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Proposition 11. Suppose the integral domain \( R \) is a local ring that is not a field. Then \( R \) is a Discrete Valuation Ring if and only if every nonzero fractional ideal of \( R \) is invertible. | Proof: If \( R \) is a D.V.R. with uniformizing parameter \( t \), then by Proposition 5 every nonzero ideal of \( R \) is of the form \( \left( {t}^{n}\right) \) for some \( n \geq 0 \) and every element \( d \) in \( R \) can be written in the form \( u{t}^{m} \) for some unit \( u \in R \) and some \( m \geq 0 \) . ... | Yes |
Proposition 12. Let \( v \) be a point on the irreducible affine curve \( C \) over \( k \) . Then \( C \) is nonsingular at \( v \) if and only if the local ring \( {\mathcal{O}}_{v, C} \) is a Discrete Valuation Ring. | Proof: Suppose first that \( v \) is nonsingular. Then \( {\dim }_{k}\left( {{\mathfrak{m}}_{v, C}/{\mathfrak{m}}_{v, C}^{2}}\right) = 1 \), and since \( {\mathcal{O}}_{v, C} \) is Noetherian, it follows from Exercise 12 in Section 1 that \( {\mathfrak{m}}_{v, C} \) is principal. Hence \( {\mathcal{O}}_{v, C} \) is a D... | Yes |
Corollary 13. An irreducible affine curve \( C \) over an algebraically closed field \( k \) is smooth if and only if its coordinate ring \( k\left\lbrack C\right\rbrack \) is integrally closed. | Proof: The curve \( C \) is smooth if and only if every localization \( {\mathcal{O}}_{v, C} \) is a D.V.R. Since \( k\left\lbrack C\right\rbrack \) has Krull dimension 1 (Exercise 11 in Section 1), the same is true for each \( {\mathcal{O}}_{v, C} \) . It then follows by Theorem 7(5) that every localization \( {\mathc... | No |
Theorem 15. Suppose \( R \) is an integral domain with fraction field \( K \neq R \) . The following are equivalent conditions for \( R \) to be a Dedekind Domain:\n\n(1) The ring \( R \) is Noetherian, integrally closed, and every nonzero prime ideal is maximal.\n\n(2) The ring \( R \) is Noetherian and for each nonze... | Proof: If \( R \) satisfies (1), then \( {R}_{P} \) is a D.V.R. by Corollary 8, so (1) implies (2). Conversely, assume each \( {R}_{P} \) is a D.V.R. Then \( R \) is integrally closed by Proposition 49 in Section 15.4 and every nonzero prime ideal is maximal by Proposition 46(3) in Section 15.4, so (2) implies (1).\n\n... | Yes |
Corollary 16. If \( {\mathcal{O}}_{K} \) is the ring of integers in an algebraic number field \( K \) then every nonzero ideal \( I \) in \( {\mathcal{O}}_{K} \) can be written uniquely as the product of powers of distinct prime ideals: | \[ I = {P}_{1}^{{e}_{1}}{P}_{2}^{{e}_{2}}\cdots {P}_{n}^{{e}_{n}} \] where \( {P}_{1},\ldots ,{P}_{n} \) are distinct prime ideals and \( {e}_{i} \geq 1 \) for \( i = 1,\ldots, n \) . | Yes |
Proposition 18. (Chinese Remainder Theorem) Suppose \( R \) is a Dedekind Domain, \( {P}_{1},{P}_{2},\ldots ,{P}_{n} \) are distinct prime ideals in \( R \) and \( {a}_{i} \geq 0 \) are integers, \( i = 1,\ldots, n \) . Then\n\n\[ R/{P}_{1}^{{a}_{1}}\cdots {P}_{n}^{{a}_{n}} \cong R/{P}_{1}^{{a}_{1}} \times R/{P}_{2}^{{... | Proof: This is immediate from Theorem 17 in Section 7.6 since the previous proposition shows that the \( {P}_{i}^{{a}_{i}} \) are pairwise comaximal ideals. | Yes |
Corollary 19. Suppose \( I \) is an ideal in the Dedekind Domain \( R \) . Then\n\n(1) there is an ideal \( J \) of \( R \) relatively prime to \( I \) such that the product \( {IJ} = \left( a\right) \) is a principal ideal,\n\n(2) if \( I \) is nonzero then every ideal in the quotient \( R/I \) is principal; equivalen... | Proof: Suppose \( I = {P}_{1}^{{e}_{1}}\cdots {P}_{n}^{{e}_{n}} \) is the prime ideal factorization of \( I \) in \( R \) . For each \( i = 1,\ldots, n \), let \( {r}_{i} \) be an element of \( {P}_{i}^{{e}_{i}} - {P}_{i}^{{e}_{i} + 1} \) . By the proposition, there is an element \( a \in R \) with \( a \equiv {r}_{i}{... | Yes |
Corollary 20. If \( R \) is a Dedekind Domain then \( R \) is a P.I.D. (i.e., \( R \) has class number 1) if and only if \( R \) is a U.F.D. | Proof: Every P.I.D. is a U.F.D., so suppose that \( R \) is a U.F.D. and let \( P \) be any prime ideal in \( R \) . Then \( P = {Ra} + {Rb} \) for some \( a \neq 0 \) and \( b \) in \( R \) by Corollary 19. We have \( \left( {a}^{\prime }\right) \subseteq P \) for one of the irreducible factors \( {a}^{\prime } \) of ... | Yes |
Proposition 21. Let \( R \) be a Dedekind Domain with fraction field \( K \). (1) Suppose \( I \) and \( J \) are two fractional ideals of \( R \). Then \( I \cong J \) as \( R \)-modules if and only if \( I \) and \( J \) differ by a nonzero principal ideal: \( I = \left( a\right) J \) for some \( 0 \neq a \in K \). | Proof: Multiplication by \( 0 \neq a \in K \) gives an \( R \)-module isomorphism from \( J \) to \( \left( a\right) J \), so if \( I = \left( a\right) J \) we have \( I \cong J \) as \( R \)-modules. For the converse, observe that we may assume \( J \neq 0 \) and then \( I \cong J \) implies \( R \cong {J}^{-1}I \). B... | Yes |
Corollary 23. A finitely generated module over a Dedekind Domain is projective if and only if it is torsion free. | Proof: We showed that a finitely generated torsion free \( R \) -module is projective in the proof of Theorem 22, so by the decomposition of \( M \) in Theorem 22, \( M \) is projective if and only if \( \operatorname{Tor}\left( M\right) \) is projective (cf. Exercise 3 in Section 10.5). To complete the proof it suffic... | No |
Proposition 1. A homomorphism \( \alpha : \mathcal{A} \rightarrow \mathcal{B} \) of cochain complexes induces group homomorphisms from \( {H}^{n}\left( \mathcal{A}\right) \) to \( {H}^{n}\left( \mathcal{B}\right) \) for \( n \geq 0 \) on their respective cohomology groups. | Proof: It is an easy exercise to show that the commutativity of (4) implies that the images and kernels at each stage of the maps in the first row are mapped to the corresponding images and kernels for the maps in the second row, thus giving a well defined map on the respective quotient (cohomology) groups. | No |
Theorem 2. (The Long Exact Sequence in Cohomology) Let \( 0 \rightarrow \mathcal{A}\overset{\alpha }{ \rightarrow }\mathcal{B}\overset{\beta }{ \rightarrow }\mathcal{C} \rightarrow 0 \) be a short exact sequence of cochain complexes. Then there is a long exact sequence of cohomology groups:\n\n\[ 0 \rightarrow {H}^{0}\... | Proof: The details of this proof are somewhat lengthy. For each \( n \) the verification that the sequence \( {H}^{n}\left( \mathcal{A}\right) \rightarrow {H}^{n}\left( \mathcal{B}\right) \rightarrow {H}^{n}\left( \mathcal{C}\right) \) is exact is a straightforward check of the definition of exactness of each map, simi... | No |
For any \( R \) -module \( A \) we have \( {\operatorname{Ext}}_{R}^{0}\left( {A, D}\right) \cong {\operatorname{Hom}}_{R}\left( {A, D}\right) \) . | Since the sequence \( {P}_{1}\overset{{d}_{1}}{ \rightarrow }{P}_{0}\overset{\epsilon }{ \rightarrow }A \rightarrow 0 \) is exact, it follows that the corresponding sequence \( 0 \rightarrow {\operatorname{Hom}}_{R}\left( {A, D}\right) \overset{\epsilon }{ \rightarrow }{\operatorname{Hom}}_{R}\left( {{P}_{0}, D}\right)... | Yes |
