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Theorem 1.11 Let \( \Gamma \) be a symmetric function of positive type on \( T \times T \) . There exists, on an appropriate probability space \( \left( {\Omega ,\mathcal{F}, P}\right) \), a centered Gaussian process whose covariance function is \( \Gamma \) . | Example Consider the case \( T = \mathbb{R} \) and let \( \mu \) be a finite measure on \( \mathbb{R} \), which is also symmetric (i.e. \( \mu \left( {-A}\right) = \mu \left( A\right) \) ). Then set, for every \( s, t \in \mathbb{R} \) ,\n\n\[ \Gamma \left( {s, t}\right) = \int {\mathrm{e}}^{\mathrm{i}\xi \left( {t - s... | Yes |
Proposition 1.13 Let \( \left( {E,\mathcal{E}}\right) \) be a measurable space, and let \( \mu \) be a \( \sigma \) -finite measure on \( \left( {E,\mathcal{E}}\right) \). There exists, on an appropriate probability space \( \left( {\Omega ,\mathcal{F}, P}\right) \), a Gaussian white noise with intensity \( \mu \). | Proof We rely on elementary Hilbert space theory. Let \( \left( {{f}_{i}, i \in I}\right) \) be a total orthonormal system in the Hilbert space \( {L}^{2}\left( {E,\mathcal{E},\mu }\right) \). For every \( f \in {L}^{2}\left( {E,\mathcal{E},\mu }\right) \), \[ f = \mathop{\sum }\limits_{{i \in I}}{\alpha }_{i}{f}_{i} \... | Yes |
Proposition 2.2 Pre-Brownian motion is a centered Gaussian process with covariance\n\n\[ K\left( {s, t}\right) = \min \{ s, t\} \overset{\text{ (not.) }}{ = }s \land t. \] | Proof By the definition of a Gaussian white noise, the variables \( {B}_{t} \) belong to a common Gaussian space, and therefore \( {\left( {B}_{t}\right) }_{t \geq 0} \) is a Gaussian process. Moreover, for every \( s, t \geq 0 \), \n\n\[ E\left\lbrack {{B}_{s}{B}_{t}}\right\rbrack = E\left\lbrack {G\left( \left\lbrack... | Yes |
Corollary 2.4 Let \( {\left( {B}_{t}\right) }_{t \geq 0} \) be a pre-Brownian motion. Then, for every choice of \( 0 = {t}_{0} < {t}_{1} < \cdots < {t}_{n} \), the law of the vector \( \left( {{B}_{{t}_{1}},{B}_{{t}_{2}},\ldots ,{B}_{{t}_{n}}}\right) \) has density \[ p\left( {{x}_{1},\ldots ,{x}_{n}}\right) = \frac{1}... | Proof The random variables \( {B}_{{t}_{1}},{B}_{{t}_{2}} - {B}_{{t}_{1}},\ldots ,{B}_{{t}_{n}} - {B}_{{t}_{n - 1}} \) are independent with respective distributions \( \mathcal{N}\left( {0,{t}_{1}}\right) ,\mathcal{N}\left( {0,{t}_{2} - {t}_{1}}\right) ,\ldots ,\mathcal{N}\left( {0,{t}_{n} - {t}_{n - 1}}\right) \). Hen... | Yes |
Proposition 2.5 Let \( B \) be a pre-Brownian motion. Then,\n\n(i) \( - B \) is also a pre-Brownian motion (symmetry property);\n\n(ii) for every \( \lambda > 0 \), the process \( {B}_{t}^{\lambda } = \frac{1}{\lambda }{B}_{{\lambda }^{2}t} \) is also a pre-Brownian motion (invariance under scaling);\n\n(iii) for every... | Proof (i) and (ii) are very easy. Let us prove (iii). With the notation of the proof of Proposition 2.3, the \( \sigma \) -field generated by \( {B}^{\left( s\right) } \) is \( \sigma \left( {\widetilde{H}}_{s}\right) \), which is independent of \( \sigma \left( {H}_{s}\right) = \sigma \left( {{B}_{r}, r \leq s}\right)... | No |
