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Then \( {\Phi }^{\varepsilon } \) satisfies\n\n(6.18)\n\n\[ {\Phi }_{t}^{\varepsilon } + \frac{\varepsilon }{2}{\Delta }_{x}{\Phi }^{\varepsilon } + \frac{q\left( x\right) }{\varepsilon }{\Phi }^{\varepsilon } = 0 \]\n\nwith \( {\Phi }^{\varepsilon }\left( {{t}_{1}, x}\right) = 1 \) . | As \( \varepsilon \rightarrow 0,{V}^{\varepsilon } = - \varepsilon \log {\Phi }^{\varepsilon } \) tends to \( {V}^{0} \), where \( {V}^{0}\left( {t, x}\right) \) is the least action for initial data \( x\left( t\right) = x \) . | No |
Proposition 7.1. If \( W\left( x\right) \) is any storage function, then \( \left( {7.4}^{\prime }\right) \) holds. | Proof. Take \( t = 0, s = T \) in (7.5) and use the fact that \( W\left( {x\left( T\right) }\right) \geq 0 \) . | No |
Proposition 7.2. Let \( W \in {C}^{1}\left( {\mathbb{R}}^{n}\right) \) satisfy \( W\left( x\right) \geq 0, W\left( 0\right) = 0 \) and \( H\left( {x,{D}_{x}W\left( x\right) }\right) \geq 0 \) for all \( x \in {\mathbb{R}}^{n} \) . Then \( W \) is a storage function. | Proof. We have\n\n\[ \frac{d}{dr}W\left( {x\left( r\right) }\right) + \mu \ell \left( {x\left( r\right) }\right) - \frac{1}{2}{\left| z\left( r\right) \right| }^{2} \leq - H\left( {x\left( r\right) ,{DW}\left( {x\left( r\right) }\right) }\right) \leq 0. \]\n\nWe integrate from \( t \) to \( s \) to obtain (7.5) | Yes |
Let \( W\left( x\right) = B{\left| x\right| }^{2}, B > 0 \) . Then\n\n\[ H\left( {x,{DW}\left( x\right) }\right) = - {2Bb}\left( x\right) \cdot x - 2{B}^{2}a\left( x\right) x \cdot x - \mu \ell \left( x\right) \geq \left( {{2Bc} - {2k}{B}^{2} - {\mu K}}\right) {\left| x\right| }^{2}, \] | with \( c, K \) as in (7.2),(7.3) and \( k > 0 \) some constant. Choose \( B < {k}^{-1}c \) . Then \( W\left( x\right) \) is a storage function if \( \mu \) is small enough, by Proposition 7.2. | Yes |
Proposition 7.3. Assume that \( \widetilde{W}\left( x\right) < \infty \) for all \( x \in {\mathbb{R}}^{n} \) . Then \( \widetilde{W} \) is a storage function. Moreover \( \widetilde{W} \leq W \) for any storage function \( W \) . | Proof. Inequality \( \left( {7.4}^{\prime }\right) \) for all \( z\left( \cdot \right) \) is equivalent to \( \widetilde{V}\left( {T, x}\right) \leq W\left( x\right) \) . Hence, \( \widetilde{W}\left( x\right) \leq W\left( x\right) \) if a storage function \( W \) exists. It remains to show that \( \widetilde{W}\left( ... | Yes |
Proposition 7.4. The linear system in (7.13) has \( H \) -infinity norm \( \leq \gamma \) if and only if (7.17) has a symmetric, nonnegative definite solution \( P \), with \( \mu = {\left( 2{\gamma }^{2}\right) }^{-1}. \) | Proof. If \( P \) is such a solution to (7.17), let \( W\left( x\right) = x \cdot {Px} \) . Then \( H\left( {x,{D}_{x}W\left( x\right) }\right) = \) 0 and \( W\left( x\right) \) is a storage function by Proposition 7.2. Conversely, if the \( H \) - infinity norm is \( \leq \gamma \), then the value function \( \widetil... | No |
Theorem 7.1. If \( \sigma \) is constant, then\n\n\[ \left| {\widetilde{V}\left( {T, x}\right) - \widetilde{V}\left( {T, y}\right) }\right| \leq \mu {c}^{-1}\begin{Vmatrix}{\ell }_{x}\end{Vmatrix} \cdot \left| {x - y}\right| \] | Proof. For any \( z\left( \cdot \right) \in \mathcal{Z}, x \) and \( y \), let \( x\left( s\right) \) be the solution to (7.1) with \( x\left( 0\right) = x \) and \( y\left( s\right) \) the solution to (7.1) with \( y\left( 0\right) = y \) . Let \( \zeta \left( s\right) = x\left( s\right) - y\left( s\right) \) . Since ... | Yes |
Theorem 8.1. Assume that \( U \) is compact, that (8.4) holds and that \( \ell \in \) \( {C}_{b}^{1}\left( {{\bar{Q}}_{0} \times U}\right) \) . Then the value function \( \Phi \) is a Lipschitz continuous viscosity solution of (8.5) with terminal data \( \Phi \left( {{t}_{1}, x}\right) = 1 \) . | Proof. As noted above, \( \Phi \) is bounded and \( \Phi \left( {t, x}\right) > 0 \), since \( \ell \) is bounded. Consider an \ | No |
Theorem 8.2. Assume that (8.4) holds, that \( \sigma = \sigma \left( {t, x}\right) \) and that \( \ell \in \) \( {C}^{1}\left( {{\bar{Q}}_{0} \times U}\right) \) . Also suppose that \( W \in {C}^{1,2}\left( {\bar{Q}}_{0}\right) \) is a solution to (8.14) such that \( \left| {{D}_{x}W\left( {t, x}\right) }\right| \leq M... | Proof. We write\n\n(8.17)\n\n\[ \bar{H}\left( {t, x,{D}_{x}W\left( {t, x}\right) }\right) = - {b}^{\underline{u}}\left( {t, x}\right) \cdot {D}_{x}W\left( {t, x}\right) - {\Lambda }^{\underline{u}}\left( {t, x}\right) \]\n\nwhere \( {\Lambda }^{\underline{u}}\left( {t, x}\right) \leq {\ell }^{\underline{u}}\left( {t, x... | Yes |
In a way similar to the stochastic linear regulator problem (Example III.8.1) let \( U = {\mathbb{R}}^{m}, f\left( {t, x, v}\right) = A\left( t\right) x + B\left( t\right) v,\sigma = \sigma \left( t\right) ,\ell \left( {t, x, v}\right) = \) \( x \cdot M\left( t\right) x + v \cdot N\left( t\right) v \), where \( M\left(... | By the same method as for the stochastic linear regulator problem, there is a solution \( W\left( {t, x}\right) \) to (8.18) with \( W\left( {{t}_{1}, x}\right) = 0 \) of the form\n\n(8.20)\n\n\[ W\left( {t, x}\right) = x \cdot \widetilde{P}\left( t\right) x + \widetilde{g}\left( t\right) ,\;{t}_{\min } < t \leq {t}_{1... | Yes |
For each \( v\left( \cdot \right) \in U \), let\n\n(9.10)\n\n\[ \n{k}^{v} = - {G}^{v}\left( {\log v}\right) + \frac{Gv}{v}.\n\]\n\nThen, for each \( \phi \) ,\n\n(9.11)\n\n\[ \n\mathop{\min }\limits_{U}\left\lbrack {-{G}^{v}\phi + {k}^{v}}\right\rbrack = {e}^{\phi }\left( {G{e}^{-\phi }}\right)\n\]\n\nThe minimum is at... | Proof. Fix \( x \in \sum \) and consider the function\n\n\[ \nF\left( y\right) = \frac{1}{v\left( x\right) }\lbrack \left( {v\phi }\right) \left( y\right) - \phi \left( x\right) v\left( y\right) + \left( {v\log v}\right) \left( y\right) \n\]\n\n\[ \n- \left( {\log v\left( x\right) }\right) v\left( y\right) - v\left( y\... | Yes |
A classical approximation of the equation (2.3) is\n\n\[ - \frac{\partial }{\partial t}{V}^{\varepsilon }\left( {t, x}\right) - \frac{\varepsilon }{2}\operatorname{tr}a\left( {t, x}\right) {D}_{x}^{2}{V}^{\varepsilon }\left( {t, x}\right) + {H}^{\varepsilon }\left( {t, x,{D}_{x}{V}^{\varepsilon }\left( {t, x}\right) }\... | Now suppose that there exists a classical solution \( {V}^{\varepsilon } \) of \( \left( {2.5}^{\varepsilon }\right) \) and (2.2), and that \( {V}^{\varepsilon } \) converges to \( V \) uniformly on \( \bar{Q} \) as \( \varepsilon \) tends zero. Then due to the stability result, Lemma II.6.2, \( V \) is a viscosity sol... | No |
Example 2.2. (Deterministic limit of controlled jump Markov processes). For \( \phi \in {C}^{1}\left( \bar{Q}\right) ,\left( {t, x}\right) \in \bar{Q} \) and \( v \in U \) let\n\n\[ \n\left( {{\mathcal{G}}_{t}\phi }\right) \left( x\right) = \mathop{\sup }\limits_{{v \in U}}\{ - b\left( {t, x, v}\right) {D\phi }\left( x... | Let \( \left\{ {X}_{s}\right\} \) be the corresponding Markov process started at \( {X}_{t} = x \) with a fixed control \( v \) . Suppose that \( O \) is convex and for \( \varepsilon \in (0,1\rbrack \), define a rescaled process \( {X}_{s}^{\varepsilon } \) by\n\n\[ \n{X}_{s}^{\varepsilon } = x + \varepsilon \left( {{... | Yes |
