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Proposition 2.12 A discrete market is arbitrage-free if and only if there exist no admissible arbitrage strategies. | Proof. We suppose that there exist no admissible arbitrage strategies and we have to show that no arbitrage opportunity exists. We prove the thesis by contradiction: we suppose there exists an arbitrage strategy \( \left( {\alpha ,\beta }\right) \) and we construct an admissible arbitrage strategy \( \left( {{\alpha }^... | Yes |
Proposition 2.16 Let \( Q \) be an EMM with numeraire \( Y \) and \( \left( {\alpha ,\beta }\right) \in \mathcal{A} \) . Then \( {\widetilde{V}}^{\left( \alpha ,\beta \right) } = \left( \frac{{V}_{n}^{\left( \alpha ,\beta \right) }}{{Y}_{n}}\right) \) is a \( Q \) -martingale: | Proof. For simplicity we only consider the case \( Y = B \) . The result is an immediate consequence of formula (2.13) which basically expresses the fact that \( \left( {\alpha ,\beta }\right) \) is self-financing if and only if \( {\widetilde{V}}^{\left( \alpha ,\beta \right) } \) is the transform of \( \widetilde{S} ... | Yes |
Proposition 2.17 (No arbitrage principle) In an arbitrage-free market, if \( \left( {\alpha ,\beta }\right) ,\left( {{\alpha }^{\prime },{\beta }^{\prime }}\right) \in \mathcal{A} \) and\n\n\[ \n{V}_{N}^{\left( \alpha ,\beta \right) } = {V}_{N}^{\left( {\alpha }^{\prime },{\beta }^{\prime }\right) }\;P\text{-a.s.,} \n\... | Proof. Since the market is arbitrage-free, there exists an EMM \( Q \) with numeraire \( Y \) . The claim follows from the fact that \( {\widetilde{V}}^{\left( \alpha ,\beta \right) },{\widetilde{V}}^{\left( {\alpha }^{\prime },{\beta }^{\prime }\right) } \) are \( Q \) - martingales with the same terminal value. Indee... | Yes |
In a discrete market \( \left( {S, B}\right) \), let \( Q \) be an EMM with numeraire \( Y \) and let \( X \) be a positive adapted process such that \( \left( \frac{{X}_{n}}{{Y}_{n}}\right) \) is a \( Q \) -martingale ( \( X \) represents the value process of another asset or strategy to be considered as the new numer... | Proof. In (2.18), \( L \mathrel{\text{:=}} \frac{d{Q}^{X}}{dQ} \) denotes the Radon-Nikodym derivative of \( {Q}^{X} \) with respect to \( Q \) and therefore\n\n\[ \n{E}^{{Q}^{X}}\left\lbrack Z\right\rbrack = {E}^{Q}\left\lbrack {ZL}\right\rbrack \]\n\nfor any random variable \( Z \) .\n\nFrom (2.18) we infer\n\n\[ \n{... | Yes |
Corollary 2.21 Under the assumptions of Theorem 2.19, for any \( n \leq N \) and \( A \in {\mathcal{F}}_{n} \), we have\n\n\[ \n{Q}^{X}\left( A\right) = {E}^{Q}\left\lbrack {\frac{{X}_{n}}{{X}_{0}}{\left( \frac{{Y}_{n}}{{Y}_{0}}\right) }^{-1}{\mathbb{1}}_{A}}\right\rbrack \n\]\n\nthat is\n\n\[ \n{\left. \frac{d{Q}^{X}}... | Proof. We have\n\n\[ \n{Q}^{X}\left( A\right) = {E}^{{Q}^{X}}\left\lbrack {\mathbb{1}}_{A}\right\rbrack = \n\]\n\n(by (2.18))\n\n\[ \n= {E}^{Q}\left\lbrack {{\mathbb{1}}_{A}\frac{{X}_{N}}{{X}_{0}}{\left( \frac{{Y}_{N}}{{Y}_{0}}\right) }^{-1}}\right\rbrack = \n\]\n\n(using that \( A \in {\mathcal{F}}_{n} \) )\n\n\[ \n= ... | Yes |
Lemma 2.23 For every EMM Q with numeraire \( B \), we have\n\n\[ \mathop{\sup }\limits_{{\left( {\alpha ,\beta }\right) \in {\mathcal{A}}_{X}^{ - }}}{\widetilde{V}}_{n}^{\left( \alpha ,\beta \right) } \leq {E}^{Q}\left\lbrack {\frac{X}{{B}_{N}} \mid {\mathcal{F}}_{n}}\right\rbrack \leq \mathop{\inf }\limits_{{\left( {\... | Proof. If \( \left( {\alpha ,\beta }\right) \in {\mathcal{A}}_{X}^{ - } \) then, by Proposition 2.16, we have\n\n\[ {\widetilde{V}}_{n}^{\left( \alpha ,\beta \right) } = {E}^{Q}\left\lbrack {{\widetilde{V}}_{N}^{\left( \alpha ,\beta \right) } \mid {\mathcal{F}}_{n}}\right\rbrack \leq {E}^{Q}\left\lbrack {\left. \frac{X... | Yes |
Theorem 2.25 Let \( X \) be a replicable derivative in an arbitrage-free market. Then for every replicating strategy \( \left( {\alpha ,\beta }\right) \in \mathcal{A} \) and for every EMM \( Q \) with numeraire \( B \), we have\n\n\[ \n{E}^{Q}\left\lbrack {\left. \frac{X}{{B}_{N}}\right| \;{\mathcal{F}}_{n}}\right\rbra... | Proof. If \( \left( {\alpha ,\beta }\right) ,\left( {{\alpha }^{\prime },{\beta }^{\prime }}\right) \in \mathcal{A} \) replicate \( X \) then they have the same terminal value and, by Proposition 2.17, they have the same value at all preceding times. Moreover, if \( \left( {\alpha ,\beta }\right) \in \mathcal{A} \) rep... | Yes |
Theorem 2.31 If the random variables \( {\mu }_{1},\ldots ,{\mu }_{N} \) are independent then the stochastic process \( S \) has the Markov property. | Proof. We have \( {}^{4} \)\n\n\[ \nE\left\lbrack {\varphi \left( {S}_{n}\right) \mid {\mathcal{F}}_{n - 1}}\right\rbrack = E\left\lbrack {\varphi \left( {{S}_{n - 1}\left( {1 + {\mu }_{n}}\right) }\right) \mid {\mathcal{F}}_{n - 1}}\right\rbrack = \n\]\n\n(applying Lemma A. 108 with \( X = 1 + {\mu }_{n}, Y = {S}_{n -... | Yes |
In the binomial model, the condition\n\n\[ d < 1 + r < u \]\n\n(2.39)\n\nis equivalent to the existence and uniqueness of the EMM Q. More precisely, if (2.39) holds then\n\n\[ q \mathrel{\\text{:=}} \\frac{1 + r - d}{u - d} \\in \\rbrack 0,1\\lbrack \]\n\n(2.40)\n\nand we have\n\n\[ Q\\left( {1 + {\\mu }_{n} = u}\\righ... | Proof. If an EMM \( Q \) exists, then by Definition 2.13 we have\n\n\[ {\\widetilde{S}}_{n - 1} = {E}^{Q}\\left\\lbrack {{\\widetilde{S}}_{n} \\mid {\\mathcal{F}}_{n - 1}}\\right\\rbrack \]\n\n(2.44)\n\nor equivalently\n\n\[ {S}_{n - 1}\\left( {1 + r}\\right) = {E}^{Q}\\left\\lbrack {{S}_{n - 1}\\left( {1 + {\\mu }_{n}... | Yes |
The binomial model is arbitrage-free and complete if and only if condition (2.39) holds. In this case the arbitrage price \( \left( {H}_{n}\right) \) of a derivative \( X \) is uniquely defined by the following risk-neutral pricing formula:\n\n\[ \n{H}_{n} = \frac{1}{{\left( 1 + r\right) }^{N - n}}{E}^{Q}\left\lbrack {... | Proof. Combining Theorem 2.33 with the Fundamental Theorems of asset pricing, we prove that the binomial model is arbitrage-free and complete if and only if condition (2.39) holds. Formula (2.46) follows from (2.27). Formula (2.47) follows from (2.46) with \( n = 0 \) and (2.42). | Yes |
Example 2.35 We suppose that the current price of a stock is \( {S}_{0} = {10} \) and that over the year the price can rise or fall within \( {20}\% \) of its initial value. We assume that the risk-free rate is \( r = 5\% \) and we determine the hedging strategy for a Call option with maturity \( T = 1 \) year and stri... | \[ \left\{ \begin{array}{l} {12\alpha } + \frac{105}{100}\beta = 2 \\ {8\alpha } + \frac{105}{100}\beta = 0 \end{array}\right. \] hence \( \alpha = \frac{1}{2} \) and \( \beta = - \frac{80}{21} \) . Then the current value of the hedging portfolio (corresponding to the arbitrage price of the option) is equal to \[ {V}_{... | Yes |
Example 2.37 (European Call option) We consider the payoff function of a European Call option with strike \( K \): \[ F\left( {S}_{N}\right) = {\left( {S}_{N} - K\right) }^{ + } = \max \left\{ {{S}_{N} - K,0}\right\} . \] | By using formula (2.47) and recalling that \( q = \frac{1 + r - d}{u - d} \), the initial price \( {C}_{0} \) of the option is given by \[ {C}_{0} = \frac{1}{{\left( 1 + r\right) }^{N}}\mathop{\sum }\limits_{{h = 0}}^{N}\left( \begin{matrix} N \\ h \end{matrix}\right) {q}^{h}{\left( 1 - q\right) }^{N - h}{\left( {u}^{h... | Yes |
