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Theorem 61.3. Let \( d : X \rightarrow X \times X \) be the diagonal map, given by \( d\left( x\right) = \) \( \left( {x, x}\right) \). Then\n\n\[ \n{d}^{ * }\left( {{\alpha }^{p} \times {\beta }^{q}}\right) = {\alpha }^{p} \cup {\beta }^{q} \n\] | Proof. Let \( {z}^{p} \) and \( {z}^{q} \) be representative cocycles for \( {\alpha }^{p} \) and \( {\beta }^{q} \), respectively. If \( T : {\Delta }_{p + q} \rightarrow X \) is a singular simplex, we compute\n\n\[ \n\left\langle {{d}^{\# }\left( {{z}^{p} \times {z}^{q}}\right), T}\right\rangle = \left\langle {{z}^{p... | Yes |
Corollary 61.4. Cup products are anticommutative. That is,\n\n\[ \n{\alpha }^{p} \cup {\beta }^{q} = {\left( -1\right) }^{pq}{\beta }^{q} \cup {\alpha }^{p}.\n\] | Proof. Let \( \lambda : X \times X \rightarrow X \times X \) be the map that reverses coordinates. Then \( \lambda \circ d = d \) . We compute\n\n\[ \n{\alpha }^{p} \cup {\beta }^{q} = {d}^{ * }\left( {{\alpha }^{p} \times {\beta }^{q}}\right) = {\left( \lambda \circ d\right) }^{ * }\left( {{\alpha }^{p} \times {\beta ... | Yes |
Theorem 61.5. In the cohomology ring \( {H}^{ * }\left( {X \times Y;R}\right) \), the following formula holds:\n\n\[ \cdot \left( {\alpha \times \beta }\right) \cup \left( {{\alpha }^{\prime } \times {\beta }^{\prime }}\right) = {\left( -1\right) }^{\left( {\dim \beta }\right) \left( {\dim {\alpha }^{\prime }}\right) }... | \[ \text{Proof. Let}{\pi }_{1} : X \times Y \rightarrow X\text{and}{\pi }_{2} : X \times Y \rightarrow Y\text{be the projection maps.} \]\n\nStep 1. We first prove the formula when \( \beta \) and \( {\alpha }^{\prime } \) are the unity elements of their respective cohomology rings. Let \( {z}^{p} \) be a cocycle repre... | Yes |
Consider the cohomology ring of \( {S}^{n} \times {S}^{m} \), where \( n, m \geq 1 \) . Let \( {\alpha }^{n} \in \) \( {H}^{n}\left( {S}^{n}\right) \) and \( {\beta }^{m} \in {H}^{m}\left( {S}^{m}\right) \) be generators. Then \( {H}^{ * }\left( {{S}^{n} \times {S}^{m}}\right) \) is free of rank 4, with basis \( 1 \tim... | In the special case \( n = m = 1 \), this space is the torus, and this is the cohomology ring we have already computed. Let us picture these cohomology classes, Represent the torus by the usual diagram, as in Figure 61.1. If \( X \) denotes the subspace \( {abca} \), then a generator \( \alpha \) for its 1-dimensional ... | Yes |
Lemma 62.2. Suppose \( K * L \) exists, and suppose \( K \) is locally finite. Then the map\n\n\[ \pi : \left| K\right| \times \left| L\right| \times I \rightarrow \left| {K * L}\right| \]\n\ndefined by\n\n\[ \pi \left( {x, y, t}\right) = \left( {1 - t}\right) x + {ty} \]\n\nis a quotient map. For each \( x \in \left| ... | Proof. The topology of \( \left| K\right| \times \left| L\right| \times I \) is coherent with the subspaces \( \sigma \times \tau \times I \), for \( \sigma \in K \) and \( \tau \in L \). To prove this fact, we apply the results of \( §{20} \). The topology of \( \left| L\right| \times I \) is coherent with the subspac... | Yes |
Corollary 62.3. Suppose \( K * L \) and \( M * N \) are defined, where \( K \) is locally finite. If \( \left| K\right| \approx \left| M\right| \) and \( \left| L\right| \approx \left| N\right| \), then \( \left| {K * L}\right| \approx \left| {M * N}\right| \). | Proof. Since \( \left| K\right| \) is locally compact, so is \( \left| M\right| \) ; therefore, \( M \) is locally finite. If \( h : \left| K\right| \rightarrow \left| M\right| \) and \( k : \left| L\right| \rightarrow \left| N\right| \) are homeomorphisms, then \( h \times k \times {i}_{l} \) induces, via the quotient... | Yes |
Lemma 62.4. Let \( J, K, L \) be complexes. Assume \( J * K \) and \( \left( {J * K}\right) * L \) exist. Then \( J * K = K * J \) and \( \left( {J * K}\right) * L = J * \left( {K * L}\right) \) . | Proof. The symmetry of the definition shows that \( J * K = K * J \) . Similarly, if \( \left( {J * K}\right) * L \) is defined, then for \( {v}_{0}\ldots {v}_{m} \in J \) and \( {w}_{0}\ldots {w}_{n} \in K \) and \( {x}_{0}\ldots {x}_{p} \in L \), the points\n\n\[ \n{v}_{0},\ldots ,{v}_{m},{w}_{0},\ldots ,{w}_{n},{x}_... | Yes |
Theorem 62.5. Assume \( K * L \) exists. If \( \left| L\right| \approx {S}^{n - 1} \), then for all \( i \) ,\n\n\[{\widetilde{H}}_{i + n}\left( {K * L}\right) \simeq {\widetilde{H}}_{i}\left( K\right)\] | Proof. In the case \( n = 1,\left| L\right| \) consists of two points and the complex \( K * L \) is just the suspension of \( K \) . The existence of the isomorphism in question is a consequence of the Mayer-Vietoris sequence. (See Theorem 25.4.)\n\nIn general; we suppose the theorem true in dimension \( n \), and pro... | Yes |
Lemma 62.6. Let \( K \) be a complex. Let \( s \) be a simplex of \( K \) . Then\n\n\[ \overline{\mathrm{{St}}}s = s * \operatorname{Lk}s \]\n\n\[ \overline{\mathrm{{St}}}s - \mathrm{{St}}s = \mathrm{{Bd}}s * \mathrm{\\;{Lk}}s. \] | Proof. The first equation is immediate from the definitions. St \( s \) is the union of all simplices of \( K \) of the form \( s * t \), and Lk \( s \) is the union of all the faces \( t \) of such simplices.\n\nOn the other hand, if a simplex of \( K \) lies in \( \overline{\mathrm{{St}}}s \) but not in \( \mathrm{{S... | Yes |
Lemma 63.1. Let \( s \) be a \( k \) -simplex of the complex \( K \) . Let \( \widehat{s} \) be its barycenter. Then\n\n\[ \n{H}_{i}\left( {\left| K\right| ,\left| K\right| - \widehat{s}}\right) \simeq \left\{ \begin{array}{ll} {\widetilde{H}}_{i - k - 1}\left( {\operatorname{Lk}s}\right) & \text{ if }\operatorname{Lk}... | Proof. If \( \operatorname{Lk}s = \varnothing \), then \( s \) is a face of no other simplex of \( K \) . Hence in \( s \) is open in \( \left| K\right| \) . It follows that\n\n\[ \n{H}_{i}\left( {\left| K\right| ,\left| K\right| - \widehat{s}}\right) \simeq {H}_{i}\left( {s, s - \widehat{s}}\right) \simeq {H}_{i}\left... | Yes |
Theorem 63.2. Let \( \left( {X, A}\right) \) be a triangulated relative homology \( n \) -manifold. Let \( s \) be a \( k \) -simplex of \( X \) not in \( A \) . If \( \mathrm{{Lk}}s \) is empty, then \( k = n \) . If \( \mathrm{{Lk}}s \) is nonempty, it has the homology of an \( n - k - 1 \) sphere, where \( n - k - 1... | Proof. Since \( s \) is not in \( A \), its barycenter \( \widehat{s} \) lies in \( X - A \) . Therefore, the local homology of \( X \) at \( \widehat{s} \) is infinite cyclic in dimension \( n \), and vanishes otherwise.\n\nIf Lk \( s \) is empty, then by the preceding lemma, \( {H}_{n}\left( {s,\operatorname{Bd}s}\ri... | Yes |
Corollary 63.3. Let \( \left( {X, A}\right) \) be a triangulated relative homology \( n \) -manifold.\n\n(a) The closure of \( X - A \) equals a union of n-simplices.\n\n(b) Every \( n - 1 \) simplex \( s \) of \( X \) not in \( A \) is a face of precisely two \( n \) - simplices of \( X \). | Proof. (a). Let \( s \) be a \( k \) -simplex of \( X \) not in \( A \) . The preceding theorem shows that \( k \leq n \) . If \( k < n \), then Lk \( s \) is a homology \( n - k - 1 \) sphere, so it contains an \( n - k - 1 \) simplex \( t \) . Then \( s \) is a face of the \( n \) -simplex \( s * t \) .\n\n(b) If \( ... | Yes |
