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Theorem 25. With the notation and hypotheses given above, the solution \( X \) of \( \left( {*4}\right) \) is given by\n\n\[ \n{X}_{t} = u\left( {{Y}_{t},{Z}_{t}}\right) \n\] | Proof. Using the F-S calculus we have\n\n\[ \nu\left( {{Y}_{t},{Z}_{t}}\right) = u\left( {{x}_{0},0}\right) + {\int }_{0}^{t}\frac{\partial u}{\partial x}\left( {{Y}_{s},{Z}_{s}}\right) \circ d{Y}_{s} + {\int }_{0}^{t}\frac{\partial u}{\partial z}\left( {{Y}_{s},{Z}_{s}}\right) \circ d{Z}_{s} \]\n\n\[ = {x}_{0} + {\int... | Yes |
Theorem 26. Let \( H \) be càdlàg, adapted, and let \( X \) be a semimartingale. Assume \( \left\lbrack {H, X}\right\rbrack \) exists. Let \( {\sigma }_{n} = \left\{ {0 = {T}_{0}^{n} \leq {T}_{1}^{n} \leq \cdots \leq {T}_{{k}_{n}}^{n}}\right\} \) be a sequence of random partitions tending to the identity. If \( H \) an... | Proof. It follows easily from the definition of \( \left\lbrack {H, X}\right\rbrack \) at the beginning of this section that \( \Delta {\left\lbrack H, X\right\rbrack }_{t} = \Delta {H}_{t}\Delta {X}_{t} \) and \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\sum \left( {{H}_{{T}_{i + 1}^{n}} - {H}_{{T}_{i}^{n}}}\rig... | Yes |
Corollary 2. Let \( X \) and \( Y \) be continuous semimartingales, and let \( {\sigma }_{n} = \) \( \left\{ {0 = {T}_{0}^{n} \leq {T}_{1}^{n} \leq \cdots \leq {T}_{{k}_{n}}^{n}}\right\} \) be a sequence of random partitions tending to the identity. Then\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\sum... | Proof. By Theorem 22 of Chap. II, any semimartingale \( Y \) has finite quadratic variation. Thus Corollary 2 is a special case of Theorem 26 (and of Corollary 1). | No |
Theorem 27. Let \( H \) be càdlàg, adapted, of finite quadratic variation, and suppose \( \mathop{\sum }\limits_{{0 < s < t}}\left| {\Delta {H}_{s}}\right| < \infty \) a.s., each \( t > 0 \) . Let \( X \) be a semimartingale, and let \( {\sigma }_{n} = {\left\{ {T}_{i}^{n}\right\} }_{0 \leq i \leq {k}_{n}} \) be a sequ... | Proof. First observe that \( {\widehat{H}}_{t} = {H}_{t} - \mathop{\sum }\limits_{{0 < s \leq t}}\Delta {H}_{s} \) defines a continuous process of finite quadratic variation and that \( \left\lbrack {\widehat{H},\bar{X}}\right\rbrack = {\left\lbrack H, X\right\rbrack }^{c} + {H}_{0}{X}_{0} \) . Next we note that\n\n\[ ... | Yes |
Theorem 29. Let \( X \) be a semimartingale and \( Y \) a continuous semimartingale, and let \( f \) be \( {\mathcal{C}}^{1} \) . Let \( \mu \) be a probability measure on \( \left\lbrack {0,1}\right\rbrack \) let \( \alpha = \int {\lambda \mu }\left( {d\lambda }\right) \) , and let \( {\sigma }_{n} = {\left\{ {T}_{i}^... | Proof. We begin by observing that\n\n\[ \mathop{\sum }\limits_{i}{\int }_{0}^{1}f\left( {{Y}_{{T}_{i}^{n}} + \lambda \left( {{Y}_{{T}_{i + 1}^{n}} - {Y}_{{T}_{i}^{n}}}\right) }\right) \mu \left( {d\lambda }\right) \left( {{X}^{{T}_{i + 1}^{n}} - {X}^{{T}_{i}^{n}}}\right) \]\n\n\[ = \mathop{\sum }\limits_{i}f\left( {Y}_... | Yes |
Theorem 31. Let \( {Z}^{j} \) be semimartingales \( \left( {1 \leq j \leq d}\right) ,{H}^{x} \) a vector of adapted processes in \( \mathbb{D} \) for each \( x \in {\mathbb{R}}^{n} \), and suppose \( \left( {x, t,\omega }\right) \mapsto {H}_{t}^{x}\left( \omega \right) \) is \( \mathcal{B} \otimes {\mathcal{B}}_{ + } \... | Proof. Let \( {X}^{0}\left( {x, t,\omega }\right) = {H}_{t}^{x}\left( \omega \right) \) and define inductively\n\n\[{X}^{n + 1}{\left( x, t,\omega \right) }^{i} = {H}_{t}^{x} + \mathop{\sum }\limits_{{j = 1}}^{d}{\int }_{0}^{t}{F}_{j}^{i}{\left( {X}^{n}\left( x,\cdot , \cdot \right) \right) }_{s - }d{Z}_{s}^{j}.\]\n\nT... | Yes |
Theorem 34. Let \( \mathbf{Z} = \left( {{Z}^{1},\ldots ,{Z}^{d}}\right) \) be a vector of independent Lévy processes, \( {\mathbf{Z}}_{0} = \mathbf{0} \), and let \( \left( {f}_{j}^{i}\right) 1 \leq j \leq d,1 \leq i \leq n \), be non-random F-S acceptable functions. \( {}^{8} \) Let \( {X}_{0} \) be as in Theorem 32 a... | Proof. We treat only the case \( n = 1 \) . By Theorem 33 \( {\left\lbrack {Z}^{i},{Z}^{j}\right\rbrack }_{t}^{c} = 0 \) if \( i \neq j \) and \( {\left\lbrack {Z}^{i},{Z}^{i}\right\rbrack }_{t}^{c} = {\alpha }_{i}t,{\alpha }_{i} \geq 0 \) . Therefore, by Theorem 22 the equation is equivalent to\n\n\[ \n{X}_{t} = {X}_{... | Yes |
Theorem 35. Let \( Z \) be a strong Markov processes with values in \( \mathbb{R},{Z}_{0} = 0 \) , such that \( Z \) is a semimartingale. Let \( f \) and \( g \) be Lipschitz functions. Let \( {X}_{0} \) be as in Theorem 32 and \( X \) be the solution of\n\n\[ \n{X}_{t} = {X}_{0} + {\int }_{0}^{t}f\left( {s-,{X}_{s - }... | Proof. First recall that \( X \) is defined on \( \bar{\Omega } = \mathbb{R} \times \Omega \) and \( Z \) is automatically extended to \( \bar{\Omega } \) as explained at the beginning of this section. The probability \( {\bar{P}}^{y} = {\varepsilon }_{y} \times P \) is such that \( {\bar{P}}^{y}\left( {{X}_{0} = y}\ri... | Yes |
Theorem 37. Let \( {H}^{x} \) be processes in \( {\mathbb{D}}^{n} \), and let \( x \mapsto {H}^{x} : {\mathbb{R}}^{n} \rightarrow {\mathbb{D}}^{n} \) be prelocally Lipschitz continuous in \( {\underline{S}}^{p} \), some \( p > n \) . Let \( F \) be an \( n \times m \) matrix of functional Lipschitz operators \( \left( ... | Proof. We recall the method of proof used to show the existence and uniqueness of a solution (Theorem 7). By stopping at a fixed time \( {t}_{0} \), we can assume the Lipschitz process is just a random variable \( K \) which is finite a.s. Then by conditioning \( {}^{13} \) we can assume without loss of generality that... | Yes |
Theorem 38. Let \( Z \) be as in (H1) and let the functions \( \left( {f}_{\alpha }^{i}\right) \) in (H2) be locally Lipschitz. Then there exists a function \( \zeta \left( {x,\omega }\right) : {\mathbb{R}}^{n} \times \Omega \rightarrow \left\lbrack {0,\infty }\right\rbrack \) such that for each \( {x\zeta }\left( {x, ... | Proof. Let \( {\Lambda }_{\ell } \) be open sets increasing to \( {\mathbb{R}}^{n} \) such that there exist \( \left( {h}_{\ell }\right) \), a sequence of \( {\mathcal{C}}^{\infty } \) functions with compact support mapping \( {\mathbb{R}}^{n} \) to \( \left\lbrack {0,1}\right\rbrack \) such that \( {h}_{\ell } = 1 \) ... | Yes |
Theorem 39. Let \( Z \) be as in (H1) and let the functions \( \left( {f}_{\alpha }^{i}\right) \) in (H2) have locally Lipschitz first partial derivatives. Then for almost all \( \omega \) there exists a function \( X\left( {t,\omega, x}\right) \) which is continuously differentiable in the open set \( \{ x \) : \( \ze... | Proof. We will give the proof in several steps. In Step 1 we will reduce the problem to one where the coefficients are globally Lipschitz. We then resolve the first system (for \( X \) ) of \( \left( D\right) \), and in Step 2 we will show that, given \( X \), there exists a \ | No |
Theorem 40. Let \( Z \) be as in (H1) and let the functions \( \left( {f}_{\alpha }^{i}\right) \) in (H2) have locally Lipschitz derivatives up to order \( N \), for some \( N,0 \leq N \leq \infty \) . Then there exists a solution \( X\left( {t, w, x}\right) \) to\n\n\[ \n{X}_{t}^{i} = {x}_{i} + \mathop{\sum }\limits_{... | Proof. If \( N = 0 \), then Theorem 40 is exactly Theorem 38. If \( N = 1 \), then Theorem 40 is Theorem 39. If \( N > 1 \), then the coefficients of equations \( \left( D\right) \) have locally Lipschitz derivatives of order \( N - 1 \) at least. Induction yields \( \left( {X, D}\right) \in {\mathcal{C}}^{N - 1} \), w... | Yes |