Theorem 2.9 (Kolmogorov’s lemma) Let \( X = {\left( {X}_{t}\right) }_{t \in I} \) be a random process indexed by a bounded interval \( I \) of \( \mathbb{R} \), and taking values in a complete metric space \( \left( {E, d}\right) \). Assume that there exist three reals \( q,\varepsilon, C > 0 \) such that, for every \(... | Proof To simplify the presentation, we take \( I = \left\lbrack {0,1}\right\rbrack \), but the proof would be the same for any bounded interval (closed or not). We fix \( \alpha \in \left( {0,\frac{\varepsilon }{q}}\right) \). The assumption of the theorem implies that, for \( a > 0 \) and \( s, t \in I \), \[ P\left( ... | Yes |
Lemma 2.10 Let \( f \) be a mapping defined on \( D \) and with values in the metric space \( \left( {E, d}\right) \) . Assume that there exists a real \( \alpha > 0 \) and a constant \( K < \infty \) such that, for every integer \( n \geq 1 \) and every \( i \in \left\{ {1,2,\ldots ,{2}^{n} - 1}\right\} \) ,\n\n\[ d\l... | Proof of Lemma 2.10 Fix \( s, t \in D \) with \( s < t \) . Let \( p \geq 1 \) be the smallest integer such that \( {2}^{-p} \leq t - s \), and let \( k \geq 0 \) be the smallest integer such that \( k{2}^{-p} \geq s \) . Then, we may write\n\n\[ s = k{2}^{-p} - {\varepsilon }_{1}{2}^{-p - 1} - \ldots - {\varepsilon }_... | Yes |
Corollary 2.11 Let \( B = {\left( {B}_{t}\right) }_{t \geq 0} \) be a pre-Brownian motion. The process \( B \) has a modification whose sample paths are continuous, and even locally Hölder continuous with exponent \( \frac{1}{2} - \delta \) for every \( \delta \in \left( {0,\frac{1}{2}}\right) \) . | Proof If \( s < t \), the random variable \( {B}_{t} - {B}_{s} \) is distributed according to \( \mathcal{N}\left( {0, t - s}\right) \) , and thus \( {B}_{t} - {B}_{s} \) has the same law as \( \sqrt{t - s}U \), where \( U \) is \( \mathcal{N}\left( {0,1}\right) \) . Consequently, for every \( q > 0 \) ,\n\n\[ E\left\l... | Yes |
Theorem 2.13 (Blumenthal’s zero-one law) The \( \sigma \) -field \( {\mathcal{F}}_{0 + } \) is trivial, in the sense that \( P\left( A\right) = 0 \) or 1 for every \( A \in {\mathcal{F}}_{0 + } \) . | Proof Let \( 0 < {t}_{1} < {t}_{2} < \cdots < {t}_{k} \) and let \( g : {\mathbb{R}}^{k} \rightarrow \mathbb{R} \) be a bounded continuous function. Also fix \( A \in {\mathcal{F}}_{0 + } \) . Then, by a continuity argument,\n\n\[ E\left\lbrack {{\mathbf{1}}_{A}g\left( {{B}_{{t}_{1}},\ldots ,{B}_{{t}_{k}}}\right) }\rig... | Yes |
Corollary 2.15 Almost surely, the function \( t \mapsto {B}_{t} \) is not monotone on any nontrivial interval. | Proof Using assertion (i) of Proposition 2.14 and the simple Markov property, we immediately get that a.s. for every rational \( q \in {\mathbb{Q}}_{ + } \), for every \( \varepsilon > 0 \), \[ \mathop{\sup }\limits_{{q \leq t \leq q + \varepsilon }}{B}_{t} > {B}_{q},\;\mathop{\inf }\limits_{{q \leq t \leq q + \varepsi... | Yes |
