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Let \( f\left( {x, v}\right) = - x + v, x \in {\mathbb{R}}^{1}, v \in \left\lbrack {-1,1}\right\rbrack \) . Then\n\n\[ f\left( {x, v}\right) \cdot x \leq - {x}^{2} + \left| v\right| \left| x\right| \leq \frac{1}{4} \] | and hence (7.16) holds. | No |
Consider the problem of minimizing\n\n\[ \frac{1}{2}{\int }_{0}^{\infty }{e}^{-s}u{\left( s\right) }^{2}{ds} \]\n\nsubject to \( \dot{x}\left( s\right) = u\left( s\right) \) and \( u\left( s\right) \in {\mathbb{R}}^{1} \) . In this case \( O = U = {\mathbb{R}}^{1} \) . | The problem is in fact an infinite horizon linear quadratic regulator problem with \( n = m = 1 \) . Since the running cost does not depend on the state, the optimal control is \( {u}^{ * }\left( s\right) \equiv 0 \) . Hence \( V\left( x\right) \equiv 0 \), which is a solution to the stationary dynamic programming equa... | Yes |
This example is related to the production planning problem described in Example 2.1. In this simple model there are two commodities; \( n = 2 \) . We assume that the running cost is given by\n\n\[ h\left( x\right) = \alpha \left| {x}_{1}\right| + \left| {x}_{2}\right| ,\;x = \left( {{x}_{1},{x}_{2}}\right) \in {\mathbb... | Clearly if there is no shortage of either commodity then the optimal production rate is zero. Also if only one of the products is in shortage then the optimal strategy is to produce that product with full capacity. Hence the optimal policy \( {\underline{u}}^{ * } \) satisfies\n\n\[ {\underline{u}}^{ * }\left( {{x}_{1}... | Yes |
Suppose that \( L = L\left( v\right) \) . Then it suffices to consider linear functions \( x\left( \cdot \right) \) in(8.2). | Indeed, given any Lipschitz continuous \( x\left( \cdot \right) \) let\n\n\[ v = \frac{1}{{t}_{1} - t}{\int }_{t}^{{t}_{1}}\dot{x}\left( s\right) {ds} \]\n\n\[ \widetilde{x}\left( s\right) = x + v\left( {s - t}\right) \]\n\nThen \( \widetilde{x}\left( t\right) = x\left( t\right) = x,\widetilde{x}\left( {t}_{1}\right) =... | Yes |
Proposition 8.1. Assume that:\n\n(a) For all \( \\left( {s, y}\\right) \\in \\bar{Q}, v \\in U,{g}_{t}\\left( {s, y}\\right) + {D}_{x}g\\left( {s, y}\\right) \\cdot v + L\\left( {s, y, v}\\right) \\geq 0 \) ;\n\n(b) For \( y \\in \\bar{O}, g\\left( {{t}_{1}, y}\\right) \\leq \\psi \\left( y\\right) \) .\n\nThen\n\n\[ V... | Proof. We take \( \\widetilde{L},\\widetilde{\\psi } \) as in (8.13), and \( \\widetilde{g} = 0 \) . Then \( V\\left( {t, x}\\right) = \\widetilde{V}\\left( {t, x}\\right) + \) \( g\\left( {t, x}\\right) \) . Since \( \\widetilde{L},\\widetilde{\\psi },\\widetilde{g} \) satisfy (8.12), \( \\widetilde{V}\\left( {t, x}\\... | Yes |
Again let \( L = L\left( v\right) \), and let \( \bar{O} \) be convex. The same argument as above shows that it suffices to minimize \( J \) in (8.11) among linear functions \( x\left( s\right) = x + v\left( {t - s}\right) \) . For \( \left( {t, x}\right) \in \bar{Q} \), the value function is \[ V\left( {t, x}\right) =... | Let us again suppose that (8.12) holds and indicate how the unknown scalar \( \lambda \) in the transversality condition can be determined. Since \( g = 0 \), the transversality condition (6.11) takes the form \[ P\left( {\tau }^{ * }\right) = {\lambda \eta }\left( {{x}^{ * }\left( {\tau }^{ * }\right) }\right), H\left... | Yes |
Example 8.2. Suppose that \( H\left( {s,\xi ,0}\right) < 0 \) for every \( \left( {s,\xi }\right) \in \left\lbrack {{t}_{0},{t}_{1}}\right) \times \partial O \) . Now \( h\left( \lambda \right) = H\left( {s,\xi ,{\lambda \eta }\left( \xi \right) }\right) \) is convex, with \( h\left( 0\right) < 0 \) and \( h\left( \lam... | By taking \( \left( {s,\xi }\right) = \left( {{\tau }^{ * },{x}^{ * }\left( {\tau }^{ * }\right) }\right) \) we get \( P\left( {\tau }^{ * }\right) \) in (8.15). | No |
Example 8.3. Let \( L\left( {x, v}\right) = \frac{1}{2}{\left| b\left( x\right) - v\right| }^{2} \), where \( b\left( \xi \right) \cdot \eta \left( \xi \right) < 0 \) for every \( \xi \in \partial O \) . This arises in a large deviations, exit problem discussed later in Chapter VII. In this example, \( H\left( {x, p}\r... | \[ \frac{1}{2}{\left| \lambda \eta \left( \xi \right) \right| }^{2} = b\left( \xi \right) \cdot \left( {{\lambda \eta }\left( \xi \right) }\right) \] Since \( \left| {\eta \left( \xi \right) }\right| = 1 \), this has two solutions: \( \lambda = 0 \) (which turns out to be irrelevant in the large deviations exit problem... | Yes |
Theorem 9.1. For each \( \left( {t, x}\right) \in {Q}_{0} \), there exists \( {u}_{R}^{ * }\left( \cdot \right) \in {\mathcal{U}}_{R}\left( t\right) \) such that \( J\left( {t, x;{u}_{R}^{ * }}\right) \leq J\left( {t, x;u}\right) \) for all \( u\left( \cdot \right) \in {\mathcal{U}}_{R}\left( t\right) \) . | Theorem 9.1 is a special case of Theorem 11.1, which is proved below. | No |
Lemma 9.2. \( \left| {{V}_{R}\left( {t, x}\right) }\right| \leq {c}_{1}\left( {{t}_{1} - t}\right) \). Moreover, there exists \( {R}_{1} \) such that \( \left| {{u}_{R}^{ * }\left( s\right) }\right| \leq {R}_{1} \) | Proof. By (9.2c), \( L\left( {s, y, v}\right) \geq - {c}_{1} \). Hence, \( {V}_{R}\left( {t, x}\right) \geq - {c}_{1}\left( {{t}_{1} - t}\right) \). On the other hand, by choosing \( u\left( s\right) \equiv 0, x\left( s\right) \equiv x \)\n\n\[ \n{V}_{R}\left( {t, x}\right) \leq {\int }_{t}^{{t}_{1}}L\left( {s, x,0}\ri... | Yes |
Lemma 10.1. Let \( \left( {t, x}\right) \) be a regular point and \( {x}^{ * }\left( \cdot \right) \) the unique minimizer of \( J \) for left endpoint \( \left( {t, x}\right) \) . If \( \left( {{\tau }_{n},{y}_{n}}\right) \rightarrow \left( {t, x}\right) \) as \( n \rightarrow \infty \) and \( {x}_{n}^{ * }\left( \cdo... | Proof. The Euler equation\n\n(10.1)\n\n\[ \n{L}_{x} = \frac{d}{ds}{L}_{v} = {L}_{vt} + {L}_{vx}{\dot{x}}_{n}^{ * } + {L}_{vv}{\ddot{x}}_{n}^{ * } \]\n\nholds, where \( {L}_{x},{L}_{v},\cdots \) are evaluated at \( \left( {s,{x}_{n}^{ * }\left( s\right) ,{\dot{x}}_{n}^{ * }\left( s\right) }\right) \) . Since \( {L}_{vv}... | Yes |
Corollary 10.1. If \( {x}^{ * }\left( \cdot \right) \) minimizes \( J \) subject to \( x\left( t\right) = x \), then \( \left( {s,{x}^{ * }\left( s\right) }\right) \) is a regular point for \( t < s < {t}_{1} \) . | Proof. When restricted to \( \left\lbrack {s,{t}_{1}}\right\rbrack ,{x}^{ * }\left( \cdot \right) \) minimizes \( J \) with left endpoint \( \left( {s,{x}^{ * }\left( s\right) }\right) \) . If \( {x}^{* * }\left( \cdot \right) \) is another minimizer of \( J \) with \( {x}^{* * }\left( s\right) = {x}^{ * }\left( s\righ... | Yes |
Theorem 10.3. Let \( \left( {t, x}\right) \) be a regular point and \( {\gamma }^{ * } = \left\{ {\left( {s,{x}^{ * }\left( s\right) }\right) : t \leq }\right. \) \( \left. {s \leq {t}_{1}}\right\} \) the unique minimizing trajectory with left endpoint \( \left( {t, x}\right) \) . If \( \left( {t, x}\right) \) is not a... | Sketch of proof. Except for one point, Theorem 10.3 is a standard result in the classical theory of first order partial differential equations. See \( \lbrack \mathrm{{CH}} \) , Chap 2], [He, Chap 3]. Hence, we shall only sketch a proof. The method of characteristics is used to construct a local solution \( W \) of the... | No |