Example 2.38 We consider a European Put option with strike \( K = \frac{5}{2} \) and value of the underlying asset \( {S}_{0} = 1 \) . We set the following values for the parameters in a three-period binomial model: | \[ u = 2,\;d = \frac{1}{2},\;r = \frac{1}{2} \] hence we obtain \[ q = \frac{1 + r - d}{u - d} = \frac{2}{3} \] First of all we construct in Figure 2.3 the binomial tree where we put the prices of the underlying asset inside the circles and the payoff of the option at maturity outside, using notation (2.66), i.e. \( {H... | Yes |
Lemma 2.39 The stochastic process \( \left( {S, A}\right) \) has the Markov property, and for every function \( f \) we have that\n\n\[ \n{E}^{Q}\left\lbrack {\varphi \left( {{S}_{n + 1},{A}_{n + 1}}\right) \mid {\mathcal{F}}_{n}}\right\rbrack = {E}^{Q}\left\lbrack {\varphi \left( {{S}_{n + 1},{A}_{n + 1}}\right) \mid ... | \[ \n= {q\varphi }\left( {u{S}_{n},{A}_{n}^{u}}\right) + \left( {1 - q}\right) \varphi \left( {d{S}_{n},{A}_{n}^{d}}\right) . \n\] | Yes |
Example 2.40 Under the assumption \( {ud} = 1 \), we have\n\n\[ \n{S}_{n, k} = \left\{ \begin{array}{ll} {d}^{n - {2k}}{S}_{0} & \text{ if }n \geq {2k} \\ {u}^{{2k} - n}{S}_{0} & \text{ if }n < {2k} \end{array}\right.\n\] | In the case of a Look-back option with fixed strike, if \( n \geq {2k} \) then \( {S}_{n, k} \leq {S}_{0} \) and\n\n\[ \n{A}_{n, k}\left( j\right) = {u}^{k - j}{S}_{0},\;j = 0,\ldots, n - k,\n\]\n\nwhile, if \( n \leq {2k} \), then \( {S}_{n, k} \geq {S}_{0} \) and\n\n\[ \n{A}_{n, k}\left( j\right) = {u}^{k - j}{S}_{0}... | No |
Lemma 2.43 If (2.84)-(2.85) hold, we have\n\n\\[ \n\\mathop{\\lim }\\limits_{{N \\rightarrow \\infty }}{q}_{N} = \\frac{1}{2} \n\\]\n\n(2.86) | Proof. By definition\n\n\\[ \n{q}_{N} = \\frac{{e}^{r{\\delta }_{N}} - {e}^{-\\sigma \\sqrt{{\\delta }_{N}} + \\beta {\\delta }_{N}}}{{e}^{\\sigma \\sqrt{{\\delta }_{N}} + \\alpha {\\delta }_{N}} - {e}^{-\\sigma \\sqrt{{\\delta }_{N}} + \\beta {\\delta }_{N}}}.\n\\]\n\n(2.87)\n\nThen, using a Taylor expansion for the e... | Yes |
Lemma 2.44 We have:\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}{E}^{{Q}_{N}}\left\lbrack {X}_{N}\right\rbrack = \left( {r - \frac{{\sigma }^{2}}{2}}\right) T \] | Proof (of Lemma 2.44). In order to prove (2.93), we compute\n\n\[ {E}^{{Q}_{N}}\left\lbrack {Y}_{1}^{\left( N\right) }\right\rbrack = {q}_{N}\left( {\sigma \sqrt{{\delta }_{N}} + \alpha {\delta }_{N}}\right) + \left( {1 - {q}_{N}}\right) \left( {-\sigma \sqrt{{\delta }_{N}} + \beta {\delta }_{N}}\right) \]\n\n\[ = \lef... | Yes |
Lemma 2.45 The sequence of random variables \( \left( {X}_{N}\right) \) defined in (2.90) converges in distribution to a random variable \( X \) that is normally distributed as in (2.95). | Proof. This result is a variation of the central limit Theorem A.146: by Lévy's Theorem A.141, it is enough to verify that the sequence \( \left( {\varphi }_{{X}_{N}}\right) \) of the corresponding characteristic functions converges pointwise. We have:\n\n\[ \n{\varphi }_{{X}_{N}}\left( \eta \right) = {E}^{{Q}_{N}}\lef... | Yes |
Corollary 2.48 (Black-Scholes formula) The following Black-Scholes formula holds:\n\n\[ \n{P}_{0} = K{e}^{-{rT}}\Phi \left( {-{d}_{2}}\right) - {S}_{0}\Phi \left( {-{d}_{1}}\right) \n\] \n\nwhere \( \Phi \) is the standard normal distribution function \n\n\[ \n\Phi \left( x\right) = \frac{1}{\sqrt{2\pi }}{\int }_{-\inf... | Proof. By (2.101), we have to prove that \n\n\[ \n{e}^{-{rT}}E\left\lbrack {\left( K - {S}_{0}{e}^{X}\right) }^{ + }\right\rbrack = K{e}^{-{rT}}\Phi \left( {-{d}_{2}}\right) - {S}_{0}\Phi \left( {-{d}_{1}}\right) , \n\] \n\nwhere \( X \) is normally distributed as in (2.102). Now,(cf. Remark A.32) \n\n\[ \nX = \left( {... | Yes |
Proposition 2.56 For every EMM \( Q \), we have\n\n\[ \mathop{\sup }\limits_{{\left( {\alpha ,\beta }\right) \in {\mathcal{A}}_{X}^{ - }}}{\widetilde{V}}_{0}^{\left( \alpha ,\beta \right) } \leq \mathop{\sup }\limits_{{\nu \in {\mathcal{T}}_{0}}}{E}^{Q}\left\lbrack {\widetilde{X}}_{\nu }\right\rbrack \leq \mathop{\inf ... | Proof. If \( \left( {\alpha ,\beta }\right) \in {\mathcal{A}}_{X}^{ - } \), there exists \( {\nu }_{0} \in {\mathcal{T}}_{0} \) such that \( {V}_{{\nu }_{0}}\left( \alpha \right) \leq {X}_{{\nu }_{0}} \) . Further, \( {\widetilde{V}}^{\left( \alpha ,\beta \right) } \) is a \( Q \) -martingale and so by the Optional sam... | Yes |
Theorem 2.57 Let \( X \) be an American derivative in an arbitrage-free and complete market. Then there exists \( \left( {\alpha ,\beta }\right) \in {\mathcal{A}}_{X}^{ + } \cap {\mathcal{A}}_{X}^{ - } \) and so we have:\ni) \( {V}_{n}^{\left( \alpha ,\beta \right) } \geq {X}_{n}, n = 0,\ldots, N \) ;\nii) there exists... | Proof. The proof is constructive and is made up of three main steps:\n\n1) we construct the smallest super-martingale \( \widetilde{H} \) greater than \( \widetilde{X} \), usually called Snell envelope of the process \( \widetilde{X} \) ;\n\n2) we use Doob's decomposition theorem to find the martingale part of the proc... | Yes |
Lemma 2.62 For any \( \nu \in {\mathcal{T}}_{0} \) we have\n\n\[ \n{E}^{Q}\left\lbrack {\widetilde{X}}_{\nu }\right\rbrack \leq {H}_{0} \]\n\nMoreover \( \nu \in {\mathcal{T}}_{0} \) is optimal for \( X \) under \( Q \) if and only if\n\n\[ \n{E}^{Q}\left\lbrack {\widetilde{X}}_{\nu }\right\rbrack = {H}_{0} \]\n | Proof. We have\n\n\[ \n{E}^{Q}\left\lbrack {\widetilde{X}}_{\nu }\right\rbrack \overset{\left( 1\right) }{ \leq }{E}^{Q}\left\lbrack {\widetilde{H}}_{\nu }\right\rbrack = {E}^{Q}\left\lbrack {\widetilde{H}}_{N}^{\nu }\right\rbrack \overset{\left( 2\right) }{ \leq }{H}_{0} \]\n\nwhere inequality (1) is a consequence of ... | Yes |
Corollary 2.63 If \( \nu \in {\mathcal{T}}_{0} \) is such that\ni) \( {\widetilde{X}}_{\nu } = {\widetilde{H}}_{\nu } \) ;\nii) \( {\widetilde{H}}^{\nu } \) is a \( Q \) -martingale;\nthen \( \nu \) is an optimal exercise strategy for \( X \) under \( Q \) . | Proof. Conditions \( i \) ) and \( {ii} \) ) imply that (1) and (2) in formula (2.152) are equalities. Consequently \( {E}^{Q}\left\lbrack {\widetilde{X}}_{\nu }\right\rbrack = {H}_{0} \) and therefore, by Lemma 2.62, \( \nu \) is optimal for \( X \) under \( Q \) . | Yes |
Proposition 2.64 The exercise strategies \( {\nu }_{\min } \) and \( {\nu }_{\max } \) are optimal for \( X \) under \( Q \) . | Proof. We show that \( {\nu }_{\min } \) and \( {\nu }_{\max } \) are optimal by verifying the conditions \( i \) ) and \( {ii} \) ) of Corollary 2.63. By definition (2.154)-(2.155) we have that\n\n\[ \n{H}_{{\nu }_{\min }} = \max \left\{ {{X}_{{\nu }_{\min }},{E}_{{\nu }_{\min }}}\right\} = {X}_{{\nu }_{\min }}, \n\]\... | Yes |
Proposition 2.65 If \( \nu \in {\mathcal{T}}_{0} \) is optimal for \( X \) under \( Q \) then\n\n\[{\nu }_{\min } \leq \nu \leq {\nu }_{\max }\] | Proof. Let us suppose that\n\n\[P\left( {\nu < {\nu }_{\min }}\right) > 0\]\n\n(2.159)\n\nWe aim at proving that \( \nu \) cannot be optimal because (1) in (2.152) is a strict inequality. Indeed, since \( P \) and \( Q \) are equivalent, from (2.159) it follows that\n\n\[Q\left( {{\widetilde{X}}_{\nu } < {\widetilde{H}... | Yes |