Proposition 4. Let \( f : A \rightarrow {A}^{\prime } \) be any homomorphism of \( R \) -modules and take projective resolutions of \( A \) and \( {A}^{\prime } \), respectively. Then for each \( n \geq 0 \) there is a lift \( {f}_{n} \) of \( f \) such that the following diagram commutes:\n\n![a9dded23-11dd-4050-9460-... | Proof: Given the two rows and map \( f \) in (8), then since \( {P}_{0} \) is projective we may lift the map \( {f\epsilon } : {P}_{0} \rightarrow {A}^{\prime } \) to a map \( {f}_{0} : {P}_{0} \rightarrow {P}_{0}^{\prime } \) in such a way that \( {\epsilon }^{\prime }{f}_{0} = {f\epsilon } \) (Proposition 30(2) in Se... | Yes |
Proposition 5. Let \( f : A \rightarrow {A}^{\prime } \) be a homomorphism of \( R \) -modules and take projective resolutions of \( A \) and \( {A}^{\prime } \) as in Proposition 4. Then for every \( n \) there is an induced group homomorphism \( {\varphi }_{n} : {\operatorname{Ext}}_{R}^{n}\left( {{A}^{\prime }, D}\r... | Proof: The existence of the map on the cohomology groups \( {\operatorname{Ext}}_{R}^{n} \) follows from Proposition 1 applied to the homomorphism of cochain complexes (9). The more difficult part is showing these maps do not depend on the choice of lifts \( {f}_{n} \) in Proposition 4. This is easily seen to be equiva... | No |
Theorem 6. The groups \( {\operatorname{Ext}}_{R}^{n}\left( {A, D}\right) \) depend only on \( A \) and \( D \), i.e., they are independent of the choice of projective resolution of \( A \) . | Proof: In the notation of Proposition 4 let \( {A}^{\prime } = A \), let \( f : A \rightarrow {A}^{\prime } \) be the identity map and let the two rows of (8) be two projective resolutions of \( A \) . For any choice of lifts of the identity map, the resulting homomorphisms on cohomology groups \( {\varphi }_{n} : {\op... | Yes |
Proposition 7. (Simultaneous Resolution) Let \( 0 \rightarrow L \rightarrow M \rightarrow N \rightarrow 0 \) be a short exact sequence of \( R \) -modules, let \( L = A \) have a projective resolution as in (6) above, and let \( N \) have a similar projective resolution where the projective modules are denoted by \( {\... | Proof: The left and right nonzero columns of (11) are exact by hypothesis. The modules in the middle column are projective (cf. Exercise 3, Section 10.5) and the row maps are the obvious ones to make each row a split exact sequence. It remains then to define the vertical maps in the middle column in such a way as to ma... | No |
Theorem 8. Let \( 0 \rightarrow L \rightarrow M \rightarrow N \rightarrow 0 \) be a short exact sequence of \( R \) -modules. Then there is a long exact sequence of abelian groups\n\n\[ 0 \rightarrow {\operatorname{Hom}}_{R}\left( {N, D}\right) \rightarrow {\operatorname{Hom}}_{R}\left( {M, D}\right) \rightarrow {\oper... | Proof: Take a simultaneous projective resolution of the short exact sequence as in Proposition 7 and take homomorphisms into \( D \) . To obtain the cohomology groups \( {\operatorname{Ext}}_{R}^{n} \) from the resulting diagram, as noted in the discussion preceding Proposition 3 we replace the lowest nonzero row in th... | Yes |
Proposition 9. For an \( R \) -module \( Q \) the following are equivalent:\n\n(1) \( Q \) is injective,\n\n(2) \( {\operatorname{Ext}}_{R}^{1}\left( {A, Q}\right) = 0 \) for all \( R \) -modules \( A \), and\n\n(3) \( {\operatorname{Ext}}_{R}^{n}\left( {A, Q}\right) = 0 \) for all \( R \) -modules \( A \) and all \( n... | Proof: We showed (2) implies (1) above, and (3) implies (2) is trivial, so it remains to show that if \( Q \) is injective then \( {\operatorname{Ext}}_{R}^{n}\left( {A, Q}\right) = 0 \) for all \( R \) -modules \( A \) and all \( n \geq 1 \) . Take a projective resolution\n\n\[ \n\cdots \rightarrow {P}_{n} \rightarrow... | Yes |
Theorem 10. Let \( 0 \rightarrow L \rightarrow M \rightarrow N \rightarrow 0 \) be a short exact sequence of \( R \) -modules. Then there is a long exact sequence of abelian groups\n\n\[ 0 \rightarrow {\operatorname{Hom}}_{R}\left( {D, L}\right) \rightarrow {\operatorname{Hom}}_{R}\left( {D, M}\right) \rightarrow {\ope... | Proof: Let \( 0 \rightarrow L \rightarrow M \rightarrow N \rightarrow 0 \) be a short exact sequence of \( R \) -modules. By taking a projective resolution of \( D \) and then applying \( {\operatorname{Hom}}_{R}\left( {\_, L}\right) ,{\operatorname{Hom}}_{R}\left( {\_, M}\right) \) and \( {\operatorname{Hom}}_{R}\left... | Yes |
Proposition 11. For an \( R \) -module \( P \) the following are equivalent:\n\n(1) \( P \) is projective,\n\n(2) \( {\operatorname{Ext}}_{R}^{1}\left( {P, B}\right) = 0 \) for all \( R \) -modules \( B \), and\n\n(3) \( {\operatorname{Ext}}_{R}^{n}\left( {P, B}\right) = 0 \) for all \( R \) -modules \( B \) and all \(... | Proof: We proved (2) implies (1) above, and (3) implies (2) is trivial, so it remains to prove that (1) implies (3). If \( P \) is a projective \( R \) -module, then the simple exact sequence\n\n\[ 0 \rightarrow P\overset{1}{ \rightarrow }P \rightarrow 0 \]\n\ngiven by the identity map on \( P \) is a projective resolu... | Yes |
Proposition 16. For a right \( R \) -module \( D \) the following are equivalent:\n\n(1) \( D \) is a flat \( R \) -module,\n\n(2) \( {\operatorname{Tor}}_{1}^{R}\left( {D, B}\right) = 0 \) for all left \( R \) -modules \( B \), and\n\n(3) \( {\operatorname{Tor}}_{n}^{R}\left( {D, B}\right) = 0 \) for all left \( R \) ... | We have defined \( {\operatorname{Tor}}_{n}^{R}\left( {A, B}\right) \) as the homology of the chain complex obtained by ten-soring a projective resolution of \( B \) on the left with \( A \) . The same groups are obtained by taking the homology of the chain complex obtained by tensoring a projective resolution of \( A ... | No |
Corollary 18. If \( A \) is an abelian group then \( A \) is torsion free if and only if \( {\mathrm{{Tor}}}_{1}\left( {A, B}\right) = 0 \) for every abelian group \( B \) (in which case \( A \) is flat as a \( \mathbb{Z} \) -module). | Proof: By the proposition, if \( A \) has no elements of finite order then we have \( {\operatorname{Tor}}_{1}\left( {A, B}\right) = {\operatorname{Tor}}_{1}\left( {t\left( A\right), B}\right) = {\operatorname{Tor}}_{1}\left( {0, B}\right) = 0 \) for every abelian group \( B \) . Conversely, if \( {\operatorname{Tor}}_... | Yes |
Proposition 20. Suppose \( {mA} = 0 \) for some integer \( m \geq 1 \) (i.e., the \( G \) -module \( A \) has exponent dividing \( m \) as an abelian group). Then\n\n\[ m{Z}^{n}\left( {G, A}\right) = m{B}^{n}\left( {G, A}\right) = m{H}^{n}\left( {G, A}\right) = 0\;\text{ for all }n \geq 0. \] | Proof: If \( f \in {C}^{n}\left( {G, A}\right) \) is an \( n \) -cochain then \( f \in A \) (if \( n = 0 \) ), in which case \( {mf} = 0 \), or \( f \) is a function from \( {G}^{n} \) to \( A \) (if \( n \geq 1 \) ), in which case \( {mf} \) is a function from \( {G}^{n} \) to \( {mA} = 0 \), so again \( {mf} = 0 \) .... | Yes |
Corollary 22. (Dimension Shifting) Suppose \( 0 \rightarrow A \rightarrow M \rightarrow C \rightarrow 0 \) is a short exact sequence of \( G \) -modules and that \( M \) is cohomologically trivial for \( G \) . Then there is an exact sequence\n\n\[ 0 \rightarrow {A}^{G} \rightarrow {M}^{G} \rightarrow {C}^{G} \rightarr... | Proof: Since \( M \) is cohomologically trivial for \( G \), the portion\n\n\[ {H}^{n}\left( {G, M}\right) \rightarrow {H}^{n}\left( {G, C}\right) \rightarrow {H}^{n + 1}\left( {G, A}\right) \rightarrow {H}^{n + 1}\left( {G, M}\right) \]\n\nof the long exact sequence in Theorem 21 reduces to\n\n\[ 0 \rightarrow {H}^{n}... | Yes |