In dimension \( n = 1 \), consider the problem of minimizing\n\n\[ J = \frac{1}{2}{\int }_{t}^{{t}_{1}}{\left| \dot{x}\left( s\right) \right| }^{2}{ds} + \psi \left( {x\left( {t}_{1}\right) }\right) . \] | From Example 8.1 it is seen that any minimizing \( {x}^{ * }\left( \cdot \right) \) is a straight line segment. Thus\n\n(10.6),\n\n\[ V\left( {t, x}\right) = \mathop{\min }\limits_{{v \in {\mathbb{R}}^{1}}}\left\lbrack {\frac{1}{2}\left( {{t}_{1} - t}\right) {v}^{2} + \psi \left( {x + \left( {{t}_{1} - t}\right) v}\rig... | Yes |
Theorem 10.4. \( E \) is a closed subset of \( {Q}_{0} \), and \( V \in {C}^{1}\left( {{\bar{Q}}_{0} \smallsetminus E}\right) \). | Proof. An easy modification of the proof of Theorem 10.3 shows that any point \( \left( {{t}_{1},{x}_{1}}\right) \) of the terminal hyperplane \( \left\{ {t}_{1}\right\} \times {\mathbb{R}}^{n} \) has a relative neighborhood \( {N}_{1} \) which contains no point of \( E \) . For \( \left( {t, x}\right) \in {N}_{1}, V\l... | Yes |
Example 10.2. Let \( H\left( {t, x, p}\right) = g\left( {t, x}\right) {\left( 1 + {\left| p\right| }^{2}\right) }^{k} \) where \( k > \frac{1}{2}, g \) and \( {D}_{x}g \) are bounded, and \( g\left( {t, x}\right) \geq c > 0 \) . It is elementary to verify (10.13). | Remark 10.4. Assumptions (10.13) imply the following, which will be used later in proving that the value function \( V \) is a viscosity solution to (10.8)-(10.9). See Theorem II.10.3. Let\n\n(10.14)\n\n\[ \n{H}_{R}\left( {t, x, p}\right) = \mathop{\max }\limits_{{\left| v\right| \leq R}}\left\lbrack {-v \cdot p - L\le... | No |
Lemma 11.1. Let \( \Lambda \) and \( {\Lambda }_{v} \) be continuous on \( \left\lbrack {t,{t}_{1}}\right\rbrack \times U \), with \( \Lambda \left( {s, \cdot }\right) \) convex for each \( s \in \left\lbrack {t,{t}_{1}}\right\rbrack \) . If \( {u}_{n}\left( \cdot \right) \rightarrow u\left( \cdot \right) \) weakly, th... | Proof. Since \( \Lambda \left( {s, \cdot }\right) \) is convex and of class \( {C}^{1} \) \[ \Lambda \left( {s, v}\right) \geq \Lambda \left( {s, u\left( s\right) }\right) + \left( {v - u\left( s\right) }\right) \cdot {\Lambda }_{v}\left( {s, u\left( s\right) }\right) \] for all \( v \in U \) . In particular, this is t... | Yes |
Lemma 11.2. Let \( {x}_{n}\left( s\right) \) be the solution to (3.2) with \( u\left( s\right) = {u}_{n}\left( s\right) \) and \( {x}_{n}\left( t\right) = x \) . Suppose that \( {u}_{n}\left( s\right) \in U \), that \( {u}_{n}\left( \cdot \right) \) tends to \( {u}^{ * }\left( \cdot \right) \) weakly and that \( {x}_{n... | Proof. By (11.1)(b) we have\n\n\[ {x}_{n}\left( s\right) = x + {\int }_{t}^{s}\left\lbrack {{f}_{1}\left( {r,{x}_{n}\left( r\right) }\right) + {f}_{2}\left( {r,{x}_{n}\left( r\right) }\right) {u}^{ * }\left( r\right) }\right\rbrack {dr} \]\n\n\[ + {\int }_{t}^{s}\left\lbrack {{f}_{2}\left( {r,{x}_{n}\left( r\right) }\r... | Yes |
Theorem 11.1. Under the assumptions (11.1) an optimal control \( {u}^{ * }\left( \cdot \right) \) exists. | Proof. Let \( {u}_{n}\left( \cdot \right) \) be a sequence in \( {\mathcal{U}}^{0}\left( t\right) \) such that \( J\left( {t, x;{u}_{n}}\right) \) tends to \( V\left( {t, x}\right) \), where \( V\left( {t, x}\right) \) is the infimum of \( J\left( {t, x;u}\right) \) among all \( u\left( \cdot \right) \in {\mathcal{U}}^... | Yes |
Lemma 11.3. Let\n\n(11.9)\n\n\[ \bar{P}\left( s\right) = {\int }_{t}^{s}{L}_{x}\left( {r,{x}^{ * }\left( r\right) ,{u}^{ * }\left( r\right) }\right) {dr}, t \leq s \leq {\tau }^{ * }.\]\n\nThen for almost all \( s \in \left\lbrack {t,{\tau }^{ * }}\right\rbrack \)\n\n(11.10)\n\n\[ \bar{P}\left( s\right) = {L}_{v}\left(... | Proof. (Sketch) We follow the classical derivation of Euler's equation. Let \( {t}_{2} < {\tau }^{ * } \) and consider any \( \xi \left( \cdot \right) \in {C}^{1}\left( {\left\lbrack {t,{t}_{2}}\right\rbrack ;{\mathbb{R}}^{n}}\right) \) such that \( \xi \left( t\right) = \xi \left( {t}_{2}\right) = 0 \) . For \( \left|... | No |
Corollary 11.1. \( {x}^{ * }\left( \cdot \right) \in {C}^{2}\left\lbrack {t,{\tau }^{ * }}\right\rbrack \) . | Proof. This is obtained in the same way as Theorem 9.2. By (9.9) we have\n\n(11.12)\n\n\[ \n{\dot{x}}^{ * }\left( s\right) = - {H}_{p}\left( {s,{x}^{ * }\left( s\right), P\left( s\right) }\right) \n\]\n\nwhere \( P\left( s\right) = C - \bar{P}\left( s\right) \) . | No |
Consider the calculus of variations problem with \( {t}_{0} = 0,{t}_{1} = \) \( 1, O = \left( {-1,1}\right) ,\Psi \equiv 0 \) and \( L\left( {t, x, v}\right) = 1 + \frac{1}{4}{v}^{2} \), i.e. the problem is to minimize\n\n\[ \n{\int }_{t}^{\tau }\left\lbrack {1 + \frac{1}{4}{\left( \dot{x}\left( s\right) \right) }^{2}}... | Using Example I.8.1, we conclude that any optimal control \( {u}^{ * } \) is constant in time. Indeed for initial condition \( \left( {t, x}\right) \in \left\lbrack {0,1}\right\rbrack \times \left\lbrack {-1,1}\right\rbrack \), an elementary calculation\n\nshows that\n\[ \n{u}^{ * }\left( s\right) = \left\{ \begin{matr... | Yes |
In the previous example \( V \) is a generalized solution of (2.1) satisfying (2.2) and (2.3). We now construct a sequence of Lipschitz continuous generalized solutions of (2.1) which also satisfy (2.2) and (2.3). | For a positive integer \( k \) define \( {h}_{k}\left( x\right) \) by\n\n\[ \n{h}_{k}\left( x\right) = \frac{1}{{2k} + 1} - \left| {x - \frac{2i}{{2k} + 1}}\right| \n\]\n\nif \( x \in \left\lbrack {\frac{{2i} - 1}{{2k} + 1},\frac{{2i} + 1}{{2k} + 1}}\right\rbrack \) for some \( i = 0, \pm 1,\ldots , \pm k \) .\n\nThen ... | Yes |
Consider a simple control problem with \( Q = \lbrack 0,1) \times \left( {-1,1}\right) \) , \( \dot{x}\left( s\right) = u\left( s\right), L \equiv 0,\Psi \left( {t, x}\right) = x \) and a control set \( U = \left\lbrack {-a, a}\right\rbrack \) with some constant \( a > 0 \) . Since the boundary data is increasing and t... | (2.4)\n\n\[ V\left( {t, x}\right) = \left\{ \begin{array}{ll} - 1, & \text{ if }x + {at} \leq a - 1 \\ x + {at} - a, & \text{ if }x + {at} \geq a - 1 \end{array}\right. \]\n\nfor \( \left( {t, x}\right) \in \bar{Q}.V \) is differentiable except at \( x + {at} = a - 1 \) and it is a generalized solution of\n\n(2.5)\n\n\... | Yes |
Let \( f, L, g \) be as in Section I.3, and \( O \) be an open subset of \( {\mathbb{R}}^{n} \). Set \( \sum = \bar{O} \) and \( \mathcal{C} = \mathcal{M}\left( \sum \right) \). We assume that \( L \) and \( g \) are also bounded from below. Fix \( {t}_{0} \leq t \leq r \leq {t}_{1}, x \in \sum, u\left( \cdot \right) \... | Since \( L, g \) and \( \psi \) are all bounded from below, \( {\mathcal{T}}_{tr}\psi \) is also bounded from below. Note that \( \mathcal{U}\left( {t, x}\right) \) is nonempty and therefore for every \( \psi \in \mathcal{C},{\mathcal{T}}_{tr}\psi \) is well-defined and belongs to \( \mathcal{C} \). Clearly \( {\mathca... | Yes |
