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Theorem 3.21 Let \( X \) be a martingale with right-continuous sample paths. Then the following properties are equivalent:\n\n(i) \( X \) is closed;\n\n(ii) the collection \( {\left( {X}_{t}\right) }_{t \geq 0} \) is uniformly integrable;\n\n(iii) \( {X}_{t} \) converges a.s. and in \( {L}^{1} \) as \( t \rightarrow \i... | Proof The fact that (i) \( \Rightarrow \) (ii) is easy: If \( Z \in {L}^{1} \), the collection of all random variables \( E\left\lbrack {Z \mid \mathcal{G}}\right\rbrack \), when \( \mathcal{G} \) varies over sub- \( \sigma \) -fields of \( \mathcal{F} \), is uniformly integrable. If (ii) holds, in particular the colle... | Yes |
Corollary 3.23 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be a martingale with right-continuous sample paths, and let \( S \leq T \) be two bounded stopping times. Then \( {X}_{S} \) and \( {X}_{T} \) are in \( {L}^{1} \) and \[ {X}_{S} = E\left\lbrack {{X}_{T} \mid {\mathcal{F}}_{S}}\right\rbrack \] | Proof Let \( a \geq 0 \) such that \( S \leq T \leq a \) . We apply Theorem 3.22 to the martingale \( {\left( {X}_{t \land a}\right) }_{t \geq 0} \) which is closed by \( {X}_{a} \). | No |
Corollary 3.24 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be a martingale with right-continuous sample paths, and let \( T \) be a stopping time.\n\n(i) The process \( {\left( {X}_{t \land T}\right) }_{t \geq 0} \) is still a martingale.\n\n(ii) Suppose in addition that the martingale \( {\left( {X}_{t}\right) }_{t ... | Proof We start with the proof of (ii). Note that \( t \land T \) is a stopping time by property (f) of stopping times. By Theorem 3.22, \( {X}_{t \land T} \) and \( {X}_{T} \) are in \( {L}^{1} \), and we also know that \( {X}_{t \land T} \) is \( {\mathcal{F}}_{t \land T} \) -measurable, hence \( {\mathcal{F}}_{t} \) ... | Yes |
Proposition 4.2 For every \( t \in (0, T\rbrack \) ,\n\n\[ \n{\int }_{0}^{t}\left| {\mathrm{\;d}a\left( s\right) }\right| = \sup \left\{ {\mathop{\sum }\limits_{{i = 1}}^{p}\left| {a\left( {t}_{i}\right) - a\left( {t}_{i - 1}\right) }\right| }\right\} \n\]\n\nwhere the supremum is over all subdivisions \( 0 = {t}_{0} <... | Proof Clearly, it is enough to treat the case \( t = T \) . The inequality \( \geq \) in the first assertion is very easy since, for any subdivision \( 0 = {t}_{0} < {t}_{1} < \cdots < {t}_{p} = T \) of \( \left\lbrack {0, T}\right\rbrack \) ,\n\n\[ \n\left| {a\left( {t}_{i}\right) - a\left( {t}_{i - 1}\right) }\right|... | Yes |
Lemma 4.3 If \( f : \left\lbrack {0, T}\right\rbrack \rightarrow \mathbb{R} \) is a continuous function, and if \( 0 = {t}_{0}^{n} < {t}_{1}^{n} < \cdots < \) \( {t}_{{p}_{n}}^{n} = T \) is a sequence of subdivisions of \( \left\lbrack {0, T}\right\rbrack \) whose mesh tends to 0, we have\n\n\[ \n{\int }_{0}^{T}f\left(... | Proof Let \( {f}_{n} \) be defined on \( \left\lbrack {0, T}\right\rbrack \) by \( {f}_{n}\left( s\right) = f\left( {t}_{i - 1}^{n}\right) \) if \( s \in \left( {{t}_{i - 1}^{n},{t}_{i}^{n}}\right\rbrack ,1 \leq i \leq {p}_{n} \), and \( {f}_{n}\left( 0\right) = f\left( 0\right) \) . Then,\n\n\[ \n\mathop{\sum }\limits... | Yes |
Proposition 4.5 Let \( A \) be a finite variation process, and let \( H \) be a progressive process such that\n\n\[ \forall t \geq 0,\forall \omega \in \Omega ,{\int }_{0}^{t}\left| {{H}_{s}\left( \omega \right) }\right| \left| {\mathrm{d}{A}_{s}\left( \omega \right) }\right| < \infty .\n\]\n\nThen the process \( H \cd... | Proof By the observations preceding the statement of Proposition 4.2, we know that the sample paths of \( H \cdot A \) are finite variation functions. It remains to verify that the process \( H \cdot A \) is adapted. To this end, it is enough to check that, if \( t > 0 \) is fixed, if \( h : \Omega \times \left\lbrack ... | Yes |
Theorem 4.8 Let \( M \) be a continuous local martingale. Assume that \( M \) is also a finite variation process (in particular \( {M}_{0} = 0 \) ). Then \( {M}_{t} = 0 \) for every \( t \geq 0 \), a.s. | Proof Set\n\n\[{\tau }_{n} = \inf \left\{ {t \geq 0 : {\int }_{0}^{t}\left| {\mathrm{\;d}{M}_{s}}\right| \geq n}\right\}\]\n\nfor every integer \( n \geq 0 \) . By Proposition 3.9, \( {\tau }_{n} \) is a stopping time (recall that \( {\int }_{0}^{t}\left| {\mathrm{\;d}{M}_{s}}\right| \) is an increasing process if \( M... | Yes |
Theorem 4.9 Let \( M = {\left( {M}_{t}\right) }_{t \geq 0} \) be a continuous local martingale. There exists an increasing process denoted by \( {\left( \langle M, M{\rangle }_{t}\right) }_{t \geq 0} \), which is unique up to indistinguishability, such that \( {M}_{t}^{2} - \langle M, M{\rangle }_{t} \) is a continuous... | Proof We start by proving the first assertion. Uniqueness is an easy consequence of Theorem 4.8. Indeed, let \( A \) and \( {A}^{\prime } \) be two increasing processes satisfying the condition given in the statement. Then the process \( {A}_{t} - {A}_{t}^{\prime } = \left( {{M}_{t}^{2} - {A}_{t}^{\prime }}\right) - \l... | Yes |
Proposition 4.11 Let \( M \) be a continuous local martingale and let \( T \) be a stopping time. Then we have a.s. for every \( t \geq 0 \) , \[ {\left\langle {M}^{T},{M}^{T}\right\rangle }_{t} = \langle M, M{\rangle }_{t \land T} \] | This follows from the fact that \( {M}_{t \land T}^{2} - \langle M, M{\rangle }_{t \land T} \) is a continuous local martingale (cf. property (c) of continuous local martingales). | No |
Proposition 4.12 Let \( M \) be a continuous local martingale such that \( {M}_{0} = 0 \) . Then we have \( \langle M, M\rangle = 0 \) if and only if \( M = 0 \) . | Proof Suppose that \( \langle M, M\rangle = 0 \) . Then \( {M}_{t}^{2} \) is a nonnegative continuous local martingale and, by Proposition 4.7 (i), \( {M}_{t}^{2} \) is a supermartingale, hence \( E\left\lbrack {M}_{t}^{2}\right\rbrack \leq \) \( E\left\lbrack {M}_{0}^{2}\right\rbrack = 0 \), so that \( {M}_{t} = 0 \) ... | Yes |
Theorem 4.13 Let \( M \) be a continuous local martingale with \( {M}_{0} \in {L}^{2} \). (i) The following are equivalent: (a) \( M \) is a (true) martingale bounded in \( {L}^{2} \). (b) \( E\left\lbrack {\langle M, M{\rangle }_{\infty }}\right\rbrack < \infty \). Furthermore, if these properties hold, the process \(... | (i) Replacing \( M \) by \( M - {M}_{0} \), we may assume that \( {M}_{0} = 0 \) in the proof. Let us first assume that \( M \) is a martingale bounded in \( {L}^{2} \). Doob’s inequality in \( {L}^{2} \) (Proposition 3.15 (ii)) shows that, for every \( T > 0 \), \[ E\left\lbrack {\mathop{\sup }\limits_{{0 \leq t \leq ... | Yes |
Proposition 4.16 Let \( B \) and \( {B}^{\prime } \) be two independent \( \left( {\mathcal{F}}_{t}\right) \) -Brownian motions. Then \( {\left\langle B,{B}^{\prime }\right\rangle }_{t} = 0 \) for every \( t \geq 0 \) . | Proof By subtracting the initial values, we may assume that \( {B}_{0} = {B}_{0}^{\prime } = 0 \) . We then observe that the process \( {X}_{t} = \frac{1}{\sqrt{2}}\left( {{B}_{t} + {B}_{t}^{\prime }}\right) \) is a martingale, as a linear combination of martingales. By checking the finite-dimensional marginals of \( X... | Yes |
