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Theorem 5.47 There exists a two-parameter stochastic process, called local time of the process \( X \) ,\n\n\[ \nL = \left\{ {{L}_{t}\left( x\right) = {L}_{t}\left( {x,\omega }\right) : {\mathbb{R}}_{ \geq 0} \times \mathbb{R} \times \Omega \rightarrow {\mathbb{R}}_{ \geq 0}}\right\} \n\]\n\nwith the following properti... | For the proof of the theorem, we refer, for example, to Karatzas-Shreve \( \left\lbrack {201}\right\rbrack \) . | No |
Proposition 5.49 Formula (5.61) is equivalent to the Black-Scholes formula (2.108) with interest rate \( r = 0 \) . | Proof. For the sake of simplicity we consider only the at-the-money case \( {S}_{0} = K \) and we leave it to the reader as an exercise to verify the general case. If \( C \) is the Black-Scholes price, by (2.108) we have\n\n\[ C = {S}_{0}\Phi \left( {d}_{1}\right) - K{e}^{-{rT}}\Phi \left( {d}_{2}\right) \]\n\nwhere \... | No |
Theorem 6.10 (Weak maximum principle) Let \( u \in {C}^{1,2}\left( {Q}_{T}\right) \cap C\left( {{Q}_{T} \cup }\right. \) \( \left. {{\partial }_{p}{Q}_{T}}\right) \) such that \( {L}_{a}u \geq 0 \) on \( {Q}_{T} \) . If \( u \leq 0 \) on \( {\partial }_{p}{Q}_{T} \), then \( u \leq 0 \) on \( {Q}_{T} \) . | Proof. First of all, by Remark 2.57 it is not restrictive to assume \( {a}_{0} > 0 \) , since we may prove the thesis for \( v \) in (6.14) with \( \alpha < {a}_{0} \) and then use the fact that \( u \) and \( v \) have the same sign.\n\nBy contradiction, we assume that \( u\left( {{t}_{0},{x}_{0}}\right) > 0 \) at som... | Yes |
Corollary 6.11 (Comparison principle) Let \( u, v \in {C}^{1,2}\left( {Q}_{T}\right) \cap C({Q}_{T} \cup \) \( \left. {{\partial }_{p}{Q}_{T}}\right) \) such that \( {L}_{a}u \leq {L}_{a}v \) in \( {Q}_{T} \) and \( u \geq v \) in \( {\partial }_{p}{Q}_{T} \) . Then \( u \geq v \) in \( {Q}_{T} \) . In particular there... | Proof. It suffices to apply the maximum principle to the function \( v - u \) . | No |
Theorem 6.12 Let \( u \in {C}^{1,2}\left( {Q}_{T}\right) \cap C\left( {{Q}_{T} \cup {\partial }_{p}{Q}_{T}}\right) \) and let us set\n\n\[ \n{a}_{1} \mathrel{\text{:=}} \max \left\{ {0, - {a}_{0}}\right\} .\n\]\n\nThen\n\n\[ \n\mathop{\sup }\limits_{{Q}_{T}}\left| u\right| \leq {e}^{{a}_{1}T}\left( {\mathop{\sup }\limi... | Proof. We first suppose that \( {a}_{0} \geq 0 \) and also that \( u \) and \( {L}_{a}u \) are bounded in \( {\partial }_{p}{Q}_{T} \) and \( {Q}_{T} \) respectively, otherwise there is nothing to prove. We consider the function\n\n\[ \nw\left( {t, x}\right) = \mathop{\sup }\limits_{{{\partial }_{p}{Q}_{T}}}\left| u\ri... | Yes |
Lemma 6.14 Let \( u \in {C}^{1,2}\left( {\mathcal{S}}_{T}\right) \cap C\left( {\overline{\mathcal{S}}}_{T}\right) \) such that\n\n\[ \left\{ \begin{array}{ll} {L}_{a}u \leq 0, & \text{ in }{\mathcal{S}}_{T}, \\ u\left( {0, \cdot }\right) \geq 0, & \text{ on }{\mathbb{R}}^{N}, \end{array}\right. \]\n\nand\n\n\[ \mathop{... | Proof. By the same argument used in the proof of Theorem 6.10 and based on Remark 2.57, it is not restrictive to assume \( {a}_{0} \geq 0 \) . Then, for fixed \( \left( {{t}_{0},{x}_{0}}\right) \in \) \( {\mathcal{S}}_{T} \) and \( \varepsilon > 0 \), we have\n\n\[ \left\{ \begin{array}{ll} {L}_{a}\left( {u + \varepsil... | Yes |
Theorem 6.18 Under Hypotheses 6.1 and 6.3, let \( u \in {C}^{1,2}\left( {\mathcal{S}}_{T}\right) \cap C\left( {\overline{\mathcal{S}}}_{T}\right) \) such that\n\n\[ \left| {u\left( {t, x}\right) }\right| \leq C{e}^{C{\left| x\right| }^{2}},\;\left( {t, x}\right) \in {\mathcal{S}}_{T}, \]\n\nfor some constant \( C \) . ... | Proof. If \( {a}_{0} \geq 0 \), then, setting\n\n\[ {w}_{ \pm } = \mathop{\sup }\limits_{{\mathbb{R}}^{N}}\left| {u\left( {0, \cdot }\right) }\right| + t\mathop{\sup }\limits_{{\mathcal{S}}_{T}}\left| {{L}_{a}u}\right| \pm u,\;\text{ in }{\mathcal{S}}_{T}, \]\n\nwe have\n\n\[ \left\{ \begin{array}{ll} {L}_{a}{w}_{ \pm ... | Yes |
Theorem 6.21 Under Hypotheses 6.1,6.3 and 6.5, if \( u \in {C}^{1,2}\left( {\mathcal{S}}_{T}\right) \) is a non-negative function such that \( {L}_{a}u \leq 0 \), then\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}\Gamma \left( {t, x;s, y}\right) u\left( {s, y}\right) {dy} \leq u\left( {t, x}\right) \n\] \n\n(6.31) \n\nfor every \... | Proof. We consider a decreasing function \( h \in C\left( \mathbb{R}\right) \) such that \( h\left( r\right) = 0 \) for \( r \geq 2 \) and \( h\left( r\right) = 1 \) for \( r \leq 1 \) . For fixed \( s \in \rbrack 0, T\lbrack \), we set\n\n\[ \n{g}_{n}\left( {s, y}\right) = u\left( {s, y}\right) h\left( \frac{\left| y\... | Yes |
Corollary 6.22 Let Hypotheses 6.1, 6.3 and 6.5 hold and suppose that \( a = 0 \) . If \( u \in {C}^{1,2}\left( {\mathcal{S}}_{T}\right) \) is a function that is bounded from below such that \( {Lu} \leq 0 \) , then (6.31) holds, i.e.\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}\Gamma \left( {t, x;s, y}\right) u\left( {s, y}\righ... | Proof. Let \( {u}_{0} = \mathop{\inf }\limits_{{\mathcal{S}}_{T}}u \) . Then, since \( a = 0 \), we have \( L\left( {u - {u}_{0}}\right) = {Lu} \leq 0 \) and, by Theorem 6.21,\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}\Gamma \left( {t, x;s, y}\right) \left( {u\left( {s, y}\right) - {u}_{0}}\right) {dy} \leq u\left( {t, x}\righ... | Yes |
Proposition 7.3 A strategy \( \left( {\alpha ,\beta }\right) \) is self-financing if and only if\n\n\[ d{\widetilde{V}}_{t}^{\left( \alpha ,\beta \right) } = {\alpha }_{t}d{\widetilde{S}}_{t} \]\n\nholds, that is\n\n\[ {\widetilde{V}}_{t}^{\left( \alpha ,\beta \right) } = {V}_{0}^{\left( \alpha ,\beta \right) } + {\int... | Proof (of Proposition 7.3). Given a strategy \( \left( {\alpha ,\beta }\right) \), we obviously have\n\n\[ {\beta }_{t}{B}_{t} = {V}_{t}^{\left( \alpha ,\beta \right) } - {\alpha }_{t}{S}_{t} \]\n\n(7.8)\n\nFurthermore\n\n\[ d{\widetilde{S}}_{t} = - r{e}^{-{rt}}{S}_{t}{dt} + {e}^{-{rt}}d{S}_{t} \]\n\n(7.9)\n\n\[ = \lef... | Yes |
Theorem 7.8 Suppose that \( \\left( {\\alpha ,\\beta }\\right) \) is a Markovian strategy and set \( f\\left( {t,{S}_{t}}\\right) = \) \( {V}_{t}^{\\left( \\alpha ,\\beta \\right) } \) . The following two conditions are equivalent:\n\ni) \( \\left( {\\alpha ,\\beta }\\right) \) is self-financing;\n\nii) \( f \) is solu... | Proof (of Theorem 7.8). [i) \( \\Rightarrow {ii} \) )] By the self-financing condition and expression (7.2) of \( S \), we have that\n\n\[ \nd{V}_{t}^{\\left( \\alpha ,\\beta \\right) } = \\left( {{\\alpha }_{t}\\mu {S}_{t} + {\\beta }_{t}r{B}_{t}}\\right) {dt} + {\\alpha }_{t}\\sigma {S}_{t}d{W}_{t} \n\]\n\n(7.16)\n\n... | Yes |
Theorem 7.13 The Black-Scholes market model is complete and arbitrage-free, this meaning that every European derivative \( F\left( {S}_{T}\right) \), with \( F \) verifying Hypothesis 7.10, is replicable in a unique way. Indeed there exists a unique strategy \( h = \left( {{\alpha }_{t},{\beta }_{t}}\right) \in \mathca... | Proof. A strategy \( \left( {\alpha ,\beta }\right) \) replicates \( F\left( {S}_{T}\right) \) if and only if:\n\ni) \( \left( {\alpha ,\beta }\right) \) is Markovian and admissible, so there exists \( f \in {C}^{1,2}\left( \left\lbrack {0, T\left\lbrack {\times {\mathbb{R}}_{ > 0}}\right. }\right) \right. \) that is l... | Yes |
Corollary 7.15 (Black-Scholes Formula) Let us assume the Black-Scholes dynamics for the underlying asset\n\n\\[ \nd{S}_{t} = \mu {S}_{t}{dt} + \sigma {S}_{t}d{W}_{t} \n\\]\n\nand let us denote by \\( r \\) the short rate. Then, if \\( K \\) is the strike price and \\( T \\) is the maturity, the following formulas for t... | Proof. The claim follows directly from the representation formula for the solution of the Cauchy problem (7.30)-(7.31) for the Black-Scholes equation (or for the heat equation, by transformation (7.22)). We are not going through the explicit computations, already carried out in Section 2.3.5. | No |