Example 3.2. (Diffusion semigroup). Set \( \sum = {\mathbb{R}}^{n} \) and \( \mathcal{C} = {C}_{p}\left( {\mathbb{R}}^{n}\right) \) . For \( \psi \in \mathcal{C}, x \in \sum \), and \( {t}_{0} \leq t < r \leq {t}_{1} \), let\n\n(3.5)\n\n\[ \left( {{\mathcal{T}}_{tr}\psi }\right) \left( x\right) = {\int }_{{\mathbb{R}}^... | The positivity of \( K \) implies \( \left( {3.2}^{\prime }\right) \) . Also the semigroup property of \( {\mathcal{T}}_{tr} \) is well known. Indeed (3.3) is proved by using the properties of the kernel \( K \) and several changes of variables in (3.5). | No |
Example 3.3. (Poisson process). Let \( \sum \) be the set of nonnegative integers, \( \mathcal{C} \) be the set of all sequences \( \{ \psi \left( i\right) : i \in \sum \} \) such that \( \left| {\psi \left( i\right) }\right| \) grows at most polynomially as \( i \) tends to infinity, and \( \lambda \left( \cdot \right... | Then it is straightforward to verify that \( {\mathcal{T}}_{tr} \) satisfies (3.1) and \( \left( {3.2}^{\prime }\right) \) . Also, for \( {t}_{0} \leq t \leq r \leq s \leq {t}_{1} \) and \( i \in \sum \) ,\n\n\[ \left( {{\mathcal{T}}_{tr}\left( {{\mathcal{T}}_{rs}\psi }\right) }\right) \left( i\right) = \mathop{\sum }\... | Yes |
Theorem 5.1. Assume (3.1), (3.2), (3.10), (3.11). Suppose that \( V \in C\left( \widehat{Q}\right) \) . Then, \( V \) is a viscosity solution of (3.12) in \( Q \) . | Proof. Let \( w \in \mathcal{D} \) and \( \left( {\bar{t},\bar{x}}\right) \in Q \) be a maximizer of the difference \( V - w \) on \( \bar{Q} \) satisfying \( V\left( {\bar{t},\bar{x}}\right) = w\left( {\bar{t},\bar{x}}\right) \) . Then, \( w \geq V \) . Using (3.2)(b) with \( \phi = w\left( {r, \cdot }\right) \) and \... | Yes |
Lemma 5.1. Suppose that \( W \in \mathcal{D} \) . Then \( W \) is a viscosity solution of (3.12) in \( Q \) if and only if it is a classical solution of (3.12). | Proof. First suppose that \( W \) is a viscosity solution. Since \( W \in \mathcal{D}, w \equiv W \) is a test function. Moreover every \( \left( {t, x}\right) \) is a maximizer and a minimizer of \( W - w \) . Hence (4.1) and (4.2) hold at every \( \left( {t, x}\right) \in Q \) with \( w = W \) . Therefore \( W \) sat... | Yes |
Lemma 6.1. Assume (6.1). Let \( {\mathcal{G}}_{t} \) be a partial differential operator as in \( \left( {4.3}\right) \left( \mathrm{i}\right) \) . Then in Definitions 4.1 and 4.2, it suffices to consider only the strict extrema of \( W - w \) . | Proof. Suppose that (4.2) holds at every strict minimum \( \left( {\bar{t},\bar{x}}\right) \in Q \) of \( W - w \) with \( W\left( {\bar{t},\bar{x}}\right) = w\left( {\bar{t},\bar{x}}\right) \) . Let \( \left( {\bar{t},\bar{x}}\right) \in Q \) be a minimum (not necessarily strict) of \( W - w \) satisfying \( W\left( {... | Yes |
Lemma 6.2. (Stability) Let \( {W}^{\varepsilon } \) be a viscosity subsolution (or a super-solution) of \n\n\[ \n- \frac{\partial }{\partial t}{W}^{\varepsilon }\left( {t, x}\right) + {F}^{\varepsilon }\left( {t, x,{D}_{x}{W}^{\varepsilon }\left( {t, x}\right) ,{D}_{x}^{2}{W}^{\varepsilon }\left( {t, x}\right) ,{W}^{\v... | Proof. Let \( w \in {C}^{\infty }\left( Q\right) \) and \( \left( {\bar{t},\bar{x}}\right) \in Q \) be a strict maximizer of \( W - w \) with \( W\left( {\bar{t},\bar{x}}\right) = w\left( {\bar{t},\bar{x}}\right) \) . Since \( {W}^{\varepsilon } \) converges to \( W \) uniformly on compact subsets of \( \bar{Q} \), the... | Yes |
Lemma 6.3. (Stability) Suppose that for every \( w,\xi \in \mathcal{D},\left( {t, x}\right) \in \bar{Q} \) and a positive, continuous function \( h \), with \( h\left( 0\right) = 0 \) ,\n\n(6.4)\n\n\[ \mathop{\lim }\limits_{\substack{{\varepsilon \downarrow 0} \\ {\left( {s, y}\right) \rightarrow \left( {t, x}\right) }... | Proof. Let \( w \in \mathcal{D} \) and \( \left( {\bar{t},\bar{x}}\right) \in Q \) be a maximizer of \( W - w \) on \( \bar{Q} \), with \( W\left( {\bar{t},\bar{x}}\right) = w\left( {\bar{t},\bar{x}}\right) \) . Set\n\n\[ \xi \left( {t, x}\right) = 1 - \exp \left( {-{\left| t - \bar{t}\right| }^{2} - {\left| x - \bar{x... | Yes |
Theorem 7.2. Assume that an optimal control \( {u}^{ * }\left( \cdot \right) \in \mathcal{U}\left( {t, x}\right) \) exists, for each \( \left( {t, x}\right) \in Q \) . Then the value function is a viscosity solution of the dynamic programming equation \( \mathrm{I}\left( {5.3}^{\prime }\right) \) in \( Q \) provided th... | Proof. An argument like the proof of (7.5) above shows that \( V \) is a viscosity subsolution of \( \mathrm{I}\left( {5.3}^{\prime }\right) \) . To show that \( V \) is a viscosity supersolution, suppose that \( w \in {C}^{\infty }\left( Q\right) \) and that \( V - w \) has a local minimum at \( \left( {\bar{t},\bar{x... | Yes |
Lemma 8.1. Let \( W \in C\left( \bar{Q}\right) \) and \( \left( {t, x}\right) \in Q \) . Then,\n\n(8.6)\n\n\[{D}^{ + }W\left( {t, x}\right) = \left\{ {\left( {q, p}\right) \in {\mathbb{R}}^{n + 1} : }\right.\]\n\n\[\left. {\mathop{\limsup }\limits_{\substack{{\left( {s, y}\right) \rightarrow \left( {t, x}\right) } \\ {... | Proof. Fix \( \left( {t, x}\right) \in Q \) . Let \( \left( {q, p}\right) \in {D}^{ + }W\left( {t, x}\right) \) and \( w \in {C}^{1}\left( Q\right) \) be a test function satisfying (8.2). Then by (8.2)(ii),\n\n\[W\left( {s, y}\right) - w\left( {s, y}\right) \leq W\left( {t, x}\right) - w\left( {t, x}\right)\]\n\nfor ev... | Yes |
Corollary 8.1. Let \( W \in C\left( \bar{Q}\right) \) . (a) \( \;{D}^{ + }W\left( {t, x}\right) \) and \( {D}^{ - }W\left( {t, x}\right) \) are convex for every \( \left( {t, x}\right) \in \bar{Q} \) . | Proof. The convexity of the sets of sub and superdifferentials follow easily from Lemma 8.1. | No |
Corollary 8.2. Let \( W \) be a viscosity subsolution (or a supersolution) of (8.1) in Q. Suppose \( W \) satisfies (8.9). Then \( W \) is a viscosity subsolution (or a supersolution, respectively) of (8.11) \[ - \frac{\partial }{\partial t}W\left( {t, x}\right) + \bar{H}\left( {t, x,{D}_{x}W\left( {t, x}\right) }\righ... | Proof. Let \( \left( {q, p}\right) \in {D}^{ + }W\left( {t, x}\right) \) for some \( \left( {t, x}\right) \in Q \) . Then (8.4) yields \[ - q + H\left( {t, x, p}\right) \leq 0. \] Also,(8.10) is satisfied. In particular, \( \left| p\right| \leq {M}_{1} \) and by (8.12), \[ - q + \bar{H}\left( {t, x, p}\right) = - q + H... | Yes |
Lemma 9.1. Let \( \widetilde{W} \) be a viscosity subsolution (or a supersolution) of (9.25) in \( Q \) . For \( \left( {t, x}\right) \in \bar{Q} \) and a constant \( \beta \), define\n\n\[ \bar{W}\left( {t, x}\right) = {e}^{\beta t}\widetilde{W}\left( {t, x}\right) \]\n\nThen \( \bar{W} \) is a viscosity subsolution (... | Proof. Let \( \bar{w} \in {C}^{1}\left( Q\right) \) and \( \left( {\bar{t},\bar{x}}\right) \) be a maximizer of \( \bar{W} - \bar{w} \) on \( \bar{Q} \), with \( \bar{W}\left( {\bar{t},\bar{x}}\right) = \bar{w}\left( {\bar{t},\bar{x}}\right) \) . Set\n\n\[ w\left( {t, x}\right) = {e}^{-{\beta t}}\bar{w}\left( {t, x}\ri... | Yes |
Theorem 10.2. ( \( Q \) bounded) Assume that \( U \) is bounded and that \( \mathrm{I}\left( {3.1}\right) \) , \( \mathrm{I}\left( {3.10}\right) ,\mathrm{I}\left( {3.11}\right) ,\left( {10.4}\right) \) hold. Then the value function \( V \) is the unique Lipschitz continuous viscosity solution to the dynamic programming... | Proof. Lemmas 10.1 and 10.2 imply that \( V \) is Lipschitz continuous. The theorem then follows from Theorem 7.1 and Corollary 9.1. | No |