Proposition 4.18 (Kunita-Watanabe) Let \( M \) and \( N \) be two continuous local martingales and let \( H \) and \( K \) be two measurable processes. Then, a.s., \[ {\int }_{0}^{\infty }\left| {H}_{s}\right| \left| {K}_{s}\right| \left| {\mathrm{d}\langle M, N{\rangle }_{s}}\right| \leq {\left( {\int }_{0}^{\infty }{... | Proof Only in this proof, we use the special notation \( \langle M, N{\rangle }_{s}^{t} = \langle M, N{\rangle }_{t} - \langle M, N{\rangle }_{s} \) for \( 0 \leq s \leq t \) . The first step of the proof is to observe that we have a.s. for every choice of the rationals \( s < t \) (and also by continuity for every rea... | No |
Proposition 4.21 Let \( 0 = {t}_{0}^{n} < {t}_{1}^{n} < \cdots < {t}_{{p}_{n}}^{n} = t \) be an increasing sequence of subdivisions of \( \left\lbrack {0, t}\right\rbrack \) whose mesh tends to 0 . Then,\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\sum }\limits_{{i = 1}}^{{p}_{n}}\left( {{X}_{{t}_{i}^{... | Proof We treat the case where \( X = Y \) and leave the general case to the reader. We have\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{{p}_{n}}{\left( {X}_{{t}_{i}^{n}} - {X}_{{t}_{i - 1}^{n}}\right) }^{2} = \mathop{\sum }\limits_{{i = 1}}^{{p}_{n}}{\left( {M}_{{t}_{i}^{n}} - {M}_{{t}_{i - 1}^{n}}\right) }^{2} + \mathop{\s... | No |
Proposition 5.3 For every \( M \in {\mathbb{H}}^{2} \) , \( \mathcal{E} \) is dense in \( {L}^{2}\left( M\right) \) . | Proof By elementary Hilbert space theory, it is enough to verify that, if \( K \in {L}^{2}\left( M\right) \) is orthogonal to \( \mathcal{E} \), then \( K = 0 \) . Assume that \( K \in {L}^{2}\left( M\right) \) is orthogonal to \( \mathcal{E} \), and set, for every \( t \geq 0 \) ,\n\n\[ \n{X}_{t} = {\int }_{0}^{t}{K}_... | Yes |
Theorem 5.4 Let \( M \in {\mathbb{H}}^{2} \) . For every \( H \in \mathcal{E} \) of the form\n\n\[ \n{H}_{s}\left( \omega \right) = \mathop{\sum }\limits_{{i = 0}}^{{p - 1}}{H}_{\left( i\right) }\left( \omega \right) {\mathbf{1}}_{\left( {t}_{i},{t}_{i + 1}\right\rbrack }\left( s\right) ,\n\]\n\nthe formula\n\n\[ \n{\l... | Proof As a preliminary observation, we note that the definition of \( H \cdot M \) when \( H \in \mathcal{E} \) does not depend on the decomposition chosen for \( H \) in the first display of the theorem. Using this remark, one then checks that the mapping \( H \mapsto H \cdot M \) is linear. We next verify that this m... | Yes |
Proposition 5.5 Let \( H \in {L}^{2}\left( M\right) \) . If \( K \) is a progressive process, we have \( {KH} \in \) \( {L}^{2}\left( M\right) \) if and only if \( K \in {L}^{2}\left( {H \cdot M}\right) \) . If the latter properties hold, \[ \left( {KH}\right) \cdot M = K \cdot \left( {H \cdot M}\right) . \] | Proof Using property (5.4), we have \[ E\left\lbrack {{\int }_{0}^{\infty }{K}_{s}^{2}{H}_{s}^{2}\mathrm{\;d}\langle M, M{\rangle }_{s}}\right\rbrack = E\left\lbrack {{\int }_{0}^{\infty }{K}_{s}^{2}\mathrm{\;d}\langle H \cdot M, H \cdot M{\rangle }_{s}}\right\rbrack , \] which gives the first assertion. For the second... | Yes |
Theorem 5.6 Let \( M \) be a continuous local martingale. For every \( H \in {L}_{\mathrm{{loc}}}^{2}\left( M\right) \) , there exists a unique continuous local martingale with initial value 0 , which is denoted by \( H \cdot M \), such that, for every continuous local martingale \( N \), \[ \langle H \cdot M, N\rangle... | Proof We may assume that \( {M}_{0} = 0 \) (in the general case, we write \( M = {M}_{0} + {M}^{\prime } \) and we just set \( H \cdot M = H \cdot {M}^{\prime } \), noting that \( \langle M, N\rangle = \left\langle {{M}^{\prime }, N}\right\rangle \) for every continuous local martingale \( N \) ). Also we may assume th... | Yes |
Proposition 5.8 Let \( X = M + V \) be the canonical decomposition of a continuous semimartingale \( X \), and let \( t > 0 \) . Let \( {\left( {H}^{n}\right) }_{n \geq 1} \) and \( H \) be locally bounded progressive processes, and let \( K \) be a nonnegative progressive process. Assume that the following properties ... | Proof The a.s. convergence\n\n\[ \n{\int }_{0}^{t}{H}_{s}^{n}\mathrm{\;d}{V}_{s}\underset{n \rightarrow \infty }{ \rightarrow }{\int }_{0}^{t}{H}_{s}\mathrm{\;d}{V}_{s} \n\]\n\nfollows from the usual dominated convergence theorem. So we just have to verify that \( {\int }_{0}^{t}{H}_{s}^{n}\mathrm{\;d}{M}_{s} \) conver... | Yes |
Proposition 5.9 Let \( X \) be a continuous semimartingale, and let \( H \) be an adapted process with continuous sample paths. Then, for every \( t > 0 \), for every sequence \( 0 = {t}_{0}^{n} < \cdots < {t}_{{p}_{n}}^{n} = t \) of subdivisions of \( \left\lbrack {0, t}\right\rbrack \) whose mesh tends to 0, we have\... | Proof For every \( n \geq 1 \), define a process \( {H}^{n} \) by\n\n\[ {H}_{s}^{n} = \left\{ \begin{array}{l} {H}_{{t}_{i}^{n}}\text{ if }{t}_{i}^{n} < s \leq {t}_{i + 1}^{n},\text{ for every }i \in \left\{ {0,1,\ldots ,{p}_{n} - 1}\right\} \\ {H}_{0}\text{ if }s = 0 \\ 0\text{ if }s > t. \end{array}\right. \]\n\nNote... | Yes |
Proposition 5.11 Let \( M \) be a continuous local martingale and, for every \( \lambda \in \mathbb{C} \), let\n\n\[ \mathcal{E}{\left( \lambda M\right) }_{t} = \exp \left( {\lambda {M}_{t} - \frac{{\lambda }^{2}}{2}\langle M, M{\rangle }_{t}}\right) \]\n\nThe process \( \mathcal{E}\left( {\lambda M}\right) \) is a com... | Proof If \( F\left( {r, x}\right) \) is a twice continuously differentiable function on \( {\mathbb{R}}^{2} \), Itô’s formula gives\n\n\[ F\left( {\langle M, M{\rangle }_{t},{M}_{t}}\right) = F\left( {0,{M}_{0}}\right) + {\int }_{0}^{t}\frac{\partial F}{\partial x}\left( {\langle M, M{\rangle }_{s},{M}_{s}}\right) \mat... | Yes |
Theorem 5.13 (Dambis-Dubins-Schwarz) Let \( M \) be a continuous local martingale such that \( \langle M, M{\rangle }_{\infty } = \infty \) a.s. There exists a Brownian motion \( {\left( {\beta }_{s}\right) }_{s \geq 0} \) such that\n\n\[ \text{a.s.}\forall t \geq 0,\;{M}_{t} = {\beta }_{ < M, M{ > }_{t}}\text{.} \] | Proof We first assume that \( {M}_{0} = 0 \) . For every \( r \geq 0 \), we set\n\n\[ {\tau }_{r} = \inf \left\{ {t \geq 0 : \langle M, M{\rangle }_{t} \geq r}\right\} \]\n\nNote that \( {\tau }_{r} \) is a stopping time by Proposition 3.9. Furthermore, we have \( {\tau }_{r} < \infty \) for every \( r \geq 0 \), on th... | Yes |
Lemma 5.14 We have a.s. for every \( 0 \leq a < b \) , \[ {M}_{t} = {M}_{a},\forall t \in \left\lbrack {a, b}\right\rbrack \Leftrightarrow \langle M, M{\rangle }_{b} = \langle M, M{\rangle }_{a}. \] | Proof of Lemma 5.14 Thanks to the continuity of sample paths of \( M \) and \( \langle M, M\rangle \) , it is enough to verify that for any fixed \( a \) and \( b \) such that \( 0 \leq a < b \), we have \[ \left\{ {{M}_{t} = {M}_{a},\forall t \in \left\lbrack {a, b}\right\rbrack }\right\} = \left\{ {\langle M, M{\rang... | No |
Proposition 5.15 Let \( M \) and \( N \) be two continuous local martingales such that \( {M}_{0} = \) \( {N}_{0} = 0 \) . Assume that\n\n(i) \( \langle M, M{\rangle }_{t} = \langle N, N{\rangle }_{t} \) for every \( t \geq 0 \), a.s.\n\n(ii) \( M \) and \( N \) are orthogonal \( \left( {\langle M, N{\rangle }_{t} = 0}... | Proof We use the notation of the proof of Theorem 5.13 and note that we have \( {\beta }_{r} = {M}_{{\tau }_{r}} \) and \( {\gamma }_{r} = {N}_{{\tau }_{r}} \), where\n\n\[ \n{\tau }_{r} = \inf \left\{ {t \geq 0 : \langle M, M{\rangle }_{t} \geq r}\right\} = \inf \left\{ {t \geq 0 : \langle N, N{\rangle }_{t} \geq r}\r... | Yes |
