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Theorem 5 Let \( {d}^{k} \) solves (6). If \( {d}^{k} \neq 0 \) then \( f\left( {{x}^{k} + \alpha {d}^{k}}\right) < f\left( {x}^{k}\right) \) holds for any \( \alpha \in (0,1\rbrack \) . | proof: Obviously, \( {d}^{k} \) is an optimal solution for\n\n\[ \operatorname{\mathbf{m} \mathbf{i} \mathbf{n} }\frac{1}{2}{\left( {x}^{k} + d\right) }^{T}Q\left( {{x}^{k} + d}\right) + {c}^{T}\left( {{x}^{k} + d}\right) \]\n\n\[ \text{s.t.}\;{a}_{i}^{T}d = 0,\;i \in {W}_{k}\text{.} \]\n\nHence\n\n\[ f\left( {x}^{k}\r... | Yes |
Lemma 1 If \( {d}^{{k}_{1}} = {d}^{{k}_{2}} = 0 \) and \( {d}^{k} \neq 0,\forall {k}_{1} < k < {k}_{2} \), then \( {W}_{{k}_{1}} \neq {W}_{{k}_{2}} \) . | Proof: If \( {k}_{2} = {k}_{1} + 1 \) then \( {W}_{{k}_{2}} = {W}_{{k}_{1}} \smallsetminus \left\{ {i}_{{k}_{1}}\right\} \neq {W}_{{k}_{1}} \) . Now we assume that \( {k}_{2} > {k}_{1} + 1 \) . We only need to prove that when \( {k}_{2} = {k}_{1} + 2 \) , outgoing index \( {i}_{{k}_{1}} \) in Step \( {k}_{1}\left( {{d}... | Yes |
Theorem 6 Let \( \{ {x}^{k}\} \) be generated by the active set method. If the vectors \( \left\lbrack {{a}_{i},\text{ }i \in \mathcal{A}\left( {x}^{k}\right) }\right\rbrack \) are linearly independent, then the algorithm terminates at a KKT point of (1) after a finite number of iterations, or the objective function of... | Proof: Suppose that the objective function \( f \) is bounded below and the infinite sequence \( \{ {x}^{k}\} \) is generated by the active set method. There exists \( {k}_{0} > 0 \) such that \( {x}^{k} = {x}^{{k}_{0}},\forall k \geq {k}_{0} \) . That is, for any \( k \geq {k}_{0},{d}^{k} = 0 \) or \( {\alpha }_{k} = ... | Yes |
The n-dimensional non-negative orthant, \( {\mathcal{R}}_{ + }^{n} = \left\{ {\mathbf{x} \in {\mathcal{R}}^{n} : \mathbf{x} \geq }\right. \) \( \mathbf{0}\} \), is a convex cone. | The dual of the cone is also \( {\mathcal{R}}_{ + }^{n} \) so that it’s self-dual. | No |
Let \( \mathbf{c} \in {\mathcal{R}}^{n} \) be given and \( \mathbf{x} \in {\mathcal{R}}_{+ + }^{n} \) . Then \( {\mathbf{c}}^{T}\mathbf{x} \) is homogeneous of degree 1 and | \[ \mathcal{P}\left( \mathbf{x}\right) = n\ln \left( {{\mathbf{c}}^{T}\mathbf{x}}\right) - \mathop{\sum }\limits_{{j = 1}}^{n}\log {x}_{j} \] is homogeneous of degree 0 , where ln is the natural logarithmic function. | No |
Example 1.9 Let \( X \in {\mathcal{S}}_{+ + }^{n} \) and \( f\left( X\right) = \ln \det \left( X\right) \) . Then | \[ \nabla f\left( X\right) = {X}^{-1} \] or \[ \frac{\partial f}{\partial {x}_{ij}} = {\left( {X}^{-1}\right) }_{ij},\forall i, j \] and \[ {\nabla }^{2}f\left( X\right) = - {X}^{-1} \oplus {X}^{-1} \] where \( \oplus \) denotes the standard Kronecker product, or \[ \frac{{\partial }^{2}f}{\partial {x}_{ij}\partial {x}... | "Yes" |
Example 1.15 Consider the bisection method to locate a root of a continuous function \( f\left( x\right) : \mathcal{R} \rightarrow \mathcal{R} \) within interval \( \left\lbrack {0,1}\right\rbrack \), where \( f\left( 0\right) > 0 \) and \( f\left( 1\right) < 0 \). The method calls the oracle to evaluate \( f\frac{1}{2... | Thus, in \( \log \left( {1/\epsilon }\right) \) steps, we must have an approximate root whose distance to the root is less than \( \epsilon \) . Therefore, the bisection method is a polynomial algorithm. | Yes |
Example 1.16 Consider the conjugate gradient algorithm for minimizing \( \frac{1}{2}{\mathbf{x}}^{T}Q\mathbf{x} + \) c. Starting from an \( {\mathbf{x}}^{0} \in {\mathcal{R}}^{n} \) and \( {\mathbf{d}}^{0} = Q{\mathbf{x}}^{0} + \mathbf{c} \), the method uses iterative formula | \[ {\mathbf{x}}^{k + 1} = {\mathbf{x}}^{k} - {\alpha }^{k}{\mathbf{d}}^{k} \] where \[ {\alpha }^{k} = \frac{{\left( {\mathbf{d}}^{k}\right) }^{T}\left( {Q{\mathbf{x}}^{k} + \mathbf{c}}\right) }{{\begin{Vmatrix}{\mathbf{d}}^{k}\end{Vmatrix}}_{Q}^{2}} \] and \[ {\mathbf{d}}^{k + 1} = Q{\mathbf{x}}^{k + 1} - {\theta }^{k... | "No" |
The dual to the LP example: | \[ \begin{array}{ll} \text{ maximize } & y \\ \text{ subject to } & y\left( \begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right) + \mathbf{s} = \left( \begin{array}{l} 2 \\ 1 \\ 1 \end{array}\right) , \\ & \mathbf{s} = \left( {{s}_{1};{s}_{2};{s}_{3}}\right) \in {K}^{ * } = {\mathcal{R}}_{ + }^{3}. \end{array} \] | Yes |
Example 2.3 Let \( P\left( {\mathbf{y} \in {\mathcal{R}}^{m}}\right) = C - \mathop{\sum }\limits_{i}^{m}{y}_{i}{A}_{i} \), where \( C \) and \( {A}_{i}, i = 1,\ldots, m \) , are given symmetric matrices. The problem of maximizes the min-eigenvalue of \( P\left( \mathbf{y}\right) \) can be cast as a (CLD) problem: | \[ \text{maximize}\;{y}_{0} \] \[ \text{subject to}{y}_{0}I + \mathop{\sum }\limits_{i}^{m}{y}_{i}{A}_{i} + S = C, S \succcurlyeq \mathbf{0}\text{.} \] Then, the dual problem would in CLP form: minimize \( \;C \bullet X \) \[ \text{subject to}{A}_{i} \bullet X = 0, i = 1,\ldots, m\text{,} \] \[ I \bullet X = 1, X \succ... | Yes |
Example 2.4 (Euclidean Facility Location). This problem is to determine the location of a facility serving \( n \) clients placed in a Euclidean space, whose known locations are denoted by \( {\mathbf{a}}_{l} \in {\mathcal{R}}^{d}, l = 1,\ldots, n \) . The location of the facility would minimize the sum of the Euclidea... | The problem can be reformulated as\n\n\[ \n\begin{matrix} \operatorname{minimize} & \mathop{\sum }\limits_{{l = 1}}^{n}{\delta }_{l} & \\ \text{ subject to } & {\mathbf{s}}_{j} + \mathbf{f} = {\mathbf{a}}_{l}, & \forall l = 1,\ldots, n, \\ & \begin{Vmatrix}{\mathbf{s}}_{l}\end{Vmatrix} \leq {\delta }_{l}, & \forall l =... | Yes |
