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Lemma 62.2.1 Let \( {r}_{j}^{m} \) denote \( j\left( \frac{T}{{2}^{m}}\right) \) where \( j \in \left\{ {0,1,\cdots ,{2}^{m}}\right\} \) . Also let \( {D}_{m} = \) \( {\left\{ {r}_{j}^{m}\right\} }_{j = 1}^{{2}^{m}} \) and \( D = { \cup }_{m = 1}^{\infty }{D}_{m} \) . Suppose \( X\left( t\right) \) satisfies\n\n\[ \lef... | Proof: Suppose \( {d}^{\prime } < d \) . Suppose first \( m = n + 1 \) . Then \( d = \left( {k + 1}\right) T{2}^{-\left( {n + 1}\right) } \) and \( {d}^{\prime } = {kT}{2}^{-\left( {n + 1}\right) } \) . Then from 62.2 .3\n\n\[ \begin{Vmatrix}{X\left( {d}^{\prime }\right) - X\left( d\right) }\end{Vmatrix} \leq {2}^{-\ga... | Yes |
Theorem 62.2.3 Suppose \( X \) is a stochastic process on \( \left\lbrack {0, T}\right\rbrack \) having values in the Banach space \( E \) . Suppose also that there exists a constant, \( C \) and positive numbers \( \alpha ,\beta ,\alpha \geq 1 \), such that\n\n\[ E\left( {\parallel X\left( t\right) - X\left( s\right) ... | Proof: The proof considers piecewise linear approximations of \( X \) which are automatically continuous. These are shown to converge to \( Y \) in \( {L}^{\alpha }\left( {\Omega ;C\left( {\left\lbrack {0, T}\right\rbrack, E}\right) }\right) \) so it follows that \( Y \) must be continuous for a.e. \( \omega \) . Final... | Yes |
Lemma 62.3.7 Define \( Q + h \) as \[ Q + h \equiv \{ \left( {t + h,\omega }\right) : \left( {t,\omega }\right) \in Q\} . \] Then if \( Q \in \mathcal{P} \), it follows that \( Q + h \in \mathcal{P} \) . | Proof: This is most easily seen through the use of the following diagram. In this diagram, \( Q \) is in \( \mathcal{P} \) so it is progressively measurable.  By definition, \( S \) in the picture is \( \mathcal{B}... | Yes |
Lemma 62.3.8 Let \( Q \in \mathcal{P} \) . Then \( {\tau }_{h}{\mathcal{X}}_{Q} \) is \( \mathcal{P} \) measurable. | Proof: If \( {\tau }_{h}{\mathcal{X}}_{Q}\left( {t,\omega }\right) = 1 \), then you need to have \( \left( {t - h,\omega }\right) \in Q \) and so \( \left( {t,\omega }\right) \in \) \( Q + h \) . Thus\n\n\[ \n{\tau }_{h}{\mathcal{X}}_{Q} = {\mathcal{X}}_{Q + h} \n\]\n\nwhich is \( \mathcal{P} \) measurable since \( Q \... | Yes |
Lemma 62.3.9 Let \( f\left( {t,\omega }\right) \) have values in a separable Banach space and suppose \( f \) is \( \mathcal{P} \) measurable. Then \( {\tau }_{h}f \) is \( \mathcal{P} \) measurable. | Proof: Taking values in a separable Banach space and being \( \mathcal{P} \) measurable, \( f \) is the pointwise limit of \( \mathcal{P} \) measurable simple functions. If \( {s}_{n} \) is one of these, then from the above lemmas, \( {\tau }_{h}{s}_{n} \) is \( \mathcal{P} \) measurable. Then, letting \( n \rightarrow... | Yes |
Proposition 62.3.12 Let \( {\mathcal{F}}_{t} \) be a filtration as above and let \( X \) be a predictable stochastic process. Then \( X \) is \( {\mathcal{F}}_{t} \) adapted. | Proof: Let \( {s}_{0} > 0 \) and define\n\n\[ \n{\mathcal{G}}_{{s}_{0}} \equiv \left\{ {S \in {\mathcal{P}}_{\infty } : {S}_{{s}_{0}} \in {\mathcal{F}}_{{s}_{0}}}\right\} \n\] \n\nwhere \n\n\[ \n{S}_{{s}_{0}} \equiv \left\{ {\omega \in \Omega : \left( {{s}_{0},\omega }\right) \in S}\right\} . \n\] \n\nIt is clear \( {\... | Yes |
Proposition 62.3.13 Let \( \mathcal{P} \) denote the predictable \( \sigma \) algebra and let \( \mathcal{R} \) denote the progressively measurable \( \sigma \) algebra. Then \( \mathcal{P} \subseteq \mathcal{R} \) . | Proof: Let \( \mathcal{G} \) denote those sets of \( \mathcal{P} \) such that they are also in \( \mathcal{R} \) . Then \( \mathcal{G} \) clearly contains the \( \pi \) system of sets \( \{ 0\} \times A, A \in {\mathcal{F}}_{0} \), and \( (s, t\rbrack \times A, A \in {\mathcal{F}}_{s} \) . Furthermore, \( \mathcal{G} \... | Yes |
Proposition 62.3.14 Let \( X\left( t\right) \) be a stochastic process having values in \( E \) a complete metric space and let it be \( {\mathcal{F}}_{t} \) adapted and left continuous. Then it is predictable. Also, if \( X\left( t\right) \) is stochastically continuous and adapted on \( \left\lbrack {0, T}\right\rbra... | Proof:Define \( {I}_{m, k} \equiv \left( {\left( {k - 1}\right) {2}^{-m}T, k{2}^{-m}T}\right\rbrack \) if \( k \geq 1 \) and \( {I}_{m,0} = \{ 0\} \) if \( k = 1 \) . Then define\n\n\[ {X}_{m}\left( t\right) \equiv \mathop{\sum }\limits_{{k = 1}}^{{2}^{m}}X\left( {T\left( {k - 1}\right) {2}^{-m}}\right) {\mathcal{X}}_{... | Yes |
Example 62.4.2 Let \( {\mathcal{F}}_{t} \) be a filtration and let \( Z \) be in \( {L}^{1}\left( {\Omega ,{\mathcal{F}}_{T}, P}\right) \) . Then let \( X\left( t\right) = E\left( {Z \mid {\mathcal{F}}_{t}}\right) . | This works because for \( s < t, E\left( {X\left( t\right) \mid {\mathcal{F}}_{s}}\right) = E\left( {E\left( {Z \mid {\mathcal{F}}_{t}}\right) \mid {\mathcal{F}}_{s}}\right) = E\left( {Z \mid {\mathcal{F}}_{s}}\right) = \) \( X\left( s\right) \) | Yes |
Proposition 62.4.3 The following statements hold for a stochastic process defined on \( \left\lbrack {0, T}\right\rbrack \times \Omega \) having values in a real separable Banach space, E.\n\n1. If \( X\left( t\right) \) is a martingale then \( \parallel X\left( t\right) \parallel, t \in \left\lbrack {0, T}\right\rbrac... | Proof:Let \( s \leq t \)\n\n\[ \parallel X\left( s\right) \parallel = \parallel E\left( {X\left( s\right) - X\left( t\right) \left| {{\mathcal{F}}_{s}) + E\left( {X\left( t\right) \mid {\mathcal{F}}_{s}}\right) }\right| }\right) \]\n\n\[ \begin{matrix} & = 0 \\ \leq & \overset{ = 0}{\overbrace{\parallel E\left( {X\left... | Yes |
Lemma 62.5.1 Let \( \\left\\{ {\\mathcal{F}}_{t}\\right\\} \) be a filtration and let \( \\{ X\\left( t\\right) \\} \) be a nonnegative valued submartingale for \( t \\in \\left\\lbrack {S, T}\\right\\rbrack \) . Then for \( \\lambda > 0 \) and any \( p \\geq 1 \\), if \( {A}_{t} \) is a \( {\\mathcal{F}}_{t} \) measur... | Proof: From Jensen's inequality,\n\n\[ \n{\\lambda }^{p}P\\left( {A}_{t}\\right) \\leq {\\int }_{{A}_{t}}X{\\left( t\\right) }^{p}{dP} \\leq {\\int }_{{A}_{t}}E{\\left( X\\left( T\\right) \\mid {\\mathcal{F}}_{t}\\right) }^{p}{dP}\n\]\n\n\[ \n\\leq {\\int }_{{A}_{t}}E\\left( {X{\\left( T\\right) }^{p} \\mid {\\mathcal{... | Yes |
Theorem 62.5.3 Let \( X\left( t\right) \) for \( t \in I = \left\lbrack {0, T}\right\rbrack \) be an \( E \) valued right continuous martingale with respect to a filtration, \( {\mathcal{F}}_{t} \) . Then for \( p \geq 1 \) , \[ P\left( \left\lbrack {\mathop{\sup }\limits_{{t \in I}}\parallel X\left( t\right) \parallel... | Proof: By Proposition 62.4.3 \( \parallel X\left( t\right) \parallel, t \in I \) is a submartingale and so from Theorem 62.5.2, it follows 62.5.21 and 62.5.22 hold. - | No |
Example 62.6.2 The first hitting time of an adapted process \( X\left( n\right) \) of a Borel set \( G \) is a stopping time. This is defined as\n\n\[ \tau \equiv \min \{ k : X\left( k\right) \in G\} \] | To see this, note that\n\n\[ \left\lbrack {\tau = n}\right\rbrack = { \cap }_{k < n}\left\lbrack {X\left( k\right) \in {G}^{C}}\right\rbrack \cap \left\lbrack {X\left( n\right) \in G}\right\rbrack \in {\mathcal{F}}_{n}. \] | Yes |
Proposition 62.6.3 For \( \tau \) a stopping time, \( {\mathcal{F}}_{\tau } \) is a \( \sigma \) algebra and if \( Y\left( k\right) \) is \( {\mathcal{F}}_{k} \) measurable for all \( k, Y\left( k\right) \) having values in a separable Banach space \( E \), then\n\n\[ \omega \rightarrow Y\left( {\tau \left( \omega \rig... | Proof: Let \( {A}_{n} \in {\mathcal{F}}_{\tau } \) . I need to show \( { \cup }_{n}{A}_{n} \in {\mathcal{F}}_{\tau } \) . In other words, I need to show that\n\n\[ { \cup }_{n}{A}_{n} \cap \left\lbrack {\tau \leq k}\right\rbrack \in {\mathcal{F}}_{k} \]\n\nThe left side equals\n\n\[ { \cup }_{n}\left( {{A}_{n} \cap \le... | Yes |