Theorem 10.3. \( \left( {Q = {Q}_{0}}\right) \) . Assume that \( H \) satisfies \( \mathrm{I}\left( {10.13}\right) \) . Then \( V \) is the unique bounded, Lipschitz continuous viscosity solution to the Hamilton-Jacobi equation I(10.8) with boundary data I(10.9). | Proof. The function \( L \) satisfies I(9.2). By Theorem I.9.3, \( V \) is bounded and Lipschitz continuous, with Lipschitz constant \( M \) in I(9.10). Moreover, by \( \mathrm{I}\left( {9.7}\right) \) there exists \( {R}_{1} \) such that \( V = {V}_{R} \) for \( R \geq {R}_{1} \), where \( {V}_{R} \) is the value func... | Yes |
Theorem 10.4. ( \( Q \) bounded). Assume that I(11.6), I(11.7) hold. Then the value function \( V \) is Lipschitz continuous on \( \bar{Q} \) . Moreover, \( V \) is the unique Lipschitz continuous viscosity solution to the Hamilton-Jacobi equation I(8.9) with boundary data \( V\left( {t, x}\right) = \Psi \left( {t, x}\... | Proof. By Theorems I.11.2 and I.11.3, there exists \( {R}_{1} \) such that \( V\left( {t, x}\right) = \) \( {V}_{R}\left( {t, x}\right) \) for all \( \left( {t, x}\right) \in \bar{Q} \) and \( R \geq {R}_{1} \), where \( {V}_{R} \) is the value function with control constraint \( \left| {u\left( s\right) }\right| \leq ... | Yes |
Theorem 12.1. \( \mathcal{U}\left( {t, x}\right) \) satisfies \( \mathrm{I}\left( {5.2}\right) \) at every \( \left( {t, x}\right) \in \bar{Q} \) . In particular, if \( f, L \) satisfy the hypotheses of Theorem 7.1 or Theorem 7.2 and the value function \( V \) is continuous on \( \bar{Q} \), then \( V \) is a viscosity... | Proof. Fix \( \left( {t, x}\right) \in Q \) and \( v \in U \) . Let \( u\left( s\right) = v \) for \( s \in \left\lbrack {t,{t}_{1}}\right\rbrack \) . Let \( x\left( s\right) \) be the solution of \( \mathrm{I}\left( {3.2}\right) \) and \( \mathrm{I}\left( {3.3}\right) \) and \( \tau \) be the exit time of \( \left( {s... | Yes |
Theorem 12.2. Suppose that \( f, L \) satisfy the hypotheses of Theorem 7.1 or Theorem 7.2. Assume that \( V \) is continuous on \( \bar{Q} \) . Then the value function \( V \) of the optimal control problem with a state constraint is a constrained viscosity solution of \( \mathrm{I}\left( {5.3}^{\prime }\right) \) . | Proof. In view of Theorem 12.1, it suffices to prove (12.5) for every smooth function \( w \in {C}^{\infty }\left( \bar{Q}\right) \) and a minimizer \( \left( {\bar{t},\bar{x}}\right) \in \left\lbrack {{t}_{0},{t}_{1}}\right) \times \partial O \) of the difference \( V - w \) with \( V\left( {\bar{t},\bar{x}}\right) = ... | Yes |
Suppose that \( W \in {C}^{2}\left( {\left\lbrack {0,1}\right\rbrack \times \left\lbrack {-1,1}\right\rbrack }\right) \) is a constrained viscosity solution of (12.6). Then (12.6) is satisfied at every interior point \( \left( {t, x}\right) \in \left( {0,1}\right) \times \left( {-1,1}\right) \) , and by continuity at e... | \[
\frac{\partial }{\partial t}W\left( {\bar{t},1}\right) \geq \frac{\partial }{\partial t}w\left( {\bar{t},1}\right)
\]
and
\[
\frac{\partial }{\partial x}W\left( {\bar{t},1}\right) \leq \frac{\partial }{\partial x}w\left( {\bar{t},1}\right)
\]
In fact for every \( p \geq \frac{\partial }{\partial x}W\left( {\bar{t}... | Yes |
Consider the equation (2.5) with boundary conditions \( \left( {13.7}\right) \left( i\right) \) \( V\left( {t,1}\right) = 1,\;t \in \left\lbrack {0,1}\right\rbrack \) \( \left( {13.7}\right) \left( {ii}\right) \) \( V\left( {t, - 1}\right) = - 1,\;t \in \left\lbrack {0,1}\right\rbrack \) and the terminal condition (2.6... | It is straightforward to check that \( V \) given by (2.4), is a viscosity solution of (2.5) in \( Q = \lbrack 0,1) \times \left( {-1,1}\right) \) and (13.7)(ii) is satisfied. Moreover, for every \( t \in \lbrack 0,1) \) , \( V\left( {t,1}\right) = \left( {-1}\right) \vee \left( {1 - a + {at}}\right) < 1. Also except a... | Yes |
Theorem 14.1. Assume (9.4) and (14.1). Let \( W \) be a viscosity subsolution and \( V \) be a viscosity supersolution of (8.1) in \( Q \) and the lateral boundary condition (9.3a), respectively. If \( Q \) is unbounded, we also assume that \( W \) and \( V \) are bounded and uniformly continuous on \( \bar{Q} \) . The... | \[ \mathop{\sup }\limits_{\bar{Q}}\left\lbrack {W - V}\right\rbrack \leq \mathop{\sup }\limits_{\bar{O}}\left( {W\left( {{t}_{1}, x}\right) - V\left( {{t}_{1}, x}\right) }\right) \vee O. \] | Yes |
Let \( \sum = {\mathbb{R}}^{1} = \) real line and \( {G\phi } = - \frac{1}{2}{\phi }^{\prime \prime } \) . Then \( - G \) generates the Markov process \( x\left( s\right) = x + w\left( s\right), s \geq 0 \), where \( w\left( s\right) \) is a standard brownian motion. Let \( \ell \left( x\right) = {x}^{2},\beta = 1 \) .... | The general solution to (3.4) is\n\n\[ \phi \left( x\right) = {x}^{2} + 1 + {c}_{1}\exp \left( {\sqrt{2}x}\right) + {c}_{2}\exp \left( {-\sqrt{2}x}\right) .\n\]\n\nSince \( \exp \left( {-s \pm \sqrt{2}x\left( s\right) }\right) \) is a martingale,(3.5) is satisfied only when \( {c}_{1} = {c}_{2} = \) 0 . The other solut... | Yes |
Theorem 8.1. Let \( W \in \mathcal{D} \) be a classical solution to (8.1) - (8.2). Then for all \( \left( {t, x}\right) \in \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \times \sum \) :\n\n(a) \( W\left( {t, x}\right) \leq J\left( {t, x;\pi }\right) \) for every admissible control system \( \pi \) .\n\n(b) If there exis... | Proof of (a): Let \( \pi \) be any admissible control system. Since \( u\left( s\right) \in U \) ,\n\n\[ \n{A}^{u\left( s\right) }W\left( {s, x\left( s\right) }\right) + L\left( {s, x\left( s\right), u\left( s\right) }\right) \geq 0. \n\]\n\nFrom (8.2) and the Dynkin formula\n\n\[ \nW\left( {t, x}\right) = {E}_{tx}\lef... | Yes |
This is a stochastic perturbation of the linear quadratic regulator problem (Example I.2.3 and I.5.1). The stochastic differential equations for \( x\left( s\right) \) are linear:\n\n(8.6)\n\n\[ \n{dx} = \left\lbrack {A\left( s\right) x\left( s\right) + B\left( s\right) u\left( s\right) }\right\rbrack {ds} + \sigma \le... | To apply the Verification Theorem 8.1, let us seek a solution \( W \) of the dynamic programming equation (8.1) which has the special form\n\n(8.8)\n\n\[ \nW\left( {t, x}\right) = x \cdot P\left( t\right) x + g\left( t\right)\n\]\n\nWhen \( t = {t}_{1} \) we must have \( W\left( {{t}_{1}, x}\right) = x \cdot {Dx} \). T... | Yes |
Consider the problem of minimizing\n\n\[ J = {E}_{tx}\left\{ {{\int }_{t}^{{t}_{1}}\left\lbrack {\frac{1}{2}{\left| u\left( s\right) \right| }^{2} - q\left( {x\left( s\right) }\right) }\right\rbrack {ds} + \psi \left( {x\left( {t}_{1}\right) }\right) }\right\} \]\nin the class of admissible systems \( \pi \). | In Example 8.2, the dynamic programming equation (8.1) becomes\n\n\[ 0 = \frac{\partial W}{\partial t} + \frac{1}{2}{\Delta }_{x}W - \frac{1}{2}{\left| {D}_{x}W\right| }^{2} - q\left( x\right) ,\]\n\nwith the terminal data (8.2). The procedure used to get (8.5) leads to minimizing \( \frac{1}{2}{\left| v\right| }^{2} +... | Yes |