Proposition 23. (Shapiro’s Lemma) For any subgroup \( H \) of \( G \) and any \( H \) -module \( A \) we have \( {H}^{n}\left( {G,{M}_{H}^{G}\left( A\right) }\right) \cong {H}^{n}\left( {H, A}\right) \) for \( n \geq 0 \) . | Proof: Let \( \cdots \rightarrow {P}_{n} \rightarrow \cdots \rightarrow {P}_{0} \rightarrow \mathbb{Z} \rightarrow 0 \) be a resolution of \( \mathbb{Z} \) by projective \( G \) -modules (for example, the standard resolution). The cohomology groups \( {H}^{n}\left( {G,{M}_{H}^{G}\left( A\right) }\right) \) are computed... | Yes |
For any \( G \) -module \( A \) the module \( {M}_{1}^{G}\left( A\right) \) is cohomologically trivial for \( G \), i.e., \( {H}^{n}\left( {G,{M}_{1}^{G}\left( A\right) }\right) = 0 \) for all \( n \geq 1 \) . | This follows immediately from the proposition applied with \( H = 1 \) together with the computation of the cohomology of the trivial group in Example 2 preceding Proposition 20. | No |
Proposition 26. Suppose \( H \) is a subgroup of \( G \) of index \( m \) . Then \( \operatorname{Cor} \circ \operatorname{Res} = m \), i.e., if \( c \) is a cohomology class in \( {H}^{n}\left( {G, A}\right) \) for some \( G \) -module \( A \), then\n\n\[ \operatorname{Cor}\left( {\operatorname{Res}\left( c\right) }\r... | Proof: This follows from the explicit formula for corestriction in Example 4 above, as follows. If \( f \in {\operatorname{Hom}}_{\mathbb{Z}H}\left( {{P}_{n}, A}\right) \) were in \( {\operatorname{Hom}}_{\mathbb{Z}G}\left( {{P}_{n}, A}\right) \), i.e., if \( f \) were also a \( G \) - module homomorphism, then \( {g}_... | Yes |
Corollary 27. Suppose the finite group \( G \) has order \( m \) . Then \( m{H}^{n}\left( {G, A}\right) = 0 \) for all \( n \geq 1 \) and any \( G \) -module \( A \) . | Proof: Let \( H = 1 \), so that \( \left\lbrack {G : H}\right\rbrack = m \), in Proposition 26. Then for any class \( c \in {H}^{n}\left( {G, A}\right) \) we have \( {mc} = \operatorname{Cor}\left( {\operatorname{Res}\left( c\right) }\right) \) . Since \( \operatorname{Res}\left( c\right) \in {H}^{n}\left( {H, A}\right... | Yes |
Corollary 28. If \( G \) is a finite group then \( {H}^{n}\left( {G, A}\right) \) is a torsion abelian group for all \( n \geq 1 \) and all \( G \) -modules \( A \) . | Proof: This is immediate from the previous corollary. | No |
Corollary 29. Suppose \( G \) is a finite group whose order is relatively prime to the exponent of the \( G \) -module \( A \) . Then \( {H}^{n}\left( {G, A}\right) = 0 \) for all \( n \geq 1 \) . In particular, if \( A \) is a finite abelian group with \( \left( {\left| G\right| ,\left| A\right| }\right) = 1 \) then \... | Proof: This follows since the abelian group \( {H}^{n}\left( {G, A}\right) \) is annihilated by \( \left| G\right| \) by the previous corollary and is annihilated by the exponent of \( A \) by Proposition 20 . | No |
Proposition 31. Let \( A \) be a \( G \) -module and let \( E \) be the semidirect product \( A \rtimes G \) . For each cocycle \( f \in {Z}^{1}\left( {G, A}\right) \) define \( {\sigma }_{f} : E \rightarrow E \) by\n\n\[ \n{\sigma }_{f}\left( \left( {a, g}\right) \right) = \left( {a + f\left( g\right), g}\right) .\n\]... | Proof: It is an exercise to see that the cocycle condition implies \( {\sigma }_{f} \) is an automorphism of \( E \) that stabilizes the chain \( 1 \trianglelefteq A \trianglelefteq E \) . Likewise one checks directly that \( {\sigma }_{{f}_{1} + {f}_{2}} = {\sigma }_{{f}_{1}} \circ {\sigma }_{{f}_{2}} \), so the map \... | No |
Corollary 38. If \( A \) is a finite abelian group and \( \left( {\left| A\right| ,\left| G\right| }\right) = 1 \) then every extension of \( G \) by \( A \) splits. | Proof: This follows immediately from Corollary 29 in Section 2. | No |
Theorem 39. (Schur’s Theorem) If \( E \) is any finite group containing a normal subgroup \( N \) whose order and index are relatively prime, then \( N \) has a complement in \( E \) . | Proof: We use induction on the order of \( E \) . Since we may assume \( N \neq 1 \), let \( p \) be a prime dividing \( \left| N\right| \) and let \( P \) be a Sylow \( p \) -subgroup of \( N \) . Let \( {E}_{0} \) be the normalizer in \( E \) of \( P \) and let \( {N}_{0} = N \cap {E}_{0} \) . By Frattini’s Argument ... | Yes |
Proposition 40. The \( F \) -algebra \( {B}_{f} \) with \( K \) -vector space basis \( {u}_{\sigma } \) in (39) and multiplication defined by (40) is a central simple \( F \) -algebra. | Proof: It remains to show that the center of \( {B}_{f} \) is \( F \) and that \( {B}_{f} \) contains no nonzero proper ideals. Suppose \( x = \mathop{\sum }\limits_{{\sigma \in G}}{\alpha }_{\sigma }{u}_{\sigma } \) is an element in the center of \( {B}_{f} \) . Then \( {x\beta } = {\beta x} \) for \( \beta \in K \) s... | Yes |
Proposition 41. The crossed product algebra for the trivial cohomology class in \( {H}^{2}\left( {G,{K}^{ \times }}\right) \) is isomorphic to the matrix algebra \( {M}_{n}\left( F\right) \) where \( n = \left\lbrack {K : F}\right\rbrack \) . | Proof: If \( \alpha \in K \) then multiplication by \( \alpha \) defines a linear transformation \( {T}_{\alpha } \) of \( K \) viewed as an \( n \) -dimensional vector space over \( F \) . Similarly, every automorphism \( \sigma \in G \) defines an \( F \) -linear transformation \( {T}_{\sigma } \) of \( K \), and we ... | Yes |
Theorem 1. (Maschke’s Theorem) Let \( G \) be a finite group and let \( F \) be a field whose characteristic does not divide \( \left| G\right| \) . If \( V \) is any \( {FG} \) -module and \( U \) is any submodule of \( V \), then \( V \) has a submodule \( W \) such that \( V = U \oplus W \) (i.e., every submodule is... | Proof: The idea of the proof of Maschke’s Theorem is to produce an \( {FG} \) -module homomorphism\n\n\[ \pi : V \rightarrow U \]\n\nwhich is a projection onto \( U \), i.e., which satisfies the following two properties:\n\n(i) \( \pi \left( u\right) = u\; \) for all \( u \in U \)\n\n(ii) \( \pi \left( {\pi \left( v\ri... | No |
Corollary 2. If \( G \) is a finite group and \( F \) is a field whose characteristic does not divide \( \left| G\right| \), then every finitely generated \( {FG} \) -module is completely reducible (equivalently, every \( F \) -representation of \( G \) of finite degree is completely reducible). | Proof: Let \( V \) be a finitely generated \( {FG} \) -module. As noted above, \( V \) is finite dimensional over \( F \), so we may proceed by induction on its dimension. If \( V \) is irreducible, it is completely reducible and the result holds. Suppose therefore that \( V \) has a proper, nonzero \( {FG} \) -submodu... | Yes |
Corollary 3. Let \( G \) be a finite group, let \( F \) be a field whose characteristic does not divide \( \left| G\right| \) and let \( \varphi : G \rightarrow {GL}\left( V\right) \) be a representation of \( G \) of finite degree. Then there is a basis of \( V \) such that for each \( g \in G \) the matrix of \( \var... | Proof: By Corollary 2 we may write \( V = {U}_{1} \oplus {U}_{2} \oplus \cdots \oplus {U}_{m} \), where \( {U}_{i} \) is an irreducible \( {FG} \) -submodule of \( V \) . Let \( {\mathcal{B}}_{i} \) be a basis of \( {U}_{i} \) and let \( \mathcal{B} \) be the union of the \( {\mathcal{B}}_{i} \) ’s. For each \( g \in G... | Yes |