Lemma 62.6.4 In the situation of Definition 62.6.1, let \( \sigma ,\tau \) be two stopping times. Then\n\n1. \( \tau \) is \( {\mathcal{F}}_{\tau } \) measurable\n\n2. \( {\mathcal{F}}_{\sigma } \cap \left\lbrack {\sigma \leq \tau }\right\rbrack \subseteq {\mathcal{F}}_{\sigma \land \tau } = {\mathcal{F}}_{\sigma } \ca... | Proof: Consider the first claim. \( \left\lbrack {\tau \leq l}\right\rbrack \cap \left\lbrack {\tau \leq m}\right\rbrack = \left\lbrack {\tau \leq \lfloor l\rfloor \land m}\right\rbrack \in {\mathcal{F}}_{\left\lbrack l\right\rbrack \land m} \subseteq \) \( {\mathcal{F}}_{m} \) and so \( \tau \) is \( {\mathcal{F}}_{\t... | Yes |
Theorem 62.6.6 Let \( \left\{ {X}_{n}\right\} \) be a submartingale. Then there exists a unique stochastic process, \( \left\{ {A}_{n}\right\} \) and martingale, \( \left\{ {M}_{n}\right\} \) such that\n\n1. \( {A}_{n}\left( \omega \right) \leq {A}_{n + 1}\left( \omega \right) ,{A}_{1}\left( \omega \right) = 0 \) ,\n\n... | Proof: Let \( {A}_{1} \equiv 0 \) and define\n\n\[ \n{A}_{n} \equiv \mathop{\sum }\limits_{{k = 2}}^{n}E\left( {{X}_{k} - {X}_{k - 1} \mid {\mathcal{F}}_{k - 1}}\right) .\n\]\n\nIt follows \( {A}_{n} \) is \( {\mathcal{F}}_{n - 1} \) measurable. Since \( \left\{ {X}_{k}\right\} \) is a submartingale, \( {A}_{n} \) is i... | Yes |
Theorem 62.6.7 Let \( \{ X\left( k\right) \} \) be a real valued submartingale with respect to the increasing sequence of \( \sigma \) algebras, \( \left\{ {\mathcal{F}}_{k}\right\} \) and let \( \sigma \leq \tau \) be two stopping times such that \( \tau \) is bounded. Then \( M\left( \tau \right) \) defined as\n\n\[ ... | Proof: That \( \omega \rightarrow X\left( {\tau \left( \omega \right) }\right) \) is integrable follows right away as in the optional sampling theorem for martingales. You just consider the finitely many values of \( \tau \) .\n\nUse Theorem 62.6.6 above to write\n\n\[ X\left( n\right) = M\left( n\right) + A\left( n\ri... | Yes |
Lemma 62.7.7 \( {\tau }_{n} \) is a stopping time \( \left( {\left\lbrack {{\tau }_{n} \leq t}\right\rbrack \in {\mathcal{F}}_{t}\text{.}}\right) {Also}{\mathcal{F}}_{\tau } \subseteq {\mathcal{F}}_{{\tau }_{n}} \) and for each \( \omega ,{\tau }_{n}\left( \omega \right) \downarrow \tau \left( \omega \right) . | Proof: Say \( t \in \left( {{t}_{k - 1}^{n},{t}_{k}^{n}}\right\rbrack \) . Then \( \left\lbrack {{\tau }_{n} \leq t}\right\rbrack = \left\lbrack {\tau \leq {t}_{k - 1}^{n}}\right\rbrack \) if \( t < {t}_{k}^{n} \) and it equals \( \left\lbrack {\tau \leq {t}_{k}^{n}}\right\rbrack \) if \( t = {t}_{k}^{n} \) . Either wa... | Yes |
Proposition 62.7.8 Let \( \left( {\Omega ,\mathcal{F}, P}\right) \) be a probability space and let \( \sigma \leq \tau \) be two stopping times with respect to a filtration, \( {\mathcal{F}}_{t} \) . Then \( {\mathcal{F}}_{\sigma } \subseteq {\mathcal{F}}_{\tau } \) . If \( X\left( t\right) \) is a right continuous sto... | Proof: Let \( A \in {\mathcal{F}}_{\sigma } \) . Then \( A \cap \left\lbrack {\sigma \leq t}\right\rbrack \in {\mathcal{F}}_{t} \) for all \( t \geq 0 \) . Since \( \sigma \leq \tau \) ,\n\n\[ A \cap \left\lbrack {\tau \leq t}\right\rbrack = \overset{ \in {\mathcal{F}}_{t}}{\overbrace{A \cap \left\lbrack {\sigma \leq t... | Yes |
Example 62.7.9 Let \( \{ \tau \left( t\right) \} \) be an increasing family of stopping times. Then \( \tau \left( t\right) \) is adapted to the \( \sigma \) algebras \( {\mathcal{F}}_{\tau \left( t\right) } \) and \( \{ \tau \left( t\right) \} \) is a submartingale adapted to these \( \sigma \) algebras. | First I need to show that a stopping time, \( \tau \) is \( {\mathcal{F}}_{\tau } \) measurable. Consider \( \left\lbrack {\tau \leq s}\right\rbrack \) . Is this in \( {\mathcal{F}}_{\tau } \) ? Is \( \left\lbrack {\tau \leq s}\right\rbrack \cap \left\lbrack {\tau \leq r}\right\rbrack \in {\mathcal{F}}_{r} \) for each ... | Yes |
Example 62.7.10 Let \( \tau \) be a stopping time and let \( X \) be continuous and adapted to the filtration \( {\mathcal{F}}_{t} \). Then for \( a > 0 \), define \( \sigma \) as\n\n\[ \sigma \left( \omega \right) \equiv \inf \{ t > \tau \left( \omega \right) : \parallel X\left( t\right) \left( \omega \right) - X\left... | To see this is so, let\n\n\[ Y\left( t\right) \left( \omega \right) = \parallel X\left( {t \vee \tau }\right) \left( \omega \right) - X\left( {\tau \left( \omega \right) }\right) \parallel \]\n\nThen \( Y\left( t\right) \) is \( {\mathcal{F}}_{t \vee \tau } \) measurable. It is desired to show that \( Y \) is \( {\math... | Yes |
Lemma 62.7.12 Let \( X\left( t\right) \) be a right continuous nonnegative submartingale such that the filtration \( \left\{ {\mathcal{F}}_{t}\right\} \) is normal. Recall this includes\n\n\[ \n{\mathcal{F}}_{t} = { \cap }_{s > t}{\mathcal{F}}_{s} \n\] \n\nAlso let \( \tau \) be a stopping time with values in \( \left\... | Proof: First of all, say \( t \in \left( {{t}_{k}^{n},{t}_{k + 1}^{n}}\right\rbrack \) . If \( t < {t}_{k + 1}^{n} \), then\n\n\[ \n\left\lbrack {{\tau }_{n} \leq t}\right\rbrack = \left\lbrack {\tau \leq {t}_{k}^{n}}\right\rbrack \in {\mathcal{F}}_{{t}_{k}^{n}} \subseteq {\mathcal{F}}_{t} \n\] \n\nand if \( t = {t}_{k... | Yes |
Lemma 62.7.13 Let \( X\left( t\right) \) be a right continuous submartingale such that the filtration \( \left\{ {\mathcal{F}}_{t}\right\} \) is normal. Recall this includes\n\n\[{\mathcal{F}}_{t} = { \cap }_{s > t}{\mathcal{F}}_{s}\]\n\nAlso let \( \tau \) be a stopping time with values in \( \left\lbrack {0, T}\right... | Proof: It was shown above that \( {\tau }_{n} \) is a stopping time. Also, \( {t}_{k}^{n} \rightarrow X\left( {t}_{k}^{n}\right) \) is a discrete submartingale. Then by Theorem 62.6.6 there is a martingale \( {t}_{k}^{n} \rightarrow M\left( {t}_{k}^{n}\right) \) and an increasing submartingale \( {t}_{k}^{n} \rightarro... | Yes |
Theorem 62.7.14 Let \( \{ M\left( t\right) \} \) be a right continuous martingale having values in \( E \) a separable real Banach space with respect to the increasing sequence of \( \sigma \) algebras, \( \left\{ {\mathcal{F}}_{t}\right\} \) which is assumed to be a normal filtration satisfying,\n\n\[ \n{\mathcal{F}}_... | Proof: Since \( M\left( t\right) \) is a martingale, \( \parallel M\left( t\right) \parallel \) is a submartingale. Let\n\n\[ \n{\tau }_{n}\left( \omega \right) \equiv \mathop{\sum }\limits_{{k = 0}}^{\infty }{2}^{-n}\left( {k + 1}\right) T{\mathcal{X}}_{{\tau }^{-1}\left( \left( {k{2}^{-n}T,\left( {k + 1}\right) T{2}^... | Yes |
Theorem 62.7.15 Let \( \{ X\left( t\right) \} \) be a right continuous submartingale with respect to the increasing sequence of \( \sigma \) algebras, \( \left\{ {\mathcal{F}}_{t}\right\} \) which is assumed to be a normal filtration, \[ {\mathcal{F}}_{t} = { \cap }_{s > t}{\mathcal{F}}_{s} \] for \( t \in \left\lbrack... | Proof: Let \[ {\tau }_{n}\left( \omega \right) \equiv \mathop{\sum }\limits_{{k \geq 0}}{2}^{-n}\left( {k + 1}\right) T{\mathcal{X}}_{{\tau }^{-1}\left( \left( {k{2}^{-n}T,\left( {k + 1}\right) T{2}^{-n}}\right\rbrack \right) }\left( \omega \right) . \] Then by Lemma 62.7.13 \( {\tau }_{n} \) is a stopping time, the fu... | Yes |
