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Theorem 9.9 Let \( {M}_{t} \) be a continuous square integrable martingale. There exists a continuous adapted increasing process \( \langle M{\rangle }_{t} \) with \( \langle M{\rangle }_{0} = 0 \) and with increasing paths such that \( {M}_{t}^{2} - \langle M{\rangle }_{t} \) is a martingale. | Proof By Jensen's inequality for conditional expectations,\n\n\[ \mathbb{E}\left\lbrack {{M}_{t}^{2} \mid {\mathcal{F}}_{s}}\right\rbrack \geq {\left( \mathbb{E}\left\lbrack {M}_{t} \mid {\mathcal{F}}_{s}\right\rbrack \right) }^{2} = {M}_{s}^{2} \]\n\nif \( s < t \), and so \( {M}_{t}^{2} \) is a submartingale. Since \... | Yes |
Lemma 10.1 The predictable \( \sigma \) -field \( \mathcal{P} \) is generated by the collection \( \mathcal{C} \) of processes of the form \( {X}_{t}\left( \omega \right) = \mathop{\sum }\limits_{{i = 1}}^{n}{K}_{i}\left( \omega \right) {1}_{\left( {a}_{i},{b}_{i}\right\rbrack }\left( t\right) \), where for each \( i,{... | Proof If \( X \in \mathcal{C} \), then \( X \) is bounded, adapted, and left continuous, hence \( X \) is a predictable process. Thus \( \mathcal{C} \subset \mathcal{P} \n\nOn the other hand, if \( Y \) is a bounded, adapted, left-continuous process, we can approximate \( Y \) by the processes\n\n\[ \n{Y}_{t}^{n}\left(... | Yes |
Proposition 10.3 With \( H \) as in (10.3) and \( N \) defined by (10.4), \( {N}_{t} \) is a continuous martingale,\n\n\[ \mathbb{E}{N}_{\infty }^{2} = \mathbb{E}{\int }_{0}^{\infty }{H}_{s}^{2}d\langle M{\rangle }_{s}, \]\n\nand\n\n\[ \langle N{\rangle }_{t} = {\int }_{0}^{t}{H}_{s}^{2}d\langle M{\rangle }_{s} \] | Proof By linearity, \( {N}_{t} \) is a continuous martingale. We have\n\n\[ \mathbb{E}{N}_{\infty }^{2} = \mathbb{E}\left\lbrack {\mathop{\sum }\limits_{j}{H}_{j}^{2}{\left( {M}_{{b}_{j}} - {M}_{{a}_{j}}\right) }^{2}}\right\rbrack \]\n\n\[ + 2\mathbb{E}\left\lbrack {\mathop{\sum }\limits_{{i < j}}{H}_{i}{H}_{j}\left( {... | Yes |
Theorem 11.2 Suppose \( {X}_{t}^{1},\ldots ,{X}_{t}^{d} \) are continuous semimartingales, \( {X}_{t} = \left( {{X}_{t}^{1},\ldots ,{X}_{t}^{d}}\right) \) , and \( f \) is a \( {C}^{2} \) function on \( {\mathbb{R}}^{d} \) . Then with probability one, | \[ f\left( {X}_{t}\right) = f\left( {X}_{0}\right) + {\int }_{0}^{t}\mathop{\sum }\limits_{{i = 1}}^{d}\frac{\partial f}{\partial {x}_{i}}\left( {X}_{s}\right) d{X}_{s}^{i} \] \[ + \frac{1}{2}{\int }_{0}^{t}\mathop{\sum }\limits_{{i, j = 1}}^{d}\frac{{\partial }^{2}f}{\partial {x}_{i}\partial {x}_{j}}\left( {X}_{s}\rig... | Yes |
Theorem 12.2 Suppose \( {M}_{t} \) is a continuous local martingale, \( {M}_{0} = 0,\langle M{\rangle }_{t} \) is strictly increasing, and \( \mathop{\lim }\limits_{{t \rightarrow \infty }}\langle M{\rangle }_{t} = \infty \), a.s. Let\n\n\[ \tau \left( t\right) = \inf \left\{ {u : \langle M{\rangle }_{u} \geq t}\right\... | Proof Let us first suppose that \( {W}_{t}^{2} \) is integrable. We have by Proposition 9.3 that\n\n\[ \mathbb{E}\left\lbrack {{W}_{t} \mid {\mathcal{F}}_{s}^{\prime }}\right\rbrack = \mathbb{E}\left\lbrack {{M}_{\tau \left( t\right) } \mid {\mathcal{F}}_{\tau \left( s\right) }}\right\rbrack = {M}_{\tau \left( s\right)... | No |
Corollary 12.5 If \( {M}_{t} \) is a square integrable martingale with respect to the minimal augmented filtration of a one-dimensional Brownian motion \( W \), then \( {M}_{t} \) has a version with continuous paths. | Proof By Corollary 3.13, \( M \) has a version with right continuous paths. By Corollary 12.4, \( M \) can be written as a stochastic integral with respect to \( W \) . But such stochastic integrals have continuous paths by Theorem 10.4. | Yes |
Theorem 12.6 Let \( {M}_{t} \) be a continuous local martingale with \( {M}_{0} = 0 \), a.s., and suppose \( 2 \leq p < \infty \) . There exists a constant \( {c}_{1} \) depending on \( p \) such that for any finite stopping time \( T \) , \[ \mathbb{E}{\left( {M}_{T}^{ * }\right) }^{p} \leq {c}_{1}\mathbb{E}\langle M{... | Proof There is nothing to prove if the left-hand side is zero, so we may assume it is positive. First suppose \( {M}_{T}^{ * } \) is bounded by a positive constant \( K \) . Note for \( p \geq 2 \) the function \( x \rightarrow {\left| x\right| }^{p} \) is \( {C}^{2} \) . By Doob’s inequalities and then Itô’s formula (... | Yes |
Theorem 12.7 Let \( {M}_{t} \) be a continuous local martingale with \( {M}_{0} = 0 \), a.s., and suppose \( 2 \leq p < \infty \) . There exists a constant \( {c}_{2} \) depending on \( p \) such that for any finite stopping time \( T \) ,\n\n\[ \mathbb{E}\langle M{\rangle }_{T}^{p/2} \leq {c}_{2}\mathbb{E}{\left( {M}_... | Proof As in the previous theorem, we may assume the left-hand side is positive. Set \( r = p/2 \) . Let us first suppose \( \langle M{\rangle }_{T} \) and \( {M}_{T}^{ * } \) are bounded by a positive constant \( K \) . Let \( {N}_{t} = {M}_{t \land T} \), so that \( \langle N{\rangle }_{\infty } = \langle M{\rangle }_... | Yes |
Theorem 12.8 Suppose \( f \in {C}^{3} \) and \( X \) is a continuous semimartingale. Then\n\n\[ f\left( {X}_{t}\right) = f\left( {X}_{0}\right) + {\int }_{0}^{t}{f}^{\prime }\left( {X}_{s}\right) \circ d{X}_{s}. \]\n | Proof By Itô’s formula applied to the function \( f \) and the definition of the Stratonovich integral, it suffices to show that\n\n\[ {\left\langle {f}^{\prime }\left( X\right), X\right\rangle }_{t} = {\int }_{0}^{t}{f}^{\prime \prime }\left( {X}_{s}\right) d\langle X{\rangle }_{s}. \]\n\n(12.13)\n\nApplying Itô’s for... | Yes |
Proposition 12.9 Suppose \( H \) and \( X \) are continuous semimartingales and \( {t}_{0} > 0 \) . Then \( {\int }_{0}^{t}{H}_{s} \circ d{X}_{s} \) is the limit in probability as \( n \rightarrow \infty \) of | Proof We write the sum as\n\n\[ \sum {H}_{k{t}_{0}/{2}^{n}}\left( {{X}_{\left( {k + 1}\right) {t}_{0}/{2}^{n}} - {X}_{k{t}_{0}/{2}^{n}}}\right) \]\n\n\[ + \frac{1}{2}\sum \left( {{H}_{\left( {k + 1}\right) {t}_{0}/{2}^{n}} - {H}_{k{t}_{0}/{2}^{n}}}\right) \left( {{X}_{\left( {k + 1}\right) {t}_{0}/{2}^{n}} - {X}_{k{t}_... | No |
Lemma 13.1 Suppose \( Y \) is a continuous local martingale with \( {Y}_{0} = 0 \) and \( {Z}_{t} = {e}^{{Y}_{t}-\langle Y{\rangle }_{t}/2} \) . If \( \langle Y{\rangle }_{t} \) is a bounded random variable for each \( t \), then \( \mathbb{E}{\left| {Z}_{t}\right| }^{p} < \infty \) for each \( p > 1 \) and each \( t \... | Proof Let us first suppose \( Y \) is bounded in absolute value by \( N \) . Since \( {Z}_{t} \geq 0 \), we have by the Cauchy-Schwarz inequality\n\n\[ \n\mathbb{E}{Z}_{t}^{p} = \mathbb{E}{e}^{p{Y}_{t} - p\langle Y{\rangle }_{t}/2} \n\]\n\n(13.2)\n\n\[ \n= \mathbb{E}\left\lbrack {{e}^{p{Y}_{t} - {p}^{2}\langle Y{\rangl... | Yes |
Lemma 13.2 Suppose \( {A}_{t} \) is a continuous increasing process adapted to a filtration \( \left\{ {\mathcal{F}}_{t}\right\} \) satisfying the usual conditions. Let \( X \) be a bounded random variable, \( H \) a bounded adapted process, \( s < t \), and \( B \in {\mathcal{F}}_{s} \). Then\n\n\[ \mathbb{E}\left\lbr... | Proof By linearity, it suffices to suppose \( X \) and \( H \) are non-negative. Let \( {A}_{r}^{\prime } = {A}_{r + s} \), \( {H}_{r}^{\prime } = {H}_{r + s} \), and \( {\mathcal{F}}_{r}^{\prime } = {\mathcal{F}}_{r + s} \). Let \( {C}_{r} = {\int }_{0}^{r}{H}_{r}^{\prime }{1}_{B}d{A}_{s}^{\prime } \), and so we must ... | Yes |
