - N \) is then a constant... | Yes |
Lemma 1. Let \( \mathbb{F} \) be a discrete time filtration and let \( C \) be a non-decreasing process with \( {C}_{0} = 0 \), and \( {C}_{k} \in {\mathcal{F}}_{k - 1} \) . Suppose there exists a constant \( N > 0 \) such that \( E\left\{ {{C}_{\infty } - {C}_{k} \mid {\mathcal{F}}_{k}}\right\} \leq N \) a.s. for all ... | Proof of Lemma 1. First observe \( E\left\{ {C}_{\infty }\right\} = E\left\{ {E\left\{ {{C}_{\infty } - {C}_{0} \mid {\mathcal{F}}_{0}}\right\} }\right\} \leq N \) . Letting \( {c}_{k} = {C}_{k + 1} - {C}_{k} \geq 0 \), we obtain by rearranging terms:\n\n\[ \n{C}_{\infty }^{2} = 2\mathop{\sum }\limits_{{k \geq 0}}\left... | Yes |
Theorem 9. Let \( A \) be an FV process, \( {A}_{0} = 0 \), and \( E\left\{ {\left| A\right| }_{\infty }\right\} < \infty \) . Then \( A \) is natural if and only if\n\n\[ E\left\{ {{\int }_{0}^{\infty }{M}_{s - }d{A}_{s}}\right\} = E\left\{ {{M}_{\infty }{A}_{\infty }}\right\} \]\n\nfor any bounded martingale \( M \) ... | Proof. By integration by parts we have\n\n\[ {\int }_{0}^{\infty }{M}_{s - }d{A}_{s} = {M}_{\infty }{A}_{\infty } - {M}_{0}{A}_{0} - {\int }_{0}^{\infty }{A}_{s - }d{M}_{s} - {\left\lbrack M, A\right\rbrack }_{\infty }. \]\n\nThen \( {M}_{0}{A}_{0} = 0 \) and letting \( {N}_{t} = {\int }_{0}^{t}{A}_{s - }d{M}_{s} \), w... | Yes |
Theorem 10. Let \( A \) be an \( {FV} \) process of integrable variation which is natural. If \( A \) is a martingale then \( A \) is identically zero. | Proof. Let \( T \) be a finite stopping time and let \( H \) be any bounded, nonnegative martingale. Then \( E\left\{ {{\int }_{0}^{T}{H}_{s - }d{A}_{s}}\right\} = 0 \), as is easily seen by approximating sums and the Dominated Convergence Theorem, since \( {\int }_{0}^{T}\left| {d{A}_{s}}\right| \in {L}^{1} \) and \( ... | Yes |
Theorem 11. Let \( A \) be an FV process of integrable variation with \( {A}_{0} = 0 \) . If \( A \) is predictable, then \( A \) is natural. | Proof. Let \( M \) be a bounded martingale. First assume \( A \) is bounded. Then the stochastic integral \( {\int }_{0}^{\infty }{A}_{s}d{M}_{s} \) exists, and it is a martingale by Theorem 29 in Chap. IV combined with, for example, Corollary 3 of Theorem 27 of Chap. II. Therefore \( E\left\{ {{\int }_{0}^{\infty }{A}... | Yes |
Theorem 12. Let \( M \) be a local martingale with paths of finite variation on compact time sets. If \( M \) is predictably measurable, then \( M \) is constant. That is, \( {M}_{t} = {M}_{0} \) for all \( t \), almost surely. | Proof. This theorem is a combination of Theorems 10 and 11. | No |
Theorem 15 (Rao’s Theorem). A quasimartingale \( X \) has a unique decomposition \( X = M + A \), where \( M \) is a local martingale and \( A \) is a predictable process with paths of locally integrable variation and \( {A}_{0} = 0 \) . | Proof. This theorem is a combination of Theorems 13 and 14. | No |