Proposition 2.16 Let \( 0 = {t}_{0}^{n} < {t}_{1}^{n} < \cdots < {t}_{{p}_{n}}^{n} = t \) be a sequence of subdivisions of \( \left\lbrack {0, t}\right\rbrack \) whose mesh tends to 0 (i.e. \( \mathop{\sup }\limits_{{1 \leq i \leq {p}_{n}}}\left( {{t}_{i}^{n} - {t}_{i - 1}^{n}}\right) \rightarrow 0 \) as \( n \rightarr... | Proof This is an immediate consequence of Proposition 1.14, writing \( {B}_{{t}_{i}^{n}} - {B}_{{t}_{i - 1}^{n}} = \) \( G\left( \left( {{t}_{i - 1}^{n},{t}_{i}^{n}}\right\rbrack \right) \), where \( G \) is the Gaussian white noise associated with \( B \) . | Yes |
Corollary 2.17 Almost surely, the function \( t \mapsto {B}_{t} \) has infinite variation on any non-trivial interval. | Proof From the simple Markov property, it suffices to consider the interval \( \left\lbrack {0, t}\right\rbrack \) for some fixed \( t > 0 \) . We use Proposition 2.16, and note that by extracting a subsequence we may assume that the convergence in this proposition holds a.s. We then observe that\n\n\[ \mathop{\sum }\l... | Yes |
Theorem 2.21 For every \( t > 0 \), set \( {S}_{t} = \mathop{\sup }\limits_{{s \leq t}}{B}_{s} \) . Then, if \( a \geq 0 \) and \( b \in ( - \infty, a\rbrack \) , we have\n\n\[ P\left( {{S}_{t} \geq a,{B}_{t} \leq b}\right) = P\left( {{B}_{t} \geq {2a} - b}\right) . \]\n\nIn particular, \( {S}_{t} \) has the same distr... | Proof We apply the strong Markov property at the stopping time\n\n\[ {T}_{a} = \inf \left\{ {t \geq 0 : {B}_{t} = a}\right\} . \]\n\nWe already saw (Proposition 2.14) that \( {T}_{a} < \infty \) a.s. Then, using the notation of Theorem 2.20, we have\n\n\[ P\left( {{S}_{t} \geq a,{B}_{t} \leq b}\right) = P\left( {{T}_{a... | Yes |
Corollary 2.22 For every \( a > 0,{T}_{a} \) has the same distribution as \( \frac{{a}^{2}}{{B}_{1}^{2}} \) and has density\n\n\[ f\left( t\right) = \frac{a}{\sqrt{{2\pi }{t}^{3}}}\exp \left( {-\frac{{a}^{2}}{2t}}\right) {\mathbf{1}}_{\{ t > 0\} }.\] | Proof Using Theorem 2.21 in the second equality, we have, for every \( t \geq 0 \),\n\n\[ P\left( {{T}_{a} \leq t}\right) = P\left( {{S}_{t} \geq a}\right) = P\left( {\left| {B}_{t}\right| \geq a}\right) = P\left( {{B}_{t}^{2} \geq {a}^{2}}\right) = P\left( {t{B}_{1}^{2} \geq {a}^{2}}\right) = P\left( {\frac{{a}^{2}}{{... | Yes |
Proposition 3.4 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be a random process with values in a metric space \( \left( {E, d}\right) \) (equipped with its Borel \( \sigma \) -field). Suppose that \( X \) is adapted and that the sample paths of \( X \) are right-continuous (i.e. for every \( \omega \in \Omega, t \map... | Proof We treat only the case of right-continuous sample paths, as the other case is similar. Fix \( t > 0 \) . For every \( n \geq 1 \) and \( s \in \left\lbrack {0, t}\right\rbrack \), define a random variable \( {X}_{s}^{n} \) by setting\n\n\[ \n{X}_{s}^{n} = {X}_{{kt}/n}\;\text{ if }s \in \lbrack \left( {k - 1}\righ... | Yes |
