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how are polynomials used in finance

$$, \(h_{ij}(x)=-\alpha_{ij}x_{i}+(1-{\mathbf{1}}^{\top}x)\gamma_{ij}\), $$ a_{ii}(x) = -\alpha_{ii}x_{i}^{2} + x_{i}(\phi_{i} + \psi_{(i)}^{\top}x) + (1-{\mathbf{1}} ^{\top}x) g_{ii}(x) $$, \(a(x){\mathbf{1}}=(1-{\mathbf{1}}^{\top}x)f(x)\), \(f_{i}\in{\mathrm {Pol}}_{1}({\mathbb {R}}^{d})\), $$ \begin{aligned} x_{i}\bigg( -\sum_{j=1}^{d} \alpha_{ij}x_{j} + \phi_{i} + \psi_{(i)}^{\top}x\bigg) &= (1 - {\mathbf{1}}^{\top}x)\big(f_{i}(x) - g_{ii}(x)\big) \\ &= (1 - {\mathbf{1}}^{\top}x)\big(\eta_{i} + ({\mathrm {H}}x)_{i}\big) \end{aligned} $$, \({\mathrm {H}} \in{\mathbb {R}}^{d\times d}\), \(x_{i}\phi_{i} = \lim_{s\to0} s^{-1}\eta_{i} + ({\mathrm {H}}x)_{i}\), $$ x_{i}\bigg(- \sum_{j=1}^{d} \alpha_{ij}x_{j} + \psi_{(i)}^{\top}x + \phi _{i} {\mathbf{1}} ^{\top}x\bigg) = 0 $$, \(x_{i} \sum_{j\ne i} (-\alpha _{ij}+\psi _{(i),j}+\alpha_{ii})x_{j} = 0\), \(\psi _{(i),j}=\alpha_{ij}-\alpha_{ii}\), $$ a_{ii}(x) = -\alpha_{ii}x_{i}^{2} + x_{i}\bigg(\alpha_{ii} + \sum_{j\ne i}(\alpha_{ij}-\alpha_{ii})x_{j}\bigg) = \alpha_{ii}x_{i}(1-{\mathbf {1}}^{\top}x) + \sum_{j\ne i}\alpha_{ij}x_{i}x_{j} $$, $$ a_{ii}(x) = x_{i} \sum_{j\ne i}\alpha_{ij}x_{j} = x_{i}\bigg(\alpha_{ik}s + \frac{1-s}{d-1}\sum_{j\ne i,k}\alpha_{ij}\bigg). satisfies a square-root growth condition, for some constant Step 6: Visualize and predict both the results of linear and polynomial regression and identify which model predicts the dataset with better results. Suppose \(j\ne i\). For \(s\) sufficiently close to 1, the right-hand side becomes negative, which contradicts positive semidefiniteness of \(a\) on \(E\). Ann. : Markov Processes: Characterization and Convergence. At this point, we have proved, on \(E\), which yields the stated form of \(a_{ii}(x)\). As mentioned above, the polynomials used in this study are Power, Legendre, Laguerre and Hermite A. 7000+ polynomials are on our. MATH Verw. 1, 250271 (2003). Let $$, \(g\in{\mathrm {Pol}}({\mathbb {R}}^{d})\), \({\mathcal {R}}=\{r_{1},\ldots,r_{m}\}\), \(f_{i}\in{\mathrm {Pol}}({\mathbb {R}}^{d})\), $$ {\mathcal {V}}(S)=\{x\in{\mathbb {R}}^{d}:f(x)=0 \text{ for all }f\in S\}. After stopping we may assume that \(Z_{t}\), \(\int_{0}^{t}\mu_{s}{\,\mathrm{d}} s\) and \(\int _{0}^{t}\nu_{s}{\,\mathrm{d}} B_{s}\) are uniformly bounded. Commun. \(Y_{t} = Y_{0} + \int_{0}^{t} b(Y_{s}){\,\mathrm{d}} s + \int_{0}^{t} \sigma(Y_{s}){\,\mathrm{d}} W_{s}\). Finally, after shrinking \(U\) while maintaining \(M\subseteq U\), \(c\) is continuous on the closure \(\overline{U}\), and can then be extended to a continuous map on \({\mathbb {R}}^{d}\) by the Tietze extension theorem; see Willard [47, Theorem15.8]. \(Z\) By LemmaF.1, we can choose \(\eta>0\) independently of \(X_{0}\) so that \({\mathbb {P}}[ \sup _{t\le\eta