Theorem 4. (Wedderburn’s Theorem) Let \( R \) be a nonzero ring with 1 (not necessarily commutative). Then the following are equivalent:\n\n(1) every \( R \) -module is projective\n\n(2) every \( R \) -module is injective\n\n(3) every \( R \) -module is completely reducible\n\n(4) the ring \( R \) considered as a left ... | Proof: A proof of Wedderburn's Theorem is outlined in Exercises 1 to 10 | No |
Lemma 7. Let \( R \) be an arbitrary nonzero ring.\n\n(1) If \( M \) and \( N \) are simple \( R \) -modules and \( \varphi : M \rightarrow N \) is a nonzero \( R \) -module homomorphism, then \( \varphi \) is an isomorphism.\n\n(2) (Schur’s Lemma) If \( M \) is a simple \( R \) -module, then \( {\operatorname{Hom}}_{R... | Proof of Lemma 7: To prove (1) note that since \( \varphi \) is nonzero, \( \ker \varphi \) is a proper submodule of \( M \) . By simplicity of \( M \) we have \( \ker \varphi = 0 \) . Similarly, the image of \( \varphi \) is a nonzero submodule of the simple module \( N \), hence \( \varphi \left( M\right) = N \) . Th... | Yes |
Proposition 8. Let \( R = {R}_{1} \times {R}_{2} \times \cdots \times {R}_{r} \), where \( {R}_{i} \) is the ring of \( {n}_{i} \times {n}_{i} \) matrices over the division ring \( {\Delta }_{i} \), for \( i = 1,2,\ldots, r \) .\n\n(1) Identify \( {R}_{i} \) with the \( {i}^{\text{th }} \) component of the direct produ... | Proof: In part (1) since multiplication in the direct product of rings is componentwise it is clear that \( {z}_{i} \) times the element \( \left( {{a}_{1},\ldots ,{a}_{r}}\right) \) of \( R \) is the \( r \) -tuple with \( {a}_{i} \) in position \( i \) and zeros elsewhere. Thus \( {R}_{i} = {z}_{i}R,{z}_{i} \) is the... | Yes |
Proposition 9. If \( \Delta \) is a division ring that is a finite dimensional vector space over an algebraically closed field \( F \) and \( F \subseteq Z\left( \Delta \right) \), then \( \Delta = F \) . | Proof: Since \( F \subseteq Z\left( \Delta \right) \), for each \( \alpha \in \Delta \) the division ring generated by \( \alpha \) and \( F \) is a field. Also, since \( \Delta \) is finite dimensional over \( F \) the field \( F\left( \alpha \right) \) is a finite extension of \( F \) . Because \( F \) is algebraical... | Yes |
(1) Let \( A \) be a finite abelian group. Every irreducible complex representation of \( A \) is 1-dimensional (i.e., is a homomorphism from \( A \) into \( {\mathbb{C}}^{ \times } \) ) and \( A \) has \( \left| A\right| \) inequivalent irreducible complex representations. Furthermore, every finite dimensional complex... | Proof: If \( A \) is abelian, \( \mathbb{C}A \) is a commutative ring. Since a \( k \times k \) matrix ring is not commutative whenever \( k > 1 \) we must have each \( {n}_{i} = 1 \) . Thus \( r = \left| A\right| ( = \) the number of conjugacy classes of \( A \) ). Since every \( \mathbb{C}A \) -module is a direct sum... | Yes |
Proposition 13. Let \( {z}_{1},\ldots ,{z}_{r} \) be the orthogonal primitive central idempotents in \( \mathbb{C}G \) labelled in such a way that \( {z}_{i} \) acts as the identity on the irreducible \( \mathbb{C}G \) -module \( {M}_{i} \), and let \( {\chi }_{i} \) be the character afforded by \( {M}_{i} \). Then\n\n... | Proof: Let \( z = {z}_{i} \) and write\n\n\[ \nz = \mathop{\sum }\limits_{{g \in G}}{\alpha }_{g}g \n\]\n\nRecall from Example 4 in this section that if \( \rho \) is the regular character of \( G \) then\n\n\[ \n\rho \left( g\right) = \left\{ \begin{array}{ll} 0 & \text{ if }g \neq 1 \\ \left| G\right| & \text{ if }g ... | Yes |
Proposition 14. If \( \psi \) is any character of \( G \) then \( \psi \left( x\right) \) is a sum of roots of \( 1 \) in \( \mathbb{C} \) and \( \psi \left( {x}^{-1}\right) = \overline{\psi \left( x\right) } \) for all \( x \in G. | Proof: Let \( \varphi \) be a representation whose character is \( \psi \), fix an element \( x \in G \) and let \( \left| x\right| = k \) . Since the minimal polynomial of \( \varphi \left( x\right) \) divides \( {X}^{k} - 1 \) (hence has distinct roots), there is a basis of the underlying vector space such that the m... | Yes |
Theorem 15. (The First Orthogonality Relation for Group Characters) Let \( G \) be a finite group and let \( {\chi }_{1},\ldots ,{\chi }_{r} \) be the irreducible characters of \( G \) over \( \mathbb{C} \) . Then with respect to the inner product \( \left( {,\text{ }}\right) \) above we have\n\n\[ \left( {{\chi }_{i},... | Proof: We have just established that the irreducible characters form an orthonormal basis for the space of class functions. If \( \theta \) is any class function, write \( \theta = \mathop{\sum }\limits_{{i = 1}}^{r}{a}_{i}{\chi }_{i} \) , for some \( {a}_{i} \in \mathbb{C} \) . It follows from linearity of the Hermiti... | Yes |
Proposition 17. If \( {\psi }_{1} \) and \( {\psi }_{2} \) are characters, then so is their product \( {\psi }_{1}{\psi }_{2} \) . | Proof: Let \( {V}_{1} \) and \( {V}_{2} \) be \( \mathbb{C}G \) -modules affording characters \( {\psi }_{1} \) and \( {\psi }_{2} \) and define \( W = {V}_{1}{ \otimes }_{\mathbb{C}}{V}_{2} \) . Since each \( g \in G \) acts as a linear transformation on \( {V}_{1} \) and \( {V}_{2} \), the action of \( g \) on simple... | No |
Proposition 2. Let \( \alpha \in \mathbb{C} \). (1) The following are equivalent: (i) \( \alpha \) is an algebraic integer, (ii) \( \alpha \) is algebraic over \( \overline{\mathbb{Q}} \) and the minimal polynomial of \( \alpha \) over \( \mathbb{Q} \) has integer coefficients, and (iii) \( \mathbb{Z}\left\lbrack \alph... | Proof: These are established in Section 15.3. (The portion of Section 15.3 consisting of integral extensions and properties of algebraic integers may be read independently from the rest of Chapter 15.) | No |
Corollary 3. For every character \( \psi \) of the finite group \( G,\psi \left( x\right) \) is an algebraic integer for all \( x \in G \) . | Proof: By Proposition 14 in Section 18.3, \( \psi \left( x\right) \) is a sum of roots of 1 . Each root of 1 is an algebraic integer, so the result follows immediately from Proposition 2(2). | Yes |
Corollary 5. The degree of each complex irreducible representation of a finite group \( G \) divides the order of \( G \), i.e., \( {\chi }_{i}\left( 1\right) \left| G\right| \) for \( i = 1,2,\ldots, r \) . | Proof: Under the notation of Proposition 4 and with \( {g}_{j} \in {\mathcal{K}}_{j} \) we have\n\n\[ \frac{\left| G\right| }{{\chi }_{i}\left( 1\right) } = \frac{\left| G\right| }{{\chi }_{i}\left( 1\right) }\left( {{\chi }_{i},{\chi }_{i}}\right) \]\n\n\[ = \mathop{\sum }\limits_{{j = 1}}^{r}\frac{\left| {\mathcal{K}... | Yes |
Lemma 6. If \( G \) is any group that has a conjugacy class \( \mathcal{K} \) and an irreducible matrix representation \( \varphi \) with character \( \chi \) such that \( \left( {\left| \mathcal{K}\right| ,\chi \left( 1\right) }\right) = 1 \), then for \( g \in \mathcal{K} \) either \( \chi \left( g\right) = 0 \) or \... | Proof: By hypothesis there exist \( s, t \in \mathbb{Z} \) such that \( s\left| \mathcal{K}\right| + {t\chi }\left( 1\right) = 1 \) . Thus\n\n\[ s\left| \mathcal{K}\right| \chi \left( g\right) + {t\chi }\left( 1\right) \chi \left( g\right) = \chi \left( g\right) . \]\n\nDivide both sides of this by \( \chi \left( 1\rig... | Yes |