Theorem 2.1 (Farkas' lemma for CLP) The following results hod.\n\ni) Consider set\n\n\[ \n{\\mathcal{F}}_{p} \\mathrel{\\text{:=}} \\{ X : \\mathcal{A}X = \\mathbf{b}, X \\in K\\} .\n\]\n\nSuppose that there exists a vector \( \\widehat{\\mathbf{y}} \) such that \( - {\\mathcal{A}}^{T}\\widehat{\\mathbf{y}} \\in {K}^{ ... | Proof.We prove the first part of i) of the theorem. It is clear that \( C \) is a convex set. To prove that \( C \) is a closed set, we need to show that if \( {\\mathbf{y}}^{k} \\mathrel{\\text{:=}} \\mathcal{A}{X}^{k} \\in {\\mathcal{R}}^{m} \) for \( {X}^{k} \\in K, k = 1,\\ldots \), converges to a vector \( \\overl... | Yes |
Theorem 2.2 (Weak duality theorem in CLP) Let \( {\mathcal{F}}_{p} \) and \( {\mathcal{F}}_{d} \) be non-empty. Then,\n\n\[ C \bullet X \geq {\mathbf{b}}^{T}\mathbf{y}\;\text{ for all }\;X \in {\mathcal{F}}_{p},\left( {\mathbf{y}, S}\right) \in {\mathcal{F}}_{d}. \] | Proof.By direct calculation\n\n\[ C \bullet X - {\mathbf{b}}^{T}\mathbf{y} = \left( {\mathop{\sum }\limits_{{i = 1}}^{m}{y}_{i}{A}_{i} + S}\right) \bullet X - {\mathbf{b}}^{T}\mathbf{y} \]\n\n\[ = \mathop{\sum }\limits_{{i = 1}}^{m}{y}_{i}\left( {{A}_{i} \bullet X}\right) + S \bullet X - {\mathbf{b}}^{T}\mathbf{y} \]\n... | Yes |
Example 2.6 The following semidefinite program has a duality gap: | \[ C = \left( \begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) ,{A}_{1} = \left( \begin{array}{lll} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right) ,{A}_{2} = \left( \begin{matrix} 0 & - 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 2 \end{matrix}\right) \]\n\nand\n\[ \mathbf{b} = \left( \begin{array}... | Yes |
1) The LP primal and dual example:\n\n\\( \\begin{aligned} \\text{ minimize } & \\left( \\begin{array}{lll} 2 & 1 & 1 \\end{array}\\right) \\mathbf{x} & \\text{ maximize } & y \\\\\n\\text{ subject to } & \\left( \\begin{array}{lll} 1 & 1 & 1 \\end{array}\\right) \\mathbf{x} = 1, & \\text{ subject to } & y\\left( \\beg... | A complementarity pair is at \\( {y}^{ * } = 1 \\) :\n\n\\[ \n{\\mathbf{x}}^{ * } = \\left( \\begin{matrix} 0 \\\\ {0.5} \\\\ {0.5} \\end{matrix}\\right) \\;\\text{ and }\\;{\\mathbf{s}}^{ * } = \\left( \\begin{array}{l} 1 \\\\ 0 \\\\ 0 \\end{array}\\right) .\n\\]\n\nNote that the primal has multiple optimal solutions. | Yes |
Theorem 2.7 Let \( {A}_{1},\ldots ,{A}_{m} \in {\mathcal{S}}^{n} \) be symmetric positive semidefinite matrices and \( r = \max \left\{ {\operatorname{rank}\left( {A}_{i}\right) }\right\} \) . Then, for any integer \( d \geq 1 \), there exists an \( \widehat{X} \succcurlyeq \mathbf{0} \) with rank at most \( d \) such ... | To prove the theorem, we first introduce Markov's Inequality\n\nLemma 2.8 Let \( h \) b | No |
Lemma 2.8 Let \( h \) be any positive valued and non-decreasing function and let \( \xi \) be a random variable. Then, for any positive value \( t \), the probability\n\n\[ \Pr \left( {\xi \geq t}\right) \leq \frac{\mathrm{E}\left\lbrack {h\left( \xi \right) }\right\rbrack }{h\left( t\right) }, t > 0. \] | Proof.Let \( {f}_{\xi } \) be the density function of \( \xi \) . Then,\n\n\[ \mathrm{E}\left\lbrack {h\left( \xi \right) }\right\rbrack = {\int }_{-\infty }^{\infty }h\left( z\right) {f}_{\xi }\left( z\right) {dz} \]\n\n\[ = {\int }_{-\infty }^{t}h\left( z\right) {f}_{\xi }\left( z\right) {dz} + {\int }_{t}^{\infty }h... | Yes |
Proposition 2.10 Let \( H \in {\mathcal{S}}^{n} \) be a symmetric positive semidefinite matrix with \( r \equiv \operatorname{rank}\left( H\right) \geq 1 \) . Then, for any \( \alpha > 1 \), we have:\n\n\[ \Pr \left( {H \bullet \widehat{X} \geq \alpha \operatorname{tr}\left( H\right) }\right) \leq r \cdot \exp \left( {... | Proof. We now consider factorization \( H = \mathop{\sum }\limits_{{k = 1}}^{r}{\mathbf{q}}_{k}{\mathbf{q}}_{k}^{T} \) . Thus, we have \( H \bullet \widehat{X} = \) \( \mathop{\sum }\limits_{{k = 1}}^{r}\mathop{\sum }\limits_{{j = 1}}^{d}{\left( {\mathbf{q}}_{k}^{T}{\mathbf{\xi }}^{j}\right) }^{2} \) . Observe that \( ... | Yes |
Theorem 2.11 Consider LP and SDP.\n\ni) (Uniqueness Theorem for Linear Programming) An LP optimal solution \( {\mathbf{x}}^{ * } \) is unique if and only if the cardinality of \( \operatorname{supp}\left( {\mathbf{x}}^{ * }\right) \) is maximal among all optimal solutions and the columns of \( {A}_{\operatorname{supp}\... | Proof.It is easy to see both conditions are necessary for LP case, since otherwise, one can find an optimal solution with a different support size. To see sufficiency, suppose there there is another optimal solution \( {\mathbf{y}}^{ * } \) such that \( {\mathbf{x}}^{ * } - {\mathbf{y}}^{ * } \neq \mathbf{0} \) . We mu... | Yes |
Proposition 3.2 Let both (CLP) and (CLD) have interior feasible solutions. Then, for any given \( 0 < \mu < \infty \), the optimizers of (CLPB) and (CLDB) exist and they are unique and in the interior of cone \( K \), respectively. | In fact, for given \( \mu \), the optimizers of (CLPB) have necessary and sufficient conditions:\n\n\[ C + \mu \nabla B\left( X\right) - {\mathcal{A}}^{T}\mathbf{y} = \mathbf{0} \]\n\n\[ \mathcal{A}X = \mathbf{b} \]\n\nLet \( S = C - {\mathcal{A}}^{T}\mathbf{y} \). Then the conditions become\n\n\[ S + \mu \nabla B\left... | Yes |