Lemma 9.1. Let \( W \in D \) be a classical solution to (9.4). Then:\n\n(a) \( \;W\left( x\right) \leq J\left( {x;\pi }\right) \) for every admissible \( \pi \) such that\n\n(9.5)\n\n\[ \mathop{\liminf }\limits_{{{t}_{1} \rightarrow \infty }}{e}^{-\beta {t}_{1}}{E}_{x}W\left( {x\left( {t}_{1}\right) }\right) \leq 0. \] | Proof of (a). Since \( u\left( s\right) \in U \),\n\n\[ - {G}^{u\left( s\right) }W\left( {x\left( s\right) }\right) - {\beta W}\left( {x\left( s\right) }\right) + L\left( {x\left( s\right), u\left( s\right) }\right) \geq 0. \]\n\nBy applying Dynkin’s formula to \( \Phi = {e}^{-{\beta t}}W \), as in the derivation of (9... | Yes |
Theorem 9.1. Let \( W \in D \) be a classical solution to (9.4). Then \( W\left( x\right) \leq \) \( {V}_{AS}\left( x\right) \) . If there exists \( {\pi }^{ * } \in {C}_{1} \) such that (9.6) holds and\n\n\[ \n{u}^{ * }\left( s\right) \in \arg \min \left\lbrack {-{G}^{v}W\left( {{x}^{ * }\left( s\right) }\right) + L\l... | Theorem 9.1 is an immediate consequence of Lemma 9.1. The control system \( {\pi }^{ * } \) is optimal in the class \( {C}_{1} \) . | No |
Consider an infinite horizon discounted stochastic linear regulator. For simplicity, we take scalar state \( x\left( s\right) \), control \( u\left( s\right) \) and brownian motion \( w\left( s\right) \). Using notations similar to Example 8.1\n\n(9.11)\n\n\[ \n{dx} = \left\lbrack {{Ax}\left( s\right) + {Bu}\left( s\ri... | Let us seek a solution of (9.4) of the form\n\n(9.14)\n\n\[ \nW\left( x\right) = K{x}^{2} + g \n\]\n\nwhere \( K > 0 \) and \( g \) are constants. A calculation gives \( K \) as the positive root of the quadratic equation\n\n(9.15)\n\n\[ \n{\beta K} = {2AK} - \frac{{B}^{2}}{N}{K}^{2} + M \n\]\n\nand \( g = {\beta }^{-1... | Yes |
Example 9.2. This is a highly simplified production planning model, in which demand is random. Let \( x\left( s\right) \) be the inventory level of some good at time \( s, u\left( s\right) \) the production rate, and \( z\left( s\right) \) the demand rate. We allow \( x\left( s\right) < 0 \) , which corresponds to unfi... | It can be shown that a classical solution \( W\left( x\right) \geq 0 \) exists. Moreover, \( W\left( {\cdot, z}\right) \) is a strictly convex function on \( {\mathbb{R}}^{1}, W\left( {x, z}\right) \rightarrow + \infty \) as \( \left| x\right| \rightarrow \infty \) and \( \partial W/\partial x \) is bounded. See [FSS, ... | Yes |
This is another simple production planning model, in which demand is fixed but random machine breakdown and repairs are allowed. It was analyzed by Akella-Kumar [AK]. We will formulate the model and summarize their results. There is one machine, which is either working or not. If the machine is working, it produces a g... | A candidate optimal Markov control policy is \[ {\underline{u}}^{ * }\left( {x,1}\right) = \left\{ \begin{array}{ll} 0 & \text{ if }{W}_{x}\left( {x,1}\right) > 0 \\ d & \text{ if }{W}_{x}\left( {x,1}\right) = 0 \\ K & \text{ if }{W}_{x}\left( {x,1}\right) < 0 \end{array}\right. \] Akella-Kumar [AK] show that (9.25) ha... | Yes |
If there exists \( {\underline{u}}^{ * } \in \mathcal{L} \) such that\n\n\[ \n{\underline{u}}^{ * }\left( {s, y}\right) \in \arg \min \left\lbrack {f\left( {s, y, v}\right) \cdot {D}_{x}W\left( {s, y}\right) }\right.\n\]\n\n(3.15)\n\n\[ \n\left. {+\frac{1}{2}\operatorname{tr}a\left( {s, y, v}\right) {D}_{x}^{2}W\left( ... | Given any reference probability system \( \nu \), let \( {x}^{ * }\left( s\right) \) be the solution to (3.13) with \( \underline{u} = {\underline{u}}^{ * } \), and\n\n(3.16)\n\n\[ \n{u}^{ * }\left( s\right) = {\underline{u}}^{ * }\left( {s,{x}^{ * }\left( s\right) }\right) \;,\;t \leq s \leq {\tau }^{ * }.\n\]\n\nThen... | No |
Let \( U \) be compact. Let \( {\underline{u}}^{ * } \) be Borel measurable and satisfy (3.15) for almost all \( \left( {s, y}\right) \in Q \) . Given \( \left( {t, x}\right) \in Q \), assume that there exist \( {\nu }^{ * } = \left( {{\Omega }^{ * },\left\{ {\mathcal{F}}_{s}^{ * }\right\} ,{P}^{ * },{w}^{ * }}\right) ... | Proof. By hypothesis, there exists \( N \in \mathcal{B}\left( Q\right) \) with \( \left( {n + 1}\right) \) -dimensional Lebesgue measure 0 such that (3.15) holds for all \( \left( {s, y}\right) \in Q \smallsetminus N \) . Let \( \chi \left( s\right) \) be the indicator function of the event \( \left( {s,{x}^{ * }\left(... | Yes |
Theorem 4.1. If assumptions (3.5) (4.1) hold, then (3.3)-(3.4) has a unique solution \( W \in {C}^{1,2}\left( Q\right) \cap C\left( \bar{Q}\right) \) . | This theorem is due to Krylov. For a proof see [Kr2, Chap. 6, esp. pp. 279 and 297]. | No |
Theorem 4.3. Let \( Q = {Q}_{0} \) and assume (3.5),(4.6). Then (4.2) with the Cauchy data \( W\left( {{t}_{1}, x}\right) = \psi \left( x\right) \) has a unique solution \( W \in {C}^{1,2}\left( {\bar{Q}}_{0}\right) \cap {C}_{p}\left( {\bar{Q}}_{0}\right) \). | This is [FR, Thm 6.2, p. 169]. The proof of existence of \( W \) given there uses PDE arguments; see [FR, p. 210]. However, uniqueness is obtained using the stochastic control interpretation of the HJB equation (3.3) and a verification theorem. | Yes |
Theorem 5.1 (Verification Theorem). Let \( W \in {C}^{2}\left( O\right) \cap {C}_{p}\left( \bar{O}\right) \) be a solution to (5.8)-(5.9). Then for every \( x \in O \) :\n\n(a) \( W\left( x\right) \leq J\left( {x;u}\right) \) for any admissible progressively measurable control process \( u\left( \cdot \right) \) such t... | Sketch of proof. We apply the Ito differential rule to \( \Phi \left( {x, t}\right) = W\left( x\right) {e}^{-{\beta t}} \) . As in the proof of Lemma 3.1, for any \( {t}_{1} < \infty \) ,\n\n\[ W\left( x\right) \leq {E}_{x}\left\{ {{\int }_{0}^{{t}_{1} \land \tau }{e}^{-{\beta s}}L\left( {x\left( s\right), u\left( s\ri... | Yes |
Let \( n = 1, O = \left( {-b, b}\right) \) and let\n\n\[ \n{dx} = u\left( s\right) {ds} + {\alpha dw}\left( s\right) ,\;\alpha > 0, \]\n\nwhere the control constraint \( \left| {u\left( s\right) }\right| \leq 1 \) is imposed. Then (5.16) becomes\n\n\[ \n\frac{{\alpha }^{2}}{2}{V}^{\prime \prime }\left( x\right) + \math... | With this in mind let \( W\left( x\right) \) be the solution to\n\n\[ \n\frac{{\alpha }^{2}}{2}{W}^{\prime \prime }\left( x\right) + {W}^{\prime }\left( x\right) + 1 = 0,\;0 < x < b, \]\n\n\[ \n{W}^{\prime }\left( 0\right) = 0, W\left( b\right) = 0. \]\n\nFor \( - b \leq x \leq 0 \), let \( W\left( x\right) = W\left( {... | Yes |
There exists \( C \) such that\n\n\[ \n{E}_{tx}{\int }_{t}^{{t}_{1}}\left| {{\Lambda }_{\rho }\left( {s,{x}_{\rho }\left( s\right) }\right) - \Lambda \left( {s, x\left( s\right) }\right) }\right| {ds} \leq C{\rho }^{-\frac{1}{2}}{\left( 1 + {\left| x\right| }^{{2k} + 1}\right) }^{\frac{1}{2}}, \]\n\nwhere \( k \) is as... | Proof. Let \( {\chi }_{\rho } \) be the indicator function of the event \( {\tau }_{\rho } < {t}_{1} \) . Using the Cauchy-Schwarz inequality\n\n\[ \n{E}_{tx}{\int }_{{\tau }_{\rho }}^{{t}_{1}}\left| {\Lambda }_{\rho }\right| {ds} \leq {E}_{tx}{\int }_{t}^{{t}_{1}}{\chi }_{\rho }\left| {\Lambda }_{\rho }\right| {ds} \]... | Yes |