Lemma 7. If \( \left| \mathcal{K}\right| \) is a power of a prime for some nonidentity conjugacy class \( \mathcal{K} \) of \( G \) , then \( G \) is not a non-abelian simple group. | Proof: Suppose to the contrary that \( G \) is a non-abelian simple group and let \( \left| \mathcal{K}\right| = {p}^{c} \) . Let \( g \in \mathcal{K} \) . If \( c = 0 \) then \( g \in Z\left( G\right) \), contrary to a non-abelian simple group having a trivial center. As above, let \( {\chi }_{1},\ldots ,{\chi }_{r} \... | Yes |
Lemma 9. If \( G \) is solvable of order \( > 1 \), then there exists \( P \trianglelefteq G \) with \( P \) a nontrivial \( p \) -group for some prime \( p \) . | Proof: This is a special case of the exercise on minimal normal subgroups of solvable groups at the end of Section 6.1. One can see this easily by letting \( P \) be a nontrivial Sylow subgroup of the last nontrivial term, \( {G}^{\left( n - 1\right) } \), in the derived series of \( G \) (where \( G \) has solvable le... | No |
Lemma 10. Let \( G \) be a group of order \( {p}_{1}^{{\alpha }_{1}}{p}_{2}^{{\alpha }_{2}}\cdots {p}_{t}^{{\alpha }_{t}} \) where \( {p}_{1},\ldots ,{p}_{t} \) are distinct primes. Suppose there are subgroups \( H \) and \( \bar{K} \) of \( G \) such that for each \( i \in \{ 1,\ldots, t\} \) , either \( {p}_{i}^{{\al... | Proof: Fix some \( i \in \{ 1,\ldots, t\} \) and suppose first that \( {p}_{i}^{{\alpha }_{i}} \) divides the order of \( H \) . Since \( {HK} \) is a disjoint union of right cosets of \( H \) and each of these right cosets has order equal to \( \left| H\right| \), it follows that \( {p}_{i}^{{\bar{\alpha }}_{i}} \) di... | Yes |
Theorem 11. Let \( H \) be a subgroup of the finite group \( G \) and let \( {g}_{1},\ldots ,{g}_{m} \) be representatives for the distinct left cosets of \( H \) in \( G \) . Let \( V \) be an \( {FH} \) -module affording the matrix representation \( \varphi \) of \( H \) of degree \( n \) . The \( {FG} \) -module \( ... | Proof: First note that \( {FG} \) is a free right \( {FH} \) -module:\n\n\[ {FG} = {g}_{1}{FH} \oplus {g}_{2}{FH} \oplus \cdots \oplus {g}_{m}{FH}. \]\n\nSince tensor products commute with direct sums (Theorem 17, Section 10.4), as abelian groups we have\n\n\[ W = {FG}{ \otimes }_{FH}V \cong \left( {{g}_{1} \otimes V}\... | Yes |
In the notation of Theorem 11, if \( \psi \) is the character afforded by \( V \) then the induced character is given by \[ {\operatorname{Ind}}_{H}^{G}\left( \psi \right) \left( g\right) = \mathop{\sum }\limits_{{i = 1}}^{m}\psi \left( {{g}_{i}^{-1}g{g}_{i}}\right) \] where \( \psi \left( {{g}_{i}^{-1}g{g}_{i}}\right)... | Proof: From the matrix of \( g \) computed above, the blocks \( \varphi \left( {{g}_{i}^{-1}g{g}_{i}}\right) \) down the diagonal of \( \Phi \left( g\right) \) are zero except when \( {g}_{i}^{-1}g{g}_{i} \in H \). Thus the trace of the block matrix \( \Phi \left( g\right) \) is the sum of the traces of the matrices \(... | Yes |
Proposition 13. Let \( G \) be a Frobenius group of order \( {q}^{a}p \), where \( p \) and \( q \) are distinct primes, such that the Frobenius kernel \( Q \) is an elementary abelian \( q \) -group of order \( {q}^{a} \) and the cyclic group \( G/Q \) acts irreducibly by conjugation on \( Q \) . Then the following ho... | Proof: Note that \( {QP} \) equals \( G \) by order consideration. By definition of a Frobenius group and because \( Q \) is abelian, \( {C}_{G}\left( h\right) = Q \) for every nonidentity element \( h \) of \( Q \) . If \( x \) were an element of order \( {pq} \), then \( {x}^{p} \) would be an element of order \( q \... | Yes |
Proposition 14. Let \( G \) be a group, let \( H \) be a subgroup of \( G \) and let \( \psi \) and \( {\psi }^{\prime } \) be characters of \( H \) . (1) (Induction of characters is additive) \( {\operatorname{Ind}}_{H}^{G}\left( {\psi + {\psi }^{\prime }}\right) = {\operatorname{Ind}}_{H}^{G}\left( \psi \right) + {\o... | It follows from part (1) of Proposition 14 that if \( \mathop{\sum }\limits_{{i = 1}}^{s}{n}_{i}{\psi }_{i} \) is any integral linear combination of characters of \( H \) with \( {n}_{i} \geq 0 \) for all \( i \) then \[ {\operatorname{Ind}}_{H}^{G}\left( {\mathop{\sum }\limits_{{i = 1}}^{s}{n}_{i}{\psi }_{i}}\right) =... | Yes |
For any \( i \in \{ 1,2,3,4\} \) let \( q = {q}_{i} \), let \( Q = {Q}_{i} \), let \( N = {N}_{i} \) and let \( p = \left| {N : Q}\right| \) . Let \( {\psi }_{1},\ldots ,{\psi }_{k} \) be the distinct irreducible characters of \( N \) of degree \( p \) . Then there are distinct irreducible characters \( {\chi }_{1},\ld... | Proof: Let \( {\alpha }_{j} = {\psi }_{1} - {\psi }_{j} \) for \( j = 2,3,\ldots, k \) so \( {\alpha }_{j} \) satisfies the hypothesis of Lemma 15. Since \( {\psi }_{1} \neq {\psi }_{j} \), by Lemma 15\n\n\[ 2 = \left| \right| {\alpha }_{j}{\left| \right| }^{2} = {\left( {\alpha }_{j},{\alpha }_{j}\right) }_{N} = {\lef... | Yes |
Lemma 17. The exceptional characters associated to \( {Q}_{i} \) are all distinct from the exceptional characters associated to \( {Q}_{j} \) for \( i \) and \( j \) distinct elements of \( \{ 1,2,3,4\} \) . | Proof: Let \( \chi \) be an exceptional character associated to \( {Q}_{i} \) and let \( \theta \) be an exceptional character associated to \( {Q}_{j} \) . By construction, there are distinct irreducible characters \( \psi \) and \( {\psi }^{\prime } \) of \( {Q}_{i} \) such that \( {\psi }^{ * } - {\psi }^{\prime * }... | Yes |
Proposition 1. Let \( I \) be a nonempty countable set and for each \( i \in I \) let \( {A}_{i} \) be a set. The cardinality of the Cartesian product is the product of the cardinalities of the sets \( {A}_{i} \), i.e., \[ \left| {\mathop{\prod }\limits_{{i \in I}}{A}_{i}}\right| = \mathop{\prod }\limits_{{i \in I}}\le... | Proof: In order to count the number of choice functions note that each \( i \in I \) may be mapped to any of the \( \left| {A}_{i}\right| \) elements of \( {A}_{i} \) and for \( i \neq j \) the values of choice functions at \( i \) and \( j \) may be chosen completely independently. Thus the number of choice functions ... | Yes |
Theorem 2. Assuming the usual (Zermelo-Fraenkel) axioms of set theory, the following are equivalent: (1) Zorn's Lemma (2) the Axiom of Choice (3) the Well Ordering Principle. | Proof: This follows from elementary set theory. We refer the reader to Real and Abstract Analysis by Hewitt and Stromberg, Springer-Verlag, 1965, Section 3 for these equivalences and some others. | No |
Proposition 1.1. The stochastic integral with respect to Brownian motion \( {\left( {B}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \), defined on simple predictable processes \( {\left( {u}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) of the form (1.2) \( {by} \)\n\n\[ \n{\int }_{0}^{\infty }{u}_{t}d{B}_{t} \mathrel{\text{:... | Proof. We start by showing that the isometry (1.4) 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 \( 0 = {t}_{0} < {t}_{1} < \cdots {t}_{n} \) :\n\n\[ \n\mathbb{E}\left\lbrack {\left( {\int }_{0}^{\infty }{u}_{t}... | Yes |