Theorem 42. For \( x \in {\mathbb{R}}^{n} \), let \( {H}^{x} \) be in \( {\mathbb{D}}^{k} \) such that they are locally bounded uniformly in \( x \) . Assume further that there exists a sequence of stopping times \( {\left( {T}_{\ell }\right) }_{\ell \geq 1} \) increasing to \( \infty \) a.s. such that \( {\begin{Vmatr... | Proof. By Theorem 5 there exists an arbitrarily large stopping time \( T \) such that \( {Z}^{T - } \in {\underline{H}}^{\infty } \) . Thus without loss of generality we can assume that \( Z \in {\underline{H}}^{\infty } \) , and that \( {H}^{x} \) is bounded by some constant \( K \), uniformly in \( x \) . Further we ... | Yes |
Theorem 43. Let \( F \) be a matrix of process Lipschitz operators and \( {X}^{x} \) the solution of \( \left( *\right) \) with initial condition \( x \), for continuous semimartingales \( {Z}^{\alpha } \) , \( 1 \leq \alpha \leq m \) . Fix \( x, y \in {\mathbb{R}}^{n} \) . For \( r \in \mathbb{R} \) there exist for ev... | Proof. Fix \( x, y \in {\mathbb{R}}^{n} \) and let \( U = {X}^{x} - {X}^{y}, V = F{\left( {X}^{x}\right) }_{ - } - F{\left( {X}^{y}\right) }_{ - } \) . Itô’s formula applies since \( U \) is never zero by weak injectivity (Theorem 36). Using the Einstein convention,\n\n\[ \parallel U{\parallel }^{r} = \parallel x - y{\... | Yes |
Theorem 46. Let \( {Z}^{\alpha } \) be continuous semimartingales, \( 1 \leq \alpha \leq m \), and \( F \) be an \( n \times m \) matrix of process Lipschitz operators. Let \( X \) be the solution of \( \left( *\right) \) . Let \( \varphi : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) be the flow \( \varphi \left( ... | Proof. As noted preceding Theorem 41, the flow \( \varphi \) is continuous from \( {\mathbb{R}}^{n} \) to \( {\mathcal{D}}^{n} \), topologized by uniform convergence on compacts; hence for a.a. \( \omega \) it is continuous from \( {\mathbb{R}}^{n} \) to \( {\mathbb{R}}^{n} \) for all \( t \) .\n\nThe flow \( \varphi \... | Yes |
Theorem 47. Let \( Y, Z \) be given \( n \times n \) matrices of semimartingales, \( H \) an \( n \times n \) matrix of locally bounded predictable processes. Then,\n\n(i) \( {Y}_{t}{Z}_{t} - {Y}_{0}{Z}_{0} = {\int }_{0}^{t}{Y}_{s - }d{Z}_{s} + {\int }_{0}^{t}\left( {d{Y}_{s}}\right) {Z}_{s - } + {\left\lbrack Y, Z\rig... | Proof. The first three identities are easily proved by calculating the entries of the matrices and using the results of Chap. II. Similarly the existence and uniqueness result for the stochastic integral equation is a simple consequence of Theorem 7. | No |
Theorem 48. Let \( Z \) be an \( n \times n \) matrix of continuous semimartingales with \( {Z}_{0} = 0 \) . Then \( \mathcal{E}\left( Z\right) \) and \( {\mathcal{E}}^{R}\left( {-Z + \left\lbrack {Z, Z}\right\rbrack }\right) \) are inverses; that is, \( \mathcal{E}\left( Z\right) {\mathcal{E}}^{R}( - Z + \left\lbrack ... | Proof. Let \( U = \mathcal{E}\left( Z\right) \) and \( V = {\mathcal{E}}^{R}\left( {-Z + \left\lbrack {Z, Z}\right\rbrack }\right) \) . Since \( {U}_{0}{V}_{0} = I \), it suffices to show that \( d\left( {{U}_{t}{V}_{t}}\right) = 0 \), all \( t > 0 \) . Note that\n\n\[ \n{dV} = \left( {-{dZ} + d\left\lbrack {Z, Z}\righ... | Yes |
Theorem 51. Let \( \left( {{Z}^{1},\ldots ,{Z}^{m}}\right) \) be continuous semimartingales and let \( \left( {f}_{\alpha }^{i}\right) \) , \( 1 \leq i \leq n,1 \leq \alpha \leq m \), be functions mapping \( {\mathbb{R}}^{n} \) to \( \mathbb{R} \), with partial derivatives of all orders, and bounded first partials. The... | Proof. Let \( \varphi \) denote the flow of \( X \) . Since \( {\left( {f}_{\alpha }^{i}\right) }_{1 \leq i \leq n,1 \leq \alpha \leq m} \) have bounded first partials, they are globally Lipschitz, and hence there are no finite explosions. Moreover since they are \( {\mathcal{C}}^{\infty } \), the flow is \( {\mathcal{... | Yes |
Theorem 52. Let \( H \) be a semimartingale and let \( Z \) be a continuous semi-martingale with \( {Z}_{0} = 0 \) . Then the solution \( {\mathcal{E}}_{H}\left( Z\right) \) of equation \( \left( {* * }\right) \) is given \( {by} \)\n\n\[ \n{\mathcal{E}}_{H}{\left( Z\right) }_{t} = \mathcal{E}{\left( Z\right) }_{t}\lef... | Proof. We use the method of \ | No |
Theorem 53. Let \( H \) be càdlàg, adapted (i.e., \( H \in \mathbb{D} \) ), and let \( Z \) be a continuous semimartingale with \( {Z}_{0} = 0 \) . Let \( {X}_{t} = {\mathcal{E}}_{H}{\left( Z\right) }_{t} \) be the solution of\n\n\[ \n{X}_{t} = {H}_{t} + {\int }_{0}^{t}{X}_{s - }d{Z}_{s} \n\]\n\nThen \( {X}_{t} = {\mat... | Proof. Let \( {Y}_{t} = {X}_{t} - {H}_{t} \) . Then \( Y \) satisfies\n\n\[ \n{Y}_{t} = {\int }_{0}^{t}{H}_{s - }d{Z}_{s} + {\int }_{0}^{t}{Y}_{s - }d{Z}_{s} \n\]\n\n\[ \n= {K}_{t} + {\int }_{0}^{t}{Y}_{s - }d{Z}_{s} \n\]\n\nwhere \( K \) is the semimartingale \( {H}_{ - } \cdot Z \) . By Theorem 52,\n\n\[ \n{Y}_{t} = ... | Yes |
Theorem 54 (Comparison Theorem). Let \( {\left( {Z}^{\alpha }\right) }_{1 \leq \alpha \leq m} \) be continuous semimartingales with \( {Z}_{0}^{\alpha } = 0 \), and let \( {F}_{\alpha } \) be process Lipschitz. Let \( A \) be a continuous, adapted process with increasing paths, strictly increasing at \( t = 0 \) . Let ... | Proof. Let \[ {U}_{t} = {X}_{t} - {Y}_{t} \] \[ {N}_{t} = {\int }_{0}^{t}\left\{ {F{\left( X\right) }_{s - } - F{\left( Y\right) }_{s - }}\right\} {\left( {X}_{s} - {Y}_{s}\right) }^{-1}{1}_{\left\{ {X}_{s} \neq {Y}_{s}\right\} }d{Z}_{s},\text{ and } \] \[ {C}_{t} = {x}_{0} - {y}_{0} + {\int }_{0 + }^{t}\left\{ {G{\lef... | Yes |
Theorem 55. Let \( {A}^{j},1 \leq j \leq m \), be \( n \times n \) matrices of càdlàg, adapted processes, and let \( U \) be the solution of \( \left( {*5}\right) \). Let \( {X}_{t}^{x} \) be the solution of \( \left( {*4}\right) \) where \( {H}_{t} = x, x \in {\mathbb{R}}^{n} \). Then \( {X}_{t}^{x} = {U}_{t}x \) and ... | Proof. Note that \( {U}_{t} \) is an \( n \times n \) matrix for each \( \left( {t,\omega }\right) \) and \( x \in {\mathbb{R}}^{n} \), so that \( {U}_{t}x \) is in \( {\mathbb{R}}^{n} \). If \( {X}_{t}^{x} = {U}_{t}x \), then since the coefficients are process Lipschitz we can apply Theorem 46 (which says that the flo... | Yes |
Theorem 56. Let \( H \) be a column vector of \( n \) semimartingales, \( {Z}^{j}(1 \leq j \leq \) m) be continuous semimartingales with \( {Z}_{0}^{j} = 0 \), and let \( {A}^{j},1 \leq j \leq m \) be \( n \times n \) matrices of processes in \( \mathbb{D} \) . Let \( U \) be the solution of equation (*5). Then the sol... | Proof. Write \( {X}^{H} \) as the matrix product \( {UY} \) . Recall that \( {U}^{-1} \) exists by Theorem 48, hence \( Y = {U}^{-1}{X}^{H} \) is a semimartingale, that we need to find explicitly. Using matrix notation throughout, we have\n\n\[ \nd\left( {UY}\right) = {dH} + \mathop{\sum }\limits_{{j = 1}}^{m}{A}_{ - }... | Yes |