Corollary 5.17 Let \( M \) be a continuous local martingale such that \( {M}_{0} = 0 \) . The condition\n\n\[ E\left\lbrack {\langle M, M{\rangle }_{\infty }^{1/2}}\right\rbrack < \infty \]\n\nimplies that \( M \) is a uniformly integrable martingale. | Proof By the case \( p = 1 \) of Theorem 5.16, the condition \( E\left\lbrack {\langle M, M{\rangle }_{\infty }^{1/2}}\right\rbrack < \infty \) implies that \( E\left\lbrack {M}_{\infty }^{ * }\right\rbrack < \infty \) . Proposition 4.7 (ii) then shows that the continuous local martingale \( M \), which is dominated by... | Yes |
Lemma 5.19 Under the assumptions of the theorem, the vector space generated by the random variables\n\n\\[ \n\\exp \\left( {\\mathrm{i}\\mathop{\\sum }\\limits_{{j = 1}}^{n}{\\lambda }_{j}\\left( {{B}_{{t}_{j}} - {B}_{{t}_{j - 1}}}\\right) }\\right)\n\\]\n\nfor any choice of \\( 0 = {t}_{0} < {t}_{1} < \\cdots < {t}_{n... | Proof It is enough to prove that, if \\( Z \\in {L}_{\\mathbb{C}}^{2}\\left( {\\Omega ,{\\mathcal{F}}_{\\infty }, P}\\right) \\) is such that\n\n\\[ \nE\\left\\lbrack {Z\\exp \\left( {\\mathrm{i}\\mathop{\\sum }\\limits_{{j = 1}}^{n}{\\lambda }_{j}\\left( {{B}_{{t}_{j}} - {B}_{{t}_{j - 1}}}\\right) }\\right) }\\right\\... | Yes |
Proposition 5.20 Assume that \( Q \) is a probability measure on \( \left( {\Omega ,\mathcal{F}}\right) \), which is absolutely continuous with respect to \( P \) on the \( \sigma \) -field \( {\mathcal{F}}_{\infty } \). For every \( t \in \left\lbrack {0,\infty }\right\rbrack \), let\n\n\[ \n{D}_{t} = {\frac{\mathrm{d... | Proof If \( A \in {\mathcal{F}}_{t} \), we have\n\n\[ \nQ\left( A\right) = {E}_{Q}\left\lbrack {\mathbf{1}}_{A}\right\rbrack = {E}_{P}\left\lbrack {{\mathbf{1}}_{A}{D}_{\infty }}\right\rbrack = {E}_{P}\left\lbrack {{\mathbf{1}}_{A}{E}_{P}\left\lbrack {{D}_{\infty } \mid {\mathcal{F}}_{t}}\right\rbrack }\right\rbrack \n... | Yes |
Proposition 5.21 Let \( D \) be a continuous local martingale taking (strictly) positive values. There exists a unique continuous local martingale \( L \) such that\n\n\[ \n{D}_{t} = \exp \left( {{L}_{t} - \frac{1}{2}\langle L, L{\rangle }_{t}}\right) = \mathcal{E}{\left( L\right) }_{t}.\n\]\n\nMoreover, \( L \) is giv... | Proof Uniqueness is an easy consequence of Theorem 4.8. Then, since \( D \) takes positive values, we can apply Itô’s formula to \( \log {D}_{t} \) (see the remark before Proposition 5.11), and we get\n\n\[ \n\log {D}_{t} = \log {D}_{0} + {\int }_{0}^{t}\frac{\mathrm{d}{D}_{s}}{{D}_{s}} - \frac{1}{2}{\int }_{0}^{t}\fra... | Yes |
Theorem 5.22 (Girsanov) Assume that the probability measures \( P \) and \( Q \) are mutually absolutely continuous on \( {\mathcal{F}}_{\infty } \). Let \( {\left( {D}_{t}\right) }_{t \geq 0} \) be the martingale with càdlàg sample paths such that, for every \( t \geq 0 \), \[ {D}_{t} = {\frac{\mathrm{d}Q}{\mathrm{\;d... | Proof The fact that \( {D}_{t} \) can be written in the form \( {D}_{t} = \mathcal{E}{\left( L\right) }_{t} \) follows from Proposition 5.21 (we are assuming that \( D \) has continuous sample paths, and we also know from Proposition 5.20 that \( D \) takes positive values). Then, let \( T \) be a stopping time and let... | Yes |
Theorem 5.23 Let \( L \) be a continuous local martingale such that \( {L}_{0} = 0 \) . Consider the following properties:\n\n(i) \( E\left\lbrack {\exp \frac{1}{2}\langle L, L{\rangle }_{\infty }}\right\rbrack < \infty \) (Novikov’s criterion);\n\n(ii) \( L \) is a uniformly integrable martingale, and \( E\left\lbrack... | Proof (i) \( \Rightarrow \) (ii) Property (i) implies that \( E\left\lbrack {\langle L, L{\rangle }_{\infty }}\right\rbrack < \infty \) hence also that \( L \) is a continuous martingale bounded in \( {L}^{2} \) (Theorem 4.13). Then,\n\n\[ \exp \frac{1}{2}{L}_{\infty } = {\left( \mathcal{E}{\left( L\right) }_{\infty }\... | Yes |
Proposition 5.24 (Cameron-Martin formula) Let \( W\left( \mathrm{{dw}}\right) \) be the Wiener measure on \( C\left( {{\mathbb{R}}_{ + },\mathbb{R}}\right) \), and let \( h \) be a function in the Cameron-Martin space \( \mathcal{H} \). Then, for every nonnegative measurable function \( \Phi \) on \( C\left( {{\mathbb{... | \[ \int W\left( {\mathrm{\;d}\mathrm{w}}\right) \Phi \left( {\mathrm{w} + h}\right) = \int W\left( {\mathrm{\;d}\mathrm{w}}\right) \exp \left( {{\int }_{0}^{\infty }\dot{h}\left( s\right) \mathrm{d}\mathrm{w}\left( s\right) - \frac{1}{2}{\int }_{0}^{\infty }\dot{h}{\left( s\right) }^{2}\mathrm{\;d}s}\right) \Phi \left(... | Yes |
Theorem 6.3 Assume that \( E \) is a Polish space equipped with its Borel \( \sigma \) -field & For every \( U \in F\left( {\mathbb{R}}_{ + }\right) \), let \( {\mu }_{U} \) be a probability measure on \( {E}^{U} \) . Assume that the collection \( \left( {{\mu }_{U}, U \in F\left( {\mathbb{R}}_{ + }\right) }\right) \) ... | Remark The uniqueness of \( \mu \) is an immediate consequence of the monotone class lemma (cf. Appendix A1). | No |
Corollary 6.4 We assume that \( E \) satisfies the assumption of the previous theorem and that \( {\left( {Q}_{t}\right) }_{t \geq 0} \) is a transition semigroup on \( E \) . Let \( \gamma \) be a probability measure on \( E \) . Then there exists a (unique) probability measure \( P \) on \( {\Omega }^{ * } \) under w... | Proof Let \( U = \left\{ {{t}_{1},\ldots ,{t}_{p}}\right\} \in F\left( {\mathbb{R}}_{ + }\right) \), with \( 0 \leq {t}_{1} < \cdots < {t}_{p} \) . We define a probability measure \( {P}^{U} \) on \( {E}^{U} \) (identified with \( {E}^{p} \) as explained above) by setting\n\n\[ \int {P}^{U}\left( {\mathrm{\;d}{x}_{1}\l... | Yes |
Lemma 6.6 Let \( X \) be a Markov process with semigroup \( {\left( {Q}_{t}\right) }_{t \geq 0} \) with respect to the filtration \( \left( {\mathcal{F}}_{t}\right) \) . Let \( h \in B\left( E\right) \) be nonnegative and let \( \lambda > 0 \) . Then the process\n\n\[ \n{\mathrm{e}}^{-{\lambda t}}{R}_{\lambda }h\left( ... | Proof The random variables \( {\mathrm{e}}^{-{\lambda t}}{R}_{\lambda }h\left( {X}_{t}\right) \) are bounded and thus in \( {L}^{1} \) . Then, for every \( s \geq 0 \) ,\n\n\[ \n{Q}_{s}{R}_{\lambda }h = {\int }_{0}^{\infty }{\mathrm{e}}^{-{\lambda t}}{Q}_{s + t}h\mathrm{\;d}t \n\]\n\nand it follows that\n\n\[ \n{\mathr... | Yes |
Proposition 6.8 Let \( \lambda > 0 \), and set \( \mathcal{R} = \left\{ {{R}_{\lambda }f : f \in {C}_{0}\left( E\right) }\right\} \) . Then \( \mathcal{R} \) does not depend on the choice \( \lambda > 0 \) . Furthermore, \( \mathcal{R} \) is a dense subspace of \( {C}_{0}\left( E\right) \) . | Proof If \( \lambda \neq \mu \), the resolvent equation gives\n\n\[ \n{R}_{\lambda }f = {R}_{\mu }\left( {f + \left( {\mu - \lambda }\right) {R}_{\lambda }f}\right) .\n\]\n\nHence any function of the form \( {R}_{\lambda }f \) with \( f \in {C}_{0}\left( E\right) \) is also of the form \( {R}_{\mu }g \) for some \( g \... | Yes |
Proposition 6.10 Let \( f \in D\left( L\right) \) and \( s > 0 \) . Then \( {Q}_{s}f \in D\left( L\right) \) and \( L\left( {{Q}_{s}f}\right) = {Q}_{s}\left( {Lf}\right) \) . | \[ \frac{{Q}_{t}\left( {{Q}_{s}f}\right) - {Q}_{s}f}{t} = {Q}_{s}\left( \frac{{Q}_{t}f - f}{t}\right) \] and using the fact that \( {Q}_{s} \) is a contraction of \( {C}_{0}\left( E\right) \), we get that \( {t}^{-1}\left( {{Q}_{t}\left( {{Q}_{s}f}\right) - {Q}_{s}f}\right) \) converges to \( {Q}_{s}\left( {Lf}\right) ... | Yes |
Proposition 6.11 If \( f \in D\left( L\right) \), we have, for every \( t \geq 0 \), \[ {Q}_{t}f = f + {\int }_{0}^{t}{Q}_{s}\left( {Lf}\right) \mathrm{d}s = f + {\int }_{0}^{t}L\left( {{Q}_{s}f}\right) \mathrm{d}s. \] | Proof Let \( f \in D\left( L\right) \). For every \( t \geq 0 \), \[ {\varepsilon }^{-1}\left( {{Q}_{t + \varepsilon }f - {Q}_{t}f}\right) = {Q}_{t}\left( {{\varepsilon }^{-1}\left( {{Q}_{\varepsilon }f - f}\right) }\right) \underset{\varepsilon \downarrow 0}{ \rightarrow }{Q}_{t}\left( {Lf}\right) . \] Moreover, the p... | Yes |