In a three-period binomial model, we consider an American Put option with payoff \( {X}_{n} = {\left( \frac{1}{2} - {S}_{n}\right) }^{ + }, n = 0,1,2,3 \) . We assume that \( u = 2, d = r = \frac{1}{2} \) and the initial price of the underlying asset is \( {S}_{0} = 1 \) . In Figure 2.10 we represent the asset prices a... | We first compute the arbitrage price process \( H \) and the minimal and maximal optimal exercise strategies. By (2.167) we have\n\n\[ \n{H}_{n} = \left\{ \begin{array}{ll} {\left( \frac{1}{2} - {S}_{3}\right) }^{ + }, & n = 3, \\ \max \left\{ {{\left( \frac{1}{2} - {S}_{n}\right) }^{ + },{E}_{n}}\right\} , & n = 0,1,2... | Yes |
Proposition 2.67 We have\n\ni) \( {H}_{n}^{A} \geq {H}_{n}^{E} \) for \( 0 \leq n \leq N \) ;\n\nii) if \( {H}_{n}^{E} \geq {X}_{n} \) for every \( n \), then\n\n\[ \n{H}_{n}^{A} = {H}_{n}^{E},\;n = 0,\ldots, N \n\]\n\nand \( \nu \equiv N \) is an optimal exercise strategy. | Proof. i) Since \( {\widetilde{H}}^{A} \) is a \( Q \) -super-martingale, we have\n\n\[ \n{\widetilde{H}}_{n}^{A} \geq {E}^{Q}\left\lbrack {{\widetilde{H}}_{N}^{A} \mid {\mathcal{F}}_{n}}\right\rbrack = {E}^{Q}\left\lbrack {{\widetilde{X}}_{N} \mid {\mathcal{F}}_{n}}\right\rbrack = {\widetilde{H}}_{n}^{E}, \n\]\n\nhenc... | Yes |
The equivalence between the prices of American and European derivatives does not hold for Call options that pay dividends and for Put options. As a simple example, let us consider an American Put option in a one-period binomial model \( \\left( {N = 1}\\right) \) with \( r > 0 \) and, for the sake of simplicity, | \[ q = \\frac{1 + r - d}{u - d} = \\frac{1}{2} \] Then \( u + d = 2\\left( {1 + r}\\right) \) and the price of the corresponding European Put option is \[ {p}_{0} = \\frac{1}{2\\left( {1 + r}\\right) }\\left( {{\\left( K - u{S}_{0}\\right) }^{ + } + {\\left( K - d{S}_{0}\\right) }^{ + }}\\right) = \] (if, for example, ... | Yes |
In the particular case of an American Put option, \( \varphi \left( S\right) = \) \( {\left( K - S\right) }^{ + } \) with maturity \( T \), some properties of the free boundary can be proved by resorting solely to arbitrage arguments. Let us put\n\n\[ \n{R}_{e}\left( t\right) = \left\{ {S \mid \left( {t, S}\right) \in ... | Indeed let \( f\left( {t, S}\right) \) be the price of the option. Then \( f\left( {t, S}\right) \) is strictly positive for every \( t \in \lbrack 0, T\lbrack \) : on the other hand, since \( \varphi \left( S\right) = 0 \) for \( S \geq K \), we have\n\n\[ \n{R}_{e}\left( t\right) \subseteq \{ S < K\} ,\;t \in \lbrack... | Yes |
Proposition 2.72 Assume that the parameter \( d \) in the binomial model is smaller than 1. The function \( x \mapsto {P}^{E}\left( x\right) \) is continuous, convex and decreasing for \( x \in {\mathbb{R}}_{ \geq 0} \) . Further,\n\n\[ \n{P}^{E}\left( 0\right) = \frac{K}{{\left( 1 + r\right) }^{N}},\;{P}^{E}\left( x\r... | Proof. We can write (2.174) more explicitly as \n\n\[ \n{P}^{E}\left( x\right) = \frac{1}{{\left( 1 + r\right) }^{N}}\mathop{\sum }\limits_{{h = 0}}^{N}{c}_{h}{\left( K - {u}^{h}{d}^{N - h}x\right) }^{ + }, \n\] \n\nwhere \( {c}_{h} = \left( \begin{matrix} N \\ h \end{matrix}\right) {q}^{h}{\left( 1 - q\right) }^{N - h... | Yes |
A first continuous time model for the price of a risky asset \( S \) is the following:\n\n\[ \n{S}_{t} = {S}_{0}\left( {1 + {\mu t}}\right) + \sigma {W}_{t},\;t \geq 0.\n\]\n\n(3.3)\n\nIn (3.3), \( {S}_{0} \) is the initial price of the asset, \( \mu \) is the expected rate of return and \( \sigma \) denotes the riskin... | From (3.4) it follows that\n\n\[ \nE\left\lbrack {S}_{t}\right\rbrack = {S}_{0}\left( {1 + {\mu t}}\right)\n\]\n\nso that the expectation of \( S \) corresponds to a risk-free deterministic dynamics. Then a Brownian motion introduces \ | Yes |
Theorem 3.14 A Brownian motion \( W \) on the space \( \left( {\Omega ,\mathcal{F}, P,\left( {\mathcal{F}}_{t}\right) }\right) \) has the Markov property with respect to \( \left( {\mathcal{F}}_{t}\right) \) and, in particular, formulas (3.7)-(3.8) hold: in a more compact form, we have\n\n\[ E\left\lbrack {\varphi \lef... | We note that (3.9) implies in particular (cf. Remark A.109) that\n\n\[ E\left\lbrack {\varphi \left( {W}_{T}\right) \mid {\mathcal{F}}_{t}}\right\rbrack = E\left\lbrack {\varphi \left( {W}_{T}\right) \mid {W}_{t}}\right\rbrack ,\;T \geq t, \]\ni.e. \( W \) is a Markov stochastic process, according to Definition 3.9.\n\... | Yes |
Lemma 3.16 For every \( H \in \mathcal{B}\left( {C\left( \left\lbrack {0, T}\right\rbrack \right) }\right) \), we have\n\n\[ \n{\widehat{X}}^{-1}\left( H\right) = \{ \omega \in \Omega \mid X\left( \omega \right) \in H\} \in \mathcal{F}, \]\n\nand therefore\n\n\[ \n\widehat{X} : \left( {\Omega ,\mathcal{F}}\right) \righ... | Proof. Since \( \mathcal{B}\left( {C\left( \left\lbrack {0, T}\right\rbrack \right) }\right) \) is generated by a countable collection of balls, it suffices to observe that, being \( X \) continuous, we have\n\n\[ \n\left\{ {X \in \bar{D}\left( {{w}_{0}, r}\right) }\right\} = \mathop{\bigcap }\limits_{{t \in \left\lbra... | Yes |
Lemma 3.19 We have that\n\n\[ \sigma \left( {{\mathbb{X}}_{t}, t \in \left\lbrack {0, T}\right\rbrack }\right) = \mathcal{B}\left( {C\left( \left\lbrack {0, T}\right\rbrack \right) }\right) \] | Proof. Given a set \( \tau = \left\{ {{t}_{1},\ldots ,{t}_{n}}\right\} \) consisting of a finite number of points in \( \left\lbrack {0, T}\right\rbrack \) and \( K = {K}_{1} \times \cdots \times {K}_{n} \) with \( {K}_{i} \in \mathcal{B}\left( {\mathbb{R}}^{N}\right), i = 1,\ldots, n \), a \ | No |
Proposition 3.22 Two processes are equivalent if and only if they have the same law. | Proof. Let \( X, Y \) be two equivalent stochastic processes defined on \( \left( {\Omega ,\mathcal{F}, P}\right) \) and \( \left( {{\Omega }^{\prime },{\mathcal{F}}^{\prime },{P}^{\prime }}\right) \) respectively. The claim follows from Proposition A.6, observing that, by assumption,\n\n\[ P\left( {X \in \mathcal{H}\l... | No |
Proposition 3.23 A Brownian motion \( W \) on the filtered probability space \( \left( {\Omega ,\mathcal{F}, P,\left( {\mathcal{F}}_{t}\right) }\right) \) verifies the following properties:\n\n1) \( W \) has independent and stationary increments, i.e. for \( 0 \leq t \leq T \) the random variable \( {W}_{T} - {W}_{t} \... | Sketch of the proof. It is easy to prove that, if \( W \) is a Brownian motion, then it verifies 1). First of all it suffices to prove the independence of the increments: if \( N = 3 \), since\n\n\[ \n\left\{ {\left( {{W}_{{t}_{2}} - {W}_{{t}_{1}}}\right) \in H}\right\} \in {\mathcal{F}}_{{t}_{2}} \n\]\nthe claim is an... | Yes |
Proposition 3.25 Let \( X, Y \) be a.s. right-continuous stochastic processes. If \( X \) is a modification of \( Y \), then \( X, Y \) are indistinguishable. In particular we can equivalently write\n\n\[{X}_{t} = {Y}_{t}\text{ a.s. for every }t\;\text{ or }\;{X}_{t} = {Y}_{t}\text{ for every }t\text{ a.s. }\] | Proof (of Proposition 3.25). It suffices to consider the case \( Y = 0 \) . Let \( F \in \mathcal{N} \) be the set in which the paths of \( X \) are not right continuous. We set\n\n\[N = \mathop{\bigcup }\limits_{{t \in {\mathbb{R}}_{ \geq 0} \cap \mathbb{Q}}}{N}_{t} \cup F\]\n\nwhere \( {N}_{t} = \left\{ {\omega \in \... | Yes |