Proposition 3.6 Write \( {\mathcal{G}}_{t} = {\mathcal{F}}_{t + } \) for every \( t \in \left\lbrack {0,\infty }\right\rbrack \) .\n\n(i) A random variable \( T : \Omega \rightarrow \left\lbrack {0,\infty }\right\rbrack \) is a stopping time of the filtration \( \left( {\mathcal{G}}_{t}\right) \) if and only if \( \{ T... | Proof\n\n(i) Suppose that \( T \) is a stopping time of the filtration \( \left( {\mathcal{G}}_{t}\right) \) . Then, for every \( t > 0 \),\n\n\[ \{ T < t\} = \mathop{\bigcup }\limits_{{q \in {\mathbb{Q}}_{ + }, q < t}}\{ T \leq q\} \in {\mathcal{F}}_{t} \]\n\nbecause \( \{ T \leq q\} \in {\mathcal{G}}_{q} \subset {\ma... | Yes |
Theorem 3.7 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be a progressive process with values in a measurable space \( \left( {E,\mathcal{E}}\right) \), and let \( T \) be a stopping time. Then the function \( \omega \mapsto {X}_{T}\left( \omega \right) \mathrel{\text{:=}} \) \( {X}_{T\left( \omega \right) }\left( \om... | Proof We use property (j) above. Let \( t \geq 0 \) . The restriction to \( \{ T \leq t\} \) of the function \( \omega \mapsto {X}_{T}\left( \omega \right) \) is the composition of the two mappings\n\n\[ \n\{ T \leq t\} \ni \omega \mapsto \left( {\omega, T\left( \omega \right) \land t}\right) \n\]\n\n\[ \n{\mathcal{F}}... | Yes |
Proposition 3.8 Let \( T \) be a stopping time and let \( S \) be an \( {\mathcal{F}}_{T} \) -measurable random variable with values in \( \left\lbrack {0,\infty }\right\rbrack \), such that \( S \geq T \). Then \( S \) is also a stopping time. | Proof For the first assertion, we write, for every \( t \geq 0 \), \[ \{ S \leq t\} = \{ S \leq t\} \cap \{ T \leq t\} \in {\mathcal{F}}_{t} \] since \( \{ S \leq t\} \) is \( {\mathcal{F}}_{T} \) -measurable. | Yes |
Proposition 3.9 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be an adapted process with values in a metric space \( \left( {E, d}\right) \). (i) Assume that the sample paths of \( X \) are right-continuous, and let \( O \) be an open subset of \( E \). Then \[ {T}_{O} = \inf \left\{ {t \geq 0 : {X}_{t} \in O}\right\} ... | Proof (i) For every \( t > 0 \), \[ \left\{ {{T}_{O} < t}\right\} = \mathop{\bigcup }\limits_{{s \in \lbrack 0, t) \cap \mathbb{Q}}}\left\{ {{X}_{s} \in O}\right\} \in {\mathcal{F}}_{t} \] and we use Proposition 3.6 (i). (ii) For every \( t \geq 0 \), \[ \left\{ {{T}_{F} \leq t}\right\} = \left\{ {\mathop{\inf }\limits... | Yes |
Proposition 3.12 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be an adapted process and let \( f : \mathbb{R} \rightarrow {\mathbb{R}}_{ + } \) be a convex function such that \( E\left\lbrack {f\left( {X}_{t}\right) }\right\rbrack < \infty \) for every \( t \geq 0 \). (i) If \( {\left( {X}_{t}\right) }_{t \geq 0} \) i... | Proof By Jensen’s inequality, we have, for \( s < t \), \[ E\left\lbrack {f\left( {X}_{t}\right) \mid {\mathcal{F}}_{s}}\right\rbrack \geq f\left( {E\left\lbrack {{X}_{t} \mid {\mathcal{F}}_{s}}\right\rbrack }\right) \geq f\left( {X}_{s}\right) . \] In the last inequality, we need the fact that \( f \) is nondecreasing... | Yes |