For any \( u \in {L}_{ad}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) we have\n\n\[ \mathbb{E}\left\lbrack {{\int }_{0}^{\infty }{u}_{s}d{B}_{s} \mid {\mathcal{F}}_{t}}\right\rbrack = {\int }_{0}^{t}{u}_{s}d{B}_{s},\;t \in {\mathbb{R}}_{ + }.\n\]\nIn particular, \( {\int }_{0}^{t}{u}_{s}d{B}_{s} \) is \( {\m... | Proof. Let \( u \in \mathcal{P} \) have 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{B}_{s} \mid {\mathcal{F}}_{t}}\right\rbrack = \mathbb{E}\left\lbr... | Yes |
Corollary 1.1. The indefinite stochastic integral \( {\left( {\int }_{0}^{t}{u}_{s}d{B}_{s}\right) }_{t \in {\mathbb{R}}_{ + }} \) of \( u \in \) \( {L}_{ad}^{2}\left( {\Omega \times {\mathbb{R}}_{ + }}\right) \) is a martingale, i.e.:\n\n\[ \mathbb{E}\left\lbrack {{\int }_{0}^{t}{u}_{\tau }d{B}_{\tau } \mid {\mathcal{... | As an immediate consequence of the above corollary we have\n\n\[ \mathbb{E}\left\lbrack {{\int }_{t}^{\infty }{u}_{\tau }d{B}_{\tau } \mid {\mathcal{F}}_{t}}\right\rbrack = 0,\;\text{ and }\;\mathbb{E}\left\lbrack {{\int }_{0}^{t}{u}_{\tau }d{B}_{\tau } \mid {\mathcal{F}}_{t}}\right\rbrack = {\int }_{0}^{t}{u}_{\tau }d... | No |
Proposition 1.3. Let \( f \in {L}^{2}\left( {\mathbb{R}}_{ + }\right) \) . The stochastic integral\n\n\[ \n{\int }_{0}^{\infty }f\left( t\right) d{B}_{t} \]\n\n is a Gaussian random variable with mean 0 and variance\n\n\[ \n{\int }_{0}^{\infty }{\left| f\left( t\right) \right| }^{2}{dt} \]\n | Proof. From the relation\n\n\[ \n\operatorname{Var}\left( {\alpha X}\right) = {\alpha }^{2}\operatorname{Var}\left( X\right) \]\n\ncf. (12.1) in Appendix A, the stochastic integral\n\n\[ \n{\int }_{0}^{\infty }f\left( t\right) d{B}_{t} \mathrel{\text{:=}} \mathop{\sum }\limits_{{k = 1}}^{n}{a}_{k}\left( {{B}_{{t}_{k}} ... | Yes |
Proposition 1.4. We have\n\n\[ \n{\left\lbrack B, B\right\rbrack }_{t} = \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\sum }\limits_{{i = 1}}^{n}{\left( {B}_{{t}_{i}^{n}} - {B}_{{t}_{i - 1}^{n}}\right) }^{2},\;t \geq 0, \n\] \n\nwhere the limit exists in \( {L}^{2}\left( \Omega \right) \) and is independent o... | Proof. As an immediate consequence of the Definition 1.3 of the stochastic integral we have\n\n\[ \n{B}_{s}\left( {{B}_{t} - {B}_{s}}\right) = {\int }_{s}^{t}{B}_{s}d{B}_{\tau },\;0 \leq s \leq t, \n\] \n\nhence\n\n\[ \n{\left\lbrack B, B\right\rbrack }_{{t}_{i}^{n}} - {\left\lbrack B, B\right\rbrack }_{{t}_{i - 1}^{n}... | Yes |
Proposition 1.5. 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. [Protter (2005)], Theorem I-28). For every subdivision\n\n\[ \n\left\{ {0 = {t}_{0}^{n} < \cdots < {t}_{n}^{n} = t}\right\} \n\]\n\nwe have, by independence of the increments of Brownian motion:\n\n\[ \n\mathbb{E}\left\lbrack {\left( t - \mathop{\sum }\limits_{{i = 1}}^{n}{\left( {B}_{{t}_{i}^{n}} - {B... | Yes |
Proposition 2.1. The Black-Scholes PDE for the price of a European call is written as\n\n\\[ \n\\frac{\\partial C}{\\partial t}\\left( {t, x}\\right) + {r}_{t}x\\frac{\\partial C}{\\partial x}\\left( {t, x}\\right) + \\frac{1}{2}{x}^{2}{\\sigma }_{t}^{2}\\frac{{\\partial }^{2}C}{\\partial {x}^{2}}\\left( {t, x}\\right)... | The solution of this PDE is given by the Black-Scholes formula\n\n\\[ \nC\\left( {t, x}\\right) = \\operatorname{Bl}\\left( {K, x,{\\widetilde{\\sigma }}_{t},{\\widetilde{r}}_{t}, T - t}\\right) \\mathrel{\\text{:=}} {x\\Phi }\\left( {d}_{1}\\right) - K{e}^{-\\left( {T - t}\\right) {\\widetilde{r}}_{t}}\\Phi \\left( {d... | Yes |
Lemma 2.1. The following statements are equivalent:\n\ni) the portfolio \( {V}_{t} \) is self-financing,\n\nii) we have\n\n\[ \n{\widetilde{V}}_{t} = {\widetilde{V}}_{0} + {\int }_{0}^{t}{\sigma }_{u}{\eta }_{u}{\widetilde{S}}_{u}d{\widehat{B}}_{u},\;t \in {\mathbb{R}}_{ + },\n\]\n\n(2.10)\n\n\n\niii) we have\n\n\[ \n{... | Proof. First, note that (2.10) is clearly equivalent to (2.11). Now, the self-financing condition (2.3) shows that\n\n\[ \nd{V}_{t} = {\zeta }_{t}d{A}_{t} + {\eta }_{t}d{S}_{t} \]\n\n\[ \n= {\zeta }_{t}{A}_{t}{r}_{t}{dt} + {\eta }_{t}{r}_{t}{S}_{t}{dt} + {\sigma }_{t}{\eta }_{t}{S}_{t}d{\widehat{B}}_{t} \]\n\n\[ \n= {r... | Yes |
Lemma 2.2. Let \( \phi \in {\mathcal{C}}_{b}^{2}\left( \mathbb{R}\right) \) . The predictable representation\n\n\[ \phi \left( {S}_{T}\right) = {\mathbb{E}}_{\mathbb{Q}}\left\lbrack {\phi \left( {S}_{T}\right) }\right\rbrack + {\int }_{0}^{T}{\xi }_{t}d{\widehat{B}}_{t} \]\n\n(2.18)\n\nis given by\n\n\[ {\xi }_{t} = {\... | Proof. Since \( {P}_{t, T}\phi \) is in \( {\mathcal{C}}^{2}\left( \mathbb{R}\right) \), we can apply the Itô formula (1.8) to the process\n\n\[ t \mapsto {P}_{t, T}\phi \left( {S}_{t}\right) = {\mathbb{E}}_{\mathbb{Q}}\left\lbrack {\phi \left( {S}_{T}\right) \mid {\mathcal{F}}_{t}}\right\rbrack \]\n\n(2.20)\n\nwhich i... | Yes |
Lemma 2.3. Let \( X \) be a centered Gaussian random variable with variance \( {v}^{2} \) . We have\n\n\[ \n{\mathbb{E}}_{\mathbb{Q}}\left\lbrack {\left( {e}^{m + X} - K\right) }^{ + }\right\rbrack = {e}^{m + \frac{{v}^{2}}{2}}\Phi \left( {v + \left( {m - \log K}\right) /v}\right) - {K\Phi }\left( {\left( {m - \log K}\... | Proof. We have\n\n\[ \n{\mathbb{E}}_{\mathbb{Q}}\left\lbrack {\left( {e}^{m + X} - K\right) }^{ + }\right\rbrack = {\int }_{-\infty }^{\infty }{\left( {e}^{m + x} - K\right) }^{ + }{e}^{-\frac{{x}^{2}}{2{v}^{2}}}\frac{dx}{\sqrt{{2\pi }{v}^{2}}}\n\]\n\n\[ \n= {\int }_{-m + \log K}^{\infty }\left( {{e}^{m + x} - K}\right... | Yes |
Proposition 2.3. Assume that \( F = {\left( {S}_{T} - K\right) }^{ + } \) . Then for \( 0 \leq t \leq T \) we have\n\n\[{\xi }_{t} = {\sigma }_{t}{\mathbb{E}}_{\mathbb{Q}}{\left\lbrack {S}_{t, T}^{x}{\mathbf{1}}_{\lbrack K,\infty \lbrack }\left( {S}_{t, T}^{x}\right) \right\rbrack }_{x = {S}_{t}}.\] | Proof. This result follows from Lemma 2.2 and the relation \( {P}_{t, T}f\left( x\right) = \) \( {\mathbb{E}}_{\mathbb{Q}}\left\lbrack {f\left( {S}_{t, T}^{x}\right) }\right\rbrack \), after approximation of \( x \mapsto {\left( x - K\right) }^{ + } \) with \( {\mathcal{C}}^{2} \) functions. \( ▱ \) | No |
Proposition 2.4. The Delta of a European call option with payoff \( F = \) \( {\left( {S}_{T} - K\right) }^{ + } \) is given by\n\n\[ \n{\eta }_{t} = \Phi \left( \frac{\log \left( {{S}_{t}/K}\right) + \left( {{\widetilde{r}}_{t} + {\widetilde{\sigma }}_{t}^{2}/2}\right) \left( {T - t}\right) }{{\widetilde{\sigma }}_{t}... | Proof. By (2.13) we have, taking \( x = {S}_{t} \) ,\n\n\[ \n{\eta }_{t} = \frac{1}{{\sigma }_{t}{S}_{t}}{e}^{-{\widetilde{r}}_{t}\left( {T - t}\right) }{\xi }_{t}\n\]\n\n\[ \n= {e}^{-{\widetilde{r}}_{t}\left( {T - t}\right) }{\mathbb{E}}_{\mathbb{Q}}\left\lbrack {\frac{{S}_{t, T}^{x}}{x}{\mathbf{1}}_{\lbrack K,\infty ... | Yes |
Property 4.1. All solutions of stochastic differential equations such as (4.3) have the Markov property. | As a consequence, the arbitrage price \( P\left( {t, T}\right) \) satisfies\n\n\[ P\left( {t, T}\right) = {\mathbb{E}}_{\mathbb{Q}}\left\lbrack {{e}^{-{\int }_{t}^{T}{r}_{s}{ds}} \mid {\mathcal{F}}_{t}}\right\rbrack \]\n\n\[ = {\mathbb{E}}_{\mathbb{Q}}\left\lbrack {{e}^{-{\int }_{t}^{T}{r}_{s}{ds}} \mid {r}_{t}}\right\... | No |