Proposition 4.1 (Stability). Let \( H \) in (4.6) be a continuous function. Then \( {V}^{ * } \) is a viscosity subsolution of (2.3) in \( Q \) and \( {V}_{ * } \) is a viscosity super-solution of (2.3) in \( Q \) . | Proof. Let \( w \in {C}^{\infty }\left( \bar{Q}\right) \) and \( \left( {\bar{t},\bar{x}}\right) \in Q \) be a strict maximizer of \( {V}^{ * } - w \) on \( \bar{Q} \) with \( {V}^{ * }\left( {\bar{t},\bar{x}}\right) = w\left( {\bar{t},\bar{x}}\right) \) . Choose a sequence\n\n\[ \left( {{t}_{\varepsilon },{x}_{\vareps... | Yes |
Proposition 6.1. Suppose that \( {V}^{\varepsilon } \in C\left( \bar{Q}\right) \) and \( {\Psi }^{\varepsilon } \in C\left( \bar{Q}\right) \) converges to \( \Psi \) uniformly on \( \bar{Q} \), as \( \varepsilon \) tends to zero. Then \( {V}^{ * } \) and \( {V}_{ * } \) are a viscosity subsolution and a viscosity super... | Proof. We already know that \( {V}^{ * } \) is a viscosity subsolution of (2.3) in \( Q \) . We continue by verifying (2.4)(a) in the viscosity sense, i.e. in the sense of the above definition. So let \( w \in {C}^{\infty }\left( \bar{Q}\right) \) and let \( \left( {\bar{t},\bar{x}}\right) \in \left\lbrack {{t}_{0},{t}... | Yes |
Theorem 8.2. Assume (8.2), (8.16) and that there exists a Lipschitz continuous viscosity solution of (2.3) in \( Q \), satisfying (2.4)(a) pointwise. Then (2.3)-(2.4) has a weak comparison principle. | Proof. A minor modification of the proof of the previous theorem yields this result. Indeed follow that proof without a change until the sixth step of the last case. Replace that step by the following:\n\n\( {\mathbf{6}}^{\prime } \) . Then \( \left( {{q}_{\gamma },{p}_{\varepsilon }}\right) \in {D}^{ + }V\left( {{s}^{... | Yes |
Theorem 8.4. Assume \( \Psi \in C\left( \bar{Q}\right) \) and let \( H \) be as in (8.18) with \( L, f \) satisfying the hypotheses of Theorem II.13.1(b). Then, (2.3)-(2.4) has a weak comparison principle. | Sketch of proof. Using Theorem II.13.1 we obtain a viscosity solution \( V \in C\left( \bar{Q}\right) \) of (2.3),(2.4). Also the assumptions on \( L \) and \( f \) imply (8.1), and the differentiability of \( \partial O \) implies (8.2). Then as in the proof of Theorem 8.1, we compare the viscosity subsolution \( W \)... | No |
Then \( {V}^{\varepsilon } \) is the unique solution of\n\n\[ - \frac{\partial }{\partial t}{V}^{\varepsilon }\left( {t, x}\right) - \frac{\varepsilon }{2}\frac{{\partial }^{2}}{\partial {x}^{2}}{V}^{\varepsilon }\left( {t, x}\right) - \frac{\partial }{\partial x}{V}^{\varepsilon }\left( {t, x}\right) = 0,\left( {t, x}... | A direct calculation gives,\n\n\[ \mathop{\lim }\limits_{{\varepsilon \downarrow 0}}{V}^{\varepsilon }\left( {t, x}\right) = \left\{ \begin{matrix} {e}^{-t + x - 1}\text{ if }\left( {t, x}\right) \in \left\lbrack {0,1}\right\rbrack \times (0,1\rbrack \\ 0\;\text{ if }\left( {t, x}\right) \in \left\lbrack {0,1}\right\rb... | No |
Lemma 10.2. For any positive constant \( M \) and \( \widetilde{\psi } \in {C}^{2}\left( \bar{O}\right) \) with \( \widetilde{\psi }\left( x\right) = 0 \) for all \( x \in \partial O \), there exists \( {K}_{M} > 0 \) satisfying\n\n(10.11)\n\n\[ \n{V}^{\varepsilon }\left( {t, x}\right) \geq M\widetilde{\psi }\left( x\r... | Proof. Set\n\n\[ \n{K}_{M} = \mathop{\sup }\limits_{{\left( {t, x}\right) \in \bar{Q}}}\left\{ {-\frac{M}{2}\operatorname{tr}a\left( {t, x}\right) {D}^{2}\widetilde{\psi }\left( x\right) + H\left( {t, x,{MD}\widetilde{\psi }\left( x\right) }\right) }\right\} \n\]\n\nand\n\n\[ \n{\bar{g}}^{\varepsilon }\left( {t, x}\rig... | Yes |
Theorem 10.1. Suppose \( \partial O \) is of class \( {C}^{3} \) . Assume (10.5). Then \( {V}^{\varepsilon } \) converges to \( V \) uniformly on compact subsets of \( \left\lbrack {{t}_{0},{t}_{1}}\right) \times \bar{O} \), as \( \varepsilon \downarrow 0 \) . | Proof. It suffices to show that for all \( \delta > 0 \), (10.19) \[ {V}^{ * }\left( {t, x}\right) \leq V\left( {t + \delta, x}\right) ,\left( {t, x}\right) \in \left\lbrack {{t}_{0},{t}_{1} - \delta }\right) \times \bar{O}, \] and (10.20) \[ V\left( {t - \delta, x}\right) \leq {V}_{ * }\left( {t, x}\right) ,\;\left( {... | Yes |
Theorem 11.1. Assume (11.3) and that (11.1)-(11.2) has a bounded, Lipschitz continuous viscosity solution \( V \) . Then a weak comparison principle holds. | An easy consequence of Theorem 11.1 is the following result stated in Section VI.6.\n\nProof of Theorem VI.6.2. Since VI(6.8) holds, the Hamiltonian \( H\left( {t, x, p}\right) \) in VI(4.8) satisfies (11.3)(b). Moreover, VI(6.8)(a)(b) imply (11.3)(a) and that \( \left| {{V}^{\epsilon }\left( {t, x}\right) }\right| \le... | Yes |
Lemma 3.1. Let \( x \in O,\widehat{v} \in U \) and \( h > 0 \) . If \( x + h\widehat{v} \in O \), then\n\n\[ V\left( x\right) - V\left( {x + h\widehat{v}}\right) \leq h\widehat{c}\left( \widehat{v}\right) \] | Proof. In view of (2.2iv), we may assume that \( \left| \widehat{v}\right| \leq 1 \) . For \( \left( {\xi \left( \cdot \right) ,\widehat{u}\left( \cdot \right) }\right) \in \) \( {\widehat{\mathcal{A}}}_{\nu } \), let\n\n\[ {\xi }_{h}\left( s\right) = \left\{ {\begin{array}{ll} 0, & s = 0 \\ \xi \left( s\right) + h, & ... | Yes |
Theorem 4.1. (Verification) Assume \( O \) is convex. Let \( W \) be a classical solution of (2.7) with the boundary condition (2.8). Then for every \( x \in \bar{O} \) (a) \( W\left( x\right) \leq J\left( {x;\xi ,\widehat{u}}\right) \) for any \( \left( {\xi \left( \cdot \right) ,\widehat{u}\left( \cdot \right) }\righ... | Proof. (a) Extend \( W \) to \( {\mathbb{R}}^{n} \) by setting \( W\left( x\right) = 0 \) for \( x \notin O \) . Since \( W\left( x\right) = 0 \) for \( x \in \partial O \) and since \( W \) is continuous on \( \bar{O} \), the extension of \( W \) is also continuous on \( {\mathbb{R}}^{n} \) . Let \( \zeta \in {C}^{\in... | Yes |
Consider a one dimensional problem with \( O = \left( {-\infty ,\infty }\right) \) , \( \widehat{f} \equiv 0,\widehat{\sigma } \equiv \sqrt{2},\widehat{c}\left( v\right) = \left| v\right| ,\widehat{K} = \{ - 1\} \) and \( \widehat{L} \) is convex. Then \( U = ( - \infty ,0\rbrack \) and the hypotheses of Lemma 3.2 are ... | We will first construct a convex, polynomially growing solution \( W \) of (4.5) and then using the Verification Theorem we will show that \( W = V \) . Now suppose that \( W \) is indeed a convex solution of (4.5). Set\n\n\[ a = \sup \left\{ {x : {W}_{x}\left( x\right) < 1}\right\} .\n\nHere \( a \) may be equal to \(... | Yes |
Consider the same problem as in the previous example but with \( \widehat{K} = \{ 1, - 1\} \) . Then \( U = \left( {-\infty ,\infty }\right) \) and (2.7) takes the form\n\n\[ \max \left\{ {V\left( x\right) - {V}_{xx}\left( x\right) - \alpha {x}^{2},\left| {{V}_{x}\left( x\right) }\right| - 1}\right\} = 0,\forall x \in ... | Following the procedure devised in Example 4.1, we look for a convex, poly-nomially growing function \( W \) and constants \( - \infty \leq a < b \leq + \infty \) satisfying\n\n\[ W\left( x\right) - {W}_{xx}\left( x\right) = \alpha {x}^{2},\;\forall a < x < b, \]\n\n\[ {W}_{x}\left( x\right) = - 1,\;\forall x \leq a, \... | Yes |