Example 3.2 The following are solutions \( Z \) (LP),(SDP) and (SOCP). | - For LP cone\n\n\[ \mathbf{z} = \mathbf{s}./\mathbf{x} \in {\mathcal{R}}^{n},\;\text{ that is }\;\mathbf{z} = \left( \begin{matrix} \frac{{s}_{1}}{{x}_{1}} \\ \vdots \\ \frac{{s}_{n}}{{x}_{n}} \end{matrix}\right) .\n\]\n\n- For SDP cone\n\n\[ Z = {X}^{.5}{\left( {X}^{.5}S{X}^{.5}\right) }^{-{.5}}{X}^{.5} \in {\mathcal... | Yes |
Example 3.3 The detailed dual Gram matrices for LP, SDP and SOCP are: | - for LP, it is\n\n\[ \n{A\Delta }{\left( \mathbf{s}\right) }^{-2}{A}^{T} \]\n\n- for SDP, it is\n\n\[ \n\mathcal{A}{S}^{-2}{\mathcal{A}}^{T} \mathrel{\text{:=}} \left( \begin{matrix} {A}_{1}{S}^{-1} \bullet {S}^{-1}{A}_{1} & \cdots & {A}_{1}{S}^{-1} \bullet {S}^{-1}{A}_{m} \\ \vdots & \ddots & \vdots \\ {A}_{m}{S}^{-1... | Yes |
Proposition 3.5 For any \( X \in \overset{ \circ }{K} \) and \( S \in {K}^{ * } \), \[ \text{i) if}K = {\mathcal{R}}_{ + }^{n}\text{or}K = {\mathcal{S}}_{ + }^{n} \] \[ {\psi }_{n}\left( {X, S}\right) \mathrel{\text{:=}} n\ln \left( {X \bullet S}\right) + B\left( X\right) + B\left( S\right) \geq n\ln \left( n\right) ; ... | Proof. We leave i) as an exercise. To prove ii) of the second-order cone, we see first \[ {\psi }_{1}\left( {\mathbf{x},\mathbf{s}}\right) \mathrel{\text{:=}} \ln \left( {\mathbf{x} \bullet \mathbf{s}}\right) + B\left( \mathbf{x}\right) + B\left( \mathbf{s}\right) = \ln \left( \frac{\mathbf{x} \bullet \mathbf{s}}{\delt... | No |
Example 3.4 Consider LP, SDP and SOCP. | - For LP, the primal, dual and primal-dual potential-reduction directions would be\n\n\[ \text{(LP-Primal)}\;{\mathbf{d}}_{\mathbf{s}} + \Delta {\left( \mathbf{x}\right) }^{-2}{\mathbf{d}}_{\mathbf{x}} = - \frac{\rho }{{\mathbf{x}}^{T}\mathbf{s}}\mathbf{s} + \Delta {\left( \mathbf{x}\right) }^{-1}\mathbf{e}\text{;} \]\... | Yes |
Lemma 3.8 Let\n\n\\[ \n{\\mu }^{k} = \\frac{{\\left( {x}^{k}\\right) }^{T}{s}^{k}}{n} = \\frac{{c}^{T}{x}^{k} - {z}^{k}}{n}\\;\\text{ and }\\;\\mu = \\frac{{\\left( {x}^{k}\\right) }^{T}s\\left( {z}^{k}\\right) }{n}. \n\\]\n\nIf\n\n\\[ \n\\begin{Vmatrix}{p\\left( {z}^{k}\\right) }\\end{Vmatrix} < \\min \\left( {\\eta \... | Proof. The proof is by contradiction.\n\ni) If the first inequality of (3.30) is not true, then \\( \\exists j \\) such that \\( {s}_{j}\\left( {z}^{k}\\right) \\leq 0 \\) and\n\n\\[ \n\\begin{Vmatrix}{p\\left( {z}^{k}\\right) }\\end{Vmatrix} \\geq 1 - \\frac{\\left( n + \\rho \\right) }{n{\\mu }^{k}}{x}_{j}{s}_{j}\\le... | Yes |
Theorem 3.9 Given \( \left( {{x}^{k},{y}^{k},{s}^{k}}\right) \in \mathcal{F} \) . Let \( \rho = \sqrt{n},{z}^{k} = {\mathbf{b}}^{T}{y}^{k},{x}^{k + 1} \) be given by (3.25), and \( {y}^{k + 1} = y\left( {z}^{k}\right) \) in (3.28) and \( {s}^{k + 1} = s\left( {z}^{k}\right) \) in (3.27). Then, either\n\n\[ \n{\psi }_{n... | Proof. If (3.29) does not hold, i.e.,\n\n\[ \n\begin{Vmatrix}{p\left( {z}^{k}\right) }\end{Vmatrix} \geq \min \left( {\eta \sqrt{\frac{n}{n + {\eta }^{2}}},1 - \eta }\right) ,\n\]\n\nthen\n\n\[ \n{\mathcal{P}}_{n + \rho }\left( {{x}^{k + 1},{z}^{k}}\right) - {\mathcal{P}}_{n + \rho }\left( {{x}^{k},{z}^{k}}\right) \leq... | Yes |
Corollary 3.10 Let \( \rho = \sqrt{n} \) . Then, Algorithm 3.1 terminates in at most \( O\left( {\sqrt{n}\ln \left( {{c}^{T}{x}^{0} - {\mathbf{b}}^{T}{y}^{0}}\right) /\epsilon }\right) \) iterations with\n\n\[ \n{c}^{T}{x}^{k} - {\mathbf{b}}^{T}{y}^{k} \leq \epsilon \n\] | Proof. In \( O\left( {\sqrt{n}\ln \left( {{\left( {x}^{0}\right) }^{T}{s}^{0}/\epsilon }\right) }\right) \) iterations\n\n\[ \n- \sqrt{n}\ln \left( {{\left( {x}^{0}\right) }^{T}{s}^{0}/\epsilon }\right) = {\psi }_{n + \rho }\left( {{x}^{k},{s}^{k}}\right) - {\psi }_{n + \rho }\left( {{x}^{0},{s}^{0}}\right) \n\]\n\n\[ ... | Yes |
Lemma 3.11 Let\n\n\\[ \n{\\mu }^{k} = \\frac{{S}^{k} \\bullet {X}^{k}}{n} = \\frac{C \\bullet {X}^{k} - {z}^{k}}{n}\\;\\text{ and }\\;\\mu = \\frac{S\\left( {z}^{k}\\right) \\bullet {X}^{k}}{n}. \n\\]\n\nIf\n\n\\[ \n\\begin{Vmatrix}{P\\left( {z}^{k}\\right) }\\end{Vmatrix} < \\min \\left( {\\beta \\sqrt{\\frac{n}{n + {... | Proof. The proof is by contradiction. For example, if the first inequality of (3.37) is not true, then \\( {\\left( {X}^{k}\\right) }^{.5}S\\left( {z}^{k}\\right) {\\left( {X}^{k}\\right) }^{.5} \\) has at least one eigenvalue less than or equal to zero, and\n\n\\[ \n\\begin{Vmatrix}{P\\left( {z}^{k}\\right) }\\end{Vma... | No |
Theorem 3.12 Given \( {X}^{k} \in {\mathcal{F}}_{p} \) and \( \left( {{y}^{k},{S}^{k}}\right) \in {\mathcal{F}}_{d} \), let \( \rho = \sqrt{n},{z}^{k} = {\mathbf{b}}^{T}{y}^{k} \) , \( {X}^{k + 1} \) be given by (??), and \( {y}^{k + 1} = y\left( {z}^{k}\right) \) in (3.34) and \( {S}^{k + 1} = S\left( {z}^{k}\right) \... | Proof. If (3.36) does not hold, i.e.,\n\n\[ \begin{Vmatrix}{P\left( {z}^{k}\right) }\end{Vmatrix} \geq \min \left( {\beta \sqrt{\frac{n}{n + {\beta }^{2}}},1 - \beta }\right) ,\]\n\nthen, since \( \psi \left( {{X}^{k + 1},{S}^{k}}\right) - \psi \left( {{X}^{k},{S}^{k}}\right) = \mathcal{P}\left( {{X}^{k + 1},{z}^{k}}\r... | Yes |
Corollary 3.13 Let \( \rho = \sqrt{n} \) . Then, Algorithm 3.2 terminates in at most \( O\left( {\sqrt{n}\ln \left( {C \bullet {X}^{0} - {\mathbf{b}}^{T}{y}^{0}}\right) /\epsilon }\right) \) iterations with\n\n\[ C \bullet {X}^{k} - {\mathbf{b}}^{T}{y}^{k} \leq \epsilon \] | Proof. In \( O\left( {\sqrt{n}\ln \left( {{S}^{0} \bullet {X}^{0}/\epsilon }\right) }\right) \) iterations\n\n\[ - \sqrt{n}\ln \left( {{S}^{0} \bullet {X}^{0}/\epsilon }\right) = \psi \left( {{X}^{k},{S}^{k}}\right) - \psi \left( {{X}^{0},{S}^{0}}\right) \]\n\n\[ \geq \sqrt{n}\ln {S}^{k} \bullet {X}^{k} + n\ln n - \psi... | Yes |