If \( M \) is a triangulated topological \( n \) -manifold, then the link of each \( k \) -simplex of \( M \) is a topological \( n - k - 1 \) sphere. | Certainly this result is true for our usual triangulations of 2-manifolds, for instance; the links of 1-simplices are 0-spheres and the links of vertices are 1-spheres. This conjecture is one of long standing, and has only recently been answered, in the negative. R. D. Edwards has found a counterexample in dimension 5.... | No |
There do exist homology \( n \) -manifolds that are not topological manifolds. | We outline the construction of one such, but the proof uses tools we have not studied.\n\nThe basic fact one needs is that there is a compact triangulated 3-manifold \( M \) that is a homology 3-sphere, but whose fundamental group \( {\pi }_{1}\left( M\right) \) does not vanish. The construction can be found in \( \lef... | No |
Let \( X \) be the 2-dimensional complex pictured in Figure 64.1. The complex sd \( X \) is indicated by dotted lines. The block dual to any 2-simplex \( \sigma \) consists of its barycenter \( \widehat{\sigma } \) alone. The block \( D\left( {ab}\right) \) dual to the 1-simplex \( s = {ab} \) consists of its barycente... | This situation will hold in general, as we shall see. | No |
Theorem 64.1. Let \( X \) be a locally finite simplicial complex that consists entirely of n-simplices and their faces. Let \( \sigma \) be a \( k \) -simplex of \( X \) . Then:\n\n(a) The dual blocks are disjoint and their union is \( \left| X\right| \) .\n\n(b) \( \bar{D}\left( \sigma \right) \) is the polytope of a ... | Proof. (a) The open simplices of \( \operatorname{sd}X \) are disjoint, and each one lies in precisely one block \( D\left( \sigma \right) \) -namely, the one such that \( \widehat{\sigma } \) is its final vertex.\n\n(b) Given \( \sigma \) of dimension \( k \), let \( {\sigma }^{\prime } \) be an \( n \) -simplex of \(... | Yes |
The group \( {H}_{l}\left( {{X}_{p},{X}_{p - 1}}\right) \) vanishes for \( l \neq p \) and is a free abelian group for \( i = p \) . A basis when \( i = p \) is obtained by choosing generators for the groups \( {H}_{p}\left( {\bar{D}\left( \sigma \right) ,\dot{D}\left( \sigma \right) }\right) \), as \( D\left( \sigma \... | The proof of (a) follows the pattern of Lemma 39.2, but is easier. Because \( \bar{D}\left( \sigma \right) \) is a cone with vertex \( \widehat{\sigma } \), its base \( \dot{D}\left( \sigma \right) \) is a deformation retract of \( \bar{D}\left( \sigma \right) - \widehat{\sigma } \), for each \( \sigma \) . These defor... | Yes |
Corollary 65.2. Let \( X \) be a compact triangulated homology \( n \) -manifold. If \( X \) is connected, then for any two n-simplices \( \sigma ,{\sigma }^{\prime } \) of \( X \), there is a sequence\n\n\[ \sigma = {\sigma }_{0},{\sigma }_{1},\ldots ,{\sigma }_{k} = {\sigma }^{\prime } \]\n\nof \( n \) -simplices of ... | Proof. Let us define \( \sigma \sim {\sigma }^{\prime } \) if there is such a sequence connecting them. This relation is clearly an equivalence relation. The sum \( \sum {\sigma }_{i} \) of the \( n \) -simplices in any one equivalence class is a cycle if its coefficient group is taken to be \( \mathbf{Z}/2 \) . For, g... | Yes |
Corollary 65.3. Let \( X \) be a compact triangulated homology \( n \) -manifold. If \( X \) is connected, then \( {H}_{n}\left( X\right) \simeq \mathbf{Z} \) for \( X \) orientable and \( {H}_{n}\left( X\right) = 0 \) for \( X \) nonorientable. | Proof. If \( X \) is orientable, then \( {H}_{n}\left( X\right) \cong {H}^{0}\left( X\right) \cong \mathbf{Z} \) . Conversely, we show that if \( {H}_{n}\left( X\right) \neq 0 \), then \( X \) is orientable. Orient the \( n \) -simplices of \( X \) arbitrarily. Suppose \( z \) is a non-trivial \( n \) -cycle of \( X \)... | Yes |
Theorem 66.1. Cap product is bilinear, and is natural with respect to continuous maps. It satisfies the boundary formula\n\n\[ \n\partial \left( {{d}^{p} \cap {c}_{p + q}}\right) = {\left( -1\right) }^{q}\left( {\delta {d}^{p} \cap {c}_{p + q}}\right) + {d}^{p} \cap \partial {c}_{p + q}, \n\]\n\nand is related to cup p... | Proof. Bilinearity is easy to check. Naturality is harder to formulate than to prove. Given a continuous map \( f : X \rightarrow Y \), consider the following diagram:\n\n\[ \n\begin{array}{l} {S}^{p}\left( X\right) \otimes {S}_{p + q}\left( X\right) \overset{ \curvearrowleft }{ \rightarrow }{S}_{q}\left( X\right) \\ {... | No |
Theorem 66.3. Let \( {\epsilon }_{ * } : {H}_{0}\left( {X;G}\right) \rightarrow G \) be the homomorphism induced by the augmentation map \( \epsilon \), which is an isomorphism if \( X \) is path connected. Then the Kronecker index equals the composite\n\n\[ \n{H}^{p}\left( {X;G}\right) \otimes {H}_{p}\left( X\right) \... | Proof. The augmentation map \( \epsilon : {C}_{0} \rightarrow \mathbf{Z} \) gives rise to a homomorphism \( {C}_{0} \otimes G \rightarrow \mathbf{Z} \otimes G \cong G \) that carries boundaries to zero. Hence it induces a homology homomorphism \( {\epsilon }_{ * } \) . We compute directly as follows: If \( T : {\Delta ... | No |
Theorem 66.4. The simplicial cap product satisfies the boundary formula and cup product formula of Theorem 66.1. It induces a homomorphism in simplicial theory\n\n\[ \n{H}^{p}\left( {K;R}\right) \otimes {H}_{p + q}\left( {K;R}\right) \rightarrow {H}_{q}\left( {K;R}\right) \n\]\n\nthat corresponds to the singular cap pr... | Proof. Let \( \eta : {C}_{p}\left( K\right) \rightarrow {S}_{p}\left( \left| K\right| \right) \) be the chain map defined in \( §{34} \) that induces an isomorphism of simplicial with singular theory. Since \( \eta \) is a monomorphism onto a direct summand, its dual \( \widetilde{\eta } \) is surjective.\n\nWe first s... | Yes |
Theorem 68.2. Let \( X \) be a compact, connected, triangulable homology \( n \) -manifold. Let \( F \) be a field; assume \( F \) equals \( \mathbf{Z}/2 \) if \( X \) is non-orientable. Let \( \Lambda \) generate; \( {H}^{n}\left( {X;F}\right) \) . There are (vector space) bases \( {\alpha }_{1},\ldots ,{\alpha }_{m} ... | Proof. In the orientable case, choose \( \Gamma \) so that the isomorphism\n\n\[ \n{H}^{n}\left( {X;F}\right) \xrightarrow[]{\cap \Gamma }{H}_{0}\left( {X;F}\right) \xrightarrow[]{{\epsilon }_{ * }}F, \n\]\ncarries \( \Lambda \) to \( 1 \in F \) . In the non-orientable case, \( {\Gamma }_{\left( 2\right) } \) is unique... | Yes |
Theorem 68.3. If \( u \) is the non-zero element of \( {H}^{1}\left( {{P}^{n};\mathbf{Z}/2}\right) \), then \( {u}^{k} \) is the non-zero element of \( {H}^{k}\left( {{P}^{n};\mathbf{Z}/2}\right) \), for \( k = 2,\ldots, n \) . Thus \( {H}^{ * }\left( {{P}^{n};\mathbf{Z}/2}\right) \) is a truncated polynomial algebra o... | Proof. We proceed by induction, beginning with \( n = 2 \) . The vector space \( {H}^{1}\left( {{P}^{2};\mathbf{Z}/2}\right) \) has dimension 1 . If \( u \) is its non-zero element, then by Theorem 68.2, \( {u}^{2} = u \cup u \) must be non-zero.\n\nSuppose the theorem is true for dimension \( n - 1 \) . The inclusion ... | Yes |
Corollary 68.4. \( {H}^{ * }\left( {{P}^{\infty };\mathbf{Z}/2}\right) \) is a polynomial algebra over \( \mathbf{Z}/2 \) with a single one-dimensional generator. | Proof. This corollary follows from the fact that inclusion \( j : {P}^{n} \rightarrow {P}^{\infty } \) induces a cohomology isomorphism (over \( \mathbf{Z}/2 \) ) in dimensions less than or equal to \( n \), and the fact that \( {j}^{ * } \) preserves cup products. | No |