Corollary 62.8.2 In the situation of Theorem 62.8.1, let \( s > 0 \) and let \( {D}_{1} \) and \( {D}_{2} \) be two countable dense subsets of \( \mathbb{R} \) . Then\n\n\[ \mathop{\lim }\limits_{{r \rightarrow s -, r \in {D}_{1}}}X\left( {r,\omega }\right) = \mathop{\lim }\limits_{{r \rightarrow s -, r \in {D}_{2}}}X\... | Proof: Let \( \left\{ {r}_{n}^{i}\right\} \) be an increasing sequence from \( {D}_{i} \) converging to \( s \) and let \( N \) be the exceptional set corresponding to the countable dense set \( {D}_{1} \cup {D}_{2} \) . Then for \( \omega \notin N \), and \( i = 1,2 \) ,\n\n\[ \mathop{\lim }\limits_{{r \rightarrow s -... | Yes |
Theorem 62.8.4 Let \( \{ X\left( t\right) \} \) be a submartingale adapted to a normal filtration \( {\mathcal{F}}_{t} \) . There exists a right continuous submartingale having left limits, \( \{ Y\left( t\right) \} \) such that \( Y\left( t\right) = X\left( t\right) \) a.e. for every \( t \in {\mathbb{Q}}^{ + } \) . F... | Proof: From Theorem 62.8.1, there exists a set of measure zero, \( N \) such that for \( \omega \notin N \), left and right limits of the following form exist. \[ \mathop{\lim }\limits_{{r \rightarrow t +, r \in \mathbb{Q}}}X\left( {r,\omega }\right) ,\;\mathop{\lim }\limits_{{r \rightarrow t -, r \in \mathbb{Q}}}X\lef... | Yes |
Lemma 62.9.1 Let \( X \) be right continuous and adapted such that the given filtration is complete in the sense that \( {\mathcal{F}}_{0} \) contains all sets \( A \) of \( \mathcal{F} \) such that \( P\left( A\right) = 0 \) . Then there exists a set of measure zero \( N \) and a \( \mathcal{F} \times \mathcal{B}\left... | Proof: Let \( \left\{ {{t}_{0}^{n},{t}_{1}^{n},\cdots ,{t}_{{m}_{n}}^{n}}\right\} \) be a partition of \( \left\lbrack {0, T}\right\rbrack \) in which \( \left| {{t}_{i}^{n} - {t}_{i - 1}^{n}}\right| < {\rho }_{n} \) where \( {\rho }_{n} \rightarrow 0 \) . Now define \( {X}_{n} \) as follows:\n\n\[ \n{X}_{n}\left( t\ri... | Yes |
Proposition 62.9.3 The stopped submartingale just defined is a submartingale. | Proof: By the optional sampling theorem for submartingales, Theorem 62.7.15, it follows that for \( s < t \) ,\n\n\[ E\left( {{X}^{{\tau }_{n}}\left( t\right) \mid {\mathcal{F}}_{s}}\right) \equiv E\left( {X\left( {t \land {\tau }_{n}}\right) \mid {\mathcal{F}}_{s}}\right) \geq X\left( {t \land {\tau }_{n} \land s}\rig... | Yes |
Theorem 62.9.4 Let \( \{ X\left( t\right) \} \) be a right continuous nonnegative submartingale adapted to the normal filtration \( {\mathcal{F}}_{t} \) for \( t \in \left\lbrack {0, T}\right\rbrack \) . Let \( p \geq 1 \) . Define\n\n\[ \n{X}^{ * }\left( t\right) \equiv \sup \{ X\left( s\right) : 0 < s < t\} ,{X}^{ * ... | Proof: The first inequality follows from Theorem 62.5.2. However, it can also be obtained a different way using stopping times.\n\nDefine the stopping time\n\n\[ \n\tau \equiv \inf \{ t > 0 : X\left( t\right) > \lambda \} \land T.\n\]\n\n(The infimum over an empty set will equal \( \infty \) .)This is a stopping time b... | Yes |
Theorem 62.9.5 Let \( \{ X\left( t\right) \} \) be a right continuous submartingale adapted to the normal filtration \( {\mathcal{F}}_{t} \) for \( t \in \left\lbrack {0, T}\right\rbrack \) and \( {X}^{ * }\left( t\right) \) defined as in Theorem 62.9.4\n\n\[ \n{X}^{ * }\left( t\right) \equiv \sup \{ X\left( s\right) :... | Proof: The function \( f\left( r\right) = {r}^{ + } \equiv \frac{1}{2}\left( {\left| r\right| + r}\right) \) is convex and increasing. Therefore, \( {X}^{ + }\left( t\right) \) is also a submartingale but this one is nonnegative. Also\n\n\[ \n\left\lbrack {{X}^{ * }\left( T\right) > \lambda }\right\rbrack = \left\lbrac... | Yes |
Lemma 62.10.1 The above \( {\tau }_{i} \) are stopping times for \( t \in \left\lbrack {0, M}\right\rbrack \) . | Proof: It is obvious that \( {\tau }_{0} \) is a stopping time because it is the minimum of \( M \) and the first hitting time of a closed set by a continuous adapted process. Consider a stopping time \( \eta \leq M \) and let\n\n\[ \sigma \equiv \inf \left\{ {t > 0 : {\left( X\left( t \vee \eta \right) - X\left( \eta ... | Yes |
Lemma 62.10.2 Let \( \{ Y\left( t\right) \} \) be a continuous submartingale adapted to a normal filtration \( {\mathcal{F}}_{t} \) for \( t \in \left\lbrack {0, M}\right\rbrack \) . Then if \( {U}_{\left\lbrack a, b\right\rbrack }^{M} \) is defined as the above upper bound to the number of upcrossings of \( \{ Y\left(... | \[ E\left( {U}_{\left\lbrack a, b\right\rbrack }^{M}\right) \leq \frac{1}{b - a}\left( {E{\left( Y\left( M\right) - a\right) }_{ + } + a - a}\right) + 1 \] \[ = \frac{1}{b - a}E\left| {Y\left( M\right) }\right| + \frac{1}{b - a}\left| a\right| + 1 \] | Yes |
Theorem 62.10.3 Let \( \{ X\left( t\right) \} \) be a continuous submartingale adapted to a normal filtration such that\n\n\[ \mathop{\sup }\limits_{t}\{ E\left( \left| {X\left( t\right) }\right| \right) \} = C < \infty . \]\n\nThen there exists \( {X}_{\infty } \in {L}^{1}\left( \Omega \right) \) such that\n\n\[ \math... | Proof: Let \( {U}_{\left\lbrack a, b\right\rbrack } \) be defined by\n\n\[ {U}_{\left\lbrack a, b\right\rbrack } = \mathop{\lim }\limits_{{M \rightarrow \infty }}{U}_{\left\lbrack a, b\right\rbrack }^{M} \]\n\nThus the random variable \( {U}_{\left\lbrack a, b\right\rbrack } \) is an upper bound for the number of upcro... | Yes |
Lemma 62.11.2 Let \( M \) be a right continuous martingale adapted to the normal filtration \( {\mathcal{F}}_{t} \) and let \( \tau \) be a stopping time. Then \( {M}^{\tau } \) is also a martingale adapted to the filtration \( {\mathcal{F}}_{t} \) . | Proof:Let \( s < t \) . By the Doob optional sampling theorem,\n\n\[ E\left( {{M}^{\tau }\left( t\right) \mid {\mathcal{F}}_{s}}\right) \equiv E\left( {M\left( {\tau \land t}\right) \mid {\mathcal{F}}_{s}}\right) = M\left( {\tau \land t \land s}\right) = {M}^{\tau }\left( s\right) . \] | Yes |
Proposition 62.12.3 The functions \( M\left( t\right) \) for each \( M \in {\mathcal{M}}_{T}^{p}\left( E\right) \) are equi integrable. | Proof: This follows because\n\n\[ \n{\int }_{\lbrack \parallel M\left( t\right) \parallel \geq \lambda \rbrack }\parallel M\left( t\right) {\parallel }^{p}{dP} \leq {\int }_{\left\lbrack \mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\parallel M\left( t\right) \parallel \geq \lambda \right\rbrack }\le... | Yes |
Lemma 63.1.1 Let \( \{ X\left( t\right) \} \) be a stochastic process adapted to the filtration \( \left\{ {\mathcal{F}}_{t}\right\} \) for \( t \geq 0 \) . Then it is a martingale for the given filtration if for every stopping time \( \sigma \) it follows\n\n\[ E\left( {X\left( t\right) }\right) = E\left( {X\left( \si... | Proof: Let \( s < t \) and \( A \in {\mathcal{F}}_{s} \) . Define a stopping time\n\n\[ \sigma \left( \omega \right) \equiv s{\mathcal{X}}_{A}\left( \omega \right) + t{\mathcal{X}}_{{A}^{C}}\left( \omega \right) \]\n\nThis is a stopping time because \( \left\lbrack {\sigma \leq l}\right\rbrack = \Omega \) if \( l \geq ... | Yes |
Lemma 63.1.2 Suppose \( {X}_{n} \rightarrow X \) in \( {L}^{1}\left( {\Omega ,\mathcal{F}, P;E}\right) \) where \( E \) is a separable Banach space. Then letting \( \mathcal{G} \) be a \( \sigma \) algebra contained in \( \mathcal{F} \) , \[ E\left( {{X}_{n} \mid \mathcal{G}}\right) \rightarrow E\left( {X \mid \mathcal... | Proof: This follows from the definitions and Theorem 61.1.1 on Page 2091. \[ {\int }_{\Omega }\parallel E\left( {X \mid \mathcal{G}}\right) - E\left( {{X}_{n} \mid \mathcal{G}}\right) \parallel {dP} = {\int }_{\Omega }\begin{Vmatrix}{E\left( {{X}_{n} - X \mid \mathcal{G}}\right) }\end{Vmatrix}{dP} \] \[ \leq {\int }_{\... | Yes |