Proposition 7.17 (No-arbitrage principle) The family \( \mathcal{A} \) does not contain arbitrage strategies. | Proof. The claim follows directly from Corollary 6.22. By contradiction, let \( \left( {\alpha ,\beta }\right) \in \mathcal{A} \), with \( {V}_{t}^{\left( \alpha ,\beta \right) } = f\left( {t,{S}_{t}}\right) \), be an arbitrage strategy: then \( f \) is lower bounded, it is a solution of the PDE (7.30) and we have that... | Yes |
Theorem 7.19 The portfolio given by (7.43) is self-financing if and only if \( g \) is solution of the differential equation\n\n\[ \frac{{\sigma }^{2}{s}^{2}}{2}{\partial }_{ss}g\left( {t, s}\right) + \left( {\mu - \sigma {\lambda }_{f}\left( {t, s}\right) }\right) s{\partial }_{s}g\left( {t, s}\right) + {\partial }_{t... | The value \( {\left( g\left( t,{S}_{t}\right) \right) }_{t < T} \) is the arbitrage price of \( G\left( {S}_{T}\right) \) and the replicating strategy is given by (7.45). | Yes |
Lemma 7.21 We have\n\n\[ \n{g}^{\prime }\left( {d}_{1}\right) = 0 \n\]\n\n(7.53)\n\nand consequently\n\n\[ \n{S}_{t}{\Phi }^{\prime }\left( {d}_{1}\right) = K{e}^{-r\left( {T - t}\right) }{\Phi }^{\prime }\left( {{d}_{1} - \sigma \sqrt{T - t}}\right) . \n\]\n\n(7.54) | Proof. It is enough to observe that\n\n\[ \n{\Phi }^{\prime }\left( x\right) = \frac{{e}^{-\frac{{x}^{2}}{2}}}{\sqrt{2\pi }}. \n\]\n\nThen\n\n\[ \n{g}^{\prime }\left( d\right) = {S}_{t}\frac{{e}^{-\frac{{d}^{2}}{2}}}{\sqrt{2\pi }} - K{e}^{-r\left( {T - t}\right) }\frac{{e}^{-\frac{{\left( d - \sigma \sqrt{T - t}\right)... | Yes |
Theorem 7.22 The following conditions are equivalent:\n\ni) \( {\left( {\alpha }_{t},{\beta }_{t}\right) }_{t \in \left\lbrack {0, T}\right\rbrack } \) is self-financing, i.e. we have\n\n\[ \n{df}\left( {t,{S}_{t},{A}_{t}}\right) = {\alpha }_{t}d{S}_{t} + {\beta }_{t}d{B}_{t} \n\] \n\nii) \( f \) is a solution of the p... | The proof of Theorem 7.22 is formally analogous to the ones of Theorems 7.8 and 7.13. Let us observe that equation (7.68) cannot be transformed into a parabolic equation by a change of variables as in the European case. In particular the results of existence and uniqueness for the Cauchy problem of Appendix A. 3 and Se... | Yes |
Example 8.12 Without any further regularity assumption on the initial datum \( \varphi \), we have \( {\partial }_{{x}_{i}}u\left( {t, x}\right) = O\left( \frac{1}{\sqrt{t}}\right) \) as \( t \rightarrow {0}^{ + } \), consistently with estimate (8.17). Indeed, for the 2-dimensional heat equation and initial datum \( \v... | \[ {\partial }_{x}u\left( {0, t}\right) = \frac{1}{\sqrt{2\pi t}}{\int }_{-\infty }^{0}\frac{y}{t}\exp \left( {-\frac{{y}^{2}}{2t}}\right) {dy} = \] (by the change of variable \( z = - \frac{{y}^{2}}{2t} \) ) \[ = - \frac{1}{\sqrt{2\pi t}}{\int }_{-\infty }^{0}{e}^{z}{dz} = - \frac{1}{\sqrt{2\pi t}}. \] | Yes |
Theorem 8.27 Under the Hypotheses 8.1, 8.3 and 8.26 there exists a strong solution \( u \) to the problem (8.24). Moreover, for every \( p \geq 1 \) and \( O \) such that \( \bar{O} \subseteq B\left( T\right) \), there exists a positive constant \( c \), depending only on \( L, O, B\left( T\right), p \) and on the \( {... | We prove Theorem 8.27 by using a classical penalization technique. Let us consider a family \( {\left( {\beta }_{\varepsilon }\right) }_{\varepsilon \in \rbrack 0,1\lbrack } \) of functions in \( {C}^{\infty }\left( \mathbb{R}\right) \) : for every \( \varepsilon > 0,{\beta }_{\varepsilon } \) is a bounded, increasing ... | No |
Lemma 8.29 There exists a barrier function for \( L \) at any point \( \left( {t, x}\right) \in \) \( {\partial }_{P}B\left( T\right) \) . | Proof. If the point belongs to the basis of the cylinder \( B\left( T\right) \), i.e. it is of the form \( \left( {0,\bar{x}}\right) \), then a barrier function is given by\n\n\[ w\left( {t, x}\right) = {e}^{t\parallel a{\parallel }_{\infty }}\left( {{\left| x - \bar{x}\right| }^{2} + {Ct}}\right) ,\]\n\nwith \( C \) a... | Yes |
Proposition 8.31 Let \( u \) be a strong solution to the problem (8.24) and \( v \) a super-solution, i.e. \( v \in {S}_{\text{loc }}^{1}\left( {B\left( T\right) }\right) \cap C\left( \overline{B\left( T\right) }\right) \) . If\n\n\[ \left\{ \begin{array}{l} \max \{ {Lv},\varphi - v\} \leq 0,\;\text{ a.e. in }B\left( T... | Proof. By contradiction, we suppose that the open set defined by\n\n\[ D \mathrel{\text{:=}} \{ z \in B\left( T\right) \mid u\left( z\right) > v\left( z\right) \} \]\n\nis not empty. Then, since \( u > v \geq \varphi \) in \( D \), we have that\n\n\[ {Lu} = 0,\;{Lv} \leq 0\;\text{ in }D, \]\n\nand \( u = v \) on \( \pa... | Yes |
Lemma 9.5 (Gronwall’s lemma) Let \( \varphi \in C\left( \left\lbrack {0, T}\right\rbrack \right) \) be such that\n\n\[ \varphi \left( t\right) \leq a + {\int }_{0}^{t}f\left( s\right) \varphi \left( s\right) {ds},\;t \in \left\lbrack {0, T}\right\rbrack ,\]\n\nwhere \( a \in \mathbb{R} \) and \( f \) is a continuous, n... | Proof. We put\n\n\[ F\left( t\right) = a + {\int }_{0}^{t}f\left( s\right) \varphi \left( s\right) {ds}. \]\n\nBy assumption, \( \varphi \leq F \) and since \( f \) is non-negative we have\n\n\[ \frac{d}{dt}\left( {{e}^{-{\int }_{0}^{t}f\left( s\right) {ds}}F\left( t\right) }\right) = {e}^{-{\int }_{0}^{t}f\left( s\rig... | Yes |
Lemma 9.9 For all \( n \in \mathbb{N} \) and \( {a}_{1},\ldots ,{a}_{n} \in \mathbb{R} \) we have\n\n\[ \n{\left( {a}_{1} + \cdots + {a}_{n}\right) }^{2} \leq n\left( {{a}_{1}^{2} + \cdots + {a}_{n}^{2}}\right) .\n\] | Proof. We have\n\n\[ \n{\left( {a}_{1} + \cdots + {a}_{n}\right) }^{2} = {a}_{1}^{2} + \cdots + {a}_{n}^{2} + 2\mathop{\sum }\limits_{{i < j}}{a}_{i}{a}_{j} \n\]\n\n\[ \n\leq {a}_{1}^{2} + \cdots + {a}_{n}^{2} + \mathop{\sum }\limits_{{i < j}}\left( {{a}_{i}^{2} + {a}_{j}^{2}}\right) \n\]\n\n\[ \n= n\left( {{a}_{1}^{2}... | Yes |
Lemma 9.10 Under the standard hypotheses i) and iii) of Definition 9.4, the functional \( \Psi \) in (9.8) is well defined from \( {\mathcal{A}}_{\mathrm{c}} \) to \( {\mathcal{A}}_{\mathrm{c}} \) . Further, there exists a constant \( {C}_{1} \) depending on \( T \) and \( K \) only, such that\n\n\[ \llbracket \Psi \le... | Proof. By the assumption of linear growth on the coefficients, we have\n\n\[ E\left\lbrack {\mathop{\sup }\limits_{{0 \leq s \leq t}}{\left| b\left( s,{X}_{s}\right) \right| }^{2}}\right\rbrack + E\left\lbrack {\mathop{\sup }\limits_{{0 \leq s \leq t}}{\left| \sigma \left( s,{X}_{s}\right) \right| }^{2}}\right\rbrack \... | Yes |
Theorem 9.14 Let \( X \) be solution of the SDE\n\n\[ \n{X}_{t} = {X}_{0} + {\int }_{0}^{t}b\left( {s,{X}_{s}}\right) {ds} + {\int }_{0}^{t}\sigma \left( {s,{X}_{s}}\right) d{W}_{s},\;t \in \left\lbrack {0, T}\right\rbrack .\n\]\n\nIf the standard hypotheses of Definition 9.4 hold and \( E\left\lbrack {\left| {X}_{0}\r... | Proof. We prove the claim in the case \( p = 1, N = 1 \) and \( {t}_{0} = 0 \) . The case \( p > 1 \) is analogous and can be proved by using the fact that \( {X}^{2p} \) is a solution of the SDE\n\n\[ \n{X}_{t}^{2p} = {X}_{0}^{2p} + {\int }_{0}^{t}\left( {{2p}{X}_{s}^{{2p} - 1}b\left( {s,{X}_{s}}\right) + p\left( {{2p... | No |