Theorem 14.2 The two-dimensional processes \( \left( {\left| W\right| ,{L}^{0}}\right) \) and \( \left( {M - W, M}\right) \) have the same law. | Proof Let \( {V}_{t} = - {N}_{t} \) in the Tanaka formula, so that\n\n\[\n\left| {W}_{t}\right| = - {V}_{t} + {L}_{t}^{0}\n\]\n\n(14.4)\n\nLet \( {S}_{t} = \mathop{\sup }\limits_{{s \leq t}}{V}_{s} \) . We will show \( {S}_{t} = {L}_{t}^{0} \) . This will prove the result, since \( V \) is a Brownian motion, and hence ... | Yes |
Theorem 15.1 Suppose \( F : \lbrack 0,\infty ) \times \mathbb{R} \rightarrow \mathbb{R} \) is a bounded function and there exists a positive real \( k \) such that\n\n\[ \left| {F\left( {t, x}\right) - F\left( {t, y}\right) }\right| \leq k\left| {x - y}\right| \]\n\nfor all \( t \geq 0 \) and all \( x, y \in \mathbb{R}... | Proof Note each \( {y}^{i}\left( t\right) \) is bounded in absolute value by \( \left| {y}_{0}\right| + t\sup \left| F\right| \) . Let \( {g}_{i}\left( t\right) = \) \( \mathop{\sup }\limits_{{s \leq t}}\left| {{y}^{i + 1}\left( s\right) - {y}^{i}\left( s\right) }\right| \) . If \( s \leq t \), then\n\n\[ \left| {{y}^{... | Yes |
We have\n\n\[ g\left( {W}_{1}\right) = \mathbb{E}g\left( {W}_{1}\right) + {\int }_{0}^{1}a\left( {s,{W}_{s}}\right) d{W}_{s},\;\text{ a.s. } \]\n\n(15.7)\n\nand\n\n\[ \mathbb{E}\left\lbrack {g\left( {W}_{1}\right) \mid {\mathcal{F}}_{s}}\right\rbrack = b\left( {s,{W}_{s}}\right) ,\;\text{ a.s. } \]\n\n(15.8) | Proof We will first prove (15.7), and we will first look at the case when \( g\left( x\right) = {e}^{iux} \) .\n\nBy Itô’s formula with the function \( f\left( x\right) = {e}^{x} \) applied to the semimartingale \( {X}_{t} = {iu}{W}_{t} + \) \( {u}^{2}t/2 \)\n\n\[ {e}^{{iu}{W}_{t} + {u}^{2}t/2} = 1 + {\int }_{0}^{t}{e}... | Yes |
Proposition 15.3 Let \( g \) be defined by (15.12) and define a and b by (15.5) and (15.6).\n\n(1) For each \( L > 0 \) and \( {s}_{0} < 1 \), a is continuously differentiable on \( \left\lbrack {0,{s}_{0}}\right\rbrack \times \left\lbrack {-L, L}\right\rbrack \) . Also, for each \( L > 0 \) and \( {s}_{0} < 1 \), a is... | Proof To start, we observe that for every \( r > 0 \) ,\n\n\[ \mathbb{E}{e}^{r\left| {W}_{1}\right| } \leq \mathbb{E}{e}^{r{W}_{1}} + \mathbb{E}{e}^{-r{W}_{1}} < \infty . \]\n\nSince \( {\left| z\right| }^{m} \leq m!{e}^{\left| z\right| } \) if \( m \) is a non-negative integer, then by the Cauchy-Schwarz inequality an... | Yes |
Corollary 15.5 Let \( W \) be a Brownian motion and let \( \left\{ {\mathcal{F}}_{t}\right\} \) be the minimal augmented filtration for \( W \) . Let \( Y \) be a random variable with \( \mathbb{E}Y = 0 \) and \( \operatorname{Var}Y < \infty \) . There exists a stopping time \( V \) with respect to \( \left\{ {\mathcal... | Proof We sketch the proof and ask you to give the details in Exercise 15.3. Define\n\n\[ \bar{\Phi }\left( {q, r}\right) = \frac{1}{{\left( a\left( r, B\left( r,{W}_{q}\left( \omega \right) \right) \right) \right) }^{2}} \]\n\nand solve the equation\n\n\[ \frac{d{\bar{\tau }}_{t}}{dt} = \Phi \left( {t,{\bar{\tau }}_{t}... | No |
Theorem 15.6\n\n\\[ \n\\mathop{\\sup }\\limits_{{i \\leq n}}\\left| {{W}_{{U}_{i}} - {W}_{i}}\\right| /\\sqrt{n}\n\\]\n\ntends to 0 in probability as \\( n \\rightarrow \\infty \\) . | Proof We will show that for each \\( \\varepsilon > 0 \\)\n\n\\[ \n\\mathop{\\limsup }\\limits_{{n \\rightarrow \\infty }}\\mathbb{P}\\left( {\\mathop{\\sup }\\limits_{{k \\leq n}}\\left| {{W}_{{U}_{k}} - {W}_{k}}\\right| > \\varepsilon \\sqrt{n}}\\right) \\leq \\varepsilon .\n\\]\n\n(15.18)\n\nSince the paths of Brown... | Yes |
Proposition 16.1 Let \( T \) be a stopping time. There exist predictable stopping times \( {S}_{1},{S}_{2},\ldots \) and a totally inaccessible stopping time \( U \) such that \( \left\lbrack {T, T}\right\rbrack = \left\lbrack {U, U}\right\rbrack \cup \left( {{ \cup }_{i = 1}^{\infty }\left\lbrack {{S}_{i},{S}_{i}}\rig... | Proof Let\n\n\[ \n{a}_{1} = \sup \{ \mathbb{P}\left( {S = T < \infty }\right) : S\text{is a predictable stopping time}\} \n\] \n\nand choose \( {S}_{1} \) to be a predictable stopping time such that \( \mathbb{P}\left( {{S}_{1} = T < \infty }\right) \geq \frac{1}{2}{a}_{1} \) . Given \( {S}_{1},\ldots ,{S}_{n} \), let ... | Yes |
Proposition 16.2 (1) The optional \( \sigma \) -field \( \mathcal{O} \) is generated by the collection of sets\n\n\[ \n\{ \lbrack S, T) : S, T\text{stopping times}\} \text{.} \n\] | Proof (1) Since \( {1}_{\lbrack S, T)} \) is a bounded right-continuous process that is adapted to \( \left\{ {\mathcal{F}}_{t}\right\} \), sets of the form \( \lbrack S, T) \) are optional. Now suppose \( X \) is a bounded adapted process with right-continuous paths. Let \( \varepsilon > 0 \), let \( {U}_{0} = 0 \), a... | No |
Proposition 16.3 (1) If \( A \) is an open set, then \( {T}_{A} \) and \( {U}_{A} \) are stopping times. | Proof (1) Since the paths of \( {X}_{t} \) are right continuous and \( A \) is open, for each \( t \) ,\n\n\[ \left( {{T}_{A} < t}\right) = { \cup }_{q \in {\mathbb{Q}}_{ + }, q < t}\left( {{X}_{t} \in A}\right) \in {\mathcal{F}}_{t}, \]\n\nwhere \( {\mathbb{Q}}_{ + } \) denotes the non-negative rationals. Thus \( {T}_... | Yes |
Theorem 16.4 If \( A \) is a Borel subset of \( \mathcal{S} \), then \( {R}_{t}\left( A\right) \in {\mathcal{F}}_{t} \) and there exists an increasing sequence of compact sets \( {K}_{n} \) contained in \( A \) such that \( \mathbb{P}\left( {{R}_{t}\left( {K}_{n}\right) }\right) \uparrow \mathbb{P}\left( {{R}_{t}\left(... | Since \( \left( {{U}_{A} \leq t}\right) = {R}_{t}\left( A\right) \), we have the following as an immediate corollary. | No |
Theorem 16.6 Suppose \( \left\{ {\mathcal{F}}_{t}\right\} \) is a filtration satisfying the usual conditions and \( X \) is a right continuous process whose jump times are totally inaccessible. If \( B \) is a Borel subset of \( \mathcal{S} \) , then \( {T}_{B} \) is a stopping time. | Proof If we let \( {Y}_{t}^{\delta } = {X}_{t + \delta } \) and \( {U}_{B}^{\delta } = \inf \left\{ {t \geq 0 : {Y}_{t}^{\delta } \in B}\right\} \), then by the above, \( {U}_{B}^{\delta } \) is a stopping time with respect to the filtration \( \left\{ {\mathcal{F}}_{t}^{\delta }\right\} \), where \( {\mathcal{F}}_{t}^... | Yes |
Theorem 16.8 There exists an increasing sequence of compacts \( {K}_{n} \) contained in B such that \( {T}_{{K}_{n}} \downarrow {T}_{B} \) | Proof Let \( {Y}_{t}^{\delta } = {X}_{t + \delta } \) and \( {U}_{B}^{\delta } = \inf \left\{ {t \geq 0 : {Y}_{t} \in B}\right\} \) . Applying the above proposition to \( {Y}_{t}^{1/m} \), for each \( m \) there exist compact sets \( {L}_{n}^{m} \), increasing in \( n \) and contained in \( B \), such that \( {U}_{{L}_... | Yes |
Theorem 16.9 If \( E \in \mathcal{O} \), then \( {D}_{E} \) is a stopping time. | The proof of this theorem is beyond the scope of this book, and we refer the reader to Dellacherie and Meyer (1978) for a proof. | No |