We claim that there exists a smooth function \( \widehat{L} \) satisfying (4.15i) \( \widehat{L}\left( x\right) = \alpha {x}^{2},\left| x\right| \leq 1, \) (4.15ii) \( \widehat{L}\left( x\right) = {10} - \frac{3}{2}{e}^{-\left( {\left| x\right| - {x}_{0}}\right) /2},\left| x\right| \geq {x}_{0}, \) (4.15iii) \( \wideha... | Indeed for \( \left| x\right| \notin \left( {1,{x}_{0}}\right) \), define \( \widehat{L}\left( x\right) \) by (4.15i) and (ii). Set \( h\left( x\right) = \widehat{L}\left( x\right) - \parallel x\left| {-1}\right| - \alpha ,\left| x\right| \notin \left( {1,{x}_{0}}\right) \) Then (4.14) yields \( h\left( {x}_{0}\right) ... | Yes |
In this example \( O = {\mathbb{R}}^{2},\widehat{f} \equiv 0,\beta = 1,\widehat{\sigma } \) is equal to \( \sqrt{2} \) times \( {a2} \times 2 \) identity matrix, \[ U = ( - \infty ,0\rbrack \times ( - \infty ,0\rbrack \] and for \( v \in \left( {{v}_{1},{v}_{2}}\right) \in U, x = \left( {{x}_{1},{x}_{2}}\right) \in O \... | Let \( W \) be the function defined by (4.9). Then \[ V\left( {{x}_{1},{x}_{2}}\right) = W\left( {x}_{1}\right) + W\left( {x}_{2}\right) \] is a classical solution of (4.16). Moreover \[ \mathcal{P} = \left( {-\infty, a}\right) \times \left( {-\infty, a}\right) \] and \( \partial \mathcal{P} \) is not differentiable. | No |
Consider the explicit finite difference scheme (3.26) in Section 3. Then | \[ \left| {{J}_{k}^{h}\left( {x;\pi }\right) }\right| \leq \parallel L\parallel \left( {{t}_{1} - t}\right) + \parallel \psi \parallel . \]\n\nThe value function \( {V}^{h}\left( {t, x}\right) \) therefore satisfies the same inequality, and in (4.5) we may take \( K = \left( {{t}_{1} - {t}_{0}}\right) \parallel L\paral... | No |
Lemma 4.1. \( {V}^{ * } \) is a viscosity subsolution of the HJB equation, and \( {V}_{ * } \) is a viscosity supersolution. | Proof. To show that \( {V}^{ * } \) is a viscosity subsolution, suppose that \( w \) is a test function such that \( {V}^{ * } - w \) has a maximum at \( \left( {\bar{t},\bar{x}}\right) \in {Q}_{0} \) . As noted earlier (Definition VII.4.2 and Remark VII.4.2) we can assume that the maximum is strict. Then there is a se... | Yes |
Theorem 4.1. Let \( {V}^{h} \) be a solution to (4.1) and (4.2). Assume that (4.3)\n- (4.6) and (4.11) hold. Then\n\n(4.12)\n\n\[ \mathop{\lim }\limits_{\substack{{\left( {s, y}\right) \rightarrow \left( {t, x}\right) } \\ {h \downarrow 0} }}{V}^{h}\left( {s, y}\right) = V\left( {t, x}\right) \]\n\nuniformly on any com... | Proof. \( V \) is a bounded, uniformly continuous viscosity solution of the HJB equation (3.2) with the terminal data (3.3). (See comment following V(9.1).) By Lemma 4.1, \( {V}^{ * } \) is a bounded uppersemicontinuous subsolution of (3.2) and by Lemma \( {4.2}{V}^{ * }\left( {{t}_{1}, x}\right) = \psi \left( x\right)... | No |
Lemma 5.1. Assume that \( V \in C\left( \bar{Q}\right) \) and that\n\n(5.5)\n\n\[ \mathop{\lim }\limits_{\substack{{\left( {s, y}\right) \rightarrow \left( {t, x}\right) } \\ {h \downarrow 0} }}{V}^{h}\left( {s, y}\right) = \Psi \left( {t, x}\right) \]\n\nuniformly with respect to \( \left( {t, x}\right) \in {\partial ... | Sketch of proof. The proof is almost the same as for Theorem 4.1, and we merely sketch it. For \( \left( {t, x}\right) \in Q \), define \( {V}^{ * }\left( {t, x}\right) \) and \( {V}_{ * }\left( {t, x}\right) \) by (4.8). As in Lemma 4.1, \( {V}^{ * } \) is a viscosity subsolution and \( {V}_{ * } \) a viscosity supers... | No |
Lemma 5.3. \( \mathop{\lim }\limits_{\substack{{\left( {s, y}\right) \rightarrow \left( {{t}_{1}, x}\right) } \\ {h \downarrow 0} }}{V}^{h}\left( {s, y}\right) = \psi \left( x\right) \) uniformly with respect to \( x \in \bar{O} \) . | Proof. Choose \( \widetilde{\psi } \in {C}^{2}\left( \bar{O}\right) \) such that \( \psi \left( x\right) < \widetilde{\psi }\left( x\right) \) for all \( x \in \bar{O} \) . Let\n\n\[ W\left( {t, x}\right) = K\left( {{t}_{1} - t}\right) + \widetilde{\psi }\left( x\right) \]\n\nFor \( K \) sufficiently large, \( W \) sat... | Yes |
Assume that the uniform ellipticity condition IV(3.5), or IV(5.10), holds. Choose any \( {v}_{1} \in U \) . The linear elliptic PDE\n\n\[ \n{G}^{{v}_{1}}\phi \left( x\right) = 1, x \in O, \n\]\n\nwith the boundary \( \phi \left( x\right) = 0, x \in \partial O \), has a positive solution \( \phi \in {C}^{2}\left( \bar{O... | In fact, \( \phi \left( x\right) \) is the mean exit time from \( O \) for the solution to the stochastic differential equation IV(5.1) with \( u\left( s\right) \equiv {v}_{1} \) and \( x\left( 0\right) = x \) . Then (5.14) holds with \( \delta \) arbitrary and \( c = 1 \) . | No |
Consider a deterministic control problem \( \left( {a\left( {x, v}\right) \equiv 0}\right) \) in which the following slightly stronger form of I(3.11) holds. Assume that for every \( \xi \in \partial O \) there exists \( v\left( \xi \right) \in U \) such that\n\n(5.15)\n\n\[ f\left( {\xi, v\left( \xi \right) }\right) \... | Let \( \phi \left( x\right) = \) dist \( \left( {x,\partial O}\right) \) . Since \( \partial O \) is a \( {C}^{3} \) manifold, \( \phi \in {C}^{2}\left( {\bar{O}}_{\delta }\right) \) for \( \delta \) small enough. Moreover, \( {D\phi }\left( \xi \right) = - \eta \left( \xi \right) \) for \( \xi \in \partial O \) . Henc... | Yes |
Lemma 5.4. Assume (5.13), (5.14). Then\n\n\\[ \n\\mathop{\\lim }\\limits_{\\substack{{\\left( {s, y}\\right) \\rightarrow \\left( {t, x}\\right) } \\\\ {h \\downarrow 0} }}{V}^{h}\\left( {s, y}\\right) = 0 \n\\]\n\nuniformly for \\( \\left( {t, x}\\right) \\in \\left\\lbrack {{t}_{0},{t}_{1}}\\right\\rbrack \\times \\p... | Proof. Let \\( {Q}_{\\delta } = \\left\\lbrack {{t}_{0},{t}_{1}}\\right) \\times {O}_{\\delta } \\), and let\n\n\\[ \nW\\left( x\\right) = {K\\phi }\\left( x\\right) + {a}_{1} \n\\]\n\nwhere \\( K \\) is to be suitably chosen and \\( {a}_{1} > 0 \\) is arbitrary. Then\n\n\\[ \n\\mathcal{H}\\left( {x,{DW},{D}^{2}W}\\rig... | Yes |
Lemma 5.5. Let \( \eta \in {C}^{2}\left( {\mathbb{R}}^{n}\right) \) with \( {D\eta } \) and \( {D}^{2}\eta \) bounded. Then there exists \( M \) such that, for all \( \alpha > 0,\left( {x, p, A}\right) \in {\mathbb{R}}^{2n} \times {\mathcal{S}}_{ + }^{n} \)\n\n\[ \mid \widetilde{\mathcal{H}}\left( {x,{p}_{i}^{ \pm },{A... | Proof. We recall the definition (3.27) of \( \widetilde{\mathcal{H}} \), and that \( f,\sigma \) are assumed bounded. Then\n\n\[ \left| {{f}_{i}^{ + }{\left( {p}_{i} + \alpha {\eta }_{{x}_{i}}\right) }^{ + } - {f}_{i}^{ + }{p}_{i}^{ + }}\right| \leq \alpha \begin{Vmatrix}{f}_{i}^{ + }\end{Vmatrix}\begin{Vmatrix}{\eta }... | Yes |
Theorem 5.3. Let \( {\bar{V}}^{h} \) be the value function for the explicit finite difference scheme with numerical cutoff. Then\n\n\[ \mathop{\lim }\limits_{\substack{{\left( {s, y}\right) \rightarrow \left( {t, x}\right) } \\ {h \downarrow 0} }}{\bar{V}}^{h}\left( {s, y}\right) = V\left( {t, x}\right) \]\n\nuniformly... | Proof. By Theorem 4.1, we need only verify that (4.11) holds uniformly on compact sets. If \( \psi \) is bounded and uniformly continuous, this follows from Lemma 5.6 with \( {\widetilde{V}}^{h} = {\bar{V}}^{h} \) and Theorem 4.2. Next, suppose that \( \psi \) is merely bounded and continuous. Let \( K \) be compact. C... | Yes |