Theorem 16. Let \( A \) be an increasing process of integrable variation, and let \( H \in \mathbb{L} \) be such that \( E\left\{ {{\int }_{0}^{t}{H}_{s}d{A}_{s}}\right\} < \infty \) . Then,\n\n\[ E\left\{ {{\int }_{0}^{t}{H}_{s}d{A}_{s}}\right\} = E\left\{ {{\int }_{0}^{t}{H}_{s}d{\widetilde{A}}_{s}}\right\} \] | Proof. Since \( A - \widetilde{A} \) is a martingale, so also is \( \int {H}_{s}d\left( {{A}_{s} - {\widetilde{A}}_{s}}\right) \), and hence it has constant expectation equal to 0 . | Yes |
Theorem 17. Let \( P\left( {\tau = 0}\right) = 0 \) and \( P\left( {\tau > t}\right) > 0 \), each \( t > 0 \) . Then the \( \mathbb{F} \) compensator \( A \) of \( \eta \), where \( {\eta }_{t} = {1}_{\{ t \geq \tau \} } \), is given by\n\n\[ \n{A}_{t} = {\int }_{0}^{\tau \land t}\frac{1}{1 - F\left( {u - }\right) }{dF... | Proof of Theorem 17. Fix \( {t}_{0} > 0 \) and let \( {\pi }_{n} \) be a sequence of partitions of \( \left\lbrack {0,{t}_{0}}\right\rbrack \) with \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\operatorname{mesh}\left( {\pi }_{n}\right) = 0 \) . Define \( {A}_{t}^{n} = \mathop{\sum }\limits_{{\pi }^{n}}E\left\{ {{... | Yes |
Corollary 2. If \( \tau \) in Theorem 17 has an exponential distribution, then \( {A}_{t} = \) \( \tau \land t \) . | Proof. Since in this case \( F \) is continuous and a distribution function, we can represent it as \( F\left( x\right) = 1 - {e}^{-\phi \left( x\right) } \) . We have that \( {A}_{t} = - \ln \left( {1 - F\left( {\tau \land t}\right) }\right) \) and equivalently we can write \( {A}_{t} = \phi \left( {\tau \land t}\righ... | Yes |
Theorem 18. Let \( N \) be a counting process without explosions, adapted to a filtration \( \mathbb{G} \) satisfying the usual hypotheses. Then the compensator of \( N \), call it \( A \), always exists. | Proof. Since \( N \) has non-decreasing paths, to ensure the existence of a compensator we need only to show that \( N \) is locally of integrable variation. Let \( {T}_{n} \) be the time of the \( n \) -th jump of \( N \) . Since \( N \) has no explosions, the times\n\n\( {T}_{n} \) increase to \( \infty \) a.s. Moreo... | Yes |
Theorem 21. Let \( A \) be an increasing process of locally integrable variation, and let \( T \) be a jump time of \( A \) which is totally inaccessible. Then its compensator \( \widetilde{A} \) is continuous at \( T \) . | Proof. Both theorems are simple consequences of Theorem 7. | No |
Theorem 22. Let \( T \) be a totally inaccessible stopping time. There exists a martingale \( M \) with paths of finite variation and with exactly one jump, of size one, occurring at time \( T \) (that is, \( {M}_{T} \neq {M}_{T - } \) on \( \{ T < \infty \} \) ). | Proof. Define\n\n\[ \n{U}_{t} = {1}_{\{ t \geq T\} }\n\]\n\nThen \( U \) is an increasing, bounded process of integrable variation, and we let \( A = \widetilde{U} \) be the compensator of \( U.A \) is continuous by Theorem 20, and \( M = U - A \) is the required martingale. | Yes |