Proposition 6.12 Let \( \lambda > 0 \) .\n\n(i) For every \( g \in {C}_{0}\left( E\right) ,{R}_{\lambda }g \in D\left( L\right) \) and \( \left( {\lambda - L}\right) {R}_{\lambda }g = g \) .\n\n(ii) If \( f \in D\left( L\right) ,{R}_{\lambda }\left( {\lambda - L}\right) f = f \) .\n\nConsequently, \( D\left( L\right) =... | Proof\n\n(i) If \( g \in {C}_{0}\left( E\right) \), we have for every \( \varepsilon > 0 \),\n\n\[{\varepsilon }^{-1}\left( {{Q}_{\varepsilon }{R}_{\lambda }g - {R}_{\lambda }g}\right) = {\varepsilon }^{-1}\left( {{\int }_{0}^{\infty }{\mathrm{e}}^{-{\lambda t}}{Q}_{\varepsilon + t}g\mathrm{\;d}t - {\int }_{0}^{\infty ... | Yes |
Corollary 6.13 The semigroup \( {\left( {Q}_{t}\right) }_{t \geq 0} \) is determined by the generator \( L \) (including also the domain \( D\left( L\right) \) ). | Proof Let \( f \) be a nonnegative function in \( {C}_{0}\left( E\right) \) . Then \( {R}_{\lambda }f \) is the unique element of \( D\left( L\right) \) such that \( \left( {\lambda - L}\right) {R}_{\lambda }f = f \) . On the other hand, knowing \( {R}_{\lambda }f\left( x\right) = \) \( {\int }_{0}^{\infty }{\mathrm{e}... | Yes |
Theorem 6.14 Let \( h, g \in {C}_{0}\left( E\right) \) . The following two conditions are equivalent:\n\n(i) \( h \in D\left( L\right) \) and \( {Lh} = g \) .\n\n(ii) For every \( x \in E \), the process\n\n\[ h\left( {X}_{t}^{x}\right) - {\int }_{0}^{t}g\left( {X}_{s}^{x}\right) \mathrm{d}s \]\n\nis a martingale, with... | Proof We first prove that (i) \( \Rightarrow \) (ii). Let \( h \in D\left( L\right) \) and \( g = {Lh} \) . By Proposition 6.11, we have then, for every \( s \geq 0 \) ,\n\n\[ {Q}_{s}h = h + {\int }_{0}^{s}{Q}_{r}g\mathrm{\;d}r \]\n\nIt follows that, for \( t \geq 0 \) and \( s \geq 0 \) ,\n\n\[ E\left\lbrack {h\left( ... | Yes |
Theorem 6.15 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be a Markov process with semigroup \( {\left( {Q}_{t}\right) }_{t \geq 0} \), with respect to the filtration \( {\left( {\overline{\mathcal{F}}}_{t}\right) }_{t \in \left\lbrack {0,\infty }\right\rbrack } \). Set \( {\widetilde{\mathcal{F}}}_{\infty } = {\mathc... | Proof Let \( {E}_{\Delta } = E \cup \{ \Delta \} \) be the Alexandroff compactification of \( E \), which is obtained by adding the point at infinity \( \Delta \) to \( E \) (and by definition the neighborhoods of \( \Delta \) are the complements of compact subsets of \( E \) ). We agree that every function \( f \in {C... | Yes |
Theorem 6.16 (Simple Markov property) Let \( {\left( {Y}_{t}\right) }_{t \geq 0} \) be a Markov process with semigroup \( {\left( {Q}_{t}\right) }_{t \geq 0} \), with respect to the filtration \( {\left( {\mathcal{F}}_{t}\right) }_{t \geq 0} \). We assume that the sample paths of \( Y \) are càdlàg. Let \( s \geq 0 \) ... | Proof As in the preceding remark, it suffices to consider the case where \( \Phi = {1}_{A} \) and\n\n\[ A = \left\{ {f \in \mathbb{D}\left( E\right) : f\left( {t}_{1}\right) \in {B}_{1},\ldots, f\left( {t}_{p}\right) \in {B}_{p}}\right\} ,\n\nwhere \( 0 \leq {t}_{1} < {t}_{2} < \cdots < {t}_{p} \) and \( {B}_{1},\ldots... | Yes |
Lemma 6.18 Let \( x \in E \) . There exists a real number \( q\left( x\right) \geq 0 \) such that the random variable \( {T}_{1} \) is exponentially distributed with parameter \( q\left( x\right) \) under \( {P}_{x} \) . Furthermore, if \( q\left( x\right) > 0,{T}_{1} \) and \( {X}_{{T}_{1}} \) are independent under \(... | Proof Let \( s, t \geq 0 \) . We have\n\n\[ \n{P}_{x}\left( {{T}_{1} > s + t}\right) = {E}_{x}\left\lbrack {{\mathbf{1}}_{\left\{ {T}_{1} > s\right\} }\Phi \left( {\left( {X}_{s + r}\right) }_{r \geq 0}\right) }\right\rbrack , \n\] \n\nwhere \( \Phi \left( f\right) = {\mathbf{1}}_{\{ f\left( r\right) = f\left( 0\right)... | Yes |
Proposition 6.19 Let \( L \) denote the generator of \( {\left( {Q}_{t}\right) }_{t \geq 0} \) . Then \( D\left( L\right) = {C}_{0}\left( E\right) = \) \( B\left( E\right) \), and, for every \( \varphi \in B\left( E\right) \), for every \( x \in E \) :\n\n- if \( q\left( x\right) = 0,{L\varphi }\left( x\right) = 0 \) ;... | Proof Let \( \varphi \in B\left( E\right) \) and \( x \in E \) . If \( q\left( x\right) = 0 \), it is trivial that \( {Q}_{t}\varphi \left( x\right) = \varphi \left( x\right) \) and so\n\n\[ \n\mathop{\lim }\limits_{{t \downarrow 0}}\frac{{Q}_{t}\varphi \left( x\right) - \varphi \left( x\right) }{t} = 0.\n\]\n\nSuppose... | Yes |
Proposition 6.20 We assume that \( q\left( y\right) > 0 \) for every \( y \in E \) . Let \( x \in E \) . Then, \( {P}_{x} \) a.s., the jump times \( {T}_{1} < {T}_{2} < {T}_{3} < \cdots \) are all finite and the sequence \( {X}_{0},{X}_{{T}_{1}},{X}_{{T}_{2}},\ldots \) is under \( {P}_{x} \) a discrete Markov chain wit... | Proof An application of the strong Markov property shows that all stopping times \( {T}_{1},{T}_{2},\ldots \) are finite \( {P}_{x} \) a.s. Then, let \( y, z \in E \), and \( {f}_{1},{f}_{2} \in B\left( {\mathbb{R}}_{ + }\right) \) . By the strong Markov property at \( {T}_{1} \) ,\n\n\[ \n{E}_{x}\left\lbrack {{\mathbf... | Yes |
Proposition 6.21 The collection \( {\left( {Q}_{t}\right) }_{t \geq 0} \) is a Feller semigroup on \( \mathbb{R} \) . Furthermore, \( {\left( {Y}_{t}\right) }_{t \geq 0} \) is a Markov process with semigroup \( {\left( {Q}_{t}\right) }_{t \geq 0} \) . | Proof Let us show that \( {\left( {Q}_{t}\right) }_{t \geq 0} \) is a transition semigroup. Let \( \varphi \in B\left( \mathbb{R}\right), s, t \geq 0 \) and \( x \in \mathbb{R} \) . Property (ii) shows that the law of \( \left( {{Y}_{t},{Y}_{t + s} - {Y}_{t}}\right) \) is the product probability measure \( {Q}_{t}\left... | Yes |
Proposition 6.22 Under the preceding assumptions, the semigroup \( {\left( {Q}_{t}\right) }_{t \geq 0} \) is Feller. Furthermore, for every \( \lambda > 0 \), and every \( x \geq 0 \) , \[ \int {Q}_{t}\left( {x,\mathrm{\;d}y}\right) {\mathrm{e}}^{-{\lambda y}} = {\mathrm{e}}^{-x{\psi }_{t}\left( \lambda \right) } \] wh... | Proof Let us start with the second assertion. If \( x, y > 0 \), the equality \( {Q}_{t}\left( {x, \cdot }\right) * \) \( {Q}_{t}\left( {y, \cdot }\right) = {Q}_{t}\left( {x + y, \cdot }\right) \) implies that \[ \left( {\int {Q}_{t}\left( {x,\mathrm{\;d}z}\right) {\mathrm{e}}^{-{\lambda z}}}\right) \left( {\int {Q}_{t... | Yes |
Theorem 7.1 Let \( \varphi \in {C}_{0}\left( {\mathbb{R}}^{d}\right) \) . For every \( t > 0 \) and \( x \in {\mathbb{R}}^{d} \), set\n\n\[ \n{u}_{t}\left( x\right) = {Q}_{t}\varphi \left( x\right) = {E}_{x}\left\lbrack {\varphi \left( {B}_{t}\right) }\right\rbrack \n\]\n\nThen, the function \( {\left( {u}_{t}\left( x\... | Proof By the remarks preceding the theorem, we already know that, for every \( t > 0,{u}_{t} \) is a \( {C}^{\infty } \) function, \( {u}_{t} \in D\left( L\right) \), and \( L{u}_{t} = \frac{1}{2}\Delta {u}_{t} \) . Let \( \varepsilon > 0 \) . By applying Proposition 6.11 to \( f = {u}_{\varepsilon } \), we get for eve... | Yes |