Theorem 4.1 [Theorem 3.9.4 [KS5]]. Given \( x \geq 0 \), and a consumption process \( c\left( \cdot \right) \) satisfying the budget constraint (4.5) with an equality, there exists a portfolio process \( \pi \left( \cdot \right) \), such that \( \left( {\pi, c}\right) \) is admissible at \( x \) . Moreover, the corresp... | The proof is an application of the martingale representation result. | No |
Corollary 4.1. The Merton portfolio selection problem is equivalent to maximizing (3.1) under the budget constraint (4.5). | We solve the equivalent problem using a Lagrange multiplier technique. For \( y \geq 0 \), consider\n\n\[ H\left( {x, y, c\left( \cdot \right) }\right) \mathrel{\text{:=}} E{\int }_{0}^{\infty }{e}^{-{\beta t}}\ell \left( {c\left( t\right) }\right) {dt} + y\left( {x - E\left\lbrack {{\int }_{0}^{\infty }{e}^{-{rt}}c\le... | Yes |
Theorem 5.2. For any reference probability system \( \nu \), and \( x \in O \) ,\n\n\[ J\left( {x;{c}^{ * },{M}_{i}^{ * }}\right) = {V}_{\nu }\left( x\right) = V\left( x\right) . \] | We refer to [ShS] for the proof. | No |
Consider the financial market described in Section 2 and suppose that the fair value is a function \( V\left( {t, s}\right) \) of the current stock price \( s \) and the time \( t \) . If this is the fair value one can buy or sell this derivative in the market for this value. Suppose that we sell it and obtain a cash a... | \[ {dx}\left( \rho \right) = x\left( \rho \right) \left\lbrack {\left( {r + \pi \left( \rho \right) \left\lbrack {\mu - r}\right\rbrack }\right) {d\rho } + \pi \left( \rho \right) {\sigma dw}\left( \rho \right) }\right\rbrack . \] with the initial condition \( x\left( t\right) = V\left( {t, s}\right) \) . Arbitrage arg... | Yes |
Theorem 7.1. In the financial market of Section 2 (without consumption), for any \( \left( {t, s, x}\right) \) the utility price of a derivative is equal to the Black-Scholes price given (6.5). | Proof. When consumption \( c\left( t\right) \) is omitted from the model in Section 2, equation (2.2) becomes (6.1). Let \( x \) and \( \pi \) be as in (6.1). For the purposes of this proof, it is more convenient to work with \( \alpha \mathrel{\text{:=}} {x\pi } \) . With this notation, (6.1) has the form\n\n(7.3)\n\n... | Yes |
Theorem 10.1 [CK]. The minimal super-replicating cost \( \bar{V}\left( {t, s}\right) \) is the value function of the standard optimal control problem,\n\n\[ \bar{V}\left( {t, s}\right) = E\left\lbrack {\exp \left( {-{\int }_{t}^{T}{\delta }_{K}\left( {\nu \left( \rho \right) }\right) {d\rho }}\right) \varphi \left( {{S... | Now this problem can be solved by dynamic programming. Indeed, an explicit solution was obtained by Broadie, Cvitanic & Soner [BCS]. Here we state it without proof. | No |
Theorem 10.2 [BCS]. The minimal super-replicating cost \( \\bar{V}\\left( {t, s}\\right) \) is equal to the Black-Scholes price with a modified pay-off \( \\widehat{\\varphi } \) given by\n\n\[ \n\\widehat{\\varphi }\\left( s\\right) \\mathrel{\\text{:=}} \\mathop{\\sup }\\limits_{{\\nu \\in \\widetilde{K}}}\\left\\{ {... | Recall that we are assuming that the interest rate is zero. One needs to discount the above formula appropriately to obtain the formula in models with non-zero interest rate. | No |
Theorem 11.1. Equation (11.6) has a unique solution \( Z \in {C}^{3}\left( {\mathbb{R}}^{1}\right) \) such that \( Z\left( \eta \right) \) is bounded and \( {Z}_{\eta }\left( \eta \right) \) tends to 0 as \( \left| \eta \right| \rightarrow \infty \) . Moreover, \( W\left( {x,\eta }\right) = \) \( {\gamma }^{-1}{x}^{\ga... | Proof. (Sketch) We first find \( {K}^{ + },{K}^{ - } \) which are constant super- and sub-solutions to (11.6). Let \( \underline{\beta } \) be as in (11.12)(ii) and choose \( \bar{\beta } \) such that \( \beta - \) \( {\gamma Q}\left( \eta \right) \leq \bar{\beta } \) for all \( \zeta \) . We take\n\n\[ \n{Z}^{ + } = \... | Yes |
Theorem 3.1. Let \( W \in {C}^{1}\left( {\bar{Q}}_{0}\right) \) be a solution to the upper Isaacs PDE (3.6) and (3.8). Moreover, let \( {\underline{u}}^{ * },{\underline{z}}^{ * } \) be Lipschitz continuous and satisfy (3.12,(3.13)). Then:\n\n(a) For every \( z\left( \cdot \right) \in {L}^{\infty }\left( {\left\lbrack ... | Proof. Let \( F\left( {t, x, u, z}\right) \) be as above, with \( {V}_{ + } \) replaced by \( W \) . Then\n\n\[ F\left( {s, x\left( s\right) ,{\underline{u}}^{ * }\left( {s, x\left( s\right) }\right), z\left( s\right) }\right) \leq - {H}_{ + }\left( {s, x\left( s\right) ,{D}_{x}W\left( {s, x\left( s\right) }\right) }\r... | Yes |
Lemma 4.1. There exists a constant \( {M}_{1} \) such that\n\n\[ \left| {{V}_{ + }\left( {t, x}\right) - {V}_{ + }\left( {t, y}\right) }\right| \leq {M}_{1}\left| {x - y}\right| \]\n\nfor all \( t \in \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack, x, y, \in {\mathbb{R}}^{n} \) . | Lemma 4.1 is proved in the same way as \( \Pi \left( {10.2}\right) \), making use of the uniform Lipschitz bounds in (3.4)(c) and (d). | No |
Theorem 5.2. The upper value function \( {V}_{ + } \) is bounded and Lipschitz continuous on \( {\bar{Q}}_{0} \) . | Proof. By the definition (4.2) and (3.4)(b)(d), (5.2) \[ \left| {{V}_{ + }\left( {t, x}\right) }\right| \leq \left( {{t}_{1} - t}\right) \parallel L\parallel + \parallel \psi \parallel \] where \( \parallel \parallel \) is the sup norm. Lemma 4.1 gives a uniform Lipschitz bound for \( {V}_{ + }\left( {t, \cdot }\right)... | Yes |
Lemma 6.1. Let \( \Phi \) be continuous on \( U \times Z \) . Then for every \( \theta > 0 \) there exists a Borel measurable function \( \zeta \) from \( U \) into \( Z \) such that\n\n(6.1)\n\n\[ \mathop{\max }\limits_{{z \in Z}}\Phi \left( {u, z}\right) < \Phi \left( {u,\zeta \left( u\right) }\right) + \theta ,\;\te... | Proof. Given \( \delta > 0 \), partition the compact set \( U \) into Borel sets \( {A}_{1},\cdots ,{A}_{M} \) of diameter less than \( \delta \) . Choose \( {u}_{i} \in {A}_{i} \) and \( {z}_{i} \) which maximizes \( \Phi \left( {{u}_{i}, \cdot }\right) \) on \( Z \) . Let \( \zeta \left( u\right) = {z}_{i} \) for all... | Yes |
Lemma 6.2. For \( w \in {C}^{1}\left( {Q}_{0}\right) ,\left( {t, x}\right) \in {Q}_{0} \)\n\n(6.3)\n\n\[ \mathop{\lim }\limits_{{h \downarrow 0}}\frac{1}{h}\left\lbrack {\mathop{\sup }\limits_{{\beta \in \Delta }}\mathop{\inf }\limits_{{u \in \mathcal{U}}}{\int }_{t}^{t + h}F\left( {s, x\left( s\right), u\left( s\right... | Proof. Since \( \left| {x\left( s\right) - x}\right| \leq \parallel G\parallel h \) for \( s \in \left\lbrack {t, t + h}\right\rbrack \),\n\n(6.4)\n\n\[ \left| {F\left( {s, x\left( s\right), u\left( s\right) ,\beta \left( u\right) \left( s\right) }\right) - F\left( {t, x, u\left( s\right) ,\beta \left( u\right) \left( ... | Yes |
Theorem 6.1. The upper value function \( {V}_{ + } \) is the unique bounded, uniformly continuous viscosity solution to the upper Isaacs PDE (3.6) with the terminal condition (3.8). | Proof. By Theorem 5.2, \( {V}_{ + } \) is bounded and Lipschitz continuous on \( {\bar{Q}}_{0} \) (hence uniformly continuous on \( {\bar{Q}}_{0} \) ). We show that \( {V}_{ + } \) is a viscosity solution to (3.6). Uniqueness follows from Corollary II.9.1. Following the notation of Section II. 3 let \( \mathcal{C} = \{... | Yes |