Lemma 3.14 Let the direction \( d = \left( {{d}_{x},{d}_{y},{d}_{s}}\right) \) be generated by equation (3.38) with \( \gamma = n/\left( {n + \rho }\right) \), and let\n\n\[ \theta = \frac{\alpha \sqrt{\min \left( {Xs}\right) }}{\begin{Vmatrix}{\left( XS\right) }^{-1/2}\left( \frac{{x}^{T}s}{\left( n + \rho \right) }e ... | Let \( v = {Xs} \) . Then, we can prove the following lemma (Exercise 3.5):\n\nLemma 3.15 Let \( v \in | No |
Theorem 3.16 Let \( \rho = O\left( \sqrt{n}\right) \) . Then, Algorithm 3.3 terminates in at most \( O\left( {\sqrt{n}\ln \left( {{\left( {x}^{0}\right) }^{T}{s}^{0}/\epsilon }\right) }\right) \) iterations with\n\n\[ \n{c}^{T}{x}^{k} - {\mathbf{b}}^{T}{y}^{k} \leq \epsilon \n\] | Once we have a pair \( \left( {X, y, S}\right) \in \mathcal{F} \) with \( \mu = S \bullet X/n \), we can apply the primal-dual Newton method to generate a new iterate \( {X}^{ + } \) and \( \left( {{y}^{ + },{S}^{ + }}\right) \) as follows: Solve for \( {d}_{X},{d}_{y} \) and \( {d}_{S} \) from the system of linear equ... | No |
Lemma 3.17 Let the direction \( {d}_{X},{d}_{y} \) and \( {d}_{S} \) be generated by equation (3.40) with \( \gamma = n/\left( {n + \rho }\right) \), and let\n\n\[ \theta = \frac{\alpha }{{\begin{Vmatrix}{V}^{-1/2}\end{Vmatrix}}_{\infty }\begin{Vmatrix}{\frac{I \bullet V}{n + \rho }{V}^{-1/2} - {V}^{1/2}}\end{Vmatrix}}... | Applying Lemma 3.15 to \( v \in {\mathcal{R}}^{n} \) as the vector of the \( n \) eigenvalues of \( V \), we can prove the following lemma: | No |
Lemma 3.20 Let \( {\mu }^{k} = \frac{{\Delta }^{k}}{n} = \frac{{\bar{z}}^{k} - {\mathbf{b}}^{T}{y}^{k}}{n},\mu = \frac{X\left( {\bar{z}}^{k}\right) \bullet {S}^{k}}{n} = \frac{C \bullet X\left( {\bar{z}}^{k}\right) - {\mathbf{b}}^{T}{y}^{k}}{n},\rho \geq \) \( n + \sqrt{n} \), and \( \alpha < 1 \) . If\n\n\[ \begin{Vma... | Proof. The proofs are by contradiction. If the first inequality is false, then \( {\left( {S}^{k}\right) }^{.5}X\left( {\bar{z}}^{k}\right) {\left( {S}^{k}\right) }^{.5} \) has at least one nonpositive eigenvalue, which by (3.52) implies that \( \begin{Vmatrix}{P\left( {\bar{z}}^{k}\right) }\end{Vmatrix} \geq 1 \) .\n\... | Yes |
Theorem 3.21\n\n\[ \Psi \left( {{X}^{k + 1},{S}^{k + 1}}\right) \leq \Psi \left( {{X}^{k},{S}^{k}}\right) - \delta \]\n\nwhere \( \delta > 1/{50} \) for a suitable \( \alpha \) . | Proof.\n\n\[ \Psi \left( {{X}^{k + 1},{S}^{k + 1}}\right) - \Psi \left( {{X}^{k},{S}^{k}}\right) = \left( {\Psi \left( {{X}^{k + 1},{S}^{k + 1}}\right) - \Psi \left( {{X}^{k + 1},{S}^{k}}\right) }\right) + \left( {\Psi \left( {{X}^{k + 1},{S}^{k}}\right) - \Psi \left( {{X}^{k},{S}^{k}}\right) }\right) .\n\nIn each iter... | Yes |
Corollary 3.22 Let \( \rho \geq n + \sqrt{n} \) and \( \Psi \left( {{X}^{0},{S}^{0}}\right) \leq \left( {\rho - n}\right) \ln \left( {{X}^{0} \bullet {S}^{0}}\right) \) . Then, the algorithm terminates in at most \( O\left( {\left( {\rho - n}\right) \ln \left( {{X}^{0} \bullet {S}^{0}/\epsilon }\right) }\right) \) iter... | Proof. In \( O\left( {\left( {\rho - n}\right) \ln \left( {{X}^{0} \bullet {S}^{0}/\epsilon }\right) }\right) \) iterations,\n\n\[ \Psi \left( {{X}^{k},{S}^{k}}\right) \leq \left( {\rho - n}\right) \ln \left( \epsilon \right) ). \]\n\nAlso,\n\n\[ \left( {\rho - n}\right) \ln \left( {C \bullet {X}^{k} - {\mathbf{b}}^{T}... | Yes |
Theorem 3.23 Each iteration of the dual algorithm uses \( O\left( {{m}^{3} + n{m}^{2} + {n}^{2}m + }\right. \) \( {n}^{3} \) ) floating point iterations. | Proof. Creating \( S \), or \( S + {\mathcal{A}}^{T}\left( {d\left( {\bar{z}}^{k}\right) }\right) \), uses matrix additions and \( O\left( {m{n}^{2}}\right) \) operations; factoring it uses \( O\left( {n}^{3}\right) \) operations. Creating the Gram matrix uses \( n{m}^{2} + 2{n}^{2}m + O\left( {nm}\right) \) operations... | Yes |
Theorem 3.24 System (HSDP) is feasible (e.g. all zeros) and any feasible solution \( \left( {y, X,\tau, S,\kappa }\right) \) is self-complementary: \( X \bullet S + {\tau \kappa } = 0 \) or\n\n\[ \left( \begin{matrix} X & 0 \\ {0}^{T} & \tau \end{matrix}\right) \left( \begin{matrix} S & 0 \\ {0}^{T} & \kappa \end{matri... | Let’s find such a feasible solution starting from \( {X}^{0} = I \succ 0,{S}^{0} = I \succ 0 \) , and \( {y}^{0} = 0 \) . Then, we formulate\n\n(HSDP) min\n\ns.t. \( \;\mathcal{A}X\; - {b\tau }\; + \bar{b}\theta \; = 0 \) ,\n\n\[ - {\mathcal{A}}^{T}y\; + {C\tau }\; - \bar{C}\theta \; \succcurlyeq 0, \]\n\n\[ {\mathbf{b... | Yes |
Theorem 3.26 Let \( \\left( {{y}^{ * },{X}^{ * },{\\tau }^{ * },{\\theta }^{ * } = 0,{S}^{ * },{\\kappa }^{ * }}\\right) \) be a max-rank solution for (HSDP). | i) (SDP) and (SDD) are (Solvable) if and only if \( {\\tau }^{ * } > 0 \) . In this case, \( {X}^{ * }/{\\tau }^{ * } \) is an optimal solution for (SDP) and \( \\left( {{y}^{ * }/{\\tau }^{ * },{S}^{ * }/{\\tau }^{ * }}\\right) \) is an optimal solution for (SDD).\n\nii) (SDP) or (SDD) is (Infeasible) if and only if \... | Yes |