Theorem 9.16 Consider \( {\mathcal{L}}_{{t}_{0}, t} \) in (9.20) for \( 0 \leq {t}_{0} < t \leq T \) and assume that the coefficients of the SDE are Lipschitz continuous in \( x \) uniformly with respect to \( t \), that is\n\n\[ \n{\left| b\left( t, x\right) - b\left( t, y\right) \right| }^{2} + {\left| \sigma \left( ... | Proof. We only consider the case \( p = 1 \) and \( {t}_{0} = 0 \) . Using Lemma 9.9 we get\n\n\[ \n\llbracket X - Y{\rrbracket }_{t}^{2} \leq 4\left( {E\left\lbrack {\left( {X}_{0} - {Y}_{0}\right) }^{2}\right\rbrack + t{\int }_{0}^{t}\llbracket b\left( {\cdot, X}\right) - b\left( {\cdot, Y}\right) {\rrbracket }_{s}^{... | Yes |
Proposition 9.24 Let \( P, Q \) be solutions of the martingale problem associated to \( {\mathcal{A}}_{t} \) with initial datum \( {x}_{0} \in {\mathbb{R}}^{N} \), i.e. such that\n\n\[ P\left( {w\left( 0\right) = {x}_{0}}\right) = Q\left( {w\left( 0\right) = {x}_{0}}\right) = 1. \]\n\nSuppose that for every \( T > 0 \)... | Proof. By Theorem 9.22, the process \( {\mathbb{X}}_{t}\left( w\right) = w\left( t\right) \) is solution to the SDE (9.25) on some extension of the space of continuous functions endowed with the probability measure \( P \) and the same result holds for \( Q \) . It follows that, if \( u \) is a solution of the problem ... | Yes |
Theorem 9.27 Consider a SDE with measurable and bounded coefficients \( b \) and \( \sigma \) . As usual, we denote by \( {\mathcal{A}}_{t} \) the related differential operator defined in (9.26). If, for all \( T > 0 \) and for all \( \varphi \in {C}_{b}\left( {\mathbb{R}}^{N}\right) \), there exists a bounded classica... | Sufficient conditions for the solvability of problem (9.29), as requested in Theorem 9.27, were given in Chapter 8. If the coefficients \( {c}_{ij},{b}_{j} \) are Hölder continuous bounded functions and the matrix \( \left( {c}_{ij}\right) \) is uniformly positive definite, then the operator \( {\mathcal{A}}_{t} + {\pa... | Yes |
Theorem 9.28 (Maximal martingale inequalities) Let \( X \) be a right-continuous super-martingale. For every \( \lambda > 0 \) we have\n\n\[ P\left( {\mathop{\sup }\limits_{{0 \leq t \leq T}}{X}_{t} \geq \lambda }\right) \leq \frac{E\left\lbrack {X}_{0}\right\rbrack + E\left\lbrack {X}_{T}^{ - }\right\rbrack }{\lambda ... | Proof. We use the notation\n\n\[ {\widehat{X}}_{t} = \mathop{\sup }\limits_{{0 \leq s \leq t}}{X}_{s} \]\n\nand, for fixed \( \lambda > 0 \), we put\n\n\[ \tau \left( \omega \right) = \inf \left\{ {t \geq 0 \mid {X}_{t}\left( \omega \right) \geq \lambda }\right\} \land T,\;\omega \in \Omega . \]\n\nThen \( \tau \) is a... | Yes |
Corollary 9.29 (Exponential inequality) Let \( W \) be a real Brownian motion and \( \sigma \in {\mathbb{L}}^{2} \) such that\n\n\[{\int }_{0}^{T}{\sigma }_{s}^{2}{ds} \leq k\;\text{ a.s. }\n\]\nfor a constant \( k \) . Then, if we put\n\n\[{X}_{t} = {\int }_{0}^{t}{\sigma }_{s}d{W}_{s}\]\n\nfor every \( \lambda > 0 \)... | Proof. We consider the quadratic variation process\n\n\[ \langle X{\rangle }_{t} = {\int }_{0}^{t}{\sigma }_{s}^{2}{ds} \]\n\nand we recall that \( {}^{6} \)\n\n\[ {Z}_{t}^{\left( \alpha \right) } = \exp \left( {\alpha {X}_{t} - \frac{{\alpha }^{2}}{2}\langle X{\rangle }_{t}}\right) \]\n\nis a continuous super-martinga... | Yes |
Corollary 9.31 Let \( W \) be a d-dimensional Brownian motion and \( \sigma \in {\mathbb{L}}^{2} \) an \( \left( {N \times d}\right) \) -matrix such that \( {}^{7} \n\n\[ \n{\int }_{0}^{T}\left| {{\sigma }_{s}{\sigma }_{s}^{ * }}\right| {ds} \leq k \n\]\n\nfor a constant \( k \) . Then, if we put\n\n\[ \n{X}_{t} = {\in... | Proof. Let us notice that, if\n\n\[ \n\mathop{\sup }\limits_{{0 \leq t \leq T}}\left| {{X}_{t}\left( \omega \right) }\right| \geq \lambda \n\]\n\nthen\n\n\[ \n\mathop{\sup }\limits_{{0 \leq t \leq T}}\left| {{X}_{t}^{i}\left( \omega \right) }\right| \geq \frac{\lambda }{\sqrt{N}} \n\]\n\n\( {}^{7} \) We recall that, if... | Yes |
Theorem 9.32 Let us consider the SDE in \( {\mathbb{R}}^{N} \)\n\n\[ \n{X}_{t} = {x}_{0} + {\int }_{0}^{t}b\left( {s,{X}_{s}}\right) {ds} + {\int }_{0}^{t}\sigma \left( {s,{X}_{s}}\right) d{W}_{s}.\n\]\n\n(9.37)\n\nWe suppose that \( \sigma \) is a bounded and measurable \( \left( {N \times d}\right) \) -matrix: in par... | Proof. By Proposition A. 56 we have\n\n\[ \nE\left\lbrack {e}^{\alpha {\bar{X}}_{T}^{2}}\right\rbrack = 1 + {\int }_{0}^{+\infty }{2\alpha \lambda }{e}^{\alpha {\lambda }^{2}}P\left( {{\bar{X}}_{T} \geq \lambda }\right) {d\lambda }, \n\]\n\nso it is enough to have a suitable estimate of \( P\left( {{\bar{X}}_{T} \geq \... | Yes |
Theorem 9.33 Suppose that the coefficients of the SDE (9.37) are measurable and satisfy the estimate (9.6) of linear growth. Then if \( X \) is a solution of (9.37), for every \( p \geq 1 \) we have\n\n\[ E\left\lbrack {\mathop{\sup }\limits_{{0 \leq t \leq T}}{\left| {X}_{t}\right| }^{p}}\right\rbrack < \infty . \]\n\... | Proof. We resort to a trick to go back to the case of a SDE with bounded coefficients. We consider the function \( f\left( x\right) = \log \left( {1 + {\left| x\right| }^{2}}\right) \) and compute the derivatives of first and second order:\n\n\[ {\partial }_{{x}_{i}}f\left( x\right) = \frac{2{x}_{i}}{1 + {\left| x\righ... | Yes |
Consider the radially symmetric function defined on \( {\mathbb{R}}^{d} \smallsetminus \{ 0\} \)\n\n\[ u\left( x\right) = \left\{ \begin{array}{ll} \log \left| x\right| & \text{ for }d = 2 \\ {\left| x\right| }^{-d + 2} & \text{ for }d \geq 3 \end{array}\right. \]\n\nwhich is a harmonic function, that is a solution of ... | The function \( u \) is usually called fundamental solution of the Laplace equation since it plays a role analogous to the Gaussian function for the heat equation. For a given a \( d \) -dimensional Brownian motion \( W \) and \( {x}_{0} \in {\mathbb{R}}^{d} \smallsetminus \{ 0\} \), we set \( B = W + {x}_{0} \) and de... | Yes |
Theorem 9.37 Let \( f \in {L}^{\infty }\left( D\right) ,\varphi \in C\left( {\partial D}\right) \) and \( a \in C\left( D\right) \) such that \( a \geq 0 \) . If \( u \in {C}^{2}\left( D\right) \cap C\left( \bar{D}\right) \) is solution to the Dirichlet problem (9.53) then, for fixed \( x \in D \) and writing for the s... | Proof. For \( \varepsilon > 0 \) small enough, let \( {D}_{\varepsilon } \) be a domain such that\n\n\[ x \in {D}_{\varepsilon },\;{\bar{D}}_{\varepsilon } \subseteq D,\;\operatorname{dist}\left( {\partial {D}_{\varepsilon },\partial D}\right) \leq \varepsilon . \]\n\nWe denote the exit time of \( {X}^{x} \) from \( {D... | Yes |
If \( a = f = 0,\left( {9.54}\right) \) can be rewritten in terms of a mean value formula. More precisely, we denote the distribution of the random variable \( {X}_{{\tau }_{x}}^{x} \) by \( {\mu }^{x} \) : then \( {\mu }^{x} \) is a probability measure on \( \partial D \) and by (9.54) we have\n\n\[ u\left( x\right) =... | The law \( {\mu }^{x} \) is usually called harmonic measure of \( \mathcal{A} \) over \( \partial D \) . In particular, let us consider the case of a Brownian motion \( {X}^{x} \) with initial point \( x \in {\mathbb{R}}^{N} \) : then \( \mathcal{A} = \frac{1}{2}\Delta \) and if \( D = B\left( {0, R}\right) \) is the E... | Yes |
The process \( {X}_{t} = \left( {{W}_{t}, - t}\right) \), where \( W \) is a real Brownian motion, is solution of the SDE\n\n\[ \left\{ \begin{array}{l} d{X}_{t}^{1} = d{W}_{t} \\ d{X}_{t}^{2} = - {dt} \end{array}\right. \] | and the corresponding characteristic operator\n\n\[ \mathcal{A} = \frac{1}{2}{\partial }_{{x}_{1}{x}_{1}} - {\partial }_{{x}_{2}} \]\n\nis the heat operator in \( {\mathbb{R}}^{2} \). | Yes |