Theorem 58. The flow \( \varphi : x \rightarrow {X}_{t}\left( {x,\omega }\right) \) of the solution \( X \) of \( \left( *\right) \) is a diffeomorphism if the collections of functions\n\n\[ x \mapsto {X}_{t}^{j}\left( {x,\omega }\right) \;x \mapsto {H}^{j}\left( {x,\omega }\right) \]\n\nare diffeomorphisms. | Proof. By Theorem 57, the solution \( \left( *\right) \) can be constructed by composition of functions of the types given in the theorem. Since the composition of diffeo-morphisms is a diffeomorphism, the theorem is proved. | Yes |
Theorem 60. Let \( {Z}^{\alpha } \) be semimartingales, \( 1 \leq \alpha \leq m \) with \( {Z}_{0}^{\alpha } = 0 \), and let \( F \) be an \( n \times m \) matrix of process Lipschitz operators with non-random Lipschitz constant \( K \) . Let \( {H}^{i} \in \mathbb{D},1 \leq i \leq n \) (càdlàg, adapted). If \( \mathop... | Proof. Let \( x, y \in {\mathbb{R}}^{n} \), and let \( {X}^{x},{X}^{y} \) denote the solutions of the above equation with initial conditions \( x, y \), respectively. Let \( u = x - y, U = {X}^{x} - {X}^{y} \) , and \( V = F{\left( {X}^{x}\right) }_{ - } - F{\left( {X}^{y}\right) }_{ - } \) . Then \( V \in \mathbb{L} \... | Yes |
Theorem 61. Let \( {\left( {Z}^{\alpha }\right) }_{1 \leq \alpha \leq m} \) be semimartingales, \( {Z}_{0}^{\alpha } = 0 \), F an \( n \times m \) matrix of process Lipschitz operators with a non-random Lipschitz constant, and \( {H}^{i} \in \mathbb{D},1 \leq i \leq n \). If \( \mathop{\sum }\limits_{{\alpha = 1}}^{m}{... | Proof. Fix \( x, y \in {\mathbb{R}}^{n}, x \neq y \), and let \( U = {X}^{x} - {X}^{y}, V = F{\left( {X}^{x}\right) }_{ - } - F{\left( {X}^{y}\right) }_{ - } \). By Theorem \( {60U} \) is never zero. As in the proof of Theorem 43, by Itô’s formula, \[ {\begin{Vmatrix}{U}_{y}\end{Vmatrix}}^{r} = \parallel x - y{\paralle... | Yes |
Theorem 63. Let \( Z \) be an \( n \times n \) matrix of semimartingales with \( {Z}_{0} = 0 \) . Suppose that \( {W}_{t} = - {Z}_{t} + {\left\lbrack Z, Z\right\rbrack }_{t}^{c} + \mathop{\sum }\limits_{{0 < s \leq t}}{\left( I + \Delta {Z}_{s}\right) }^{-1}{\left( \Delta {Z}_{s}\right) }^{2} \) is a well-defined semim... | Proof. Let \( U = \mathcal{E}\left( W\right), V = {\mathcal{E}}^{R}\left( Z\right) \), and \( {J}_{t} = \mathop{\sum }\limits_{{0 < s \leq t}}{\left( I + \Delta {Z}_{s}\right) }^{-1}{\left( \Delta {Z}_{s}\right) }^{2} \).\n\nThen,\n\n\[ {dU} = {U}_{ - }\left( {-{dZ} + d{\left\lbrack Z, Z\right\rbrack }^{c} + {dJ}}\righ... | Yes |
Theorem 64. Let \( {\left( {Z}^{\alpha }\right) }_{1 \leq \alpha \leq m} \) be semimartingales with \( {Z}_{0} = 0 \), and let \( f \) be a matrix of coefficients satisfying Hypotheses (H2) and (H3). Let \( X \) be the unique solution of\n\n\[ \n{X}_{t} = x + {\int }_{0}^{t}f\left( {X}_{s - }\right) d{Z}_{s} \n\]\n\n\(... | Proof. By Theorem 39 (in Sect. 7) the Jacobian matrix \( D \) satisfies the right stochastic exponential equation\n\n\[ \n{D}_{t} = I + {\int }_{0}^{t}\left( {\frac{\partial {f}_{\alpha }^{i}}{\partial {x}_{k}}\left( {X}_{s - }\right) d{Z}_{s}^{\alpha }}\right) {D}_{s - } \n\]\n\nand the matrix semimartingale different... | Yes |
Theorem 65. Let \( Z \) and \( f \) be as given in Hypotheses (H1),(H2), and (H3), and let \( X \) be the solution of \[ {X}_{t} = x + {\int }_{0}^{t}f\left( {X}_{s - }\right) d{Z}_{s} \] \( \left( *\right) \) The flow of \( X \) is a.s. a diffeomorphism of \( {\mathbb{R}}^{n} \) (resp. trajectories of \( X \) from dif... | Proof. Recall the processes \( {X}_{{T}_{j}}^{j}\left( {x,\omega }\right) \) and \( {X}_{t}^{n + 1}\left( {x,\omega }\right) \) defined in Theorem 57, and the linkage operator \( {H}^{j}\left( x\right) = x + f\left( x\right) \Delta {Z}_{{T}_{j}} \), defined immediately preceding Theorem 57. By hypothesis the linkage op... | Yes |
Theorem 68 (Gronwall’s Inequality). Let \( \alpha \) be a function from \( {\mathbb{R}}_{ + } \) to itself, and suppose\n\n\[ \alpha \left( s\right) \leq c + k{\int }_{0}^{s}\alpha \left( r\right) {dr} < \infty \]\n\nfor \( 0 \leq s \leq t \) . Then \( \alpha \left( t\right) \leq c{e}^{kt} \) . Moreover if \( c = 0 \) ... | The proof is simple and usually taught in elementary differential equations courses. | No |
Theorem 70. For \( k \geq 1 \) assume that \( \sigma \) is differentiable \( k \) times and that all partial derivatives of \( \sigma \) of order greater than or equal to one are bounded. Assume also that \( Z \) has moments of order \( {kp} \) with \( p \geq 2 \) . With the notation defined right before this theorem, ... | Proof. We recall from Theorem 39 that \( {X}^{x} \) together with \( {X}^{x,\left( k\right) } \) constitute the unique solution of the system of equations\n\n\[{X}_{t}^{i} = {x}_{i} + \mathop{\sum }\limits_{{\alpha = 1}}^{m}{\int }_{0}^{t}{\sigma }_{\alpha }^{i}\left( {X}_{s - }\right) d{Z}_{s}^{\alpha }\]\n\n\( \left(... | No |
Theorem 71. Let \( Z \) be a continuous semimartingale, \( \sigma \) a continuous function, and suppose there exists a (path-by-path) unique, non-exploding solution to the equation\n\n\[ \n{X}_{t} = {X}_{0} + {\int }_{0}^{t}\sigma \left( {X}_{s}\right) {X}_{s}d{Z}_{s} \n\]\n\nwith \( {X}_{0} > 0 \) almost surely. Let \... | Proof of Theorem 71. Define \( {T}_{n} = \inf \left\{ {t > 0 : {X}_{t} = 1/n}\right. \) or \( \left. {{X}_{t} = {X}_{0} \vee n}\right\} \) , and note that \( P\left( {{T}_{n} > 0}\right) = 1 \) because \( P\left( {{X}_{0} > 0}\right) = 1 \) and \( Z \) is continuous. Using Itô’s formula up to time \( {T}_{n} \) we have... | Yes |
Theorem 72. Let \( Z \) be a semimartingale and let \( \sigma \) be a bounded, continuous function such that there exists a unique non-exploding solution of\n\n\[ \n{X}_{t} = {X}_{0} + {\int }_{0}^{t}\sigma \left( {X}_{s - }\right) {X}_{s - }d{Z}_{s} \n\]\n\n\( \textit{with }{X}_{0} > 0\textit{ almost surely. If in add... | Proof. \( \parallel \sigma {\parallel }_{\infty } \) refers to the \( {L}^{\infty } \) norm of \( \sigma \) . The proof is similar to the proof of Theorem 71. Using Itô's formula gives\n\n\[ \n\ln \left( \left| {X}_{t}\right| \right) = \ln \left( {X}_{0}\right) + {\int }_{0}^{t}\sigma \left( {X}_{s - }\right) d{Z}_{s} ... | Yes |
Theorem 73. Let \( \left( {\Omega ,\mathcal{F},\mathbb{F}, P}\right) \) be a filtered probability space with \( Z \) a càdlàg semimartingale defined on the space. Let \( f \) be a Borel function which is never 0 and is such that for every \( x \in \mathbb{R} \) the equation\n\n\[ d{X}_{t} = f\left( {X}_{t - }\right) d{... | Proof. Let \( {\Omega }^{\prime } \) be the function space of càdlàg functions defined on \( {\mathbb{R}}_{ + } \) with \( {X}_{t}\left( \omega \right) = \omega \left( t\right) \), the projection operator. Let \( {\mathbb{F}}^{\prime } \) be the canonical filtration and let \( \left( {\theta }_{t}^{\prime }\right) \) b... | Yes |
Theorem 2. Let \( X \) be a semimartingale with decomposition \( X = M + A \) and let \( \mathcal{G} \) be a \( \sigma \) -algebra independent of the local martingale term \( M \) . Let \( \mathbb{H} \) denote the filtration obtained by expanding \( \mathbb{F} \) with the one \( \sigma \) -algebra \( \mathcal{G} \) (th... | Proof. Since the local martingale \( M \) remains a local martingale under \( \mathbb{H} \), the theorem follows. | No |