Corollary 4.12 Let \( X = {V}^{T}V \succ 0, d\left( X\right) \leq e,{A}_{i} \bullet X = {b}_{i}\left( {i = 1,\ldots, m}\right), D = \) \( \operatorname{diag}\left( {\sqrt{{x}_{11}},\ldots ,\sqrt{{x}_{nn}}}\right) \), and \( \widehat{x} = {D\sigma }\left( {{V}^{T}u}\right) \) where \( u \) with \( \parallel u\parallel =... | \[ \mathop{\lim }\limits_{{X \rightarrow \underline{X}}}{\mathrm{E}}_{u}\left( {q\left( \widehat{x}\right) }\right) = \mathop{\lim }\limits_{{X \rightarrow \underline{X}}}\frac{2}{\pi }Q \bullet \left( {D\arcsin \left\lbrack {{D}^{-1}X{D}^{-1}}\right\rbrack D}\right) \leq \frac{2}{\pi }\underline{p} + \left( {1 - \frac... | Yes |
Theorem 4.13 Let \( \widehat{x} \) be randomly generated from \( \underline{X} \) . Then\n\n\[ \frac{{\mathrm{E}}_{u}q\left( \widehat{x}\right) - \underline{q}}{\bar{q} - \underline{q}} \leq \frac{\pi }{2} - 1 < 4/7 \] | Proof. Since\n\n\[ \bar{p} \geq \bar{q} \geq \frac{2}{\pi }\bar{p} + \left( {1 - \frac{2}{\pi }}\right) \underline{p} \geq \left( {1 - \frac{2}{\pi }}\right) \bar{p} + \frac{2}{\pi }\underline{p} \geq \underline{q} \geq \underline{p}, \]\n\nwe have, from Corollary 4.12,\n\n\[ \frac{{\mathrm{E}}_{u}q\left( \widehat{x}\r... | Yes |
Lemma 4.17 Assume (4.17) and (4.18) hold. Then, for any given \( \gamma \geq \alpha /(4 - \) \( \beta ) \), if random variable \( z\left( \gamma \right) \) meets its expectation, i.e., \( z\left( \gamma \right) \geq \alpha + {\gamma \beta } \), then\n\n\[ w\left( \widetilde{S}\right) \geq 2\left( {\sqrt{\gamma \left( {... | Proof. Suppose\n\n\[ w\left( S\right) = \lambda {w}^{ * }\;\text{ and }\;\left| S\right| = {\delta n},\]\n\nwhich from (4.19) and \( z\left( \gamma \right) \geq \alpha + {\gamma \beta } \) implies that\n\n\[ \lambda \geq \alpha + {\gamma \beta } - {4\gamma \delta }\left( {1 - \delta }\right) \]\n\nApplying (4.13) we se... | Yes |
Lemma 4.18 For any given \( 0 \leq \theta \leq 1 \) in our rounding method, inequalities (4.17) and (4.18) hold for\n\n\[ \alpha = \alpha \left( \theta \right) \;\text{ and }\;\beta = \beta \left( \theta \right) . \] | Proof. Since \( 1 - \frac{2}{\pi }\arcsin \left( \theta \right) \geq 0 \) for any \( 0 \leq \theta \leq 1 \), from definition (4.22) we have\n\n\[ \mathrm{E}\left\lbrack {w\left( S\right) }\right\rbrack = \frac{1}{4}\mathop{\sum }\limits_{{i, j}}{w}_{ij}\left( {1 - \frac{2}{\pi }\arcsin \left( {\bar{X}}_{ij}\right) }\r... | Yes |
If there are \( {2n} + n\left( {n + 1}\right) /2 \) distance pairs each of which has an accurate distance measure and other distance bounds are feasible. Then, the minimal value of \( \alpha = 0 \) in (5.6). Moreover, if (5.6) has a unique minimal solution\n\n\[ \n\\bar{Z} = \\left( \\begin{matrix} I & \\bar{X} \\\\ {\... | Proof. Let \( {X}^{ * } \) be the true locations of the \( n \) points, and\n\n\[ \n{Z}^{ * } = \\left( \\begin{matrix} I & {X}^{ * } \\\\ {\\left( {X}^{ * }\\right) }^{T} & {\\left( {X}^{ * }\\right) }^{T}{X}^{ * } \\end{matrix}\\right) .\n\]\n\nThen \( {Z}^{ * } \) and \( \alpha = 0 \) is a feasible solution for (5.6... | Yes |
Proposition 5.2 Let the noisy measurements\n\n\[ \n{\widehat{d}}_{ij} = {d}_{ij} + {\epsilon }_{i} + {\epsilon }_{j},\forall i \neq j \]\n\nand\n\n\[ \n{\widehat{d}}_{kj} = {d}_{kj} + {\epsilon }_{j},\forall k, j \]\n\nwhere \( {d}_{ij} \) are the true distances and \( {\epsilon }_{j} \) are independent random errors w... | Proof. Since \( \alpha = 0 \), we have, for all \( i, j, k \) ,\n\n\[ \n{\bar{Y}}_{ii} - 2{\bar{Y}}_{ij} + {\bar{Y}}_{jj} = {\left( {d}_{ij} + {\epsilon }_{i} + {\epsilon }_{j}\right) }^{2} \]\n\n\[ \n{\bar{Y}}_{jj} - 2{\bar{x}}_{j}^{T}{a}_{k} + {\begin{Vmatrix}{a}_{k}\end{Vmatrix}}^{2} = {\left( {d}_{kj} + {\epsilon }... | Yes |
Proposition 5.5 If every sensor point is connected, directly or indirectly, to an anchor point in (5.8), then any solution to (5.9) must be bounded, that is, \( {Y}_{jj} \) is bounded for all \( j = 1,\ldots, n \) . | Proof. If sensor point \( {x}_{j} \) is connected to an anchor point \( {a}_{k} \), then we have:\n\n\[ \n{\begin{Vmatrix}{x}_{j}\end{Vmatrix}}^{2} - 2{a}_{k}^{T}{x}_{j} + {\begin{Vmatrix}{a}_{k}\end{Vmatrix}}^{2} \leq {Y}_{jj} - 2{a}_{k}^{T}{x}_{j} + {\begin{Vmatrix}{a}_{k}\end{Vmatrix}}^{2} = {\bar{d}}_{kj}^{2} \n\]\... | Yes |
Proposition 5.7 Suppose that the network is connected. Furthermore, suppose that \( {y}_{ij} < 0 \) for all \( \left( {i, j}\right) \in {N}_{x} \), and that \( {w}_{ki} < 0 \) for some \( \left( {k, i}\right) \in {N}_{a} \), with \( {N}_{a} \neq \varnothing \) . Then, \( {U}_{22} \) is positive definite, i.e. it has ra... | Proof. Since \( A \) and \( D \) are positive semidefinite, we have \( {x}^{T}{U}_{22}x \geq 0 \) for all \( x \in {R}^{n} \) . We now show that there is no \( x \in {R}^{n} \smallsetminus \{ 0\} \) such that \( {x}^{T}{Ax} = {x}^{T}{Dx} = \) 0 . Suppose to the contrary that we have such an \( x \) . Then, since \( D \... | Yes |
Theorem 5.8 If a problem (graph) contains a subproblem (subgraph) that is strongly localizable, then the submatrix solution corresponding to the subproblem in the SDP solution has rank \( d \) . That is, the SDP relaxation computes a solution that localizes all possibly localizable unknown points. | Proof. Let the subproblem have \( {n}_{s} \) unknown points and they are indexed as \( 1,\ldots ,{n}_{s} \) . Since it is strongly localizable, an optimal dual slack matrix \( {U}_{s} \) of the SDP relaxation for the subproblem has rank \( {n}_{s} \) . Then in the dual problem of the SDP relaxation for the whole proble... | Yes |