Lemma 3.31 Every right-continuous and adapted process is progressively measurable. | Proof. Let \( X \) be right continuous and adapted. For fixed \( t \) and \( n \in \mathbb{N} \), we set \( {X}_{t}^{\left( n\right) } = {X}_{t} \) and\n\n\[ \n{X}_{s}^{\left( n\right) } = {X}_{\frac{k + 1}{{2}^{n}}t},\;\text{ for }s \in \left\lbrack {\frac{k}{{2}^{n}}t,\frac{k + 1}{{2}^{n}}t\lbrack }\right. \text{,}\;... | Yes |
Example 3.35 Let \( M \) be a martingale and \( X \) an adapted and bounded process. Then we have\n\n\[ E\left\lbrack {{M}_{T}{X}_{t}}\right\rbrack = E\left\lbrack {{M}_{t}{X}_{t}}\right\rbrack ,\;t \leq T. \] | Indeed\n\n\[ E\left\lbrack {{M}_{t}{X}_{t}}\right\rbrack = E\left\lbrack {E\left\lbrack {{M}_{T} \mid {\mathcal{F}}_{t}}\right\rbrack {X}_{t}}\right\rbrack = E\left\lbrack {E\left\lbrack {{M}_{T}{X}_{t} \mid {\mathcal{F}}_{t}}\right\rbrack }\right\rbrack = E\left\lbrack {{M}_{T}{X}_{t}}\right\rbrack . \] | Yes |
Proposition 3.37 If \( W \) is a Brownian motion on \( \left( {\Omega ,\mathcal{F}, P,\left( {\mathcal{F}}_{t}\right) }\right) \) and \( \sigma \in \mathbb{R} \) , then\ni) \( {W}_{t} \) ;\nii) \( {W}_{t}^{2} - t \) ;\niii) \( \exp \left( {\sigma {W}_{t} - \frac{{\sigma }^{2}}{2}t}\right) \nare continuous \( {\mathcal{... | Proof. i) By Hölder's inequality,\n\n\[ E{\left\lbrack \left| {W}_{t}\right| \right\rbrack }^{2} \leq E\left\lbrack {W}_{t}^{2}\right\rbrack = t \]\n\nand so \( W \) is integrable. Further, for \( 0 \leq s \leq t \) we have\n\n\[ E\left\lbrack {{W}_{t} \mid {\mathcal{F}}_{s}}\right\rbrack = E\left\lbrack {{W}_{t} - {W}... | No |
Theorem 3.38 (Doob’s inequality) Let \( M \) be a right continuous martin- \( {\\text{gale}}^{6} \) and \( p > 1 \) . Then for every \( T \)\n\n\[ E\\left\\lbrack {\\mathop{\\sup }\\limits_{{t \\in \\left\\lbrack {0, T}\\right\\rbrack }}{\\left| {M}_{t}\\right| }^{p}}\\right\\rbrack \\leq {q}^{p}E\\left\\lbrack {\\left... | Proof. We denote by \( {\\left( {t}_{n}\\right) }_{n \\geq 0} \) an enumeration of the rational numbers in the interval \( \\lbrack 0, T\\left\\lbrack \\right. \) with \( {t}_{0} = 0 \), i.e.\n\n\[ \\mathbb{Q} \\cap \\left\\lbrack {0, T\\left\\lbrack { = \\left\{ {{t}_{0},{t}_{1},\\ldots }\\right\} .}\\right. }\\right.... | Yes |
If \( {M}_{t} = E\left\lbrack {Z \mid {\mathcal{F}}_{t}}\right\rbrack \) is the martingale in Example 3.33 with \( Z \in {L}^{2}\left( {\Omega, P}\right) \), then using Doob’s and Jensen’s inequalities we get | \[ E\left\lbrack {\mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}{\left| {M}_{t}\right| }^{2}}\right\rbrack \leq {4E}\left\lbrack {\left| {M}_{T}\right| }^{2}\right\rbrack \leq {4E}\left\lbrack {\left| Z\right| }^{2}\right\rbrack . \] | Yes |
Proposition 3.50 The random variable \( \tau \) is a stopping time if and only if\n\n\[ \n\{ \tau < t\} \in {\mathcal{F}}_{t}\n\]\n\n(3.35)\n\nfor every \( t > 0 \) . Consequently we also have \( \{ \tau = t\} ,\{ \tau \geq t\} ,\{ \tau > t\} \in {\mathcal{F}}_{t} \) . | Proof. If \( \tau \) is a stopping time, then\n\n\[ \n\{ \tau < t\} = \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}\left\{ {\tau \leq t - \frac{1}{n}}\right\}\n\]\n\nwith \( \left\{ {\tau \leq t - \frac{1}{n}}\right\} \in {\mathcal{F}}_{t - \frac{1}{n}} \subseteq {\mathcal{F}}_{t} \) . Conversely, for every \( \varepsi... | Yes |
Proposition 3.51 Let \( \tau ,{\tau }_{1} \) be stopping times. Then also\n\n\[ \tau \land {\tau }_{1} = \min \left\{ {\tau ,{\tau }_{1}}\right\} \;\text{ and }\;\tau \vee {\tau }_{1} = \max \left\{ {\tau ,{\tau }_{1}}\right\} \]\n\nare stopping times. | Proof. It suffices to observe that\n\n\[ \left\{ {\min \left\{ {\tau ,{\tau }_{1}}\right\} \leq t}\right\} = \{ \tau \leq t\} \cup \left\{ {{\tau }_{1} \leq t}\right\} \]\n\n\[ \left\{ {\max \left\{ {\tau ,{\tau }_{1}}\right\} \leq t}\right\} = \{ \tau \leq t\} \cap \left\{ {{\tau }_{1} \leq t}\right\} \] | Yes |
Corollary 3.53 Let \( X = {\left( {X}_{t}\right) }_{t \in {\mathbb{R}}_{ \geq 0}} \) be a stochastic process in \( {\mathbb{R}}^{N} \), right-continuous and \( {\mathcal{F}}_{t} \) -adapted and let \( \bar{H} \) be a closed set in \( {\mathbb{R}}^{N} \). We put\n\n\[ \nI\left( \omega \right) = \left\{ {t \geq 0 \mid {X... | Proof. We consider the sequence of open sets in \( {\mathbb{R}}^{N} \)\n\n\[ \n{H}_{n} = \left\{ {x \in {\mathbb{R}}^{N} \mid \operatorname{dist}\left( {x, H}\right) < \frac{1}{n}}\right\} ,\;n \in \mathbb{N},\n\]\n\nwhere \( \operatorname{dist}\left( {\cdot, H}\right) \) is the Euclidean distance from \( H \). The cla... | Yes |
Theorem 3.56 (Doob’s optional sampling Theorem) Let \( M \) be a right-continuous martingale and let \( {\tau }_{1},{\tau }_{2} \) be stopping times such that \( {\tau }_{1} \leq {\tau }_{2} \leq T \) a.s., with \( T > 0 \) . Then\n\n\[ \n{M}_{{\tau }_{1}} = E\left\lbrack {{M}_{{\tau }_{2}} \mid {\mathcal{F}}_{{\tau }_... | Proof. Let \( \left( {\tau }_{1, n}\right) ,\left( {\tau }_{2, n}\right) \) be sequences of discrete stopping times, constructed as in Remark 3.55, approximating \( {\tau }_{1} \) and \( {\tau }_{2} \) respectively. By the continuity assumption,\n\n\[ \n\mathop{\lim }\limits_{{n \rightarrow \infty }}{M}_{{\tau }_{i, n}... | Yes |
Theorem 3.58 Let \( X \) be a stochastic process on the space \( \left( {\Omega ,\mathcal{F}, P,\left( {\mathcal{F}}_{t}\right) }\right) \) and let \( \tau \) be an a.s. bounded stopping time. We consider the stopped process \( {X}^{\tau } \) defined by\n\n\[ \n{X}_{t}^{\tau }\left( \omega \right) = {X}_{t \land \tau \... | Proof. i) The function\n\n\[ \n\varphi : \left\lbrack {0, t}\right\rbrack \times \Omega \rightarrow \left\lbrack {0, t}\right\rbrack \times \Omega ,\;\varphi \left( {s,\omega }\right) = \left( {s \land \tau \left( \omega \right) ,\omega }\right) ,\n\]\n\nis measurable with respect to the product \( \sigma \) -algebra \... | Yes |
Lemma 3.61 If \( g \in \mathrm{{BV}} \cap C\left( \left\lbrack {a, b}\right\rbrack \right) \), then\n\n\[ \n{V}_{\left\lbrack a, b\right\rbrack }\left( g\right) = \mathop{\lim }\limits_{{\left| \varsigma \right| \rightarrow 0}}{V}_{\left\lbrack a, b\right\rbrack }\left( {g,\varsigma }\right) .\n\]\n\n(3.46) | Proof. Let us first recall that \( \left| \varsigma \right| \) denotes the refinement parameter of the partition \( \varsigma \), defined in (3.42). By contradiction, if (3.46) were not true, then there would exist a partition \( \varsigma = \left\{ {{t}_{0},\ldots ,{t}_{N}}\right\} \) of \( \left\lbrack {a, b}\right\r... | Yes |
Example 3.62 The function \( g : \left\lbrack {0,2}\right\rbrack \rightarrow \mathbb{R} \), identically zero except for \( t = 1 \) where \( g\left( 1\right) = 1 \), is such that \( {V}_{\left\lbrack 0,2\right\rbrack }\left( g\right) = 2 \) . | On the other hand, \[ {V}_{\left\lbrack 0,2\right\rbrack }\left( {g,\varsigma }\right) = 0 \] for every partition \( \varsigma \) not containing 1 . Therefore (3.46) is not true for a generic \( g \in \operatorname{BV}\left( \left\lbrack {a, b}\right\rbrack \right) \) . | No |