Theorem 68.5. If \( f : {S}^{n} \rightarrow {S}^{m} \) is continuous and antipode-preserving, then \( n \leq m \) . | Proof. Let \( {a}_{n} = \left( {1,0,\ldots ,0}\right) \in {S}^{n} \) ; we call it the \ | No |
Theorem 68.6 (The Borsuk-Ulam theorem). If \( h : {S}^{n} \rightarrow {\mathbf{R}}^{n} \) is a continuous map. then \( h\left( x\right) = h\left( {-x}\right) \) for at least one \( x \in {S}^{a} \) . | Proof. If \( h\left( x\right) \neq h\left( {-x}\right) \) for all \( x \in {S}^{n} \), then the function \( f : {S}^{n} \rightarrow {S}^{n - 1} \) defined by:\n\n\[ f\left( x\right) = \frac{h\left( x\right) - h\left( {-x}\right) }{\parallel h\left( x\right) - h\left( {-x}\right) \parallel } \]\n\nis continuous and anti... | Yes |
Theorem 68.7 (The baby ham sandwich theorem). Let \( {A}_{1} \) and \( {A}_{2} \) be two bounded measurable subsets of \( {\mathbf{R}}^{2} \). There is a line in \( {\mathbf{R}}^{2} \) that bisects both \( {A}_{1} \) and \( {A}_{2} \). | Proof. Suppose \( {A}_{1} \) and \( {A}_{2} \) lie in the plane \( {\mathbf{R}}^{2} \times 1 \) in \( {\mathbf{R}}^{3} \). For each unit vector \( \overrightarrow{u} \) in \( {\mathbf{R}}^{3} \), consider the plane \( P\left( \overrightarrow{u}\right) \) through the origin perpendicular to \( \overrightarrow{u} \). Let... | Yes |
Lemma 69.2. Let \( X \) be orientable. Then the Bockstein homomorphisms commute with the Poincaré duality isomorphism, up to sign. | Proof. Let \( \phi : {C}^{k}\left( X\right) \rightarrow {D}_{n - k}\left( X\right) \) be the Poincaré duality isomorphism of Theorem 65.1. Consider the commutative diagram\n\n\[\n\begin{array}{l} 0 \rightarrow {C}^{k}\left( X\right) \otimes G \rightarrow {C}^{k}\left( X\right) \otimes {G}^{\prime } \rightarrow {C}^{k}\... | Yes |
Theorem 69.3. Let \( X = L\left( {n, k}\right) \) . Let \( \Lambda \) generate the infinite cyclic group \( {H}^{3}\left( X\right) \) ; let \( {\Lambda }_{\left( n\right) } \) denote its image in \( {H}^{3}\left( {X;\mathbf{Z}/n}\right) \) under the coefficient homomorphism induced by \( \mathbf{Z} \rightarrow \mathbf{... | Proof. Let \n\n\[ \n{\beta }_{ * } : {H}_{2}\left( {X;\mathbf{Z}/n}\right) \rightarrow {H}_{1}\left( {X;\mathbf{Z}/n}\right) \n\] \n\nbe the homology Bockstein associated with the given coefficient sequence. Since the Bockstein homomorphisms commute with Poincaré duality up to sign, it suffices to show that in homology... | Yes |
Theorem 69.4. For \( X = L\\left( {n, k}\\right) \) and \( Y = L\\left( {n, l}\\right) \) to have the same homotopy type, it is necessary that\n\n\[ k \\equiv \\pm {a}^{2}l\\;\\left( {\\;\\operatorname{mod}\\,n}\\right) \]\n\nfor some a. | Proof. Let \( f : X \\rightarrow Y \) be a homotopy equivalence. Let \( \\Lambda \) generate \( {H}^{3}\\left( X\\right) \) and let \( {\\Lambda }^{\\prime } \) generate \( {H}^{3}\\left( Y\\right) \) . Let \( u \\in {H}^{1}\\left( {X;\\mathbf{Z}/n}\\right) \) and \( v \\in {H}^{1}\\left( {Y;\\mathbf{Z}/n}\\right) \) b... | Yes |
Lemma 70.1. Let \( A \) be a full subcomplex of the finite simplicial complex \( X \) . Let \( C \) consist of all simplices of \( X \) that are disjoint from \( \left| A\right| \) . Then \( \left| A\right| \) is a deformation retract of \( \left| X\right| - \left| C\right| \), and \( \left| C\right| \) is a deformatio... | Proof,’ First, we note that each vertex of \( X \) belongs to either \( A \) or \( C \) . Second, we note that \( C \) is a full subcomplex of \( X \) : If the vertices of \( \sigma \) are in \( C \), then the simplex \( \dot{\sigma } \) cannot intersect \( \left| A\right| \), so it is in \( C \) . Third, we note that ... | Yes |
Corollary 70.3. Let \( \\left( {X, A}\\right) \) be a compact triangulated relative homology \( n \) -manifold, with \( \\left| X\\right| - \\left| A\\right| \) connected. If \( \\sigma \) and \( {\\sigma }^{\\prime } \) are two n-simplices of \( X \) not in \( A \), there is a sequence\n\n\[ \n\\sigma = {\\sigma }_{0}... | Proof. We define two \( n \) -simplices of \( X \) not in \( A \) to be equivalent if there is such a sequence joining them. The sum of the members of any one equivalence class is a relative cycle of \( \\left( {X, A}\\right) \) with \( \\mathbf{Z}/2 \) coefficients. We conclude that there is only one equivalence class... | Yes |
Corollary 70.4. Let \( \left( {X, A}\right) \) be a compact triangulated relative homology \( n \) -manifold. Assume \( \left| X\right| - \left| A\right| \) is connected. Then \( {H}_{n}\left( {X, A}\right) \cong \mathbf{Z} \) if \( \left( {X, A}\right) \) is orientable and \( {H}_{n}\left( {X, A}\right) = 0 \) if \( \... | Proof. The proof follows the pattern of Corollary 65.3. | No |
Theorem 70.6. Let \( \\left( {X, A}\\right) \) be a compact triangulable relative homology \( n \) - manifold. If \( \\left( {X, A}\\right) \) is orientable, let \( \\Gamma \) denote an orientation class for \( \\left( {X, A}\\right) \) ; then the following diagram commutes, up to a sign depending on \( k \) and \( n \... | Proof. Choose a triangulation of \( \\left( {X, A}\\right) \) . Then consider the diagram  Here \( \\eta \) is the chain map carrying simplicial chains to singular chains; the unlabelled maps are induced by inclusion... | Yes |
Theorem 70.7 (Poincaré-Lefschetz duality). Let \( M \) be a compact triangu-lable \( n \) -manifold with boundary, such that \( \mathrm{{Bd}}M \) has a product neighborhood in \( M \) . \( I{f}^{\prime }\left( {M,\mathrm{{Bd}}M}\right) \) is orientable, let \( \Gamma \in {H}_{n}\left( {M,\mathrm{{Bd}}M}\right) \) be an... | Proof. We first prove that inclusion induces an isomorphism \[ {j}_{ * } : {H}_{n - k}\left( {M - \operatorname{Bd}M;G}\right) \rightarrow {H}_{n - k}\left( {M;G}\right) . \] Let \( h : \operatorname{Bd}M \times \lbrack 0,1) \rightarrow M \) be a product neighborhood of the boundary. Let \[ N = M - h\left( {\operatorna... | Yes |
Corollary 71.2 (The polyhedral Jordan curve theorem). Let \( n > 0 \) . If \( A \) is a subset of \( {S}^{n} \) that is homeomorphic to \( {S}^{n - 1} \), and if \( \left( {{S}^{n}, A}\right) \) is triangulable, then \( {S}^{u} - A \) has precisely two path components, of which \( A \) is the common boundary. | Proof. \( A \) is a proper subset of \( {S}^{n} \), since \( {S}^{n - 1} \) is not homeomorphic to \( {S}^{n} \) . The fact that \( {S}^{n} - A \) has exactly two path components follows from the fact that \( {\widetilde{H}}^{n - 1}\left( A\right) \) is infinite cyclic because \( A \approx {S}^{n - 1} \), and the fact ... | Yes |
Lemma 72.2. Let \( D \) be a polyhedron in the compact triangulable space \( X \) . Then there are arbitrarily small neighborhoods \( U \) of \( D \) such that:\n\n(1) \( \bar{U} \) and \( {X}_{t} - U \) are polyhedra in \( X \) .\n\n(2) The following inclusions are homotopy equivalences:\n\n\[ D \rightarrow U \rightar... | Proof. It suffices to show that if \( D \) is the polytope of a full subcomplex of \( X \), then \( U = \operatorname{St}\left( {D,\operatorname{sd}X}\right) \) satisfies the requirements of the lemma.\n\nThe preceding lemma shows that the inclusions \( D \rightarrow U \rightarrow \bar{U} \) are homotopy equivalences. ... | Yes |