Theorem 23 (Le Jan’s Theorem). Let \( T \) be a stopping time and let \( H \) be an integrable random variable such that \( E\left\{ {H \mid {\mathcal{F}}_{T - }}\right\} = 0 \) on \( \{ T < \infty \} \) . Then the right continuous martingale \( {H}_{t} = E\left\{ {H \mid {\mathcal{F}}_{t}}\right\} \) is zero on \( \lb... | Proof. Since the martingale \( {\left( {H}_{t}\right) }_{t \geq 0} \) is right continuous it suffices to show that \( {H}_{t}{1}_{\{ t < T\} } = 0 \) almost surely for all \( t \) . Let \( \Lambda \in {\mathcal{F}}_{t} \) . Then \( \Lambda \cap \{ t < T\} \) belongs both to \( {\mathcal{F}}_{T - } \) and also to \( {\m... | Yes |
Theorem 24. Let \( T \) be a predictable stopping time and let \( A \) be increasing, predictable, and locally of integrable variation. Then \( {A}_{T} \) and \( \Delta {A}_{T} \) are each \( {\mathcal{F}}_{T - } \) measurable. | Proof. Let \( {S}_{n} \) be a sequence of stopping times announcing \( T \) . Then \( {A}_{{S}_{n}} \in \) \( {\mathcal{F}}_{{S}_{n}} \subset {\mathcal{F}}_{T - } \) for each \( n \) . Since \( {A}_{T - } = \mathop{\lim }\limits_{{n \rightarrow \infty }}{A}_{{S}_{n}} \) we have that \( {A}_{T - } \in \) \( {\mathcal{F}... | Yes |
Theorem 26. A classical semimartingale is a semimartingale. | Proof. Let \( X \) be a classical semimartingale. Then \( {X}_{t} = {M}_{t} + {A}_{t} \) where \( M \) is a local martingale and \( A \) is an \( {FV} \) process. The process \( A \) is a semimartingale by Theorem 7 of Chap. II, and \( M \) is decomposable by the corollary of Theorem 25, hence also a semimartingale (Th... | Yes |
Theorem 27. A càdlàg quasimartingale is a semimartingale. | Proof. By Theorem 15 a quasimartingale is a classical semimartingale. Hence it is a semimartingale by Theorem 26. | Yes |
Theorem 28. A càdlàg supermartingale is a semimartingale. | Proof. Since a local semimartingale is a semimartingale (corollary to Theorem 6 of Chap. II), it suffices to show that for a supermartingale \( X \), the stopped process \( {X}^{t} \) is a semimartingale. However for a partition \( \tau \) of \( \left\lbrack {0, t}\right\rbrack \) , \[ E\left\{ {\mathop{\sum }\limits_{... | Yes |
Theorem 29. Let \( M \) be a local martingale and let \( H \in \mathbb{L} \). Then the stochastic integral \( H \cdot M \) is again a local martingale. | Proof. A local martingale is a semimartingale by the corollary of Theorem 25 and Theorem 9 of Chap. II; thus \( H \cdot M \) is defined. By the Fundamental Theorem of Local Martingales (Theorem 25) for \( \beta > 0 \) we can write \( M = \) \( N + A \) where \( N, A \) are local martingales, the jumps of \( N \) are bo... | Yes |
Theorem 30. If \( X \) is a special semimartingale, then its decomposition \( X = \) \( M + A \) with \( A \) predictable is unique. | Proof. Let \( X = N + B \) be another such decomposition. Then \( M - N = B - A \) , hence \( B - A \) is an \( {FV} \) process which is a local martingale. Moreover, \( B - A \) is predictable, and hence constant by Theorem 12. Since \( {B}_{0} - {A}_{0} = 0 \), we conclude \( B = A \). | Yes |