If \( \sigma = 0 \), the characteristic operator is a first-order differential operator | The corresponding SDE is actually deterministic and becomes\n\n\[ {X}_{t}^{x} = x + {\int }_{0}^{t}b\left( {X}_{s}^{x}\right) {ds} \]\n\ni.e. \( X \) is an integral curve of the vector field \( b \) :\n\n\[ \frac{d}{dt}{X}_{t} = b\left( {X}_{t}\right) \]\n\nNote that\n\n\[ \frac{d}{dt}u\left( {X}_{t}\right) = \left\lan... | Yes |
Theorem 9.44 Let \( f \in {L}^{\infty }\left( Q\right) ,\varphi \in C\left( {{\partial }_{p}Q}\right) \) and \( a \in C\left( Q\right) \) such that\n\n\[ \n{a}_{0} \mathrel{\text{:=}} \inf a \]\n\nis finite. If \( u \in {C}^{2}\left( Q\right) \cap C\left( \bar{Q}\right) \) is a solution of the problem (9.57) then, for ... | Proof. The proof is analogous to that of Theorem 9.37. | No |
Theorem 9.45 (Feynman-Kač formula) Let \( u \in {C}^{2}\left( {\mathcal{S}}_{T}\right) \cap C\left( {\overline{\mathcal{S}}}_{T}\right) \) be a solution of the Cauchy problem (9.58) where \( a \in C\left( {\mathcal{S}}_{T}\right) \) is such that \( {a}_{0} = \) \( \inf a > - \infty \) . Assume that \( i \) ), ii) and a... | Proof. If \( {\tau }_{R} \) denotes the exit time of \( X \) from the Euclidean ball with radius \( R \), by Theorem 9.44 we have\n\n\[ u\left( {t, x}\right) = E\left\lbrack {{e}^{-{\int }_{t}^{T \land {\tau }_{R}}a\left( {s,{X}_{s}}\right) {ds}}u\left( {T \land {\tau }_{R},{X}_{T \land {\tau }_{R}}}\right) }\right\rbr... | Yes |
Theorem 9.47 (Itô formula) If \( f = f\left( {t, x}\right) \in {S}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{N}}\right) \) and \( {\left( \nabla f\right) }^{2} \in \) \( {L}^{q}\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{N}}\right) \) with \( p, q > 1 + \frac{N}{2} \), then we h... | \[ f\left( {t,{X}_{t}}\right) = f\left( {0,{X}_{0}}\right) + {\int }_{0}^{t}{Lf}\left( {s,{X}_{s}}\right) {ds} + {\int }_{0}^{t}\nabla f\left( {s,{X}_{s}}\right) \cdot \sigma \left( {s,{X}_{s}}\right) d{W}_{s}. \] | No |
Proposition 9.49 Under the assumptions of Theorem 9.48, suppose that the function \( \varphi \) and the coefficients of the SDE (9.64) are Lipschitz continuous in \( x \) uniformly with respect to \( t \) over \( {\mathcal{S}}_{T} \) . Further, let the coefficient a be a constant or \( \varphi \) be bounded. Then the s... | Proof. Let us first consider the case \( a \) is a constant. The claim follows from the general inequality\n\n\[ \left| {\mathop{\sup }\limits_{\tau }F\left( \tau \right) - \mathop{\sup }\limits_{\tau }G\left( \tau \right) }\right| \leq \mathop{\sup }\limits_{\tau }\left| {F\left( \tau \right) - G\left( \tau \right) }\... | Yes |
If \( N = d, B = 0 \) and \( b,\sigma \) are constant with \( \sigma \) non degenerate, \( L \) in (9.75) is the parabolic operator with constant coefficients\n\n\[ L = \frac{1}{2}\mathop{\sum }\limits_{{i, j = 1}}^{N}{\left( \sigma {\sigma }^{ * }\right) }_{ij}{\partial }_{{x}_{i}{x}_{j}} + \mathop{\sum }\limits_{{i =... | Then, since \( {e}^{tB} \) is the identity matrix, by (9.78)-(9.79) we have\n\n\[ {m}_{x}\left( t\right) = x + {tb},\;\mathcal{C}\left( t\right) = {t\sigma }{\sigma }^{ * }, \]\n\nand by (9.80) the fundamental solution of \( L \) is given by (see also Appendix A.3.2)\n\n\[ \Gamma \left( {t, x;T, y}\right) = \frac{{\lef... | Yes |
Example 9.53 The SDE in \( {\mathbb{R}}^{2} \)\n\n\[ \left\{ \begin{array}{l} d{X}_{t}^{1} = d{W}_{t} \\ d{X}_{t}^{2} = {X}_{t}^{1}{dt} \end{array}\right. \]\n\nis the simplified version of the Langevin equation [231] that describes the motion of a particle in the phase space: \( {X}_{t}^{1} \) and \( {X}_{t}^{2} \) re... | Since \( {B}^{2} = 0 \), the matrix \( B \) is nilpotent and\n\n\[ {e}^{tB} = \left( \begin{array}{ll} 1 & 0 \\ t & 1 \end{array}\right) \]\n\nMoreover, if we put \( x = \left( {{x}_{1},{x}_{2}}\right) \), using notation (9.78)-(9.79) we have\n\n\[ {m}_{x}\left( t\right) = {e}^{tB}x = \left( {{x}_{1},{x}_{2} + t{x}_{1}... | Yes |
Let \( B \) and \( \sigma \) be as in Example 9.53: then \( v \) has real values and problem (9.88) becomes\n\n\[ \left\{ \begin{array}{l} {\gamma }_{1}^{\prime }\left( t\right) = v\left( t\right) \\ {\gamma }_{2}^{\prime }\left( t\right) = {\gamma }_{1}\left( t\right) \\ \gamma \left( 0\right) = x \end{array}\right. \... | Proof (of Theorem 9.55). We recall that by (9.79) we have\n\n\[ \mathcal{C}\left( T\right) = {e}^{TB}M\left( T\right) {e}^{T{B}^{ * }}, \]\n\nwith \( M \) as in (9.86). Since the exponential matrices are non-degenerate, \( \mathcal{C}\left( T\right) \) is positive definite if and only if \( M\left( T\right) \) is as su... | Yes |
Example 9.59 In Example 9.53, we have\n\n\\[ \sigma = \\left( \\begin{array}{l} 1 \\ 0 \\end{array}\\right) ,\\;{B\\sigma } = \\left( \\begin{array}{ll} 0 & 0 \\ 1 & 0 \\end{array}\\right) \\left( \\begin{array}{l} 1 \\ 0 \\end{array}\\right) = \\left( \\begin{array}{l} 0 \\ 1 \\end{array}\\right) ,\\]\n\ntherefore \\(... | Proof (of Theorem 9.58). We recall the Cayley-Hamilton theorem: let\n\n\\[ p\\left( \\lambda \\right) = \\det \\left( {A - \\lambda {I}_{N}}\\right) = {\\lambda }^{N} + {a}_{1}{\\lambda }^{N - 1} + \\cdots + {a}_{N - 1}\\lambda + {a}_{N} \\]\n\nbe the characteristic polynomial of an \\( \\left( {N \\times N}\\right) \\... | Yes |
Proposition 9.62 Kalman and Hörmander conditions are equivalent. | Proof. It is enough to notice that, for \( i = 1,\ldots, d \) ,\n\n\[ \n\left\lbrack {{\partial }_{{x}_{i}}, Y}\right\rbrack = \mathop{\sum }\limits_{{k = 1}}^{N}{b}_{ki}{\partial }_{{x}_{k}} \n\] \n\nis the \( i \) -th column of the matrix \( B \) . Further, \( \left\lbrack {\left\lbrack {{\partial }_{{x}_{i}}, Y}\rig... | Yes |
Example 9.64 (Brownian bridge) Let \( b \in \mathbb{R} \) . We consider the 1-dimensional SDE \[ {dB} = \frac{b - {B}_{t}}{1 - t}{dt} + d{W}_{t} \] whose solution, at least for \( t < 1 \), is given by \[ {B}_{t} = {B}_{0}\left( {1 - t}\right) + {bt} + \left( {1 - t}\right) {\int }_{0}^{t}\frac{d{W}_{s}}{1 - s}. \] | Then we have \[ E\left\lbrack {B}_{t}\right\rbrack = {B}_{0}\left( {1 - t}\right) + {bt} \] and, by Itô isometry, \[ \operatorname{var}\left( {B}_{t}\right) = {\left( 1 - t\right) }^{2}{\int }_{0}^{t}\frac{ds}{{\left( 1 - s\right) }^{2}} = t\left( {1 - t}\right) . \] We note that \[ \mathop{\lim }\limits_{{t \rightarro... | Yes |
We consider the following model for the motion of a particle with friction: speed and position are described by the pair \( {X}_{t} = \left( {{V}_{t},{P}_{t}}\right) \), solution of the linear SDE \[ \left\{ \begin{array}{l} d{V}_{t} = - \mu {V}_{t}{dt} + {\sigma d}{W}_{t} \\ d{P}_{t} = {V}_{t}{dt} \end{array}\right. \... | We can easily check that the Kalman condition is verified. Further, it is immediate to prove by induction that, for every \( n \in \mathbb{N} \), we have that \[ {B}^{n} = \left( \begin{matrix} {\left( -\mu \right) }^{n} & 0 \\ {\left( -\mu \right) }^{n - 1} & 0 \end{matrix}\right) \] and so \[ {e}^{tB} = {I}_{2} + \ma... | Yes |
Lemma 10.1 If there exists a constant \( C \) such that\n\n\[{\int }_{0}^{T}{\left| {\lambda }_{t}\right| }^{2}{dt} \leq C\;\text{ a.s. }\](10.3)\nthen \( {Z}^{\lambda } \) in (10.1) is a martingale such that\n\n\[E\left\lbrack {\mathop{\sup }\limits_{{0 \leq t \leq T}}{\left( {Z}_{t}^{\lambda }\right) }^{p}}\right\rbr... | Proof. We put\n\n\[{\widehat{Z}}_{T} = \mathop{\sup }\limits_{{0 \leq t \leq T}}{Z}_{t}^{\lambda }\](10.4)\n\nFor every \( \zeta > 0 \), we have\n\n\[P\left( {{\widehat{Z}}_{T} \geq \zeta }\right) \leq P\left( {\mathop{\sup }\limits_{{0 \leq t \leq T}}\exp \left( {-{\int }_{0}^{t}{\lambda }_{s} \cdot d{W}_{s}}\right) \... | Yes |