Theorem 16.10 If \( X \) is an optional process taking values in \( \mathcal{S} \) and \( B \) is a Borel subset of \( \mathcal{S} \) , then \( {U}_{B} \) and \( {T}_{B} \) are stopping times. | Proof Since \( B \) is a Borel subset of \( \mathcal{S} \) and \( X \) is an optional process, then \( {1}_{B}\left( {X}_{t}\right) \) is also an optional process. \( {U}_{B} \) is then the debut of the set \( E = \left\{ {\left( {s,\omega }\right) : {1}_{B}\left( {{X}_{s}\left( \omega \right) }\right) = 1}\right\} \),... | No |
Corollary 16.14 (1) If \( X \) and \( Y \) are optional processes such that \( \mathbb{P}\left( {{X}_{T} = {Y}_{T}}\right) = 1 \) for every finite stopping time \( T \), then \( X \) and \( Y \) are indistinguishable: \( \mathbb{P}\left( {{X}_{t} = {Y}_{t}}\right. \) for all \( \left. t\right) = 1 \) . | Proof We prove (1), the proof of (2) being similar. Let \( F = \left\{ {\left( {t,\omega }\right) : {X}_{t}\left( \omega \right) \neq {Y}_{t}\left( \omega \right) }\right\} \) . Then \( F \) is an optional set, and if \( \mathbb{P}\left( {\pi \left( F\right) }\right) > 0 \), there exists a stopping time \( U \) with \(... | Yes |
Proposition 16.16 Let \( {X}_{t} \) be a predictable process with paths that are right continuous with left limits. If \( a \in \mathbb{R} \) and \( T = \inf \left\{ {t > 0 : {X}_{t} \geq a}\right\} \), then \( T \) is a predictable stopping time. | Proof The set \( A = \left\{ {\left( {t,\omega }\right) : {X}_{t}\left( \omega \right) \geq a}\right\} \) is a predictable set. Since \( {X}_{t} \) is right continuous, \( \lbrack T,\infty ) = A \cup \left( {T,\infty }\right) \in \mathcal{P} \) by Proposition 16.2, and so \( \left\lbrack {T, T}\right\rbrack = \lbrack T... | Yes |
Theorem 16.17 Let \( X \) be a bounded process that is \( \mathcal{H} \) measurable. There exists a unique optional process \( {}^{o}X \) such that\n\n\[ \n{}^{o}{X}_{T}{1}_{\left( T < \infty \right) } = \mathbb{E}\left\lbrack {{X}_{T}{1}_{\left( T < \infty \right) } \mid {\mathcal{F}}_{T}}\right\rbrack \n\]\n\nfor all... | Proof of Theorem 16.17 The uniqueness is immediate from Corollary 16.14. We look at existence. If \( {X}_{t}\left( \omega \right) = {1}_{F}\left( \omega \right) {1}_{\lbrack a, b)}\left( t\right) \) where \( F \in {\mathcal{F}}_{\infty } \), we set \( {}^{o}{X}_{t} \) equal to \( \mathbb{E}\left\lbrack {{1}_{F} \mid {\... | Yes |
Theorem 16.18 Let \( X \) be a bounded measurable process. There exists a unique predictable process \( {}^{p}X \), called the predictable projection of \( X \), such that\n\n\[ \mathbb{E}\left\lbrack {{}^{p}{X}_{T};T < \infty }\right\rbrack = \mathbb{E}\left\lbrack {{X}_{T};T < \infty }\right\rbrack \]\n\nfor every pr... | Proof Uniqueness is as before. If \( {X}_{t} = {1}_{F}\left( \omega \right) {1}_{(a, b\rbrack }\left( t\right) \), we let \( {}^{p}{X}_{t} = {1}_{(a, b\rbrack }\left( t\right) {Z}_{t - }\left( \omega \right) \), where \( {Z}_{t - } \) denotes the left-hand limit of \( {Z}_{t} \) at time \( t \) and \( {Z}_{t} \) is the... | Yes |
Lemma 16.19 Suppose \( T \) is a predictable stopping time predicted by stopping times \( {T}_{n} \) . Then \( {\mathcal{F}}_{T - } = \mathop{\bigvee }\limits_{{n = 1}}^{\infty }{\mathcal{F}}_{{T}_{n}} \) . | Proof If \( X \) is left continuous, adapted, and bounded, then \( {X}_{T} = \lim {X}_{{T}_{m}} \) and \( {X}_{{T}_{m}} \in {\mathcal{F}}_{{T}_{m}} \subset \) \( \mathop{\bigvee }\limits_{n}{\mathcal{F}}_{{T}_{n}} \), so \( {X}_{T} \in \mathop{\bigvee }\limits_{n}{\mathcal{F}}_{{T}_{n}} \) . An argument using the monot... | No |
Corollary 16.20 Suppose \( T \) is a predictable stopping time. If \( M \) is a uniformly integrable martingale with right-continuous paths, then\n\n\[ \mathbb{E}\left\lbrack {{M}_{T} \mid {\mathcal{F}}_{T - }}\right\rbrack = {M}_{T - } \] | Proof If \( {X}_{t} = {M}_{t - } \), then \( X \) is left continuous, hence predictable, so \( {M}_{T - } = {X}_{T} \) is \( {\mathcal{F}}_{T - } \) measurable by the definition of \( {\mathcal{F}}_{T - } \) and a limit argument. Suppose the sequence \( {T}_{n} \) predicts \( T \) . If \( A \in {\mathcal{F}}_{{T}_{m}} ... | No |
Corollary 16.21 Let \( S \) be a predictable stopping time, \( M \) a square integrable martingale, and \( {N}_{t} = \Delta {M}_{S}{1}_{\left( t \geq S\right) } \) . Then \( {N}_{t} \) is a square integrable martingale. | Proof Since \( \left| {N}_{t}\right| \leq 2\mathop{\sup }\limits_{{s \geq 0}}\left| {M}_{s}\right|, N \) is square integrable. We will show \( N \) is a martingale by showing \( \mathbb{E}{N}_{T} = 0 \) for all bounded stopping times \( T \), and then appealing to Proposition 9.5.\n\nIf \( T \) is a bounded stopping ti... | Yes |
Proposition 16.22 Let \( \left\{ {\mathcal{F}}_{t}\right\} \) be the minimal augmented filtration of a Brownian motion. If \( T \) is a stopping time with respect to \( \left\{ {\mathcal{F}}_{t}\right\} \), then \( T \) is a predictable stopping time. | Proof Let \( T \) be a stopping time for Brownian motion. Let \( g \) be a continuous strictly increasing function from \( \left\lbrack {0,\infty }\right\rbrack \) to \( \left\lbrack {0,1}\right\rbrack \), e.g., \( g\left( s\right) = \left( {2/\pi }\right) \arctan s \) . Let \( {M}_{t} \) be the right-continuous modifi... | Yes |
Proposition 16.23 Suppose \( {A}_{t} \) is an increasing process such that\n\n(1) \( \Delta {A}_{T} = 0 \) whenever \( T \) is a totally inaccessible stopping time, and\n\n(2) \( \Delta {A}_{T} \) is \( {\mathcal{F}}_{T - } \) measurable whenever \( T \) is a predictable stopping time.\n\nThen \( A \) is predictable. | Proof Let \( {U}_{mi} \) be the \( i \) th time \( \left| {\Delta {A}_{t}}\right| \in \left( {{2}^{-m},{2}^{-m + 1}}\right\rbrack \) . The \( {U}_{mi} \) are predictable stopping times by Exercise 16.5. We decompose each \( {U}_{mi} \) as in Proposition 16.1. Since \( A \) does not jump at totally inaccessible times, n... | No |
Theorem 16.24 Suppose \( \mu \) is a bounded positive measure on \( \mathcal{H} \) such that \( \mu \left( X\right) = 0 \) whenever \( X = 0 \) . Then there exists a unique right-continuous increasing process \( A \) with \( {A}_{0} = 0 \), a.s., such that \( \mu = {\mu }_{A} \) . | Proof First, uniqueness. If \( \mu = {\mu }_{A} = {\mu }_{B} \), let \( t > 0 \) and let \( C \) be the set of \( \omega \) ’s where \( {A}_{t}\left( \omega \right) > {B}_{t}\left( \omega \right) + \varepsilon \) . Then \( {\mu }_{A}\left( {\left\lbrack {0, t}\right\rbrack \times C}\right) \geq {\mu }_{B}\left( {\left\... | Yes |
Theorem 16.25 Suppose \( A \) is right continuous, \( {A}_{0} = 0 \), a.s., and \( {\mu }_{A}\left( X\right) = {\mu }_{A}\left( {{}^{o}X}\right) \) for every bounded \( \mathcal{H} \) measurable process \( X \) . Then \( {A}_{t} \) is optional. | Proof Since \( {A}_{t} \) is right continuous, we need only show that \( {A}_{t} \) is adapted. Fix \( t \) and let \( Y \) be a bounded \( {\mathcal{F}}_{\infty } \) measurable random variable,\n\n\[ Z = Y - \mathbb{E}\left\lbrack {Y \mid {\mathcal{F}}_{t}}\right\rbrack \]\n\nand \( {X}_{s}\left( \omega \right) = {1}_... | Yes |