Let \( n = 1, U = Z = \left\lbrack {-1,1}\right\rbrack \) and \( \dot{x}\left( s\right) = \left| {u\left( s\right) - z\left( s\right) }\right| \), where \( \cdot = d/{ds} \). The game payoff is \( P\left( {t, x;u, z}\right) = \psi \left( {x\left( {t}_{1}\right) }\right) \), where \( \psi \in {C}^{1}\left( {\mathbb{R}}^... | We obtain \( {H}_{ + }\left( p\right) = - p \) and \( {H}_{ - }\left( p\right) = 0 \) if \( p \geq 0 \). By explicitly solving (3.6) and (3.8), \( {V}_{ + }\left( {t, x}\right) = \psi \left( {x + {t}_{1} - t}\right) \). The optimal control policy for the minimizer is \( {\underline{u}}^{ * }\left( {t, x}\right) = 0 \) ... | Yes |
Lemma 7.1. There exists \( {M}_{1} \) such that\n\n\[ \left| {{V}_{+R}\left( {t, x}\right) - {V}_{+R}\left( {t, y}\right) }\right| \leq {M}_{1}\left| {x - y}\right| \]\n\nfor all \( t \in \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack, x, y \in {\mathbb{R}}^{n} \) . | Proof. Let \( {\beta }_{0}\left( u\right) \left( s\right) = 0 \) for all \( u\left( \cdot \right) \in \mathcal{U}, s \in \left\lbrack {t,{t}_{1}}\right\rbrack \) . Then\n\n\[ - \left( {{t}_{1} - t}\right) \parallel \ell \parallel \leq \mathop{\inf }\limits_{{u \in \mathcal{U}}}P\left( {t, x;u,{\beta }_{0}}\right) \leq ... | Yes |
Theorem 7.1. (a) There exists \( {R}_{1} \) such that \( {V}_{+R}\left( {t, x}\right) = {V}_{ + }\left( {t, x}\right) \) for every \( R \geq {R}_{1} \) . | Proof. Let us prove (a). Part (b) follows from (a) by the same proof as the Theorem VI.6.1. Let\n\n\[ \n{H}_{+R}\left( {t, x, p}\right) = - \mathop{\min }\limits_{{v \in U}}\mathop{\max }\limits_{{\left| \xi \right| \leq R}}\left\lbrack {p \cdot \left( {f\left( {t, x, v}\right) + \sigma \left( {t, x, v}\right) \xi }\ri... | Yes |
Theorem 7.2. \( {V}^{\varepsilon }\left( {t, x}\right) \) tends to \( {V}_{ + }\left( {t, x}\right) \) uniformly on compact subsets of \( {\bar{Q}}_{0} \) . | Proof. By Corollary VI.8.1, \( {V}^{\varepsilon } \) is a viscosity solution of\n\n(7.7)\n\n\[ \n- \frac{\partial {V}^{\varepsilon }}{\partial t} + {\overline{\mathcal{H}}}_{\varepsilon }\left( {t, x,{D}_{x}{V}^{\varepsilon },{D}_{x}^{2}{V}^{\varepsilon }}\right) = 0 \n\] \n\nwith \( {\overline{\mathcal{H}}}_{\varepsil... | Yes |
Lemma 8.1. For \( t = {r}_{i}, r = {r}_{j} \) with \( i < j \), let \( {\mathcal{U}}_{{\pi }_{1}} = {\mathcal{U}}_{\pi }\left( {t, r}\right) ,{\Delta }_{1} = \Delta \left( {t, r}\right) \) . Then\n\n\[ \n{V}_{ + }^{\pi }\left( {t, x}\right) = \mathop{\sup }\limits_{{{\beta }_{1} \in {\Delta }_{1}}}\mathop{\inf }\limits... | Lemma 8.1 is a discrete time version of the dynamic programming principle for differential games. It is proved in exactly the same way as Theorem 5.1, taking \( t = {r}_{i}, r = {r}_{j} \) and considering only controls \( u\left( \cdot \right) \in {\mathcal{U}}_{\pi }\left( {t,{t}_{1}}\right) ,{u}_{1}\left( \cdot \righ... | No |
Consider any partition of \( \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \) into subintervals \( \left\lbrack {{r}_{j},{r}_{j + 1}}\right) \) , with endpoints in a finite set \( \pi \) as in Section 8. For \( t = {r}_{i},{\mathcal{U}}_{\pi }\left( {t,{t}_{1}}\right) \) consists of functions \( u\left( \cdot \right) \) ... | To see this, given any \( \beta \in \Delta \left( {t,{t}_{1}}\right) \), choose \( {u}_{i} \in U \) arbitrarily and let \( \widehat{u}\left( s\right) = {u}_{i},\widehat{z}\left( s\right) = \beta \left( \widehat{u}\right) \left( s\right) \) for \( s \in \left\lbrack {{r}_{i},{r}_{i + 1}}\right) \) . Proceeding by induct... | Yes |
In this example, \( \alpha \) is progressive but not strictly progressive. Let \( \alpha \left( z\right) \left( s\right) = \phi \left( {z\left( s\right) }\right) \) where \( \phi \) is a non constant Borel measurable function from \( Z \) into \( U \) . There is a partition \( U = {U}_{1} \cup {U}_{2} \) into Borel set... | Let \( \beta \left( u\right) \left( s\right) = \chi \left( {u\left( s\right) }\right) \) . Then \( \left( {\alpha \circ \beta }\right) \left( z\right) \left( s\right) = \left( {\phi \circ \chi }\right) \left( {z\left( s\right) }\right) \) . If (9.1) holds, then \( \widehat{u}\left( s\right) \) is a fixed point of \( \p... | Yes |
Theorem 9.1. \( W = {V}_{ + } \), where \( {V}_{ + } \) is the upper value function. | The following argument shows that \( {V}_{ + } \leq W \) . Given \( \theta > 0 \) there exists \( \beta \in \Delta \) such that (see (4.2)) (9.3) \[ P\left( {t, x;u,\beta \left( u\right) }\right) \geq {V}_{ + }\left( {t, x}\right) - \theta \] for every \( u\left( \cdot \right) \in \mathcal{U} \) . Given any \( \alpha \... | Yes |
For \( t = {r}_{i}, i = 0,1,\cdots, N - 1 \) ,\n\n\[\n\text{(9.4)}{V}_{ + }^{\pi }\left( {{r}_{i}, x}\right) = \mathop{\inf }\limits_{{u \in U}}\mathop{\sup }\limits_{{{z}_{1} \in {\mathcal{Z}}_{1}}}\left\lbrack {{\int }_{{r}_{i}}^{{r}_{i + 1}}L\left( {s, x\left( s\right), u,{z}_{1}\left( s\right) }\right) {ds} + {V}_{... | Proof. Denote by \( \left( *\right) \) the expression in brackets. In Lemma 8.1, take \( t = {r}_{i} \) , \( r = {r}_{i + 1} \) . Since \( u\left( s\right) = u \) is a constant on \( \left\lbrack {{r}_{i},{r}_{i + 1}}\right) \), chosen without knowing \( {z}_{1}\left( \cdot \right) \)\n\n\[{V}_{ + }^{\pi }\left( {{r}_{... | Yes |
Lemma 9.2. For \( t = {r}_{i}, i = 0,1,\cdots, N - 1 \) ,\n\n\[ \n{V}_{ + }^{\pi }\left( {t, x}\right) = \mathop{\inf }\limits_{{\alpha \in {\Gamma }_{\pi }}}\mathop{\sup }\limits_{{z \in \mathcal{Z}}}P\left( {t, x;\alpha \left( z\right), z}\right) .\n\] | Proof. Denote the right side of (9.5) by \( {W}^{\pi }\left( {t, x}\right) \) . Then \( {V}_{ + }^{\pi } \leq {W}^{\pi } \) by the argument used above to show that \( {V}_{ + } \leq W \) . We show that \( {W}^{\pi } \leq {V}_{ + }^{\pi } \) as follows. Given \( \left( {t, x}\right) \) and \( \theta > 0 \), with \( t = ... | Yes |
Corollary 6.8 (Sufficiency in Theorem 6.6): If \( N = D \) and \( \sigma \left( t\right) \) is nonsingular for Lebesgue-almost-every \( t \in \left\lbrack {0, T}\right\rbrack \) almost surely, then the financial market is complete. | Proof. We verify the condition of Proposition 6.2. Let \( B \) be an \( \mathcal{F}\left( T\right) \) -measurable random variable satisfying (6.3), and define the Lévy \( {P}_{0} \) -martingale\n\n\[ \n{M}_{0}\left( t\right) = {E}_{0}\left\lbrack {\left. \frac{B}{{S}_{0}\left( T\right) }\right| \;\mathcal{F}\left( t\ri... | Yes |
Example 3.1 (Forward contract to purchase a stock that pays no dividends): Suppose the contract is to purchase one share of the first stock, i.e., \( B = {S}_{1}\left( T\right) \) . If the first stock pays no dividends and \( \sigma \left( \cdot \right) \) satisfies the Novikov condition (1.5.17) with \( n = 1 \), then... | \[ {V}^{FC}\left( {t;q}\right) = {S}_{1}\left( t\right) - q{S}_{0}\left( t\right) \cdot {E}_{0}\left\lbrack {1/{S}_{0}\left( T\right) \mid \mathcal{F}\left( t\right) }\right\rbrack ,\;0 \leq t \leq T. \] (3.3) If in addition \( {S}_{0}\left( T\right) \) is nonrandom, the hedging portfolio is particularly simple. The ag... | Yes |
Example 3.3 (Forward price of a stock that pays no dividends): If \( B = \) \( {S}_{1}\left( T\right) \), the first stock pays no dividends, and \( \sigma \left( \cdot \right) \) satisfies the Novikov condition (1.5.17) with \( n = 1 \), then (3.3) and (3.5) yield | \[ f\left( t\right) = \frac{{S}_{1}\left( t\right) /{S}_{0}\left( t\right) }{{E}_{0}\left\lbrack {1/{S}_{0}\left( T\right) \mid \mathcal{F}\left( t\right) }\right\rbrack },\;0 \leq t \leq T. \] | Yes |