Corollary 63.1.3 Let \( X, Y \) be in \( {L}^{2}\left( {\Omega ,\mathcal{F}, P;H}\right) \) where \( H \) is a separable Hilbert space and let \( X \) be \( \mathcal{G} \) measurable where \( \mathcal{G} \subseteq \mathcal{F} \). Then \[ E\left( {\left( {X, Y}\right) \mid \mathcal{G}}\right) = \left( {X, E\left( {Y \mi... | Proof: First let \( X = a{\mathcal{X}}_{B} \) where \( B \in \mathcal{G} \). Then for \( A \in \mathcal{G} \), \[ {\int }_{A}E\left( {\left( {a{\mathcal{X}}_{B}, Y}\right) \mid \mathcal{G}}\right) {dP} = {\int }_{A}{\mathcal{X}}_{B}E\left( {\left( {a, Y}\right) \mid \mathcal{G}}\right) {dP} = {\int }_{A}{\mathcal{X}}_{... | Yes |
Proposition 63.2.2 If \( M\left( t\right) \) is a continuous local martingale (submartingale) for a normal filtration as above, \( M\left( 0\right) = 0 \), then there exists a localizing sequence \( {\tau }_{n} \) such that for each \( n \) the stopped martingale(submartingale) \( {M}^{{\tau }_{n}} \) is uniformly boun... | Proof: First consider the claim about \( {M}^{\tau } \) being a martingale (submartingale) when \( M \) is. By optional sampling theorem,\n\n\[ E\left( {{M}^{\tau }\left( t\right) \mid {\mathcal{F}}_{s}}\right) = E\left( {M\left( {\tau \land t}\right) \mid {\mathcal{F}}_{s}}\right) = M\left( {\tau \land t \land s}\righ... | Yes |
Corollary 63.2.3 Let \( {\mathcal{F}}_{t} \) be a normal filtration and let \( A\left( t\right), B\left( t\right) \) be adapted to \( {\mathcal{F}}_{t} \) , continuous, and increasing with \( A\left( 0\right) = B\left( 0\right) = 0 \) and suppose \( A\left( t\right) - B\left( t\right) \equiv M\left( t\right) \) is a lo... | Proof: Let \( \left\{ {\tau }_{n}\right\} \) be a localizing sequence for \( M \) . For given \( n \), consider the martingale, \[ {M}^{{\tau }_{n}}\left( t\right) = {A}^{{\tau }_{n}}\left( t\right) - {B}^{{\tau }_{n}}\left( t\right) \] Then from Lemma 63.1.5, it follows \( {M}^{{\tau }_{n}}\left( t\right) = 0 \) for a... | Yes |
Lemma 63.2.4 Let \( X\left( t\right) \) be continuous and adapted to a normal filtration \( {\mathcal{F}}_{t} \) and let \( \eta \) be a stopping time. Then if \( K \) is a closed set, \[ \tau \equiv \inf \{ t > \eta : X\left( t\right) \in K\} \] is also a stopping time. | Proof: First consider \( Y\left( t\right) = X\left( {t \vee \eta }\right) - X\left( \eta \right) \) . I claim that \( Y\left( t\right) \) is adapted to \( {\mathcal{F}}_{t} \) . Consider \( U \) and open set and \( \left\lbrack {Y\left( t\right) \in U}\right\rbrack \) . Is it in \( {\mathcal{F}}_{t} \) ? We know it is ... | Yes |
Corollary 63.2.6 In the situation of Proposition 63.2.5 let \( \tau \) be a stopping time. Then\n\n\[ \left\lbrack {M}^{\tau }\right\rbrack = {\left\lbrack M\right\rbrack }^{\tau } \] | Proof:\n\n\[ {\left\lbrack M\right\rbrack }^{\tau }\left( t\right) + {N}_{1}\left( t\right) = {\left( \parallel M{\parallel }^{2}\right) }^{\tau }\left( t\right) = {\begin{Vmatrix}{M}^{\tau }\end{Vmatrix}}^{2}\left( t\right) = \left\lbrack {M}^{\tau }\right\rbrack \left( t\right) + {N}_{2}\left( t\right) \]\n\nwhere \(... | Yes |
Lemma 63.3.2 The following hold for the covariation.\n\n\[ \left\lbrack M\right\rbrack = \left\lbrack {M, M}\right\rbrack \]\n\n\[ \left\lbrack {M, N}\right\rbrack = \text{ local martingale } + \frac{1}{4}\left( {\parallel M + N{\parallel }^{2} - \parallel M - N{\parallel }^{2}}\right) \]\n\n\[ = \left( {M, N}\right) +... | Proof: From the definition of covariation,\n\n\[ \left\lbrack M\right\rbrack = \parallel M{\parallel }^{2} - {\mathcal{N}}_{1} \]\n\n\[ \left\lbrack {M, M}\right\rbrack = \frac{1}{4}\left( {\left\lbrack {M + M}\right\rbrack - \left\lbrack {M - M}\right\rbrack }\right) = \frac{1}{4}\left( {\parallel M + M{\parallel }^{2... | Yes |
Corollary 63.3.3 Let \( M, N \) be two continuous local martingales, \( M\left( 0\right) = N\left( 0\right) = \) 0, as in Proposition 63.2.5. Then \( \left\lbrack {M, N}\right\rbrack \) is of bounded variation and \[ {\left( M, N\right) }_{H} - \left\lbrack {M, N}\right\rbrack \] is a local martingale. Also for \( \tau... | Proof: Since \( \left\lbrack {M, N}\right\rbrack \) is the difference of increasing functions, it is of bounded variation. \[ {\left( M, N\right) }_{H} - \left\lbrack {M, N}\right\rbrack = \overset{{\left( M, N\right) }_{H}}{\overbrace{\frac{1}{4}\left( {\parallel M + N{\parallel }^{2} - \parallel M - N{\parallel }^{2}... | Yes |
Theorem 63.4.3 Let \( \{ M\left( t\right) \} \) be a continuous \( H \) valued martingale which is uniformly bounded, \( M\left( 0\right) = 0 \), where \( H \) is a separable Hilbert space and \( t \in \left\lbrack {0, T}\right\rbrack \) . Then if \( F \) is a function of the sort described in the good lambda inequalit... | Proof: Using Corollary 63.3.3, let\n\n\[ N\left( t\right) \equiv {\begin{Vmatrix}M\left( t\right) - {M}^{\tau }\left( t\right) \end{Vmatrix}}^{2} - \left\lbrack {M - {M}^{\tau }}\right\rbrack \left( t\right) \]\n\n\[ = {\begin{Vmatrix}M\left( t\right) - {M}^{\tau }\left( t\right) \end{Vmatrix}}^{2} - \left\lbrack M\rig... | Yes |
Theorem 63.4.4 Let \( \{ M\left( t\right) \} \) be a continuous \( H \) valued local martingale, \( M\left( 0\right) = \) 0, where \( H \) is a separable Hilbert space and \( t \in \left\lbrack {0, T}\right\rbrack \) . Then if \( F \) is a function of the sort described in the good lambda inequality, that is,\n\n\[ F\l... | Proof: Let \( \left\{ {\tau }_{n}\right\} \) be an increasing localizing sequence for \( M \) such that \( {M}^{{\tau }_{n}} \) is uniformly bounded. Such a localizing sequence exists from Proposition 63.2.2. Then from Theorem 63.4.3 there exist constants \( c, C \) independent of \( {\tau }_{n} \) such that\n\n\[ c{\i... | Yes |
Corollary 63.4.5 Let \( \{ M\left( t\right) \} \) be a continuous \( H \) valued local martingale and let \( \varepsilon ,\delta \in \left( {0,\infty }\right) \) . Then there is a constant \( C \), independent of \( \varepsilon ,\delta \) such that\n\n\[ P\left( \left\lbrack {\overset{{M}^{ * }\left( T\right) }{\overbr... | Proof: Let the stopping time \( \tau \) be defined by\n\n\[ \tau \equiv \inf \left\{ {t > 0 : {\left\lbrack M\right\rbrack }^{1/2}\left( t\right) > \delta }\right\} \]\nThen\n\n\[ P\left( \left\lbrack {{M}^{ * } \geq \varepsilon }\right\rbrack \right) = P\left( {\left\lbrack {{M}^{ * } \geq \varepsilon }\right\rbrack \... | Yes |
Proposition 63.4.6 The space \( {M}_{T}^{2}\left( H\right) \) is a Hilbert space. Here \( H \) is a separable Hilbert space. | Proof: We already know from Proposition 62.12.2 that this space is a Banach space. It is only necessary to exhibit an equivalent norm which makes it a Hilbert space. However, you can let \( F\left( \lambda \right) = {\lambda }^{2} \) in the Burkholder Davis Gundy theorem and obtain for \( M \in {M}_{T}^{2}\left( H\righ... | Yes |
What is the quadratic variation of the Wiener process? | The quadratic variation of the Wiener process is just \( t \) . This is because if \( A \in {\mathcal{F}}_{s}, s < t \)\n\n\[ E\left( {{\mathcal{X}}_{A}\left( {{\left| W\left( t\right) \right| }^{2} - t}\right) }\right) = \]\n\n\[ E\left( {{\mathcal{X}}_{A}\left( {{\left| W\left( t\right) - W\left( s\right) \right| }^{... | Yes |
Lemma 63.5.4 Let \( M, N \) be continuous local martingales, \( M\left( 0\right) = N\left( 0\right) = 0 \) having values in a separable Hilbert space, U. Then\n\n\[{\left\lbrack M + N\right\rbrack }^{1/2} \leq \left( {{\left\lbrack M\right\rbrack }^{1/2} + {\left\lbrack N\right\rbrack }^{1/2}}\right)\]\n\n\( \left( {63... | Proof: Since \( \left( {M, N}\right) \rightarrow \left\lbrack {M, N}\right\rbrack \) is bilinear and satisfies\n\n\[ \left\lbrack {M, N}\right\rbrack = \left\lbrack {N, M}\right\rbrack \]\n\n\[ \left\lbrack {{aM} + b{M}_{1}, N}\right\rbrack = a\left\lbrack {M, N}\right\rbrack + b\left\lbrack {{M}_{1}, N}\right\rbrack \... | Yes |