Theorem 31. Let \( X \) be a classical semimartingale with bounded jumps. Then \( X \) is a special semimartingale. | Proof. Let \( {X}_{t} = {X}_{0} + {M}_{t} + {A}_{t} \) be a decomposition of \( X \) with \( {M}_{0} = {A}_{0} = 0 \) , \( M \) a local martingale, and \( A \) an \( {FV} \) process. By Theorem 25 we can then also write\n\n\[ \n{X}_{t} = {X}_{0} + {N}_{t} + {B}_{t} \n\]\n\nwhere \( N \) is a local martingale with bound... | Yes |
Theorem 34. Let \( M \) be a local martingale and let \( {M}_{t}^{ * } = \mathop{\sup }\limits_{{s \leq t}}\left| {M}_{s}\right| \) . Then the increasing process \( {M}^{ * } \) is locally integrable. | Proof. Without loss of generality assume \( {M}_{0} = 0 \) . Let \( {T}_{n} \) be a sequence of stopping times increasing to \( \infty \) such that \( {M}^{{T}_{n}} \) is a uniformly integrable martingale for each \( n \) . Since we can replace \( {T}_{n} \) with \( {T}_{n} \land n \) if necessary, without loss of gene... | Yes |
Theorem 35 (Girsanov-Meyer Theorem). Let \( P \) and \( Q \) be equivalent. Let \( X \) be a classical semimartingale under \( P \) with decomposition \( X = M + A \) . Then \( X \) is also a classical semimartingale under \( Q \) and has a decomposition \( X = L + C \), where\n\n\[ \n{L}_{t} = {M}_{t} - {\int }_{0}^{t... | Proof. Recall that by Theorem 2 of Chap. II it is trivial that \( X \) is a \( Q \) semi-martingale. We need to show it is a classical semimartingale, with the above decomposition being valid.\n\nSince \( M \) and \( Z \) are \( P \) local martingales, they are semimartingales (corollary of Theorem 26) and\n\n\[ \n\int... | Yes |
Theorem 37. Let \( X \) be a \( P \) local martingale with \( {X}_{0} = 0 \) . Let \( Q \) be another probability absolutely continuous with respect to \( P \), and let \( {Z}_{t} = E\left\{ {\left. \frac{dQ}{dP}\right| \;{\mathcal{F}}_{t}}\right\} \) . Assume that \( \langle X, Z\rangle \) exists for \( P \) . Then \(... | Proof. \( {Z}_{0} = E\{ Z\} = 1 \), so if we let \( {R}_{n} = \inf \left\{ {t > 0 : {Z}_{t} \leq 1/n}\right\} \), then \( {R}_{n} \) increase to \( \infty, Q \) -a.s., and the process \( \frac{1}{{Z}_{t - }} \) is bounded on \( \left\lbrack {0,{R}_{n}}\right\rbrack \) . By Theorem 36, \( {X}_{t}^{{R}_{n}} - {\int }_{0}... | Yes |
Theorem 38 (Lenglart-Girsanov Theorem). Let \( X \) be a \( P \) local martingale with \( {X}_{0} = 0 \) . Let \( Q \) be a probability absolutely continuous with respect to \( P \), and let \( {Z}_{t} = {E}_{P}\left\{ {\left. \frac{dQ}{dP}\right| \;{\mathcal{F}}_{t}}\right\}, R = \inf \left\{ {t > 0 : {Z}_{t} = 0,{Z}_... | Proof. Let \( {R}_{n} = \inf \left\{ {t > 0 : {Z}_{t} \leq \frac{1}{n}}\right\} \) . (Recall that \( {Z}_{0} = 1 \), and also note that it is possible that \( {R}_{n} = R \) .) Then both \( {X}^{{R}_{n}} \) and \( {Z}^{{R}_{n}} \) are \( P \) local martingales. Also note that \( {A}_{t}^{{R}_{n}} = {\int }_{0}^{t}\frac... | Yes |