Proposition 3.13 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be a submartingale or a supermartingale. Then, for every \( t > 0 \) ,\n\n\[ \mathop{\sup }\limits_{{0 \leq s \leq t}}E\left\lbrack \left| {X}_{s}\right| \right\rbrack < \infty \] | Proof It is enough to treat the case where \( {\left( {X}_{t}\right) }_{t \geq 0} \) is a submartingale. Since \( {\left( {X}_{t}\right) }^{ + } \) is also a submartingale, we have for every \( s \in \left\lbrack {0, t}\right\rbrack \) ,\n\n\[ E\left\lbrack {\left( {X}_{s}\right) }^{ + }\right\rbrack \leq E\left\lbrack... | Yes |
Proposition 3.14 Let \( {\left( {M}_{t}\right) }_{t \geq 0} \) be a square integrable martingale (that is, \( {M}_{t} \in {L}^{2} \) for every \( t \geq 0 \) ). Let \( 0 \leq s < t \) and let \( s = {t}_{0} < {t}_{1} < \cdots < {t}_{p} = t \) be a subdivision of the interval \( \left\lbrack {s, t}\right\rbrack \) . The... | Proof For every \( i = 1,\ldots, p \),\n\n\[ E\left\lbrack {{\left( {M}_{{t}_{i}} - {M}_{{t}_{i - 1}}\right) }^{2} \mid {\mathcal{F}}_{s}}\right\rbrack = E\left\lbrack {E\left\lbrack {{\left( {M}_{{t}_{i}} - {M}_{{t}_{i - 1}}\right) }^{2} \mid {\mathcal{F}}_{{t}_{i - 1}}}\right\rbrack \mid {\mathcal{F}}_{s}}\right\rbra... | Yes |
Theorem 3.17 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be a supermartingale, and let \( D \) be a countable dense subset of \( {\mathbb{R}}_{ + } \) . (i) For almost every \( \omega \in \Omega \), the restriction of the function \( s \mapsto {X}_{s}\left( \omega \right) \) to the set \( D \) has a right-limit \[ {X... | Proof (i) Fix \( T \in D \) . By the remark following Proposition 3.15, we have \[ \mathop{\sup }\limits_{{s \in D \cap \left\lbrack {0, T}\right\rbrack }}\left| {X}_{s}\right| < \infty ,\;\text{ a.s. } \] As in the proof of Proposition 3.15, we can choose a sequence \( {\left( {D}_{m}\right) }_{m \geq 1} \) of finite ... | Yes |
Theorem 3.18 Assume that the filtration \( \left( {\mathcal{F}}_{t}\right) \) is right-continuous and complete. Let \( X = {\left( {X}_{t}\right) }_{t \geq 0} \) be a supermartingale, such that the function \( t \rightarrow E\left\lbrack {X}_{t}\right\rbrack \) is right-continuous. Then \( X \) has a modification with ... | Proof Let \( D \) be a countable dense subset of \( {\mathbb{R}}_{ + } \) as in Theorem 3.17. Let \( N \) be the negligible set defined in (3.3). We set, for every \( t \geq 0 \) ,\n\n\[ \n{Y}_{t}\left( \omega \right) = \left\{ \begin{array}{ll} {X}_{t + }\left( \omega \right) & \text{ if }\omega \notin N \\ 0 & \text{... | Yes |
Theorem 3.19 Let \( X \) be a supermartingale with right-continuous sample paths. Assume that the collection \( {\left( {X}_{t}\right) }_{t \geq 0} \) is bounded in \( {L}^{1} \). Then there exists a random variable \( {X}_{\infty } \in {L}^{1} \) such that\n\n\[ \mathop{\lim }\limits_{{t \rightarrow \infty }}{X}_{t} =... | Proof Let \( D \) be a countable dense subset of \( {\mathbb{R}}_{ + } \). From the proof of Theorem 3.17, we have, for every \( T \in D \) and \( a < b \),\n\n\[ E\left\lbrack {{M}_{ab}^{X}\left( {D \cap \left\lbrack {0, T}\right\rbrack }\right) }\right\rbrack \leq \frac{1}{b - a}E\left\lbrack {\left( {X}_{T} - a\righ... | Yes |