Lemma 5.10 There exists an integer \( r \geq k \) such that we can compute, in polynomial time, \( r \) nonnegative reals \( {\lambda }_{1},{\lambda }_{2},\cdots ,{\lambda }_{r} \) and \( r \) orthogonal unit vectors \( {v}_{1},{v}_{2},\cdots ,{v}_{r} \) such that\n\n(i). \( \mathop{\sum }\limits_{{i = 1}}^{r}{\lambda ... | Proof. We solve the semidefinte program (5.20), and let \( {X}^{ * } \) be an optimal solution of (5.20). We claim that the rank of \( {X}^{ * } \), say \( r \), is at least \( k \) . This follows from the fact that \( \operatorname{tr}\left( {X}^{ * }\right) = k \) and \( I - {X}^{ * } \succcurlyeq 0 \) . In other wor... | Yes |
Lemma 5.12 For \( i = 1,2,\cdots, k,\begin{Vmatrix}{x}_{i}\end{Vmatrix} = 1 \) . | Proof. Recall that \( \left\langle {{v}_{l},{v}_{j}}\right\rangle = 0 \) for any \( l \neq j \) and \( \begin{Vmatrix}{v}_{j}\end{Vmatrix} = 1 \) . By definition,\n\n\[ \n{\begin{Vmatrix}{x}_{i}\end{Vmatrix}}^{2} = \left\langle {{x}_{i},{x}_{i}}\right\rangle \n\] \n\n\[ \n= \left\langle {\frac{\mathop{\sum }\limits_{{j... | Yes |
Lemma 5.14 Let \( \phi \) be a random vector whose entries are independent and either 1 or -1 with probability .5 each way. Then, for any vector \( e \) and \( \beta > 0 \) , | \[ \operatorname{prob}\left\{ {\langle \phi, e{\rangle }^{2} > \beta \parallel e{\parallel }^{2}}\right\} < 2 \cdot \exp \left( {-\frac{\beta }{2}}\right) . \] | Yes |
Lemma 5.15 For each \( i = 1,2,\cdots, k \) and each \( p \in P \), we have\n\n\[ \operatorname{prob}\left\{ {{\left\langle p,{x}_{i}\right\rangle }^{2} > {12}\log \left( n\right) \cdot {C}_{ip}}\right\} < \frac{2}{{n}^{3}}. \]\n | Proof. Given \( i \) and \( p \), define a \( \left| {I}_{i}\right| \) dimensional vector \( e \) such that its entries are \( \sqrt{{\lambda }_{j}}\left\langle {p,{v}_{j}}\right\rangle, j \in {I}_{i} \), respectively. Furthermore, we define the vector \( {\left. \phi \right| }_{{I}_{i}} \) whose entries are those of \... | Yes |
Theorem 5.16 We can computed in polynomial time, \( a\left( {d - k}\right) \) -flat such that, with probability at least \( 1 - \frac{2}{n} \), the distance between any point \( p \in P \) and \( F \) is at most \( \sqrt{{12}\log \left( n\right) } \cdot {R}_{k}\left( P\right) \) . | Proof. For given \( i = 1,2,\cdots, k \) and \( p \in P \), consider the event\n\n\[ \n{B}_{ip} = \left\{ {\phi \mid {\left\langle p,{x}_{i}\right\rangle }^{2} > {12}\log \left( n\right) \cdot {C}_{ip}}\right\} \n\]\n\nand \( B = { \cup }_{i, p}{B}_{ip} \) . The probability that the event \( B \) happens is bounded by\... | Yes |
Theorem 8.2 Given any initial point in \( {\mathcal{N}}_{\infty }^{ - }\left( \eta \right) \) for any constant \( \eta \in \left( {0,1}\right) \) , Algorithm 8.1, with any constant \( 0 < \gamma < 1 \), will terminate in \( O\left( {{n}^{\frac{r + 1}{2r}}\log \left( {{\left( {x}^{0}\right) }^{T}{s}^{0}/\epsilon }\right... | As we can see that if \( r = n \) and \( n \) increases, the iteration complexity of the algorithm tends to \( O\left( {\sqrt{n}\log \left( {{\left( {x}^{0}\right) }^{T}{s}^{0}/\epsilon }\right) }\right) \) asymptotically. Furthermore, a popular choice for \( \gamma \) in practice is not a constant but \( \gamma = O\le... | No |
Theorem 8.3 Let \( B \) be an optimal basis for \( {LP}\left( {A,{b}^{k},{c}^{k}}\right) \) . Then, there is \( 0 < \bar{t} < \infty \) such that \( B \) must be also an optimal basis for the original \( {LP}\left( {A, b, c}\right) \) when \( {\left( {x}^{k}\right) }^{T}{s}^{k} \leq {2}^{-\bar{t}} \) . Furthermore, if ... | This advocates for an application of the above basis identification procedure to the perturbed problem (8.18), since an optimal complementary solution to problem (8.18) is known, and it will be an optimal basis for (LP) when problem (8.18 is near (LP). | No |
If \( f : {\mathbb{R}}^{2} \rightarrow \mathbb{R} \) is a differentiable function, then\n\n\[ \n{D}_{\mathbf{u}}f\left( {\mathbf{x}}_{0}\right) = \frac{\mathrm{d}f}{\mathrm{\;d}\mathbf{u}}\left( {\mathbf{x}}_{0}\right) = {\left. \frac{\partial f}{\partial x}\right| }_{0}u + {\left. \frac{\partial f}{\partial y}\right| ... | ## Proof:\n\nIn light of Definition 2.4, \( f \) differentiable means\n\n\[ \n\frac{f\left( {{\mathbf{x}}_{0} + t\mathbf{u}}\right) - f\left( {\mathbf{x}}_{0}\right) }{t} = \frac{1}{t}\left( {{\left. f\left( {\mathbf{x}}_{0}\right) + \mathop{\sum }\limits_{{i = 1}}^{n}\frac{\partial f}{\partial {x}_{i}}\right| }_{0}t{u... | Yes |
Suppose \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) is continuous and \( \frac{\partial f}{\partial {x}_{i}}, i = 1,2,\ldots, n \) exist and are continuous in \( {S}_{r}\left( \mathbf{x}\right) \subset {D}_{f} \) and that both \( \frac{{\partial }^{2}f}{\partial {x}_{i}\partial {x}_{j}} \) and \( \frac{{\partial ... | (For the standard proof, see a standard text book such as [1] or [2].) | No |
Let \( f : {D}_{f} \subset {\mathbb{R}}^{2} \rightarrow \mathbb{R} \) be a function with continuous derivatives of order \( 0,1,2 \), and 3 (that is, \( f \) is a class \( {C}^{3} \) function) in some neighbourhood of a point \( \mathbf{a} \in {D}_{f} \) . Then,\n\n\[ f\left( {a + h, b + k}\right) = f\left( {a, b}\righ... | Using Taylor polynomials of order 2 (see Equation 2.7) results in a considerable simplification. The difficulties of critical point problems involving more complex functions are reduced to the level featured in Examples 3.1 and 3.2,\n\nsince the function approximations are algebraic.\n\nThe Taylor polynomial approximat... | Yes |
