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The job of an actuary is to gather and analyze data that will help them determine the probability of a catastrophic event occurring, such as a death or financial loss, and the expected impact of the event. \(\mathrm{BESQ}(\alpha)\) \((Y^{2},W^{2})\) Fac. A polynomial in one variable (i.e., a univariate polynomial) with constant coefficients is given by a_nx^n+.+a_2x^2+a_1x+a_0. \(t<\tau\), where Soc. The coefficient in front of \(x_{i}^{2}\) on the left-hand side is \(-\alpha_{ii}+\phi_{i}\) (recall that \(\psi_{(i),i}=0\)), which therefore is zero. For (ii), first note that we always have \(b(x)=\beta+Bx\) for some \(\beta \in{\mathbb {R}}^{d}\) and \(B\in{\mathbb {R}}^{d\times d}\). The degree of a polynomial in one variable is the largest exponent in the polynomial. The simple polynomials used are x, x 2, , x k. We can obtain orthogonal polynomials as linear combinations of these simple polynomials. 1655, pp. The condition \({\mathcal {G}}q=0\) on \(M\) for \(q(x)=1-{\mathbf{1}}^{\top}x\) yields \(\beta^{\top}{\mathbf{1}}+ x^{\top}B^{\top}{\mathbf{1}}= 0\) on \(M\). (eds.) \(b:{\mathbb {R}}^{d}\to{\mathbb {R}}^{d}\) Lecture Notes in Mathematics, vol. Polynomials are also "building blocks" in other types of mathematical expressions, such as rational expressions. Econom. Therefore, the random variable inside the expectation on the right-hand side of(A.2) is strictly negative on \(\{\rho<\infty\}\). The dimension of an ideal \(I\) of \({\mathrm{Pol}} ({\mathbb {R}}^{d})\) is the dimension of the quotient ring \({\mathrm {Pol}}({\mathbb {R}}^{d})/I\); for a definition of the latter, see Dummit and Foote [16, Sect. If a person has a fixed amount of cash, such as $15, that person may do simple polynomial division, diving the $15 by the cost of each gallon of gas. Given any set of polynomials \(S\), its zero set is the set. given by. 34, 15301549 (2006), Ging-Jaeschke, A., Yor, M.: A survey and some generalizations of Bessel processes. J. There exists an \end{aligned}$$, \(\lim_{t\uparrow\tau}Z_{t\wedge\rho_{n}}\), \(2 {\mathcal {G}}p - h^{\top}\nabla p = \alpha p\), \(\alpha\in{\mathrm{Pol}}({\mathbb {R}}^{d})\), $$ \log p(X_{t}) = \log p(X_{0}) + \frac{\alpha}{2}t + \int_{0}^{t} \frac {\nabla p^{\top}\sigma(X_{s})}{p(X_{s})}{\,\mathrm{d}} W_{s} $$, \(b:{\mathbb {R}}^{d}\to{\mathbb {R}}^{d}\), \(\sigma:{\mathbb {R}}^{d}\to {\mathbb {R}}^{d\times d}\), \(\|b(x)\|^{2}+\|\sigma(x)\|^{2}\le\kappa(1+\|x\|^{2})\), \(Y_{t} = Y_{0} + \int_{0}^{t} b(Y_{s}){\,\mathrm{d}} s + \int_{0}^{t} \sigma(Y_{s}){\,\mathrm{d}} W_{s}\), $$ {\mathbb {P}}\bigg[ \sup_{s\le t}\|Y_{s}-Y_{0}\| < \rho\bigg] \ge1 - t c_{1} (1+{\mathbb {E}} [\| Y_{0}\|^{2}]), \qquad t\le c_{2}. that satisfies. Polynomials are an important part of the "language" of mathematics and algebra. This happens if \(X_{0}\) is sufficiently close to \({\overline{x}}\), say within a distance \(\rho'>0\). Substituting into(I.2) and rearranging yields, for all \(x\in{\mathbb {R}}^{d}\). The assumption of vanishing local time at zero in