Theorem 3.64 A real function has bounded variation if and only if it is a difference of two monotone increasing functions. | Proof. As a consequence of the triangular inequality we have\n\n\[ \n{V}_{\left\lbrack a, b\right\rbrack }\left( {{g}_{1} + {g}_{2}}\right) \leq {V}_{\left\lbrack a, b\right\rbrack }\left( {g}_{1}\right) + {V}_{\left\lbrack a, b\right\rbrack }\left( {g}_{2}\right) \n\] \n\nand so from Example 3.60-i) it follows that th... | Yes |
Consider the interval \( \left\lbrack {0,1}\right\rbrack \) and the BV function \( g\left( t\right) = \) \( {\mathbb{1}}_{\{ T\} }\left( t\right), t \in \left\lbrack {0,1}\right\rbrack \), where \( T \in \rbrack 0,1\lbrack \) . If \( \varsigma \) is a partition of \( \left\lbrack {0,1}\right\rbrack \), we have two case... | \[ {\int }_{0}^{1}u\left( t\right) {dg}\left( t\right) = 0 \] On the other hand, if \( T \in \varsigma = \left( {{t}_{0},\ldots ,{t}_{N}}\right) \), say \( T = {t}_{n} \), then \[ S\left( {u, g,\varsigma ,\tau }\right) = u\left( {\tau }_{n}\right) \left( {g\left( {t}_{n}\right) - g\left( {t}_{n - 1}\right) }\right) + u... | Yes |
Theorem 3.70 (Itô formula) Let \( F \in {C}^{1}\left( {\left\lbrack {a, b}\right\rbrack \times \mathbb{R}}\right) \) and \( g \in \mathrm{{BV}} \cap C\left( \left\lbrack {a, b}\right\rbrack \right) \) . Then we have\n\n\[ F\left( {b, g\left( b\right) }\right) - F\left( {a, g\left( a\right) }\right) = {\int }_{a}^{b}\le... | Proof (of Theorem 3.70). For every \( \varsigma \in {\mathcal{P}}_{\left\lbrack a, b\right\rbrack } \), we have\n\n\[ F\left( {b, g\left( b\right) }\right) - F\left( {a, g\left( a\right) }\right) = \mathop{\sum }\limits_{{k = 1}}^{N}\left( {F\left( {{t}_{k}, g\left( {t}_{k}\right) }\right) - F\left( {{t}_{k - 1}, g\lef... | Yes |
Proposition 3.73 If \( g \in \mathrm{{BV}} \cap C\left( \left\lbrack {0, t}\right\rbrack \right) \) then\n\n\[ \mathop{\lim }\limits_{{\left| \varsigma \right| \rightarrow 0}}{V}_{t}^{\left( 2\right) }\left( {g,\varsigma }\right) = 0 \] | Proof. The function \( g \) is uniformly continuous on \( \left\lbrack {0, t}\right\rbrack \), consequently for every \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that\n\n\[ \left| {g\left( {t}_{k}\right) - g\left( {t}_{k - 1}\right) }\right| \leq \varepsilon \]\n\nfor every \( \varsigma = \left\{ {{t}_{0},... | Yes |
Theorem 3.74 If \( W \) is a Brownian motion, then we have\n\n\[ \mathop{\lim }\limits_{{\left| \varsigma \right| \rightarrow 0}}{V}_{t}^{\left( 2\right) }\left( {W,\varsigma }\right) = t\;\text{ in }{L}^{2}\left( {\Omega, P}\right) . \] | Proof. To unburden notations, for fixed \( t > 0 \) and the partition\n\n\[ \varsigma = \left\{ {{t}_{0},\ldots ,{t}_{N}}\right\} \in {\mathcal{P}}_{\left\lbrack 0, t\right\rbrack } \]\n\nwe set\n\n\[ {\Delta }_{k} = {W}_{{t}_{k}} - {W}_{{t}_{k - 1}}\;\text{ and }\;{\delta }_{k} = {t}_{k} - {t}_{k - 1} \]\n\nfor \( k =... | No |
Example 4.4 Integrating the simple process \( u = {\mathbb{1}}_{\rbrack 0, t\rbrack } \), we get | \[ {W}_{t} = {\int }_{0}^{t}d{W}_{s} \] | No |
For every \( u \in {\mathbb{L}}^{2} \) there exists a sequence \( \left( {u}^{n}\right) \) of simple processes in \( {\mathbb{L}}^{2} \) such that\n\n\[ \mathop{\lim }\limits_{{n \rightarrow + \infty }}E\left\lbrack {{\int }_{0}^{T}{\left( {u}_{t} - {u}_{t}^{n}\right) }^{2}{dt}}\right\rbrack = \mathop{\lim }\limits_{{n... | In particular an approximating sequence is defined by\n\n\[ {u}^{n} = \mathop{\sum }\limits_{{k = 1}}^{{{2}^{n} - 1}}\left( {\frac{1}{{t}_{k} - {t}_{k - 1}}{\int }_{{t}_{k - 1}}^{{t}_{k}}{u}_{s}{ds}}\right) {\mathbb{1}}_{\left\rbrack {t}_{k},{t}_{k + 1}\right\rbrack }, \]\n\n(4.14)\n\nwhere \( {t}_{k} \mathrel{\text{:=... | No |
Theorem 4.11 For every \( u, v \in {\mathbb{L}}^{2},\alpha \in \mathbb{R} \) and \( 0 \leq a < b < c \), we have: (1) linearity:\n\n\[{\int }_{0}^{a}\left( {\alpha {u}_{t} + \beta {v}_{t}}\right) d{W}_{t} = \alpha {\int }_{0}^{a}{u}_{t}d{W}_{t} + \beta {\int }_{0}^{a}{v}_{t}d{W}_{t}\]\n\n(2) additivity:\n\n\[{\int }_{a... | Proof. The theorem can be proved by taking the limit in the analogous relations that hold for the integral of simple stochastic processes: the details are left as an exercise. | No |
Corollary 4.13 If \( u \in {\mathbb{L}}^{2} \) and for a fixed positive \( T \) we have\n\n\[{\int }_{0}^{T}{u}_{t}d{W}_{t} = 0\]\n\nthen \( u \) is \( \left( {m \otimes P}\right) \) -equivalent to the null process on \( \left\lbrack {0, T}\right\rbrack \times \Omega \), that is\n\n\[\left\{ {\left( {t,\omega }\right) ... | Proof. The thesis follows from Itô isometry, since we have\n\n\[0 = E\left\lbrack {\left( {\int }_{0}^{T}{u}_{t}d{W}_{t}\right) }^{2}\right\rbrack = E\left\lbrack {{\int }_{0}^{T}{u}_{t}^{2}{dt}}\right\rbrack .\] | Yes |
Corollary 4.14 Let \( F \in \mathcal{F} \) and let \( u, v \in {\mathbb{L}}^{2} \) be modifications on \( F \), i.e. \( {u}_{t}\left( \omega \right) = {v}_{t}\left( \omega \right) \) for almost all \( \omega \in F \) and for every \( t \in \left\lbrack {0, T}\right\rbrack \) . If\n\n\[ \n{X}_{t} = {\int }_{0}^{t}{u}_{s... | Proof. Let us consider the approximation by simple processes \( {u}^{n},{v}^{n} \) in \( {\mathbb{L}}^{2} \) defined in (4.14). By construction \( {u}^{n} \) and \( {v}^{n} \) are modifications on \( F \) for every \( n \) . Hence it follows directly that, if\n\n\[ \n{X}_{t}^{n} = {\int }_{0}^{t}{u}_{s}^{n}d{W}_{s},\;{... | Yes |
Example 4.15 Let us consider a process of the form\n\n\[ \n{S}_{t} = {S}_{0} + {\int }_{0}^{t}\mu \left( s\right) {ds} + {\int }_{0}^{t}\sigma \left( s\right) d{W}_{s} \n\]\n\nwhere \( {S}_{0} \in \mathbb{R} \) and \( \mu ,\sigma \in {L}^{2}\left( \left\lbrack {0, T}\right\rbrack \right) \) are deterministic functions.... | \[ \nE\left\lbrack {S}_{t}\right\rbrack = {S}_{0} + {\int }_{0}^{t}\mu \left( s\right) {ds} \n\]\n\nand\n\n\[ \n\operatorname{var}\left( {S}_{t}\right) = E\left\lbrack {\left( {S}_{t} - {S}_{0} - {\int }_{0}^{t}\mu \left( s\right) ds\right) }^{2}\right\rbrack = \n\]\n\n(by Itô isometry)\n\n\[ \n= {\int }_{0}^{t}\sigma ... | Yes |
Given \( u \in {\mathbb{L}}^{2} \), the process \[ {X}_{t} = {\int }_{0}^{t}{u}_{s}d{W}_{s},\;t \geq 0, \] is \( {L}^{2} \) -continuous at every point. | Indeed, if \( t > {t}_{0} \), \[ E\left\lbrack {\left( {X}_{t} - {X}_{{t}_{0}}\right) }^{2}\right\rbrack = E\left\lbrack {\left( {\int }_{{t}_{0}}^{t}{u}_{s}d{W}_{s}\right) }^{2}\right\rbrack = \] (by Itô isometry) \[ = {\int }_{{t}_{0}}^{t}E\left\lbrack {u}_{s}^{2}\right\rbrack {ds} \rightarrow 0,\;\text{ as }t \right... | Yes |
Proposition 4.20 Let \( u \in {\mathbb{L}}^{2} \) be an \( {L}^{2} \) -continuous process on \( \left\lbrack {0, T}\right\rbrack \) . If we put\n\n\[ \n{u}^{\left( \varsigma \right) } = \mathop{\sum }\limits_{{k = 1}}^{N}{u}_{{t}_{k - 1}}{\mathbb{1}}_{\left\rbrack {t}_{k - 1},{t}_{k}\right\rbrack }\n\]\n\nwhere \( \var... | Proof. For every \( \varepsilon > 0 \), there exists \( {}^{2}{\delta }_{\varepsilon } > 0 \) such that, if \( \left| \varsigma \right| < {\delta }_{\varepsilon } \), then we have\n\n\[ \n{\int }_{0}^{T}E\left\lbrack {\left( {u}_{t} - {u}_{t}^{\left( \varsigma \right) }\right) }^{2}\right\rbrack {dt} = \mathop{\sum }\l... | Yes |