Theorem 72.3 (Lefschetz duality). Let \( \\left( {X, A}\\right) \) be a compact triangulable relative homology \( n \) -manifold. There is a function assigning to each polyhedron \( D \) in \( \\left( {X, A}\\right) \) that contains \( A \), an isomorphism\n\n\[ \n{\\lambda }_{D} : {H}^{k}\\left( {X, D;G}\\right) \\rig... | Proof. Let \( \\Gamma \) be an orientation class for \( \\left( {X, A}\\right) \) in the orientable case; let it denote an orientation class over \( \\mathbf{Z}/2 \) otherwise.\n\nStep 1. If \( U \) is any neighborhood of \( A \) such that \( \\widetilde{U} \) is a polyhedron in \( \\left( {X, A}\\right) \), let \( {X}... | No |
Corollary 72.4 (Alexander duality). Let \( n \) be fixed. There is a function assigning to each proper nonempty polyhedron \( A \) in \( {S}^{n} \), an isomorphism\n\n\[ \n{\alpha }_{A} : {\widetilde{H}}^{k}\left( A\right) \rightarrow {\widetilde{H}}_{n - k - 1}\left( {{S}^{n} - A}\right) .\n\]\n\nThis assignment is na... | Proof. The case \( k < n - 1 \) follows by noting that \( {\alpha }_{A} \) may be defined as the composite of the isomorphisms\n\n\[ \n{\widetilde{H}}^{k}\left( A\right) \overset{{\delta }^{ * }}{ \rightarrow }{H}^{k + 1}\left( {{S}^{n}, A}\right) \overset{{\lambda }_{A}}{ \rightarrow }{H}_{n - k - 1}\left( {{S}^{n} - ... | Yes |
Consider the family whose elements are open coverings \( \mathcal{A} \) of a topological space \( X \) . We make this family into a directed set by declaring \( \mathcal{A} < \mathcal{B} \) if \( \mathcal{B} \) is a refinement of \( \mathcal{A} \) . This means that for each element \( B \) of \( \mathcal{B} \), there i... | This means that for each element \( B \) of \( \mathcal{B} \), there is at least one element \( A \) of \( \mathcal{A} \) containing it. (1) and (2) are immediate; to check (3), we note that given open coverings \( \mathcal{A} \) and \( \mathcal{B} \) of \( X \), the collection\n\n\[ \mathcal{D} = \{ A \cap B \mid A \i... | No |
Let \( \left\{ {{G}_{\alpha },{f}_{\alpha \beta }}\right\} \) be a direct system of abelian groups and homomorphisms; let \( H \) be an abelian group. If for each \( \alpha \), one has a homomorphism \( {\phi }_{\alpha } : {G}_{\alpha } \rightarrow H \), and if \( {\phi }_{\beta } \circ {f}_{\alpha \beta } = {\phi }_{\... | \[ \underline{\Phi } : \mathop{\lim }\limits_{ \rightarrow }{G}_{\alpha } \rightarrow H \] It is a special case of the preceding construction, in which the second direct system consists of the single group \( H \) . | Yes |
Example 4. Suppose one has a sequence of abelian groups and homomorphisms\n\n\[ \n{G}_{1}\overset{{f}_{1}}{ \rightarrow }{G}_{2}\overset{{f}_{2}}{ \rightarrow }{G}_{3} \rightarrow \cdots \n\]\n\nIt becomes a direct system with index set \( J = {\mathbf{Z}}_{ + } \) if we define\n\n\[ \n{f}_{mn} = {f}_{n - 1} \circ {f}_... | Define\n\n\[ \n{\phi }_{n} : {G}_{n} \rightarrow H \n\]\n\nby the equation \( {\phi }_{n}\left( m\right) = m/{2}^{n} \) . Then one checks that \( {\phi }_{n} \circ {f}_{n - 1} = {\phi }_{n - 1} \) . It is easy to check that \( \Phi \) is both injective and surjective. | No |
Lemma 73.1. Suppose one is given a direct system \( \\left\\{ {{G}_{\\alpha },{f}_{\\alpha \\beta }}\\right\\} \) of abelian groups and homomorphisms, indexed by the directed set \( J \) . If \( {J}_{0} \) is cofinal in \( J \) , then \( {J}_{0} \) is a directed set, and inclusion induces an isomorphism\n\n\[ \n{\\unde... | Proof. The axioms for a directed set are easy to check. Given \( \\alpha ,\\beta \) in \( {J}_{0} \) , they have an upper bound in \( J \), and hence an upper bound in \( {J}_{\\mathrm{o}} \) . Let \( \\phi : {J}_{\\mathrm{o}} \\rightarrow J \) be the inclusion map, and for each \( \\alpha \\in {J}_{0} \), let \( {\\ph... | Yes |
Theorem 73.2. Let \( K \) be a simplicial complex. Then\n\n\[{\check{H}}^{k}\left( {\left| K\right| ;G}\right) \simeq {H}^{k}\left( {K;G}\right) .\]\n\nThe same is true for reduced cohomology. | Proof. For any complex \( K \), let \( \mathcal{A}\left( K\right) \) be the covering of \( \left| K\right| \) by the open stars of its vertices. The vertex correspondence \( {f}_{K} \) that assigns to the vertex \( v \) of \( K \), the vertex St \( v \) of \( N\left( {\mathcal{A}\left( K\right) }\right) \), defines an ... | Yes |
Corollary 74.2. Let \( n > 1 \) . Let \( M \) be a compact connected triangulable \( n - 1 \) manifold; suppose \( h : M \rightarrow {S}^{n} \) is an imbedding. Then \( M \) is orientable, and \( {S}^{n} - h\left( M\right) \) has precisely two path components, of which \( h\left( M\right) \) is the common boundary. | Proof. By Alexander duality, we know that \( {\check{H}}^{n - 1}\left( M\right) \cong \) \( {\widetilde{H}}_{0}\left( {{S}^{n} - h\left( M\right) }\right) \) . Since \( M \) is triangulable, its Čech and simplicial cohomology groups are isomorphic. If \( M \) were non-orientable, we would have \( {H}^{n - 1}\left( M\ri... | Yes |
Proposition 1.2.3. Let \( {\left( {u}_{k}\right) }_{k \in \mathbb{N}} \in {L}^{2}\left( {\Omega \times \mathbb{N}}\right) \) be a predictable square-integrable process. We have\n\n\[ \mathbb{E}\left\lbrack {J\left( u\right) \mid {\mathcal{F}}_{k}}\right\rbrack = J\left( {u{\mathbf{1}}_{\left\lbrack 0, k\right\rbrack }}... | Proof. In case \( {\left( {u}_{k}\right) }_{k \in \mathbb{N}} \) has finite support in \( \mathbb{N} \) it suffices to note that\n\n\[ \mathbb{E}\left\lbrack {J\left( u\right) \mid {\mathcal{F}}_{k}}\right\rbrack = \mathbb{E}\left\lbrack {\mathop{\sum }\limits_{{i = 0}}^{k}{u}_{i}{Y}_{i} \mid {\mathcal{F}}_{k}}\right\r... | Yes |
Corollary 1.2.4. The indefinite stochastic integral \( {\left( J\left( u{\mathbf{1}}_{\left\lbrack 0, k\right\rbrack }\right) \right) }_{k \in \mathbb{N}} \) is a discrete time martingale with respect to \( {\left( {\mathcal{F}}_{n}\right) }_{n \geq - 1} \) . | Proof. We have\n\n\[ \mathbb{E}\left\lbrack {J\left( {u{\mathbf{1}}_{\left\lbrack 0, k + 1\right\rbrack }}\right) \mid {\mathcal{F}}_{k}}\right\rbrack = \mathbb{E}\left\lbrack {\mathbb{E}\left\lbrack {J\left( {u{\mathbf{1}}_{\left\lbrack 0, k + 1\right\rbrack }}\right) \mid {\mathcal{F}}_{k + 1} \mid {\mathcal{F}}_{k}}... | Yes |
Proposition 1.3.2. The multiple stochastic integral \( {J}_{n}\left( {f}_{n}\right) \) of \( {f}_{n} \in {\ell }^{2}{\left( \mathbb{N}\right) }^{\circ n} \) , \( n \geq 1 \), is defined as\n\n\[ \n{J}_{n}\left( {f}_{n}\right) = \mathop{\sum }\limits_{{\left( {{i}_{1},\ldots ,{i}_{n}}\right) \in {\Delta }_{n}}}{f}_{n}\l... | Proof. Note that we have\n\n\[ \n{J}_{n}\left( {f}_{n}\right) = n!\mathop{\sum }\limits_{{0 \leq {i}_{1} < \cdots < {i}_{n}}}{f}_{n}\left( {{i}_{1},\ldots ,{i}_{n}}\right) {Y}_{{i}_{1}}\cdots {Y}_{{i}_{n}} \n\]\n\n\[ \n= n!\mathop{\sum }\limits_{{{i}_{n} = 0}}^{\infty }\mathop{\sum }\limits_{{0 \leq {i}_{n - 1} < {i}_{... | Yes |
Lemma 1.3.3. For all \( n \geq 1 \) we have\n\n\[ \mathbb{E}\left\lbrack {{J}_{n}\left( {f}_{n}\right) \mid {\mathcal{F}}_{k}}\right\rbrack = {J}_{n}\left( {{f}_{n}{\mathbf{1}}_{{\left\lbrack 0, k\right\rbrack }^{n}}}\right) \]\n\n\( k \in \mathbb{N},{f}_{n} \in {\ell }^{2}{\left( \mathbb{N}\right) }^{\circ n}. \) | Proof. This lemma can be proved in two ways, either as a consequence of Proposition 1.2.3 and Proposition 1.3.2 or via the following direct argument, noting that for all \( m = 0,\ldots, n \) and \( {g}_{m} \in {\ell }^{2}{\left( \mathbb{N}\right) }^{\circ m} \) we have:\n\n\[ \mathbb{E}\left\lbrack {\left( {{J}_{n}\le... | Yes |