LemmaA.1(i) cannot be replaced by the zero volatility condition \(\nu =0\) on \(\{Z=0\}\), even if the strictly positive drift condition is retained. Probably the most important application of Taylor series is to use their partial sums to approximate functions . Google Scholar, Forman, J.L., Srensen, M.: The Pearson diffusions: a class of statistically tractable diffusion processes. Since \(a \nabla p=0\) on \(M\cap\{p=0\}\) by (A1), condition(G2) implies that there exists a vector \(h=(h_{1},\ldots ,h_{d})^{\top}\) of polynomials such that, Thus \(\lambda_{i} S_{i}^{\top}\nabla p = S_{i}^{\top}a \nabla p = S_{i}^{\top}h p\), and hence \(\lambda_{i}(S_{i}^{\top}\nabla p)^{2} = S_{i}^{\top}\nabla p S_{i}^{\top}h p\). . The left-hand side, however, is nonnegative; so we deduce \({\mathbb {P}}[\rho<\infty]=0\). and assume the support These terms each consist of x raised to a whole number power and a coefficient. Oliver & Boyd, Edinburgh (1965), MATH Let \(E_{Y}\)-valued solutions to(4.1) with driving Brownian motions \end{aligned}$$, $$ \mathrm{Law}(Y^{1},Z^{1}) = \mathrm{Law}(Y,Z) = \mathrm{Law}(Y,Z') = \mathrm{Law}(Y^{2},Z^{2}), $$, $$ \|b_{Z}(y,z) - b_{Z}(y',z')\| + \| \sigma_{Z}(y,z) - \sigma_{Z}(y',z') \| \le \kappa\|z-z'\|. Springer, Berlin (1998), Book $$, \({\mathcal {V}}( {\mathcal {R}})={\mathcal {V}}(I)\), \(S\subseteq{\mathcal {I}}({\mathcal {V}}(S))\), $$ I = {\mathcal {I}}\big({\mathcal {V}}(I)\big). The strict inequality appearing in LemmaA.1(i) cannot be relaxed to a weak inequality: just consider the deterministic process \(Z_{t}=(1-t)^{3}\). \(X\) . The theorem is proved. (x-a)^2+\frac{f^{(3)}(a)}{3! Some differential calculus gives, for \(y\neq0\), for \(\|y\|>1\), while the first and second order derivatives of \(f(y)\) are uniformly bounded for \(\|y\|\le1\). Then by LemmaF.2, we have \({\mathbb {P}}[ \inf_{u\le\eta} Z_{u} > 0]<1/3\) whenever \(Z_{0}=p(X_{0})\) is sufficiently close to zero. A Taylor series approximation uses a Taylor series to represent a number as a polynomial that has a very similar value to the number in a neighborhood around a specified \(x\) value: \[f(x) = f(a)+\frac {f'(a)}{1!} By symmetry of \(a(x)\), we get, Thus \(h_{ij}=0\) on \(M\cap\{x_{i}=0\}\cap\{x_{j}\ne0\}\), and, by continuity, on \(M\cap\{x_{i}=0\}\). For \(s\) sufficiently close to 1, the right-hand side becomes negative, which contradicts positive semidefiniteness of \(a\) on \(E\). Let \(Q^{i}({\mathrm{d}} z;w,y)\), \(i=1,2\), denote a regular conditional distribution of \(Z^{i}\) given \((W^{i},Y^{i})\). and The zero set of the family coincides with the zero set of the ideal \(I=({\mathcal {R}})\), that is, \({\mathcal {V}}( {\mathcal {R}})={\mathcal {V}}(I)\). $$, $$ {\mathbb {P}}\bigg[ \sup_{t\le\varepsilon}\|Y_{t}-Y_{0}\| < \rho\bigg]\ge 1-\rho ^{-2}{\mathbb {E}}\bigg[\sup_{t\le\varepsilon}\|Y_{t}-Y_{0}\|^{2}\bigg]. Finance and Stochastics If \(i=j\), we get \(a_{jj}(x)=\alpha_{jj}x_{j}^{2}+x_{j}(\phi_{j}+\psi_{(j)}^{\top}x_{I} + \pi _{(j)}^{\top}x_{J})\) for some \(\alpha_{jj}\in{\mathbb {R}}\), \(\phi_{j}\in {\mathbb {R}}\), \(\psi _{(j)}\in{\mathbb {R}}^{m}\), \(\pi_{(j)}\in{\mathbb {R}}^{n}\) with \(\pi _{(j),j}=0\). As mentioned above, the polynomials used in this study are Power, Legendre, Laguerre and Hermite A. Then the law under \(\overline{\mathbb {P}}\) of \((W,Y,Z)\) equals the law of \((W^{1},Y^{1},Z^{1})\), and the law under \(\overline{\mathbb {P}}\) of \((W,Y,Z')\) equals the law of \((W^{2},Y^{2},Z^{2})\). Hence \(\beta_{j}> (B^{-}_{jI}){\mathbf{1}}\) for all \(j\in J\). \(Y^{1}_{0}=Y^{2}_{0}=y\) Equ. and such that the operator Polynomials an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable (s). The following argument is a version of what is sometimes called McKeans argument; see Mayerhofer etal. Learn more about Institutional subscriptions. Defining \(c(x)=a(x) - (1-x^{\top}Qx)\alpha\), this shows that \(c(x)Qx=0\) for all \(x\in{\mathbb {R}}^{d}\), that \(c(0)=0\), and that \(c(x)\) has no linear part. Taylor Polynomials. The time-changed process \(Y_{u}=p(X_{\gamma_{u}})\) thus satisfies, Consider now the \(\mathrm{BESQ}(2-2\delta)\) process \(Z\) defined as the unique strong solution to the equation, Since \(4 {\mathcal {G}}p(X_{t}) / h^{\top}\nabla p(X_{t}) \le2-2\delta\) for \(t<\tau(U)\), a standard comparison theorem implies that \(Y_{u}\le Z_{u}\) for \(u< A_{\tau(U)}\); see for instance Rogers and Williams [42, TheoremV.43.1]. Let It gives necessary and sufficient conditions for nonnegativity of certain It processes. Then over Let \((W^{i},Y^{i},Z^{i})\), \(i=1,2\), be \(E\)-valued weak solutions to (4.1), (4.2) starting from \((y_{0},z_{0})\in E\subseteq{\mathbb {R}}^{m}\times{\mathbb {R}}^{n}\). : On a property of the lognormal distribution. This finally gives. Positive semidefiniteness requires \(a_{jj}(x)\ge0\) for all \(x\in E\). As \(f^{2}(y)=1+\|y\|\) for \(\|y\|>1\), this implies \({\mathbb {E}}[ \mathrm{e}^{\varepsilon' \| Y_{T}\|}]<\infty\). \(A\in{\mathbb {S}}^{d}\) Then define the equivalent probability measure \({\mathrm{d}}{\mathbb {Q}}=R_{\tau}{\,\mathrm{d}}{\mathbb {P}}\), under which the process \(B_{t}=Y_{t}-\int_{0}^{t\wedge\tau}\rho(Y_{s}){\,\mathrm{d}} s\) is a Brownian motion. Financial Planning o Polynomials can be used in financial planning. At this point, we have proved, on \(E\), which yields the stated form of \(a_{ii}(x)\). Part of Springer Nature. $$, \(\frac{\partial^{2} f(y)}{\partial y_{i}\partial y_{j}}\), $$ \mu^{Z}_{t} \le m\qquad\text{and}\qquad\| \sigma^{Z}_{t} \|\le\rho, $$, $$ {\mathbb {E}}\left[\varPhi(Z_{T})\right] \le{\mathbb {E}}\left[\varPhi (V)\right] $$, \({\mathbb {E}}[\mathrm{e} ^{\varepsilon' V^{2}}] <\infty\), \(\varPhi (z) = \mathrm{e}^{\varepsilon' z^{2}}\), \({\mathbb {E}}[ \mathrm{e}^{\varepsilon' Z_{T}^{2}}]<\infty\), \({\mathbb {E}}[ \mathrm{e}^{\varepsilon' \| Y_{T}\|}]<\infty\), $$ {\mathrm{d}} Y_{t} = \widehat{b}_{Y}(Y_{t}) {\,\mathrm{d}} t + \widehat{\sigma}_{Y}(Y_{t}) {\,\mathrm{d}} W_{t}, $$, \(\widehat{b}_{Y}(y)=b_{Y}(y){\mathbf{1}}_{E_{Y}}(y)\), \(\widehat{\sigma}_{Y}(y)=\sigma_{Y}(y){\mathbf{1}}_{E_{Y}}(y)\), \({\mathrm{d}} Y_{t} = \widehat{b}_{Y}(Y_{t}) {\,\mathrm{d}} t + \widehat{\sigma}_{Y}(Y_{t}) {\,\mathrm{d}} W_{t}\), \((y_{0},z_{0})\in E\subseteq{\mathbb {R}}^{m}\times{\mathbb {R}}^{n}\), \(C({\mathbb {R}}_{+},{\mathbb {R}}^{d}\times{\mathbb {R}}^{m}\times{\mathbb {R}}^{n}\times{\mathbb {R}}^{n})\), $$ \overline{\mathbb {P}}({\mathrm{d}} w,{\,\mathrm{d}} y,{\,\mathrm{d}} z,{\,\mathrm{d}} z') = \pi({\mathrm{d}} w, {\,\mathrm{d}} y)Q^{1}({\mathrm{d}} z; w,y)Q^{2}({\mathrm{d}} z'; w,y). This proves \(a_{ij}(x)=-\alpha_{ij}x_{i}x_{j}\) on \(E\) for \(i\ne j\), as claimed. Exponents are used in Computer Game Physics, pH and Richter Measuring Scales, Science, Engineering, Economics, Accounting, Finance, and many other disciplines. This is a preview of subscription content, access via your institution. Camb. 333, 151163 (2007), Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Polynomial can be used to keep records of progress of patient progress. 435445. Polynomials in one variable are algebraic expressions that consist of terms in the form axn a x n where n n is a non-negative ( i.e. It has the following well-known property. Verw. J. Probab. Contemp. Share Cite Follow answered Oct 22, 2012 at 1:38 ILoveMath 10.3k 8 47 110 J. R. Stat. Pick \(s\in(0,1)\) and set \(x_{k}=s\), \(x_{j}=(1-s)/(d-1)\) for \(j\ne k\). $$, $$ {\mathbb {P}}_{z}[\tau_{0}>\varepsilon] = \int_{\varepsilon}^{\infty}\frac {1}{t\varGamma (\widehat{\nu})}\left(\frac{z}{2t}\right)^{\widehat{\nu}} \mathrm{e}^{-z/(2t)}{\,\mathrm{d}} t, $$, \({\mathbb {P}}_{z}[\tau _{0}>\varepsilon]=\frac{1}{\varGamma(\widehat{\nu})}\int _{0}^{z/(2\varepsilon )}s^{\widehat{\nu}-1}\mathrm{e}^{-s}{\,\mathrm{d}} s\), $$ 0 \le2 {\mathcal {G}}p({\overline{x}}) < h({\overline{x}})^{\top}\nabla p({\overline{x}}). Polynomials in finance! Since \(\rho_{n}\to \infty\), we deduce \(\tau=\infty\), as desired. This directly yields \(\pi_{(j)}\in{\mathbb {R}}^{n}_{+}\). MATH Or one variable. The right-hand side is a nonnegative supermartingale on \([0,\tau)\), and we deduce \(\sup_{t<\tau}Z_{t}<\infty\) on \(\{\tau <\infty \}\), as required. \(Z_{0}\ge0\), \(\mu\) $$, \(\rho=\inf\left\{ t\ge0: Z_{t}<0\right\}\), \(\tau=\inf \left\{ t\ge\rho: \mu_{t}=0 \right\} \wedge(\rho+1)\), $$ {\mathbb {E}}[Z^{-}_{\tau\wedge n}] = {\mathbb {E}}\big[Z^{-}_{\tau\wedge n}{\boldsymbol{1}_{\{\rho< \infty\}}}\big] \longrightarrow{\mathbb {E}}\big[ Z^{-}_{\tau}{\boldsymbol{1}_{\{\rho < \infty\}}}\big] \qquad(n\to\infty). 16-34 (2016). \(V\), denoted by \({\mathcal {I}}(V)\), is the set of all polynomials that vanish on \(V\). We need to identify \(\phi_{i}\) and \(\psi _{(i)}\). Hence, for any \(0<\varepsilon' <1/(2\rho^{2} T)\), we have \({\mathbb {E}}[\mathrm{e} ^{\varepsilon' V^{2}}] <\infty\). \(M\) For example, the set \(M\) in(5.1) is the zero set of the ideal\(({\mathcal {Q}})\). Writing the \(i\)th component of \(a(x){\mathbf{1}}\) in two ways then yields, for all \(x\in{\mathbb {R}}^{d}\) and some \(\eta\in{\mathbb {R}}^{d}\), \({\mathrm {H}} \in{\mathbb {R}}^{d\times d}\). is a Brownian motion. $$, \(\tau_{E}=\inf\{t\colon X_{t}\notin