Proposition 16.27 Let \( {A}_{t} \) be an adapted increasing process with \( {A}_{0} = 0 \), a.s. Then \( {A}_{t} - {\widetilde{A}}_{t} \) is a martingale. | Proof Let \( s < t \), let \( B \in {\mathcal{F}}_{s} \), define\n\n\[ S\left( \omega \right) = \left\{ {\begin{array}{ll} s, & \omega \in B, \\ \infty , & \omega \notin B, \end{array}\;\text{ and }\;T\left( \omega \right) = \left\{ \begin{array}{ll} t, & \omega \in B, \\ \infty , & \omega \notin B. \end{array}\right. ... | No |
Proposition 16.28 If \( M \) is a predictable uniformly integrable martingale with paths that are right continuous with left limits, then \( M \) is continuous. | Proof Let \( \varepsilon > 0 \) and let \( T = \inf \left\{ {t : \left| {\Delta {M}_{t}}\right| > \varepsilon }\right\} .T \) is a predictable stopping time by Exercise 16.2. By Corollary 16.20, \( \mathbb{E}\left\lbrack {{M}_{T} \mid {\mathcal{F}}_{T - }}\right\rbrack = {M}_{T - } \) . By the definition of \( {\mathca... | No |
Theorem 16.29 Suppose \( {Z}_{t} \) is a submartingale of class \( D \) with paths that are right continuous with left limits and such that \( {Z}_{0} = 0 \), a.s. Then \( {Z}_{t} = {M}_{t} + {A}_{t} \), where \( {M}_{t} \) is a uniformly integrable right-continuous martingale with \( {M}_{0} = 0 \), a.s., and \( {A}_{... | Proof We start with uniqueness. If \( {Z}_{t} = {M}_{t} + {A}_{t} = {N}_{t} + {B}_{t} \), then \( {M}_{t} - {N}_{t} = {B}_{t} - {A}_{t} \), and so \( {M}_{t} - {N}_{t} \) is a predictable uniformly integrable martingale. By Proposition \( {16.28},{M}_{t} - {N}_{t} \) is a continuous martingale. Since \( {M}_{t} - {N}_{... | Yes |
Proposition 16.30 The process \( A \) is continuous if and only if \( \mathbb{E}{Z}_{{T}_{n}} \rightarrow \mathbb{E}{Z}_{T} \) whenever \( {T}_{n} \uparrow T \) and \( {T}_{n} < T \) on \( \left( {T > 0}\right) \) . | Proof Let \( T \) be a predictable stopping time predicted by the sequence \( {T}_{n} \) . Since we know \( \mathbb{E}\left\lbrack {{A}_{\infty } - {A}_{{T}_{n}}}\right\rbrack = \mathbb{E}\left\lbrack {{Z}_{\infty } - {Z}_{{T}_{n}}}\right\rbrack \), then taking limits,\n\n\[ \mathbb{E}\left\lbrack {{A}_{\infty } - {A}_... | Yes |
Corollary 16.31 Let \( S \) be a totally inaccessible stopping time, \( Y \) a non-negative bounded random variable that is \( {\mathcal{F}}_{S} \) measurable, and \( {A}_{t} = Y{1}_{\left( t \geq S\right) } \) . Let \( \widetilde{A} \) be the compensator of \( A \) . Then \( \widetilde{A} \) has continuous paths. | Proof Let \( T \) be a stopping time and let \( {T}_{n} \) be stopping times increasing to \( T \) . If we have \( \mathbb{P}\left( {T = S}\right) = 0 \), then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{A}_{{T}_{n}} = {A}_{T} \), a.s., since \( A \) jumps only at time \( S \) . If \( \mathbb{P}\left( {T = S}\ri... | Yes |
Proposition 16.33 Let \( U \) be a stopping time, \( Y \) a non-negative integrable random variable that is \( {\mathcal{F}}_{U} \) measurable. Let \( {N}_{t} \) be the right-continuous version of \( \mathbb{E}\left\lbrack {Y \mid {\mathcal{F}}_{t}}\right\rbrack \) . Suppose there exists \( K > 0 \) such that \( {N}_{t... | Proof As in the proof of Proposition 16.32, it suffices to show\n\n\[ \mathbb{E}\left\lbrack {{A}_{\infty } - {A}_{T - };T < \infty }\right\rbrack \leq K\mathbb{P}\left( {T < \infty }\right) ,\]\n\n(16.14)\n\nwhere \( \lambda > 0 \) and \( T = \inf \left\{ {t : {A}_{t} \geq \lambda }\right\} \) . Since \( A \) is a pre... | Yes |
Lemma 17.1 If \( {A}_{t} = {B}_{t} - {C}_{t} \), where \( {B}_{t} \) and \( {C}_{t} \) are increasing right-continuous processes with \( {B}_{0} = {C}_{0} = 0 \), a.s., and in addition \( B \) and \( C \) are bounded, then\n\n\[ \mathbb{E}\mathop{\sup }\limits_{{t \geq 0}}{\widetilde{A}}_{t}^{2} < \infty \] | Proof By Proposition 16.32, \( \mathbb{E}{\widetilde{B}}_{\infty }^{2} < \infty \) and \( \mathbb{E}{\widetilde{C}}_{\infty }^{2} < \infty \), and so\n\n\[ \mathbb{E}\mathop{\sup }\limits_{{t \geq 0}}{\widetilde{A}}_{t}^{2} \leq \mathbb{E}\left\lbrack {2\mathop{\sup }\limits_{{t \geq 0}}{\widetilde{B}}_{t}^{2} + 2\math... | Yes |
Lemma 17.2 Suppose \( {A}_{t} \) is a bounded increasing right-continuous process with \( {A}_{0} = 0 \) , a.s., \( {\widetilde{A}}_{t} \) is the compensator of \( A \), and \( {M}_{t} = {A}_{t} - {\widetilde{A}}_{t} \) . Suppose \( {N}_{t} \) is a right continuous square integrable martingale such that \( \left( {\Del... | Proof By Lemma 17.1, \( M \) is square integrable. Suppose\n\n\[ H\left( {s,\omega }\right) = K\left( \omega \right) {1}_{(a, b\rbrack }\left( s\right) \]\n\nwith \( K \) being \( {\mathcal{F}}_{a} \) measurable. Since \( {M}_{t} \) is of bounded variation, we have (this is a Lebesgue-Stieltjes integral here)\n\n\[ \ma... | Yes |
Proposition 17.4 \( {M}_{t}^{2} - {\left\lbrack M\right\rbrack }_{t} \) is a martingale. | Proof By the orthogonality lemma and (17.1) it is easy to see that\n\n\[ \langle M{\rangle }_{t} = {\left\langle {M}^{c}\right\rangle }_{t} + \mathop{\sum }\limits_{i}{\left\langle {M}_{i}\right\rangle }_{t} \]\n\nSince \( {M}_{t}^{2} - \langle M{\rangle }_{t} \) is a martingale, we need only show \( {\left\lbrack M\ri... | No |
Theorem 17.5 Suppose \( {X}_{t} = {M}_{t} + {A}_{t} \), where \( {M}_{t} \) is a square integrable martingale and \( {A}_{t} \) is a process with paths of bounded variation whose total variation is integrable. Suppose \( f \) is \( {C}^{2} \) on \( \mathbb{R} \) with bounded first and second derivatives. Then\n\n\[ f\l... | Proof The proof will be given in several steps. Set\n\n\[ S\left( t\right) = {\int }_{0}^{t}{f}^{\prime }\left( {X}_{s - }\right) d{X}_{s},\;Q\left( t\right) = \frac{1}{2}{\int }_{0}^{t}{f}^{\prime \prime }\left( {X}_{s - }\right) d{\left\langle {X}^{c}\right\rangle }_{s}, \]\n\nand\n\n\[ J\left( t\right) = \mathop{\su... | No |
Lemma 17.6 (1) The sum of two local martingales is a local martingale. | Proof (1) If the sequence \( {S}_{n} \) reduces \( M \) and the sequence \( {T}_{n} \) reduces \( N \), then \( {S}_{n} \land {T}_{n} \) will reduce \( M + N \) . | No |
Lemma 17.7 (1) If \( T \) strongly reduces \( M \) and \( S \leq T \), then \( S \) strongly reduces \( M \) . | Proof (1) Note \( \mathbb{E}\left\lbrack {\left| {M}_{S}\right| \mid {\mathcal{F}}_{s}}\right\rbrack \leq \mathbb{E}\left\lbrack {\left| {M}_{T}\right| \mid {\mathcal{F}}_{s}}\right\rbrack \) by Jensen’s inequality, hence \( S \) strongly reduces \( M \) . | Yes |
Lemma 17.8 If \( M \) is a local martingale with \( {M}_{0} = 0 \), then there exist stopping times \( {T}_{n} \uparrow \infty \) that strongly reduce \( M \) . | Proof Let \( {R}_{n} \uparrow \infty \) be a sequence reducing \( M \) . Let\n\n\[ \n{S}_{nm} = {R}_{n} \land \inf \left\{ {t : \mathbb{E}\left\lbrack {\left| {M}_{{R}_{n}}\right| \mid {\mathcal{F}}_{t}}\right\rbrack \geq m}\right\} .\n\]\n\nArrange the stopping times \( {S}_{nm} \) into a single sequence \( \left\{ {U... | Yes |
Corollary 17.11 If \( {X}_{t} = \left( {{X}_{t}^{1},\ldots ,{X}_{t}^{d}}\right) \) is a process taking values in \( {\mathbb{R}}^{d} \) such that each component is a semimartingale, and \( f \) is a \( {C}^{2} \) function on \( {\mathbb{R}}^{d} \), then | \[ f\left( {X}_{t}\right) = f\left( {X}_{0}\right) + {\int }_{0}^{t}\mathop{\sum }\limits_{{i = 1}}^{d}\frac{\partial f}{\partial {x}_{i}}\left( {X}_{s - }\right) d{X}_{s}^{i} \] \[ + \frac{1}{2}{\int }_{0}^{t}\mathop{\sum }\limits_{{i, j = 1}}^{d}\frac{{\partial }^{2}f}{\partial {x}_{i}\partial {x}_{j}}\left( {X}_{s -... | Yes |