Theorem 3 (Itô’s Theorem for Lévy Processes). The Lévy process \( Z \) is an \( \mathbb{H} \) semimartingale. If moreover \( E\left\{ \left| {Z}_{t}\right| \right\} < \infty \), all \( t \geq 0 \), then the process | Proof. We begin by assuming \( E\left\{ {Z}_{t}^{2}\right\} < \infty \), each \( t > 0 \) . Without loss of generality we can further assume \( E\left\{ {Z}_{t}\right\} = 0 \) . Since \( Z \) has independent increments, we know \( Z \) is an \( \mathbb{F} \) martingale. Let \( 0 \leq s < t \leq 1 \) be rationals with \... | No |
Theorem 4. Let \( M \) be an \( \mathbb{F} \) local martingale and suppose \( M \) is a semimartingale in an expanded filtration \( \mathbb{H} \). Then \( M \) is a special semimartingale in \( \mathbb{H} \). | Proof. First recall that any local martingale is a special semimartingale. In particular the process \( {M}_{t}^{ * } = \mathop{\sup }\limits_{{s \leq t}}\left| {M}_{s}\right| \) is locally integrable (see Theorem 33 of Chap. III), and this of course remains locally integrable in the expanded filtration \( \mathbb{H} \... | Yes |
Theorem 5. Let \( M \) be an \( \mathbb{F} \) local martingale, and let \( H \) be predictable such that \( {\int }_{0}^{t}{H}_{s}^{2}d{\left\lbrack M, M\right\rbrack }_{s} \) is locally integrable. Suppose \( \mathbb{H} \) is an expansion of \( \mathbb{F} \) such that \( M \) is an \( \mathbb{H} \) semimartingale. The... | Proof. First assume that \( E\left\{ {{\int }_{0}^{\infty }{H}_{s}^{2}d{\left\lbrack M, M\right\rbrack }_{s}}\right\} < \infty \), which implies that \( H \cdot M \) (where \( H \cdot M \) denotes the stochastic integral process \( {\left( {\int }_{0}^{t}{H}_{s}d{M}_{s}\right) }_{t \geq 0} \) ) is a square integrable m... | Yes |
Theorem 7. Let \( M \) be a local martingale defined on the standard space of canonical Brownian motion. Let \( \mathbb{H} \) be the minimal expanded filtration containing \( {B}_{1} \) and satisfying the usual hypotheses. Then \( M \) is an \( \mathbb{H} \) semimartingale if and only if the integral \( {\int }_{0}^{1}... | Proof. By the Martingale Representation Theorem we have that every \( \mathbb{F} \) local martingale \( M \) has a representation \( {M}_{t} = {M}_{0} + {\int }_{0}^{t}{H}_{s}d{B}_{s} \), where \( H \) is predictable and \( {\int }_{0}^{t}{H}_{s}^{2}{ds} < \infty \) a.s., each \( t > 0 \) . By Theorem 5 we know that \(... | Yes |
Theorem 8. Let \( {X}^{n} \) be a sequence of real-valued random variables. Then \( {X}^{n} \) converges to 0 in probability if and only if \( \mathop{\lim }\limits_{{n \rightarrow \infty }}E\left\{ {\min \left( {1,\left| {X}^{n}\right| }\right) }\right\} = 0 \) . | A proof of Theorem 8 can be found in textbooks on probability (see for example [109]). | No |
Theorem 9. Let \( L \) be a random variable with values in a standard Borel space. Then there exists a regular conditional distribution \( {Q}_{t}\left( {\omega ,{dx}}\right) \) which is a version of \( E\left\{ {{1}_{\{ L \in {dx}\} } \mid {\mathcal{F}}_{t}}\right\} \) . | For a proof of Theorem 9 the reader can see, for example, Breiman [23, page 79]. | No |
Theorem 11. Let \( L \) be a random variable with values in a standard Borel space \( \left( {\mathbb{E},\mathcal{E}}\right) \), and let \( {Q}_{t}\left( {\omega ,{dx}}\right) \) denote the regular conditional distribution of \( L \) given \( {\mathcal{F}}_{t} \), each \( t \geq 0 \) . Then there exists for each \( t \... | Proof. It suffices to show that the existence of \( {\eta }_{t} \) for each \( t > 0 \) implies the existence of \( \eta \) with the right properties; we will show that the distribution measure of \( L \) is such an \( \eta \) . As in the proof of Theorem 10 let \( \left( {x,\omega }\right) \mapsto {q}_{t}\left( {x,\om... | Yes |
Corollary 1 (Independence). Let \( L \) be independent of the filtration \( \mathbb{F} \) . Then every \( \mathbb{F} \) semimartingale is also an \( \mathbb{H}\left( L\right) \) semimartingale. | Proof. Since \( L \) is independent of \( {\mathcal{F}}_{t}, E\left\{ {g\left( L\right) \mid {\mathcal{F}}_{t}}\right\} = E\{ g\left( L\right) \} \) for any bounded, Borel function \( g \) . Therefore\n\n\[ E\left\{ {g\left( L\right) \mid {\mathcal{F}}_{t}}\right\} = {\int }_{\mathbb{E}}{Q}_{t}\left( {\omega ,{dx}}\rig... | Yes |
Corollary 2 (Countably-valued random variables). Let \( L \) be a random variable taking on only a countable number of values. Then every \( \mathbb{F} \) semi-martingale is also an \( \mathbb{H}\left( L\right) \) semimartingale. | Proof. Let \( L \) take on the values \( {\alpha }_{1},{\alpha }_{2},{\alpha }_{3},\ldots \) . The distribution of \( L \) is given by \( \eta \left( {dx}\right) = \mathop{\sum }\limits_{{i = 1}}^{\infty }P\left( {L = {\alpha }_{i}}\right) {\varepsilon }_{{\alpha }_{i}}\left( {dx}\right) \), where \( {\varepsilon }_{{\... | Yes |
Corollary 3 (Jacod’s Countable Expansion). Let \( \mathcal{A} = \left( {{A}_{1},{A}_{2},\ldots }\right) \) be a sequence of events such that \( {A}_{i} \cap {A}_{j} = \varnothing, i \neq j \), all in \( \mathcal{F} \), and such that \( \mathop{\bigcup }\limits_{{i = 1}}^{\infty }{A}_{i} = \Omega \) . Let \( \mathbb{H} ... | Proof. Define \( L = \mathop{\sum }\limits_{{i = 1}}^{\infty }{2}^{-i}{1}_{{A}_{i}} \) . Then \( \mathbb{H} = \mathbb{H}\left( L\right) \) and we need only to apply the preceding corollary. | Yes |
Theorem 14. Let \( Y \) be a random variable with \( E\{ \left| Y\right| \} < \infty \) . A right continuous version of the martingale \( {Y}_{t} = E\left\{ {Y \mid {\mathcal{F}}_{t}^{L}}\right\} \) is given by the formula\n\n\[ \n{Y}_{t} = \frac{1}{{Z}_{t}}{}^{o}\left( {Y{1}_{\{ t < L\} }}\right) + Y{1}_{\{ t \geq L\}... | Proof. Let \( {\mathcal{O}}^{L} \) denote the optional \( \sigma \) -algebra on \( {\mathbb{R}}_{ + } \times \Omega \), corresponding to the filtration \( {\mathbb{F}}^{L} \) . On \( \lbrack 0, L),{\mathcal{O}}^{L} \) coincides with the trace of \( \mathcal{O} \) on \( \lbrack 0, L) \) . (By \( \mathcal{O} \) we mean o... | Yes |
Theorem 16. \( L \) is an honest time if and only if there exists an optional set \( \Lambda \subset \left\lbrack {0,\infty }\right\rbrack \times \Omega \) such that \( L\left( \omega \right) = \sup \{ t \leq \infty : \left( {t,\omega }\right) \in \Lambda \} \) . | Proof. The end of an optional set is always an honest random variable. Indeed, on \( \{ L \leq t\} \), the random variable \( L \) coincides with the end of the set \( \Lambda \cap (\left\lbrack {0, t}\right\rbrack \times \) \( \Omega ) \), which is \( {\mathcal{F}}_{t} \) measurable.\n\nFor the converse we suppose \( ... | Yes |
Theorem 17. Let \( L \) be an honest time. Define\n\n\[ \n{\mathcal{G}}_{t} = \left\{ {\Gamma : \Gamma = \left( {A\cap \{ L > t\} }\right) \cup \left( {B\cap \{ L \leq t\} }\right) \text{ for some }A, B \in {\mathcal{F}}_{t}}\right\} \]\n\nThen \( \mathbb{G} = {\left( {\mathcal{G}}_{t}\right) }_{t \geq 0} \) constitute... | Proof. Let \( s < t \) and take \( H \in {\mathcal{G}}_{s} \), of the form \( \left( {A\cap \{ L > s\} }\right) \cup \left( {B\cap \{ L \leq s\} }\right) \) with \( A, B \in {\mathcal{F}}_{s} \) . We will show that \( H \in {\mathcal{G}}_{t} \), which shows that the collection \( \mathbb{G} \) is filtering to the right... | No |