E\}\le\tau\), \(\int_{0}^{t}{\boldsymbol{1}_{\{p(X_{s})=0\} }}{\,\mathrm{d}} s=0\), $$ \begin{aligned} \log& p(X_{t}) - \log p(X_{0}) \\ &= \int_{0}^{t} \left(\frac{{\mathcal {G}}p(X_{s})}{p(X_{s})} - \frac {1}{2}\frac {\nabla p^{\top}a \nabla p(X_{s})}{p(X_{s})^{2}}\right) {\,\mathrm{d}} s + \int_{0}^{t} \frac {\nabla p^{\top}\sigma(X_{s})}{p(X_{s})}{\,\mathrm{d}} W_{s} \\ &= \int_{0}^{t} \frac{2 {\mathcal {G}}p(X_{s}) - h^{\top}\nabla p(X_{s})}{2p(X_{s})} {\,\mathrm{d}} s + \int_{0}^{t} \frac{\nabla p^{\top}\sigma(X_{s})}{p(X_{s})}{\,\mathrm{d}} W_{s} \end{aligned} $$, $$ V_{t} = \int_{0}^{t} {\boldsymbol{1}_{\{X_{s}\notin U\}}} \frac{1}{p(X_{s})}|2 {\mathcal {G}}p(X_{s}) - h^{\top}\nabla p(X_{s})| {\,\mathrm{d}} s. $$, \(E {\cap} U^{c} {\cap} \{x:\|x\| {\le} n\}\), $$ \varepsilon_{n}=\min\{p(x):x\in E\cap U^{c}, \|x\|\le n\} $$, $$ V_{t\wedge\sigma_{n}} \le\frac{t}{2\varepsilon_{n}} \max_{\|x\|\le n} |2 {\mathcal {G}}p(x) - h^{\top}\nabla p(x)| < \infty. Finance Assessment of present value is used in loan calculations and company valuation. , Note that \(E\subseteq E_{0}\) since \(\widehat{b}=b\) on \(E\). Let It is used in many experimental procedures to produce the outcome using this equation. Let \(X\) and \(\tau\) be the process and stopping time provided by LemmaE.4. 31.1. Also, the business owner needs to calculate the lowest price at which an item can be sold to still cover the expenses. \(\mu>0\) \(E\) Example: 21 is a polynomial. Martin Larsson. Ackerer, D., Filipovi, D.: Linear credit risk models. Indeed, the known formulas for the moments of the lognormal distribution imply that for each \(T\ge0\), there is a constant \(c=c(T)\) such that \({\mathbb {E}}[(Y_{t}-Y_{s})^{4}] \le c(t-s)^{2}\) for all \(s\le t\le T, |t-s|\le1\), whence Kolmogorovs continuity lemma implies that \(Y\) has a continuous version; see Rogers and Williams [42, TheoremI.25.2]. Thus, choosing curves \(\gamma\) with \(\gamma'(0)=u_{i}\), (E.5) yields, Combining(E.4), (E.6) and LemmaE.2, we obtain. Taking \(p(x)=x_{i}\), \(i=1,\ldots,d\), we obtain \(a(x)\nabla p(x) = a(x) e_{i} = 0\) on \(\{x_{i}=0\}\). \(\sigma\) \(\nu=0\). If there are real numbers denoted by a, then function with one variable and of degree n can be written as: f (x) = a0xn + a1xn-1 + a2xn-2 + .. + an-2x2 + an-1x + an Solving Polynomials Thus we may find a smooth path \(\gamma_{i}:(-1,1)\to M\) such that \(\gamma _{i}(0)=x\) and \(\gamma_{i}'(0)=S_{i}(x)\). We now focus on the converse direction and assume(A0)(A2) hold. The first can approximate a given polynomial. MATH Finally, let \(\alpha\in{\mathbb {S}}^{n}\) be the matrix with elements \(\alpha_{ij}\) for \(i,j\in J\), let \(\varPsi\in{\mathbb {R}}^{m\times n}\) have columns \(\psi_{(j)}\), and \(\varPi \in{\mathbb {R}} ^{n\times n}\) columns \(\pi_{(j)}\). Accounting To figure out the exact pay of an employee that works forty hours and does twenty hours of overtime, you could use a polynomial such as this: 40h+20 (h+1/2h) \(y\in E_{Y}\). MATH This completes the proof of the theorem. In view of (C.4) and the above expressions