Corollary 4.1.4 The average value of function \( f \) over \( I \) is | \[ \frac{1}{\text{ length }I} \cdot {\int }_{I}f\left( x\right) \mathrm{d}x = \frac{{\int }_{I}f\left( x\right) \mathrm{d}x}{{\int }_{I}1\mathrm{\;d}x}. \] | Yes |
Corollary 4.1.3 \( {\iint }_{R}1\mathrm{\;d}A \) is the area of \( R \) . | true not just for rectangles!! | No |
Let \( x\left( {u, v}\right) \) and \( y\left( {u, v}\right) \) be a bijective [one-to-one and onto] and \( {C}^{1} \) transformation of \( E \) in the uv-plane onto \( D \) in the xy-plane. If \( f\left( {x, y}\right) \) is integrable in \( D \) then \( f\left( {x\left( {u, v}\right), y\left( {u, v}\right) }\right) = ... | \[ {\iint }_{D}f\left( {x, y}\right) \mathrm{d}A = {\iint }_{E}F\left( {u, v}\right) \left| \frac{\partial \left( {x, y}\right) }{\partial \left( {u, v}\right) }\right| \mathrm{d}{A}^{\prime }.\] | Yes |
Suppose \( I \) is an open connected interval in \( \mathbb{R} \). The vector \( \mathbf{r}\left( t\right) \) is a continuous vector function of \( t \in I \), with continuous first derivatives (that is, is a \( {C}^{1} \) function) if all of its component functions are \( {C}^{1} \) functions of \( t \in I \). | As with all derivatives, the derivative vector is defined in terms of a converged limiting process, much the same as described in Chapter 1, but now applied to each component of \( \mathbf{r}\left( \mathbf{t}\right) \), and the results are recombined.\n\nThe preceding definition means that\n\n\[ \mathop{\lim }\limits_{... | Yes |
Suppose \( \mathbf{f}\left( \mathbf{x}\right) = \left( {{f}_{1}\left( \mathbf{x}\right) ,{f}_{2}\left( \mathbf{x}\right) ,{f}_{3}\left( \mathbf{x}\right) }\right) \) is a conservative field with potential \( \phi \left( \mathbf{x}\right) \) defined on an open connected domain, D. Then\n\n\[{\int }_{\Gamma }\mathbf{f} \... | A proof of the theorem, and our consequential statements, follows readily from our earlier finding that for a conservative field \( \mathbf{f} \cdot \mathrm{d}\mathbf{r} \) is a perfect differential, \( \mathrm{d}\phi \) . The integral of this perfect differential results in the difference in the values of the potentia... | Yes |
Corollary 5.2.1 For the conditions of Theorem 5.2, let \( {\Gamma }_{1} \) and \( {\Gamma }_{2} \) be two curves in \( D \) that join \( {\mathbf{r}}_{a} \) and \( {\mathbf{r}}_{b} \) (Figure 5.27). Then\n\n\[ \n{\int }_{{\Gamma }_{1}}\mathbf{f} \cdot \mathrm{d}\mathbf{r} = {\int }_{{\Gamma }_{2}}\mathbf{f} \cdot \math... | \[ \n\Rightarrow \;{\int }_{{\Gamma }_{1}}\mathbf{f} \cdot \mathrm{d}\mathbf{r} - {\int }_{{\Gamma }_{2}}\mathbf{f} \cdot \mathrm{d}\mathbf{r} = 0 \n\]\n\n\[ \n\Rightarrow \;{\int }_{{\Gamma }_{1}}\mathbf{f} \cdot \mathrm{d}\mathbf{r} + {\int }_{-{\Gamma }_{2}}\mathbf{f} \cdot \mathrm{d}\mathbf{r} = 0 \n\]\n\n\[ \n\Rig... | Yes |
Corollary 4.1.1 (2D version) for \( f \geq 0 \) gives the geometric interpretation of a double integral of \( f \) as\n\n\( {\iint }_{D}f\left( {x, y}\right) \mathrm{d}A = \) volume of body under \( f \) over \( D \) (a planar subset of \( {\mathbb{R}}^{2} \) ). | If, however, \( f\left( {x, y}\right) \) were interpreted as, say, a charge/unit area (surface charge density) defined on \( {\mathbb{R}}^{2} \) then the integral would give the total charge on the planar region \( D \) in the \( {xy} \) -plane: \( {\iint }_{D}f\left( {x, y}\right) \mathrm{d}A = Q \) (Figure 5.29). | No |
Theorem 5.3\n\nSuppose \( \\mathbf{f} : {\\mathbb{R}}^{3} \\rightarrow {\\mathbb{R}}^{3} \) is a \( {C}^{1} \) vector field defined on and within a domain \( V \) which is bounded by a piecewise smooth closed surface \( S \) which in turn has a continuously varying outward unit normal \( N \) . Then\n\n\[ \n{\\oiint }_... | ## Sketch proof of Gauss's theorem\n\nWe start by splitting the surface and volume integrals into their component terms: \n\nIt is always possible to treat \( S \) as the union of piecewise smooth surfaces and \( V \... | Yes |
Green's theorem:\n\nLet \( \\mathbf{f}\\left( {x, y}\\right) = {f}_{1}\\left( {x, y}\\right) {\\mathbf{e}}_{1} + {f}_{2}\\left( {x, y}\\right) {\\mathbf{e}}_{2} \) be a smooth 2D field defined on and within the positively oriented simple closed boundary \( \\Gamma \) of a closed and bounded region \( D \\subset {\\math... | Sketch proof of Green's theorem (for a simply-connected domain)\n\nSuppose \( D \) is \( x \) -simple and \( y \) -simple. As indicated in Figure 5.52 functions \( {\\phi }_{1},{\\phi }_{2},{\\psi }_{1} \), and \( {\\psi }_{2} \) can be found such that\n\n\[ \nD = \\left\\{ {\\left( {x, y}\\right) : a \\leq x \\leq b,{... | Yes |
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.: | \[ \mathbb{E}\left\lbrack {{\int }_{0}^{t}{u}_{\tau }d{B}_{\tau } \mid {\mathcal{F}}_{s}}\right\rbrack = {\int }_{0}^{s}{u}_{\tau }d{B}_{\tau },\;0 \leq s \leq t. \] | Yes |
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 inde... | 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}_{... | 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\niii) we have\n\n\[ \n{V}_{t} = {V}_{... | 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 |