Lemma 10.3 Assume that \( {Z}^{\lambda } \) in (10.1) is a \( P \) -martingale and \( Q \) is the probability measure defined in (10.5). Then a process \( {\left( {M}_{t}\right) }_{t \in \left\lbrack {0, T}\right\rbrack } \) is a \( Q \) - martingale if and only if \( {\left( {M}_{t}{Z}_{t}^{\lambda }\right) }_{t \in \... | Proof. Since \( {Z}^{\lambda } \) is strictly positive and adapted, it is clear that \( M \) is adapted if and only if \( M{Z}^{\lambda } \) is adapted. Moreover, since \( {Z}^{\lambda } \) is a \( P \) -martingale, \( M \) is \( Q \) -integrable if and only if \( M{Z}^{\lambda } \) is \( P \) -integrable: indeed\n\n\[... | Yes |
Lemma 10.8 The collection of random variables of the form\n\n\[ \varphi \left( {{W}_{{t}_{1}},\ldots ,{W}_{{t}_{n}}}\right) \]\n\nwith \( \varphi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) ,{t}_{k} \in \left\lbrack {0, T}\right\rbrack \) for \( k = 1,\ldots, n \) and \( n \in \mathbb{N} \), is dense in \( {L}^... | Proof. We consider a countable dense subset \( {\left\{ {t}_{n}\right\} }_{n \in \mathbb{N}} \) of \( \left\lbrack {0, T}\right\rbrack \) and we define the discrete filtration\n\n\[ {\mathcal{F}}_{n} \mathrel{\text{:=}} \sigma \left( {{W}_{{t}_{1}},\ldots ,{W}_{{t}_{n}}}\right) ,\;n \in \mathbb{N}; \]\n\nwe observe tha... | Yes |
Lemma 10.9 The space of the linear combinations of random variables of the form\n\n\\[ \n{Z}^{\lambda } = \exp \\left( {-{\\int }_{0}^{T}\\lambda \\left( t\\right) \\cdot d{W}_{t} - \\frac{1}{2}{\\int }_{0}^{T}{\\left| \\lambda \\left( t\\right) \\right| }^{2}{dt}}\\right) ,\n\\]\n\nwhere \\( \\lambda \\) is a function... | Proof. We prove the claim by verifying that, if\n\n\\[ \n{\\left\\langle X,{Z}^{\lambda }\\right\\rangle }_{{L}^{2}\\left( \\Omega \\right) } = {\\int }_{\\Omega }X{Z}^{\lambda }{dP} = 0\n\\]\n\n(10.11)\n\nfor every \\( \\lambda \\in {L}^{\infty }\\left( \\left\\lbrack {0, T}\\right\\rbrack \\right) \\), then \\( X = 0... | Yes |
Theorem 10.11 Let \( {\left( {M}_{t}\right) }_{t \in \left\lbrack {0, T}\right\rbrack } \) be a \( {\mathcal{F}}^{W} \) -martingale such that\n\n\[ \n{M}_{T} \in {L}^{2}\left( {\Omega ,{\mathcal{F}}_{T}^{W}}\right)\n\]\n\nThen there exists a unique (up to \( \left( {m \otimes P}\right) \) -equivalence) process \( u \in... | Proof. Since \( {M}_{T} \in {L}^{2}\left( {\Omega ,{\mathcal{F}}_{T}^{W}}\right) \), by Theorem 10.7 there exists \( u \in {\mathbb{L}}^{2}\left( {\mathcal{F}}^{W}\right) \)\n\nsuch that\n\[ \n{M}_{T} = {M}_{0} + {\int }_{0}^{T}{u}_{s} \cdot d{W}_{s}\n\]\n\nFor a fixed \( t \leq T \), taking the conditional expectation... | Yes |
Theorem 10.12 Let \( {\left( {M}_{t}\right) }_{t \in \left\lbrack {0, T}\right\rbrack } \) be a \( {\mathcal{F}}^{W} \) -local martingale. Then there exists a unique (up to \( \left( {m \otimes P}\right) \) -equivalence) process \( u \in {\mathbb{L}}_{\text{loc }}^{2}\left( {\mathcal{F}}^{W}\right) \) such that\n\n\[ \... | Proof. Uniqueness follows from Proposition 5.3, that is from the uniqueness of the representation of an Itô process. Regarding the existence, we assume at first that \( M \) is continuous: by Remark 4.38, there exists a localizing sequence \( \left( {\tau }_{n}\right) \) such that \( \left( {M}^{{\tau }_{n}}\right) \) ... | Yes |
Theorem 10.13 Under the assumptions of Girsanov’s Theorem 10.5, if \( M \) is a local martingale in \( \left( {\Omega ,\mathcal{F}, Q,\left( {\mathcal{F}}_{t}^{W}\right) }\right) \), then there exists a unique (up to \( \left( {m \otimes P}\right) \) -equivalence) \( u \in {\mathbb{L}}_{\text{loc }}^{2}\left( {\mathcal... | Proof. As usual we can always use the localization argument as in the proof of Theorem 10.12, so it is enough to consider the case \( M \) is a martingale. We note that, since \( M \) is a \( Q \) -martingale with respect to \( {\mathcal{F}}^{W} \) which is the natural filtration for \( W \) and not for \( {W}^{\lambda... | Yes |
Theorem 10.14 (Change of drift) Let \( Q \) be a probability measure equivalent to \( P \) . The Radon-Nikodym derivative of \( Q \) with respect to \( P \) is an exponential martingale | Proof. We denote by \( Z \) the density process of \( Q \) with respect to \( P \) (cf. Example 3.34):\n\n\[ {Z}_{t} = {E}^{P}\left\lbrack {\left. {\frac{dQ}{dP} \mid {\mathcal{F}}_{t}^{W}}\right| \; = \frac{dQ}{dP}{\left. \right| }_{{\mathcal{F}}_{t}^{W}},\;t \in \left\lbrack {0, T}\right\rbrack }\right\rbrack \]\n\nS... | Yes |
Theorem 10.20 (Change of drift with correlation) For any probability measure \( Q \) equivalent to \( P \) there exists a process \( \lambda \in {\mathbb{L}}_{\text{loc }}^{2} \) such that\n\n\[{\left. \frac{dQ}{dP}\right| }_{{\mathcal{F}}_{t}^{W}} = {Z}_{t}\;\text{ and }\;d{Z}_{t} = - {Z}_{t}{\lambda }_{t} \cdot d{W}_... | Proof. By the martingale representation Theorem 10.12 for the standard Brownian motion \( \bar{W} \), there exists a \( d \) -dimensional process \( \bar{\lambda } \in {\mathbb{L}}_{\text{loc }}^{2}\left( {\mathcal{F}}^{W}\right) \) such that \( {\left. \frac{dQ}{dP}\right| }_{{\mathcal{F}}_{t}^{W}} = {Z}_{t} \) and\n\... | Yes |
Theorem 10.26 Formulas (10.37)-(10.38) establish a one-to-one correspondence between EMMs and market prices of risk. The dynamics, under an EMM \( Q \), of the asset prices is given by\n\n\[ d{S}_{t}^{i} = {r}_{t}{S}_{t}^{i}{dt} + {\sigma }_{t}^{i}{S}_{t}^{i}d{W}_{t}^{\lambda, i}, \]\n\n(10.42)\n\nwhere \( {W}^{\lambda... | Proof. We have already proved, by using Theorem 10.20, that any EMM \( Q \) defines a market price of risk \( \lambda \) such that (10.42) holds.\n\nConversely, if \( \lambda \) is a market price of risk, we consider the process \( Z \) in (10.38) and using that \( Z \) is a \( P \) -martingale, we define the measure \... | Yes |
Theorem 10.27 (Existence of an EMM) Assume that the processes\n\n\[ \n{\lambda }_{t}^{i} = \frac{{\mu }_{t}^{i} - {r}_{t}}{{\sigma }_{t}^{i}},\;i = 1,\ldots, N \]\n\nverify the integrability condition (10.3), that is\n\n\[ \n{\int }_{0}^{T}{\left| {\lambda }_{t}^{i}\right| }^{2}{dt} \leq C\;\text{ a.s. } \]\n\n(10.45)\... | Proof. By Theorem 10.26, in order to show that an EMM exists, it suffices to construct a market price of risk. Let \( \lambda \in {\mathbb{L}}^{2} \) be any \( d \) -dimensional process with the first \( N \) components defined by (10.41) and such that estimate (10.45) holds for any \( i = 1,\ldots, d \) . By Lemma 10.... | Yes |
In the Black-Scholes market model \( N = d = 1 \) and the coefficients \( r,\mu ,\sigma \) are constant. In this case the market price of risk is uniquely determined by equation (10.41) and we have\n\n\[ \lambda = \frac{\mu - r}{\sigma } \] | which corresponds to the value found in Section 7.3.4. By Theorem 10.26, the process\n\n\[ {W}_{t}^{\lambda } = {W}_{t} + {\lambda t},\;t \in \left\lbrack {0, T}\right\rbrack \]\n\nis a Brownian motion under the measure \( Q \) defined by\n\n\[ \frac{dQ}{dP} = \exp \left( {-\lambda {W}_{T} - \frac{{\lambda }^{2}}{2}T}\... | Yes |
Example 10.32 In a market model where the number of risky assets is equal to the dimension of the Brownian motion, i.e. \( N = d \), the drift condition (10.41) determines the process \( \lambda \) univocally. Therefore, under the assumptions of Theorem 10.27 we have that the EMM Q exists and is unique. As usual the \(... | \[ d{\widetilde{S}}_{t}^{i} = {\sigma }_{t}^{i}{\widetilde{S}}_{t}^{i}d{W}_{t}^{\lambda, i},\;i = 1,\ldots, N \] where \( {W}^{\lambda } \) is the \( Q \) -Brownian motion defined by \( d{W}_{t} = d{W}_{t}^{\lambda } - {\lambda }_{t}{dt} \). | Yes |