Lemma 1.5.1. For all \( n \in \mathbb{N} \) we have\n\n\[ \n{L}^{0}\left( {\Omega ,{\mathcal{F}}_{n}}\right) = \left( {{\mathcal{H}}_{0} \oplus \cdots \oplus {\mathcal{H}}_{n + 1}}\right) \bigcap {L}^{0}\left( {\Omega ,{\mathcal{F}}_{n}}\right) .\n\] | Proof. It suffices to note that \( {\mathcal{H}}_{l} \cap {L}^{0}\left( {\Omega ,{\mathcal{F}}_{n}}\right) \) has dimension \( \left( \begin{matrix} n + 1 \\ l \end{matrix}\right) ,1 \leq l \leq \) \( n + 1 \) . More precisely it is generated by the orthonormal basis\n\n\[ \n\left\{ {{Y}_{{k}_{1}}\cdots {Y}_{{k}_{l}} =... | Yes |
Proposition 1.5.3. We have the identity\n\n\[ \n{L}^{2}\left( \Omega \right) = {\bigoplus }_{n = 0}^{\infty }{\mathcal{H}}_{n} \n\] | Proof. It suffices to show that \( \mathcal{S} \) is dense in \( {L}^{2}\left( \Omega \right) \) . Let \( F \) be a bounded random variable. Relation (1.5.3) of Lemma 1.5.1 shows that \( \mathbb{E}\left\lbrack {F \mid {\mathcal{F}}_{n}}\right\rbrack \in \mathcal{S} \) . The martingale convergence theorem, cf. e.g. Theo... | Yes |
Corollary 1.6.3. A random variable \( F : \Omega \rightarrow \mathbb{R} \) is \( {\mathcal{F}}_{n} \) -measurable if and only if\n\n\[ \n{D}_{k}F = 0 \n\] \n\nfor all \( k > n \) . | If \( F \) has the form \( F = f\left( {{X}_{0},\ldots ,{X}_{n}}\right) \), we may also write\n\n\[ \n{D}_{k}F = \sqrt{{p}_{k}{q}_{k}}\left( {{F}_{k}^{ + } - {F}_{k}^{ - }}\right) ,\;k \in \mathbb{N}, \n\] \n\nwith\n\n\[ \n{F}_{k}^{ + } = f\left( {{X}_{0},\ldots ,{X}_{k - 1}, + 1,{X}_{k + 1},\ldots ,{X}_{n}}\right) , \... | Yes |
Proposition 1.6.4. Let \( F, G : \Omega \rightarrow \mathbb{R} \) . We have\n\n\[ \n{D}_{k}\left( {FG}\right) = F{D}_{k}G + G{D}_{k}F - \frac{{X}_{k}}{\sqrt{{p}_{k}{q}_{k}}}{D}_{k}F{D}_{k}G,\;k \in \mathbb{N}.\n\] | Proof. Let \( {F}_{ + }^{k}\left( \omega \right) = F\left( {\omega }_{ + }^{k}\right) ,{F}_{ - }^{k}\left( \omega \right) = F\left( {\omega }_{ - }^{k}\right), k \geq 0 \) . We have\n\n\[ \n{D}_{k}\left( {FG}\right) = \sqrt{{p}_{k}{q}_{k}}\left( {{F}_{ + }^{k}{G}_{ + }^{k} - {F}_{ - }^{k}{G}_{ - }^{k}}\right)\n\]\n\n\[... | Yes |
Proposition 1.7.1. For all \( F \in \mathcal{S} \) we have\n\n\[ F = \mathbb{E}\left\lbrack F\right\rbrack + \mathop{\sum }\limits_{{k = 0}}^{\infty }\mathbb{E}\left\lbrack {{D}_{k}F \mid {\mathcal{F}}_{k - 1}}\right\rbrack {Y}_{k} \]\n\n\[ = \mathbb{E}\left\lbrack F\right\rbrack + \mathop{\sum }\limits_{{k = 0}}^{\inf... | Proof. The formula is obviously true for \( F = {J}_{0}\left( {f}_{0}\right) \) . Given \( n \geq 1 \), as a consequence of Proposition 1.3.2 above and Lemma 1.3.3 we have:\n\n\[ {J}_{n}\left( {f}_{n}\right) = n\mathop{\sum }\limits_{{k = 0}}^{\infty }{J}_{n - 1}\left( {{f}_{n}\left( {*, k}\right) {\mathbf{1}}_{{\left\... | Yes |
Lemma 1.7.2. The operator\n\n\\[ \n{L}^{2}\left( \Omega \right) \rightarrow {L}^{2}\left( {\Omega \times \mathbb{N}}\right) \n\\]\n\n\\[ \nF \mapsto {\left( \mathbb{E}\left\lbrack {D}_{k}F \mid {\mathcal{F}}_{k - 1}\right\rbrack \right) }_{k \in \mathbb{N}} \n\\]\n\nis bounded with norm equal to one. | Proof. Let \\( F \in \mathcal{S} \\) . From Relation (1.7.1) and the isometry formula (1.2.2) for the stochastic integral operator \\( J \\) we get\n\n\\[ \n{\\begin{Vmatrix}\\mathbb{E}\left\\lbrack D.F \mid {\\mathcal{F}}_{\\cdot - 1}\\right\\rbrack \\end{Vmatrix}}_{{L}^{2}\left( {\\Omega \\times \\mathbb{N}}\\right) ... | Yes |
Corollary 1.7.3. The Clark formula of Proposition 1.7.1 extends to any \( F \in {L}^{2}\left( \Omega \right) \) . | Proof. Since \( F \mapsto \mathbb{E}\left\lbrack {D.F \mid {\mathcal{F}}_{\cdot - 1}}\right\rbrack \) is bounded from Lemma 1.7.2, the Clark formula extends to \( F \in {L}^{2}\left( \Omega \right) \) by a standard Cauchy sequence argument. | Yes |
Corollary 1.7.4. Let \( a \in \mathbb{N} \) and \( F \in {L}^{2}\left( \Omega \right) \) . We have\n\n\[ F = \mathbb{E}\left\lbrack {F \mid {\mathcal{F}}_{a}}\right\rbrack + \mathop{\sum }\limits_{{k = a + 1}}^{\infty }\mathbb{E}\left\lbrack {{D}_{k}F \mid {\mathcal{F}}_{k - 1}}\right\rbrack {Y}_{k} \]\n\n(1.7.3)\n\nan... | Proof. From Proposition 1.2.3 and the Clark formula (1.7.1) of Proposition 1.7.1 we have\n\n\[ \mathbb{E}\left\lbrack {F \mid {\mathcal{F}}_{a}}\right\rbrack = \mathbb{E}\left\lbrack F\right\rbrack + \mathop{\sum }\limits_{{k = 0}}^{a}\mathbb{E}\left\lbrack {{D}_{k}F \mid {\mathcal{F}}_{k - 1}}\right\rbrack {Y}_{k} \]\... | Yes |
Proposition 1.7.5. Let \( {\left( {M}_{n}\right) }_{n \in \mathbb{N}} \) be a martingale in \( {L}^{2}\left( \Omega \right) \) with respect to \( {\left( {\mathcal{F}}_{n}\right) }_{n \in \mathbb{N}} \) . There exists a predictable process \( {\left( {u}_{k}\right) }_{k \in \mathbb{N}} \) locally in \( {L}^{2}\left( {\... | Proof. Let \( k \geq 1 \) . From Corollaries 1.6.3 and 1.7.4 we have:\n\n\[ \n{M}_{k} = \mathbb{E}\left\lbrack {{M}_{k} \mid {\mathcal{F}}_{k - 1}}\right\rbrack + \mathbb{E}\left\lbrack {{D}_{k}{M}_{k} \mid {\mathcal{F}}_{k - 1}}\right\rbrack {Y}_{k}\n\]\n\n\[ \n= {M}_{k - 1} + \mathbb{E}\left\lbrack {{D}_{k}{M}_{k} \m... | Yes |
Proposition 1.8.2. The operator \( \delta \) is adjoint to \( D \) : | \[ \mathbb{E}\left\lbrack {\langle {DF}, u{\rangle }_{{\ell }^{2}\left( \mathbb{N}\right) }}\right\rbrack = \mathbb{E}\left\lbrack {{F\delta }\left( u\right) }\right\rbrack ,\;F \in \mathcal{S}, u \in \mathcal{U}. \] Proof. We consider \( F = {J}_{n}\left( {f}_{n}\right) \) and \( {u}_{k} = {J}_{m}\left( {{g}_{m + 1}\l... | Yes |
Corollary 1.8.5. If \( {\left( {u}_{k}\right) }_{k \in \mathbb{N}} \) satisfies \( {D}_{k}{u}_{k} = 0 \), i.e. \( {u}_{k} \) does not depend on \( {X}_{k}, k \in \mathbb{N} \), then \( \delta \left( u\right) \) coincides with the (discrete time) stochastic integral | \[ \delta \left( u\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }{Y}_{k}{u}_{k} \] provided the series converges in \( {L}^{2}\left( \Omega \right) \) . If moreover \( {\left( {u}_{k}\right) }_{k \in \mathbb{N}} \) is predictable and square-summable we have the isometry \[ \mathbb{E}\left\lbrack {\delta {\left( u\r... | Yes |
Proposition 1.9.1. For any \( F \in \mathcal{S} \) we have\n\n\[ \n{LF} = {\delta DF} = \mathop{\sum }\limits_{{k = 0}}^{\infty }{Y}_{k}\left( {{D}_{k}F}\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }\sqrt{{p}_{k}{q}_{k}}{Y}_{k}\left( {{F}_{k}^{ + } - {F}_{k}^{ - }}\right) ,\n\] | Proof. Note that \( {D}_{k}{D}_{k}F = 0, k \in \mathbb{N} \), and use Relation (1.8.1) of Proposition 1.8.3. | No |
Lemma 1.9.2. Let the probability kernel \( {Q}_{t}\left( {\widetilde{\omega },{d\omega }}\right) \) be defined by\n\n\[ \mathbb{E}\left\lbrack {\left. \frac{d{Q}_{t}\left( {\widetilde{\omega }, \cdot }\right) }{d\mathbb{P}}\right| \;{\mathcal{F}}_{N}}\right\rbrack \left( \omega \right) = {q}_{t}^{N}\left( {\widetilde{\... | Proof. Since \( {L}^{2}\left( {\Omega ,{\mathcal{F}}_{N}}\right) \) has finite dimension \( {2}^{N + 1} \), it suffices to consider functionals of the form \( F = {Y}_{{k}_{1}}\cdots {Y}_{{k}_{n}} \) with \( 0 \leq {k}_{1} < \cdots < {k}_{n} \leq N \) . By Relation (1.4.5) we have for \( \omega \in \Omega, k \in \mathb... | Yes |
Proposition 1.9.3. The process \( {\left( X\left( t\right) \right) }_{t \in {\mathbb{R}}_{ + }} = {\left( {\left( {X}_{k}\left( t\right) \right) }_{k \in \mathbb{N}}\right) }_{t \in {\mathbb{R}}_{ + }} \) is the Ornstein-Uhlenbeck process associated to \( {\left( {P}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \), i.e. we ... | Proof. By construction of \( {\left( X\left( t\right) \right) }_{t \in {\mathbb{R}}_{ + }} \) in Relations (1.9.2)-(1.9.5) we have\n\n\[ \n\mathbb{P}\left( {{X}_{k}\left( t\right) = 1 \mid {X}_{k}\left( 0\right) }\right) = {p}_{k}\left( {1 + {\mathrm{e}}^{-t}{Y}_{k}\left( 0\right) \sqrt{\frac{{q}_{k}}{{p}_{k}}}}\right)... | Yes |