Assume that \( L,\widetilde{L},{\widetilde{L}}_{x} \) are continuous, and that \( L - \widetilde{L},{\widetilde{L}}_{x} \) are bounded on \( {\bar{Q}}_{0} \times U \) . Then\n\n(6.12)\n\n\[ \n{E}_{tx}{\int }_{t}^{{t}_{1}}\left| {\Lambda \left( {s, x\left( s\right) }\right) - \widetilde{\Lambda }\left( {s,\widetilde{x}\... | Proof. We have\n\n\[ \n\left| {\Lambda \left( {s, x\left( s\right) }\right) - \widetilde{\Lambda }\left( {s,\widetilde{x}\left( s\right) }\right) }\right| \leq \left| {\Lambda \left( {s, x\left( s\right) }\right) - \widetilde{\Lambda }\left( {s, x\left( s\right) }\right) }\right| + \left| {\widetilde{\Lambda }\left( {s... | Yes |
Lemma 6.3. If \( {L}_{x} \) is continuous and bounded on \( {\bar{Q}}_{0} \times U \), then\n\n\[ \left| {V\left( {t, x}\right) - {V}^{\epsilon }\left( {t, x}\right) }\right| \leq K{\epsilon }^{\frac{1}{2}}. \] | Proof. To obtain (6.17) from (6.16) there is the minor difficulty that \( V \) is defined in (2.10) using 4-tuples \( \nu \), while \( {V}^{\epsilon } \) is defined in terms of 5- tuples \( \mu \) . However, given \( \mu = \left( {\Omega ,\left\{ {\mathcal{F}}_{s}\right\}, P, w,{w}_{1}}\right) \), let \( \nu = \left( {... | Yes |
Lemma 7.2. Assume (3.5) and (4.5). Let \( W \) be as in Theorem 4.2. Then \( W = V = {V}_{\nu } \) for every \( \nu \), and property (DP) holds. | Proof. By Dynkin's formula\n\n(7.11)\n\n\[ W\left( {t, x}\right) \leq {E}_{tx}\left\{ {{\int }_{t}^{\theta }L\left( {s, x\left( s\right), u\left( s\right) }\right) {ds} + W\left( {\theta, x\left( \theta \right) }\right) }\right\} \]\n\nfor every \( \nu, u\left( \cdot \right) \in {\mathcal{A}}_{tv} \) and \( {\mathcal{F... | Yes |
Corollary 7.1. Assume (6.1),(6.3), and that \( \psi \) is bounded and uniformly continuous. Then the conclusions of Theorem 7.1 are true. | Proof. (a) Suppose first that \( \psi \in {C}_{b}^{2}\left( {\mathbb{R}}^{n}\right) \) . We replace \( L \) by \( \widetilde{L} = L - {G}^{v}\psi \) and \( \psi \) by terminal data 0 as in the discussion following (6.2).\n\n(b) If \( \psi \) is bounded and uniformly continuous, then a standard smoothing technique (Appe... | Yes |
Lemma 8.1. There exists \( {M}_{1} \) such that\n\n(8.2)\n\n\[ \left| {{\Delta }_{x}J}\right| \leq {M}_{1}\left( {1 + {\left| x\right| }^{\ell }}\right) \]\n\nfor every direction \( \xi \) and \( 0 < h \leq 1 \) . The constant \( {M}_{1} \) depends on \( {C}_{1},{C}_{2},{C}_{4},\ell \) and \( {t}_{1} - {t}_{0} \) . | Proof. Given \( \left( {t,{x}^{0}}\right) \in {Q}_{0} \) let \( x\left( s\right) \) be the solution of (2.1) with \( x\left( t\right) = {x}^{0} \) ; and let \( {x}_{h}\left( s\right) \) be the solution of (2.1) with \( {x}_{h}\left( t\right) = {x}^{0} + {h\xi } \) . Also, let \( {\Delta x}\left( s\right) = \) \( \frac{... | Yes |
There exists \( {M}_{2} \) such that\n\n\[ \left| {{\Delta }_{t}\bar{J}}\right| \leq {M}_{2}\left( {1 + {\left| x\right| }^{k} + {\left| x\right| }^{\ell }}\right) \]\n\nfor \( 0 < h \leq {t}_{1} - t \) . The constant \( {M}_{2} \) depends on \( {C}_{i}, i = 1,\cdots ,4, k,\ell \) and \( {t}_{1} - {t}_{0} \) where \( k... | Proof. Let\n\n\[ \widetilde{f}\left( {s, y, v}\right) = f\left( {s + h, y, v}\right) ,\widetilde{\sigma }\left( {s, y, v}\right) = \sigma \left( {s + h, y, v}\right) .\n\nThen\n\n\[ \parallel \widetilde{f} - f\parallel \leq \begin{Vmatrix}{f}_{t}\end{Vmatrix}h \leq {C}_{1}h,\;\parallel \widetilde{\sigma } - \sigma \par... | Yes |
Theorem 8.1. Assume (6.1), (6.2) and (8.1). If \( V \) is differentiable at \( \left( {t, x}\right) \in {Q}_{0} \), then\n\n\[ \text{(a)}\left| {{D}_{x}V\left( {t, x}\right) }\right| \leq {M}_{1}\left( {1 + {\left| x\right| }^{\ell }}\right) \]\n\n\[ \text{(b)}\left| {{V}_{t}\left( {t, x}\right) }\right| \leq {M}_{2}\l... | Proof. Use (8.2) and let \( h \rightarrow {0}^{ + } \) . Then\n\n\[ \left| {{V}_{\xi }\left( {t, x}\right) }\right| \leq {M}_{1}\left( {1 + {\left| x\right| }^{\ell }}\right) \]\n\nwhere \( {V}_{\xi } \) denotes the derivative of \( V\left( {t, \cdot }\right) \) in direction \( \xi \) . Since \( \xi \) is arbitrary, th... | No |
Theorem 9.2. Assume (6.1),(6.2),(8.1),(9.1). Let \( Q \subset {Q}_{0} \) be an open set such that \( V \in {C}^{1,2}\left( Q\right) \) and \( V \) satisfies (3.3) in \( Q \) . Then there exists \( {M}_{4} \) such that, for all \( \left( {t, x}\right) \in Q \) and \( v \in U \) ,\n\n(9.10)\n\[ \left| {{A}^{v}V\left( {t,... | Proof. We recall that\n\n\[ {A}^{v}V = {V}_{t} + \frac{1}{2}\operatorname{tr}a{D}_{x}^{2}V + f \cdot {D}_{x}V. \]\n\nBy (6.1c) and (8.14)\n\n\[ \left| {V}_{t}\right| \leq {M}_{2}\left( {1 + {\left| x\right| }^{k} + {\left| x\right| }^{\ell }}\right) \]\n\n(9.11)\n\n\[ \left| f\right| \left| {{D}_{x}V}\right| \leq {M}_{... | Yes |
Lemma 10.1. Let \( {\Psi }_{n} \in {L}_{\ell {oc}}^{1}\left( {Q}_{0}\right) \) be such that, for every compact set \( B \subset {Q}_{0},\left| {{\Psi }_{n}\left( {t, x}\right) }\right| \leq {K}_{B} \) for all \( \left( {t, x}\right) \in B \) and \( n = 1,2,\cdots \) . Then:\n\n(a) There is a subsequence \( {\Psi }_{{n}... | Lemma 10.1 follows from [Ro, Thm 17, p. 202]. | No |
Lemma 10.2. Let \( {V}_{n} \in {C}^{1}\left( {Q}_{0}\right) \) for \( n = 1,2,\cdots \) . Assume that, for every compact set \( B \subset \) int \( {Q}_{0},{V}_{n} \rightarrow V \) uniformly on \( B \) as \( n \rightarrow \infty \) and that \( \left| {{V}_{nt}\left( {t, x}\right) }\right| + \left| {{D}_{x}{V}_{n}\left(... | Proof. This is an easy consequence of Lemma 10.1, together with (10.1), (10.2). | No |
Lemma 10.3. Assume that:\n\n(a) \( {f}_{n},{\sigma }_{n} \) are such that (10.7) holds;\n\n(b) \( {V}_{n} \in {C}^{1,2}\left( {Q}_{0}\right) \) ;\n\n(c) For every compact set \( B \subset {Q}_{0},{V}_{n} \rightarrow V \) uniformly on \( B \) and \( \left| {{A}_{n}^{v}{V}_{n}\left( {t, x}\right) }\right| \leq \) \( {K}_... | Proof. For any \( \Phi \in {C}_{0}^{\infty }\left( {Q}_{0}\right) \)\n\n\[ \n{\int }_{{Q}_{0}}\left( {{A}_{n}^{v}{V}_{n}}\right) {\Phi dxdt} = {\int }_{{Q}_{0}}{V}_{n}{\left( {A}_{n}^{v}\right) }^{ * }{\Phi dxdt} \n\]\n\n\[ \n\mathop{\lim }\limits_{{n \rightarrow \infty }}{\int }_{{Q}_{0}}{V}_{n}{\left( {A}_{n}^{v}\rig... | Yes |
Theorem 10.1. Let \( V \) be the value function. Then \( {A}^{v}V \) exists in the generalized sense. Moreover, \( V \) is a generalized subsolution of the HJB equation. | Proof. The same kind of approximations used in the proof of Theorem 6.1 provide \( {f}_{n},{\sigma }_{n},{L}_{n} \) with the following properties: (a) \( {f}_{n},{\sigma }_{n},{L}_{n} \) converge uniformly on compact sets to \( f,\sigma, L \) as \( n \rightarrow \infty \) ; (b) the partial derivatives of \( {f}_{n},{\s... | Yes |
Lemma 10.4. If \( V\left( {t, \cdot }\right) \) is differentiable at \( x \), then \( {D}_{x}{V}_{n}\left( {t, x}\right) \rightarrow {D}_{x}V\left( {t, x}\right) \) as \( n \rightarrow \infty \) . | Proof. Consider any direction \( \xi \in {\mathbb{R}}^{n}\left( {\left| \xi \right| = 1}\right) \) . Then Taylor’s formula with remainder gives, since \( {V}_{n} \in {C}^{1,2}\left( {Q}_{0}\right) \) ,\n\n\[ \n{V}_{n}\left( {t, x + {h\xi }}\right) = {V}_{n}\left( {t, x}\right) + h{D}_{x}{V}_{n}\left( {t, x}\right) \cdo... | Yes |