Example 3.4 (Forward price of a stock with nonrandom dividend rate): If \( B = {S}_{1}\left( T\right) \), the dividend rate process \( {\delta }_{1}\left( \cdot \right) \) is nonrandom and the processes \( {\sigma }_{11}\left( \cdot \right) ,\ldots ,{\sigma }_{1N}\left( \cdot \right) \) are uniformly bounded, then the ... | To see this, observe from (1.5.16) that the process \( \frac{{S}_{1}\left( t\right) }{{S}_{0}\left( t\right) }\exp \left\{ {{\int }_{0}^{t}{\delta }_{1}\left( u\right) {du}}\right\} \) is a \( {P}_{0} \) -martingale, so the numerator of (3.4) is\n\n\[ {E}_{0}\left\lbrack {{S}_{1}\left( T\right) /{S}_{0}\left( T\right) ... | Yes |
Example 3.5 (Forward price of a stock with nonrandom dividend payments, when the money market is nonrandom): If \( B = {S}_{1}\left( T\right) \), if the dividend-payment \( \rho \left( \cdot \right) \triangleq {\delta }_{1}\left( \cdot \right) {S}_{1}\left( \cdot \right) \) and money-market prices \( {S}_{0}\left( \cdo... | From (3.4) we have\n\n\[ f\left( t\right) = {S}_{0}\left( T\right) \left\lbrack {\frac{{S}_{1}\left( t\right) }{{S}_{0}\left( t\right) } - {\int }_{t}^{T}\frac{\rho \left( u\right) }{{S}_{0}\left( u\right) }{du}}\right\rbrack . \]\n\n(3.8) | No |
Corollary 3.9 (Forward-futures spread): Under the conditions of Theorem 3.7, we have\n\n\\[\nf\\left( t\\right) = \\varphi \\left( t\\right) + \\frac{{\\operatorname{Cov}}_{0}\\left\\lbrack {B,1/{S}_{0}\\left( T\\right) \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack }{{E}_{0}\\left\\lbrack {1/{S}_{0}\\left( T\\ri... | Proof. Because\n\n\\[\n{\\operatorname{Cov}}_{0}\\left\\lbrack {X, Y \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack = {E}_{0}\\left\\lbrack {{XY} \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack - {E}_{0}\\left\\lbrack {X \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack \\cdot {E}_{0}\\left\\lbrack {Y \\m... | Yes |
A European call option on the first stock in our market is the ECC given by \( C\left( t\right) = 0,\;0 \leq t < T \) and \( C\left( T\right) = {\left( {S}_{1}\left( T\right) - q\right) }^{ + } \) . The nonrandom constant \( q > 0 \) is called the strike price, and \( T \) is the expiration date. The random variable \(... | With \( \varphi \left( x\right) \triangleq {\left( {x}_{1} - q\right) }^{ + } \), the Gaussian integration in (4.6) can be carried out explicitly, to yield\n\n\[ \n{u}^{ECC}\left( {s,{x}_{1};q}\right) = \left\{ \begin{array}{ll} {x}_{1}{e}^{-{\delta }_{1}s}\Phi \left( {{\rho }_{ + }\left( {s,{x}_{1};q}\right) }\right) ... | Yes |
The European put option confers to its holder the right to sell a stock at a future time at a prespecified price. We model a put on the first stock as the ECC with \( C\left( t\right) = 0 \) for \( 0 \leq t < T \) and \( C\left( T\right) = {\left( q - {S}_{1}\left( T\right) \right) }^{ + } \) . Because \( {\left( q - {... | First note from Remark 1.5.11 that\n\n\[ \n{e}^{-\left( {r - {\delta }_{1}}\right) t}{S}_{1}\left( t\right) = {S}_{1}\left( 0\right) + {\int }_{0}^{t}{e}^{-\left( {r - {\delta }_{1}}\right) u}\left\lbrack {d{S}_{1}\left( u\right) - \left( {r - {\delta }_{1}}\right) {S}_{1}\left( u\right) {du}}\right\rbrack \n\]\n\n\[ \... | Yes |
Consider an ECC of the form \( C\left( t\right) = 0 \) for \( 0 \leq t < T \) and \( C\left( T\right) = G\left( \omega \right) \) , where \( G : C\left( \left\lbrack {0, T}\right\rbrack \right) \rightarrow \mathbb{R} \) is a functional satisfying under \( {P}_{0} \) the conditions (E.4)-(E.6) of Appendix E. Then from t... | \[ {V}^{ECC}\left( t\right) = {e}^{-r\left( {T - t}\right) }{E}_{0}\left\lbrack {G\left( {W}_{0}\right) \mid \mathcal{F}\left( t\right) }\right\rbrack \] \[ = {e}^{-r\left( {T - t}\right) }{E}_{0}G\left( {W}_{0}\right) \] \[ + {e}^{-r\left( {T - t}\right) }{\int }_{0}^{t}{E}_{0}\left\lbrack {\partial G\left( {{W}_{0};(... | Yes |
Theorem 6.7 (McKean (1965)): Under the assumption (6.7), the value process for a perpetual American call option is given by\n\n\[ \n{V}^{AC}\left( {t;\infty }\right) = g\left( {S\left( t\right) }\right) ,\;0 \leq t < \infty ,\n\]\n\nwhere the function \( g \) is\n\n\[ \ng\left( x\right) = \left\{ \begin{array}{ll} \lef... | Proof. Itô's rule for convex functions (e.g., Karatzas and Shreve (1991), Theorem 3.6.22 and Problem 3.6.7(i)) implies\n\n\[ \nd\left( {{e}^{-{rt}}g\left( {S\left( t\right) }\right) }\right) = {e}^{-{rt}}S\left( t\right) {g}^{\prime }\left( {S\left( t\right) }\right) {\sigma d}{W}_{0}\left( t\right) - {e}^{-{rt}}\left(... | Yes |
Theorem 7.3 (Maximization of the expected utility from consumption): Let Assumptions 2.1 and 7.1 hold, let \( x \in \left( {{\mathcal{X}}_{1}\left( \infty \right) ,\infty }\right) \) be given, and define\n\n\[ \n{c}_{1}\left( t\right) \triangleq {I}_{1}\left( {t,{\mathcal{Y}}_{1}\left( x\right) {H}_{0}\left( t\right) }... | It is now not difficult to see that the value function \( {V}_{1} \) is given by (cf. (6.21))\n\n\[ \n{V}_{1}\left( x\right) = \left\{ \begin{array}{ll} {G}_{1}\left( {{\mathcal{Y}}_{1}\left( x\right) }\right) , & x > {\mathcal{X}}_{1}\left( \infty \right) , \\ E{\int }_{0}^{T}{U}_{1}\left( {t,\bar{c}\left( t\right) }\... | Yes |
Example 7.5 (Subsistence consumption): Suppose\n\n\[ \n{U}_{1}\left( c\right) = \left\{ \begin{array}{ll} \log \left( {c - \bar{c}}\right) , & \bar{c} < c < \infty , \\ - \infty , & - \infty < c \leq \bar{c}, \end{array}\right. \n\]\n\nwhere \( \bar{c} \) is a positive constant that consumption must exceed at all times... | If \( {S}_{0}\left( \cdot \right) \) is deterministic, we can derive the optimal portfolio explicitly. Under this condition,\n\n\[ \n{X}_{1}\left( t\right) = \frac{T - t}{T{H}_{0}\left( t\right) }\left( {x - \bar{c}{h}_{1}}\right) + \bar{c}{S}_{0}\left( t\right) \left( {{\int }_{t}^{T}\frac{du}{{S}_{0}\left( u\right) }... | Yes |
Example 7.9 (Portfolio insurance): Suppose\n\n\\[ \n{U}_{2}\\left( x\\right) = \\left\\{ \\begin{array}{ll} \\log \\left( {x - \\bar{x}}\\right) , & \\bar{x} < x < \\infty , \\\\ - \\infty , & - \\infty < x \\leq \\bar{x}, \\end{array}\\right.\n\\]\n\nwhere \\( \\bar{x} \\) is a positive constant below which terminal w... | We have \\( {\\mathcal{Y}}_{2}\\left( x\\right) = 1/\\left( {x - \\bar{x}{h}_{2}}\\right) \\) for \\( x > \\bar{x}{h}_{2} \\), and the optimal consumption and wealth processes are \\( {c}_{2}\\left( t\\right) \\equiv 0 \\) and\n\n\\[ \n{X}_{2}\\left( t\\right) = \\frac{1}{{H}_{0}\\left( t\\right) }\\left\\{ {x - \\bar{... | Yes |
Theorem 8.11 (Hamilton-Jacobi-Bellman equation): Under Assumptions 8.1 and 8.2, the value function \( V\\left( {t, x}\\right) \) of (8.29),(8.30) is of class \( {C}^{1,2} \) on the set \( D \) of (8.10), continuous on the set \( \\{ \\left( {t, x}\\right) \\in \) \( \\left\\lbrack {0, T}\\right\\rbrack \\times \\left( ... | Proof. Differentiating (8.9) and (8.29) and using the formula (8.35), we obtain for \( \\left( {t, x}\\right) \\in D \) ,\n\n\[ {\\mathcal{X}}_{t}\\left( {t,\\mathcal{Y}\\left( {t, x}\\right) }\\right) + {\\mathcal{X}}_{y}\\left( {t,\\mathcal{Y}\\left( {t, x}\\right) }\\right) {\\mathcal{Y}}_{t}\\left( {t, x}\\right) =... | Yes |