Theorem 18. Let \( X \) be a square integrable martingale for the \( \mathbb{F} \) filtration. Then \( X \) is a semimartingale for the \( \mathbb{G} \) filtration if \( L \) is an honest time. Moreover \( X \) has a \( \mathbb{G} \) decomposition\n\n\[ \n{X}_{t} = \left\{ {{X}_{t} - {\int }_{0}^{t \land L}\frac{1}{{Z}... | Proof of Theorem 18. We first observe that without loss of generality we can assume \( {X}_{0} = 0 \) . Let \( H \) be a bounded \( \mathbb{F} \) predictable process. We define stochastic integrals at the random time \( L \) by\n\n\[ \n{\int }_{0}^{L}{H}_{s}d{X}_{s} = {\left( H \cdot X\right) }_{L},\;{\int }_{L}^{\inft... | Yes |
Theorem 19. Let \( X \) be a semimartingale for the \( \mathbb{F} \) filtration. Then \( X \) is a semimartingale for the \( \mathbb{G} \) filtration if \( L \) is an honest time. | Proof. Let \( X = M + A \) be a decomposition of \( X \) in the \( \mathbb{F} \) filtration, where \( M \) is a local martingale and \( A \) is a finite variation process. Since \( A \) remains a finite variation process in the larger \( \mathbb{G} \) filtration, we need only concern ourselves with \( M \) . By the Fun... | Yes |
Let \( B \) be a standard Brownian motion and let \( X \) be the unique solution of the stochastic differential equation\n\n\[ \n{X}_{t} = {X}_{0} + {\int }_{0}^{t}\sigma \left( {X}_{s}\right) d{B}_{s} + {\int }_{0}^{t}b\left( {X}_{s}\right) {ds} \n\]\n\nfor \( 0 \leq t \leq 1 \), where \( \sigma \) and \( b \) are Lip... | Proof. We first note that \( B \) is an \( \left( {\mathbb{F},\widetilde{\mathbb{H}}}\right) \) reversible semimartingale as we saw in the example on page 367. We have that \( \left\lbrack {\sigma \left( X\right), B}\right\rbrack = {\int }_{0}^{t}{\sigma }^{\prime }\left( {X}_{s}\right) \sigma \left( {X}_{s}\right) {ds... | Yes |
Problem 4. The most celebrated example is the stochastic solution of the Dirichlet problem:\n\nGiven a (reasonable) domain \( U \) in \( {\mathbf{R}}^{n} \) and a continuous function \( f \) on\n\nthe boundary of \( U,\partial U \) . Find a function \( \widetilde{f} \) continuous on the closure \( \bar{U} \) of \( U \)... | In 1944 Kakutani proved that the solution could be expressed in terms of Brownian motion (which will be constructed in Chapter 2): \( \widetilde{f}\left( x\right) \) is the expected value of \( f \) at the first exit point from \( U \) of the Brownian motion starting at \( x \in U \) . | Yes |
Theorem 2.2.3 (Kolmogorov's continuity theorem). Suppose that the 连续鞅与Brownian运动 process \( X = {\left\{ {X}_{t}\right\} }_{t \geq 0} \) satisfies the following condition: For all \( T > 0 \) there exist positive constants \( \alpha ,\beta, D \) such that Pag Ag - 个特例 \[ E\left\lbrack {\left| {X}_{t} - {X}_{s}\right| }... | For a proof see for example Stroock and Varadhan (1979, p. 51). | No |
Example 3.1.1. Choose\n\n\\[ \n{\\phi }_{1}\\left( {t,\\omega }\\right) = \\mathop{\\sum }\\limits_{{j \\geq 0}}{B}_{j \\cdot {2}^{-n}}\\left( \\omega \\right) \\cdot {\\mathcal{X}}_{\\left\\lbrack j \\cdot {2}^{-n},\\left( j + 1\\right) {2}^{-n}\\right) }\\left( t\\right) \n\\]\n\n\\[ \n{\\phi }_{2}\\left( {t,\\omega ... | This only reflects the fact that the variations of the paths of \\( {B}_{t} \\) are too big to enable us to define the integral (3.1.6) in the Riemann-Stieltjes sense. In fact, one can show that the paths \\( t \\rightarrow {B}_{t} \\) of Brownian motion are nowhere differentiable, almost surely (a.s.). (See Breiman (1... | Yes |
Lemma 3.1.5 (The Itô isometry). If \( \phi \left( {t,\omega }\right) \) is bounded and elementary then | Proof. Put \( \Delta {B}_{j} = {B}_{{t}_{j + 1}} - {B}_{{t}_{j}} \) . Then\n\n\[ E\left\lbrack {{e}_{i}{e}_{j}\Delta {B}_{i}\Delta {B}_{j}}\right\rbrack = \left\{ \begin{matrix} 0 & \text{ if } & i \neq j \\ E\left\lbrack {e}_{j}^{2}\right\rbrack \cdot \left( {{t}_{j + 1} - {t}_{j}}\right) & \text{ if } & i = j \end{ma... | Yes |
Corollary 3.1.8. If \( f\left( {t,\omega }\right) \in \mathcal{V}\left( {S, T}\right) \) and \( {f}_{n}\left( {t,\omega }\right) \in \mathcal{V}\left( {S, T}\right) \) for \( n = 1,2,\ldots \) and \( E\left\lbrack {{\int }_{S}^{T}{\left( {f}_{n}\left( t,\omega \right) - f\left( t,\omega \right) \right) }^{2}{dt}}\right... | \[ {\int }_{S}^{T}{f}_{n}\left( {t,\omega }\right) d{B}_{t}\left( \omega \right) \rightarrow {\int }_{S}^{T}f\left( {t,\omega }\right) d{B}_{t}\left( \omega \right) \;\text{ in }{L}^{2}\left( P\right) \text{ as }n \rightarrow \infty . \] | Yes |
Assume \( {B}_{0} = 0 \) . Then\n\n\[ \n{\int }_{0}^{t}{B}_{s}d{B}_{s} = \frac{1}{2}{B}_{t}^{2} - \frac{1}{2}t \n\] | Proof. Put \( {\phi }_{n}\left( {s,\omega }\right) = \sum {B}_{j}\left( \omega \right) \cdot {\mathcal{X}}_{\left\lbrack {t}_{j},{t}_{j + 1}\right) }\left( s\right) \), where \( {B}_{j} = {B}_{{t}_{j}} \) . Then\n\n\[ \nE\left\lbrack {{\int }_{0}^{t}{\left( {\phi }_{n} - {B}_{s}\right) }^{2}{ds}}\right\rbrack = E\left\... | No |
Theorem 3.2.1. Let \( f, g \in \mathcal{V}\left( {0, T}\right) \) and let \( 0 \leq S < U < T \) . Then\n\n(i) \( {\int }_{S}^{T}{fd}{B}_{t} = {\int }_{S}^{U}{fd}{B}_{t} + {\int }_{U}^{T}{fd}{B}_{t} \) for a.a. \( \omega \)\n\n(ii) \( {\int }_{S}^{T}\left( {{cf} + g}\right) d{B}_{t} = c \cdot {\int }_{S}^{T}{fd}{B}_{t}... | Proof. This clearly holds for all elementary functions, so by taking limits we obtain this for all \( f, g \in \mathcal{V}\left( {0, T}\right) \) . | No |
Example 3.2.3. Brownian motion \( {B}_{t} \) in \( {\mathbf{R}}^{n} \) is a martingale w.r.t. the \( \sigma \) - algebras \( {\mathcal{F}}_{t} \) generated by \( \left\{ {{B}_{s};s \leq t}\right\} \) | \[ E{\left\lbrack \left| {B}_{t}\right| \right\rbrack }^{2} \leq E\left\lbrack {\left| {B}_{t}\right| }^{2}\right\rbrack = {\left| {B}_{0}\right| }^{2} + {nt}\;\text{ and if }s \geq t\text{ then } \]\n\[ E\left\lbrack {{B}_{s} \mid {\mathcal{F}}_{t}}\right\rbrack = E\left\lbrack {{B}_{s} - {B}_{t} + {B}_{t} \mid {\math... | Yes |
Corollary 3.2.6. Let \( f\left( {t,\omega }\right) \in \mathcal{V}\left( {0, T}\right) \) for all \( T \) . Then\n\n\[ \n{M}_{t}\left( \omega \right) = {\int }_{0}^{t}f\left( {s,\omega }\right) d{B}_{s} \]\n\nis a martingale w.r.t. \( {\mathcal{F}}_{t} \) and\n\n\[ \nP\left\lbrack {\mathop{\sup }\limits_{{0 \leq t \leq... | Proof. This follows from (3.2.2), the a.s. \( t \) -continuity of \( {M}_{t} \) and the martingale inequality (Theorem 3.2.4), combined with the Itô isometry (3.1.14). | No |
Theorem 4.1.2 (The 1-dimensional Itô formula).\n\nLet \( {X}_{t} \) be an Itô process given by\n\n\[ d{X}_{t} = {udt} + {vd}{B}_{t} \]\n\nLet \( g\left( {t, x}\right) \in {C}^{2}\left( {\lbrack 0,\infty }\right) \times \mathbf{R}) \) (i.e. \( g \) is twice continuously differentiable on \( \lbrack 0,\infty ) \times \ma... | where \( {\left( d{X}_{t}\right) }^{2} = \left( {d{X}_{t}}\right) \cdot \left( {d{X}_{t}}\right) \) is computed according to the rules\n\n\[ {dt} \cdot {dt} = {dt} \cdot d{B}_{t} = d{B}_{t} \cdot {dt} = 0,\;d{B}_{t} \cdot d{B}_{t} = {dt}. \] | Yes |
Let us return to the integral\n\n\[ I = {\int }_{0}^{t}{B}_{s}d{B}_{s}\;\text{ from Chapter }3. \] | Choose \( {X}_{t} = {B}_{t} \) and \( g\left( {t, x}\right) = \frac{1}{2}{x}^{2} \) . Then\n\n\[ {Y}_{t} = g\left( {t,{B}_{t}}\right) = \frac{1}{2}{B}_{t}^{2}. \]\n\nThen by Itô's formula,\n\n\[ d{Y}_{t} = \frac{\partial g}{\partial t}{dt} + \frac{\partial g}{\partial x}d{B}_{t} + \frac{1}{2}\frac{{\partial }^{2}g}{\pa... | Yes |