Theorem 39. Let \( M \) be a continuous local martingale. Then\n\n\[ E\left\{ {e}^{\frac{1}{2}{M}_{t}}\right\} \leq E{\left\{ {e}^{\frac{1}{2}{\left\lbrack M, M\right\rbrack }_{t}}\right\} }^{1/2}. \]\n | Proof.\n\n\[ {\left( \mathcal{E}\left( M\right) \right) }^{\frac{1}{2}} = {\left( {e}^{{M}_{t} - \frac{1}{2}{\left\lbrack M, M\right\rbrack }_{t}}\right) }^{\frac{1}{2}} \]\n\n\[ = {e}^{\frac{1}{2}{M}_{t}}{\left( {e}^{-\frac{1}{2}{\left\lbrack M, M\right\rbrack }_{t}}\right) }^{\frac{1}{2}} \]\nwhich implies that\n\n\[... | Yes |
Theorem 40 (Kazamaki’s Criterion). Let \( M \) be a continuous local martingale. Suppose \( \mathop{\sup }\limits_{T}E\left\{ {e}^{\left( \frac{1}{2}{M}_{T}\right) }\right\} < \infty \), where the supremum is taken over all bounded stopping times. Then \( \mathcal{E}\left( M\right) \) is a uniformly integrable martinga... | Proof. Let \( 0 < a < 1 \), and \( p > 1 \) be such that \( \frac{\sqrt{p}}{\left( \sqrt{p} - 1\right) } < \frac{1}{a} \) . Our hypothesis combined with the preceding lemma imply that \( \mathcal{E}\left( {aM}\right) \) is an \( {L}^{q} \) bounded martingale, where \( \frac{1}{p} + \frac{1}{q} = 1 \), which in turn imp... | Yes |
Theorem 41 (Novikov’s Criterion). Let \( M \) be a continuous local martingale, and suppose that\n\n\[ E\left\{ {e}^{\frac{1}{2}{\left\lbrack M, M\right\rbrack }_{\infty }}\right\} < \infty \]\n\nThen \( \mathcal{E}\left( M\right) \) is a uniformly integrable martingale. | Proof. By Theorem 39 we have \( E\left\{ {e}^{\frac{1}{2}{M}_{T}}\right\} \leq E{\left\{ {e}^{\frac{1}{2}{\left\lbrack M, M\right\rbrack }_{T}}\right\} }^{\frac{1}{2}} \), and we need only to apply Kazamaki's criterion (Theorem 40). | Yes |
Theorem 42. Let \( W \) be a standard Brownian motion on \( \left( {\Omega ,\mathcal{F},\mathbb{F}, P}\right) \), and let \( H \in \mathbb{L} \) be bounded. Let\n\n\[ \n{X}_{t} = {\int }_{0}^{t}{H}_{s}{ds} + {W}_{t} \n\]\n\nand define \( Q \) by \( \frac{dQ}{dP} = \exp \left\{ {{\int }_{0}^{T} - {H}_{s}d{W}_{s} - \frac... | Proof. Let \( {Z}_{T} = \exp \left\{ {{\int }_{0}^{T} - {H}_{s}d{W}_{s} - \frac{1}{2}{\int }_{0}^{T}{H}_{s}^{2}{ds}}\right\} \) . Then if \( {Z}_{t} = E\left\{ {{Z}_{T} \mid {\mathcal{F}}_{t}}\right\} \) we know by Theorem 37 of Chap. II that \( Z \) satisfies the equation\n\n\[ \n{Z}_{t} = 1 - {\int }_{0}^{t}{Z}_{s - ... | Yes |
Lemma 2. There exists a law \( Q \) equivalent to \( P \) such that \( {X}_{t} \in {L}^{1}\left( {dQ}\right) \) , \( 0 \leq t \leq {u}_{0} \) | Proof of Lemma 2. Let \( Y = \mathop{\sup }\limits_{{0 \leq t \leq {u}_{0}}}\left| {X}_{t}\right| \) . Since \( X \) has càdlàg paths, \( Y < \infty \) a.s. Moreover if \( D \) is a countable dense subset of \( \left\lbrack {0,{u}_{0}}\right\rbrack \), then \( Y = \mathop{\sup }\limits_{{t \in D}}\left| {X}_{t}\right| ... | Yes |