Proposition 7.3 Let \( u \) be harmonic on the domain \( D \) . Let \( {D}^{\prime } \) be a bounded subdomain of \( D \) whose closure is contained in \( D \), and consider the stopping time \( T \mathrel{\text{:=}} \inf \left\{ {t \geq 0 : {B}_{t} \notin {D}^{\prime }}\right\} \) . Then, for every \( x \in {D}^{\prim... | Proof Since \( {D}^{\prime } \) is bounded, both \( u \) and \( \nabla u \) are bounded on \( {D}^{\prime } \), and we also know that \( {P}_{x}\left( {T < \infty }\right) = 1 \) for every \( x \in {D}^{\prime } \) . It follows from (7.1) that \( u\left( {B}_{t \land T}\right) \) is a (true) martingale, and in particul... | Yes |
Proposition 7.4 (Mean value property) Suppose that \( u \) is harmonic on the domain \( D \) . Then, for every \( x \in D \) and for every \( r > 0 \) such that the closed ball of radius \( r \) centered at \( x \) is contained in \( D \), we have\n\n\[ u\left( x\right) = \int {\sigma }_{x, r}\left( {\mathrm{\;d}y}\rig... | Proof First observe that, if \( {T}_{1} = \inf \left\{ {t \geq 0 : \left| {B}_{t}\right| = 1}\right\} \), the distribution of \( {B}_{{T}_{1}} \) under \( {P}_{0} \) is invariant under all vector isometries of \( {\mathbb{R}}^{d} \) (by the invariance properties of Brownian motion stated at the end of Chap. 2) and ther... | Yes |
Lemma 7.5 Let \( u \) be a locally bounded and measurable function on \( D \) that satisfies the mean value property. Then \( u \) is harmonic on \( D \) . | Proof Fix \( {r}_{0} > 0 \) and let \( {D}^{\prime } \) be the open subset of \( D \) consisting of all points whose distance to \( {D}^{c} \) is greater than \( {r}_{0} \) . It is enough to prove that \( u \) is twice continuously differentiable and \( {\Delta u} = 0 \) on \( {D}^{\prime } \) . Let \( h : \mathbb{R} \... | Yes |
Let \( D \) be a bounded domain, and write \( T = \inf \left\{ {t \geq 0 : {B}_{t} \notin D}\right\} \) for the exit time of Brownian motion from \( D \) . (i) Let \( g \) be a continuous function on \( \partial D \), and let \( u \) be a solution of the Dirichlet problem in \( D \) with boundary condition \( g \) . Th... | (i) Fix \( x \in D \) and \( {\varepsilon }_{0} > 0 \) such that the ball of radius \( {\varepsilon }_{0} \) centered at \( x \) is contained in \( D \) . For every \( \varepsilon \in \left( {0,{\varepsilon }_{0}}\right) \), let \( {D}_{\varepsilon } \) be the connected component containing \( x \) of the open set cons... | Yes |
Theorem 7.8 (Solution of the Dirichlet problem) Let \( D \) be a bounded domain in \( {\mathbb{R}}^{d} \). Assume that \( D \) satisfies the exterior cone condition at every \( y \in \partial D \). Then, for every continuous function \( g \) on \( \partial D \), the formula\n\n\[ u\left( x\right) = {E}_{x}\left\lbrack ... | Proof Thanks to Proposition 7.7 (ii), we only need to verify that, for every fixed \( y \in \partial D \)\n\n\[ \mathop{\lim }\limits_{{x \rightarrow y, x \in D}}u\left( x\right) = g\left( y\right) \]\n\n(7.2)\n\nLet \( \varepsilon > 0 \). Since \( g \) is continuous, we can find \( \delta > 0 \) such that we have \( \... | Yes |
Lemma 7.9 Under the exterior cone condition, we have for every \( y \in \partial D \) and every \( \eta > 0 \)\n\n\[ \mathop{\lim }\limits_{{x \rightarrow y, x \in D}}{P}_{x}\left( {T > \eta }\right) = 0. \]\n | Proof For every \( u \in {\mathbb{R}}^{d} \) with \( \left| u\right| = 1 \) and every \( \gamma \in \left( {0,1}\right) \), consider the circular cone\n\n\[ \mathcal{C}\left( {u,\gamma }\right) \mathrel{\text{:=}} \left\{ {z \in {\mathbb{R}}^{d} : z \cdot u > \left( {1 - \gamma }\right) \left| z\right| }\right\} \]\n\n... | Yes |
Lemma 7.11 For every fixed \( y \in \partial {\mathcal{B}}_{1} \), the function \( x \mapsto K\left( {x, y}\right) \) is harmonic on \( {\mathcal{B}}_{1} \) . | Proof Set \( {K}_{y}\left( x\right) = K\left( {x, y}\right) \) for \( x \in {\mathcal{B}}_{1} \) . Then \( {K}_{y} \) is a \( {C}^{\infty } \) function on \( {\mathcal{B}}_{1} \) . Moreover a (somewhat tedious) direct calculation left to the reader shows that \( \Delta {K}_{y} = 0 \) on \( {\mathcal{B}}_{1} \) . | No |
Lemma 7.12 Let \( 0 \leq {r}_{1} < {r}_{2} \) be two real numbers and let \( h : \left( {{r}_{1},{r}_{2}}\right) \rightarrow \mathbb{R} \) be a measurable function. The function \( u\left( x\right) = h\left( \left| x\right| \right) \) is harmonic on the domain \( \left\{ {x \in {\mathbb{R}}^{d} : {r}_{1} < \left| x\rig... | Proof Suppose that \( u\left( x\right) = h\left( \left| x\right| \right) \) is harmonic on \( \left\{ {x \in {\mathbb{R}}^{d} : {r}_{1} < \left| x\right| < {r}_{2}}\right\} \) . Then \( u \) is twice continuously differentiable and so is \( h \) . From the expression of the Laplacian of a radial function, we get that \... | Yes |
Lemma 7.13 For every \( x \in {\mathcal{B}}_{1} \) , \[ {\int }_{\partial {\mathcal{B}}_{1}}K\left( {x, y}\right) {\sigma }_{1}\left( {\mathrm{\;d}y}\right) = 1 \] | Proof For every \( x \in {\mathcal{B}}_{1} \), set \[ F\left( x\right) = {\int }_{\partial {\mathcal{B}}_{1}}K\left( {x, y}\right) {\sigma }_{1}\left( {\mathrm{\;d}y}\right) . \] Then the preceding lemma implies that \( F \) is harmonic on \( {\mathcal{B}}_{1} \). Indeed, if \( x \in {\mathcal{B}}_{1} \) and \( r < 1 -... | Yes |
Theorem 7.14 Let \( g \) be a continuous function on \( \partial {\mathcal{B}}_{1} \) . The unique solution of the Dirichlet problem in \( {\mathcal{B}}_{1} \) with boundary condition \( g \) is given by \[ u\left( x\right) = {\int }_{\partial {\mathcal{B}}_{1}}g\left( y\right) K\left( {x, y}\right) {\sigma }_{1}\left(... | Proof The very same arguments as in the beginning of the proof of Lemma 7.13 show that \( u \) is harmonic on \( {\mathcal{B}}_{1} \) . To verify the boundary condition, fix \( {y}_{0} \in \partial {\mathcal{B}}_{1} \) . For every \( \delta > 0 \), the explicit form of the Poisson kernel shows that, if \( x \in {\mathc... | Yes |
Corollary 7.15 Let \( T = \inf \left\{ {t \geq 0 : {B}_{t} \notin {\mathcal{B}}_{1}}\right\} \) . For every \( x \in {\mathcal{B}}_{1} \), the distribution of \( {B}_{T} \) under \( {P}_{x} \) has density \( K\left( {x, y}\right) \) with respect to \( {\sigma }_{1}\left( {\mathrm{\;d}y}\right) \) . | This is immediate since, by combining Proposition 7.7 (i) with Theorem 7.14, we get that, for any continuous function \( g \) on \( \partial {\mathcal{B}}_{1} \) ,\n\n\[ \n{E}_{x}\left\lbrack {g\left( {B}_{T}\right) }\right\rbrack = {\int }_{\partial {\mathcal{B}}_{1}}g\left( y\right) K\left( {x, y}\right) {\sigma }_{1... | Yes |
Proposition 7.16 Suppose that \( x \neq 0 \), and let \( \varepsilon \) and \( R \) be such that \( 0 < \varepsilon < \left| x\right| < R \) . Then,\n\n\[ \n{P}_{x}\left( {{U}_{\varepsilon } < {U}_{R}}\right) = \left\{ \begin{array}{l} \frac{\log R - \log \left| x\right| }{\log R - \log \varepsilon }\text{ if }d = 2, \... | Proof Write \( {D}_{\varepsilon, R} \) for the annulus \( \left\{ {y \in {\mathbb{R}}^{d} : \varepsilon < \left| y\right| < R}\right\} \) . Let \( u\left( x\right) \) be the function defined for \( x \in {D}_{\varepsilon, R} \) that appears in the right-hand side of (7.4). By Lemma 7.12, \( u \) is harmonic on \( {D}_{... | Yes |
Theorem 7.18 Let \( \Phi : \mathbb{C} \rightarrow \mathbb{C} \) be a nonconstant holomorphic function. For every \( t \geq 0 \), set\n\n\[ \n{C}_{t} = {\int }_{0}^{t}{\left| {\Phi }^{\prime }\left( {B}_{s}\right) \right| }^{2}\mathrm{\;d}s \]\n\nLet \( z \in \mathbb{C} \) . There exists a complex Brownian motion \( \Ga... | Proof Let \( g \) and \( h \) stand respectively for the real and imaginary parts of \( \Phi \) . Since \( g \) and \( h \) are harmonic, an application of Itô’s formula gives under \( {P}_{z} \),\n\n\[ \ng\left( {B}_{t}\right) = g\left( z\right) + {\int }_{0}^{t}\frac{\partial g}{\partial x}\left( {B}_{s}\right) \math... | Yes |