What is\n\n\[ \n{\int }_{0}^{t}{sd}{B}_{s}? \n\] | From classical calculus it seems reasonable that a term of the form \( t{B}_{t} \) should appear, so we put\n\n\[ \ng\left( {t, x}\right) = {tx} \n\]\n\nand\n\n\[ \n{Y}_{t} = g\left( {t,{B}_{t}}\right) = t{B}_{t} \n\]\n\nThen by Itô's formula,\n\n\[ \nd{Y}_{t} = {B}_{t}{dt} + {td}{B}_{t} + 0 \n\]\n\ni.e.\n\n\[ \nd\left... | Yes |
Theorem 4.2.1 (The general Itô formula).\n\nLet\n\n\[ \n{dX}\left( t\right) = {udt} + {vdB}\left( t\right) \]\n\nbe an \( n \) -dimensional Itô process as above. Let \( g\left( {t, x}\right) = \left( {{g}_{1}\left( {t, x}\right) ,\ldots ,{g}_{p}\left( {t, x}\right) }\right) \) be a \( {C}^{2} \) map from \( \lbrack 0,\... | The proof is similar to the 1-dimensional version (Theorem 4.1.2) and is omitted. | No |
Let \( B = \left( {{B}_{1},\ldots ,{B}_{n}}\right) \) be Brownian motion in \( {\mathbf{R}}^{n}, n \geq 2 \) , and consider\n\n\[ R\left( {t,\omega }\right) = \left| {B\left( {t,\omega }\right) }\right| = {\left( {B}_{1}^{2}\left( t,\omega \right) + \cdots + {B}_{n}^{2}\left( t,\omega \right) \right) }^{\frac{1}{2}}, \... | \[ {dR} = \mathop{\sum }\limits_{{i = 1}}^{n}\frac{{B}_{i}d{B}_{i}}{R} + \frac{n - 1}{2R}{dt}. \]\n\nThe process \( R \) is called the \( n \) -dimensional Bessel process because its generator (Chapter 7) is the Bessel differential operator \( {Af}\left( x\right) = \frac{1}{2}{f}^{\prime \prime }\left( x\right) + \frac... | No |
Lemma 4.3.1. Fix \( T > 0 \) . The set of random variables\n\n\[ \left\{ {\phi \left( {{B}_{{t}_{1}},\ldots ,{B}_{{t}_{n}}}\right) ;{t}_{i} \in \left\lbrack {0, T}\right\rbrack ,\phi \in {C}_{0}^{\infty }\left( {\mathbf{R}}^{n}\right), n = 1,2,\ldots }\right\} \]\n\nis dense in \( {L}^{2}\left( {{\mathcal{F}}_{T}, P}\r... | Proof. Let \( {\left\{ {t}_{i}\right\} }_{i = 1}^{\infty } \) be a dense subset of \( \left\lbrack {0, T}\right\rbrack \) and for each \( n = 1,2,\ldots \) let \( {\mathcal{H}}_{n} \) be the \( \sigma \) -algebra generated by \( {B}_{{t}_{1}}\left( \cdot \right) ,\ldots ,{B}_{{t}_{n}}\left( \cdot \right) \) . Then clea... | Yes |
Theorem 4.3.4 (The martingale representation theorem).\n\nLet \( B\left( t\right) = \left( {{B}_{1}\left( t\right) ,\ldots ,{B}_{n}\left( t\right) }\right) \) be n-dimensional. Suppose \( {M}_{t} \) is an \( {\mathcal{F}}_{t}^{\left( n\right) } \) - martingale (w.r.t. \( P \) ) and that \( {M}_{t} \in {L}^{2}\left( P\r... | Proof \( \left( {n = 1}\right) \) . By Theorem 4.3.3 applied to \( T = t, F = {M}_{t} \), we have that for all \( t \) there exists a unique \( {f}^{\left( t\right) }\left( {s,\omega }\right) \in {L}^{2}\left( {{\mathcal{F}}_{t}, P}\right) \) such that\n\n\[ \n{M}_{t}\left( \omega \right) = E\left\lbrack {M}_{t}\right\... | Yes |
Let us return to the population growth model in Chapter 1:\n\n\\[ \n\\frac{d{N}_{t}}{dt} = {a}_{t}{N}_{t},\\;{N}_{0}\\text{ given } \n\\]\n\nwhere \\( {a}_{t} = {r}_{t} + \\alpha {W}_{t},{W}_{t} = \\) white noise, \\( \\alpha = \\) constant.\n\nLet us assume that \\( {r}_{t} = r = \\) constant. By the Itô interpretatio... | To evaluate the integral on the left hand side we use the Itô formula for the function\n\n\\[ \ng\\left( {t, x}\\right) = \\ln x;\\;x > 0 \n\\]\n\nand obtain\n\n\\[ \nd\\left( {\\ln {N}_{t}}\\right) = \\frac{1}{{N}_{t}} \\cdot d{N}_{t} + \\frac{1}{2}\\left( {-\\frac{1}{{N}_{t}^{2}}}\\right) {\\left( d{N}_{t}\\right) }^... | Yes |
Theorem 5.1.2 (The law of iterated logarithm).\n\n\[ \lim \mathop{\sup }\limits_{{t \rightarrow \infty }}\frac{{B}_{t}}{\sqrt{{2t}\log \log t}} = 1\text{ a.s. } \] | For a proof we refer to Lamperti (1977), §22. | No |
Let us return to the equation in Problem 2 of Chapter 1:\n\n\[ L{Q}_{t}^{\prime \prime } + R{Q}_{t}^{\prime } + \frac{1}{C}{Q}_{t} = {F}_{t} = {G}_{t} + \alpha {W}_{t}. \] | We introduce the vector\n\n\[ X = X\left( {t,\omega }\right) = \left( \begin{array}{l} {X}_{1} \\ {X}_{2} \end{array}\right) = \left( \begin{array}{l} {Q}_{t} \\ {Q}_{t}^{\prime } \end{array}\right) \;\text{ and obtain }\n\n\[ \left\{ \begin{array}{l} {X}_{1}^{\prime } = {X}_{2} \\ L{X}_{2}^{\prime } = - R{X}_{2} - \fr... | Yes |
Choose \( X = B,1 \) -dimensional Brownian motion, and\n\n\[ g\left( {t, x}\right) = {e}^{ix} = \left( {\cos x,\sin x}\right) \in {\mathbf{R}}^{2}\;\text{ for }x \in \mathbf{R}. \]\n\nThen\n\n\[ Y = g\left( {t, X}\right) = {e}^{iB} = \left( {\cos B,\sin B}\right) \]\n\nis by Itô's formula again an Itô process.\n\nIts c... | \[ \left( {5.1.13}\right) \]\n\nOr, in matrix notation,\n\n\[ {dY} = - \frac{1}{2}{Ydt} + {KYdB},\;\text{ where }K = \left( \begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}\right) . \] | Yes |
Theorem 5.2.1. (Existence and uniqueness theorem for stochastic differential equations).\n\nLet \( T > 0 \) and \( b\left( {\cdot , \cdot }\right) : \left\lbrack {0, T}\right\rbrack \times {\mathbf{R}}^{n} \rightarrow {\mathbf{R}}^{n},\sigma \left( {\cdot , \cdot }\right) : \left\lbrack {0, T}\right\rbrack \times {\mat... | Proof of Theorem 5.2.1. The uniqueness follows from the Itô isometry (Corollary 3.1.7) and the Lipschitz property (5.2.2): Let \( {X}_{1}\left( {t,\omega }\right) = {X}_{t}\left( \omega \right) \) and \( {X}_{2}\left( {t,\omega }\right) = {\widehat{X}}_{t}\left( \omega \right) \) be solutions with initial values \( Z,\... | Yes |
Lemma 5.3.1. If \( b \) and \( \sigma \) satisfy the conditions of Theorem 5.2.1 then we have\n\nA solution (weak or strong) of (5.2.3) is weakly unique . | Sketch of proof. Let \( \left( {\left( {{\widetilde{X}}_{t},{\widetilde{B}}_{t}}\right) ,{\widetilde{\mathcal{H}}}_{t}}\right) \) and \( \left( {\left( {{\widehat{X}}_{t},{\widehat{B}}_{t}}\right) ,{\widehat{\mathcal{H}}}_{t}}\right) \) be two weak solutions. Let \( {X}_{t} \) and \( {Y}_{t} \) be the strong solutions ... | Yes |
Consider the 1-dimensional stochastic differential equation\n\n\\[ \nd{X}_{t} = \\operatorname{sign}\\left( {X}_{t}\\right) d{B}_{t};\\;{X}_{0} = 0.\n\\]\n\n\\( \\left( {5.3.1}\\right) \\)\n\nwhere\n\n\\[ \n\\operatorname{sign}\\left( x\\right) = \\left\\{ \\begin{array}{ll} + 1 & \\text{ if }x \\geq 0 \\ - 1 & \\text{... | Note that here \\( \\sigma \\left( {t, x}\\right) = \\sigma \\left( x\\right) = \\operatorname{sign}\\left( x\\right) \\) does not satisfy the Lipschitz condition (5.2.2), so Theorem 5.2.1 does not apply. Indeed, the equation (5.3.1) has no strong solution. To see this, let \\( {\\widehat{B}}_{t} \\) be a Brownian moti... | No |
Lemma 6.1.1. Let \( \mathcal{H} \subset \mathcal{F} \) be a \( \sigma \) -algebra and let \( X \in {L}^{2}\left( P\right) \) be \( \mathcal{F} \) - measurable. Put \( \mathcal{N} = \left\{ {Y \in {L}^{2}\left( P\right) ;Y}\right. \) is \( \mathcal{H} \) -measurable \( \} \) and let \( {\mathcal{P}}_{\mathcal{N}} \) den... | Proof. Recall (see Appendix B) that \( E\left\lbrack {X \mid \mathcal{H}}\right\rbrack \) is by definition the \( P \) -unique function from \( \Omega \) to \( \mathbf{R} \) such that (i) \( E\left\lbrack {X \mid \mathcal{H}}\right\rbrack \) is \( \mathcal{H} \) -measurable (ii) \( {\int }_{A}E\left\lbrack {X \mid \mat... | Yes |