Proposition 4.22 Given \( u \in {\mathbb{L}}^{2}\left( {\mathcal{F}}_{t}\right) \), we set\n\n\[ \n{X}_{t} = {\int }_{0}^{t}{u}_{s}d{W}_{s},\;t \in \left\lbrack {0, T}\right\rbrack .\n\]\n\n(4.23)\n\nIf \( \tau \) is an \( \left( {\mathcal{F}}_{t}\right) \) -stopping time such that \( 0 \leq \tau \leq T \) a.s. then \(... | Proof. It is clear that, by definition of stopping time, the process \( \left( {{u}_{t}{\mathbb{1}}_{\{ t \leq \tau \} }}\right) \) belongs to \( {\mathbb{L}}^{2} \) and in particular is adapted. We put\n\n\[ \nY = {\int }_{0}^{T}{u}_{s}{\mathbb{1}}_{\{ s \leq \tau \} }d{W}_{s}\n\]\n\nand we prove that\n\n\[ \n{X}_{\ta... | Yes |
Corollary 4.23 Let \( {t}_{0} \in \left\lbrack {0, T\left\lbrack \right. }\right. \) and \( \tau \in \left\lbrack {{t}_{0}, T}\right\rbrack \) be a stopping time. If \( u, v \in {\mathbb{L}}^{2} \) then we have\n\n\[ E\left\lbrack {{\int }_{{t}_{0}}^{\tau }{u}_{t}d{W}_{t} \mid {\mathcal{F}}_{{t}_{0}}}\right\rbrack = 0 ... | Proof. By (4.24) we have\n\n\[ {\int }_{{t}_{0}}^{\tau }{u}_{t}d{W}_{t} = {\int }_{{t}_{0}}^{T}{u}_{t}{\mathbb{1}}_{\{ t \leq \tau \} }d{W}_{t} \]\n\nwith \( {u}_{t}{\mathbb{1}}_{\{ t \leq \tau \} } \in {\mathbb{L}}^{2} \) and so the claim follows from Theorem 4.11. | Yes |
Proposition 4.24 Let \( X \) be as in (4.27) with \( u \in {\mathbb{L}}^{2} \) . Then for any \( t > 0 \) , there exists the limit\n\n\[ \mathop{\lim }\limits_{\substack{{\left| \varsigma \right| \rightarrow 0} \\ {\varsigma \in {\mathcal{P}}_{\left\lbrack 0, t\right\rbrack }} }}\mathop{\sum }\limits_{{k = 1}}^{N}{\lef... | Proof. If \( u \) is a simple \( {\mathbb{L}}^{2} \) -process,(4.28) can be proved by proceeding as in Theorem 3.74. In general the claim follows approximating \( X \) by integrals of simple processes.\n\nNext we verify that \( {X}^{2} - \langle X\rangle \) is a martingale. For every \( 0 \leq s < t \) we have\n\n\[ E\... | Yes |
Theorem 4.26 (Doob-Meyer decomposition theorem) For every \( M = \) \( {\left( {M}_{t}\right) }_{t \in \left\lbrack {0, T}\right\rbrack } \in {\mathcal{M}}_{\mathrm{c}}^{2}\left( {\mathcal{F}}_{t}\right) \) there exists a unique (up to indistinguishability) increasing continuous process \( A \) such that \( {A}_{0} = 0... | The proof of Theorem 4.26 is based on a discrete approximation procedure: we observe that, if \( \left( {M}_{n}\right) \) is a real discrete martingale, then the process \( \left( {A}_{n}\right) \) defined by \( {A}_{0} = 0 \) and \[ {A}_{n} = \mathop{\sum }\limits_{{k = 1}}^{n}{\left( {M}_{k} - {M}_{k - 1}\right) }^{2... | Yes |
A particularly important case is when\n\n\[ \n{X}_{t} = {\int }_{0}^{t}{u}_{s}d{W}_{s},\;{Y}_{t} = {\int }_{0}^{t}{v}_{s}d{W}_{s}, \n\]\n\nwith \( u, v \in {\mathbb{L}}^{2} \) . Then, proceeding as in Proposition 4.24, we can show that\n\n\[ \n{X}_{t}{Y}_{t} - {\int }_{0}^{t}{u}_{s}{v}_{s}d{W}_{s} \n\]\n\nis a martinga... | Proceeding as in Theorem 3.74, we can also directly prove that\n\n\[ \n{\int }_{0}^{t}{u}_{s}{v}_{s}d{W}_{s} = \mathop{\lim }\limits_{\substack{{\left| \varsigma \right| \rightarrow 0} \\ {\varsigma \in {\mathcal{P}}_{\left\lbrack 0, t\right\rbrack }} }}\mathop{\sum }\limits_{{k = 1}}^{N}\left( {{X}_{{t}_{k}} - {X}_{{t... | Yes |
We go back to Example 4.15 and consider\n\n\[ \n{S}_{t} = {S}_{0} + {\int }_{0}^{t}\mu \left( s\right) {ds} + {\int }_{0}^{t}\sigma \left( s\right) d{W}_{s} \n\]\n\nwith \( \mu ,\sigma \in {L}^{2}\left( \left\lbrack {0, T}\right\rbrack \right) \) deterministic functions. We proved that\n\n\[ \n\operatorname{var}\left( ... | Now we observe that the process \( {S}_{0} + {\int }_{0}^{t}\mu \left( s\right) {ds} \) is continuous and has bounded variation by Example 3.60-iii). Therefore, according to Definition 4.31 and formula (4.29), we have\n\n\[ \n\langle S{\rangle }_{t} = \operatorname{var}\left( {S}_{t}\right) ,\;t \in \left\lbrack {0, T}... | Yes |
Every stochastic process that is progressively measurable and has a.s. continuous paths belongs to \( {\mathbb{L}}_{\text{loc }}^{2} \) . In particular \( \exp \left( {W}_{t}^{4}\right) \), where \( W \) is a Brownian motion, belongs to \( {\mathbb{L}}_{\text{loc }}^{2} \) . | I) Given \( u \in {\mathbb{L}}_{\text{loc }}^{2} \), the process \( {}^{7} \)\n\[ \n{A}_{t} = {\int }_{0}^{t}{u}_{s}^{2}{ds},\;t \in \left\lbrack {0, T}\right\rbrack \n\]\n\nis continuous and adapted to the filtration. Indeed it is enough to observe that \( u \) can be approximated pointwise by a sequence of simple and... | Yes |
Proposition 4.39 If \( M \in {\mathcal{M}}_{\mathrm{c},\text{ loc }} \) and\n\n\[ \mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| {M}_{t}\right| \in {L}^{1}\left( {\Omega, P}\right) \]\n\nthen \( M \) is a martingale. In particular every bounded \( {}^{8}M \in {\mathcal{M}}_{\mathrm{c},\text{ lo... | Proof. The claim follows directly from (4.41), applying the dominated convergence theorem for conditional expectation. | No |
Proposition 4.40 Every continuous non-negative local martingale \( M \) is also a super-martingale. Further, if\n\n\[ E\left\lbrack {M}_{T}\right\rbrack = E\left\lbrack {M}_{0}\right\rbrack \]\n\n(4.42)\n\nthen \( {\left( {M}_{t}\right) }_{0 \leq t \leq T} \) is a martingale. | Proof. Applying Fatou's lemma for conditional expectation to (4.41), we get\n\n\[ {M}_{s} \geq E\left\lbrack {{M}_{t} \mid {\mathcal{F}}_{s}}\right\rbrack ,\;0 \leq s \leq t \leq T \]\n\n(4.43)\n\nand this proves the first part of the claim.\n\nBy taking the expectation in the previous relation we get\n\n\[ E\left\lbra... | Yes |
Proposition 4.41 If \( M \in {\mathcal{M}}_{\mathrm{c},\text{ loc }} \) and \( \tau \) is a stopping time, then also \( {M}_{t \land \tau } \in \) \( {\mathcal{M}}_{\mathrm{c},\mathrm{{loc}}} \) . | Proof. If \( \left( {\tau }_{n}\right) \) is a localizing sequence for \( M \) and \( {X}_{t} = {M}_{t \land \tau } \), we have\n\n\[ \n{X}_{t \land {\tau }_{n}} = {M}_{\left( {t \land \tau }\right) \land {\tau }_{n}} = {M}_{\left( {t \land {\tau }_{n}}\right) \land \tau }.\n\]\n\nConsequently, by Theorem 3.58 and sinc... | Yes |
Theorem 4.42 We have:\n\ni) if \( u \in {\mathbb{L}}^{2} \), then \( X \in {\mathcal{M}}_{\mathrm{c}}^{2} \) ;\n\nii) if \( u \in {\mathbb{L}}_{\text{loc }}^{2} \), then \( X \in {\mathcal{M}}_{\mathrm{c},\text{ loc }} \) and a localizing sequence for \( X \) is given by\n\n\[{\tau }_{n} = \inf \left\{ {t \in \left\lbr... | Proof. We only prove \( {ii} \) ). We saw at the beginning of Paragraph 4.4 that \( \left( {\tau }_{n}\right) \) in (4.45) is an increasing sequence of stopping times such that \( {\tau }_{n} \rightarrow T \) a.s. for \( n \rightarrow \infty \) .\n\nBy Definition 4.35, on \( {F}_{k} = \left\{ {{A}_{T} \leq k}\right\} \... | Yes |
Given \( u \in {\mathbb{L}}_{\text{loc }}^{2} \), let \( X \) and \( A \) be the processes in (4.44). Then \( {X}^{2} - A \) is a continuous local martingale: \( A \) is called quadratic variation process of \( X \) and we write \( A = \langle X\rangle \) . | Let us consider the localizing sequence \( \left( {\tau }_{n}\right) \) for \( X \) defined in Theorem 4.42. We proved that (cf. (4.46))\n\n\[ \n{X}_{t \land {\tau }_{n}} = {\int }_{0}^{t}{u}_{s}{\mathbb{1}}_{\left\{ s \leq {\tau }_{n}\right\} }d{W}_{s} \n\]\n\nwith \( {u}_{s}{\mathbb{1}}_{\left\{ s \leq {\tau }_{n}\ri... | Yes |