Corollary 17.12 If \( X \) and \( Y \) are semimartingales of the above form, | Proof Apply Theorem 17.10 with \( f\left( x\right) = {x}^{2} \) . Since in this case\n\n\[ f\left( {X}_{s}\right) - f\left( {X}_{s - }\right) - {f}^{\prime }\left( {X}_{s - }\right) \Delta {X}_{s} = \Delta {X}_{s}^{2}, \]\n\nwe obtain\n\n\[ {X}_{t}^{2} = {X}_{0}^{2} + 2{\int }_{0}^{t}{X}_{s - }d{X}_{s} + {\left\lbrack ... | Yes |
Theorem 17.13 Let \( {X}_{t} \) be a semimartingale. Define\n\n\[ \n{Z}_{t} = {Z}_{0}\exp \left( {{X}_{t} - \frac{1}{2}{\left\langle {X}^{c}\right\rangle }_{t}}\right) \mathop{\prod }\limits_{{0 \leq s \leq t}}\left( {1 + \Delta {X}_{s}}\right) {e}^{-\Delta {X}_{s}}.\n\]\n\nThen \( {Z}_{t} \) is a semimartingale, \( \m... | Proof Since the product of finitely many functions of bounded variation which are purely discontinuous will give a function of the same type and in each finite interval there are only finitely many jumps of \( {X}_{t} \) of size larger in absolute value than \( 1/2 \), it suffices to consider\n\n\[ \n{V}_{t}^{\prime } ... | Yes |
Theorem 17.14 Suppose \( X \) is a local martingale with respect to \( \mathbb{P} \). Then \( {X}_{t} - {D}_{t} \) is a local martingale with respect to \( \mathbb{Q} \), where\n\n\[ \n{D}_{t} = {\int }_{0}^{t}\frac{1}{{M}_{s}}d{\left\lbrack X, M\right\rbrack }_{s} = {\int }_{0}^{t}\frac{{M}_{s - }}{{M}_{s}}d{\left\lbr... | Proof Exercise 17.6 tells us that it suffices to show that \( {M}_{t}\left( {{X}_{t} - {D}_{t}}\right) \) is a local martingale with respect to \( \mathbb{P} \). By Corollary 17.12,\n\n\[ \nd{\left( M\left( X - D\right) \right) }_{t} = {\left( X - D\right) }_{t - }d{M}_{t} + {M}_{t - }d{X}_{t} - {M}_{t - }d{D}_{t}\n\]\... | No |
Corollary 18.3 Let \( {\mathcal{F}}_{t} \) and \( {N}_{t}\left( {A}_{k}\right) \) be as in Theorem 18.2. Suppose \( {Y}_{t} \) is a process with paths that are right continuous with left limits such that \( {Y}_{t} - {Y}_{s} \) is independent of \( {\mathcal{F}}_{s} \) whenever \( s < t \) and \( {Y}_{t} - {Y}_{s} \) h... | Proof The law of \( {Y}_{0} \) is the same as that of \( {Y}_{t} - {Y}_{t} \), so \( {Y}_{0} = 0 \), a.s. By the fact that \( Y \) has stationary and independent increments,\n\n\[ \mathbb{E}{e}^{{iu}{Y}_{s + t}} = \mathbb{E}{e}^{{iu}{Y}_{s}}\mathbb{E}{e}^{{iu}\left( {{Y}_{s + t} - {Y}_{s}}\right) } = \mathbb{E}{e}^{{iu... | Yes |
Lemma 19.3 Suppose \( {P}_{t} \) are Markov transition probabilities. If \( f \) is Borel measurable and either non-negative or bounded, then \( {P}_{t}f \) is non-negative (respectively, bounded) and Borel measurable and\n\n\[ \n{P}_{t}f\left( x\right) = {\mathbb{E}}^{x}f\left( {X}_{t}\right) ,\;x \in \mathcal{S}.\n\]... | Proof Using (19.7) and Definition 19.1(2), the Borel measurability and (19.9) hold when \( f \) is the indicator of a set \( A \) . By linearity they hold for simple functions, and then using monotone convergence they hold for non-negative functions. Using linearity again, we have measurability and (19.9) holding for \... | Yes |
Proposition 19.4 Let \( W \) be a Brownian motion as defined by Definition 2.1, let \( {W}_{t}^{x} = x + {W}_{t} \) , and let \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right) \) be defined by (19.2) and (19.3). If \( f \) is bounded and Borel measurable, | Proof We will first prove\n\n\[ \n{\mathbb{E}}^{x}\left\lbrack {f\left( {X}_{t + s}\right) \mid {\mathcal{F}}_{s}}\right\rbrack = {\mathbb{E}}^{{X}_{s}}f\left( {X}_{t}\right) \n\]\n\n(19.16)\n\nwhen \( f\left( x\right) = {e}^{i\alpha x} \) . Using independent increments and the fact that \( {W}_{t + s} - {W}_{s} \) has... | Yes |
Proposition 19.5 Ifs, \( t > 0 \) and \( x, z \in \mathbb{R} \), then\n\n\[ \n{\int }_{y \in \mathbb{R}}\frac{1}{\sqrt{2\pi t}}{e}^{-{\left( y - x\right) }^{2}/{2t}}\frac{1}{\sqrt{2\pi s}}{e}^{-{\left( z - y\right) }^{2}/{2s}}{dy} \n\]\n\n(19.18)\n\n\[ \n= \frac{1}{\sqrt{{2\pi }\left( {s + t}\right) }}{e}^{-{\left( z -... | Proof This is a well-known property of the Gaussian density, but we can derive (19.18) from Proposition 19.4. Let \( f \) be continuous with compact support. Taking expectations in (19.15),\n\n\[ \n{\mathbb{E}}^{x}f\left( {X}_{t + s}\right) = {\mathbb{E}}^{x}\left\lbrack {{\mathbb{E}}^{{X}_{s}}f\left( {X}_{t}\right) }\... | Yes |
Proposition 20.3 Let \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right) \) be a Markov process and suppose that (20.4) holds. If Assumption 20.1 holds and \( f \) is a bounded Borel measurable function, then\n\n\[ \n{\mathbb{E}}^{x}\left\lbrack {f\left( {X}_{s + t}\right) \mid {\mathcal{F}}_{s}}\right\rbrack = {\mathbb{E}}^{{... | Proof We start with (20.5). By linearity, we have\n\n\[ \n{\mathbb{E}}^{x}\left\lbrack {f\left( {X}_{s + t}\right) \mid {\mathcal{F}}_{s}^{0}}\right\rbrack = {\mathbb{E}}^{{X}_{s}}f\left( {X}_{t}\right) ,\;{\mathbb{P}}^{x}\text{-a.s.,}\n\]\n\n(20.8)\n\nwhen \( f \) is a simple random variable, then by monotone converge... | Yes |
Proposition 20.5 Let \( \\left( {{X}_{t},{\\mathbb{P}}^{x}}\\right) \) be a Markov process and suppose (20.11) holds. Suppose \( Y = \\mathop{\\prod }\\limits_{{i = 1}}^{n}{f}_{i}\\left( {X}_{{t}_{i} - s}\\right) \), where the \( {f}_{i} \) are bounded, Borel measurable, and \( s \\leq {t}_{1} \\leq \\cdots \\leq {t}_{... | Proof We will prove this by induction on \( n \) . The case \( n = 1 \) is (20.11), so we suppose the equality holds for \( n \) and prove it for \( n + 1 \) .\n\nLet \( V = \\mathop{\\prod }\\limits_{{j = 2}}^{{n + 1}}{f}_{j}\\left( {X}_{{t}_{j} - {t}_{1}}\\right) \) and \( h\\left( y\\right) = {\\mathbb{E}}^{y}V \) .... | Yes |
Theorem 20.6 Let \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right) \) be a Markov process and suppose (20.11) holds. Suppose \( Y \) is bounded and measurable with respect to \( {\mathcal{F}}_{\infty } \) . Then\n\n\[ \n{\mathbb{E}}^{x}\left\lbrack {Y \circ {\theta }_{s} \mid {\mathcal{F}}_{s}}\right\rbrack = {\mathbb{E}}^{{... | Proof If in Proposition 20.5 we take \( {f}_{j}\left( x\right) = {1}_{{A}_{j}}\left( x\right) \) for Borel measurable \( {A}_{j} \), we have\n\n\[ \n{\mathbb{E}}^{x}\left\lbrack {{1}_{B} \circ {\theta }_{s} \mid {\mathcal{F}}_{s}}\right\rbrack = {\mathbb{E}}^{{X}_{s}}{1}_{B}\n\]\n\n(20.14)\n\nwhen \( B = \left\{ {\omeg... | Yes |
Proposition 20.7 Let \( \\left( {{X}_{t},{\mathbb{P}}^{x}}\\right) \) be a Markov process with respect to \( \\left\\{ {\\mathcal{F}}_{t}\\right\\} \). Let \( {\\mathcal{F}}_{t}^{0} \) and \( {\\mathcal{F}}_{t} \) be defined by (20.2) and (20.3). Then \( {\\mathcal{F}}_{t} = {\\mathcal{F}}_{t}^{0} \) for each \( t \\ge... | Proof Let \( {Y}_{1} = \\mathop{\\prod }\\limits_{{i = 1}}^{n}{f}_{i}\\left( {X}_{{t}_{i}}\\right) \) and \( {Y}_{2} = \\mathop{\\prod }\\limits_{{j = 1}}^{m}{g}_{j}\\left( {X}_{{u}_{j}}\\right) \), where \( {t}_{1} < \\cdots < {t}_{n} \\leq s \) and \( 0 \\leq {u}_{1} < \\cdots < {u}_{m} \) and the \( {f}_{j} \) and \... | Yes |