Lemma 10.5. \( {A}_{0n}{V}_{n} \) tends to \( {A}_{0}V \) weakly* in \( {L}_{\ell {oc}}^{\infty }\left( {Q}_{0}\right) \), where \( {A}_{0}V \) exists in the generalized sense. | Proof. \( {A}_{0n}{V}_{n}\left( {t, x}\right) \) is uniformly bounded on each compact set \( B \subset {Q}_{0} \) since \( {A}_{0n} = {A}_{n}^{v} - {f}_{n} \cdot {D}_{x} \) and \( {A}_{n}^{v}{V}_{n},{f}_{n} \cdot {D}_{x}{V}_{n} \) are uniformly bounded on compact sets. The proof is then the same as for Lemma 10.3. | No |
Theorem 10.2. In the semilinear case \( \left( {\sigma = \sigma \left( {t, x}\right) }\right) \) the value function \( V \) is a generalized solution the HJB equation (10.12). | Proof. We use the same approximations \( {f}_{n},{\sigma }_{n},{L}_{n} \) to \( f,\sigma, L \) as in the proof of Theorem 10.1. In the semilinear case, (10.9) becomes\n\n\[ 0 = - {A}_{0n}{V}_{n} + {H}_{n}\left( {t, x,{D}_{x}{V}_{n}}\right) ,\text{ where } \]\n\n\[ {A}_{0n}\Phi = {\Phi }_{t} + \frac{1}{2}\operatorname{t... | Yes |
Lemma 10.5. \( J\left( {t,\cdot , \cdot }\right) \) is convex on \( {\mathbb{R}}^{n} \times {\mathcal{A}}_{t\nu } \) . | Proof. Let \( {x}^{0},{x}^{1} \in {\mathbb{R}}^{n} \) and \( {u}_{0}\left( \cdot \right) ,{u}_{1}\left( \cdot \right) \in {\mathcal{A}}_{t\nu } \) . For \( 0 \leq \lambda \leq 1 \), let\n\n\[ \n{x}^{\lambda } = \left( {1 - \lambda }\right) {x}^{0} + \lambda {x}^{1},{u}_{\lambda }\left( \cdot \right) = \left( {1 - \lamb... | Yes |
Lemma 10.6. \( V\left( {t, \cdot }\right) \) is convex on \( {\mathbb{R}}^{n} \) . | Proof. Given \( {x}^{0},{x}^{1} \in {\mathbb{R}}^{n} \), we define \( {x}^{\lambda } \) as above. Given \( \delta > 0 \), choose \( {u}_{0}\left( \cdot \right) ,{u}_{1}\left( \cdot \right) \) such that\n\n\[ J\left( {t,{x}^{i},{u}_{i}}\right) < V\left( {t,{x}^{i}}\right) + \delta \text{ for }i = 0,1. \]\n\nBy Lemma 10.... | Yes |
Corollary 3.1. Suppose that \( V \in C\left( \bar{Q}\right) \) satisfies (2.1). Then \( V \) a viscosity solution of the dynamic programming equation\n\n(3.6)\n\n\[ - \frac{\partial }{\partial t}V\left( {t, x}\right) + \left( {{\mathcal{G}}_{t}V\left( {t, \cdot }\right) }\right) \left( x\right) = 0\text{ in }Q. \] | Proof. We have shown that the two parameter family of nonlinear operators \( \left\{ {\mathcal{T}}_{t{t}_{1}}\right\} \) satisfy \( \Pi \left( {3.1}\right) ,\Pi \left( {3.2}\right) ,\Pi \left( {3.11}\right) \) . Hence the viscosity property of \( V \) follows from Theorem II.5.1 and Remark II.6.1. | No |
Lemma 5.1. On \( \bar{Q},{W}^{\varepsilon } \) and \( {W}_{\varepsilon } \) are semiconvex, and semiconcave, respectively. Moreover, \( {W}^{\varepsilon } \) and \( {W}_{\varepsilon } \) converge to \( W \) uniformly on \( \bar{Q} \) as \( \varepsilon \rightarrow 0 \) . | Proof. We claim that\n\n\[ \n{w}^{\varepsilon }\left( {t, x}\right) = {W}^{\varepsilon }\left( {t, x}\right) + \frac{1}{2{\varepsilon }^{2}}\left( {{t}^{2} + {\left| x\right| }^{2}}\right) \n\] \n\nis convex on every convex subset of \( \bar{Q} \) . Indeed let \( \left( {t + h, x + z}\right) ,\left( {t - h, x - z}\righ... | Yes |
Lemma 5.2. Fix \( \varepsilon > 0 \) . For each \( \left( {\bar{t},\bar{x}}\right) \in {Q}_{\varepsilon {k}_{0}} \), there exists \( \left( {\bar{s},\bar{y}}\right) \in Q \) such that\n\n\[ \n{D}^{+\left( {1,2}\right) }{W}^{\varepsilon }\left( {\bar{t},\bar{x}}\right) \subset {D}^{+\left( {1,2}\right) }W\left( {\bar{s}... | Proof. Let \( \left( {\bar{s},\bar{y}}\right) \) be a maximizer of (5.1) at \( \left( {\bar{t},\bar{x}}\right) \) . Then,\n\n\[ \nW\left( {\bar{s},\bar{y}}\right) - \frac{1}{2{\varepsilon }^{2}}\left( {{\left| \bar{t} - \bar{s}\right| }^{2} + {\left| \bar{x} - \bar{y}\right| }^{2}}\right) \geq W\left( {\bar{t},\bar{x}}... | Yes |
Lemma 6.1. Assume \( \mathcal{H} \) in (4.1) is continuous. Then, every continuous viscosity subsolution \( W \) of (4.1) satisfies (6.1). Also, every continuous viscosity supersolution \( V \) of (4.1) satisfies (6.2). | Proof. This is an immediate consequence of the continuity of \( \mathcal{H} \) and (4.2) or (4.3). Indeed, we simply take\n\n\[ \nC\left( M\right) = \sup \{ \left| {\mathcal{H}\left( {t, x, p, A, w}\right) }\right| : \parallel \left( {t, x, p, A, w}\right) \parallel \leq M\} .\n\] | Yes |
Lemma 7.1. Let \( \mathcal{H} \) be as in (2.12). Assume \( \operatorname{IV}\left( {2.2}\right) \) and that \( L\left( {\cdot ,\cdot, v}\right) \in \) \( {C}^{1}\left( \bar{Q}\right) \) . Then, there exists a continuous function \( \omega : \lbrack 0,\infty ) \rightarrow \lbrack 0,\infty ) \) that satisfies \( \omega ... | Proof. Recall that\n\n\[ \mathcal{H}\left( {t, x, p, A}\right) = \mathop{\sup }\limits_{{v \in U}}\left\{ {-\frac{1}{2}\operatorname{tr}\left( {\sigma {\sigma }^{\prime }}\right) \left( {t, x, v}\right) A - f\left( {t, x, v}\right) \cdot p - L\left( {t, x, v}\right) }\right\} .\n\nSet \( {p}_{\alpha } = \alpha \left( {... | Yes |
Corollary 8.1. (Uniqueness). Assume (7.1). Then there is at most one viscosity solution \( V \in C\left( \bar{Q}\right) \) of (4.1) in \( Q \) satisfying the boundary and terminal conditions\n\n(8.8a)\n\n\[ V\left( {t, x}\right) = g\left( {t, x}\right) ,\;\left( {t, x}\right) \in \left\lbrack {{t}_{0},{t}_{1}}\right) \... | In particular, in (4.1) we may take \( V \) to be the value function of the stochastic control problem defined in Chapter IV. Then (4.1) becomes the dynamic programming equation IV(3.3). If \( V \in C\left( \bar{Q}\right) \) satisfies (8.8) and the dynamic programming principle (2.2), then it is the unique viscosity so... | Yes |
Lemma 9.1. Let \( W \in C\left( {\bar{Q}}_{0}\right) \) be a viscosity subsolution (or supersolution) of \( \operatorname{IV}\left( {3.3}\right) \) in \( {Q}_{0} \) . Then \( \widehat{W} \) is a viscosity subsolution (or supersolution, respectively) of (9.3) in \( {Q}_{R} \) . | Proof. Suppose that \( \widehat{w} \in {C}^{\infty }\left( {\bar{Q}}_{R}\right) \) and \( \widehat{W} - \widehat{w} \) has a maximum at \( \left( {\bar{t},\bar{x}}\right) \in \) \( {Q}_{R} \), satisfying \( \left( {\widehat{W} - \widehat{w}}\right) \left( {\bar{t},\bar{x}}\right) = 0 \) . Set \( w\left( {t, x}\right) =... | Yes |
Let \( F\left( \mathcal{J}\right) = \exp \left( {\rho \mathcal{J}}\right) ,\rho \neq 0 \), where exp denotes the exponential function. Then \( {r}_{F}\left( \mathcal{J}\right) = \left| \rho \right| \) is constant. | This exponential function \( F \) is the one which will be considered in this chapter. The certainty equivalent expectation is then\n\n\[ \n{\mathcal{E}}^{0}\left( \mathcal{J}\right) = {\rho }^{-1}\log {E}^{0}\left\lbrack {\exp \left( {\rho \mathcal{J}}\right) }\right\rbrack \]\n\nAs \( \rho \rightarrow 0 \), the right... | No |
Lemma 3.1. Assume (3.8),(3.11) and that the expectation \( E\left( \mathcal{J}\right) \) exists. Then\n\n(3.13)\n\n(a) \( \;{\mathcal{E}}^{0}\left( \mathcal{J}\right) \geq E\left( {\mathcal{J} - Z}\right) \)\n\n(b) \( \;{\mathcal{E}}^{0}\left( \mathcal{J}\right) = E\left( {\mathcal{J} - Z}\right) \) if \( \mathcal{J} -... | Proof. This is immediate from Jensen's inequality. See (2.8) above. \( ▱ \) | No |