Theorem 7.19 Let \( z \in \mathbb{C} \smallsetminus \{ 0\} \) and write \( z = \exp \left( {r + \mathrm{i}\theta }\right) \) where \( r \in \mathbb{R} \) and \( \theta \in \) \( ( - \pi ,\pi \rbrack \) . There exist two independent linear Brownian motions \( \beta \) and \( \gamma \) that start respectively from \( r \... | Proof The \ | No |
Proposition 7.22 Consider the planar Brownian motion B started from \( z \neq 0 \) . Then, for every \( a > 0 \) , \[ \mathop{\lim }\limits_{{t \rightarrow \infty }}P\left( {\mathop{\min }\limits_{{0 \leq s \leq t}}\left| {B}_{s}\right| \leq {t}^{-a/2}}\right) = \frac{1}{1 + a}. \] | Proof Without loss of generality, we take \( z = 1 \) . We keep the notation introduced in the proofs of Lemma 7.21 and Theorem 7.20. We observe that \[ \log \left( {\mathop{\min }\limits_{{0 \leq s \leq t}}\left| {B}_{s}\right| }\right) = \mathop{\min }\limits_{{0 \leq s \leq t}}{\beta }_{{H}_{s}} = \mathop{\min }\lim... | Yes |
Theorem 8.3 Under the preceding assumptions, pathwise uniqueness holds for \( E\left( {\sigma, b}\right) \), and, for every choice of the filtered probability space \( \left( {\Omega ,\mathcal{F},\left( {\mathcal{F}}_{t}\right), P}\right) \) and of the \( \left( {\mathcal{F}}_{t}\right) \) -Brownian motion \( B \), for... | Proof For the sake of simplicity, we consider only the case \( d = m = 1 \) . The reader will be able to check that the general case follows from exactly the same arguments, at the cost of a heavier notation. Let us start by proving pathwise uniqueness. We consider (on the same filtered probability space, with the same... | Yes |
Lemma 8.4 (Gronwall’s lemma) Let \( T > 0 \) and let \( g \) be a nonnegative bounded measurable function on \( \left\lbrack {0, T}\right\rbrack \) . Assume that there exist two constants \( a \geq 0 \) and \( b \geq 0 \) such that, for every \( t \in \left\lbrack {0, T}\right\rbrack \) ,\n\n\[ g\left( t\right) \leq a ... | Proof of the lemma By iterating the condition on \( g \), we get,\n\n\[ g\left( t\right) \leq a + a\left( {bt}\right) + {b}^{2}{\int }_{0}^{t}\mathrm{\;d}s{\int }_{0}^{s}\mathrm{\;d}{rg}\left( r\right) \]\n\n\[ \leq a + a\left( {bt}\right) + a\frac{{\left( bt\right) }^{2}}{2} + \cdots + a\frac{{\left( bt\right) }^{n}}{... | Yes |
Corollary 8.8 Suppose that \( {\left( {X}_{t}\right) }_{t \geq 0} \) solves \( E\left( {\sigma, b}\right) \) on a filtered probability space \( \left( {\Omega ,\mathcal{F},\left( {\mathcal{F}}_{t}\right), P}\right) \) . Then \( {\left( {X}_{t}\right) }_{t \geq 0} \) satisfies the strong Markov property: If \( T \) is a... | Proof It suffices to apply Theorem 6.17. Alternatively, we could also argue in a similar manner as in the proof of Theorem 8.6, letting the stopping time \( T \) play the same role as the deterministic time \( s \) in the latter proof, and using the strong Markov property of Brownian motion. | No |
Proposition 9.2 Let \( X \) be a continuous semimartingale and \( a \in \mathbb{R} \). There exists an increasing process \( {\left( {L}_{t}^{a}\left( X\right) \right) }_{t \geq 0} \) such that the following three identities hold:\n\n\[ \left| {{X}_{t} - a}\right| = \left| {{X}_{0} - a}\right| + {\int }_{0}^{t}\operato... | Proof We apply Proposition 9.1 to the convex function \( f\left( x\right) = \left| {x - a}\right| \), noting that \( {f}_{ - }^{\prime }\left( x\right) = \operatorname{sgn}\left( {x - a}\right) \). It follows from Proposition 9.1 that the process \( {\left( {L}_{t}^{a}\left( X\right) \right) }_{t \geq 0} \) defined by\... | Yes |
Proposition 9.3 Let \( X \) be a continuous semimartingale and let \( a \in \mathbb{R} \) . Then a.s. the random measure \( {\mathrm{d}}_{s}{L}_{s}^{a}\left( X\right) \) is supported on \( \left\{ {s \geq 0 : {X}_{s} = a}\right\} \) . | Proof Set \( {W}_{t} = \left| {{X}_{t} - a}\right| \) and note that (9.5) gives \( \langle W, W{\rangle }_{t} = \langle X, X{\rangle }_{t} \) since \( \left| {\operatorname{sgn}\left( x\right) }\right| = 1 \) for every \( x \in \mathbb{R} \) . By applying Itô’s formula to \( {\left( {W}_{t}\right) }^{2} \), we get\n\n\... | Yes |
Theorem 9.4 The process \( \left( {{L}^{a}\left( X\right), a \in \mathbb{R}}\right) \) with values in \( C\left( {{\mathbb{R}}_{ + },{\mathbb{R}}_{ + }}\right) \) has a càdlàg modification, which we consider from now on and for which we keep the same notation \( \left( {{L}^{a}\left( X\right), a \in \mathbb{R}}\right) ... | The proof of the theorem relies on Tanaka's formula and the following technical lemma. | No |
Theorem 9.6 (Generalized Itô formula) Let \( f \) be a difference of convex functions on \( \mathbb{R} \) . Then, for every \( t \geq 0 \) , \[ f\left( {X}_{t}\right) = f\left( {X}_{0}\right) + {\int }_{0}^{t}{f}_{ - }^{\prime }\left( {X}_{s}\right) \mathrm{d}{X}_{s} + \frac{1}{2}{\int }_{\mathbb{R}}{L}_{t}^{a}\left( X... | Proof By linearity, it suffices to treat the case when \( f \) is convex. Furthermore, by simple \ | No |
Corollary 9.7 (Density of occupation time formula) We have almost surely, for every \( t \geq 0 \) and every nonnegative measurable function \( \varphi \) on \( \mathbb{R} \) , \[ {\int }_{0}^{t}\varphi \left( {X}_{s}\right) \mathrm{d}\langle X, X{\rangle }_{s} = {\int }_{\mathbb{R}}\varphi \left( a\right) {L}_{t}^{a}\... | Proof Fix \( t \geq 0 \) and consider a nonnegative continuous function \( \varphi \) on \( \mathbb{R} \) with compact support. Let \( f \) be a twice continuously differentiable function on \( \mathbb{R} \) such that \( {f}^{\prime \prime } = \varphi \) . Note that \( f \) is convex since \( \varphi \geq 0 \) . By com... | Yes |
Corollary 9.8 If \( X \) is of the form \( {X}_{t} = {X}_{0} + {V}_{t} \), where \( V \) is a finite variation process, then \( {L}_{t}^{a}\left( X\right) = 0 \) for all \( a \in \mathbb{R} \) and \( t \geq 0 \) . | Proof From the density of occupation time formula and the fact that \( \langle X, X\rangle = 0 \) , we get \( {\int }_{\mathbb{R}}\varphi \left( a\right) {L}_{t}^{a}\left( X\right) \mathrm{d}a = 0 \) for any nonnegative measurable function \( \varphi \), and the desired result follows. | No |
Proposition 9.9 Let \( X \) be a continuous semimartingale. Then a.s. for every \( a \in \mathbb{R} \) and \( t \geq 0 \) , \[ {L}_{t}^{a}\left( X\right) = \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}\frac{1}{\varepsilon }{\int }_{0}^{t}{\mathbf{1}}_{\left\{ a \leq {X}_{s} \leq a + \varepsilon \right\} }\mathrm{... | Proof By the density of occupation time formula, \[ \frac{1}{\varepsilon }{\int }_{0}^{t}{\mathbf{1}}_{\left\{ a \leq {X}_{s} \leq a + \varepsilon \right\} }\mathrm{d}\langle X, X{\rangle }_{s} = \frac{1}{\varepsilon }{\int }_{a}^{a + \varepsilon }{L}_{t}^{b}\left( X\right) \mathrm{d}b, \] and the result follows from t... | Yes |
Corollary 9.10 Let \( p \geq 1 \) . There exists a constant \( {C}_{p} \) such that, for any continuous semimartingale \( X \) with canonical decomposition \( X = M + V \), we have for every \( a \in \mathbb{R} \) and \( t \geq 0 \) ,\n\n\[ E\left\lbrack {\left( {L}_{t}^{a}\left( X\right) \right) }^{p}\right\rbrack \le... | Proof This readily follows from the bound of Lemma 9.5, using the approximation of \( {L}_{t}^{a}\left( X\right) \) in Proposition 9.9 and Fatou’s lemma. | No |