Proposition 20.8 Let \( \\left( {{X}_{t},{\\mathbb{P}}^{x}}\\right) \) be a Markov process with respect to \( \\left\\{ {\\mathcal{F}}_{t}\\right\\} \) . If \( A \\in {\\mathcal{F}}_{0} \), then for each \( x,{\\mathbb{P}}^{x}\\left( A\\right) \) is equal to 0 or 1 . | Proof Suppose \( A \\in {\\mathcal{F}}_{0} \) . Under \( {\\mathbb{P}}^{x},{X}_{0} = x \), a.s., and then\n\n\[ \n{\\mathbb{P}}^{x}\\left( A\\right) = {\\mathbb{E}}^{{X}_{0}}{1}_{A} = {\\mathbb{E}}^{x}\\left\\lbrack {{1}_{A} \\circ {\\theta }_{0} \\mid {\\mathcal{F}}_{0}}\\right\\rbrack = {1}_{A} \\circ {\\theta }_{0} ... | Yes |
Theorem 20.9 Suppose \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right) \) is a Markov process with respect to \( \left\{ {\mathcal{F}}_{t}\right\} \), that Assumption 20.1 holds, and that \( T \) is finite stopping time. If \( Y \) is bounded and measurable with respect to \( {\mathcal{F}}_{\infty } \), then | Proof Following the proofs of Section 20.2, it is enough to prove\n\n\[ \n{\mathbb{E}}^{x}\left\lbrack {f\left( {X}_{T + t}\right) \mid {\mathcal{F}}_{T}}\right\rbrack = {\mathbb{E}}^{{X}_{T}}f\left( {X}_{t}\right) \n\]\n\n(20.15)\n\nfor \( f \) bounded. We can obtain this by a limit argument if we have (20.15) for \( ... | Yes |
Proposition 20.10 If \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right) \) is a strong Markov process and Assumption 20.1 holds, then \( {X}_{t} \) is quasi-left continuous. | Proof First suppose \( T \) is bounded, \( {T}_{n} \) increases to \( T, Y = \mathop{\lim }\limits_{{n \rightarrow \infty }}{X}_{{T}_{n}} \), and \( f \) and \( g \) are bounded and continuous. If \( {T}_{n} = T \) for some \( n \), then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}g\left( {X}_{{T}_{n} + t}\right)... | Yes |
Proposition 21.2 If \( A < \infty \), then\n\n\[ \mathop{\sup }\limits_{{x \in D}}{\mathbb{P}}^{x}\left( {{\int }_{0}^{S}f\left( {X}_{s}\right) {ds} \geq {2kA}}\right) \leq {2}^{-k}. \]\n\n(21.3) | Proof Let \( {B}_{t} = {\int }_{0}^{t \land S}f\left( {X}_{s}\right) {ds} \). This is a special case of what is known as an additive functional; see Section 22.3. Let \( {U}_{1} = \inf \left\{ {t : {B}_{t} \geq {2A}}\right\} \), and let \( {U}_{i + 1} = {U}_{i} + {U}_{1} \circ {\theta }_{{U}_{i}} \). To explain this fo... | Yes |
Proposition 21.3 Let \( W \) be a one-dimensional Brownian motion. If \( T \) is a finite stopping time and \( a < b \), then\n\n\[ \mathbb{P}\left( {{W}_{T + t} \in \left\lbrack {a, b}\right\rbrack \mid {\mathcal{F}}_{T}}\right) \leq \frac{b - a}{\sqrt{2\pi t}},\;\text{ a.s. } \] | Proof Let \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right) \) be a one-dimensional Brownian motion. If \( y \in \mathbb{R} \), then\n\n\[ {\mathbb{P}}^{y}\left( {{X}_{t} \in \left\lbrack {a, b}\right\rbrack }\right) = {\mathbb{P}}^{0}\left( {{X}_{t} \in \left\lbrack {a - y, b - y}\right\rbrack }\right) \]\n\n(21.4)\n\n\[ = ... | Yes |
Lemma 21.4 Let \( \\left( {{X}_{t},{\\mathbb{P}}^{x}}\\right) \) be a strong Markov process with state space \( \\mathcal{S} \) . For all \( x \\in \\mathcal{S} \) and all \( \\lambda \\geq 0 \) ,\n\n\[ \n{\\mathbb{P}}^{x}\\left( {\\mathop{\\sup }\\limits_{{s \\leq t}}d\\left( {{X}_{s}, x}\\right) \\geq \\lambda }\\rig... | Proof Let us use the notation\n\n\[ \nF\\left( {t,\\lambda }\\right) = \\mathop{\\sup }\\limits_{{s \\leq t}}\\mathop{\\sup }\\limits_{{y \\in \\mathcal{S}}}{\\mathbb{P}}^{y}\\left( {d\\left( {{X}_{s},{X}_{0}}\\right) \\geq \\lambda }\\right) .\n\]\n\n(21.5)\n\n\n\nWrite \( S = \\inf \\left\\{ {t : d\\left( {{X}_{t},{X... | No |
Proposition 21.5 Let \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right) \) be a strong Markov process. With \( F\left( {t,\lambda }\right) \) defined as in (21.5), suppose\n\n\[ \frac{F\left( {t,\lambda }\right) }{t} \rightarrow 0 \]\n\n(21.7)\n\nas \( t \rightarrow 0 \) for each \( \lambda > 0 \) . Then \( {X}_{t} \) has con... | Proof Suppose \( \lambda ,{t}_{0} > 0 \) and \( X \) has a jump of size larger than \( {4\lambda } \) at some time before \( {t}_{0} \) with positive probability, that is,\n\n\[ {\mathbb{P}}^{x}\left( {\mathop{\sup }\limits_{{t \leq {t}_{0}}}d\left( {{X}_{t - },{X}_{t}}\right) \geq {4\lambda }}\right) > 0, \]\n\nwhere ... | Yes |
Proposition 21.6 Define\n\n\[ h\left( x\right) = {\mathbb{E}}^{x}f\left( {X}_{{\tau }_{D}}\right) \]\n\nand \( {\mathcal{F}}_{s}^{\prime } = {\mathcal{F}}_{s \land {\tau }_{D}} \) . Then for each \( x, h\left( {X}_{t \land {\tau }_{D}}\right) \) is a martingale under \( {\mathbb{P}}^{x} \) with respect to the filtratio... | Proof Let \( s < t \) . Consider a path \( \omega \) starting at \( x \) and continuing until it exits \( D \) at time \( {\tau }_{D}\left( \omega \right) \) . If we have \( u \leq {\tau }_{D} \) and we cut off the first \( u \) time units of the path, we have a path going from \( {X}_{u}\left( \omega \right) \) and pr... | Yes |
Proposition 21.8 Suppose there exists a cone \( V \) with vertex \( y \in \partial D \) such that \( V \cap B\left( {y, r}\right) \subset \) \( {D}^{c} \) for some \( r > 0 \) . Then \( y \) is regular for \( {D}^{c} \) . | Proof By translation and rotation of the coordinates, we may suppose \( y = 0 \) and \( V = {\widetilde{V}}_{a} \) for some \( a \) . Then for each \( t \) ,\n\n\[ \n{\mathbb{P}}^{0}\left( {{\tau }_{D} \leq t}\right) \geq {\mathbb{P}}^{0}\left( {{X}_{t} \in {D}^{c}}\right) \geq {\mathbb{P}}^{0}\left( {{X}_{t} \in V \ca... | Yes |
Proposition 22.1 If \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right) \) is a strong Markov process and (22.2) holds, then \( \left( {{\widehat{X}}_{t},{\mathbb{P}}^{x}}\right) \) satisfies the Markov and strong Markov properties. | Proof As in Section 20.2, we need to show\n\n\[ \n{\mathbb{E}}^{x}\left\lbrack {f\left( {\widehat{X}}_{t}\right) \circ {\theta }_{T} \mid {\mathcal{F}}_{T}}\right\rbrack = {\mathbb{E}}^{{\widehat{X}}_{T}}f\left( {\widehat{X}}_{t}\right) ,\;{\mathbb{P}}^{x}\text{-a.s. } \n\] \n\nIf \( A \in {\mathcal{F}}_{T} \) ,\n\n\[ ... | Yes |
Proposition 22.2 Suppose \( \\left( {{X}_{t},{\\mathbb{P}}^{x}}\\right) \) is a strong Markov process and that \( h \) is non-negative and invariant. Then \( \\left( {{X}_{t},{\\mathbb{P}}_{h}^{x}}\\right) \) forms a strong Markov process. | Proof Suppose \( A \\in {\\mathcal{F}}_{s} \) and \( h\\left( x\\right) \\neq 0 \) . (We leave consideration of the case where \( h\\left( x\\right) = 0 \) to the reader.) Then\n\n\[ \n{\\mathbb{E}}_{h}^{x}\\left\\lbrack {f\\left( {X}_{t + s}\\right) ;A}\\right\\rbrack = \\frac{{\\mathbb{E}}^{x}\\left\\lbrack {f\\left(... | No |
Proposition 22.3 Let \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right) \) be a strong Markov process and \( {A}_{t} \) an additive functional. With \( B \) defined as above, \( \left( {{X}_{t}^{\prime },{\mathbb{P}}^{x}}\right) \) is also a strong Markov process. | Proof We verify the strong Markov property. Let \( {\mathcal{F}}_{t}^{\prime } = {\mathcal{F}}_{{B}_{t}} \) . Then if \( T \) is a stopping time for \( {\mathcal{F}}_{t}^{\prime } \), we have\n\n\[ \n{\mathbb{E}}^{x}\left\lbrack {f\left( {X}_{T + t}^{\prime }\right) \mid {\mathcal{F}}_{T}^{\prime }}\right\rbrack = {\ma... | Yes |