Lemma 3.2. Assume that \( \left| {z\left( s\right) }\right| \leq K\left( {1 + \left| {x\left( s\right) }\right| }\right) \) for some constant \( K \) . Then (3.8), (3.11) hold. | Proof. An estimate for solutions to stochastic differential equations (Appendix (D.13)) implies that there exist positive constants \( k, C \) such that\n\n\[ \n{E}^{0}\left\lbrack {\exp \left( {k{\left| x\left( s\right) \right| }^{2}}\right) }\right\rbrack \leq C \n\]\n\nfor \( t \leq s \leq {t}_{1} \) . Therefore, \(... | Yes |
Proposition 3.1. Let \( \widetilde{\Phi } \in {C}^{1,2}\left( {Q}_{0}\right) \cap C\left( {\bar{Q}}_{0}\right) \) be a positive, bounded solution to (3.16)-(3.17) and assume that \( \ell \) is bounded above. Then \( \widetilde{\Phi }\left( {t, x}\right) = \Phi \left( {t, x}\right) \) . | Proof. For \( R > 0 \), let \( {Q}_{R} = \left\lbrack {{t}_{0},{t}_{1} - {R}^{-1}}\right) \times {O}_{R} \), where \( {O}_{R} = \{ x : \left| x\right| < R\} \) . Let \( {\tau }_{R} \) be the exit time of \( \left( {s, x\left( s\right) }\right) \) from \( {Q}_{R} \) . By the Feynman-Kac formula (Appendix (D.15)) and (3.... | Yes |
Lemma 3.3. Assume (3.23) and let \( z\left( s\right) \) be as in (3.21). Then \( \left( {3.8}^{\prime }\right) \) and (3.11) hold. Moreover, \( \mathcal{J} - \zeta \left( {t}_{1}\right) = W\left( {t, x}\right) \) P-almost surely. | Proof. By (3.2)(ii) and (3.23), \( \left| {z\left( s\right) }\right| \leq K\left( {1 + \left| {x\left( s\right) }\right| }\right) \) for some \( K \) . By Lemma 3.2,(3.8) and (3.11) hold. Since \( W \) satisfies (3.19), the Ito differential rule gives\n\n\[ {dW}\left( {s, x\left( s\right) }\right) = - \left\lbrack {\fr... | Yes |
Theorem 3.1. Assume that (3.2) and either of the following two assumptions (3.25) or (3.26) hold. Then \( \Phi \in {C}^{1,2}\left( {\bar{Q}}_{0}\right) \) is a solution to (3.16)-(3.17) with \( \Phi \) and \( {D}_{x}\Phi \) bounded. | When (3.25) holds, an existence theorem for linear, uniformly parabolic PDEs implies that (3.16)-(3.17) has a solution \( \widetilde{\Phi } \in {C}^{1,2}\left( {\bar{Q}}_{0}\right) \) with \( \widetilde{\Phi } \) and \( {D}_{x}\widetilde{\Phi } \) bounded. See [LSU, Chapt. 4, Thm. 9.1]. By Proposition 3.1, \( \Phi = \w... | Yes |
Theorem 4.1. Let \( W \in {C}^{1,2}\left( {\bar{Q}}_{0}\right) \) be a solution to (3.19)-(3.20) such that \( \left| {{D}_{x}W\left( {t, x}\right) }\right| \leq M\left( {1 + \left| x\right| }\right) \) for some constant \( M \) . Let\n\n(4.5)\n\n\[{\underline{z}}^{ * }\left( {s, x}\right) = {\sigma }^{\prime }\left( {s... | Proof. The linear growth condition on \( {D}_{x}W\left( {t, x}\right) \) implies that \( W \in {C}_{p}\left( {\bar{Q}}_{0}\right) \) with \( p = 2 \) . By Corollary IV.3.1, \( W\left( {t, x}\right) \geq J\left( {t, x;z}\right) \) with equality when \( z\left( s\right) = {z}^{ * }\left( s\right) \), Lemmas 3.1(b) and 3.... | Yes |
Using the notation in (3.1), let\n\n\[ \n{dx} = A\left( s\right) x\left( s\right) {ds} + {\rho }^{-\frac{1}{2}}\sigma \left( s\right) d{w}^{0}\left( s\right) \]\n\nwith \( x\left( t\right) = x \) and\n\n\[ \n\ell \left( {t, x}\right) = - x \cdot M\left( t\right) x \]\n\n\[ \n\psi \left( x\right) = - x \cdot {Dx} \]\n\n... | In III(8.6) we now have \( B\left( s\right) = \sigma \left( s\right), u\left( s\right) \) replaced by \( z\left( s\right) \) and \( \sigma \left( s\right) {dw}\left( s\right) \) replaced by \( {\rho }^{-\frac{1}{2}}\sigma \left( s\right) {dw}\left( s\right) \) . In III(8.7), \( J \) is now replaced by \( \widetilde{J} ... | Yes |
In Example 4.1, suppose that the assumption that \( M\left( t\right) \) and \( D \) are nonnegative definite does not hold. Then the stochastic linear regulator problem is of indefinite sign. | The case \( \rho < 0 \) . In this case we replace \( \rho \) by \( \left| \rho \right| \) in the SDE (4.1). The logarithmic transformation \( V = {\rho }^{-1}\log \Phi \) transforms the linear PDE (3.16) into\n\n(4.11)\n\n\[ - \frac{\partial V}{\partial t} - \frac{1}{2\left| \rho \right| }\operatorname{tr}a\left( {t, x... | Yes |
Theorem 5.1. Let \( \Phi \left( {t, x}\right) \) be as in (3.14), with \( \mathcal{J} \) as in (5.1). Then \( \Phi \in \) \( {C}^{1,2}\left( Q\right) \cap C\left( \bar{Q}\right) \) and \( \Phi \) satisfies (3.16) in \( Q \) . Moreover, \( {D}_{x}\Phi \) is bounded on \( Q \) and continuous on \( \bar{Q} \smallsetminus ... | An existence theorem for linear, uniformly parabolic PDEs implies that (3.16) with boundary data \( \Phi \left( {t, x}\right) = \exp \left\lbrack {{\rho \Psi }\left( {t, x}\right) }\right\rbrack ,\left( {t, x}\right) \in {\partial }^{ * }Q \) has a solution \( \widetilde{\Phi } \) with the properties stated in Theorem ... | Yes |
Let \( \ell = 0, g = 0 \) . The PDE (3.16) becomes \( {A\Phi } = 0 \), and (3.14) becomes\n\n(5.3)\n\n\[ \Phi \left( {t, x}\right) = {P}_{tx}^{0}\left( {\tau < {t}_{1}}\right) + {E}_{tx}^{0}\left\{ {\exp \left\lbrack {{\rho \psi }\left( {x\left( {t}_{1}\right) }\right) }\right\rbrack ;\tau = {t}_{1}}\right\} . \] | The first term on the right side is the exit probability. If we formally set \( \psi \left( x\right) = - \infty \), then the second term is 0 . We will return to the exit probability problem in Section 6 and again in Section VII.10. | No |
Lemma 6.1. There exists \( {z}_{R}^{ * }\left( \cdot \right) \in {\mathcal{Z}}_{R}\left( t\right) \) such that \( J\left( {t, x;{z}_{R}^{ * }}\right) = {V}_{R}^{0}\left( {t, x}\right) \) . | Lemma 6.1 is a special case of Theorem I.11.1. | No |
Lemma 6.2. There exists \( {R}_{1} \) such that \( \left| {{z}_{R}^{ * }\left( s\right) }\right| \leq {R}_{1} \) for all \( R \) . | Proof. Let \( {x}^{ * }\left( s\right) \) be the solution to (6.6) with \( z\left( s\right) = {z}_{R}^{ * }\left( s\right) \) and \( {x}^{ * }\left( {t}_{1}\right) = x \) . Let \( P\left( s\right) \) be the solution to\n\n(6.10)\n\n\[ \frac{dP}{ds} = - {P}^{\prime }\left( s\right) {F}_{x}\left( {s,{x}^{ * }\left( s\rig... | Yes |
Theorem 6.1. Assume (3.2) and (6.8) and let \( {R}_{1} \) be as in Lemma 6.2. Then:\n\n(a) \( {V}_{R}^{0}\left( {t, x}\right) = {V}^{0}\left( {t, x}\right) \) if \( R \geq {R}_{1} \) ;\n\n(b) \( {V}^{0} \) is bounded and Lipschitz continuous on \( {\bar{Q}}_{0} \) ;\n\n(c) \( {V}^{0} \) is a viscosity solution to \( \l... | Proof. Part (a) is immediate from Lemma 6.2. From Remark II.10.1 \( {V}_{R}^{0} \) is Lipschitz. Moreover, \( {V}_{R}^{0} \) is bounded since \( \ell \) and \( \psi \) are bounded. Hence,(b) follows from (a). By Theorem II.7.1 and part (a), for \( R \geq {R}_{1},{V}^{0} \) is a viscosity solution of\n\n(6.12)\n\n\[ - \... | Yes |
Theorem 6.2. Assume (3.2) and that \( \ell ,\psi \) are bounded and uniformly continuous on \( {\bar{Q}}_{0},{\mathbb{R}}^{n} \) respectively. Then \( {V}^{\varepsilon }\left( {t, x}\right) \) tends to \( {V}^{0}\left( {t, x}\right) \) uniformly on compact subsets of \( {\bar{Q}}_{0} \) . | Theorem 6.2 will be proved in Section VII. 11 using viscosity solution methods. It can also be obtained by probabilistic methods, based on the Freidlin-Wentzell theory of large deviations. If \( \sigma \) is constant, there is another rather easy proof based on the characterization in Section 4 of \( {V}^{\varepsilon }... | No |
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