Theorem 63.5.5 The integral is well defined and has a continuous version which is a local martingale. Furthermore it satisfies the Ito isometry,\n\n\[ E\left( {\\begin{Vmatrix}{\\int }_{0}^{t}fdM\\end{Vmatrix}}_{U}^{2}\\right) = {\\int }_{\\Omega }{\\int }_{0}^{t}f{\\left( s\\right) }^{2}d\\left\\lbrack M\\right\\rbrac... | Proof: It is clear the definition is well defined because if \( \\left\{ {f}_{n}\\right\} \) and \( \\left\{ {g}_{n}\\right\} \) are two sequences of elementary functions converging to \( f \) in \( {L}^{2}\\left( {\\Omega ;{L}^{2}\\left( {\\left\\lbrack {0, T}\\right\\rbrack ,\\nu \\left( \\cdot \\right) }\\right) }\\... | Yes |
Lemma 63.6.1 The \( {k}^{\text{th }} \) term in the above sum is a martingale and the integral is also a martingale. | Proof: Let \( \sigma \) be a stopping time with two values. Then\n\n\[ E\left( {{f}_{k}\left( {M\left( {\sigma \land {t}_{k + 1}}\right) - M\left( {\sigma \land {t}_{k}}\right) }\right) }\right) \]\n\n\[ = E\left( {E\left( {{f}_{k}\left( {M\left( {\sigma \land {t}_{k + 1}}\right) - M\left( {\sigma \land {t}_{k}}\right)... | Yes |
Theorem 63.6.4 Let \( H \) be a Hilbert space and suppose \( \left( {M,{\mathcal{F}}_{t}}\right), t \in \left\lbrack {0, T}\right\rbrack \) is a uniformly bounded continuous martingale with values in \( H \) . Also let \( {\left\{ {t}_{k}^{n}\right\} }_{k = 1}^{{m}_{n}} \) be a sequence of partitions satisfying\n\n\[ \... | Proof: First suppose \( M \) is uniformly bounded.\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{{{m}_{n} - 1}}{\left| M\left( t \land {t}_{k + 1}^{n}\right) - M\left( t \land {t}_{k}^{n}\right) \right| }_{H}^{2} \]\n\n\[ = \mathop{\sum }\limits_{{k = 0}}^{{{m}_{n} - 1}}{\left| M\left( t \land {t}_{k + 1}^{n}\right) \right| }... | Yes |
Theorem 63.7.1 Let \( \left\{ {X}_{n}\right\} \) be a submartingale. Then there exists a unique stochastic process, \( \left\{ {A}_{n}\right\} \) and martingale, \( \left\{ {M}_{n}\right\} \) such that\n\n1. \( {A}_{n}\left( \omega \right) \leq {A}_{n + 1}\left( \omega \right) ,{A}_{1}\left( \omega \right) = 0 \) ,\n\n... | Proof: Let \( {A}_{1} \equiv 0 \) and define\n\n\[ \n{A}_{n} \equiv \mathop{\sum }\limits_{{k = 2}}^{n}E\left( {{X}_{k} - {X}_{k - 1} \mid {\mathcal{F}}_{k - 1}}\right) .\n\]\n\nIt follows \( {A}_{n} \) is \( {\mathcal{F}}_{n - 1} \) measurable. Since \( \left\{ {X}_{k}\right\} \) is a submartingale, \( {A}_{n} \) is i... | Yes |
Lemma 63.7.3 Let a stochastic process, \( \\left\\{ {A}_{n}\\right\\} \) be natural. Then for every martingale, \( \\left\\{ {M}_{n}\\right\\} \) , | Proof: Start with the right side.\n\n\[ E\\left( {\\mathop{\\sum }\\limits_{{j = 1}}^{{n - 1}}{M}_{j}\\left( {{A}_{j + 1} - {A}_{j}}\\right) }\\right) = E\\left( {\\mathop{\\sum }\\limits_{{j = 2}}^{n}{M}_{j - 1}{A}_{j} - \\mathop{\\sum }\\limits_{{j = 1}}^{{n - 1}}{M}_{j}{A}_{j}}\\right) \]\n\n\[ = E\\left( {\\mathop{... | Yes |
Lemma 63.7.5 Let \( f \) be right continuous. Then \( f \) is Borel measurable. Also, if the limit from the left exists, then \( {f}_{ - }\left( x\right) \equiv f{\left( x\right) }_{ - } \equiv \mathop{\lim }\limits_{{y \rightarrow x - }}f\left( y\right) \) is also Borel measurable. If \( A \) is an increasing right co... | Proof: For \( x \in {f}^{-1}\left( \left( {a,\infty }\right) \right) \), denote by \( {I}_{x} \) the union of all intervals containing \( x \) such that \( f\left( y\right) \) is larger than \( a \) for all \( y \) in the interval. Since \( f \) is right continuous, each \( {I}_{x} \) has positive length. Now if \( {I}... | Yes |
Lemma 63.7.8 Let \( \left( {\Omega ,\mathcal{F}, P}\right) \) be a probability space and let \( \mathcal{G} \) be a \( \sigma \) algebra contained in \( \mathcal{F} \) . Suppose also that \( \left\{ {f}_{n}\right\} \) is a sequence in \( {L}^{1}\left( \Omega \right) \) which converges weakly to \( f \) in \( {L}^{1}\le... | Proof:First note that if \( h \in {L}^{\infty }\left( {\Omega ,\mathcal{F}}\right) \), then \( E\left( {h \mid \mathcal{G}}\right) \in {L}^{\infty }\left( {\Omega ,\mathcal{G}}\right) \) because if \( A \in \mathcal{G}, \) \[ {\int }_{A}\left| {E\left( {h \mid \mathcal{G}}\right) }\right| {dP} \leq {\int }_{A}E\left( {... | Yes |
Example 63.7.10 Let \( \{ M\left( t\right) \} \) be a continuous martingale. Then \( \{ M\left( t\right) \} \) is of class DL. | To show this, let \( a > 0 \) be given and let \( T \) be a stopping time bounded by \( a \) . Then by the optional sampling theorem, \( M\left( 0\right), M\left( T\right), M\left( a\right) \) is a martingale and so\n\n\[ E\left( {M\left( a\right) \mid {\mathcal{F}}_{T}}\right) = M\left( T\right) \]\n\nand so by Jensen... | Yes |
Example 63.7.11 Let \( \{ X\left( t\right) \} \) be a nonnegative submartingale with \( t \rightarrow E\left( {X\left( t\right) }\right) \) right continuous so \( \{ X\left( t\right) \} \) can be considered right continuous. Then \( \{ X\left( t\right) \} \) is \( {DL} \) . | To show this, let \( T \) be a stopping time bounded by \( a > 0 \) . Then by the optional sampling theorem,\n\n\[ \n{\int }_{\left\lbrack X\left( T\right) \geq \lambda \right\rbrack }X\left( T\right) {dP} \leq {\int }_{\left\lbrack X\left( T\right) \geq \lambda \right\rbrack }X\left( a\right) {dP} \n\]\n\nand now by T... | Yes |
Example 63.7.16 Suppose \( \{ M\left( t\right) \} \) is a continuous martingale. Assume\n\n\[ \mathop{\sup }\limits_{{t \in \left\lbrack {0, a}\right\rbrack }}\parallel M\left( t\right) {\parallel }_{{L}^{2}\left( \Omega \right) } < \infty \]\n\nThen \( \{ \parallel M\left( t\right) \parallel \} \) is a submartingale a... | By Example 63.7.11, this is DL. Then there exists a unique Doob Meyer decomposition,\n\n\[ \parallel M\left( t\right) {\parallel }^{2} = Y\left( t\right) + \langle \parallel M\left( t\right) \parallel \rangle \]\n\nwhere \( Y\left( t\right) \) is a martingale and \( \{ \langle \parallel M\left( t\right) \parallel \rang... | Yes |
Lemma 63.8.2 Let \( \{ X\left( t\right) \} \) be a real martingale adapted to the filtration \( {\mathcal{F}}_{t} \) for \( t \in \) \( \left\lbrack {a, b}\right\rbrack \) some interval such that for all \( t \in \left\lbrack {a, b}\right\rbrack, E\left( {X{\left( t\right) }^{2}}\right) < \infty \) . Then \( \left\{ {X... | Proof: Suppose first \( \left\{ {X{\left( t\right) }^{2} - t}\right\} \) is a real martingale. Then since \( \{ X\left( t\right) \} \) is a martingale,\n\n\[ E\left( {{\left( X\left( t\right) - X\left( s\right) \right) }^{2} \mid {\mathcal{F}}_{s}}\right) = E\left( {X{\left( t\right) }^{2} - {2X}\left( t\right) X\left(... | Yes |
Lemma 64.1.2 There exists a sequence, \( {\left\{ {\xi }_{k}\right\} }_{k = 1}^{\infty } \) of random variables such that\n\n\[ \mathcal{L}\left( {\xi }_{k}\right) = N\left( {0,1}\right) \]\n\nand \( {\left\{ {\xi }_{k}\right\} }_{k = 1}^{\infty } \) is independent. | Proof: Let \( {i}_{1} < {i}_{2}\cdots < {i}_{n} \) be positive integers and define\n\n\[ {\mu }_{{i}_{1}\cdots {i}_{n}}\left( {{F}_{1} \times \cdots \times {F}_{n}}\right) \equiv \frac{1}{{\left( \sqrt{2\pi }\right) }^{n}}{\int }_{{F}_{1} \times \cdots \times {F}_{n}}{e}^{-{\left| \mathbf{x}\right| }^{2}/2}{dx}. \]\n\n... | Yes |
Theorem 64.2.1 Let \( W\left( t\right) \) be a Wiener process. Then there exists a set of measure \( 0, N \) such that for all \( \omega \notin N \) , \[ t \rightarrow W\left( {t,\omega }\right) \] is nowhere differentiable. | Proof: Let \( \left\lbrack {0, a}\right\rbrack \) be an interval. If for some \( \omega, t \rightarrow W\left( {t,\omega }\right) \) is differentiable at some \( s \), then for some \( n, p > 0 \) , \[ \left| \frac{W\left( {t,\omega }\right) - W\left( {s,\omega }\right) }{t - s}\right| \leq p \] whenever \( \left| {t -... | Yes |