Theorem 4.47 (Burkholder-Davis-Gundy’s inequalities) For any \( p > \) 0 there exist two positive constants \( {\lambda }_{p},{\Lambda }_{p} \) such that\n\n\[ \n{\lambda }_{p}E\left\lbrack {\langle M{\rangle }_{\tau }^{p}}\right\rbrack \leq E\left\lbrack {\mathop{\sup }\limits_{{t \in \left\lbrack {0,\tau }\right\rbra... | For the proof we refer, for example, to Theorem 3.3.28 in [201]. | No |
Corollary 4.48 If \( u \in {\mathbb{L}}_{\text{loc }}^{2} \) and\n\n\[ E\left\lbrack {\left( {\int }_{0}^{T}{u}_{t}^{2}dt\right) }^{\frac{1}{2}}\right\rbrack < \infty \]\n\nthen the process\n\n\[ {\int }_{0}^{t}{u}_{s}d{W}_{s},\;t \in \left\lbrack {0, T}\right\rbrack \] \nis a martingale. | Proof. First of all we observe that, by Hölder's inequality, we have\n\n\[ E\left\lbrack {\left( {\int }_{0}^{T}{u}_{t}^{2}dt\right) }^{\frac{1}{2}}\right\rbrack \leq E{\left\lbrack {\int }_{0}^{T}{u}_{t}^{2}dt\right\rbrack }^{\frac{1}{2}} \]\n\nand so condition (4.48) is weaker than the integrability condition in the ... | Yes |
Theorem 5.5 (Itô formula) Let \( f \in {C}^{2}\left( \mathbb{R}\right) \) and let \( W \) be a real Brownian motion. Then \( f\left( W\right) \) is an Itô process and we have\n\n\[ \n{df}\left( {W}_{t}\right) = {f}^{\prime }\left( {W}_{t}\right) d{W}_{t} + \frac{1}{2}{f}^{\prime \prime }\left( {W}_{t}\right) {dt}. \n\] | Proof. The great news of the Itô formula with respect to (3.53) is the presence of a \ | No |
Theorem 5.9 (Itô formula) Let \( X \) be the Itô process in (5.9) and \( f = \) \( f\left( {t, x}\right) \in {C}^{1,2}\left( {\mathbb{R}}^{2}\right) \) . Then the stochastic process\n\n\[ \n{Y}_{t} = f\left( {t,{X}_{t}}\right) \n\]\n\nis an Itô process and we have\n\n\[ \n{df}\left( {t,{X}_{t}}\right) = {\partial }_{t}... | Remark 5.10 Since, by Corollary 5.2, we have\n\n\[ \nd\langle X{\rangle }_{t} = {\sigma }_{t}^{2}{dt} \n\]\n\nFormula (5.11) can be written more explicitly as follows\n\n\[ \n{df} = \left( {{\partial }_{t}f + {\mu }_{t}{\partial }_{x}f + \frac{1}{2}{\sigma }_{t}^{2}{\partial }_{xx}f}\right) {dt} + {\sigma }_{t}{\partia... | Yes |
If \( f\left( {t, x}\right) = {tx} \) and \( X = W \) is a Brownian motion, we have \[ d\left( {t{W}_{t}}\right) = {W}_{t}{dt} + {td}{W}_{t} \] | In integral form we get \[ t{W}_{t} = {\int }_{0}^{t}{W}_{s}{ds} + {\int }_{0}^{t}{sd}{W}_{s} \] | Yes |
Given \( u \in {\mathbb{L}}_{\text{loc }}^{2} \), we set\n\n\[ d{Y}_{t} = {u}_{t}d{W}_{t} - \frac{1}{2}{u}_{t}^{2}{dt} \]\n\nand consider the process \( {e}^{Y} \) . | By the Itô formula we have\n\n\[ d{e}^{{Y}_{t}} = {e}^{{Y}_{t}}d{Y}_{t} + \frac{1}{2}{e}^{{Y}_{t}}d\langle Y{\rangle }_{t} = {u}_{t}{e}^{{Y}_{t}}d{W}_{t} \]\n\nTherefore \( {e}^{Y} \) is a local martingale, called exponential martingale. By Proposition 4.40, since it is a positive process, \( {e}^{Y} \) is also a super... | Yes |
Proposition 5.13 If \( \mu \in {L}^{1} \) and \( \sigma \in {L}^{2} \) are deterministic functions, then the process defined by\n\n\[ d{S}_{t} = \mu \left( t\right) {dt} + \sigma \left( t\right) d{W}_{t} \]\n\nhas normal distribution with\n\n\[ E\left\lbrack {S}_{t}\right\rbrack = {S}_{0} + {\int }_{0}^{t}\mu \left( s\... | Proof. By Theorem A. 89 and recalling Example 4.15, it is enough to prove that, for every \( t \), we have\n\n\[ E\left\lbrack {e}^{{i\xi }{S}_{t}}\right\rbrack = \exp \left( {{i\xi }\left( {{S}_{0} + {\int }_{0}^{t}\mu \left( s\right) {ds}}\right) - \frac{{\xi }^{2}}{2}{\int }_{0}^{t}{\sigma }^{2}\left( s\right) {ds}}... | No |
Lemma 5.16 Let \( W = \left( {{W}^{1},\ldots ,{W}^{d}}\right) \) be a \( d \) -dimensional Brownian motion on \( \left( {\Omega ,\mathcal{F}, P,\left( {\mathcal{F}}_{t}\right) }\right) \) . Then, for every \( i = 1,\ldots, d \) we have\n\n1) \( {W}^{i} \) is a real Brownian motion on \( \left( {\Omega ,\mathcal{F}, P,\... | Proof. The claim follows from the fact that, for \( x = \left( {{x}_{1},\ldots ,{x}_{d}}\right) \in {\mathbb{R}}^{d} \) and \( h > 0 \), we have\n\n\[ \Gamma \left( {h, x}\right) \mathrel{\text{:=}} \frac{1}{{\left( 2\pi h\right) }^{\frac{d}{2}}}\exp \left( {-\frac{{\left| x\right| }^{2}}{2h}}\right) = \mathop{\prod }\... | Yes |
Given an \( \left( {N \times d}\right) \) -dimensional matrix \( \alpha \) with constant real entries, we set\n\n\[ \varrho = \alpha {\alpha }^{ * }.\]\n\nEvidently \( \varrho = \left( {\varrho }^{ij}\right) \) is an \( \left( {N \times N}\right) \) -dimensional matrix, symmetric, positive semi-definite and \( {\varrho... | By Remark A.93, we have that\n\n\[ {B}_{t} \sim {\mathcal{N}}_{\mu, t\varrho } \] \n\nand in particular\n\n\[ \operatorname{Cov}\left( {B}_{t}\right) = t\varrho \] \n\n(5.28) \n\ni.e.\n\[ E\left\lbrack {\left( {{B}_{t}^{i} - {\mu }_{i}}\right) \left( {{B}_{t}^{j} - {\mu }_{j}}\right) }\right\rbrack = t{\varrho }^{ij}. ... | Yes |
Lemma 5.18 For every \( u, v \in {\mathbb{L}}^{2}\left( {\mathcal{F}}_{t}\right) ,{t}_{0} < t \) and \( i \neq j \), we have\n\n\[ E\left\lbrack {{\int }_{{t}_{0}}^{t}{u}_{s}d{W}_{s}^{i}{\int }_{{t}_{0}}^{t}{v}_{s}d{W}_{s}^{j} \mid {\mathcal{F}}_{{t}_{0}}}\right\rbrack = 0. \] | Proof. By an approximation argument it is enough to consider simple \( u, v \) : since the proof is similar to that of Theorem 4.5, we employ analogous notations. We have\n\n\[ E\left\lbrack {{\int }_{{t}_{0}}^{t}{u}_{s}d{W}_{s}^{i}{\int }_{{t}_{0}}^{t}{v}_{s}d{W}_{s}^{j} \mid {\mathcal{F}}_{{t}_{0}}}\right\rbrack \]\n... | Yes |
Lemma 5.22 Consider an Itô process \( X \) of the form (5.31) and set\n\n\[ \mathcal{C} = \sigma {\sigma }^{ * } \]\n\n(5.32)\n\nThen we have\n\n\[ {\left\langle {X}^{i},{X}^{j}\right\rangle }_{t} = {\int }_{0}^{t}{\mathcal{C}}_{s}^{ij}{ds},\;t \geq 0 \]\n\n(5.33)\n\nor, in differential notation,\n\n\[ d\langle X{\rang... | In practice, given two Itô processes \( X, Y \) in \( {\mathbb{R}}^{N} \), the computation of \( \langle X, Y{\rangle }_{t} \) can be handled by applying the following \ | No |
Example 5.23 For \( X = \left( {{X}^{1},{X}^{2}}\right) \) defined by\n\n\[ d{X}_{t}^{1} = {\mu }_{t}{dt} + {\alpha }_{t}d{W}_{t}^{1} + {\beta }_{t}d{W}_{t}^{2} \]\n\n\[ d{X}_{t}^{2} = {\nu }_{t}{dt} + {\gamma }_{t}d{W}_{t}^{1} + {\delta }_{t}d{W}_{t}^{2} \]\n\nwe have\n\n\[ \sigma = \left( \begin{array}{ll} \alpha & \... | Then we have\n\n\[ d{\left\langle {X}^{1}\right\rangle }_{t} = d{\left\langle {X}^{1},{X}^{1}\right\rangle }_{t} = \left( {{\alpha }_{t}^{2} + {\beta }_{t}^{2}}\right) {dt},\;d{\left\langle {X}^{1},{X}^{2}\right\rangle }_{t} = \left( {{\alpha }_{t}{\gamma }_{t} + {\beta }_{t}{\delta }_{t}}\right) {dt}. \] | Yes |
Theorem 5.25 Let \( X \) be an Itô process of the form (5.31) and \( f = f\left( {t, x}\right) \in \) \( {C}^{1,2}\left( {\mathbb{R} \times {\mathbb{R}}^{N}}\right) \) . Then\n\n\[ \n{df} = {\partial }_{t}{fdt} + \nabla f \cdot d{X}_{t} + \frac{1}{2}\mathop{\sum }\limits_{{i, j = 1}}^{N}{\partial }_{{x}_{i}{x}_{j}}{fd}... | In compact form, if we put \( \mathcal{C} = \sigma {\sigma }^{ * } \) and recall Lemma 5.22, then formula (5.34) becomes\n\n\[ \n{df} = \left( {\frac{1}{2}\mathop{\sum }\limits_{{i, j = 1}}^{N}{\mathcal{C}}_{t}^{ij}{\partial }_{{x}_{i}{x}_{j}}f + {\mu }_{t} \cdot \nabla f + {\partial }_{t}f}\right) {dt} + \nabla f \cdo... | Yes |