In the Heston stochastic volatility model [165], there is an underlying asset \( \left( {N = 1}\right) \) whose volatility is a stochastic process that is driven by a second real Brownian motion \( \left( {d = 2}\right) \) . More precisely, we assume that\n\n\[ d{S}_{t} = \mu {S}_{t}{dt} + \sqrt{{\nu }_{t}}{S}_{t}d{W}_... | By the Itô formula, the solution of (10.47) is\n\n\[ {S}_{t} = {S}_{0}\exp \left( {{\int }_{0}^{t}\sqrt{{\nu }_{s}}d{W}_{s}^{1} + {\int }_{0}^{t}\left( {\mu - \frac{{\nu }_{s}}{2}}\right) {ds}}\right) . \] | Yes |
We consider a market which consists of two geometric Brownian motions\n\n\[ \nd{S}_{t}^{i} = {\mu }^{i}{S}_{t}^{i}{dt} + {\sigma }^{i}{S}_{t}^{i}d{W}_{t},\;i = 1,2, \]\n\nwhere \( W \) is a real Brownian motion: in this case \( N = 2 > d = 1 \) . The drift condition (10.41) takes the form:\n\n\[ \n\left\{ \begin{array}... | Then the value \( V \) of the portfolio verifies\n\n\[ \nd{V}_{t} = {\alpha }_{t}^{1}d{S}_{t}^{1} + {\alpha }_{t}^{2}d{S}_{t}^{2} + r\left( {{V}_{t} - {\alpha }_{t}^{1}{S}_{t}^{1} - {\alpha }_{t}^{2}{S}_{t}^{2}}\right) {dt} \]\n\n\[ \n= \frac{{\mu }^{1} - r}{{\sigma }^{1}}{dt} + d{W}_{t} - \frac{{\mu }^{2} - r}{{\sigma... | Yes |
Proposition 10.39 A strategy \( \left( {\alpha ,\beta }\right) \) is self-financing if and only if\n\n\[ d{\widetilde{V}}_{t}^{\left( \alpha ,\beta \right) } = {\alpha }_{t} \cdot d{\widetilde{S}}_{t} \]\n\n(10.57)\n\nMoreover, a self-financing strategy \( \left( {\alpha ,\beta }\right) \) is determined by its initial ... | Proof. We have\n\n\[ d{\widetilde{V}}_{t}^{\left( \alpha ,\beta \right) } = {e}^{-{\int }_{0}^{t}{r}_{s}{ds}}\left( {-{r}_{t}{V}_{t}^{\left( \alpha ,\beta \right) }{dt} + d{V}_{t}^{\left( \alpha ,\beta \right) }}\right) = \]\n\n(by the self-financing property (10.55))\n\n\[ = {e}^{-{\int }_{0}^{t}{r}_{s}{ds}}\left( {-{... | Yes |
Corollary 10.40 Let \( Q \) be an EMM with associated \( Q \) -Brownian motion \( {W}^{\lambda } = \left( {{W}^{\lambda ,1},\ldots ,{W}^{\lambda, d}}\right) \) defined by (10.37)-(10.39). For any self-financing strategy \( \left( {\alpha ,\beta }\right) \), we have\n\n\[ \n{\widetilde{V}}_{t}^{\left( \alpha ,\beta \rig... | Proof. The thesis follows from (10.57) and Theorem 4.42. | No |
Proposition 10.41 If \( Q \) is an EMM in \( \mathcal{Q} \) and \( \left( {\alpha ,\beta }\right) \) be a self-financing strategy such that\n\n\[ \n{\alpha }^{i}{\sigma }^{i} \in {\mathbb{L}}^{2}\left( {\Omega, P}\right) ,\;i = 1,\ldots, N, \n\]\n\nthen \( {\widetilde{V}}^{\left( \alpha ,\beta \right) } \) is a strict ... | Proof. By (10.58) and Corollary 4.48, if\n\n\[ \n{E}^{Q}\left\lbrack {\left( {\int }_{0}^{T}{\left( {\alpha }_{t}^{i}{\sigma }_{t}^{i}{\widetilde{S}}_{t}^{i}\right) }^{2}dt\right) }^{\frac{1}{2}}\right\rbrack < \infty \n\]\n\nfor every \( i = 1,\ldots, N \), then \( {\widetilde{V}}^{\left( \alpha ,\beta \right) } \) is... | Yes |
Corollary 10.43 (No-arbitrage principle) If an EMM in \( \mathcal{Q} \) exists and \( \left( {\alpha ,\beta }\right) ,\left( {{\alpha }^{\prime },{\beta }^{\prime }}\right) \) are admissible self-financing strategies such that\n\n\[ \n{V}_{T}^{\left( \alpha ,\beta \right) } = {V}_{T}^{\left( {\alpha }^{\prime },{\beta ... | Proof. If \( Q \in \mathcal{Q} \) exists and \( \left( {\alpha ,\beta }\right) ,\left( {{\alpha }^{\prime },{\beta }^{\prime }}\right) \) are admissible, then \( {\widetilde{V}}^{\left( \alpha ,\beta \right) },{\widetilde{V}}^{\left( {\alpha }^{\prime },{\beta }^{\prime }\right) } \) are \( Q \) -martingales with the s... | Yes |
Lemma 10.48 Let \( X \) be a European derivative. For every EMM \( Q \in \mathcal{Q} \) and \( t \in \left\lbrack {0, T}\right\rbrack \) we have\n\n\[ \mathop{\sup }\limits_{{\left( {\alpha ,\beta }\right) \in {\mathcal{A}}_{X}^{ - }}}{V}_{t}^{\left( \alpha ,\beta \right) } \leq {E}^{Q}\left\lbrack {{e}^{-{\int }_{t}^{... | Proof. If \( \left( {\alpha ,\beta }\right) \in {\mathcal{A}}_{X}^{ - } \), then \( {\widetilde{V}}^{\left( \alpha ,\beta \right) } \) is a \( Q \) -martingale for any \( Q \in \mathcal{Q} \) : thus we have\n\n\[ {V}_{t}^{\left( \alpha ,\beta \right) } = {E}^{Q}\left\lbrack {{e}^{-{\int }_{t}^{T}{r}_{s}{ds}}{V}_{T}^{\l... | Yes |
Theorem 10.49 Let \( X \) be a replicable European derivative. For every replicating strategy \( \left( {\alpha ,\beta }\right) \in \mathcal{A} \) and for every EMM \( Q \in \mathcal{Q} \), we have\n\n\[ \n{H}_{t} \mathrel{\text{:=}} {V}_{t}^{\left( \alpha ,\beta \right) } = {E}^{Q}\left\lbrack {{e}^{-{\int }_{t}^{T}{r... | Proof. If \( \left( {\alpha ,\beta }\right) \in \mathcal{A} \) replicates \( X \), then \( \left( {\alpha ,\beta }\right) \in {\mathcal{A}}_{X}^{ - } \cap {\mathcal{A}}_{X}^{ + } \) and by Lemma 10.48 we have\n\n\[ \n{E}^{Q}\left\lbrack {{e}^{-{\int }_{t}^{T}{r}_{s}{ds}}X \mid {\mathcal{F}}_{t}^{W}}\right\rbrack = {V}_... | Yes |
Theorem 10.50 When \( N = d \), the market model \( \left( {S, B}\right) \) in \( \left( {10.25}\right) - \left( {10.27}\right) \) is complete, that is every European derivative is replicable. Moreover there exists only one EMM. | Proof. The uniqueness of the EMM has been already pointed out in Example 10.32: it follows from the fact that, when \( N = d \), the drift condition (10.41) determines uniquely the market price of risk.\n\nNext we denote by \( Q \) the EMM and by \( {W}^{\lambda } \) the associated \( Q \) -Brownian motion. We define t... | Yes |
We consider the Heston stochastic volatility model of Example 10.33. The price of the risky asset \( S \) is given by the system of SDEs (10.47)-(10.48). In this case the market price of risk is a 2-dimensional process \( \lambda = \left( {{\lambda }^{1},{\lambda }^{2}}\right) \) with \( {\lambda }_{t}^{1} \) determine... | \[ {\lambda }_{t}^{1} = \frac{\mu - r}{\sqrt{{\nu }_{t}}} \] (10.70) As already mentioned, \( \lambda \) is generally not unique and a natural choice for the second component of the market price of risk is \[ {\lambda }_{t}^{2} = \frac{a{\nu }_{t} + b}{\sqrt{{\nu }_{t}}} \] (10.71) with \( a, b \in \mathbb{R} \) : then... | Yes |
Theorem 10.55 Let \( f \) be the solution of the Cauchy problem\n\n\[ \left\{ \begin{array}{ll} {Lf} = 0, & \text{ in }\rbrack 0, T\left\lbrack {\times {\mathbb{R}}_{ > 0}^{N},}\right. \\ f\left( {T, \cdot }\right) = F, & \text{ on }{\mathbb{R}}_{ > 0}^{N}, \end{array}\right. \] \n\nwhere \n\n\[ {Lf}\left( {t, s}\right... | Proof. The claim is a consequence of the existence results for the Cauchy problem in Paragraph 8.1 and of the Feynman-Kač formula: they can be applied directly after the transformation \( s = {e}^{x} \). More precisely, for \( i = 1,\ldots, N \) , we set\n\n\[ {X}_{t}^{i} = \log {S}_{t}^{i},\;\widetilde{\sigma }\left( ... | Yes |
Theorem 10.58 Let \( Q \) be an EMM with numeraire \( B \) and let \( U \) be a \( Q \) -price process. Consider the probability measure \( {Q}^{U} \) on \( \left( {\Omega ,\mathcal{F}}\right) \) defined by \( {}^{9} \n\n\[ \n\frac{d{Q}^{U}}{dQ} = \frac{{U}_{T}{B}_{0}}{{B}_{T}{U}_{0}} \n\] \n\n(10.82) \n\nThen for any ... | Proof. We first prove that for any \( X \in {L}^{1}\left( {\Omega ,{Q}^{U}}\right) \) we have \n\n\[ \n{E}^{{Q}^{U}}\left\lbrack {X \mid {\mathcal{F}}_{t}^{W}}\right\rbrack = {E}^{Q}\left\lbrack {\frac{D\left( {t, T}\right) }{{D}^{U}\left( {t, T}\right) }X \mid {\mathcal{F}}_{t}^{W}}\right\rbrack ,\;t \in \left\lbrack ... | Yes |
Lemma 10.62 Let \( U, V \) be two positive Itô processes of the form\n\n\[ d{U}_{t} = \left( \cdots \right) {dt} + {\sigma }_{t}^{U} \cdot d{W}_{t} \]\n\n\[ d{V}_{t} = \left( \cdots \right) {dt} + {\sigma }_{t}^{V} \cdot d{W}_{t} \]\n\nwhere \( W \) is a correlated d-dimensional Brownian motion and \( {\sigma }^{U},{\s... | Proof. The thesis follows directly by the Itô formula, since we have\n\n\[ d\frac{{V}_{t}}{{U}_{t}} = \frac{d{V}_{t}}{{U}_{t}} - \frac{{V}_{t}d{U}_{t}}{{U}_{t}^{2}} + \frac{{V}_{t}}{{U}_{t}^{3}}d\langle U, U{\rangle }_{t} - \frac{1}{{U}_{t}^{2}}d\langle U, V{\rangle }_{t} \] | Yes |