Lemma 64.3.3 Let \( E \) be a real separable Banach space. Then there exists an \( E \) valued stochastic process, \( W\left( t\right) \) such that \( \mathcal{L}\left( {W\left( t\right) }\right) \) and \( \mathcal{L}\left( {W\left( t\right) - W\left( s\right) }\right) \) are Gaussian measures and the increments, \( \{... | \[ {\int }_{\Omega }\parallel W\left( t\right) - W\left( s\right) {\parallel }^{\alpha }{dP} = {\int }_{E}\parallel x{\parallel }^{\alpha }d{\mu }_{W\left( t\right) - W\left( s\right) } \] \[ = {\int }_{E}\parallel x{\parallel }^{\alpha }d{\mu }_{W\left( {t - s}\right) } = {\int }_{E}\parallel x{\parallel }^{\alpha }d{... | Yes |
Lemma 64.4.2 Let \( \{ W\left( t\right) \} \) be a stochastic process having values in a separable Banach space which has the property that if \( {t}_{1} < {t}_{2}\cdots < {t}_{n} \), then the increments,\n\n\[ \left\{ {W\left( {t}_{k}\right) - W\left( {t}_{k - 1}\right) }\right\} \]\n\nare independent and integrable a... | Proof: Consider first the claim,64.4.14. To begin with I show that if \( A \in {\mathcal{F}}_{s} \) then for all \( \varepsilon \) small enough that \( t > s + \varepsilon ,{}^{1} \)\n\n\[ {\int }_{\Omega }{\mathcal{X}}_{A}g\left( {W\left( t\right) - W\left( {s + \varepsilon }\right) }\right) {dP} = P\left( A\right) {\... | Yes |
Theorem 64.4.3 Let \( \{ W\left( t\right) \} \) be a Wiener process having values in a separable Banach space as described in Theorem 64.3.4. There exists a set of measure \( 0, N \) such that for \( \omega \notin N \), the sum in 64.3.11 converges uniformly to \( W\left( {t,\omega }\right) \) on any interval, \( \left... | Proof: By Lemma 64.4.2 the independence of the increments imply\n\n\[ \mathop{\sum }\limits_{{k = m}}^{n}{\psi }_{k}\left( t\right) {e}_{k} \]\n\nis a martingale and so by Theorem 62.5.3,\n\n\[ P\left( \left\lbrack {\mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| \right| \mathop{\sum }\limits_{{... | Yes |
Lemma 64.4.4 Let \( \\left\\{ {\\zeta }_{k}\\right\\} \) be a sequence of random variables having values in a separable real Banach space, \( E \) whose distributions are symmetric. Letting \( {S}_{k} \\equiv \) \( \\mathop{\\sum }\\limits_{{i = 1}}^{k}{\\zeta }_{i} \), suppose \( \\left\\{ {S}_{{n}_{k}}\\right\\} \) c... | Apply this lemma to the situation in which the Banach space, \( E \) is \( C\\left( {\\left\\lbrack {0, T}\\right\\rbrack ;E}\\right) \) and \( {\\zeta }_{k} = {\\psi }_{k}{e}_{k} \). Then you can conclude uniform convergence of the partial sums,\n\n\[ \n\\mathop{\\sum }\\limits_{{k = 1}}^{m}{\\psi }_{k}\\left( t\\righ... | No |
Theorem 64.5.2 Let \( U \) be a real separable Hilbert space and let \( J : {U}_{0} \rightarrow U \) be a Hilbert Schmidt operator where \( {U}_{0} \) is a real separable Hilbert space. Then let \( \left\{ {g}_{k}\right\} \) be a complete orthonormal basis for \( {U}_{0} \) and define for \( t \in \left\lbrack {0, T}\r... | Proof: First it is necessary to show the series converges in \( {L}^{2}\left( {\Omega ;U}\right) \) for each \( t \) . For convenience I will consider the series for \( W\left( t\right) - W\left( s\right) \) . (Always, it is assumed \( t > s \) .) Then since \( {\psi }_{k}\left( t\right) - {\psi }_{k}\left( s\right) \)... | Yes |
Theorem 64.5.4 Suppose \( \{ W\left( t\right) \} \) is a \( Q \) Wiener process in \( U \), a real separable Hilbert space. Then letting\n\n\[ \nQ = \mathop{\sum }\limits_{{k = 1}}^{\infty }{\lambda }_{k}{e}_{k} \otimes {e}_{k} \n\]\n\nwhere the \( \left\{ {e}_{k}\right\} \) are orthonormal, \( {\lambda }_{k} \geq 0 \)... | Proof: First of all, the fact that \( W\left( t\right) \) has values in \( U \) and that \( \left\{ {e}_{k}\right\} \) is an orthonormal basis implies the sum in 64.5.28 converges for each \( \omega \) . Consider\n\n\[ \nE\left( {\exp \left( {{ir}{\psi }_{k}\left( t\right) }\right) }\right) = E\left( {\exp \left( {{ir}... | Yes |
Lemma 64.5.5 Let \( \\left\\{ {\\zeta }_{k}\\right\\} \) be a sequence of random variables having values in a separable real Banach space, \( E \) whose distributions are symmetric. Letting \( {S}_{k} \\equiv \) \( \\mathop{\\sum }\\limits_{{i = 1}}^{k}{\\zeta }_{i} \), suppose \( \\left\\{ {S}_{{n}_{k}}\\right\\} \) c... | Then in fact, \[ {S}_{k}\\left( \\omega \\right) \\rightarrow S\\left( \\omega \\right) \\text{ a.e. }\\omega \] | Yes |
Corollary 64.6.2 The map \( h \rightarrow W\left( h\right) \) is linear. Also, \( \{ W\left( h\right) : h \in H\} \) is a closed subspace of \( {L}^{2}\left( {\Omega ,\mathcal{F}, P}\right) \) where \( \mathcal{F} = \sigma \left( {W\left( h\right) : h \in H}\right) \) . | Proof: This follows from the above description.\n\n\[ E\left( {\left\lbrack W\left( g + h\right) - \left( W\left( g\right) + W\left( h\right) \right) \right\rbrack }^{2}\right) = E\left( {W{\left( g + h\right) }^{2}}\right) \]\n\n\[ + E\left( {\left( W\left( g\right) + W\left( h\right) \right) }^{2}\right) - {2E}\left(... | Yes |
Lemma 64.6.3 Let \( X \geq 0 \) and measurable. Also define a finite measure on \( \mathcal{B}\left( {\mathbb{R}}^{p}\right) \)\n\n\[ \nu \left( B\right) \equiv {\int }_{\Omega }X{\mathcal{X}}_{B}\left( \mathbf{Y}\right) {dP} \]\n\nThen let \( f : {\mathbb{R}}^{p} \rightarrow \lbrack 0,\infty ) \) be Borel measurable. ... | Proof: First say \( X = {\mathcal{X}}_{D} \) and replace \( f\left( \mathbf{Y}\right) \) with \( {\mathcal{X}}_{{\mathbf{Y}}^{-1}\left( B\right) } \) . Then\n\n\[ {\int }_{\Omega }{\mathcal{X}}_{D}{\mathcal{X}}_{{\mathbf{Y}}^{-1}\left( B\right) }{dP} = P\left( {D \cap {\mathbf{Y}}^{-1}\left( B\right) }\right) \]\n\n\[ ... | Yes |
Lemma 64.6.4 Each \( {e}^{W\left( h\right) } \) is in \( {L}^{p}\left( \Omega \right) \) for every \( h \in H \) and for every \( p \geq 1 \) . In fact,\n\n\[{\int }_{\Omega }{\left( {e}^{W\left( h\right) }\right) }^{p}{dP} = {\int }_{\Omega }{e}^{W\left( {ph}\right) }{dP} = {e}^{\frac{1}{2}{\left| ph\right| }_{H}^{2}}... | Proof: It suffices to verify this for all positive integers \( p \) . Let \( p \) be such an integer. Note that from the linearity of \( W,{\left( {e}^{W\left( h\right) }\right) }^{p} = {e}^{{pW}\left( h\right) } = {e}^{W\left( {ph}\right) } \) and so it suffices to verify that for each \( h \in H,{e}^{W\left( h\right)... | Yes |
Lemma 65.1.3 Let \( f, g \in {L}^{2}\left( {\Omega ;H}\right) \) and suppose \( g \) is \( \mathcal{G} \) measurable and \( f \) is \( \mathcal{F} \) measurable where \( \mathcal{F} \supseteq \mathcal{G} \) . Then\n\n\[ E\left( {{\left( f, g\right) }_{H} \mid \mathcal{G}}\right) = {\left( E\left( f \mid \mathcal{G}\rig... | Proof: Let \( A \in \mathcal{G} \) . Let \( \left\{ {g}_{n}\right\} \) be a sequence of simple functions, measurable with respect to \( \mathcal{G} \) ,\n\n\[ {g}_{n}\left( \omega \right) \equiv \mathop{\sum }\limits_{{k = 1}}^{{m}_{n}}{a}_{k}^{n}{\mathcal{X}}_{{E}_{k}^{n}}\left( \omega \right) \]\n\nwhich converges in... | Yes |
Lemma 65.1.4 Let \( J : {U}_{0} \rightarrow U \) be a Hilbert Schmidt operator and let \( W\left( t\right) \) be the resulting Wiener process\n\n\[ W\left( t\right) = \mathop{\sum }\limits_{{k = 1}}^{\infty }{\psi }_{k}\left( t\right) J{g}_{k} \]\n\nwhere \( \left\{ {g}_{k}\right\} \) is an orthonormal basis for \( {U}... | Proof: For simplicity, write \( \Delta {W}_{k}\left( t\right) \) for \( W\left( {t \land {t}_{k + 1}}\right) - W\left( {t \land {t}_{k}}\right) \) and \( {\Delta }_{k}\left( t\right) = \) \( \left( {t \land {t}_{k + 1}}\right) - \left( {t \land {t}_{k}}\right) \) . If \( \Phi \left( {t}_{k}\right) \) were a constant, t... | Yes |