Theorem 1. The space of \( {\mathcal{H}}^{2} \) semimartingales is a Banach space. | Proof. The space is clearly a normed linear space and it is easy to check that \( \parallel \cdot {\parallel }_{{\mathcal{H}}^{2}} \) is a norm (recall that \( E\left\{ {\bar{N}}_{\infty }^{2}\right\} = E\left\{ {\left\lbrack \bar{N},\bar{N}\right\rbrack }_{\infty }\right\} \), and therefore \( \parallel X{\parallel }_... | Yes |
Theorem 2. For \( X \in {\mathcal{H}}^{2} \) the space \( \mathbf{b}\mathbb{L} \) is dense in \( \mathbf{b}\mathcal{P} \) under \( {d}_{X}\left( {\cdot , \cdot }\right) \) . | Proof. We use the Monotone Class Theorem. Define\n\n\( \mathcal{A} = \left\{ {H \in \mathbf{b}\mathcal{P} : \text{ for any }\varepsilon > 0,\text{ there exists }J \in \mathbf{b}\mathbb{L}\text{ such that }{d}_{X}\left( {H, J}\right) < \varepsilon }\right\} .\)\n\nTrivially \( \mathcal{A} \) contains bL. If \( {H}^{n} \... | Yes |
Theorem 3. Let \( X \in {\mathcal{H}}^{2} \) and \( {H}^{n} \in \mathbf{b}\mathbb{L} \) such that \( {H}^{n} \) is Cauchy under \( {d}_{X} \) . Then \( {H}^{n} \cdot X \) is Cauchy in \( {\mathcal{H}}^{2} \) . | Proof. Since \( {\begin{Vmatrix}{H}^{n} \cdot X - {H}^{m} \cdot X\end{Vmatrix}}_{{\mathcal{H}}^{2}} = {d}_{X}\left( {{H}^{n},{H}^{m}}\right) \), the theorem is immediate. | Yes |
Theorem 4. Let \( X \in {\mathcal{H}}^{2} \) and \( H \in \mathbf{b}\mathcal{P} \) . Suppose \( {H}^{n} \in \mathbf{b}\mathbb{L} \) and \( {J}^{m} \in \mathbf{b}\mathbb{L} \) are two sequences such that \( \mathop{\lim }\limits_{n}{d}_{X}\left( {{H}^{n}, H}\right) = \mathop{\lim }\limits_{m}{d}_{X}\left( {{J}^{m}, H}\r... | Proof. Let \( Y = \mathop{\lim }\limits_{{n \rightarrow \infty }}{H}^{n} \cdot X \) and \( Z = \mathop{\lim }\limits_{{m \rightarrow \infty }}{J}^{m} \cdot X \), where the limits are taken in \( {\mathcal{H}}^{2} \) . For \( \varepsilon > 0 \), by taking \( n \) and \( m \) large enough we have\n\n\[ \parallel Y - Z{\p... | Yes |
Theorem 5. Let \( X \) be a semimartingale in \( {\mathcal{H}}^{2} \) . Then\n\n\[ E\left\{ {\left( \mathop{\sup }\limits_{t}\left| {X}_{t}\right| \right) }^{2}\right\} \leq 8\parallel X{\parallel }_{{\mathcal{H}}^{2}}^{2} \] | Proof. For a process \( H \), let \( {H}^{ * } = \mathop{\sup }\limits_{t}\left| {H}_{t}\right| \) . Let \( X = \bar{N} + \bar{A} \) be the canonical decomposition of \( X \) . Then\n\n\[ {X}^{ * } \leq {\bar{N}}^{ * } + {\int }_{0}^{\infty }\left| {d{\bar{A}}_{s}}\right| \]\n\nDoob's maximal quadratic inequality (Theo... | Yes |
Theorem 6. Let \( X, Y \in {\mathcal{H}}^{2} \) and \( H, K \in \mathbf{b}\mathcal{P} \) . Then\n\n\[ \left( {H + K}\right) \cdot X = H \cdot X + K \cdot X \]\n\nand\n\n\[ H \cdot \left( {X + Y}\right) = H \cdot X + H \cdot Y \] | Proof. One need only check that it is possible to take a sequence \( {H}^{n} \in \mathbf{b}\mathbb{L} \) that approximates \( H \) in both \( {d}_{X} \) and \( {d}_{Y} \) . | No |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.