for \(\nabla f(y)\) and \(\frac{\partial^{2} f(y)}{\partial y_{i}\partial y_{j}}\), these are bounded, for some constants \(m\) and \(\rho\). Mark. Math. For each \(q\in{\mathcal {Q}}\), Consider now any fixed \(x\in M\). o Assessment of present value is used in loan calculations and company valuation. Polynomials can be used to extract information about finite sequences much in the same way as generating functions can be used for infinite sequences. where the MoorePenrose inverse is understood. Math. be a continuous semimartingale of the form. Narrowing the domain can often be done through the use of various addition or scaling formulas for the function being approximated. 2. \(B\) }(x-a)^3+ \cdots.\] Taylor series are extremely powerful tools for approximating functions that can be difficult to compute . $$, \(\widehat{b} :{\mathbb {R}}^{d}\to{\mathbb {R}}^{d}\), $$ \widehat{\mathcal {G}}f = \frac{1}{2}\operatorname{Tr}( \widehat{a} \nabla^{2} f) + \widehat{b} ^{\top} \nabla f $$, \(\widehat{\mathcal {G}}f={\mathcal {G}}f\), \(c:{\mathbb {R}}^{d}\to {\mathbb {R}}^{d}\), $$ c=0\mbox{ on }E \qquad \mbox{and}\qquad\nabla q^{\top}c = - \frac {1}{2}\operatorname{Tr}\big( (\widehat{a}-a) \nabla^{2} q \big) \mbox{ on } M\mbox{, for all }q\in {\mathcal {Q}}. Polynomials are used in the business world in dozens of situations. Hence, as claimed. 35, 438465 (2008), Gallardo, L., Yor, M.: A chaotic representation property of the multidimensional Dunkl processes. $$, \(4 {\mathcal {G}}p(X_{t}) / h^{\top}\nabla p(X_{t}) \le2-2\delta\), \(C=\sup_{x\in U} h(x)^{\top}\nabla p(x)/4\), $$ \begin{aligned} &{\mathbb {P}}\Big[ \eta< A_{\tau(U)} \text{ and } \inf_{u\le\eta} Z_{u} = 0\Big] \\ &\ge{\mathbb {P}}\big[ \eta< A_{\tau(U)} \big] - {\mathbb {P}}\Big[ \inf_{u\le\eta } Z_{u} > 0\Big] \\ &\ge{\mathbb {P}}\big[ \eta C^{-1} < \tau(U) \big] - {\mathbb {P}}\Big[ \inf_{u\le \eta} Z_{u} > 0\Big] \\ &= {\mathbb {P}}\bigg[ \sup_{t\le\eta C^{-1}} \|X_{t} - {\overline{x}}\| < \rho \bigg] - {\mathbb {P}}\Big[ \inf_{u\le\eta} Z_{u} > 0\Big] \\ &\ge{\mathbb {P}}\bigg[ \sup_{t\le\eta C^{-1}} \|X_{t} - X_{0}\| < \rho/2 \bigg] - {\mathbb {P}} \Big[ \inf_{u\le\eta} Z_{u} > 0\Big], \end{aligned} $$, \({\mathbb {P}}[ \sup _{t\le\eta C^{-1}} \|X_{t} - X_{0}\| <\rho/2 ]>1/2\), \({\mathbb {P}}[ \inf_{u\le\eta} Z_{u} > 0]<1/3\), \(\|X_{0}-{\overline{x}}\| <\rho'\wedge(\rho/2)\), $$ 0 = \epsilon a(\epsilon x) Q x = \epsilon\big( \alpha Qx + A(x)Qx \big) + L(x)Qx. Financ. The following auxiliary result forms the basis of the proof of Theorem5.3. is satisfied for some constant \(C\). To prove(G2), it suffices by Lemma5.5 to prove for each\(i\) that the ideal \((x_{i}, 1-{\mathbf {1}}^{\top}x)\) is prime and has dimension \(d-2\). $$, \(\tau=\inf\{t\ge0:\mu_{t}\ge0\}\wedge1\), \(0\le{\mathbb {E}}[Z_{\tau}] = {\mathbb {E}}[\int_{0}^{\tau}\mu_{s}{\,\mathrm{d}} s]<0\), \({\mathrm{d}}{\mathbb {Q}}={\mathcal {E}}(-\phi B)_{1}{\,\mathrm{d}} {\mathbb {P}}\), $$ Z_{t}=\int_{0}^{t}(\mu_{s}-\phi\nu_{s}){\,\mathrm{d}} s+\int_{0}^{t}\nu_{s}{\,\mathrm{d}} B^{\mathbb {Q}}_{s}.