Lemma 1.9.4. For \( F \in \operatorname{Dom}\left( D\right) \) we have\n\n\[ \n{\begin{Vmatrix}{P}_{t}u\end{Vmatrix}}_{{L}^{\infty }\left( {\Omega ,{\ell }^{2}\left( \mathbb{N}\right) }\right) } \leq \parallel u{\parallel }_{{L}^{\infty }\left( {\Omega ,{\ell }^{2}\left( \mathbb{N}\right) }\right) },\;t \in {\mathbb{R}... | Proof. As a consequence of the representation formula (1.9.7) we have \( \mathbb{P}\left( {d\widetilde{\omega }}\right) \) - a.s.:\n\n\[ \n{\begin{Vmatrix}{P}_{t}u\end{Vmatrix}}_{{\ell }^{2}\left( \mathbb{N}\right) }^{2}\left( \widetilde{\omega }\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }{\left| {P}_{t}{u}_{k}\... | Yes |
Proposition 1.10.1. For all \( F, G \in {L}^{2}\left( \Omega \right) \) such that \( \mathbb{E}\left\lbrack {\parallel {DF}{\parallel }_{{\ell }^{2}\left( \mathbb{N}\right) }^{2}}\right\rbrack < \infty \) we have\n\n\[ \operatorname{Cov}\left( {F, G}\right) = \mathbb{E}\left\lbrack {\mathop{\sum }\limits_{{k = 0}}^{\in... | Proof. This identity is a consequence of the Clark formula (1.7.1):\n\n\[ \operatorname{Cov}\left( {F, G}\right) = \mathbb{E}\left\lbrack {\left( {F - \mathbb{E}\left\lbrack F\right\rbrack }\right) \left( {G - \mathbb{E}\left\lbrack G\right\rbrack }\right) }\right\rbrack \]\n\n\[ = \mathbb{E}\left\lbrack {\left( {\math... | Yes |
Theorem 1.10.3. Let \( n \in \mathbb{N} \) and \( F, G \in {L}^{2}\left( \Omega \right) \) . We have\n\n\( \operatorname{Cov}\left( {F, G}\right) \)\n\n\( \left( {1.10.4}\right) \)\n\n\[ = \mathop{\sum }\limits_{{d = 1}}^{n}{\left( -1\right) }^{d + 1}\mathbb{E}\left\lbrack {\mathop{\sum }\limits_{\left\{ 1 \leq {k}_{1}... | Proof. Take \( F = G \) . For \( n = 0,\left( {1.10.4}\right) \) is a consequence of the Clark formula. Let \( n \geq 1 \) . Applying Lemma 1.7.4 to \( {D}_{{k}_{n}}\cdots {D}_{{k}_{1}}F \) with \( a = {k}_{n} \) and \( b = {k}_{n + 1} \) , and summing on \( \left( {{k}_{1},\ldots ,{k}_{n}}\right) \in {\Delta }_{n} \),... | Yes |
Proposition 1.10.7. If \( F, G \in {L}^{2}\left( \Omega \right) \) are non-decreasing then \( F \) and \( G \) are non-negatively correlated: | \[ \operatorname{Cov}\left( {F, G}\right) \geq 0 \] | No |
Proposition 1.11.3. We have \( \mathbb{E}\left\lbrack {\mathrm{e}}^{\alpha \left| F\right| }\right\rbrack < \infty \) for all \( \alpha > 0 \), and \( \mathbb{E}\left\lbrack {\mathrm{e}}^{\alpha {F}^{2}}\right\rbrack < \infty \) for all \( \alpha < 1/\left( {2{K}^{2}}\right) \) . | Proof. Let \( \lambda < c/e \) . The bound (1.11.2) implies\n\n\[ \mathbb{E}\left\lbrack {e}^{\alpha \left| F\right| }\right\rbrack = {\int }_{0}^{\infty }\mathbb{P}\left( {{e}^{\alpha \left| F\right| } \geq t}\right) {dt} \]\n\n\[ = {\int }_{-\infty }^{\infty }\mathbb{P}\left( {\alpha \left| F\right| \geq y}\right) {e... | Yes |
Lemma 1.12.1. The entropy has the tensorization property, i.e. if \( F, G \) are sufficiently integrable independent random variables we have\n\n\[ \operatorname{Ent}\left\lbrack {FG}\right\rbrack = \mathbb{E}\left\lbrack {F\operatorname{Ent}\left\lbrack G\right\rbrack }\right\rbrack + \mathbb{E}\left\lbrack {G\operato... | Proof. We have\n\n\[ \operatorname{Ent}\left\lbrack {FG}\right\rbrack = \mathbb{E}\left\lbrack {{FG}\log \left( {FG}\right) }\right\rbrack - \mathbb{E}\left\lbrack {FG}\right\rbrack \log \mathbb{E}\left\lbrack {FG}\right\rbrack \]\n\n\[ = \mathbb{E}\left\lbrack {{FG}\left( {\log F + \log G}\right) }\right\rbrack - \mat... | Yes |
Theorem 1.12.2. Let \( F \in \operatorname{Dom}\left( D\right) \) with \( F > \eta \) a.s. for some \( \eta > 0 \). We have \[ \operatorname{Ent}\left\lbrack F\right\rbrack \leq \mathbb{E}\left\lbrack {\frac{1}{F}\parallel {DF}{\parallel }_{{\ell }^{2}\left( \mathbb{N}\right) }^{2}}\right\rbrack . \] | Proof. Assume that \( F \) is \( {\mathcal{F}}_{N} \) -measurable and let \( {M}_{n} = \mathbb{E}\left\lbrack {F \mid {\mathcal{F}}_{n}}\right\rbrack ,0 \leq n \leq N \). Using Corollary 1.6.3 and the Clark formula (1.7.1) we have \[ {M}_{n} = {M}_{-1} + \mathop{\sum }\limits_{{k = 0}}^{n}{u}_{k}{Y}_{k},\;0 \leq n \leq... | Yes |
Theorem 1.12.4. Let \( F \) be \( {\mathcal{F}}_{N} \) -measurable. We have\n\n\[ \n\operatorname{Ent}\left\lbrack {\mathrm{e}}^{F}\right\rbrack \leq \mathbb{E}\left\lbrack {{\mathrm{e}}^{F}\mathop{\sum }\limits_{{k = 0}}^{N}{p}_{k}{q}_{k}\left( {{\nabla }_{k}F{\mathrm{e}}^{{\nabla }_{k}F} - {\mathrm{e}}^{{\nabla }_{k}... | Clearly, (1.12.7) is better than (1.12.6), (1.12.4) and (1.12.3). It also improves (1.12.2) from the bound\n\n\[ \nx{\mathrm{e}}^{x} - {\mathrm{e}}^{x} + 1 \leq {\left( {\mathrm{e}}^{x} - 1\right) }^{2},\;x \in \mathbb{R}\n\]\nwhich implies\n\n\[ \n{\mathrm{e}}^{F}\left( {\nabla F{\mathrm{e}}^{\nabla F} - {\mathrm{e}}^... | Yes |
Lemma 1.12.5. For any \( 0 \leq p \leq 1, t \in \mathbb{R}, a \in \mathbb{R}, q = 1 - p \) ,\n\n\[ \n{pt}{\mathrm{e}}^{t} + {qa}{\mathrm{e}}^{a} - \left( {p{\mathrm{e}}^{t} + q{\mathrm{e}}^{a}}\right) \log \left( {p{\mathrm{e}}^{t} + q{\mathrm{e}}^{a}}\right) \n\]\n\n\[ \n\leq {pq}\left( {q{\mathrm{e}}^{a}\left( {\left... | Proof. Set\n\n\[ \ng\left( t\right) = {pq}\left( {q{\mathrm{e}}^{a}\left( {\left( {t - a}\right) {\mathrm{e}}^{t - a} - {\mathrm{e}}^{t - a} + 1}\right) + p{\mathrm{e}}^{t}\left( {\left( {a - t}\right) {\mathrm{e}}^{a - t} - {\mathrm{e}}^{a - t} + 1}\right) }\right) \n\]\n\n\[ \n- {pt}{\mathrm{e}}^{t} - {qa}{\mathrm{e}... | Yes |
Proposition 1.14.3. Assume that the portfolio \( {\left( {\eta }_{k},{\zeta }_{k}\right) }_{0 \leq k \leq N} \) is self-financing. Then we have the decomposition\n\n\[ \n{V}_{n} = {V}_{-1}\mathop{\prod }\limits_{{k = 0}}^{n}\left( {1 + {r}_{k}}\right) + \mathop{\sum }\limits_{{i = 0}}^{n}{\eta }_{i}{S}_{i - 1}\sqrt{{p}... | Proof. Under the self-financing assumption we have\n\n\[ \n{V}_{i} - {V}_{i - 1} = {\zeta }_{i}\left( {{A}_{i} - {A}_{i - 1}}\right) + {\eta }_{i}\left( {{S}_{i} - {S}_{i - 1}}\right) \n\]\n\n\[ \n= {r}_{i}{\zeta }_{i}{A}_{i - 1} + \left( {{a}_{i}{\mathbf{1}}_{\left\{ {X}_{i} = - 1\right\} } + {b}_{i}{\mathbf{1}}_{\lef... | Yes |
Proposition 1.14.4. Given \( F \in {L}^{2}\left( {\Omega ,{\mathcal{F}}_{N}}\right) \), let\n\n\[ \n{\eta }_{n} = \frac{1}{{S}_{n - 1}\sqrt{{p}_{n}{q}_{n}}\left( {{b}_{n} - {a}_{n}}\right) }{\mathbb{E}}^{ * }\left\lbrack {{D}_{n}F \mid {\mathcal{F}}_{n - 1}}\right\rbrack \mathop{\prod }\limits_{{k = n + 1}}^{N}{\left( ... | Proof. Let \( {\left( {\eta }_{k}\right) }_{-1 \leq k \leq N} \) be defined by (1.14.6) and \( {\eta }_{-1} = 0 \), and consider the process \( {\left( {\zeta }_{n}\right) }_{0 \leq n \leq N} \) defined by\n\n\[ \n{\zeta }_{-1} = \frac{{\mathbb{E}}^{ * }\left\lbrack F\right\rbrack }{{S}_{-1}}\mathop{\prod }\limits_{{k ... | Yes |
Proposition 2.3.2. The law of \( {T}_{n} \) has the density \( t \mapsto {\lambda }^{n}{\mathrm{e}}^{-{\lambda t}}\frac{{t}^{n - 1}}{\left( {n - 1}\right) !} \) on \( {\mathbb{R}}_{ + } \) , \( n \geq 1 \) . | Proof. We have\n\n\[ \mathbb{P}\left( {{T}_{1} > t}\right) = \mathbb{P}\left( {{N}_{t} = 0}\right) = {\mathrm{e}}^{-{\lambda t}},\;t \in {\mathbb{R}}_{ + }, \]\n\nand by induction, assuming that\n\n\[ \mathbb{P}\left( {{T}_{n - 1} > t}\right) = \lambda {\int }_{t}^{\infty }{\mathrm{e}}^{-{\lambda s}}\frac{{\left( \lamb... | Yes |