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\n\n\[ \n{X}_{S} = E\left\lbrack {{X}_{T} \mid {\mathcal{F}}_{S}}\right\rbrack \n\] | 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 \( t \in (0, T\rbrack \) , \[ {\int }_{0}^{t}\left| {\mathrm{\;d}a\left( s\right) }\right| = \sup \left\{ {\mathop{\sum }\limits_{{i = 1}}^{p}\left| {a\left( {t}_{i}\right) - a\left( {t}_{i - 1}\right) }\right| }\right\} \] where the supremum is over all subdivisions \( 0 = {t}_{0} < {t}_{1} <... | 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 \( \left\lbrack {0, T}\right\rbrack \) , \[ \left| {a\left( {t}_{i}\right) - a\left( {t}_{i - 1}\right) }\right| = \l... | 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\[ \n{\tau }_{n} = \inf \left\{ {t \geq 0 : {\int }_{0}^{t}\left| {\mathrm{\;d}{M}_{s}}\right| \geq n}\right\} \n\] \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 ... | 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 \( t \geq 0 \) ,\n\n\[{\left\langle {M}^{T},{M}^{T}\right\rangle }_{t} = \langle M, M{\rangle }_{t \land T}\] | 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 \[ {D}_{t} = {\frac{\mathrm{d}Q}{\... | Proof If \( A \in {\mathcal{F}}_{t} \), we have \[ Q\left( A\right) = {E}_{Q}\left\lbrack {\mathbf{1}}_{A}\right\rbrack = {E}_{P}\left\lbrack {{\mathbf{1}}_{A}{D}_{\infty }}\right\rbrack = {E}_{P}\left\lbrack {{\mathbf{1}}_{A}{E}_{P}\left\lbrack {{D}_{\infty } \mid {\mathcal{F}}_{t}}\right\rbrack }\right\rbrack \] and,... | 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\[ \n\int {P}^{U}\left( {\mathrm{\;d}{x}_{1}... | 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 \( x \geq 0 \) ,\n\n\[ \n\int {Q}_{t}\left( {x,\mathrm{\;d}y}\right) {\mathrm{e}}^{-{\lambda y}} = {\mathrm{e}}^{-x{\psi }_{t}\left( \lambda \right) }\... | 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\n\n\[ \n\left( {\int {Q}_{t}\left( {x,\mathrm{\;d}z}\right) {\mathrm{e}}^{-{\lambda z}}}\right) \left( {\int {... | 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\[ \n{C}_{t} = {\int }_{0}^{t}{\left| {\Phi }^{\prime }\left( {B}_{s}\right) \right| }^{2}\mathrm{\;d}s \n\]\n\nLet \( z \in \mathbb{C} \) . There exists a complex Brownian motion \( \... | 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) \mat... | 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 |
Theorem 8.5 Under the assumptions of the preceding theorem, there exists, for every \( x \in \mathbb{R} \), a mapping \( {F}_{x} : C\left( {{\mathbb{R}}_{ + },{\mathbb{R}}^{m}}\right) \rightarrow C\left( {{\mathbb{R}}_{ + },{\mathbb{R}}^{d}}\right) \), which is measurable when \( C\left( {{\mathbb{R}}_{ + },{\mathbb{R}... | Proof Again we consider only the case \( d = m = 1 \) . Let \( \mathcal{N} \) be the class of all \( W \) -negligible sets in \( C\left( {{\mathbb{R}}_{ + },\mathbb{R}}\right) \), and, for every \( t \in \left\lbrack {0,\infty }\right\rbrack \), set\n\n\[{\mathcal{G}}_{t} = \sigma \left( {\mathrm{w}\left( s\right) ,0 \... | 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. | Yes |
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. | Yes |
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 |
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