For all sufficiently integrable random variables \( F \) we have\n\n\[ \n{\mathbb{E}}_{\mathbb{P}}\left\lbrack {F{e}^{-{\int }_{t}^{T}{r}_{s}{ds}} \mid {\mathcal{F}}_{t}}\right\rbrack = P\left( {t, T}\right) {\mathbb{E}}_{\widetilde{\mathbb{P}}}\left\lbrack {F \mid {\mathcal{F}}_{t}}\right\rbrack ,\;0 \leq t \leq T.\n\... | Proof. Indeed \( {}^{1} \), for all bounded and \( {\mathcal{F}}_{t} \) -measurable random variables \( G \) we have\n\n\[ \n{\mathbb{E}}_{\mathbb{P}}\left\lbrack {{GF}{e}^{-{\int }_{t}^{T}{r}_{s}{ds}}}\right\rbrack = P\left( {0, T}\right) {\mathbb{E}}_{\widetilde{\mathbb{P}}}\left\lbrack {G{e}^{{\int }_{0}^{t}{r}_{s}{... | Yes |
Lemma 7.1. We have\n\n\[ \n\frac{d{\widetilde{\mathbb{P}}}_{\mid {\mathcal{F}}_{t}}}{d{\mathbb{P}}_{\mid {\mathcal{F}}_{t}}} = \frac{{e}^{-{\int }_{t}^{T}{r}_{s}{ds}}}{P\left( {t, T}\right) },\;0 \leq t \leq T.\n\] | Proof. Rewrite (7.3) as\n\n\[ \n{\mathbb{E}}_{\widetilde{\mathbb{P}}}\left\lbrack {F \mid {\mathcal{F}}_{t}}\right\rbrack = {\mathbb{E}}_{\mathbb{P}}\left\lbrack {\left. {F\frac{{e}^{-{\int }_{t}^{T}{r}_{s}{ds}}}{P\left( {t, T}\right) }}\right| \;{\mathcal{F}}_{t}}\right\rbrack ,\;0 \leq t \leq T,\n\]\n\nfor all \( F \... | Yes |
Proposition 7.2. For all \( S, T \geq 0 \), the process\n\n\[ t \mapsto \frac{P\left( {t, S}\right) }{P\left( {t, T}\right) },\;0 \leq t \leq S \land T \]\n\n is an \( {\mathcal{F}}_{t} \) -martingale under \( \widetilde{\mathbb{P}} \), provided it is integrable. | Proof. For all bounded and \( {\mathcal{F}}_{s} \) -measurable random variables \( F \), from Relation (7.6) we have: \( {}^{2} \)\n\n\[ {\mathbb{E}}_{\widetilde{\mathbb{P}}}\left\lbrack {F\frac{P\left( {t, S}\right) }{P\left( {t, T}\right) }}\right\rbrack = {\mathbb{E}}_{\mathbb{P}}\left\lbrack {F\frac{{e}^{-{\int }_{... | Yes |
Proposition 7.3. The process\n\n\[ \n{\widetilde{B}}_{t} \mathrel{\text{:=}} {B}_{t} - {\int }_{0}^{t}{\zeta }_{s}{ds},\;0 \leq t \leq T \n\]\n\n(7.10)\n\nis a standard Brownian motion under \( \widetilde{\mathbb{P}} \) . | Proof. Letting\n\n\[ \n\Psi \left( t\right) = {\mathbb{E}}_{\mathbb{P}}\left\lbrack {\left. \frac{d\widetilde{\mathbb{P}}}{d\mathbb{P}}\right| \;{\mathcal{F}}_{t}}\right\rbrack \n\]\n\n\[ \n= \frac{1}{P\left( {0, T}\right) }{\mathbb{E}}_{\mathbb{P}}\left\lbrack {{e}^{-{\int }_{0}^{T}{r}_{s}{ds}} \mid {\mathcal{F}}_{t}}... | Yes |
Proposition 7.4. We have\n\n\[ \n{\mathbb{E}}_{\widetilde{\mathbb{P}}}\left\lbrack {\left. \frac{d\mathbb{P}}{d\widetilde{\mathbb{P}}}\right| \;{\mathcal{F}}_{t}}\right\rbrack = \frac{P\left( {0, T}\right) }{P\left( {t, T}\right) }\exp \left( {{\int }_{0}^{t}{r}_{s}{ds}}\right) \;0 \leq t \leq T \n\] | Proof. For all bounded and \( {\mathcal{F}}_{t} \) -measurable random variables \( F \) we have, using (7.4) and the characterization (12.4) of conditional expectation in Appendix A,\n\n\[ \n{\mathbb{E}}_{\widetilde{\mathbb{P}}}\left\lbrack {F\frac{d\mathbb{P}}{d\widetilde{\mathbb{P}}}}\right\rbrack = {\mathbb{E}}_{\ma... | Yes |
For any \( {\mathcal{F}}_{T} \) -measurable integrable random variable \( F \) we have\n\n\[ \mathbb{E}\left\lbrack {F{\mathbf{1}}_{\{ \tau > T\} } \mid {\mathcal{G}}_{t}}\right\rbrack = {\mathbf{1}}_{\{ \tau > t\} }\mathbb{E}\left\lbrack {F\exp \left( {-{\int }_{t}^{T}{\lambda }_{u}{du}}\right) \mid {\mathcal{F}}_{t}}... | Proof. By (9.4) we have\n\n\[ \frac{\mathbb{P}\left( {\tau > T \mid {\mathcal{F}}_{T}}\right) }{\mathbb{P}\left( {\tau > t \mid {\mathcal{F}}_{t}}\right) } = \frac{\exp \left( {-{\int }_{0}^{T}{\lambda }_{u}{du}}\right) }{\exp \left( {-{\int }_{0}^{t}{\lambda }_{u}{du}}\right) } = \exp \left( {-{\int }_{t}^{T}{\lambda ... | Yes |
Give a justification for the fact that\n\n\[ E\left\lbrack {\exp \left( {-{\int }_{t}^{T}\left( {{r}_{u} + {\lambda }_{u}}\right) {du}}\right) \mid {\mathcal{F}}_{t}}\right\rbrack \]\n\ncan be written as a function \( F\left( {t,{r}_{t},{\lambda }_{t}}\right) \) of \( t,{r}_{t} \) and \( {\lambda }_{t}, t \in \left\lbr... | \[ = {\mathbf{1}}_{\{ \tau > t\} }E\left\lbrack {\exp \left( {-{\int }_{t}^{T}\left( {{r}_{u} + {\lambda }_{u}}\right) {du}}\right) \mid {\mathcal{F}}_{t}}\right\rbrack . \] | No |
Proposition 10.1. For \( i = 1,\ldots, n \), let\n\n\[ \n{B}_{t}^{i} \mathrel{\text{:=}} {B}_{t} - {\int }_{0}^{t}{\zeta }_{i}\left( s\right) {ds},\;0 \leq t \leq {T}_{i},\n\]\n\n(10.4)\n\nthen \( {\left( {B}_{t}^{i}\right) }_{t \in \left\lbrack {0,{T}_{i}}\right\rbrack } \) is a standard Brownian motion under \( {\mat... | Proof. Letting\n\n\[ \n{\Phi }_{i}\left( t\right) = \mathbb{E}\left\lbrack {\left. \frac{d{\mathbb{P}}_{i}}{d\mathbb{P}}\right| \;{\mathcal{F}}_{t}}\right\rbrack = \frac{P\left( {t,{T}_{i}}\right) }{P\left( {0,{T}_{i}}\right) }{e}^{-{\int }_{0}^{t}{r}_{s}{ds}},\;0 \leq t \leq {T}_{i},\n\]\n\nwe have \( d{\Phi }_{i}\lef... | Yes |
Proposition 10.2. For all \( 1 \leq i, j \leq n \) we have\n\n\[ \n{\mathbb{E}}_{i}\left\lbrack {\left. \frac{d{\mathbb{P}}_{j}}{d{\mathbb{P}}_{i}}\right| \;{\mathcal{F}}_{t}}\right\rbrack = \frac{P\left( {0,{T}_{i}}\right) }{P\left( {0,{T}_{j}}\right) }\frac{P\left( {t,{T}_{j}}\right) }{P\left( {t,{T}_{i}}\right) }\;0... | Proof. For all bounded and \( {\mathcal{F}}_{t} \) -measurable random variables \( F \) we have \( {}^{1} \)\n\n\[ \n{\mathbb{E}}_{i}\left\lbrack {F\frac{d{\mathbb{P}}_{j}}{d{\mathbb{P}}_{i}}}\right\rbrack = \mathbb{E}\left\lbrack {F\frac{d{\mathbb{P}}_{j}}{d\mathbb{P}}}\right\rbrack \n\]\n\n\[ \n= \frac{1}{P\left( {0,... | Yes |
Proposition 10.3. The discounted annuity numeraire\n\n\\[ t \\mapsto {e}^{-{\\int }_{0}^{t}{r}_{s}{ds}}P\\left( {t,{T}_{i},{T}_{j}}\\right) ,\\;0 \\leq t \\leq {T}_{i}, \\]\n\nis a martingale under \\( \\mathbb{P},1 \\leq i < j \\leq n \\) . | Proof. This result can be recovered by linearity and the fact that \\( t \\mapsto \\) \\( {e}^{-{\\int }_{0}^{t}{r}_{s}{ds}}P\\left( {t,{T}_{k}}\\right) \\) is a martingale for all \\( k = i,\\ldots, j \\) . Alternatively, by standard arguments, given that \\( {\\delta }_{k} = {T}_{k + 1} - {T}_{k}, k = i,\\ldots, j - ... | Yes |
Proposition 10.4. We have\n\n\[ S\\left( {t,{T}_{i},{T}_{j}}\\right) = \\frac{P\\left( {t,{T}_{i}}\\right) - P\\left( {t,{T}_{j}}\\right) }{P\\left( {t,{T}_{i},{T}_{j}}\\right) },\\;0 \\leq t \\leq {T}_{i},\\;1 \\leq i < j \\leq n. \] | Proof. By definition of the forward LIBOR \( L\\left( {t, T, S}\\right) \) we have\n\n\[ P\\left( {t,{T}_{k}}\\right) - P\\left( {t,{T}_{k + 1}}\\right) - \\left( {{T}_{k + 1} - {T}_{k}}\\right) P\\left( {t,{T}_{k + 1}}\\right) L\\left( {t,{T}_{k},{T}_{k + 1}}\\right) = 0, \]\n\nhence by summation on \( k = i,\\ldots, ... | Yes |
Proposition 10.5. We have\n\n\[ \frac{d{\mathbb{P}}_{i, j \mid {\mathcal{F}}_{t}}}{d{\mathbb{P}}_{\mid {\mathcal{F}}_{t}}} = {e}^{-{\int }_{t}^{{T}_{i}}{r}_{s}{ds}}\frac{P\left( {{T}_{i},{T}_{i},{T}_{j}}\right) }{P\left( {t,{T}_{i},{T}_{j}}\right) },\;0 \leq t \leq {T}_{i + 1}. \] | Proof. It suffices to show that\n\n\[ {\mathbb{E}}_{i, j}\left\lbrack {F \mid {\mathcal{F}}_{t}}\right\rbrack = \mathbb{E}\left\lbrack {\left. {F{e}^{-{\int }_{t}^{T}{r}_{s}{ds}}\frac{P\left( {{T}_{i},{T}_{i},{T}_{j}}\right) }{P\left( {t,{T}_{i},{T}_{j}}\right) }}\right| \;{\mathcal{F}}_{t}}\right\rbrack \]\n\n(10.16)\... | Yes |