Proposition 2.3.3. The conditional density of \( \left( {{T}_{1},\ldots ,{T}_{n}}\right) \) given that \( {T}_{n + 1} = T \) is \[ \left( {{t}_{1},\ldots ,{t}_{n}}\right) \mapsto \frac{n!}{{T}^{n}}{\mathbf{1}}_{\left\{ 0 \leq {t}_{1} < \cdots < {t}_{n} \leq T\right\} }.\] | Moreover we have \[ \mathbb{E}\left\lbrack {f\left( {\frac{{T}_{1}}{{T}_{n + 1}},\ldots ,\frac{{T}_{n}}{{T}_{n + 1}}}\right) g\left( {T}_{n + 1}\right) }\right\rbrack = {\lambda }^{n + 1}{\int }_{0}^{\infty }{\mathrm{e}}^{-\lambda {t}_{n + 1}}{\int }_{0}^{{t}_{n + 1}}\cdots {\int }_{0}^{{t}_{2}}f\left( {\frac{{t}_{1}}{... | Yes |
Corollary 2.3.5. The standard Poisson process \( {\left( {N}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) defined by (2.3.3) has independent increments which are distributed according to the Poisson law, i.e. for all \( 0 \leq {t}_{0} \leq {t}_{1} < \cdots < {t}_{n} \) , \n\n\[ \n\left( {{N}_{{t}_{1}} - {N}_{{t}_{0}},\ld... | Proof. Letting \n\n\[ \nf = \mathop{\sum }\limits_{{k = 1}}^{n}{\alpha }_{k}{\mathbf{1}}_{\left( {t}_{k - 1},{t}_{k}\right\rbrack } \n\] \n\nfrom Proposition 2.3.4 we get \n\n\[ \n\mathbb{E}\left\lbrack {\exp \left( {i\mathop{\sum }\limits_{{k = 1}}^{n}{\alpha }_{k}\left( {{N}_{{t}_{k}} - {N}_{{t}_{k - 1}}}\right) }\ri... | Yes |
Proposition 2.3.6. For any \( f \in {L}^{1}\left( {{\Delta }_{n},{\mathrm{e}}^{-{s}_{n}}d{s}_{1}\cdots d{s}_{n}}\right) \) we have\n\n\[ \mathbb{E}\left\lbrack {f\left( {{T}_{1},\ldots ,{T}_{n}}\right) \mid {\mathcal{F}}_{t}}\right\rbrack \]\n\n(2.3.5)\n\n\[ = {\int }_{t}^{\infty }{\mathrm{e}}^{-\left( {{s}_{n} - t}\ri... | Proof. Apply Relation (2.3.4) using the fact that for fixed \( t > 0,{\left( {N}_{s} - {N}_{t}\right) }_{s \geq t} \) is a standard Poisson process independent of \( {\mathcal{F}}_{t} \). | No |
Proposition 2.3.7. Given that \( \left\{ {{N}_{T} \geq n}\right\} \), the jump times \( \left( {{T}_{1},\ldots ,{T}_{n}}\right) \) of the Poisson process \( {\left( {N}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) are independent uniformly distributed random variables on \( \left\lbrack {0, T}\right\rbrack \) . | Proof. For all \( n \geq 1 \) and \( f \in {\mathcal{C}}_{c}\left( {\left\lbrack 0, T\right\rbrack }^{n}\right) \) we have\n\n\[ \mathbb{E}\left\lbrack {f\left( {{T}_{1},\ldots ,{T}_{n}}\right) }\right\rbrack = {\lambda }^{n}{\int }_{0}^{T}{\mathrm{e}}^{-\lambda {t}_{n}}{\int }_{0}^{{t}_{n}}\cdots {\int }_{0}^{{t}_{2}}... | Yes |
Proposition 2.4.2. For any \( t \in \left\lbrack {0, T}\right\rbrack \) we have\n\n\[ \mathbb{E}\left\lbrack {\exp \left( {{i\alpha }\left( {{X}_{T} - {X}_{t}}\right) }\right) }\right\rbrack = \exp \left( {\lambda \left( {T - t}\right) {\int }_{-\infty }^{\infty }\left( {{\mathrm{e}}^{iy\alpha } - 1}\right) \nu \left( ... | Proof. Since \( {N}_{t} \) has a Poisson distribution with parameter \( t > 0 \) and is independent of \( {\left( {Y}_{k}\right) }_{k \geq 1} \), for all \( \alpha \in \mathbb{R} \) we have by conditioning:\n\n\[ \mathbb{E}\left\lbrack {\exp \left( {{i\alpha }\left( {{X}_{T} - {X}_{t}}\right) }\right) }\right\rbrack = ... | Yes |
Proposition 2.5.4. The definition (2.5.3) of the stochastic integral with respect to the normal martingale \( {\left( {M}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) on simple predictable processes extends to \( u \in {L}_{ad}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) via the (conditional) isometry formula\... | Proof. We start by showing that the isometry (2.5.4) holds for the simple predictable process \( u = \mathop{\sum }\limits_{{i = 1}}^{n}{G}_{i}{\mathbf{1}}_{\left( {t}_{i - 1},{t}_{i}\right\rbrack } \), with \( s = {t}_{0} < {t}_{1} < \cdots {t}_{n} \) :\n\n\[ \mathbb{E}\left\lbrack {{\left( {\int }_{0}^{\infty }{u}_{t... | Yes |
Proposition 2.5.5. Let \( A \in \mathcal{F} \) and \( u \in {L}_{ad}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) such that\n\n\[ \n{u}_{s}\left( \omega \right) = 0,\;{\mathbf{1}}_{A}\left( \omega \right) {ds} \times \mathbb{P}\left( {d\omega }\right) - \text{ a.e. } \n\]\n\nThen\n\n\[ \n{\int }_{0}^{\infty }... | Proof. Consider the sequence \( {\left( {u}_{s}^{n}\right) }_{s \in {\mathbb{R}}_{ + }}, n \in \mathbb{N} \), of simple predictable processes defined as\n\n\[ \n{u}^{n} = \mathop{\sum }\limits_{{i = 1}}^{n}{\mathbf{1}}_{\left( {t}_{i - 1}^{n},{t}_{i}^{n}\right\rbrack }\frac{1}{{t}_{i - 1}^{n} - {t}_{i - 2}^{n}}{\int }_... | Yes |
For all \( T > 0 \), the indefinite integral process\n\n\[ \n{\left( {\int }_{0}^{t}{u}_{s}d{M}_{s}\right) }_{t \in \left\lbrack {0, T}\right\rbrack } \n\]\n\nhas a measurable version in \( {L}_{ad}^{2}\left( {\Omega \times \left\lbrack {0, T}\right\rbrack }\right) \) . | Proof. Let \( {\left( {u}^{n}\right) }_{n \in \mathbb{N}} \) be a sequence of simple predictable processes converging to \( u \) in \( {L}^{2}\left( {\Omega \times \left\lbrack {0, T}\right\rbrack }\right) \) . We have\n\n\[ \n\mathbb{E}\left\lbrack {{\int }_{0}^{T}{\left( {\int }_{0}^{t}{u}_{s}d{M}_{s} - {\int }_{0}^{... | Yes |
Proposition 2.5.7. For any \( u \in {L}_{ad}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) we have\n\n\[ \mathbb{E}\left\lbrack {{\int }_{0}^{\infty }{u}_{s}d{M}_{s} \mid {\mathcal{F}}_{t}}\right\rbrack = {\int }_{0}^{t}{u}_{s}d{M}_{s},\;t \in {\mathbb{R}}_{ + }.\n\]\n\nIn particular, \( {\int }_{0}^{t}{u}_{s}... | Proof. Let \( u \in \mathcal{U} \) of the form \( u = G{\mathbf{1}}_{(a, b\rbrack } \), where \( G \) is bounded and \( {\mathcal{F}}_{a} \) -measurable.\n\ni) If \( 0 \leq a \leq t \) we have\n\n\[ \mathbb{E}\left\lbrack {{\int }_{0}^{\infty }{u}_{s}d{M}_{s} \mid {\mathcal{F}}_{t}}\right\rbrack = \mathbb{E}\left\lbrac... | Yes |
Corollary 2.5.11. Let \( T > 0 \) and let \( {\left( {u}_{t}\right) }_{t \in \left\lbrack {0, T}\right\rbrack } \in {L}^{2}\left( {\Omega \times \left\lbrack {0, T}\right\rbrack }\right) \) be an adapted process with a uniformly càdlàg version \( {\left( {\bar{u}}_{t}\right) }_{t \in \left\lbrack {0, T}\right\rbrack } ... | Proof. It suffices to check that Condition (2.5.8) holds under the hypothesis (2.5.9). | No |
Lemma 2.7.2. For all \( {f}_{n} \in {L}^{2}{\left( {\mathbb{R}}_{ + }\right) }^{\circ n}, n \geq 1 \), we have\n\n\[ \mathbb{E}\left\lbrack {{I}_{n}\left( {f}_{n}\right) \mid {\mathcal{F}}_{t}}\right\rbrack = {I}_{n}\left( {{f}_{n}{\mathbf{1}}_{{\left\lbrack 0, t\right\rbrack }^{n}}}\right) ,\;t \in {\mathbb{R}}_{ + }.... | Proof. Since the indefinite Itô integral is a martingale from (2.7.2) and Proposition 2.5.7 we have\n\n\[ \mathbb{E}\left\lbrack {{I}_{n}\left( {f}_{n}\right) \mid {\mathcal{F}}_{t}}\right\rbrack = n!\mathbb{E}\left\lbrack {{\int }_{0}^{\infty }{\int }_{0}^{{t}_{n}}\cdots {\int }_{0}^{{t}_{2}}{f}_{n}\left( {{t}_{1},\ld... | Yes |
Proposition 2.9.2. We have\n\n\[ \n{\left\lbrack M, M\right\rbrack }_{t} = \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\sum }\limits_{{i = 1}}^{n}{\left( {M}_{{t}_{i}^{n}} - {M}_{{t}_{i - 1}^{n}}\right) }^{2},\;t \geq 0, \]\n\nwhere the limit exists in \( {L}^{2}\left( \Omega \right) \) and is independent of... | Proof. We have\n\n\[ \n{\left\lbrack M, M\right\rbrack }_{{t}_{i}^{n}} - {\left\lbrack M, M\right\rbrack }_{{t}_{i - 1}^{n}} = {M}_{{t}_{i}^{n}}^{2} - {M}_{{t}_{i - 1}^{n}}^{2} - 2{\int }_{{t}_{i - 1}^{n}}^{{t}_{i}^{n}}{M}_{s}d{M}_{s} \]\n\n\[ \n= {\left( {M}_{{t}_{i}^{n}} - {M}_{{t}_{i - 1}^{n}}\right) }^{2} + 2{\int ... | Yes |