Lemma 65.3.1 Let \( \Phi : \left\lbrack {0, T}\right\rbrack \times \Omega \rightarrow E \), be \( \mathcal{B}\left( \left\lbrack {0, T}\right\rbrack \right) \times \mathcal{F} \) measurable and suppose\n\n\[ \Phi \in K \equiv {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega ;E}\right), p \geq 1 \]\n\nThen ... | Proof: For \( t \in \mathbb{R} \) let \( {\gamma }_{n}\left( t\right) \equiv k/{2}^{n},{\delta }_{n}\left( t\right) \equiv \left( {k + 1}\right) /{2}^{n} \), where \( t \in \left( {k/{2}^{n},\left( {k + 1}\right) /{2}^{n}}\right\rbrack \) , and \( {2}^{-n} < T/4 \) . Also suppose \( \Phi \) is defined to equal 0 on \( ... | Yes |
Proposition 65.3.2 Let \( \Phi \in {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega, E}\right), p \geq 1 \), be progressively measurable. Then there exists a sequence of elementary functions which converges to \( \Phi \) in \[ {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega, E}\right) . | Proof: By Lemma 65.3.1 there exists a sequence of step functions \[ {\Phi }_{k}^{l}\left( t\right) = \mathop{\sum }\limits_{{j = 1}}^{{m}_{k}}\Phi \left( {t}_{j - 1}^{k}\right) {\mathcal{X}}_{\left( {t}_{j - 1}^{k},{t}_{j}^{k}\right\rbrack }\left( t\right) \] which converges to \( \Phi \) in \( {L}^{p}\left( {\left\lbr... | Yes |
Proposition 65.3.3 Suppose \( f \geq 0 \) is progressively measureable and \( {\mathcal{F}}_{t} \) is a filtration. Then\n\n\[ \n\omega \rightarrow {\int }_{a}^{t}f\left( {s,\omega }\right) {ds} \n\]\n\nis \( {\mathcal{F}}_{t} \) adapted. | Proof: This follows right away from the fact \( f \) is \( \mathcal{B}\left( \left\lbrack {a, t}\right\rbrack \right) \times {\mathcal{F}}_{t} \) measurable. This is just product measure and so the integral from \( a \) to \( t \) is \( {\mathcal{F}}_{t} \) measurable. See also Proposition 62.3.5. | Yes |
Lemma 65.4.3 Let \( A \in \mathcal{L}\left( {U, U}\right) \) be a bounded linear transformation defined on \( U \) a separable real Hilbert space. There exists a one to one Hilbert Schmidt operator \( J : {AU} \rightarrow {U}_{1} \) where \( {U}_{1} \) is a separable real Hilbert space. In fact you can take \( {U}_{1} ... | Proof: Let \( {\alpha }_{k} > 0 \) and \( \mathop{\sum }\limits_{{k = 1}}^{\infty }{\alpha }_{k}^{2} < \infty \) . Then let \( {\left\{ {g}_{k}\right\} }_{k = 1}^{L} \) be an orthonormal basis for \( {AU} \), the inner product and norm given in Definition 65.4.1 above, and let\n\n\[ \n{Jx} \equiv \mathop{\sum }\limits_... | Yes |
Lemma 65.5.1 Let \( \Phi \in {L}^{2}\left( {\left\lbrack {a, T}\right\rbrack \times \Omega ;{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, H}\right) }\right) \) and suppose also that \( \Phi \) is progressively measurable with respect to the usual filtration associated with the Wiener process\n\n\[ \nW\left( t\right) = \mathop{\... | Proof: First, why is \( \Phi \circ {J}^{-1} \in {L}^{2}\left( {\left\lbrack {a, T}\right\rbrack \times \Omega ;{\mathcal{L}}_{2}\left( {J{Q}^{1/2}U, H}\right) }\right) \) ? This follows from the observation that \( A \) is Hilbert Schmidt if and only if \( {A}^{ * } \) is Hilbert Schmidt. In fact, the Hilbert Schmidt n... | Yes |
Theorem 65.5.3 The stochastic integral 65.5.7 is well defined. It also is a continuous martingale and does not depend on the choice of \( J \) and \( {U}_{1} \) . Furthermore, | Proof: First of all, it is obvious that it is well defined in the sense that the same stochastic process is obtained from two different sequences of elementary functions. This follows from the isometry of Proposition 65.1.5 with \( {U}_{1} \) in place of \( U \) and \( {Q}^{1/2}U \) in place of \( {U}_{0} \) . Thus if ... | Yes |
Corollary 65.5.4 Let \( \Phi ,\Psi \in {L}^{2}\left( {\left\lbrack {a, T}\right\rbrack \times \Omega ;{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, H}\right) }\right) \) and suppose they are both progressively measurable. Then\n\n\[ E\left( {\left( {\int }_{a}^{t}\Phi dW,{\int }_{a}^{t}\Psi dW\right) }_{H}\right) = E\left( {{\i... | Proof: First note that\n\n\[ {\left( {\int }_{a}^{t}\Phi dW,{\int }_{a}^{t}\Psi dW\right) }_{H} = \frac{1}{4}\left\lbrack {{\left| {\int }_{a}^{t}\left( \Phi + \Psi \right) dW\right| }_{H}^{2} - {\left| {\int }_{a}^{t}\left( \Phi - \Psi \right) dW\right| }_{H}^{2}}\right\rbrack \]\n\nand so from the above theorem,\n\n\... | Yes |
Lemma 65.10.2 Suppose \( \Phi \) is \( {\mathcal{L}}_{2}\left( {{Q}^{1/2}U, H}\right) \) progressively measurable and\n\n\[ P\left( \left\lbrack {{\int }_{a}^{T}\parallel \Phi {\parallel }_{{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, H}\right) }^{2}{ds} < \infty }\right\rbrack \right) = 1.\]\n\nDefine\n\n\[ {\tau }_{n}\left( ... | Proof: It follows from Proposition 62.7.5 that \( {\tau }_{n} \) is a stopping time because it is the first hitting time of a closed set by an adapted continuous process.\n\nIt remains to verify the two claims. There exists a set of measure \( 0, N \) such that for \( \omega \notin N \)\n\n\[ {\int }_{a}^{T}\parallel \... | Yes |
Lemma 65.10.4 The above definition is well defined. For all \( \omega \) not in a set of measure zero, \[ {\int }_{a}^{t}{\Phi dW}\left( \omega \right) \equiv \mathop{\lim }\limits_{{n \rightarrow \infty }}{\int }_{a}^{t}{\mathcal{X}}_{\left\lbrack a,{\tau }_{n}\right\rbrack }{\Phi dW}\left( \omega \right) \] the funct... | Proof: Let \( \left\{ {\tau }_{n}\right\} \) be a sequence of stopping times as described in 1 and 2 of Lemma 65.10.2. Such a sequence exists by Lemma 65.10.2. It makes sense to define the random variable \[ {\int }_{a}^{t}{\mathcal{X}}_{\left\lbrack a,{\tau }_{n}\right\rbrack }{\Phi dW} \] Now what if both \( {\tau }_... | Yes |
Lemma 65.10.5 Let \( \Phi \) be progressively measurable and suppose there exists the localizing sequence described above. Then if \( \sigma \) is a stopping time,\n\n\[{\int }_{a}^{t \land \sigma }{\Phi dW}\left( s\right) = {\int }_{a}^{t}{\mathcal{X}}_{\left\lbrack a,\sigma \right\rbrack }{\Phi dW}\left( s\right)\] | Proof: Let \( \left\{ {\tau }_{n}\right\} \) be the localizing sequence described above for which, when the local martingale is stopped, it results in a martingale, (satisfying 1 and 2 on Page 2373). Then by definition,\n\n\[{\int }_{a}^{t \land \sigma }{\Phi dW}\left( s\right) \equiv \mathop{\lim }\limits_{{n \rightar... | Yes |
Corollary 65.11.1 Suppose \( \Phi \) is \( {\mathcal{L}}_{2}\left( {{Q}^{1/2}U, H}\right) \) progressively measurable and has the localizing sequence with the two properties in Lemma 65.10.2. Then the quadratic variation, \( \left\lbrack {\Phi \cdot W}\right\rbrack \) is given by the formula\n\n\[ \left\lbrack {\Phi \c... | Proof: By the above discussion, \( {\int }_{a}^{t}{\Phi dW} \) is a local martingale. Let \( \left\{ {\tau }_{n}\right\} \) be a localizing sequence for which the stopped local martingale is a martingale and \( \Phi {\mathcal{X}}_{\left\lbrack a,{\tau }_{n}\right\rbrack } \) is in \( {L}^{2}\left( {\left\lbrack {a, T}\... | Yes |
Lemma 65.11.2 Let \( \Phi ,{\Phi }_{n} \) all be in \( {L}^{2}\left( {\left\lbrack {0, T}\right\rbrack ,{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, H}\right) }\right) \) off some set of measure zero. These are all progressively measurable. Thus there are all stochastically square integrable.\n\n\[ P\left( {{\int }_{0}^{T}\par... | Proof: Define stopping times\n\n\[ {\tau }_{np} \equiv \inf \left\{ {t \in \left\lbrack {0, T}\right\rbrack : {\int }_{0}^{t}{\begin{Vmatrix}{\Phi }_{n}\end{Vmatrix}}^{2}{ds} > p}\right\} \]\n\nLet \( {\tau }_{p} \) be similar but defined with reference to \( \Phi \) . Then by Ito isometry,\n\n\[ E\left( {\left| {\int ... | Yes |