Lemma 23.1 If \( f \) is excessive, there exist functions \( {g}_{n} \geq 0 \) such that \( U{g}_{n} \) increases up to \( f \) , where \( U{g}_{n} \) is defined by (23.1). | Proof Let \( {g}_{n} = n\left( {f - {P}_{1/n}f}\right) \) . Since \( f \) is excessive, then \( {g}_{n} \geq 0 \) . We have\n\n\[ U{g}_{n} = n{\int }_{0}^{\infty }{P}_{s}{fds} - n{\int }_{0}^{\infty }{P}_{s + \left( {1/n}\right) }{fds} \]\n\n\[ = n{\int }_{0}^{1/n}{P}_{s}{fds} \]\n\nwhich is less than \( f \) and incre... | Yes |
Proposition 23.2 (1) If \( f \) is excessive, \( T \) is a finite stopping time, and \( h\left( x\right) = {\mathbb{E}}^{x}f\left( {X}_{T}\right) \) , then \( h \) is excessive. | Proof (1) First suppose \( f = {Ug} \) for some non-negative function \( g \) . Then\n\n\[ h\left( x\right) = {\mathbb{E}}^{x}{Ug}\left( {X}_{T}\right) = {\mathbb{E}}^{x}{\mathbb{E}}^{{X}_{T}}{\int }_{0}^{\infty }g\left( {X}_{s}\right) {ds} \]\n\n(23.2)\n\n\[ = {\mathbb{E}}^{x}{\int }_{0}^{\infty }g\left( {X}_{s + T}\r... | Yes |
Proposition 23.3 Let \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right) \) be a strong Markov process. If \( f \) is excessive, then for each \( x, f\left( {X}_{t}\right) \) is right continuous with left limits \( {\mathbb{P}}^{x} \) almost surely. | For a proof, we refer the reader to Blumenthal and Getoor (1968), Theorem II.2.12 or to Exercise 23.8. | No |
Proposition 23.4 Suppose that \( g \) is non-negative, bounded, and continuous and that Assumption 20.1 holds. Let \( {g}_{0} = g \), let \( {T}_{n} = \left\{ {k/{2}^{n} : 0 \leq k \leq n{2}^{n}}\right\} \), and define\n\n\[ \n{g}_{n}\left( x\right) = \mathop{\max }\limits_{{t \in {T}_{n}}}{P}_{t}{g}_{n - 1}\left( x\ri... | Proof Since \( {g}_{n}\left( x\right) \geq {P}_{0}{g}_{n - 1}\left( x\right) = {\mathbb{E}}^{x}{g}_{n - 1}\left( {X}_{0}\right) = {g}_{n - 1}\left( x\right) \), the sequence \( {g}_{n}\left( x\right) \) is increasing. Call the limit \( H\left( x\right) \) .\n\nWe first show \( H \) is lower semicontinuous. If \( {g}_{n... | Yes |
Corollary 23.6 Suppose there exists a Borel set \( A \) such that \( h \) is an excessive majorant of \( g \) , where \( h\left( x\right) = {\mathbb{E}}^{x}g\left( {X}_{{\tau }_{A}}\right) \) and \( {\tau }_{A} = \inf \left\{ {t : {X}_{t} \notin A}\right\} \) . Then \( {g}^{ * }\left( x\right) = h\left( x\right) \) . | Proof Let \( G \) be the least excessive majorant of \( g \) . Then \( h\left( x\right) \geq G\left( x\right) \) . However, \[ h\left( x\right) = {\mathbb{E}}^{x}g\left( {X}_{{\tau }_{A}}\right) \leq \mathop{\sup }\limits_{T}{\mathbb{E}}^{x}g\left( {X}_{T}\right) = {g}^{ * }\left( x\right) = G\left( x\right) \] by Theo... | Yes |
Corollary 23.7 Suppose \( g \) is continuous and \( G \), the least excessive majorant of \( g \), is lower semicontinuous. Let \( D \) be the continuation region, suppose \( {\tau }_{D} < \infty \), a.s., and let \( h\left( x\right) = \) \( {\mathbb{E}}^{x}g\left( {X}_{{\tau }_{D}}\right) \) . If \( h \geq g \), then ... | Proof Note \( D = \{ x : g\left( x\right) < G\left( x\right) \} = { \cup }_{a < b}\left\lbrack {\left( {g\left( x\right) < a}\right) \cap \left( {G\left( x\right) > b}\right) }\right\rbrack \), where the union is over all pairs of real numbers \( a < b \) . Since \( G \) is lower semicontinuous and \( g \) is continuou... | Yes |
Theorem 24.3 Suppose \( \sigma \) and \( b \) are Lipschitz functions, but not necessarily bounded. Then there exists a pathwise solution to (24.1) and this solution is pathwise unique. | Proof Let \( {\sigma }_{n} \) and \( {b}_{n} \) be bounded Lipschitz functions that agree with \( \sigma \) and \( b \), respectively, on \( \left\lbrack {-n, n}\right\rbrack \) . Let \( {X}_{n} \) be the unique pathwise solution to (24.1) with \( \sigma \) and \( b \) replaced by \( {\sigma }_{n} \) and \( {b}_{n} \),... | No |
The unique pathwise solution to\n\n\[ d{X}_{t} = A{X}_{t}d{W}_{t} + B{X}_{t}{dt} \] | is\n\n\[ {X}_{t} = {X}_{0}{e}^{A{W}_{t} + \left( {B - {A}^{2}/2}\right) t}. \] | Yes |
Proposition 24.7 Suppose \( {Y}_{t} \) is the square of a Bessel process of order \( v \) . Suppose \( {Y}_{0} = y \) . The following hold with probability one.\n\n(1) If \( v > 2 \) and \( y > 0,{Y}_{t} \) never hits 0 .\n\n(2) If \( v = 2 \) and \( y > 0,{Y}_{t} \) hits every neighborhood of 0, but never hits the poi... | Proof We prove (2). An application of Itô's formula with the process being the square of a Bessel process of order 2 and the function being \( \log x \) shows that \( \log {Y}_{t} \) is a martingale up until the first hitting time of 0 ; cf. Exercise 21.1. The quadratic variation of \( \log {Y}_{t} \) is \( {\int }_{0}... | No |
Proposition 25.2 Suppose \( \sigma \) and \( b \) are bounded Lipschitz functions and \( {x}_{0} \in {\mathbb{R}}^{d} \) . Then weak uniqueness holds for (25.1). | Proof For notational simplicity we consider the case of dimension one. Suppose \( \left( {X, W,\mathbb{P}}\right) \) and \( \left( {{X}^{\prime },{W}^{\prime },{\mathbb{P}}^{\prime }}\right) \) are two weak solutions to (25.1). Let \( {X}_{0}\left( t\right) = {x}_{0} \) and define \( {X}_{i + 1}\left( t\right) \) by\n\... | Yes |
Proposition 25.3 If \( \sigma \) is Borel measurable and there exist \( {c}_{2} > {c}_{1} > 0 \) such that \( {c}_{1} \leq \) \( \sigma \left( x\right) \leq {c}_{2} \) for all \( x \), then weak existence and weak uniqueness hold for (25.6). | Proof We consider only uniqueness, leaving existence as Exercise 25.1. Suppose \( \left( {X, W,\mathbb{P}}\right) \) and \( \left( {{X}^{\prime },{W}^{\prime },{\mathbb{P}}^{\prime }}\right) \) are two weak solutions. Then \( {X}_{t} \) is a martingale, and as in Section 12.2, if we set\n\n\[ \n{A}_{t} = {\int }_{0}^{t... | No |
Lemma 26.1 Suppose \( {X}_{t}^{\left( j\right) }, j = 1,2 \), are two continuous processes such that\n\n\[ \mathbb{E}\exp \left( {-{\int }_{0}^{1}f\left( s\right) {X}_{s}^{\left( 1\right) }{ds}}\right) = \mathbb{E}\exp \left( {-{\int }_{0}^{1}f\left( s\right) {X}_{s}^{\left( 2\right) }{ds}}\right) \]\n\nwhenever \( f \... | Proof Let \( \varphi \) be a non-negative continuous function with support in \( \left\lbrack {0,1}\right\rbrack \) such that \( {\int }_{0}^{1}\varphi \left( x\right) {dx} = 1 \), and let \( {\varphi }_{\varepsilon }\left( x\right) = {\varepsilon }^{-1}\varphi \left( {x/\varepsilon }\right) \), so that the sequence \(... | Yes |
Theorem 27.1 \( {N}_{t}\left( \cdot \right) \) is a Poisson point process. | Proof If \( {N}_{t}\left( B\right) \) is not infinite, then it has right-continuous paths that increase at most 1 at any given time. The main step will be to show that \( {N}_{t}\left( B\right) \) has stationary increments and \( {N}_{t}\left( B\right) - {N}_{s}\left( B\right) \) is independent of the \( \sigma \) -fie... | Yes |
Proposition 27.2 If\n\n\[ A = \\left\\{ {f \\in \\mathcal{E} : \\mathop{\\sup }\\limits_{t}\\left| {f\\left( t\\right) }\\right| > a}\\right\\} \]\n\nthen \( m\\left( A\\right) = 1/a \) . | Proof Let \( U = \\inf \\left\\{ {t : \\left| {W}_{t}\\right| = a}\\right\\} \) and \( V = \\inf \\left\\{ {t > U : {W}_{t} = 0}\\right\\} \) . Since \( \\left| {W}_{t}\\right| - {L}_{t}^{0} \) is a martingale by Theorem 14.1, then \( {\\mathbb{E}}^{0}\\left| {W}_{t \\land U}\\right| = {\\mathbb{E}}^{0}{L}_{t \\land U}... | Yes |