Proposition 2.9.3. The quadratic variation of Brownian motion \( {\left( {B}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) is\n\n\[ \n{\left\lbrack B, B\right\rbrack }_{t} = t,\;t \in {\mathbb{R}}_{ + }.\n\] | Proof. (cf. e.g. [121], Theorem I-28). For every subdivision \( \left\{ {0 = {t}_{0}^{n} < \cdots < }\right. \) \( \left. {{t}_{n}^{n} = t}\right\} \) we have\n\n\[ \n\mathbb{E}\left\lbrack {\left( t - \mathop{\sum }\limits_{{i = 1}}^{n}{\left( {B}_{{t}_{i}^{n}} - {B}_{{t}_{i - 1}^{n}}\right) }^{2}\right) }^{2}\right\r... | Yes |
Proposition 2.10.2. Assume that \( {\left( {M}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) is a normal martingale in \( {L}^{4} \) having the predictable representation property. Then \( {\left( {M}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) satisfies the structure equation (2.10.1), i.e. there exists a square-integrabl... | Proof. Since \( {\left( {\left\lbrack M, M\right\rbrack }_{t} - t\right) }_{t \in {\mathbb{R}}_{ + }} \) is a martingale and \( {\left( {M}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) has the chaos representation property, Proposition 2.6.2 shows the existence of a square-integrable adapted process \( {\left( {\phi }_{t... | Yes |
Proposition 2.11.3. Under the Assumption 2.11.1, for all \( u \in {L}_{ad}^{4}(\Omega \times \) \( \left. {\mathbb{R}}_{ + }\right) \cap {L}^{4}\left( {\Omega ,{L}^{2}\left( {\mathbb{R}}_{ + }\right) }\right) \) we have \( {\int }_{0}^{\infty }{u}_{s}d{M}_{s} \in {L}^{4}\left( \Omega \right) \) and\n\n\[{\left( {\int }... | Proof. For simple predictable processes \( {\left( {u}_{s}\right) }_{s \in {\mathbb{R}}_{ + }} \) of the form (2.11.3), formula (2.11.6) follows from (2.10.2). It is extended to \( u \in {L}_{ad}^{4}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \cap \) \( {L}^{4}\left( {\Omega ,{L}^{2}\left( {\mathbb{R}}_{ + }\right... | Yes |
Corollary 2.11.4. Let \( u \in {L}^{\infty }\left( {\mathbb{R}}_{ + }\right) \) and \( v \in {L}^{4}\left( {\mathbb{R}}_{ + }\right) \). Then under Assumption 2.11.1 we have for all \( n \geq 1 \) and \( t \in {\mathbb{R}}_{ + } \):\n\n\[ \n{I}_{1}\left( {u{\mathbf{1}}_{\left\lbrack 0, t\right\rbrack }}\right) {I}_{n}\... | Proof. Applying Proposition 2.11.3 to \( {u}_{s} \) and \( {v}_{s}{I}_{n - 1}\left( {{\mathbf{1}}_{{\left\lbrack 0, s\right\rbrack }^{n - 1}}{v}^{\otimes \left( {n - 1}\right) }}\right) \), we have\n\n\[ \n{I}_{1}\left( {u{\mathbf{1}}_{\left\lbrack 0, t\right\rbrack }}\right) {I}_{n}\left( {{\mathbf{1}}_{{\left\lbrack ... | Yes |
Proposition 2.12.1. Assume that \( \phi \in {L}_{ad}^{\infty }\left( {\left\lbrack {0, T}\right\rbrack \times \Omega }\right) \) . Let \( {\left( {X}_{t}\right) }_{t \in \left\lbrack {0, T}\right\rbrack } \) be a process given by\n\n\[ \n{X}_{t} = {X}_{0} + {\int }_{0}^{t}{u}_{s}d{M}_{s} + {\int }_{0}^{t}{v}_{s}{ds},\;... | Proof. We prove the formula in the case where \( f \) does not depend on time and \( {v}_{s} = 0, s \in {\mathbb{R}}_{ + } \), since this generalization can be done using standard calculus arguments. Assume now that \( u \in \mathcal{U} \) is a simple predictable process of the form \( u = G{\mathbf{1}}_{\left\lbrack a... | Yes |
For any \( u \in {L}_{ad}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \), the equation\n\n\[ \n{Z}_{t} = 1 + {\int }_{0}^{t}{Z}_{{s}^{ - }}{u}_{s}d{M}_{s},\;t \in \left\lbrack {0, T}\right\rbrack \n\]\n\n\( \left( {2.13.1}\right) \)\n\nhas a unique solution \( {\left( {\xi }_{t}\left( u\right) \right) }_{t \in ... | \[ \n{\xi }_{t}\left( u\right) = \exp \left( {{\int }_{0}^{t}{u}_{s}d{M}_{s} - \frac{1}{2}{\int }_{0}^{t}{u}_{s}^{2}{\mathbf{1}}_{\left\{ {\phi }_{s} = 0\right\} }{ds}}\right) \mathop{\prod }\limits_{{s \in {J}_{M}^{t}}}\left( {1 + {u}_{s}{\phi }_{s}}\right) {\mathrm{e}}^{-{u}_{s}{\phi }_{s}}, \n\]\n\n\( \left( {2.13.2... | Yes |
Proposition 2.13.2. Given \( u \in {L}^{\infty }\left( \left\lbrack {0, T}\right\rbrack \right) ,{\xi }_{T}\left( u\right) \) can be represented as\n\n\[ \n{\xi }_{T}\left( u\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{1}{n!}{I}_{n}\left( {{u}^{\otimes n}{\mathbf{1}}_{{\left\lbrack 0, T\right\rbrack }^{n}}}... | Proof. Letting\n\n\[ \n{Z}_{t}^{n} = 1 + \mathop{\sum }\limits_{{k = 1}}^{n}\frac{1}{k!}{I}_{k}\left( {{u}^{\otimes k}{\mathbf{1}}_{{\left\lbrack 0, t\right\rbrack }^{k}}}\right) \n\]\n\nwe have\n\n\[ \n1 + {\int }_{0}^{t}{u}_{\tau }{Z}_{\tau }^{n}d{M}_{\tau } = 1 + \mathop{\sum }\limits_{{k = 1}}^{{n + 1}}\frac{1}{k!}... | Yes |
Proposition 3.2.3. For all \( t \in {\mathbb{R}}_{ + } > 0 \) and \( F \in {\mathbb{D}}_{2,1}\left( {\lbrack t,\infty }\right) ) \) we have\n\n\[ \mathbb{E}\left\lbrack {F \mid {\mathcal{F}}_{t}}\right\rbrack = \mathbb{E}\left\lbrack F\right\rbrack + {\int }_{0}^{t}\mathbb{E}\left\lbrack {{D}_{s}F \mid {\mathcal{F}}_{s... | Proof. This is a direct consequence of (3.2.1) and Proposition 2.5.7. | No |
Lemma 3.2.4. For all \( t \in {\mathbb{R}}_{ + } \) and \( F \in {\mathbb{D}}_{2,1}\left( {\lbrack t,\infty }\right) ) \) we have\n\n\[ \mathbb{E}\left\lbrack {\left( \mathbb{E}\left\lbrack F \mid {\mathcal{F}}_{t}\right\rbrack \right) }^{2}\right\rbrack = {\left( \mathbb{E}\left\lbrack F\right\rbrack \right) }^{2} + \... | Proof. From the Itô isometry (2.5.4) and Relation 3.2.2 we have\n\n\[ \mathbb{E}\left\lbrack {\left( \mathbb{E}\left\lbrack F \mid {\mathcal{F}}_{t}\right\rbrack \right) }^{2}\right\rbrack = \mathbb{E}\left\lbrack {\left( \mathbb{E}\left\lbrack F\right\rbrack + {\int }_{0}^{t}\mathbb{E}\left\lbrack {D}_{s}F \mid {\math... | Yes |
Lemma 3.2.5. The operator\n\n\[ \nF \mapsto {\left( \mathbb{E}\left\lbrack {D}_{t}F \mid {\mathcal{F}}_{t}\right\rbrack \right) }_{t \in {\mathbb{R}}_{ + }} \]\n\ndefined on \( \mathcal{S} \) extends to a continuous operator from \( {L}^{2}\left( \Omega \right) \) into \( {L}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }... | Proof. This follows from the bound\n\n\[ \n\parallel \mathbb{E}\left\lbrack {{D}_{ \cdot }F \mid {\mathcal{F}}_{ \cdot }}\right\rbrack {\parallel }_{{L}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) }^{2} = \parallel F{\parallel }_{{L}^{2}\left( \Omega \right) }^{2} - {\left( \mathbb{E}\left\lbrack F\right\rbrack... | Yes |
Proposition 3.2.7. For all \( F \in \operatorname{Dom}\left( D\right) \) we have\n\n\[ \operatorname{Var}\left( F\right) \leq \parallel {DF}{\parallel }_{{L}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) }^{2}. \] | Proof. From Lemma 3.2.4 we have\n\n\[ \operatorname{Var}\left( F\right) = \mathbb{E}\left\lbrack {\left| F - E\left\lbrack F\right\rbrack \right| }^{2}\right\rbrack \]\n\n\[ = \mathbb{E}\left\lbrack {\left( {\int }_{0}^{\infty }\mathbb{E}\left\lbrack {D}_{t}F \mid {\mathcal{F}}_{t}\right\rbrack d{M}_{t}\right) }^{2}\ri... | Yes |
Corollary 3.2.8. Under the Clark formula Assumption 3.2.1 the martingale \( {\left( {M}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) has the predictable representation property. | Proof. Definition 2.6.1 is satisfied because \( \mathcal{S} \) is dense in \( {L}^{2}\left( \Omega \right) \) and the process \( {\left( \mathbb{E}\left\lbrack {D}_{t}F \mid {\mathcal{F}}_{t}\right\rbrack \right) }_{t \in {\mathbb{R}}_{ + }} \) in (3.2.1) can be approximated by a sequence in \( \mathcal{P} \) from Prop... | Yes |
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