Consider the case that \( r\left( \cdot \right) = \) \( r > 0,\theta \left( \cdot \right) = \theta \neq 0 \), and \( \sigma \left( \cdot \right) = \sigma \) are constants, and \( A\left( \cdot \right) \equiv 0 \) . Set \( \gamma = \frac{1}{2}\parallel \theta {\parallel }^{2} > 0 \) . Assume that\n\n\[ \n{U}_{1}\left( {... | The functions of Theorem 8.12 can be computed explicitly, following Karatzas, Lehoczky, and Shreve (1987), as follows. Denote by \( {\lambda }_{ + } \) and \( {\lambda }_{ - } \) the respective positive and negative roots of the quadratic equation \( \gamma {\lambda }^{2} - \) \( \left( {r - \alpha - \gamma }\right) \l... | Yes |
Theorem 9.20 (Hamilton-Jacobi-Bellman equation): Let Assumptions 9.9 and 9.13 hold. Then the value function \( {V}_{\infty } \) is twice continuously differentiable on \( \left( {{\mathcal{X}}_{\infty }\left( \infty \right) ,\infty }\right) \) and satisfies the Hamilton-Jacobi-Bellman equation of dynamic programming,\n... | Proof. Equations (9.16), (9.29), and (9.40) imply\n\n\[ {V}_{\infty }^{\prime }\left( x\right) = {\mathcal{Y}}_{\infty }\left( x\right) ,\;{V}_{\infty }^{\prime \prime }\left( x\right) = {\mathcal{Y}}_{\infty }^{\prime }\left( x\right) ,\;x > {\mathcal{X}}_{\infty }\left( \infty \right) . \]\n\nWe may thus rewrite the ... | Yes |
In a continuous-time capital asset pricing model with an underlying \( N \) -dimensional Markov state process, the risk premia of assets can be computed theoretically from their covariances with a set of \( N + 1 \) mutual funds. | Breeden (1979) shows that rather than using the set of all covariances, one can in principle compute risk premia from the covariance of assets with the consumption process of an optimally behaving investor. Like the simple mean-variance capital asset pricing model, this consumption-based capital asset pricing model doe... | No |
Theorem 6.4 (Uniqueness of the equilibrium market): Assume that (6.4) holds. Then the equilibrium money market process \( {S}_{0}\left( \cdot \right) \), the state price density process \( {H}_{0}\left( \cdot \right) \), and the market price of risk process \( \theta \left( \cdot \right) \), are uniquely determined, as... | Proof. The uniqueness of \( {H}_{0}\left( \cdot \right) \) follows from Corollary 5.4, Theorem 6.1, and the initial condition \( {H}_{0}\left( 0\right) = 1 \) . The uniqueness of \( {\widehat{c}}_{1}\left( \cdot \right) ,\ldots ,{\widehat{c}}_{K}\left( \cdot \right) \) also follows from Theorem 6.1. The semimartingale ... | Yes |
Example 7.1 (Logarithmic utility with subsistence consumption): Let \( {U}_{k}\left( c\right) = \log \left( {c - {\bar{c}}_{k}}\right) \), for \( c > {c}_{k}, k = 1,\ldots, K \), where each \( {\bar{c}}_{k} \) is a nonnegative constant. Then | \[ {U}^{\prime }\left( {c;\Lambda }\right) = \mathcal{H}\left( {c;\Lambda }\right) = \frac{1}{c - \bar{c}}\mathop{\sum }\limits_{{k = 1}}^{K}{\lambda }_{k},\;c > \bar{c}. \] We normalize \( \Lambda \) by setting \( \mathop{\sum }\limits_{{k = 1}}^{K}{\lambda }_{k} = \epsilon \left( 0\right) - \bar{c} \), a strictly pos... | Yes |
Example 7.2 (Power utility with subsistence consumption): Let \( {U}_{k}\left( c\right) = \) \( \frac{1}{p}{\left( c - {\bar{c}}_{k}\right) }^{p} \) for \( c > {\bar{c}}_{k}, k = 1,\ldots, K \), where \( p < 1, p \neq 0 \), and each \( {\bar{c}}_{k} \) is a nonnegative constant. Then | \[ {U}^{\prime }\left( {c;\underset{ \sim }{\Lambda }}\right) = \mathcal{H}\left( {c;\underset{ \sim }{\Lambda }}\right) = {\left\lbrack \frac{\mathop{\sum }\limits_{{k = 1}}^{K}{\lambda }_{k}^{\frac{1}{1 - p}}}{c - \bar{c}}\right\rbrack }^{1 - p},\;c > \bar{c}. \] We normalize \( \Lambda \) by setting \( \mathop{\sum ... | Yes |
Example 7.5 (Constant aggregate endowment): If the aggregate endowment \( \epsilon > \bar{c} \) is constant, then the unique vector \( \Lambda \) satisfying the normalization \( \mathcal{H}\left( {\epsilon ;\Lambda }\right) = 1 \) is \[ \underset{ \sim }{\Lambda } = \left( {\frac{1}{{U}_{1}^{\prime }\left( {\widehat{c}... | Constant aggregate endowment implies \( \nu \left( \cdot \right) \equiv 0,\xi \left( \cdot \right) \equiv 0,\rho \left( \cdot \right) \equiv 0 \) in (2.2), and the local time of \( \epsilon \left( \cdot \right) \) at every point is zero. Therefore, the equilibrium market coefficients (6.20)-(6.22) are \[ r\left( t\righ... | Yes |
Example 7.6 \( \left( {K = 2,{U}_{1}\left( c\right) = \log c,{U}_{2}\left( c\right) = \sqrt{c}\text{.}}\right) : \) In this case, we have | \[ {U}^{\prime }\left( {c;\Lambda }\right) = \mathcal{H}\left( {c;\Lambda }\right) = \frac{{\lambda }_{1}}{2c}\left\lbrack {1 + \sqrt{1 + c{\left( \frac{{\lambda }_{2}}{{\lambda }_{1}}\right) }^{2}}}\right\rbrack ,\;c > 0, \] and the optimal consumption rates become \[ {\widehat{c}}_{1}\left( t\right) = \frac{{2\epsilo... | Yes |
Example 7.8 (Ergodic aggregate endowment): Let us suppose that each agent \( k \) has utility function \( {U}_{k} \) with \( {\bar{c}}_{k} = 0 \) and \( {U}_{k}^{\prime }\left( 0\right) = \infty \), so \( \bar{c} = 0 \) . Let us further suppose that the aggregate endowment process \( \epsilon \left( \cdot \right) \) is... | Then the diffusion process \( \epsilon \left( \cdot \right) \) is ergodic with invariant measure \( m\left( {dc}\right) /m\left( \mathcal{I}\right) \) (cf. Proposition 5.5.22 and Exercise 5.5.40 in Karatzas and Shreve (1991)). | Yes |
We consider one stock \( S\left( \cdot \right) = {S}_{1}\left( \cdot \right) \) driven by a single Brownian motion, we assume (7.1)-(7.3), and we denote \( {\sigma }_{11} \) by \( \sigma \) . A European call option corresponds to \( \varphi \left( x\right) = {\left( x - q\right) }^{ + } \), where \( q \geq 0 \) is the ... | \[ \widehat{\varphi }\left( x\right) = \left\{ \begin{array}{ll} {\left( \frac{\beta - 1}{q}\right) }^{\beta - 1}{\left( \frac{x}{\beta }\right) }^{\beta }, & \text{ if }0 < x \leq \frac{\beta q}{\beta - 1}, \\ x - q, & \text{ if }x \geq \frac{\beta q}{\beta - 1}. \end{array}\right. \] (7.25) The function \( \widehat{\... | Yes |
Example 7.4 (European put option): We assume again (7.1)-(7.3) and consider one stock. A European put option corresponds to \( \varphi \left( x\right) = {\left( q - x\right) }^{ + } \) , where \( q \geq 0 \) . We consider again \( K = \left\lbrack {\alpha ,\beta }\right\rbrack \) with \( - \infty \leq \alpha \leq 0 \le... | \[ \widehat{\varphi }\left( x\right) = \left\{ \begin{array}{ll} q - x, & \text{ if }0 < x \leq \frac{\alpha q}{\alpha - 1}, \\ {\left( \frac{\left| \alpha - 1\right| }{q}\right) }^{\alpha - 1}{\left( \frac{x}{\left| \alpha \right| }\right) }^{\alpha }, & \text{ if }x \geq \frac{\alpha q}{\alpha - 1} \end{array}\right.... | Yes |
Example 8.8 (Incomplete market): Consider the case \( K = \{ p \in \) \( \left. {{\mathbb{R}}^{N};{p}_{M + 1} = \cdots = {p}_{N} = 0}\right\} \) of Example 4.1(iii), where there are only \( M \) stocks available for investment, but these are driven by the \( N \) -dimensional Brownian motion \( W\left( \cdot \right) \)... | \[ \zeta \left( \nu \right) + {p}^{\prime }\nu = 0,\;\forall p \in K,\;\nu \in \widetilde{K}. \] | No |