Theorem 9.12 (Trotter) There exists a (unique) process \( {\left( {L}_{t}^{a}\left( B\right) \right) }_{a \in \mathbb{R}, t \geq 0} \), whose sample paths are continuous functions of the pair \( \left( {a, t}\right) \), such that, for every fixed \( a \in \mathbb{R},{\left( {L}_{t}^{a}\left( B\right) \right) }_{t \geq ... | Proof The first assertion follows by applying Theorem 9.4 and Corollary 9.7 to \( X = B \), noting that \( \langle B, B{\rangle }_{t} = t \) . We have already seen that the inclusion (9.17) holds with probability one if \( a \) is fixed, hence simultaneously for all rationals, a.s. A continuity argument allows us to ge... | Yes |
Proposition 9.13 (i) Let \( a \in \mathbb{R} \smallsetminus \{ 0\} \) and \( {T}_{a} \mathrel{\text{:=}} \inf \left\{ {t \geq 0 : {B}_{t} = a}\right\} \) . Then \( {L}_{{T}_{a}}^{0}\left( B\right) \) has an exponential distribution with mean \( 2\left| a\right| \) . | (i) By simple scaling and symmetry arguments, it is enough to take \( a = 1 \) . We then observe that \( {L}_{\infty }^{0}\left( B\right) = \infty \) a.s. Indeed, the scaling argument of the preceding proof shows that \( {L}_{\infty }^{0}\left( B\right) \) has the same distribution as \( \lambda {L}_{\infty }^{0}\left(... | Yes |
Theorem 9.14 (Lévy) The two processes \( {\left( {S}_{t},{S}_{t} - {B}_{t}\right) }_{t \geq 0} \) and \( {\left( {L}_{t}^{0}\left( B\right) ,\left| {B}_{t}\right| \right) }_{t \geq 0} \) have the same distribution. | Proof By Tanaka’s formula, for every \( t \geq 0 \) ,\n\n\[ \left| {B}_{t}\right| = - {\beta }_{t} + {L}_{t}^{0}\left( B\right) \]\n\n(9.18)\n\nwhere\n\n\[ {\beta }_{t} = - {\int }_{0}^{t}\operatorname{sgn}\left( {B}_{s}\right) \mathrm{d}{B}_{s} \]\n\nSince \( \langle \beta ,\beta {\rangle }_{t} = t \), Theorem 5.12 en... | Yes |
Proposition 9.15 We have a.s.\n\n\[ \n\\left\\{ {t \geq 0 : {B}_{t} = 0}\\right\\} = \\left\\{ {{\\tau }_{s} : s \geq 0}\\right\\} \\cup \\left\\{ {{\\tau }_{s - } : s \in D}\\right\\} \n\]\n\nwhere \( D \) is the countable set of jump times of \( {\\left( {\\tau }_{s}\\right) }_{s \geq 0} \) . | Proof We know from (9.17) that a.s.\n\n\[ \n\\operatorname{supp}\\left( {{\\mathrm{d}}_{t}{L}_{t}^{0}\\left( B\\right) }\\right) \\subset \\left\\{ {t \geq 0 : {B}_{t} = 0}\\right\\} .\n\]\n\nIt follows that any time \( t \) of the form \( t = {\\tau }_{s} \) or \( t = {\\tau }_{s - } \) must belong to the zero set of ... | Yes |
Proposition 1.2. Suppose \( \left\{ {X}_{n}\right\} \) converges in \( {L}^{p} \) to \( X \in {L}^{p} \) for some \( p \in \lbrack 1,\infty ) \) . Then for any sub- \( \sigma \) -field \( \mathcal{G} \) of \( \mathcal{F},\left\{ {E\left( {{X}_{n} \mid \mathcal{G}}\right) }\right\} \) converges in \( {L}^{p} \) to \( E\... | Proof. This follows by Jensen's inequality for conditional expectations (Theorem 9.1.4 of Chung [11]), which implies\n\n\[ E\left( {\left| E\left( {X}_{n} \mid \mathcal{G}\right) - E\left( X \mid \mathcal{G}\right) \right| }^{p}\right) \leq E\left( {E\left( {{\left| {X}_{n} - X\right| }^{p} \mid \mathcal{G}}\right) }\r... | Yes |
Theorem 1.4. Let \( p \in \lbrack 1,\infty ) \) and \( M \) be a right continuous \( {L}^{p} \) -martingale. Then for each \( t \) and \( c \geq 0 \) ,\n\n(1.3)\n\n\[ \n{c}^{p}P\left( {\mathop{\sup }\limits_{{0 \leq s \leq t}}\left| {M}_{s}\right| \geq c}\right) \leq E\left( {{\left| {M}_{t}\right| }^{p};\mathop{\sup }... | Proof. Inequality (1.3) follows by applying the discrete parameter Theorem 9.4.1 of Chung [11] to the submartingale \( {\left| M\right| }^{p} \) evaluated at finitely many time points, and then taking the limit as these points become dense in \( \left\lbrack {0, t}\right\rbrack \) . In a similar fashion, (1.4) follows ... | Yes |
Theorem 1.6. (Doob’s Stopping Theorem). Let \( p \in \lbrack 1,\infty ) \) and \( M \) be a right continuous \( {L}^{p} \) -bounded martingale. If \( p = 1 \), suppose \( M \) is also uniformly integrable. Let \( {M}_{\infty } \) be as in Theorem 1.5. Suppose \( \Gamma \subset {\mathbb{R}}_{ + } \) and \( \left\{ {{\ta... | Proof. By Theorem 1.5, \( \left\{ {{M}_{t},{\mathcal{F}}_{t}, t \in \left\lbrack {0,\infty }\right\rbrack }\right\} \) is an \( {L}^{p} \) -martingale and (1.6) holds for all \( t \) . It follows from Theorem 4 of Chung [12, p. 30] that for any two optional times \( \eta \leq \sigma ,{M}_{\eta } \) and \( {M}_{\sigma }... | Yes |
Corollary 1.7. Let \( p \in \lbrack 1,\infty ) \) and \( M \) be a right continuous \( {L}^{p} \) -martingale.\n\n(i) If \( \Gamma \subset {\mathbb{R}}_{ + } \) and \( \left\{ {{\tau }_{t}, t \in \Gamma }\right\} \) is an increasing family of optional times such that \( \mathop{\sup }\limits_{{t \in \Gamma }}{\tau }_{t... | Proof. For part (i), since \( {M}_{t \land T} = E\left( {{M}_{T} \mid {\mathcal{F}}_{t}}\right) \) and \( {M}_{T} \in {L}^{p} \), it follows that \( \left\{ {{M}_{t \land T},{\mathcal{F}}_{t}, t \in {\mathbb{R}}_{ + }}\right\} \) satisfies the hypotheses of Theorem 1.6. The conclusion in (i) then follows since \( {\tau... | No |
Proposition 1.8. Let \( p \in \lbrack 1,\infty ) \) and \( M \) be a local \( {L}^{p} \) -martingale with a localizing sequence \( \left\{ {\tau }_{k}\right\} \) . If for each \( t \geq 0 \) we have\n\n(1.8)\n\n\[ \left\{ {{\left| {M}_{t \land {\tau }_{k}}\right| }^{p}, k \in \mathbb{N}}\right\} \;\text{is uniformly in... | Proof. Suppose (1.8) is true. Then for \( t = 0 \), we have \( \left| {M}_{0}\right| \in {L}^{p} \) , and consequently \( \left\{ {{M}_{0},{\mathcal{F}}_{t}, t \in {\mathbb{R}}_{ + }}\right\} \) is an \( {L}^{p} \) -martingale. It follows by addition with (1.7) that \( \left\{ {{M}_{t \land {\tau }_{k}},{\mathcal{F}}_{... | Yes |
Proposition 1.9. Suppose \( M \) is a continuous local martingale and let \( {\tau }_{k} = \inf \left\{ {t > 0 : \left| {{M}_{t} - {M}_{0}}\right| > k}\right\} \) for each \( k \in \mathbb{N}. \) Then, for each \( p \in \lbrack 1,\infty ), \) \( M \) is a local \( {L}^{p} \) -martingale and \( \left\{ {\tau }_{k}\right... | Proof. Let \( \left\{ {\sigma }_{n}\right\} \) be a localizing sequence for \( M \), so that \( \left\{ {{M}_{t \land {\sigma }_{n}} - {M}_{0}}\right. \) , \( \left. {t \in {\mathbb{R}}_{ + }}\right\} \) is a continuous martingale. Then by Corollary 1.7(ii), \[ \left\{ {{M}_{t \land {\tau }_{k} \land {\sigma }_{n}} - {... | Yes |
Lemma 2.1. Stochastic intervals of the form \( \left\lbrack {0,\tau }\right\rbrack \) and \( (\eta ,\tau \rbrack \) are predictable. | Proof. Since \( (\eta ,\tau \rbrack = \left\lbrack {0,\tau }\right\rbrack \smallsetminus \left\lbrack {0,\eta }\right\rbrack \), it suffices to prove that a stochastic interval of the form \( \left\lbrack {0,\tau }\right\rbrack \) is predictable. For this we use a standard approximation of \( \tau \) by a decreasing se... | Yes |
For \( X \in \mathcal{E} \) we have the isometry\n\n\[ E\left\{ {\left( \int XdM\right) }^{2}\right\} = {\int }_{{R}_{ + } \times \Omega }{\left( X\right) }^{2}d{\mu }_{M} \] | Proof. Let \( X \in \mathcal{E} \) be expressed in the form (2.6) where the predictable rectangles \( {R}_{j} \equiv \left( {{s}_{j},{t}_{j}}\right\rbrack \times {F}_{j} \) for \( 1 \leq j \leq n \) and \( \{ 0\} \times {F}_{0k} \) for \( 1 \leq k \leq m \) are disjoint. Then by (2.7) we have\n\n\[ {\left( \int XdM\rig... | Yes |