What is the best linear estimate \( \widehat{X} \) of \( X \) based on \( \left\{ {{Z}_{j};j \leq k}\right\} \) ? More precisely, let\n\n\[ \mathcal{L} = \mathcal{L}\left( {Z, k}\right) = \left\{ {{c}_{1}{Z}_{1} + \cdots + {c}_{k}{Z}_{k};{c}_{1},\ldots ,{c}_{k} \in \mathbf{R}}\right\} .\n\]\n\nThen we want to find\n\n\... | We use the Gram-Schmidt procedure to obtain random variables \( {A}_{1},{A}_{2},\ldots \) such that\n\n(i) \( E\left\lbrack {{A}_{i}{A}_{j}}\right\rbrack = 0 \) for \( i \neq j \)\n\n(ii) \( \mathcal{L}\left( {A, k}\right) = \mathcal{L}\left( {Z, k}\right) \) for all \( k \) .\n\nThen\n\n\[ {\widehat{X}}_{k} = \mathop{... | Yes |
Lemma 6.2.2. Let \( X,{Z}_{s};s \leq t \) be random variables in \( {L}^{2}\left( P\right) \) and assume that\n\n\[ \left( {X,{Z}_{{s}_{1}},{Z}_{{s}_{2}},\ldots ,{Z}_{{s}_{n}}}\right) \in {\mathbf{R}}^{n + 1} \]\n\nhas a normal distribution for all \( {s}_{1},{s}_{2},\ldots ,{s}_{n} \leq t, n \geq 1 \) . Then\n\n\[ {\m... | Proof. Put \( \check{X} = {\mathcal{P}}_{\mathcal{L}}\left( X\right) ,\widetilde{X} = X - \check{X} \) . Then we claim that \( \widetilde{X} \) is independent of \( \mathcal{G} \) : Recall that a random variable \( \left( {{Y}_{1},\ldots ,{Y}_{k}}\right) \in {\mathbf{R}}^{k} \) is normal iff \( {c}_{1}{Y}_{1} + \cdots ... | Yes |
Lemma 6.2.3.\n\n\\[ \n{M}_{t} = \\left\\lbrack \\begin{matrix} {X}_{t} \\ {Z}_{t} \\end{matrix}\\right\\rbrack \\in {\\mathbf{R}}^{2}\\;\\text{ is a Gaussian process }.\n\\] | Proof. We may regard \\( {M}_{t} \\) as the solution of a 2-dimensional linear stochastic differential equation of the form\n\n\\[ \nd{M}_{t} = H\\left( t\\right) {M}_{t}{dt} + K\\left( t\\right) d{B}_{t},{M}_{0} = \\left\\lbrack \\begin{matrix} {X}_{0} \\ 0 \\end{matrix}\\right\\rbrack ;\n\\]\n\n\\( \\left( {6.2.9}\\r... | Yes |
Lemma 6.2.4. \( \mathcal{L}\left( {Z, T}\right) = \left\{ {{c}_{0} + {\int }_{0}^{T}f\left( t\right) d{Z}_{t};f \in {L}^{2}\left\lbrack {0, T}\right\rbrack ,{c}_{0} \in \mathbf{R}}\right\} \) . | Proof. Denote the right hand side by \( \mathcal{N}\left( {Z, T}\right) \) . It is enough to show that\n\na) \( \mathcal{N}\left( {Z, T}\right) \subset \mathcal{L}\left( {Z, T}\right) \)\n\nb) \( \mathcal{N}\left( {Z, T}\right) \) contains all linear combinations of the form\n\n\[ {c}_{0} + {c}_{1}{Z}_{{t}_{1}} + \cdot... | Yes |
Lemma 6.2.5. (i) \( {N}_{t} \) has orthogonal increments | Proof. (i): If \( s < t \) and \( Y \in \mathcal{L}\left( {Z, s}\right) \) we have\n\n\[ E\left\lbrack {\left( {{N}_{t} - {N}_{s}}\right) Y}\right\rbrack = E\left\lbrack {\left( {{\int }_{s}^{t}G\left( r\right) \left( {{X}_{r} - {\widehat{X}}_{r}}\right) {dr} + {\int }_{s}^{t}D\left( r\right) d{V}_{r}}\right) Y}\right\... | Yes |
Lemma 6.2.7.\n\n\[ \n{\widehat{X}}_{t} = E\left\lbrack {X}_{t}\right\rbrack + {\int }_{0}^{t}\frac{\partial }{\partial s}E\left\lbrack {{X}_{t}{R}_{s}}\right\rbrack d{R}_{s}. \n\] | Proof. From Lemma 6.2.4 we know that\n\n\[ \n{\widehat{X}}_{t} = {c}_{0}\left( t\right) + {\int }_{0}^{t}g\left( s\right) d{R}_{s}\;\text{ for some }g \in {L}^{2}\left\lbrack {0, t}\right\rbrack ,{c}_{0}\left( t\right) \in \mathbf{R}. \n\]\n\nTaking expectations we see that \( {c}_{0}\left( t\right) = E\left\lbrack {\w... | Yes |
The solution \( {\widehat{X}}_{t} = E\left\lbrack {{X}_{t} \mid {\mathcal{G}}_{t}}\right\rbrack \) of the 1-dimensional linear filtering problem\n\n(linear system) \( \;d{X}_{t} = F\left( t\right) {X}_{t}{dt} + C\left( t\right) d{U}_{t};F\left( t\right), C\left( t\right) \in \mathbf{R} \)\n\n\( \left( {6.2.3}\right) \)... | \[ d{\widehat{X}}_{t} = \left( {F\left( t\right) - \frac{{G}^{2}\left( t\right) S\left( t\right) }{{D}^{2}\left( t\right) }}\right) {\widehat{X}}_{t}{dt} + \frac{G\left( t\right) S\left( t\right) }{{D}^{2}\left( t\right) }d{Z}_{t};\;{\widehat{X}}_{0} = E\left\lbrack {X}_{0}\right\rbrack \]\n\n\( \left( {6.2.28}\right) ... | Yes |
Example 6.2.9 (Noisy observations of a constant process). Consider the simple case\n\n(system) \( \;d{X}_{t} = 0 \), i.e. \( {X}_{t} = {X}_{0};E\left\lbrack {X}_{0}\right\rbrack = 0, E\left\lbrack {X}_{0}^{2}\right\rbrack = {a}^{2} \)\n\n(observations) \( \;d{Z}_{t} = {X}_{t}{dt} + {md}{V}_{t};{Z}_{0} = 0 \)\n\n(corres... | First we solve the corresponding Riccati equation for\n\n\[ S\left( t\right) = E\left\lbrack {\left( {X}_{t} - {\widehat{X}}_{t}\right) }^{2}\right\rbrack \text{:} \]\n\n\[ \frac{dS}{dt} = - \frac{1}{{m}^{2}}{S}^{2},\;S\left( 0\right) = {a}^{2} \]\n\n i.e.\n\[ S\left( t\right) = \frac{{a}^{2}{m}^{2}}{{m}^{2} + {a}^{2}t... | Yes |
Example 6.2.10 (Noisy observations of a Brownian motion). If we modify the preceding example slightly, so that\n\n\\[ \n\\text{(system)}\\;d{X}_{t} = {cd}{U}_{t};E\\left\\lbrack {X}_{0}\\right\\rbrack = 0, E\\left\\lbrack {X}_{0}^{2}\\right\\rbrack = {a}^{2}, c\\text{constant} \n\\]\n\n(observations) \\( \\;d{Z}_{t} = ... | For simplicity let us put \\( a = 0, m = c = 1 \\) . Then\n\n\\[ \nS\\left( t\\right) = \\frac{\\exp \\left( {2t}\\right) - 1}{\\exp \\left( {2t}\\right) + 1} = \\tanh \\left( t\\right) . \n\\]\n\nThe equation for \\( {\\widehat{X}}_{t} \\) is\n\n\\[ \nd{\\widehat{X}}_{t} = - \\tanh \\left( t\\right) {\\widehat{X}}_{t}... | Yes |
Suppose we want to estimate the value of a (constant) parameter \( \theta \), based on observations \( {Z}_{t} \) satisfying the model\n\n\[ d{Z}_{t} = {\theta M}\left( t\right) {dt} + N\left( t\right) d{B}_{t}, \]\n\nwhere \( M\left( t\right), N\left( t\right) \) are known functions. | In this case the stochastic differential equation for \( \theta \) is of course\n\n\[ {d\theta } = 0, \]\n\nso the Riccati equation for \( S\left( t\right) = E\left\lbrack {\left( \theta - {\widehat{\theta }}_{t}\right) }^{2}\right\rbrack \) is\n\n\[ \frac{dS}{dt} = - {\left( \frac{M\left( t\right) S\left( t\right) }{N... | Yes |
Now consider the system\n\n\[ \nd{X}_{t} = F{X}_{t}{dt} + {Cd}{U}_{t};\;F, C\text{ constants } \neq 0 \n\]\n\nwith observations\n\n\[ \nd{Z}_{t} = G{X}_{t}{dt} + {Dd}{V}_{t};\;G, D\text{ constants } \neq 0. \n\] | The corresponding Riccati equation\n\n\[ \n{S}^{\prime } = {2FS} - \frac{{G}^{2}}{{D}^{2}}{S}^{2} + {C}^{2},\;S\left( 0\right) = {a}^{2} \n\]\n\nhas the solution\n\n\[ \nS\left( t\right) = \frac{{\alpha }_{1} - K{\alpha }_{2}\exp \left( \frac{\left( {{\alpha }_{2} - {\alpha }_{1}}\right) {G}^{2}t}{{D}^{2}}\right) }{1 -... | Yes |
Theorem 6.3.1 (The Multi-Dimensional Kalman-Bucy Filter).\n\nThe solution \( {\widehat{X}}_{t} = E\left\lbrack {{X}_{t} \mid {\mathcal{G}}_{t}}\right\rbrack \) of the multi-dimensional linear filtering problem\n\n(linear system)\n\[ d{X}_{t} = F\left( t\right) {X}_{t}{dt} + C\left( t\right) d{U}_{t} \]\n\[ F\left( t\ri... | satisfies the stochastic differential equation\n\[ d{\widehat{X}}_{t} = \left( {F - S{G}^{T}{\left( D{D}^{T}\right) }^{-1}G}\right) {\widehat{X}}_{t}{dt} + S{G}^{T}{\left( D{D}^{T}\right) }^{-1}d{Z}_{t};\;{\widehat{X}}_{0} = E\left\lbrack {X}_{0}\right\rbrack \]\n\n\( \left( {6.3.3}\right) \)\n\nwhere \( S\left( t\righ... | Yes |