Example 10.64 (Exchange option) We consider an exchange option whose payoff is\n\n\[ \n{\left( {S}_{T}^{1} - {S}_{T}^{2}\right) }^{ + } \n\]\n\nwhere the two stocks \( {S}^{1},{S}^{2} \) are modeled as geometric Brownian motions:\n\n\[ \nd{S}_{t}^{i} = {\mu }^{i}{S}_{t}^{i}{dt} + {\sigma }^{i}{S}_{t}^{i}d{W}_{t}^{i},\;... | By the results in Section 10.2.6 the market is complete, the martingale measure is unique and by the pricing formula (10.84) of Theorem 10.58, the arbitrage price \( H \) of the exchange option under the EMM \( {Q}^{2} \) with numeraire \( {S}^{2} \) , is given by\n\n\[ \n{H}_{t} = {E}^{{Q}^{2}}\left\lbrack {\frac{{S}_... | Yes |
In the Black-Scholes model, let us denote by \( {W}^{B} \) and \( {W}^{S} \) the Brownian motions with numeraires \( B \) and \( S \) respectively. Then by (10.101) we have\n\n\[ d{W}_{t}^{B} = d{W}_{t}^{S} + {\sigma dt} \] | In particular the dynamics of \( S \) under \( {W}^{S} \) is given by\n\n\[ d{S}_{t} = r{S}_{t}{dt} + \sigma {S}_{t}d{W}_{t}^{B} \]\n\n\[ = r{S}_{t}{dt} + \sigma {S}_{t}\left( {d{W}_{t}^{S} + {\sigma dt}}\right) \]\n\n\[ = \left( {r + {\sigma }^{2}}\right) {S}_{t}{dt} + \sigma {S}_{t}d{W}_{t}^{S} \] | Yes |
Theorem 10.67 The Q-risk neutral price of a Call option with underlying \( S \), strike \( K \) and maturity \( T \) is given by\n\n\[ \n{C}_{0} = {S}_{0}{Q}^{S}\left( {{S}_{T} \geq K}\right) - {Kp}\left( {0, T}\right) {Q}^{T}\left( {{S}_{T} \geq K}\right) , \n\]\n\n(10.102)\n\nwhere \( {Q}^{S} \) and \( {Q}^{T} \) den... | For the practical use of this formula we have to determine the distribution of \( S \) under the new martingale measures. We first recall the dynamics of \( S \) under the EMM \( Q \) with numeraire \( B \) and related Brownian motion \( {W}^{Q} = \)\n\n\[ \n\left( {{W}^{Q,1},\ldots ,{W}^{Q, d}}\right) \n\]\n\nwith cor... | Yes |
Theorem 10.69 The implied volatility generated by the CEV model (10.111), with \( \beta \in \rbrack 0,1\lbrack \), is approximated by the following formula:\n\n\[ \n{\sigma }_{\mathrm{{CEV}}}\left( {{S}_{t}, T, K}\right) = \frac{\sqrt{{\alpha }_{t, T}}}{{F}_{t}^{1 - \beta }}\left( {1 + \frac{\left( {1 - \beta }\right) ... | Proof. The proof proceeds in some steps.\n\nFirst step. We consider the pricing problem\n\n\[ \n\left\{ \begin{array}{l} {\partial }_{t}u\left( {t, s}\right) + \frac{{\sigma }^{2}\left( t\right) {A}^{2}\left( s\right) }{2}{\partial }_{ss}u\left( {t, s}\right) + {rs}{\partial }_{s}u\left( {t, s}\right) = 0,\;t \in \rbra... | Yes |
Theorem 11.2 There exists a unique strong solution \( f \) of the obstacle problem\n\n\[ \left\{ \begin{array}{ll} \max \left\{ {{L}_{\mathrm{{BS}}}f,\psi - f}\right\} = 0, & \text{ in }\rbrack 0, T\left\lbrack {\times {\mathbb{R}}_{ > 0},}\right. \\ f\left( {T, \cdot }\right) = \psi \left( {T, \cdot }\right) , & \text... | Proof. With the change of variables\n\n\[ u\left( {t, x}\right) = f\left( {t,{e}^{x}}\right) ,\;\varphi \left( {t, x}\right) = \psi \left( {t,{e}^{x}}\right) \]\n\nproblem (11.3) is equivalent to the obstacle problem\n\n\[ \left\{ \begin{array}{ll} \max \{ {Lu},\varphi - u\} = 0, & \text{ in }\rbrack 0, T\lbrack \times... | Yes |
Proposition 11.3 A strategy \( h = \left( {\alpha ,\beta }\right) \) is self-financing if and only if\n\n\[ d{\widetilde{V}}_{t}\left( h\right) = {\alpha }_{t}\left( {d{\widetilde{S}}_{t} + q{\widetilde{S}}_{t}{dt}}\right) \]\n\ni.e.\n\n\[ {\widetilde{V}}_{t}\left( h\right) = {V}_{0}\left( h\right) + {\int }_{0}^{t}{\a... | Proof. The proof is analogous to that of Proposition 7.3, the only difference being the term due to the dividend. Formula (11.6) follows from (11.2). | No |
Lemma 11.4 Let \( {h}^{1},{h}^{2} \in \mathcal{A} \) be two self-financing strategies such that\n\n\[ \n{V}_{\tau }\left( {h}^{1}\right) \leq {V}_{\tau }\left( {h}^{2}\right) \n\]\n\nfor some \( \tau \in {\mathcal{T}}_{T} \) . Then\n\n\[ \n{V}_{0}\left( {h}^{1}\right) \leq {V}_{0}\left( {h}^{2}\right) \n\] | Proof. The claim is an immediate consequence of (11.7), of the martingale property of both \( \widetilde{V}\left( {h}^{1}\right) \) and \( \widetilde{V}\left( {h}^{2}\right) \) and of Doob’s optional sampling theorem, Theorem 3.56. | No |
Proposition 11.5 If \( {h}_{1} \in {\mathcal{A}}_{\psi }^{ - } \) and \( {h}_{2} \in {\mathcal{A}}_{\psi }^{ + } \) then we have\n\n\[ \n{V}_{0}\left( {h}^{1}\right) \leq {V}_{0}\left( {h}^{2}\right) \n\]\n\nFurther, for every \( {h}_{1},{h}_{2} \in {\mathcal{A}}_{\psi }^{ - } \cap {\mathcal{A}}_{\psi }^{ + } \) we hav... | By Theorem 11.7, there exists a strategy \( \bar{h} \in {\mathcal{A}}_{\psi }^{ + } \cap {\mathcal{A}}_{\psi }^{ - } \) : then, by Proposition 11.5 the following definition is well-posed. | No |
Theorem 11.7 Let \( f \) be the strong solution to the obstacle problem (11.3). The self-financing strategy \( h = \left( {\alpha ,\beta }\right) \) defined by\n\n\[ \n{V}_{0}\left( h\right) = f\left( {0,{S}_{0}}\right) ,\;{\alpha }_{t} = {\partial }_{S}f\left( {t,{S}_{t}}\right) \n\]\n\nbelongs to \( {\mathcal{A}}_{\p... | Proof. The idea is to use the Itô formula to compute the stochastic differential of \( f\left( {t,{S}_{t}}\right) \) and to separate the martingale part from the drift part of the process. We recall that, by definition of strong solution, (cf. Definition 8.20), \( f \in {S}_{\text{loc }}^{p}\left( {\left\lbrack {0, T}\... | Yes |
Proposition 11.8 We have\n\n\[ C\left( {T,{S}_{0}, K, r, q}\right) = P\left( {T, K,{S}_{0}, q, r}\right) . \] | Proof. We set\n\n\[ {Z}_{t} = {e}^{\sigma {W}_{t} - \frac{{\sigma }^{2}}{2}t} \]\n\nand recall that \( Z \) is a \( Q \) -martingale with unitary mean. Moreover, the process\n\n\[ {\widetilde{W}}_{t} = {W}_{t} - {\sigma t} \]\n\nis a Brownian motion with respect to the measure \( \widetilde{Q} \) defined by\n\n\[ \frac... | Yes |
Proposition 11.10 The following properties hold:\ni) for all \( S \in {\mathbb{R}}_{ > 0} \), the function \( T \mapsto P\left( {T, S}\right) \) is increasing. In other words, if we fix the parameters of the option, the price of the Put option decreases as we get closer to maturity;\nii) for all \( T \in \left\lbrack {... | Proof. i) is trivial. ii) is an immediate consequence of (11.15), of the monotonicity and convexity properties of the payoff function and of the fact that those properties are preserved by taking the supremum over all stopping times: indeed, if \( \left( {g}_{\tau }\right) \) is a family of increasing and convex functi... | Yes |
Lemma 11.11 For all \( \left( {t, S}\right) \in \lbrack 0, T\left\lbrack {\times {\mathbb{R}}_{ > 0}}\right. \), we have that\n\n\[ \n{L}_{\mathrm{{BS}}}f\left( {t, S}\right) = \left( {{qS} - {rK}}\right) {\mathbb{1}}_{\left\{ S \leq {S}^{ * }\left( t\right) \right\} }.\n\]\n\n(11.16)\n\nIn particular \( {L}_{\mathrm{{... | Proof. First of all let us observe that \( {S}^{ * }\left( t\right) < K \) : indeed if we had \( {S}^{ * }\left( t\right) \geq K \) then it should hold that\n\n\[ \nf\left( {t, K}\right) = \psi \left( {t, K}\right) = 0\n\]\n\nand this is absurd (since \( f > 0 \) by (11.4)). Then by the convexity of \( S \mapsto \) \( ... | Yes |
Theorem 11.13 There exists a unique strong solution \( f \) of the obstacle problem\n\[ \left\{ \begin{array}{ll} \max \{ {Lf},\psi - f\} = 0, & \text{ in }\rbrack 0, T\left\lbrack {\times {\mathbb{R}}_{ > 0}^{d},}\right. \\ f\left( {T, \cdot }\right) = \psi \left( {T, \cdot }\right) , & \text{ on }{\mathbb{R}}_{ > 0}^... | Proof. By the change of variable \( S = {e}^{x} \), the claim is direct consequence of Theorems 8.21, 9.48 and of Proposition 9.49. | No |