Let \( W \) be a \( d \) -dimensional Brownian motion and \( f = f\left( {t, x}\right) \in {C}^{1,2}\left( {\mathbb{R} \times {\mathbb{R}}^{d}}\right) \) . Then we have\n\n\[ \n{df}\left( {t,{W}_{t}}\right) = \left( {{\partial }_{t}f\left( {t,{W}_{t}}\right) + \frac{1}{2}\mathop{\sum }\limits_{{i = 1}}^{d}{\partial }_{... | \[ \n= \left( {{\partial }_{t}f\left( {t,{W}_{t}}\right) + \frac{1}{2}\bigtriangleup f\left( {t,{W}_{t}}\right) }\right) {dt} + \nabla f\left( {t,{W}_{t}}\right) \cdot d{W}_{t} \n\]\n\n(5.36)\n\nwhere \( \bigtriangleup \) denotes the Laplace operator in \( {\mathbb{R}}^{d} \) . | Yes |
Let \( B = \left( {{B}^{1},\ldots ,{B}^{N}}\right) = \) \( {\alpha W} \) be a correlated Brownian motion with correlation matrix \( \varrho = \alpha {\alpha }^{ * } \) . We consider the Itô processes in \( \mathbb{R} \)\n\n\[ d{X}_{t}^{i} = {\mu }_{t}^{i}{dt} + {\sigma }_{t}^{i}d{B}_{t}^{i},\;i = 1,\ldots, N. \]\n\nThe... | \[ {df} = \left( {\frac{1}{2}\mathop{\sum }\limits_{{i, j = 1}}^{N}{\varrho }^{ij}{\sigma }_{t}^{i}{\sigma }_{t}^{j}{\partial }_{{x}_{i}{x}_{j}}f + \mathop{\sum }\limits_{{i = 1}}^{N}{\mu }_{t}^{i}{\partial }_{{x}_{i}}f + {\partial }_{t}f}\right) {dt} + \mathop{\sum }\limits_{{i = 1}}^{N}{\partial }_{{x}_{i}}f{\sigma }... | Yes |
Example 5.28 Let \( \\left( {{W}^{1},{W}^{2}}\\right) \) be a 2-dimensional Brownian motion and\n\n\[ f\\left( {t,{x}_{1},{x}_{2}}\\right) = {x}_{1}{x}_{2} \]\n\nThen\n\n\[ d\\left( {{W}^{1}{W}^{2}}\\right) = {W}^{1}d{W}^{2} + {W}^{2}d{W}^{1} \] | Further, for \( f\\left( {t,{x}_{1},{x}_{2}}\\right) = {x}_{1}^{2}{x}_{2} \) we have\n\n\[ d\\left( {{\\left( {W}^{1}\\right) }^{2}{W}^{2}}\\right) = {\\left( {W}^{1}\\right) }^{2}d{W}^{2} + 2{W}^{1}{W}^{2}d{W}^{1} + {W}^{2}{dt}. \] | No |
We consider an Itô process with \( N = 2 \) and \( d = 1 \) :\n\n\[ d{X}_{t}^{i} = {\mu }_{t}^{i}{dt} + {\sigma }_{t}^{i}d{W}_{t},\;i = 1,2. \] | In this case\n\n\[ d\left( {{X}_{t}^{1}{X}_{t}^{2}}\right) = {X}_{t}^{1}d{X}_{t}^{2} + {X}_{t}^{2}d{X}_{t}^{1} + \frac{1}{2}\left( {d{\left\langle {X}^{1},{X}^{2}\right\rangle }_{t} + d{\left\langle {X}^{2},{X}^{1}\right\rangle }_{t}}\right) \]\n\n\[ = {X}_{t}^{1}d{X}_{t}^{2} + {X}_{t}^{2}d{X}_{t}^{1} + {\sigma }_{t}^{... | Yes |
Given \( \xi \in {\mathbb{R}}^{N} \), we consider the process \( {Z}_{t}^{\xi } = \exp \left( {{\int }_{0}^{t}\xi \cdot {\sigma }_{s}d{W}_{s} - \frac{1}{2}{\int }_{0}^{t}\left\langle {{\mathcal{C}}_{s}\xi ,\xi }\right\rangle {ds}}\right) \) | By the Itô formula we get \( d{Z}_{t}^{\xi } = {Z}_{t}^{\xi }\xi \cdot d{X}_{t} = {Z}_{t}^{\xi }\xi \cdot {\sigma }_{t}d{W}_{t} \) and so \( {Z}^{\xi } \) is a positive local martingale, called exponential martingale (consistently with the 1-dimensional case, treated in Example 5.12). | No |
Proposition 5.32 If \( \mu \in {L}^{1} \) and \( \sigma \in {L}^{2} \) are deterministic functions, then the process defined by\n\n\[ d{S}_{t} = \mu \left( t\right) {dt} + \sigma \left( t\right) d{W}_{t},\;{S}_{0} = x \in \mathbb{R}, \]\n\nhas multi-normal distribution with\n\n\[ E\left\lbrack {S}_{t}\right\rbrack = x ... | The proof is analogous to that of the one-dimensional case and is therefore left as an exercise. | No |
Theorem 5.33 Let \( X \) be a continuous process in \( {\mathbb{R}}^{d} \) on \( \left( {\Omega ,\mathcal{F}, P,\left( {\mathcal{F}}_{t}\right) }\right) \) such that \( {X}_{0} = 0 \) a.s. If for every \( \xi \in {\mathbb{R}}^{d} \) the process\n\n\[ \n{Z}_{t}^{\xi } = {e}^{{i\xi } \cdot {X}_{t} + \frac{{\left| \xi \ri... | Proof. We just have to verify that:\n\ni) \( {X}_{t} - {X}_{s} \) has normal distribution \( {\mathcal{N}}_{0,\left( {t - s}\right) {I}_{d}} \) ;\n\nii) \( {X}_{t} - {X}_{s} \) is independent of \( {\mathcal{F}}_{s} \) .\n\nBy (5.40) we have that\n\n\[ \nE\left\lbrack {{e}^{{i\xi } \cdot \left( {{X}_{t} - {X}_{s}}\righ... | Yes |
Corollary 5.35 Let \( \alpha = \left( {{\alpha }^{1},\ldots ,{\alpha }^{d}}\right) \) a progressively measurable process in \( {\mathbb{R}}^{d} \) such that\n\n\[{\left| {\alpha }_{t}\right| }^{2} = \mathop{\sum }\limits_{{i = 1}}^{d}{\left( {\alpha }_{t}^{i}\right) }^{2} = 1\;t \geq 0,\text{ a.s. }\n\]\nand let \( W \... | Proof. By assumption \( \alpha \in {\mathbb{L}}^{2} \) and so \( B \) is a continuous martingale. Further, we have that\n\n\[\langle B{\rangle }_{t} = {\int }_{0}^{t}{\left| {\alpha }_{s}\right| }^{2}{ds} = t\]\n\nTherefore the hypotheses of Theorem 5.34 are verified and this concludes the proof. | Yes |
Theorem 5.37 Let \( f \in {W}_{\text{loc }}^{2, p}\left( {\mathbb{R}}^{N}\right) \) with \( p > 1 + \frac{N}{2} \) . Then we have\n\n\[ f\left( {W}_{t}\right) = f\left( 0\right) + {\int }_{0}^{t}\nabla f\left( {W}_{s}\right) \cdot d{W}_{s} + \frac{1}{2}{\int }_{0}^{t}\bigtriangleup f\left( {W}_{s}\right) {ds}. \] | The proof of the theorem is based upon the following lemmas. | No |
Lemma 5.38 Let\n\n\\[ \Gamma \\left( {t, x}\\right) = \\frac{1}{{\\left( 2\\pi t\\right) }^{\\frac{N}{2}}}\\exp \\left( {-\\frac{{\\left| x\\right| }^{2}}{2t}}\\right) ,\\;t > 0, x \\in {\\mathbb{R}}^{N}, \\]\n\nbe the density of the \\( N \\) -dimensional Brownian motion. Then\n\n\\[ \Gamma \\in {L}^{q}\\left( {\\rbra... | Proof. We have\n\n\\[ {\\int }_{0}^{T}{\\int }_{{\\mathbb{R}}^{N}}{\\Gamma }^{q}\\left( {t, x}\\right) {dxdt} = {\\int }_{0}^{T}{\\int }_{{\\mathbb{R}}^{N}}\\frac{1}{{\\left( 2\\pi t\\right) }^{\\frac{Nq}{2}}}\\exp \\left( {-\\frac{q{\\left| x\\right| }^{2}}{2t}}\\right) {dxdt} = \\]\n\n(by the change of variables \\( ... | Yes |
Lemma 5.39 Assume that \( f \in {W}^{2, p}\left( {\mathbb{R}}^{N}\right) \), with \( p > 1 + \frac{N}{2} \) . Then \( f \) is (Hölder) continuous and we have:\ni) if \( p \leq N \) then \( {\left| \nabla f\right| }^{2} \in {L}^{q}\left( {\mathbb{R}}^{N}\right) \) for some \( q > 1 + \frac{N}{2} \);\nii) if \( p > N \) ... | Proof. If \( p \geq N \) the thesis follows from the Sobolev-Morrey embedding Theorem A.168. If \( 1 + \frac{N}{2} < p < N \) then necessarily \( N > 2 \) and, again by Theorem A. 168, we have \( \nabla \widetilde{f} \in {L}^{2q}\left( {\mathbb{R}}^{N}\right) \) with\n\n\[ \n{2q} = \frac{pN}{N - p} = \frac{N}{\frac{N}{... | Yes |
Theorem 5.42 There exists a two-parameter stochastic process\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 properties:\n\n i) \( {L}_{t}\left( x\right) \) is \... | For the proof of Theorem 5.42 we refer, for example, to Karatzas-Shreve [201], p. 207. | No |
Theorem 5.45 (Tanaka formula) For every \( K \in \mathbb{R} \) we have\n\n\[ \n{\left( {W}_{t} - K\right) }^{ + } = {\left( {W}_{0} - K\right) }^{ + } + {\int }_{0}^{t}{\mathbb{1}}_{\lbrack K, + \infty \lbrack }\left( {W}_{s}\right) d{W}_{s} + \frac{1}{2}{L}_{t}\left( K\right) .\n\] | Proof (of Theorem 5.45). We construct a regularizing sequence for \( f\left( x\right) = \) \( {\left( x - K\right) }^{ + } \) using the mollifiers \( {\varrho }_{n} \) :\n\n\[ \n{f}_{n}\left( x\right) = {\int }_{\mathbb{R}}{\varrho }_{n}\left( {x - y}\right) {\left( y - K\right) }^{ + }{dy}.\n\]\n\nWe recall that, by T... | Yes |
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