Lemma 29.1 The best mean square error estimate of \( f\left( {X}_{t}\right) \) over the class of \( {\mathcal{F}}_{t}^{Z} \) measurable random variables is\n\n\[ Y = \mathbb{E}\left\lbrack {f\left( {X}_{t}\right) \mid {\mathcal{F}}_{t}^{Z}}\right\rbrack \] | Proof By our assumptions on \( f \), the random variable \( V = f\left( {X}_{t}\right) \) is in \( {L}^{2}\left( \mathbb{P}\right) \) . Let \( Y \) be the best mean square estimator. The collection \( \mathcal{M} \) of \( {L}^{2} \) random variables which are \( {\mathcal{F}}_{t}^{Z} \) measurable is a linear subspace ... | Yes |
Proposition 29.2 \( {N}_{t} \) is a Brownian motion with respect to the filtration \( \left\{ {\mathcal{F}}_{t}^{Z}\right\} \) . | Proof We will show that \( {N}_{t} \) is a continuous martingale with respect to the filtration \( \left\{ {\mathcal{F}}_{t}^{Z}\right\} \) whose quadratic variation is \( t \), and then our result follows from Lévy’s theorem (Theorem 12.1). That \( {N}_{t} \) is continuous is obvious, and \( \langle N{\rangle }_{t} = ... | No |
Lemma 29.4 If \( {Y}_{t} - {\int }_{0}^{t}{H}_{s}{ds} \) is a martingale with respect to \( \left\{ {\mathcal{F}}_{t}^{X}\right\} \), then \( {\widehat{Y}}_{t} - {\int }_{0}^{t}{\widehat{H}}_{s}{ds} \) is a martingale with respect to \( \left\{ {\mathcal{F}}_{t}^{Z}\right\} \) . | Proof Since \( {\mathcal{F}}_{s}^{Z} \subset {\mathcal{F}}_{s}^{X} \) ,\n\n\[ \mathbb{E}\left\lbrack {{\widehat{Y}}_{t} - {\widehat{Y}}_{s} - {\int }_{s}^{t}{\widehat{H}}_{r}{dr} \mid {\mathcal{F}}_{s}^{Z}}\right\rbrack \]\n\n\[ = \mathbb{E}\left\lbrack {\mathbb{E}\left\lbrack {{Y}_{t} \mid {\mathcal{F}}_{t}^{Z}}\right... | No |
Theorem 29.5 Let \( {M}_{t} = f\left( {X}_{t}\right) - f\left( {X}_{0}\right) - {\int }_{0}^{t}{A}_{s}{ds} \) be a martingale with respect to \( \left\{ {\mathcal{F}}_{t}^{X}\right\} \) and write \( {F}_{s} \) for \( f\left( {X}_{s}\right) \) . Suppose \( \langle M, W{\rangle }_{t} = {\int }_{0}^{t}{D}_{s}{ds} \) . The... | Proof By Lemma 29.4,\n\n\[{L}_{t} = {\widehat{F}}_{t} - {\widehat{F}}_{0} - {\int }_{0}^{t}{\widehat{A}}_{s}{ds}\]\n\n(29.7)\n\nis a martingale with respect to \( \left\{ {\mathcal{F}}_{t}^{Z}\right\} \) and by Theorem 29.3, there exists \( {H}_{s} \) such that\n\n\[{L}_{t} = {\int }_{0}^{t}{H}_{s}d{N}_{s}\]\n\n(29.8)\... | Yes |
Theorem 29.6 \( {V}_{t} \) solves the deterministic ordinary differential equation\n\n\[ \frac{d{V}_{t}}{dt} = 1 + {2B}{V}_{t} - {C}^{2}{V}_{t}^{2},\;{V}_{0} = \operatorname{Var}{X}_{0} \] | Proof of Theorem 29.6 By Itô’s formula, if \( f \in {C}^{2} \), \n\n\[ f\left( {X}_{t}\right) - f\left( {X}_{0}\right) = {\mathcal{F}}^{X}\text{-martingale } + {\int }_{0}^{t}\left\lbrack {\frac{1}{2}{f}^{\prime \prime }\left( {X}_{s}\right) + B{X}_{s}{f}^{\prime }\left( {X}_{s}\right) }\right\rbrack {ds}. \]\n\nBy the... | No |
Theorem 30.4 If a sequence of probability measures on a metric space \( \mathcal{S} \) is tight, there is a subsequence that converges weakly to a probability measure on \( \mathcal{S} \) . | Proof Suppose first that the metric space \( \mathcal{S} \) is compact. Then \( C\left( \mathcal{S}\right) \), the collection of continuous functions on \( \mathcal{S} \), is a separable metric space when furnished with the supremum norm; this is Exercise 30.1. Let \( \left\{ {f}_{i}\right\} \) be a countable collectio... | No |
Proposition 30.6 \( {d}_{\mathcal{M}} \) is a metric on \( \mathcal{M} \) . | Proof We start with symmetry, that is, that \( {d}_{\mathcal{M}}\left( {\mathbb{Q},\mathbb{P}}\right) = {d}_{\mathcal{M}}\left( {\mathbb{P},\mathbb{Q}}\right) \) . Let \( \alpha \) be any real number larger than \( {d}_{\mathcal{M}}\left( {P, Q}\right) \) . If \( H \) is closed, then \( {H}_{\alpha } = \{ x : d\left( {... | Yes |
Theorem 32.1 Suppose the \( {X}_{n} \) are continuous real-valued processes. Suppose for each \( \varepsilon \) and \( \eta > 0 \) there exist \( {n}_{0}, A \), and \( \delta \) (depending on \( \varepsilon \) and \( \eta \) ) such that if \( n \geq {n}_{0} \), then\n\n\[ \mathbb{P}\left( {{\omega }_{{X}_{n}}\left( \de... | Proof Since each \( {X}_{i} \) is a continuous process, then for each \( i,\mathbb{P}\left( {{\omega }_{{X}_{i}}\left( \delta \right) \geq \varepsilon }\right) \rightarrow 0 \) as \( \delta \rightarrow 0 \) by dominated convergence. Hence, given \( \varepsilon \) and \( \eta \) we can, by taking \( \delta \) smaller if... | Yes |
Example 33.2 Let's see what this expansion is in the case of Brownian motion. If we define\n\n\[ \langle f, g{\rangle }_{CM} = {\int }_{0}^{1}{f}^{\prime }\left( r\right) {g}^{\prime }\left( r\right) {dr} \]\n\n(33.5)\n\nfor \( f \) and \( g \) whose first derivatives are in \( {L}^{2}\left( \left\lbrack {0,1}\right\rb... | \[ \langle \Gamma \left( {s, \cdot }\right) ,\Gamma \left( {t, \cdot }\right) {\rangle }_{CM} = {\int }_{0}^{1}{1}_{\lbrack 0, s)}\left( r\right) {1}_{\lbrack 0, t)}\left( r\right) {dr} = s \land t \]\n\n\[ = \Gamma \left( {s, t}\right) \text{,} \]\n\nand we see that we have identified the reproducing kernel Hilbert sp... | Yes |
In the case of Brownian motion, \( \operatorname{Var}\left( {{X}_{t} - {X}_{s}}\right) = \left| {t - s}\right| \), so that \( d\left( {s, t}\right) = \) \( {\left| s - t\right| }^{1/2} \) . If \( T \) is the interval \( \left\lbrack {0,1}\right\rbrack \), then the set of intervals of length \( {\varepsilon }^{2} \) and... | Therefore \( N\left( \varepsilon \right) \leq c/{\varepsilon }^{2} \) , implying \( \log N\left( \varepsilon \right) \leq c\log \left( {1/\varepsilon }\right) \), which satisfies (33.6). This and Theorem 2.4 gives a construction of Brownian motion. | Yes |
Example 33.5 We look at fractional Brownian motion. Let \( H \in \left( {0,2}\right) .H \) is known as the Hurst index, where \( H = 1 \) corresponds to Brownian motion. Define\n\n\[ \Gamma \left( {s, t}\right) = {\left| s\right| }^{H} + {\left| t\right| }^{H} - {\left| s - t\right| }^{H}. \]\n\nThis leads to \( d\left... | Open intervals of length \( {\varepsilon }^{2/H} \) are \( \varepsilon \) -balls, and it takes \( c{\varepsilon }^{-2/H} \) of them to cover \( \left\lbrack {0,1}\right\rbrack \) . Therefore again \( N\left( \varepsilon \right) \leq c\log \left( {1/\varepsilon }\right) \), and (33.6) applies. | No |
Example 33.6 Here is our first example of a Gaussian process where \( T \) is not a subset of \( \lbrack 0,\infty ) \) . We construct a Brownian sheet, \( X\left( {{t}_{1},{t}_{2}}\right) \), where the points \( \left( {{t}_{1},{t}_{2}}\right) \in {\left\lbrack 0,1\right\rbrack }^{2} \) . More generally we can consider... | One motivation for this formula is to identify the point \( \left( {{t}_{1},{t}_{2}}\right) \) with the rectangle \( {R}_{t} \) whose lower left corner is at the origin and whose upper right corner is at \( \left( {{t}_{1},{t}_{2}}\right) \) . Then the covariance of \( {X}_{s} \) and \( {X}_{t} \) is the area of \( {R}... | Yes |
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