Theorem 7.1.2 (The Markov property for Itô diffusions).\n\nLet \( f \) be a bounded Borel function from \( {\mathbf{R}}^{n} \) to \( \mathbf{R} \) . Then, for \( t, h \geq 0 \)\n\n\[ \n{E}^{x}{\left\lbrack f\left( {X}_{t + h}\right) \mid {\mathcal{F}}_{t}^{\left( m\right) }\right\rbrack }_{\left( \omega \right) } = {E}... | Proof. Since, for \( r \geq t \), \n\n\[ \n{X}_{r}\left( \omega \right) = {X}_{t}\left( \omega \right) + {\int }_{t}^{r}b\left( {X}_{u}\right) {du} + {\int }_{t}^{r}\sigma \left( {X}_{u}\right) d{B}_{u}, \n\] \n\nwe have by uniqueness \n\n\[ \n{X}_{r}\left( \omega \right) = {X}_{r}^{t,{X}_{t}}\left( \omega \right) \n\]... | Yes |
Let \( U \subset {\mathbf{R}}^{n} \) be open. Then the first exit time\n\n\[ \n{\tau }_{U} \mathrel{\text{:=}} \inf \left\{ {t > 0;{X}_{t} \notin U}\right\} \n\]\n\nis a stopping time w.r.t. \( \left\{ {\mathcal{M}}_{t}\right\} \) | since\n\n\[ \n\left\{ {\omega ;{\tau }_{U} \leq t}\right\} = \mathop{\bigcap }\limits_{m}\mathop{\bigcup }\limits_{\substack{{r \in \mathbf{Q}} \\ {r < t} }}\left\{ {\omega ;{X}_{r} \notin {K}_{m}}\right\} \in {\mathcal{M}}_{t} \n\]\n\nwhere \( \left\{ {K}_{m}\right\} \) is an increasing sequence of closed sets such th... | Yes |
Theorem 7.2.4 (The strong Markov property for Itô diffusions).\n\nLet \( f \) be a bounded Borel function on \( {\mathbf{R}}^{n},\tau \) a stopping time w.r.t. \( {\mathcal{F}}_{t}^{\left( m\right) } \) , \( \tau < \infty \) a.s. Then\n\n\[ \n{E}^{x}\left\lbrack {f\left( {X}_{\tau + h}\right) \mid {\mathcal{F}}_{\tau }... | Proof. We try to imitate the proof of the Markov property (Theorem 7.1.2). For a.a. \( \omega \) we have that \( {X}_{r}^{\tau, x}\left( \omega \right) \) satisfies\n\n\[ \n{X}_{\tau + h}^{\tau, x} = x + {\int }_{\tau }^{\tau + h}b\left( {X}_{u}^{\tau, x}\right) {du} + {\int }_{\tau }^{\tau + h}\sigma \left( {X}_{u}^{\... | Yes |
Lemma 7.3.2. Let \( {Y}_{t} = {Y}_{t}^{x} \) be an Itô process in \( {\mathbf{R}}^{n} \) of the form\n\n\[ \n{Y}_{t}^{x}\left( \omega \right) = x + {\int }_{0}^{t}u\left( {s,\omega }\right) {ds} + {\int }_{0}^{t}v\left( {s,\omega }\right) d{B}_{s}\left( \omega \right) \n\]\n\nwhere \( B \) is \( m \) -dimensional. Let ... | Proof. Put \( Z = f\left( Y\right) \) and apply Itô’s formula (To simplify the notation we suppress the index \( t \) and let \( {Y}_{1},\ldots ,{Y}_{n} \) and \( {B}_{1},\ldots ,{B}_{m} \) denote the coordinates of \( Y \) and \( B \), respectively)\n\n\[ \n{dZ} = \mathop{\sum }\limits_{i}\frac{\partial f}{\partial {x... | Yes |
Theorem 7.3.3. Let \( {X}_{t} \) be the Itô diffusion\n\n\[ d{X}_{t} = b\left( {X}_{t}\right) {dt} + \sigma \left( {X}_{t}\right) d{B}_{t}. \]\n\nIf \( f \in {C}_{0}^{2}\left( {\mathbf{R}}^{n}\right) \) then \( f \in {\mathcal{D}}_{A} \) and\n\n\[ {Af}\left( x\right) = \mathop{\sum }\limits_{i}{b}_{i}\left( x\right) \f... | Proof. This follows from Lemma 7.3.2 (with \( \tau = t \) ) and the definition of \( A \) . | No |
Let \( B \) denote 1-dimensional Brownian motion and let \( X = \left( \begin{array}{l} {X}_{1} \\ {X}_{2} \end{array}\right) \) be the solution of the stochastic differential equation\n\n\[ \left\{ \begin{array}{ll} d{X}_{1} = {dt}; & {X}_{1}\left( 0\right) = {t}_{0} \\ d{X}_{2} = {dB}; & {X}_{2}\left( 0\right) = {x}_... | From now on we will, unless otherwise stated, let \( A = {A}_{X} \) denote the generator of the Itô diffusion \( {X}_{t} \) . We let \( L = {L}_{X} \) denote the differential operator given by the right hand side of (7.3.3). From Theorem 7.3.3 we know that \( {A}_{X} \) and \( {L}_{X} \) coincide on \( {C}_{0}^{2}\left... | No |
Consider \( n \) -dimensional Brownian motion \( B = \left( {{B}_{1},\ldots ,{B}_{n}}\right) \) starting at \( a = \left( {{a}_{1},\ldots ,{a}_{n}}\right) \in {\mathbf{R}}^{n}\left( {n \geq 1}\right) \) and assume \( \left| a\right| < R \) . What is the expected value of the first exit time \( {\tau }_{K} \) of \( B \)... | Choose an integer \( k \) and apply Dynkin’s formula with \( X = B,\tau = {\sigma }_{k} = \) \( \min \left( {k,{\tau }_{K}}\right) \), and \( f \in {C}_{0}^{2} \) such that \( f\left( x\right) = {\left| x\right| }^{2} \) for \( \left| x\right| \leq R \) : \[ {E}^{a}\left\lbrack {f\left( {B}_{{\sigma }_{k}}\right) }\rig... | Yes |
Lemma 7.5.3. If \( x \) is not a trap for \( {X}_{t} \), then there exists an open set \( U \ni x \) such that\n\n\[ \n{E}^{x}\left\lbrack {\tau }_{U}\right\rbrack < \infty \n\] | Proof. See Lemma 5.5 p. 139 in Dynkin (1965 I). | No |
Theorem 7.5.4. Let \( f \in {C}^{2} \) . Then \( f \in {\mathcal{D}}_{\mathcal{A}} \) and\n\n\[ \mathcal{A}f = \mathop{\sum }\limits_{i}{b}_{i}\frac{\partial f}{\partial {x}_{i}} + \frac{1}{2}\mathop{\sum }\limits_{{i, j}}{\left( \sigma {\sigma }^{T}\right) }_{ij}\frac{{\partial }^{2}f}{\partial {x}_{i}\partial {x}_{j}... | Proof. As before we let \( L \) denote the operator defined by the right hand side of (7.5.2). If \( x \) is a trap for \( \left\{ {X}_{t}\right\} \) then \( \overline{\mathcal{A}}f\left( x\right) = 0 \) . Choose a bounded open set \( V \) such that \( x \in V \) . Modify \( f \) to \( {f}_{0} \) outside \( V \) such t... | Yes |
The characteristic operator of the process \( Y = \left( \begin{array}{l} {Y}_{1} \\ {Y}_{2} \end{array}\right) \) from Example 5.1.4 satisfying the stochastic differential equations (5.1.13), i.e.\n\n\[ \left\{ \begin{array}{l} d{Y}_{1} = - \frac{1}{2}{Y}_{1}{dt} - {Y}_{2}{dB} \\ d{Y}_{2} = - \frac{1}{2}{Y}_{2}{dt} + ... | This is because \( {dY} = - \frac{1}{2}{Ydt} + {KYdB} \), where\n\n\[ K = \left( \begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}\right) \]\n\nso that\n\n\[ {dY} = b\left( Y\right) {dt} + \sigma \left( Y\right) {dB} \]\n\nwith\n\n\[ b\left( {{y}_{1},{y}_{2}}\right) = \left( \begin{matrix} - \frac{1}{2}{y}_{1} \\ - \frac{1}... | Yes |
Example 7.5.6. Let \( D \) be an open subset of \( {\mathbf{R}}^{n} \) such that \( {\tau }_{D} < \infty \) a.s. \( {Q}^{x} \) for all \( x \) . Let \( \phi \) be a bounded, measurable function on \( \partial D \) and define\n\n\[ \n\widetilde{\phi }\left( x\right) = {E}^{x}\left\lbrack {\phi \left( {X}_{{\tau }_{D}}\r... | \[ \n{E}^{x}\left\lbrack {\widetilde{\phi }\left( {X}_{{\tau }_{U}}\right) }\right\rbrack = {E}^{x}\left\lbrack {{E}^{{X}_{{\tau }_{U}}}\left\lbrack {\phi \left( {X}_{{\tau }_{D}}\right) }\right\rbrack }\right\rbrack = {E}^{x}\left\lbrack {\phi \left( {X}_{{\tau }_{D}}\right) }\right\rbrack = \widetilde{\phi }\left( x\... | Yes |
Theorem 8.1.1 (Kolmogorov's backward equation). Let \( f \in {C}_{0}^{2}\left( {\mathbf{R}}^{n}\right) \). a) Define \[ u\left( {t, x}\right) = {E}^{x}\left\lbrack {f\left( {X}_{t}\right) }\right\rbrack \] \( \left( {8.1.2}\right) \) Then \( u\left( {t, \cdot }\right) \in {\mathcal{D}}_{A} \) for each \( t \) and \[ \f... | Proof. a) Let \( g\left( x\right) = u\left( {t, x}\right) \). Then since \( t \rightarrow u\left( {t, x}\right) \) is differentiable we have \[ \frac{{E}^{x}\left\lbrack {g\left( {X}_{r}\right) }\right\rbrack - g\left( x\right) }{r} = \frac{1}{r} \cdot {E}^{x}\left\lbrack {{E}^{{X}_{r}}\left\lbrack {f\left( {X}_{t}\rig... | Yes |
Lemma 8.1.3. \( {R}_{\alpha }g \) is a bounded continuous function. | Proof. Since \( {R}_{\alpha }g\left( x\right) = {\int }_{0}^{\infty }{e}^{-{\alpha t}}{E}^{x}\left\lbrack {g\left( {X}_{t}\right) }\right\rbrack {dt} \), we see that Lemma 8.1.3 is a direct consequence of the next result: | No |
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