Proposition 2.2. Given \( F \in {L}^{2}\left( \Omega \right) \), let\n\n\[ \n{\eta }_{t} = \frac{\exp \left( {-{\int }_{t}^{T}{r}_{s}{ds}}\right) }{{\sigma }_{t}{S}_{t}}{\xi }_{t} \]\n\n(2.13)\n\n\[ \n{\zeta }_{t} = \frac{\exp \left( {-{\int }_{t}^{T}{r}_{u}{du}}\right) {\mathbb{E}}_{\mathbb{Q}}\left\lbrack {F \mid {\m... | Proof. Applying (2.14) and (2.15) at \( t = 0 \) we get\n\n\[ \n{\mathbb{E}}_{\mathbb{Q}}\left\lbrack F\right\rbrack \exp \left( {-{\int }_{0}^{T}{r}_{u}{du}}\right) = {\zeta }_{0}{A}_{0} + {\eta }_{0}{S}_{0} = {V}_{0}, \]\n\nhence from (2.14) again, the definition (2.13) of \( {\eta }_{t} \) and (2.12), we obtain\n\n\... | 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}\righ... | Yes |
Proposition 2.3. Assume that \( F = {\left( {S}_{T} - K\right) }^{ + } \) . Then for \( 0 \leq t \leq T \) we have\n\n\[ \n{\xi }_{t} = {\sigma }_{t}{\mathbb{E}}_{\mathbb{Q}}{\left\lbrack {S}_{t, T}^{x}{\mathbf{1}}_{\lbrack K,\infty \lbrack }\left( {S}_{t, T}^{x}\right) \right\rbrack }_{x = {S}_{t}}. \n\] | 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. \( ▱ \) | Yes |
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 |
Proposition 7.1. 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 ... | 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\nis 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\[ \n{\mathbb{E}}_{\widetilde{\mathbb{P}}}\left\lbrack {\left. \frac{d\mathbb{P}}{d\widetilde{\mathbb{P}}}\right| \;{\mathcal{F}}_{t}}\right\rbrack = \frac{P\left( {0, T}\right) }{P\left( {t, T}\right) }\exp \left( {{\int }_{0}^{t}{r}_{s}{ds}}\right) \;0 \leq t \leq T \n\]\n\nand the process... | 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 |
Lemma 9.1. ([Guo et al. (2007)]) 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... | 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 |
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 \( {\ma... | 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}\le... | 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,{T... | Yes |
Proposition 10.3. The discounted annuity numeraire\n\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\\]\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 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 the natural filtration.\n\nNote that for all square-integrable random variable \( F \) the process \( {\left( \mathbb{E}\left\lbrack F \mid {\mathcal{F}}_{t}\right\rbrack \right) }_{t ... | No |
We have\n\n\[ d{r}_{t} = - a{r}_{0}{e}^{-{at}}{dt} + {\varphi }^{\prime }\left( t\right) {dt} + d{X}_{t} \] | \[ = - a{r}_{0}{e}^{-{at}}{dt} + \theta \left( t\right) {dt} - a{\int }_{0}^{t}\theta \left( u\right) {e}^{-a\left( {t - u}\right) }{dudt} - a{X}_{t}{dt} + {\sigma d}{B}_{t} \]\n\n\[ = - a{r}_{0}{e}^{-{at}}{dt} + \theta \left( t\right) {dt} - {a\varphi }\left( t\right) {dt} - a{X}_{t}{dt} + {\sigma d}{B}_{t} \]\n\n\[ =... | Yes |
Assume that the current stock price is \( \$ {280} \), and after three months the stock price may either rise to \( \$ {320} \) or decline to \( \$ {260} \) . We shall find the rational price of a 3-month European call option with strike price \( K = \$ {280} \), provided that the simple risk-free interest rate \( r \)... | Suppose that the subjective probability of the price rise is 0.2 , and that of the fall is 0.8 ; these assumptions correspond, loosely, to a so-called bear market. Note that the word subjective means that we take the point of view of a particular individual.\n\n---\n\nGenerally speaking, the two parties involved in an ... | Yes |
Proposition 1.4.2 The spot market \( \mathcal{M} = \left( {S, B,\Phi }\right) \) is arbitrage-free if and only if the discounted stock price process \( {S}^{ * } \) admits a martingale measure \( {\mathbb{P}}^{ * } \) equivalent to \( \mathbb{P} \) . In this case, the arbitrage price at time 0 of any contingent claim \... | Proof. We know already that the martingale measure for \( {S}^{ * } \) equivalent to \( \mathbb{P} \) exists if and only if the unique solution \( {p}_{ * } \) of equation (1.5) satisfies \( 0 < {p}_{ * } < 1 \) . Suppose there is no equivalent martingale measure for \( {S}^{ * } \) ; for instance, assume that \( {p}_{... | Yes |
Corollary 1.4.1 Under the assumptions of Proposition 1.4.2, for any contingent claim \( X \), which settles at time \( T \), we have that\n\n\[ \n{\pi }_{0}\left( X\right) = {S}_{0}{\mathbb{E}}_{\overline{\mathbb{P}}}\left( {{S}_{T}^{-1}X}\right) \n\]\n\n(1.18) | Proof. We note that the Radon-Nikodým density of \( \overline{\mathbb{P}} \) with respect to \( {\mathbb{P}}^{ * } \) equals\n\n\[ \n{\eta }_{T}\left( \omega \right) \mathrel{\text{:=}} \frac{d\overline{\mathbb{P}}}{d{\mathbb{P}}^{ * }}\left( \omega \right) = \left\{ \begin{array}{ll} \frac{\bar{p}}{{p}_{ * }} = \frac{... | Yes |
Corollary 1.4.2 The following put-call parity relationship is valid\n\n\[ \n{C}_{0} - {P}_{0} = {S}_{0} - {\left( 1 + r\right) }^{-1}K.\n\]\n\n(1.19) | Proof. The formula is an immediate consequence of equality (1.3) and the pricing formula (1.14) applied to the claim \( {S}_{T} - K \) . | No |
Proposition 1.5.1 Assume that the one-period, two-state security market model \( \left( {S, B,\Phi }\right) \) is arbitrage-free. Then the forward price at time 0 for the settlement date \( T \) of one share of stock \( S \) equals \( {F}_{S}\left( {0, T}\right) = \left( {1 + r}\right) {S}_{0} \) . | Proof. We shall apply the martingale method of Proposition 1.4.2. By applying formulas (1.14) and (1.22), we get\n\n\[ \n{\pi }_{0}\left( X\right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{\widehat{r}}^{-1}X}\right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {S}_{T}^{ * }\right) - {\widehat{r}}^{-1}K = {S}_{0} - {\wid... | Yes |
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