Proposition 10.6. For all \( 1 \leq i < j \leq n \) and \( 1 \leq k \leq n \) we have\n\n\[ \n{\mathbb{E}}_{i, j}\left\lbrack {\left. \frac{d{\mathbb{P}}_{k}}{d{\mathbb{P}}_{i, j}}\right| \;{\mathcal{F}}_{t}}\right\rbrack = \frac{P\left( {0,{T}_{i},{T}_{j}}\right) }{P\left( {0,{T}_{k}}\right) }\frac{P\left( {t,{T}_{k}}... | Proof. For all bounded and \( {\mathcal{F}}_{t} \) -measurable random variables \( F \) we have\n\n\[ \n{\mathbb{E}}_{i, j}\left\lbrack {F\frac{d{\mathbb{P}}_{k}}{d{\mathbb{P}}_{i, j}}}\right\rbrack = \mathbb{E}\left\lbrack {F\frac{d{\mathbb{P}}_{k}}{d\mathbb{P}}}\right\rbrack \n\]\n\n\[ \n= \frac{1}{P\left( {0,{T}_{k}... | Yes |
Proposition 11.2. The swaption price\n\n\[ P\\left( {t,{T}_{i},{T}_{j}}\\right) {\\mathbb{E}}_{i, j}\\left\\lbrack {{\\left( S\\left( {T}_{i},{T}_{i},{T}_{j}\\right) - \\kappa \\right) }^{ + } \\mid {\\mathcal{F}}_{t}}\\right\\rbrack \]\n\ncan be approximated by\n\n\[ P\\left( {t,{T}_{i},{T}_{j}}\\right) \\mathrm{{Bl}}... | Proof. We refer to Chapter 1 of [Schoenmakers (2005)] for a more rigorous treatment. Here we simply note that this approximation can be derived as follows:\n\n\[ {dS}\\left( {t,{T}_{i},{T}_{j}}\\right) = d\\left( {\\frac{1}{P\\left( {t,{T}_{i},{T}_{j}}\\right) }\\mathop{\\sum }\\limits_{{k = i}}^{{j - 1}}\\left( {{T}_{... | Yes |
Proposition 12.2. Every integrable process \( {\left( {X}_{t}\right) }_{t \in {\mathbb{R}}_{ + }} \) with centered independent increments is a martingale with respect to the filtration \[ {\mathcal{F}}_{t} \mathrel{\text{:=}} \sigma \left( {{X}_{u} : u \leq t}\right) ,\;t \in {\mathbb{R}}_{ + }, \] it generates, called... | Note that for all square-integrable random variable \( F \) the process \( {\left( \mathbb{E}\left\lbrack F \mid {\mathcal{F}}_{t}\right\rbrack \right) }_{t \in {\mathbb{R}}_{ + }} \) is a martingale, due to the relation \[ \mathbb{E}\left\lbrack {\mathbb{E}\left\lbrack {F \mid {\mathcal{F}}_{t}}\right\rbrack \mid {\ma... | No |
We have\n\n\\[ d{r}_{t} = - a{r}_{0}{e}^{-{at}}{dt} + {\varphi }^{\prime }\\left( t\\right) {dt} + d{X}_{t} \\]\n\n\\[ = - a{r}_{0}{e}^{-{at}}{dt} + \\theta \\left( t\\right) {dt} - a{\\int }_{0}^{t}\\theta \\left( u\\right) {e}^{-a\\left( {t - u}\\right) }{dudt} - a{X}_{t}{dt} + {\\sigma d}{B}_{t} \\]\n\n\\[ = - a{r}_... | By standard arguments we find\n\n\\[ - {xF}\\left( {t, x}\\right) + \\left( {\\theta \\left( t\\right) - {ax}}\\right) \\frac{\\partial F}{\\partial x}\\left( {t, x}\\right) + \\frac{1}{2}{\\sigma }^{2}\\frac{{\\partial }^{2}F}{\\partial {x}^{2}}\\left( {t, x}\\right) + \\frac{\\partial F}{\\partial t}\\left( {t, x}\\r... | Yes |
Consider a simple harmonic oscillator, in which a forcing term \( u\left( s\right) \) is taken as the control. Let \( {x}_{1}\left( s\right) ,{x}_{2}\left( s\right) \) denote respectively the position and velocity at time \( s \) . Then\n\n\[ \frac{d}{ds}{x}_{1}\left( s\right) = {x}_{2}\left( s\right) \]\n\n(2.4)\n\n\[... | We require that \( u\left( s\right) \in U \), where \( U \) is a closed interval. For instance, if \( U = \) \( \left\lbrack {-a, a}\right\rbrack \) with \( a < \infty \), then the bound \( \left| {u\left( s\right) }\right| \leq a \) is imposed on the forcing term.\n\nLet us consider the problem of controlling the simp... | Yes |
Example 2.3. We will now describe the linear quadratic regulator problem (LQRP). Due to the simplicity of its solution, it has been applied to a large number of engineering problems. Let \( x\left( s\right) \in {R}^{n}, u\left( s\right) \in {R}^{m} \) satisfy\n\n(2.6)\n\n\[ \frac{d}{ds}x\left( s\right) = A\left( s\righ... | The solution to this problem will be discussed in Example 5.1. | No |
For any initial condition \( \left( {t, x}\right) \in \bar{Q} \) and \( r \in \left\lbrack {t,{t}_{1}}\right\rbrack \) , \[ V\left( {t, x}\right) = \mathop{\inf }\limits_{{u\left( \cdot \right) \in \mathcal{U}\left( {t, x}\right) }}\left\lbrack {{\int }_{t}^{r \land \tau }L\left( {s, x\left( s\right), u\left( s\right) ... | The above identity is called the dynamic programming principle. It is the basis of the solution technique developed by Bellman in the 1950's [Be]. An interesting observation is that an optimal control \( {u}^{ * }\left( \cdot \right) \in \mathcal{U}\left( {t, x}\right) \) minimizes (4.3) at every \( r \) . Hence to det... | Yes |
Theorem 5.1. \( \left( {Q = {Q}_{0}}\right) \) . Let \( W \in {C}^{1}\left( {\bar{Q}}_{0}\right) \) satisfy (5.3) and (5.5). Then:\n\n(5.6)\n\n\[ W\left( {t, x}\right) \leq V\left( {t, x}\right) ,\forall \left( {t, x}\right) \in {\bar{Q}}_{0}. \]\n\nMoreover, if there exists \( {u}^{ * }\left( \cdot \right) \in {\mathc... | Proof of Theorem 5.1. Consider any \( u\left( \cdot \right) \in {\mathcal{U}}^{0}\left( t\right) \) . Using multivariate calculus and the dynamic programming equation (5.3), we obtain\n\n(5.8)\n\n\[ W\left( {{t}_{1}, x\left( {t}_{1}\right) }\right) = W\left( {t, x}\right) + {\int }_{t}^{{t}_{1}}\left\lbrack {\frac{\par... | Yes |
Consider the linear quadratic regulator problem (LQRP) described in Example 2.3. In this example, \( O = {\mathbb{R}}^{n}, U = {\mathbb{R}}^{m},\mathcal{U}\left( {t, x}\right) = {\mathcal{U}}^{0}\left( t\right) \) , and \[ f\left( {t, x, v}\right) = A\left( t\right) x + B\left( t\right) v \] \[ L\left( {t, x, v}\right)... | We guess that the solution of (5.11) and (5.14) is a quadratic function in \( x \) . So, let \[ W\left( {t, x}\right) = x \cdot P\left( t\right) x \] for some symmetric matrix \( P\left( t\right) \) . We substitute \( W\left( {t, x}\right) \) into (5.11) to obtain \[ - \frac{\partial }{\partial t}W\left( {t, x}\right) ... | Yes |
Theorem 5.2. Let \( W \in {C}^{1}\left( \bar{Q}\right) \) satisfy (5.3),(5.18) and (5.19). Then\n\n\[ \nW\left( {t, x}\right) \leq V\left( {t, x}\right) \text{ for all }\left( {t, x}\right) \in \bar{Q}.\n\]\n\nMoreover, suppose that there exists \( {u}^{ * }\left( \cdot \right) \in {\mathcal{U}}^{0}\left( t\right) \) s... | The proof of Theorem 5.2 is almost the same as for Theorem 5.1. In (5.8) the integral is now from \( t \) to the exit time \( \tau \), and \( W\left( {\tau, x\left( \tau \right) }\right) \) is on the left side. By (5.18) and (5.19), \( W\left( {\tau, x\left( \tau \right) }\right) \leq \Psi \left( {\tau, x\left( \tau \r... | Yes |
For any initial condition \( x \in \bar{O} \) and \( r \geq 0 \)\n\n\[ V\left( x\right) = \mathop{\inf }\limits_{{\mathcal{U}}_{x}}\left\lbrack {{\int }_{0}^{r \land \tau }{e}^{-{\beta s}}L\left( {x\left( s\right), u\left( s\right) }\right) {ds}}\right.\n\left. {+{e}^{-{\beta \tau }}g\left( {x\left( \tau \right) }\righ... | We continue with the proof of a verification theorem. Let \( W \in {C}^{1}\left( \bar{O}\right) \) satisfy the stationary dynamic programming equation (7.10) and the boundary conditions (7.11). As in the proof of Theorem 5.1, using the state dynamics and (7.10) we calculate that\n\n\[ {e}^{-{\beta r}}W\left( {x\left( r... | No |
Lemma 7.1. Suppose that \( \beta > 0 \), that\n\n(7.16)\n\n\[ \sup \{ f\left( {x, v}\right) \cdot x : x \in \bar{O}, v \in U\} < \infty ,\]\n\nand that \( \left| {W\left( x\right) }\right| \leq C\left( {1 + {\left| x\right| }^{m}}\right) \) for some constants \( C, m \) . Then (7.14) holds. | Proof. We calculate that\n\n(7.17)\n\n\[ \frac{d}{ds}\left( {\left| x\left( s\right) \right| }^{2}\right) = {2f}\left( {x\left( s\right), u\left( s\right) }\right) \cdot x\left( s\right) \]\n\nfor any control \( u\left( \cdot \right) \) . By (7.16)\n\n\[ {\left| x\left( s\right) \right| }^{2} \leq {\left| x\right| }^{2... | No |
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