Example 10.2 (Prohibition of short-selling): We consider a market with constant coefficients and one stock, i.e., \( N = 1 \) . When short-selling is prohibited (Example 9.7(ii), \( K = \left\lbrack {0,\infty ),{K}_{ - } = ( - \infty ,0}\right\rbrack \) ), we have \( \widetilde{K} = \lbrack 0,\infty ) \) , \( \zeta \le... | \[ \check{\varphi }\left( x\right) = \mathop{\inf }\limits_{{\nu \geq 0}}\varphi \left( {x{e}^{-\nu }}\right) ,\;\forall x > 0. \] For a European call, we have \[ \check{\varphi }\left( x\right) = \mathop{\inf }\limits_{{\nu \geq 0}}{\left( x{e}^{-\nu } - q\right) }^{ + } = 0,\;\forall x > 0, \] and the lower hedging v... | Yes |
Example 10.3 (Prohibition of borrowing): We consider again a market with constant coefficients and one stock, i.e., \( N = 1 \) . When borrowing from the money market is prohibited (Example 9.7(vi), \( K = ( - \infty ,1\rbrack ,{K}_{ - } = \) \( \lbrack 1,\infty )) \), we have \( \widetilde{K} = ( - \infty ,0\rbrack ,\... | For a European call option, we have\n\n\[ \check{\varphi }\left( x\right) = \mathop{\inf }\limits_{{\nu \leq 0}}{\left( x - q{e}^{\nu }\right) }^{ + } = {\left( x - q\right) }^{ + },\;\forall x > 0; \]\nthe hedge of a long position in a European call option does not borrow from the money market and is thus unaffected b... | Yes |
Proposition 5.1 (Weak Duality): Suppose (5.2) and (5.3) hold. Then, for any given \( y \in \left( {0,\infty }\right) \) and with \( x = {\mathcal{X}}_{{\nu }_{y}}\left( y\right) \), (i) there exists an optimal consumption and portfolio-proportion process pair \( \left( {\widehat{c},\widehat{p}}\right) \in {\mathcal{A}}... | Proof. First, let us note that (5.3), (3.10), and (3.27) imply \[ V\left( {x;K}\right) \leq \widetilde{V}\left( y\right) + {xy},\forall y > 0,\forall x > 0. \] (5.5) Now fix \( y \in \left( {0,\infty }\right) \), let \( x = {\mathcal{X}}_{{\nu }_{y}}\left( y\right) \), and note that the assumption \( {\widetilde{V}}_{{... | Yes |
Example 6.7 (Utility functions of power type): Fix \( \beta \in \left( {-\infty ,1}\right) \smallsetminus \{ 0\} \) and assume\n\n\[ \n{U}_{1}\left( {t, x}\right) = {U}_{2}\left( x\right) = \frac{1}{\beta }{x}^{\beta },\;0 \leq t \leq T, x > 0.\n\] | We know from Example 3.8.13 that\n\n\[ \n{\mathcal{X}}_{\widehat{\nu }}\left( {t, y}\right) = k\left( t\right) {y}^{\frac{1}{\beta - 1}},\;{G}_{\widehat{\nu }}\left( {t, y}\right) = \frac{1}{\beta }k\left( t\right) {y}^{\frac{\beta }{\beta - 1}},\n\]\n\n\[ \n\widetilde{V}\left( {t, y}\right) = \frac{1 - \beta }{\beta }... | Yes |
Example 7.2 (Logarithmic utility, incomplete market): \( {U}_{1}\left( {t, x}\right) = \) \( {U}_{2}\left( x\right) = \log x \) for every \( \left( {t, x}\right) \in \left\lbrack {0, T}\right\rbrack \times \left( {0,\infty }\right) \). | This is Example 4.2, specialized to the case of an incomplete market; i.e., \( K \) given by (7.1). The expression in (4.23) to be minimized over \( \xi \in {\mathbb{R}}^{L} \) is\n\n\[ \frac{1}{2}{\begin{Vmatrix}\widetilde{\theta }\left( t\right) + {\rho }^{\prime }\left( t\right) \left( a\left( t\right) + \xi - r\lef... | Yes |
Example 7.3 (Logarithmic utility, incomplete market, short-selling prohibited): \( \;{U}_{1}\left( {t, x}\right) = {U}_{2}\left( x\right) = \log x \) for every \( \left( {t, x}\right) \in \left\lbrack {0, T}\right\rbrack \times \left( {0,\infty }\right) \) . In contrast to Example 7.2, we now take\n\n\[ K = \left\{ {p ... | As in the previous example, the minimizing \( \xi \) is \( \widehat{\xi }\left( t\right) = r\left( t\right) {1}_{L} - a\left( t\right) \) . The minimizing \( \eta \geq 0 \) can be solved from the Kuhn-Tucker conditions\n\n\[ {\left( \widetilde{\sigma }\left( t\right) {\widetilde{\sigma }}^{\prime }\left( t\right) \righ... | Yes |
Example 8.5 (Logarithmic utilities, general coefficients): In the special case \( {U}_{1}\left( {t, x}\right) = {U}_{2}\left( x\right) = \log x,0 \leq t \leq T, x > 0 \), we have \( {\widetilde{U}}_{1}\left( {t, y}\right) = {\widetilde{U}}_{2}\left( y\right) = \) \( - \left( {1 + \log y}\right), y > 0 \) (see Example 4... | \[ {\widetilde{V}}_{\nu }\left( y\right) = - \left( {T + 1}\right) \left( {1 + \log y}\right) - {\int }_{0}^{T}E\left\lbrack {\log {H}_{\nu }\left( t\right) }\right\rbrack {dt} - E\left\lbrack {\log {H}_{\nu }\left( T\right) }\right\rbrack . \] For \( \nu \left( \cdot \right) \in \mathcal{D} \), we have \[ - E\left\lbr... | Yes |
Lemma 1.2. Let \( \mathcal{A} \subseteq \mathcal{F} \subseteq {2}^{\Omega } \) . Suppose \( \mathcal{A} \) is a \( \pi \) -system and \( \mathcal{F} \) is a \( \lambda \) -system. Then \( \sigma \left( \mathcal{A}\right) \subseteq \mathcal{F} \) . | Proof. Let\n\n\[ \widehat{\mathcal{G}} \triangleq \bigcap \{ \mathcal{G} \supseteq \mathcal{A} \mid \mathcal{G}\text{ is a }\lambda \text{-system }\} \subseteq \mathcal{F}. \]\n\nThen \( \widehat{\mathcal{G}} \) is the smallest \( \lambda \) -system containing \( \mathcal{A} \) . Set\n\n\[ \widehat{\mathcal{F}} \triang... | Yes |
Example 1.3. Let \( \Omega = \left\lbrack {0,1}\right\rbrack ,\mathcal{F} \) the set of all Lebesgue measurable sets in \( \left\lbrack {0,1}\right\rbrack \), and \( \mathbf{P} \) the Lebesgue measure on \( \left\lbrack {0,1}\right\rbrack \) . Then one can show that \( \left( {\Omega ,\mathcal{F},\mathbf{P}}\right) \) ... | \[ \mathbf{P}\left( A\right) = {\int }_{\{ \omega \in \Omega \mid x\left( \omega \right) \geq 0\} }d\mathbf{P}\left( \omega \right) . \] | Yes |
Proposition 1.5. Let \( \Omega \) be a Polish space and \( \mathcal{F} = \mathcal{B}\left( \Omega \right) \) . Then:\n\n(i) \( \left( {\Omega ,\mathcal{F}}\right) \) is standard, and it is countably determined.\n\n(ii) For any \( {\Omega }^{\prime } \in \mathcal{F} \), let \( {\mathcal{F}}^{\prime } = {\Omega }^{\prime... | See Parthasarathy [1, pp. 133-134] and Ikeda-Watanabe [1, p. 13] for a proof and descriptions. | No |
Lemma 1.6. Let \( \mathcal{A} \) be a \( \pi \) -system on \( \Omega \) . Let \( \mathcal{H} \) be a linear space of functions from \( \Omega \) to \( \mathbb{R} \) such that\n\n\[ \left\{ \begin{array}{l} 1 \in \mathcal{H};\;{I}_{A} \in \mathcal{H},\;\forall A \in \mathcal{A}; \\ {\varphi }_{i} \in \mathcal{H},0 \leq ... | Proof. Set\n\n\[ \mathcal{F} = \left\{ {A \subseteq \Omega \mid {I}_{A} \in \mathcal{H}}\right\} \]\n\nThen \( \mathcal{F} \) is a \( \lambda \) -system containing \( \mathcal{A} \) . Thus, by Lemma 1.2, \( \sigma \left( \mathcal{A}\right) \subseteq \mathcal{F} \) .\n\nNow, for any \( \sigma \left( \mathcal{A}\right) \... | Yes |
Proposition 1.8. Let \( \mathcal{G} \) be a sub- \( \sigma \) -field of \( \mathcal{F} \) . Then\n\n(i) Map \( E\left( {\cdot \mid \mathcal{G}}\right) : {L}_{\mathcal{F}}^{1}\left( {\Omega ;{\mathbb{R}}^{m}}\right) \rightarrow {L}_{\mathcal{G}}^{1}\left( {\Omega ;{\mathbb{R}}^{m}}\right) \) is linear and bounded.\n\n(i... | Proofs of the above results are straightforward by the definitions. | No |
Proposition 1.10. Let \( \left( {\Omega ,\mathcal{F},\mathbf{P}}\right) \) be a standard probability space and \( \left( {S,\mathcal{B}}\right) \) a measurable space. Let \( \xi : \left( {\Omega ,\mathcal{F}}\right) \rightarrow \left( {S,\mathcal{B}}\right) \) be a random variable and \( {\mathbf{P}}_{\xi } \equiv \mat... | For proofs of Propositions 1.9 and 1.10, see Parthasarathy [1, pp. 145- \( {150}\rbrack \) . | No |
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