Lemma 65.14.2 Suppose \( t \rightarrow X\\left( t\\right) \) is weakly continuous into \( H \) for a.e. \( \\omega \), and that \( X \) is adapted. Then the \( {\\tau }_{n} \) described above is a stopping time. | Proof: Let \( B \\equiv \\{ x \\in H : \\left| x\\right| > n\\} \) . Then the complement of \( B \) is a closed convex set. It follows that \( {B}^{C} \) is also weakly closed. Hence \( B \) must be weakly open. Now \( t \\rightarrow X\\left( t\\right) \) is adapted as a function mapping into the topological space cons... | Yes |
Lemma 65.14.3 Let \( X\left( s\right) - {X}_{k}^{l}\left( s\right) \equiv {\Delta }_{k}\left( s\right) \) . Here \( Z \in {L}^{2}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega ,{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, H}\right) }\right) \) and let \( X \in {L}^{2}\left( {\left\lbrack {0, T}\right\rbrack \times \Ome... | Proof: Let \( k \) denote a subsequence for which \( {X}_{k}^{l} \) also converges pointwise to \( X \) . The existence of the integral follows from Lemma 65.14.2. From the assumption of weak continuity, \( \mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| {X\left( t\right) }\right| \leq C\left( \... | Yes |
For an elementary function \( Y \), and a stopping time \( \tau \) having values in the \( \left\{ {t}_{i}\right\} \), the points of discontinuity of \( Y \), it follows that \( {\mathcal{X}}_{\left\lbrack 0,\tau \right\rbrack }Y \) is also an elementary function and | \[ {\int }_{0}^{t \land \tau }\left( {Y,{dM}}\right) = {\int }_{0}^{t}\left( {{\mathcal{X}}_{\left\lbrack 0,\tau \right\rbrack }Y,{dM}}\right) = {\int }_{0}^{t}\left( {Y, d{M}^{\tau }}\right) \]\n\nProof: Consider the second equal sign.\n\n\[ {\int }_{0}^{t \land \tau }\left( {Y,{dM}}\right) = \mathop{\sum }\limits_{{i... | Yes |
Lemma 66.0.5 Let \( M \) be a local martingale on \( \left\lbrack {0, T}\right\rbrack \) where \( M\left( 0\right) = 0 \) and \( M \) is continuous. Let \( 0 < r < s < T \) and consider \( \left( {Y,\left( {{M}^{{\tau }_{p}s} - {M}^{{\tau }_{p}r}}\right) \left( t\right) }\right) \) where \( Y{\left( {M}^{{\tau }_{p}}\r... | Proof: To save notation, \( M \) is written in place of \( {M}^{{\tau }_{p}} \) . It is clear that \( \left( {Y,\left( {{M}^{s} - {M}^{r}}\right) \left( t\right) }\right) = 0 \) if \( t \leq r \) . Is it a martingale?\n\n\[ E\left( \left( {Y,\left( {{M}^{s} - {M}^{r}}\right) \left( t\right) }\right) \right) = E\left( {... | Yes |
Lemma 66.0.6 Let \( \parallel Y\left( t\right) \parallel {\left( {M}^{{\tau }_{p}}\right) }^{ * } \in {L}^{2}\left( \Omega \right) \) for each \( t \), where \( Y \) is an elementary function and let \( {\tau }_{p} \) be a stopping time for which \( {M}^{{\tau }_{p}} \) is a \( {L}^{2} \) martingale. Then | \[ E\left( {\left| {\int }_{0}^{t}\left( Y, d{M}^{{\tau }_{p}}\right) \right| }^{2}\right) \leq E\left( {{\int }_{0}^{t}\parallel Y{\parallel }_{H}^{2}d{\left\lbrack M\right\rbrack }^{{\tau }_{p}}}\right) \] | Yes |
Lemma 66.0.9 The above definition is well defined. Also, \( {\int }_{0}^{t}\left( {Y, d{M}^{{\tau }_{p}}}\right) \) is a continuous martingale. The inequality\n\n\[ E\left( {\left| {\int }_{0}^{t}\left( Y, d{M}^{{\tau }_{p}}\right) \right| }^{2}\right) \leq E\left( {{\int }_{0}^{t}\parallel Y{\parallel }_{H}^{2}d{\left... | Proof: First of all, why does the limit even exist? From Lemma 66.0.6,\n\n\[ E\left( {\left| {\int }_{0}^{t}\left( {Y}^{n}, d{M}^{{\tau }_{p}}\right) - {\int }_{0}^{t}\left( {Y}^{m}, d{M}^{{\tau }_{p}}\right) \right| }^{2}\right) \leq E\left( {{\int }_{0}^{T}{\begin{Vmatrix}{Y}^{n} - {Y}^{m}\end{Vmatrix}}_{H}^{2}d{\lef... | Yes |
Lemma 66.0.12 Let \( Y \) be an elementary function. Then if \( \tau \) is any stopping time, then off a set of measure zero, \[ {\int }_{0}^{t \land \tau }\left( {Y, d{M}^{{\tau }_{p}}}\right) = {\int }_{0}^{t}\left( {{\mathcal{X}}_{\left\lbrack 0,\tau \right\rbrack }Y, d{M}^{{\tau }_{p}}}\right) = {\int }_{0}^{t}\lef... | Proof: It remains to prove the second equation. \[ {\int }_{0}^{t \land \tau }\left( {Y, d{M}^{{\tau }_{p}}}\right) \equiv \mathop{\sum }\limits_{{i = 0}}^{{m - 1}}\left( {{Y}_{i},{M}^{{\tau }_{p}}\left( {t \land {t}_{i + 1} \land \tau }\right) - {M}^{{\tau }_{p}}\left( {t \land {t}_{i} \land \tau }\right) }\right) \] ... | Yes |
Lemma 66.0.13 Let \( Y \in \mathcal{G} \) . Then for any stopping time \( \tau \) , \[ {\int }_{0}^{t \land \tau }\left( {Y, d{M}^{{\tau }_{p}}}\right) = {\int }_{0}^{t}\left( {{\mathcal{X}}_{\left\lbrack 0,\tau \right\rbrack }Y, d{M}^{{\tau }_{p}}}\right) = {\int }_{0}^{t}\left( {Y, d{M}^{\tau \land {\tau }_{p}}}\righ... | Proof: From Lemma 66.0.9, there exists a sequence of elementary functions \( {Y}^{n} \) such that \( t \rightarrow {\int }_{0}^{t}\left( {{Y}^{n},{dM}}\right) \) converges uniformly to \( t \rightarrow {\int }_{0}^{t}\left( {Y, d{M}^{{\tau }_{p}}}\right) \) on \( \left\lbrack {0, T}\right\rbrack \) for each \( \omega \... | Yes |
Theorem 66.0.15 The above definition is well defined. Also this makes \( {\int }_{0}^{t}\left( {Y,{dM}}\right) \) a local martingale. In particular,\n\n\[ \n{\int }_{0}^{t \land {\tau }_{p}}\left( {Y,{dM}}\right) = {\int }_{0}^{t}\left( {{\mathcal{X}}_{\left\lbrack 0,{\tau }_{p}\right\rbrack }Y, d{M}^{{\tau }_{p}}}\rig... | Proof: Suppose for some \( \omega, t < {\tau }_{p} < {\tau }_{q} \) . Let \( \omega \) be such that both \( {\tau }_{p},{\tau }_{q} \) are larger than \( t \) . Then for all \( \omega \), and \( \tau \) a stopping time, \n\n\[ \n{\int }_{0}^{t \land \tau }\left( {{\mathcal{X}}_{\left\lbrack 0,{\tau }_{q}\right\rbrack }... | Yes |
Lemma 66.0.20 The above definition is well defined. Also, \( {\int }_{0}^{t}{\left\langle Y, d{M}^{{\tau }_{p}}\right\rangle }_{{W}^{\prime }, W} \) is a continuous martingale. The inequality | \[ E\left( {\left| {\int }_{0}^{t}{\left\langle Y, d{M}^{{\tau }_{p}}\right\rangle }_{{W}^{\prime }, W}\right| }^{2}\right) \leq E\left( {{\int }_{0}^{t}\parallel Y{\parallel }_{{W}^{\prime }}^{2}d{\left\lbrack M\right\rbrack }^{{\tau }_{p}}}\right) \] is also valid. For any sequence of elementary functions \( \left\{ ... | Yes |
Lemma 67.2.1 Suppose \( {\eta }_{j} \) are real random variables \( E\left( {\eta }_{j}^{2}\right) < \infty \), such that \( {\eta }_{k} \) is measurable with respect to \( {\mathcal{G}}_{j} \) for all \( j > k \) where \( \left\{ {\mathcal{G}}_{k}\right\} \) is increasing. Then\n\n\[ E\left( {\left\lbrack \mathop{\sum... | Proof: First consider a mixed term \( i < k \) .\n\n\[ E\left( {\left( {{\eta }_{i} - E\left( {{\eta }_{i} \mid {\mathcal{G}}_{i}}\right) }\right) \left( {{\eta }_{k} - E\left( {{\eta }_{k} \mid {\mathcal{G}}_{k}}\right) }\right) }\right) \]\n\nThis equals\n\n\[ E\left( {{\eta }_{i}{\eta }_{k}}\right) - E\left( {{\eta ... | Yes |
Theorem 67.6.1 Let \( \Phi \) be a progressively measurable process having values in \( {\mathcal{L}}_{2}\left( {{Q}^{1/2}U, H}\right) \) which is stochastically integrable in \( \left\lbrack {0, T}\right\rbrack \) because\n\n\[ P\left( \left\lbrack {{\int }_{0}^{T}\parallel \Phi {\parallel }_{{\mathcal{L}}_{2}\left( {... | The dependence of \( F \) on \( \omega \) is suppressed.\n\nThat last term is interesting and can be written differently. Let \( \left\{ {g}_{j}\right\} \) be an orthonormal basis for \( {Q}^{1/2}U \) . Then this integrand equals\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{L}{\left( {F}_{XX}\left( s, X\left( s\right) \right... | Yes |
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