Proposition 11.14 The strategy \( h = \left( {\alpha ,\beta }\right) \) is self-financing if and only if\n\n\[ \n{\widetilde{V}}_{t}\left( h\right) = {V}_{0}\left( h\right) + \mathop{\sum }\limits_{{i = 1}}^{d}{\int }_{0}^{t}{\alpha }_{s}^{i}d{\widetilde{S}}_{s}^{i} + \mathop{\sum }\limits_{{i = 1}}^{d}{\int }_{0}^{t}{... | Proof. The proof is analogous to that of Proposition 7.3. The second equality in (11.22) follows from (11.20). | No |
Theorem 11.16 Let \( f \) be the solution to the obstacle problem (11.21). The self-financing strategy \( \bar{h} = \left( {\alpha ,\beta }\right) \) defined by\n\n\[ \n{V}_{0}\left( \bar{h}\right) = f\left( {0,{S}_{0}}\right) ,\;{\alpha }_{t} = \nabla f\left( {t,{S}_{t}}\right) ,\n\]\n\nbelongs to \( {\mathcal{A}}_{\p... | Furthermore, an optimal exercise strategy is defined by\n\n\[ \n{\tau }_{0} = \inf \left\{ {t \in \left\lbrack {0, T}\right\rbrack \mid f\left( {t,{S}_{t}}\right) = \psi \left( {t,{S}_{t}}\right) }\right\}\n\]\n\nand we have that\n\n\[ \n{V}_{0}\left( \bar{h}\right) = {E}^{Q}\left\lbrack {{e}^{-{\int }_{0}^{{\tau }_{0}... | Yes |
Proposition 12.1 (Regularity) The solution \( X \) in (12.6) is such that\n\n\[ \left| {{X}_{t} - {X}_{s}}\right| \leq {K}_{1}\left| {t - s}\right| ,\;t, s \in \left\lbrack {0, T}\right\rbrack \]\n\nwith \( {K}_{1} \) depending on \( K \) in (12.2), \( T \) and \( {X}_{0} \) only. | Proof. By definition, if \( s < t \), we have\n\n\[ \left| {{X}_{t} - {X}_{s}}\right| = \left| {{\int }_{s}^{t}\mu \left( {u,{X}_{u}}\right) {du}}\right| \leq \left( {t - s}\right) \mathop{\max }\limits_{{u \in \left\lbrack {0, T}\right\rbrack }}\left| {\mu \left( {u,{X}_{u}}\right) }\right| .\n\nThe claim follows by t... | Yes |
Proposition 12.2 (Consistency) Let \( Y \) be a Lipschitz continuous function on \( \left\lbrack {0, T}\right\rbrack \) with Lipschitz constant \( {K}_{1} \) . For every \( t \in \left\lbrack {0, T}\right\rbrack \)\n\n\[ \left| {{L}_{t}Y - {L}_{t}^{\delta }Y}\right| \leq {C\delta } \]\n\n(12.11)\n\nwhere the constant \... | Proof. It suffices to consider the case \( t = {t}_{n} \) . We have\n\n\[ \left| {{L}_{{t}_{n}}Y - {L}_{{t}_{n}}^{\delta }Y}\right| = \left| {\mathop{\sum }\limits_{{k = 1}}^{n}{\int }_{{t}_{k - 1}}^{{t}_{k}}\left( {\mu \left( {s,{Y}_{s}}\right) - \mu \left( {{t}_{k - 1},{Y}_{{t}_{k - 1}}}\right) }\right) {ds}}\right| ... | Yes |
Proposition 12.3 (Stability - Maximum principle) Let \( X, Y \) be continuous functions on \( \left\lbrack {0, T}\right\rbrack \) . Then\n\n\[ \mathop{\max }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| {{X}_{t} - {Y}_{t}}\right| \leq {e}^{KT}\left( {\left| {{X}_{0} - {Y}_{0}}\right| + \mathop{\max }\limits... | Proof. Since\n\n\[ {X}_{t} - {Y}_{t} = {X}_{0} - {Y}_{0} + {L}_{t}^{\delta }X - {L}_{t}^{\delta }Y \]\n\n\[ + {\int }_{0}^{t}\mathop{\sum }\limits_{{n = 1}}^{N}\left( {\mu \left( {{t}_{n - 1},{X}_{{t}_{n - 1}}}\right) - \mu \left( {{t}_{n - 1},{Y}_{{t}_{n - 1}}}\right) }\right) {\mathbb{1}}_{\left\rbrack {t}_{n - 1},{t... | Yes |
Theorem 12.5 Let \( X \) and \( {X}^{\delta } \) be the solutions of \( {L}_{t}X = 0 \) and \( {L}_{t}^{\delta }{X}^{\delta } = 0 \) , respectively, with the same initial datum \( {X}_{0} = {X}_{0}^{\delta } \) . There exists a constant \( C \) depending only on \( T, K \) in (12.2) and \( {X}_{0} \) such that\n\n\[ \m... | Proof. By the maximum principle we have\n\n\[ \mathop{\max }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| {{X}_{t} - {X}_{t}^{\delta }}\right| \leq {e}^{KT}\mathop{\max }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| {{L}_{t}^{\delta }X - {L}_{t}^{\delta }{X}^{\delta }}\right| = {e}^{KT}\mathop{\... | Yes |
Proposition 12.7 (Consistency) Let \( Y \in {\mathcal{A}}_{\mathrm{c}} \) such that\n\n\[ E\left\lbrack {\mathop{\sup }\limits_{{s \in \left\lbrack {t,{t}^{\prime }}\right\rbrack }}{\left| {Y}_{s} - {Y}_{t}\right| }^{2}}\right\rbrack \leq {K}_{1}\left( {{t}^{\prime } - t}\right) ,\;0 \leq t < {t}^{\prime } \leq T. \]\n... | Proof. We have\n\n\[ {L}_{t}Y - {L}_{t}^{\delta }Y = {\int }_{0}^{t}\underset{ \mathrel{\text{:=}} {Z}_{s}^{\mu }}{\underbrace{\mathop{\sum }\limits_{{n = 1}}^{N}\left( {\mu \left( {{t}_{n - 1},{Y}_{{t}_{n - 1}}}\right) - \mu \left( {s,{Y}_{s}}\right) }\right) {\mathbb{1}}_{\left. \rbrack {t}_{n - 1},{t}_{n}\right\rbra... | Yes |
Proposition 12.8 (Stability - Maximum principle) There exists a constant \( {C}_{0} \), depending only on \( K \) and \( T \) such that, for every pair of processes \( X, Y \in {\mathcal{A}}_{\mathrm{c}} \), we have\n\n\[ \llbracket X - Y{\rrbracket }_{T}^{2} \leq {C}_{0}\left( {E\left\lbrack {\left| {X}_{0} - {Y}_{0}\... | Proof. Since\n\n\[ {X}_{t} - {Y}_{t} = {L}_{t}^{\delta }X - {L}_{t}^{\delta }Y + {X}_{0} - {Y}_{0} \]\n\n\[ + {\int }_{0}^{t}\underset{ \mathrel{\text{:=}} {Z}_{s}^{\mu }}{\underbrace{\mathop{\sum }\limits_{{n = 1}}^{N}\left( {\mu \left( {{t}_{n - 1},{X}_{{t}_{n - 1}}}\right) - \mu \left( {{t}_{n - 1},{Y}_{{t}_{n - 1}}... | Yes |
Theorem 12.9 There exists a constant \( C \) depending only on \( K, T \) and \( E\left\lbrack {X}_{0}^{2}\right\rbrack \), such that\n\n\[ \n{\left\lbrack X - {X}^{\delta }\right\rbrack }_{T} \leq C\sqrt{\delta }\n\] | Proof. By the maximum principle, Proposition 12.8, we have\n\n\[ \n{\left\lbrack X - {X}^{\delta }\right\rbrack }_{T}^{2} \leq {C}_{0}{\left\lbrack {L}^{\delta }X - {L}^{\delta }{X}^{\delta }\right\rbrack }_{T}^{2} = {C}_{0}{\left\lbrack {L}^{\delta }X - LX\right\rbrack }_{T}^{2} \leq {C\delta }\n\]\n\nwhere \( C \) de... | Yes |
Lemma 12.10 If \( v \) is four times differentiable in a convex neighborhood of the point \( y \), then the following estimates hold:\n\n\[ \left| {{D}_{\delta }^{ + }v\left( y\right) - {v}^{\prime }\left( y\right) }\right| \leq \delta \frac{\sup \left| {v}^{\prime \prime }\right| }{2} \]\n\n(12.28)\n\n\[ \left| {{D}_{... | Proof. Taking the Taylor series expansion of \( v \) with initial point \( y \), we get\n\n\[ v\left( {y + \delta }\right) = v\left( y\right) + {v}^{\prime }\left( y\right) \delta + \frac{1}{2}{v}^{\prime \prime }\left( \widehat{y}\right) {\delta }^{2}, \]\n\n(12.32)\n\n\[ v\left( {y - \delta }\right) = v\left( y\right... | Yes |
Proposition 12.12 (Discrete maximum principle) Let \( g \) be a function defined on the grid \( {G}^{\left( \tau ,\delta \right) } \) such that\n\n\[ \left\{ \begin{array}{ll} {L}^{\left( \tau ,\delta \right) }g \geq 0, & \text{ on }{G}^{\left( \tau ,\delta \right) }, \\ {g}_{n,0} \leq 0,{g}_{n, M + 1} \leq 0, & n = 0,... | Proof. We observe that \( {L}^{\left( \tau ,\delta \right) }g \geq 0 \) on \( {G}^{\left( \tau ,\delta \right) } \) if and only if\n\n\[ {g}_{n, i} \leq {g}_{n + 1, i}\left( {1 - \frac{2\tau }{{\delta }^{2}}}\right) + \left( {{g}_{n + 1, i + 1} + {g}_{n + 1, i - 1}}\right) \frac{\tau }{{\delta }^{2}} \]\n\nfor \( n = 0... | Yes |
Theorem 12.13 Let \( u \) be a solution of problem (12.41) and let us suppose that \( {\partial }_{xxxx}u \) and \( {\partial }_{tt}u \) are bounded. If condition (12.43) holds, then there exists a positive constant \( C \) such that, for every \( \delta > 0 \n\[ \mathop{\max }\limits_{{G}^{\left( \tau ,\delta \right) ... | Proof. Firstly we observe that, by Lemma 12.10 combined with condition (12.43), we have\n\n\[ \left| {\left( {L}^{\left( \tau ,\delta \right) }u\right) }_{n, i}\right| = \left| {{\left( {L}^{\left( \tau ,\delta \right) }u\right) }_{n, i} - {\left( Lu\right) }_{n + 1, i}}\right| \leq C{\delta }^{2}, \]\n\n(12.44)\n\nwit... | Yes |
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