Lemma 2.4. The set of \( \mathcal{R} \) -simple processes \( \mathcal{E} \) is dense in the Hilbert space \( {\mathcal{L}}^{2} \) . | Proof. Since \( \mathcal{P} \) is generated by the ring \( \mathcal{A} \) and \( {\mu }_{M} \) is \( \sigma \) -finite, then for each \( \varepsilon > 0 \), and \( A \in \mathcal{P} \) such that \( {\mu }_{M}\left( A\right) < \infty \), there is \( {A}_{1} \in \mathcal{A} \) such that \( {\mu }_{M}\left( {{A\Delta }{A}... | Yes |
Theorem 2.5. Let \( X \in {\Lambda }^{2}\left( {\mathcal{P}, M}\right) \) and for each \( t \) let \( {Y}_{t} = \int {1}_{\left\lbrack 0, t\right\rbrack }{XdM} \) . Then \( Y = \left\{ {{Y}_{t}, t \in {\mathbb{R}}_{ + }}\right\} \) is a zero-mean \( {L}^{2} \) -martingale and there is a version of \( Y \) with all path... | Proof. Let \( n \in {IN} \) . Then \( {1}_{\left\lbrack 0, n\right\rbrack }X \in {\mathcal{L}}^{2} \) and by Lemma 2.4 there is a sequence \( \left\{ {{X}^{k}, k \in \mathbb{N}}\right\} \) in \( \mathcal{E} \) which converges to \( {1}_{\left\lbrack 0, n\right\rbrack }X \) in \( {\mathcal{L}}^{2} \) . It follows that f... | Yes |
Theorem 2.6. Suppose the hypotheses of Theorem 2.5 hold and \( M \) has continuous paths. Then there is a version of \( Y \) with continuous paths. | Proof. We first show that for each \( n \in {IN} \) there is a continuous version \( {Z}^{n} \) of \( \left\{ {{Y}_{t}, t \in \left\lbrack {0, n}\right\rbrack }\right\} \) . For \( j < k \) and \( {Y}^{j},{Y}^{k} \) as in the above proof, \( {Y}^{k} - {Y}^{j} \) is a continuous \( {L}^{2} \) -martingale and thus by the... | Yes |
Theorem 2.7. Let \( X \in {\Lambda }^{2}\left( {\mathcal{P}, M}\right) \) and let \( Y \) denote the right continuous stochastic integral process \( \left\{ {{\int }_{\left\lbrack 0, t\right\rbrack }{XdM}, t \in {\mathbb{R}}_{ + }}\right\} \) . Then the following properties hold.\n\n(i) For \( s < t \) in \( {\mathbb{R... | Proof. For \( s < t \) in \( {\mathbb{R}}_{ + } \) and \( Z \in {\mathcal{F}}_{s},{1}_{(s, t\rbrack }Z \in \mathcal{P} \) follows by linearity and a monotone class argument from the fact that \( {1}_{(s, t\rbrack \times G} \in \mathcal{P} \) for \( G \in {\mathcal{F}}_{s} \) . Then, since \( X \in \mathcal{P},{1}_{(s, ... | Yes |
Corollary 2.8. Let \( s < t \) in \( {\mathbb{R}}_{ + }, F \in {\mathcal{F}}_{s} \), and \( \tau \) be an optional time. Then we have a.s.:\n\n\[ \int {1}_{\left\lbrack 0,\tau \right\rbrack }{1}_{(s, t\rbrack \times F}{dM} = {1}_{F}\left( {{M}_{t \land \tau } - {M}_{s \land \tau }}\right) \] | Proof. Let \( X = {1}_{(s, t\rbrack \times F} \) . Then,\n\n\[ \int {1}_{\left\lbrack 0, u\right\rbrack }{XdM} = {1}_{F}\left( {{M}_{t \land u} - {M}_{s \land u}}\right) \]\n\nThe right side of the above equality is right continuous in \( u \) and therefore may be used as the right continuous version \( {\int }_{\left\... | Yes |
Lemma 2.9. Let \( \tau \) and \( \eta \) be optional times such that \( {M}^{\tau } = \left\{ {{M}_{t \land \tau } - }\right. \) \( \left. {{M}_{0}, t \in {\mathbb{R}}_{ + }}\right\} \) and \( {M}^{\eta } = \left\{ {{M}_{t \land \eta } - {M}_{0}, t \in {\mathbb{R}}_{ + }}\right\} \) are right continuous \( {L}^{2} \) -... | Proof. Since the predictable rectangles generate \( \mathcal{P} \), it suffices to prove (2.24) when \( A \) is a predictable rectangle. Clearly both sides of (2.24) are zero when \( A = \{ 0\} \times {F}_{0} \) for some \( {F}_{0} \in {\mathcal{F}}_{0} \) . On the other hand, if \( A = (s, t\rbrack \times F \) for som... | Yes |
Lemma 2.10. Let \( \tau \) and \( \eta \) be optional times such that \( {M}^{\tau } = \left\{ {{M}_{t \land \tau } - }\right. \) \( \left. {{M}_{0}, t \in {\mathbb{R}}_{ + }}\right\} \) and \( {M}^{\eta } = \left\{ {{M}_{t \land \eta } - {M}_{0}, t \in {\mathbb{R}}_{ + }}\right\} \) are right continuous \( {L}^{2} \) ... | Proof. To prove (2.25), it is equivalent to prove the processes \( \left\{ {{Y}_{t \land \tau \land \eta }^{\tau }, t \geq 0}\right\} \) and \( \left\{ {{Y}_{t \land \tau \land \eta }^{\eta }, t \geq 0}\right\} \) are indistinguishable, and since these are right continuous, it suffices to prove\n\n(2.26)\n\n\[ {Y}_{t \... | Yes |
Theorem 2.11. Let \( M \) be a continuous local martingale and \( X \) be a continuous adapted process. Then \( X \in \Lambda \left( {\mathcal{P}, M}\right) \) and \( \left\{ {{\int }_{0}^{t}{XdM}, t \in {\mathbb{R}}_{ + }}\right\} \) is a continuous local martingale. | If \( M \) is a right continuous \( {L}^{2} \) -martingale and \( X \in {\Lambda }^{2}\left( {\mathcal{P}, M}\right) \), the above definition of \( {\int }_{\left\lbrack 0, t\right\rbrack }{XdM} \) is consistent with that given in the previous section because the integrals are unchanged if \( M \) is replaced by \( M -... | Yes |
(ii) If for each \( n \geq 1,{C}_{n} \) is a compact subset of \( {\mathbb{R}}_{ + } \) and \( {C}_{n} \supset {C}_{n + 1} \) for all \( n \), then\n\n\[\n\mathop{\lim }\limits_{{n \rightarrow \infty }} \uparrow \left( {\inf {C}_{n}}\right) = \inf \left( {\mathop{\bigcap }\limits_{{n = 1}}^{\infty }{C}_{n}}\right) .\n\... | Proof. We prove part (ii) only. If there is an \( n \) such that \( {C}_{n} = \varnothing \), then \( \inf {C}_{n} = \infty \) and the result reduces to \( \infty = \infty \) . On the other hand, suppose none of the \( {C}_{n} \) is empty, let \( C = \mathop{\bigcap }\limits_{{n = 1}}^{\infty }{C}_{n} \) and \( {t}_{n}... | Yes |
(i) If \( {A}_{j} \subset {\mathbb{R}}_{ + } \times \Omega \) for \( 1 \leq j \leq k \leq \infty \), then\n\n\[ \mathop{\inf }\limits_{{1 \leq j \leq k}}{D}_{{A}_{j}} = {D}_{{ \cup }_{j = 1}^{k}{A}_{j}} \] | Proof. This is an immediate consequence of Lemma 2.13. | No |
Lemma 2.15. If \( A \in \mathcal{A} \), then \( {D}_{A} \) is optional and has finite range. | Proof. Since \( A \in \mathcal{A} \), there are finitely many disjoint predictable rectangles \( {R}_{1},\ldots ,{R}_{n} \), such that \( A = \mathop{\bigcup }\limits_{{j = 1}}^{n}{R}_{j} \) . By Lemma 2.14(i),\n\n\[ \n{D}_{A} = \mathop{\min }\limits_{{1 \leq j \leq n}}{D}_{{R}_{j}} \n\]\n\nIf \( R = (s, t\rbrack \time... | Yes |
Theorem 3.1. Any left continuous adapted process is predictable. | Proof. Suppose \( X \) is a left continuous adapted process. For each \( n \) let\n\n\[ \n{X}_{s}^{n} = \left\{ \begin{array}{ll} {X}_{0} & \text{ if }s = 0 \\ {X}_{k{2}^{-n}} & \text{ if }s \in \left( {k{2}^{-n},\left( {k + 1}\right) {2}^{-n}}\right\rbrack \text{ for }k \in {\mathbb{N}}_{0}. \end{array}\right. \n\]\n\... | Yes |
Theorem 3.2. \( \sigma \) (l.c.r.l.) \( \subset \sigma \) (c.) | Proof. It suffices to prove that any bounded l.c.r.l. adapted process \( X \) is a pointwise limit on \( {\mathbb{R}}_{ + } \times \Omega \) of continuous adapted processes. Extend the definition of \( {X}_{s} \) to negative values of \( s \) by setting \( {X}_{s} = {X}_{0} \) for \( s < 0 \) . For each fixed \( n \in ... | Yes |
Lemma 3.3. Let \( X \) be a right continuous adapted process and \( \tau \) and \( \eta \) be optional times. Then\n\n\[ \text{(i)}\;{X}_{\tau }{1}_{\{ \tau < \infty \} } \in {\mathcal{F}}_{\tau }\text{,}\]\n\n(ii) For any random variable \( Y \in {\mathcal{F}}_{\tau } \), the process \( Y{1}_{\lbrack \tau ,\eta )} \) ... | Proof. Part (i) is proved in Chung [12; Theorem 10, p. 19 and p. 15]. Since any \( Y \in {\mathcal{F}}_{\tau } \) can be expressed as a pointwise limit of \( {\mathcal{F}}_{\tau } \) -simple functions, it suffices to prove (ii) for \( Y = {1}_{\Lambda } \) where \( \Lambda \in {\mathcal{F}}_{\tau } \) . Assuming \( Y \... | Yes |
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