C^{-1}} \|X_{t} - X_{0}\| <\rho/2 ]>1/2\). The following two examples show that the assumptions of LemmaA.1 are tight in the sense that the gap between (i) and (ii) cannot be closed. For(ii), note that \({\mathcal {G}}p(x) = b_{i}(x)\) for \(p(x)=x_{i}\), and \({\mathcal {G}} p(x)=-b_{i}(x)\) for \(p(x)=1-x_{i}\). That is, \(\phi_{i}=\alpha_{ii}\). Sending \(m\) to infinity and applying Fatous lemma gives the result. $$, $$ Z_{u} = p(X_{0}) + (2-2\delta)u + 2\int_{0}^{u} \sqrt{Z_{v}}{\,\mathrm{d}}\beta_{v}. Also, = [1, 10, 9, 0, 0, 0] is also a degree 2 polynomial, since the zero coefficients at the end do not count. $$, $$ A_{t} = \int_{0}^{t} {\boldsymbol{1}_{\{X_{s}\notin U\}}} \frac{1}{p(X_{s})}\big(2 {\mathcal {G}}p(X_{s}) - h^{\top}\nabla p(X_{s})\big) {\,\mathrm{d}} s $$, \(\rho_{n}=\inf\{t\ge0: |A_{t}|+p(X_{t}) \ge n\}\), $$\begin{aligned} Z_{t} &= \log p(X_{0}) + \int_{0}^{t} {\boldsymbol{1}_{\{X_{s}\in U\}}} \frac {1}{2p(X_{s})}\big(2 {\mathcal {G}}p(X_{s}) - h^{\top}\nabla p(X_{s})\big) {\,\mathrm{d}} s \\ &\phantom{=:}{}+ \int_{0}^{t} \frac{\nabla p^{\top}\sigma(X_{s})}{p(X_{s})}{\,\mathrm{d}} W_{s}. 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 \(M\cap\{x_{i}=0\}\). with the spectral decomposition Polynomial can be used to keep records of progress of patient progress. - \(\{Z=0\}\) \(K\) \(\widehat{b} :{\mathbb {R}}^{d}\to{\mathbb {R}}^{d}\) For instance, a polynomial equation can be used to figure the amount of interest that will accrue for an initial deposit amount in an investment or savings account at a given interest rate. Moreover, fixing \(j\in J\), setting \(x_{j}=0\) and letting \(x_{i}\to\infty\) for \(i\ne j\) forces \(B_{ji}>0\). These quantities depend on\(x\) in a possibly discontinuous way. Appl. We have not been able to exhibit such a process. If the ideal \(I=({\mathcal {R}})\) satisfies (J.1), then that means that any polynomial \(f\) that vanishes on the zero set \({\mathcal {V}}(I)\) has a representation \(f=f_{1}r_{1}+\cdots+f_{m}r_{m}\) for some polynomials \(f_{1},\ldots,f_{m}\). This proves \(a_{ij}(x)=-\alpha_{ij}x_{i}x_{j}\) on \(E\) for \(i\ne j\), as claimed. 30, 605641 (2012), Stieltjes, T.J.: Recherches sur les fractions continues. $$, \(\tau=\inf\{t\ge0:\mu_{t}\ge0\}\wedge1\), \(0\le{\mathbb {E}}[Z_{\tau}] = {\mathbb {E}}[\int_{0}^{\tau}\mu_{s}{\,\mathrm{d}} s]<0\), \({\mathrm{d}}{\mathbb {Q}}={\mathcal {E}}(-\phi B)_{1}{\,\mathrm{d}} {\mathbb {P}}\), $$ Z_{t}=\int_{0}^{t}(\mu_{s}-\phi\nu_{s}){\,\mathrm{d}} s+\int_{0}^{t}\nu_{s}{\,\mathrm{d}} B^{\mathbb {Q}}_{s}. , As in the proof of(i), it is enough to consider the case where \(p(X_{0})>0\). , The proof of Theorem4.4 follows along the lines of the proof of the YamadaWatanabe theorem that pathwise uniqueness implies uniqueness in law; see Rogers and Williams [42, TheoremV.17.1]. It remains to show that \(\alpha_{ij}\ge0\) for all \(i\ne j\). Many of us are familiar with this term and there would be some who are not.Some people use polynomials in their heads every day without realizing it, while others do it more consciously. be two Furthermore, the drift vector is always of the form \(b(x)=\beta +Bx\), and a brief calculation using the expressions for \(a(x)\) and \(b(x)\) shows that the condition \({\mathcal {G}}p> 0\) on \(\{p=0\}\) is equivalent to(6.2). This relies on (G2) and(A1). Sminaire de Probabilits XI. It also implies that \(\widehat{\mathcal {G}}\) satisfies the positive maximum principle as a linear operator on \(C_{0}(E_{0})\). $$, \(\beta^{\top}{\mathbf{1}}+ x^{\top}B^{\top}{\mathbf{1}}= 0\), \(\beta^{\top}{\mathbf{1}}+ x^{\top}B^{\top}{\mathbf{1}} =\kappa(1-{\mathbf{1}}^{\top}x)\), \(B^{\top}{\mathbf {1}}=-\kappa {\mathbf{1}} =-(\beta^{\top}{\mathbf{1}}){\mathbf{1}}\), $$ \min\Bigg\{ \beta_{i} + {\sum_{j=1}^{d}} B_{ji}x_{j}: x\in{\mathbb {R}}^{d}_{+}, {\mathbf{1}} ^{\top}x = {\mathbf{1}}, x_{i}=0\Bigg\} \ge0, $$, $$ \min\Biggl\{ \beta_{i} + {\sum_{j\ne i}} B_{ji}x_{j}: x\in{\mathbb {R}}^{d}_{+}, {\sum_{j\ne i}} x_{j}=1\Biggr\} \ge0. In: Dellacherie, C., et al. J. Then by Its formula and the martingale property of \(\int_{0}^{t\wedge\tau_{m}}\nabla f(X_{s})^{\top}\sigma(X_{s}){\,\mathrm{d}} W_{s}\), Gronwalls inequality now yields \({\mathbb {E}}[f(X_{t\wedge\tau_{m}})\, |\,{\mathcal {F}} _{0}]\le f(X_{0}) \mathrm{e}^{Ct}\). . 131, 475505 (2006), Hajek, B.: Mean stochastic comparison of diffusions. $$, \(f,g\in {\mathrm{Pol}}({\mathbb {R}}^{d})\),, We have, where we recall that \(\rho\) is the radius of the open ball \(U\), and where the last inequality follows from the triangle inequality provided \(\|X_{0}-{\overline{x}}\|\le\rho/2\). have the same law. Courier Corporation, North Chelmsford (2004), Wong, E.: The construction of a class of stationary Markoff processes. This right-hand side has finite expectation by LemmaB.1, so the stochastic integral above is a martingale. Let \(X\) and \(\tau\) be the process and stopping time provided by LemmaE.4. This establishes(6.4). Example: 21 is a polynomial. Wiley, Hoboken (2005), Filipovi, D., Mayerhofer, E., Schneider, P.: Density approximations for multivariate affine jump-diffusion processes. For each \(i\) such that \(\lambda _{i}(x)^{-}\ne0\), \(S_{i}(x)\) lies in the tangent space of\(M\) at\(x\). Google Scholar, Stoyanov, J.: Krein condition in probabilistic moment problems. on and assume the support This process starts at zero, has zero volatility whenever \(Z_{t}=0\), and strictly positive drift prior to the stopping time \(\sigma\), which is strictly positive. , Note that \(E\subseteq E_{0}\) since \(\widehat{b}=b\) on \(E\). Indeed, non-explosion implies that either \(\tau=\infty\), or \({\mathbb {R}}^{d}\setminus E_{0}\neq\emptyset\) in which case we can take \(\Delta\in{\mathbb {R}}^{d}\setminus E_{0}\). For this we observe that for any \(u\in{\mathbb {R}}^{d}\) and any \(x\in\{p=0\}\), In view of the homogeneity property, positive semidefiniteness follows for any\(x\). The research leading to these results has received funding from the European Research Council under the European Unions Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n.307465-POLYTE. Replacing \(x\) by \(sx\), dividing by \(s\) and sending \(s\) to zero gives \(x_{i}\phi_{i} = \lim_{s\to0} s^{-1}\eta_{i} + ({\mathrm {H}}x)_{i}\), which forces \(\eta _{i}=0\), \({\mathrm {H}}_{ij}=0\) for \(j\ne i\) and \({\mathrm {H}}_{ii}=\phi _{i}\). Condition(G1) is vacuously true, so we prove (G2). The proof of(ii) is complete. Nonetheless, its sign changes infinitely often on any time interval \([0,t)\) since it is a time-changed Brownian motion viewed under an equivalent measure. MathSciNet In Section 2 we outline the construction of two networks which approximate polynomials. \(Y\) This proves the result. Let of \(Y^{1}\), \(Y^{2}\) on \(d\)-dimensional Brownian motion This process satisfies \(Z_{u} = B_{A_{u}} + u\wedge\sigma\), where \(\sigma=\varphi_{\tau}\). But all these elements can be realized as \((TK)(x)=K(x)Qx\) as follows: If \(i,j,k\) are all distinct, one may take, and all remaining entries of \(K(x)\) equal to zero. \(\widehat{\mathcal {G}}f={\mathcal {G}}f\) This happens if \(X_{0}\) is sufficiently close to \({\overline{x}}\), say within a distance \(\rho'>0\). In order to maintain positive semidefiniteness, we necessarily have \(\gamma_{i}\ge0\). and Reading: Average Rate of Change. J. Since uniqueness in law holds for \(E_{Y}\)-valued solutions to(4.1), LemmaD.1 implies that \((W^{1},Y^{1})\) and \((W^{2},Y^{2})\) have the same law, which we denote by \(\pi({\mathrm{d}} w,{\,\mathrm{d}} y)\). For \(i\ne j\), this is possible only if \(a_{ij}(x)=0\), and for \(i=j\in I\) it implies that \(a_{ii}(x)=\gamma_{i}x_{i}(1-x_{i})\) as desired. 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. The strict inequality appearing in LemmaA.1(i) cannot be relaxed to a weak inequality: just consider the deterministic process \(Z_{t}=(1-t)^{3}\). In economics we learn that profit is the difference between revenue (money coming in) and costs (money going out). For (ii), first note that we always have \(b(x)=\beta+Bx\) for some \(\beta \in{\mathbb {R}}^{d}\) and \(B\in{\mathbb {R}}^{d\times d}\). (1) The individual summands with the coefficients (usually) included are called monomials (Becker and Weispfenning 1993, p. 191), whereas the . \(c_{1},c_{2}>0\) By the above, we have \(a_{ij}(x)=h_{ij}(x)x_{j}\) for some \(h_{ij}\in{\mathrm{Pol}}_{1}(E)\). It follows from the definition that \(S\subseteq{\mathcal {I}}({\mathcal {V}}(S))\) for any set \(S\) of polynomials. Toulouse 8(4), 1122 (1894), Article where the MoorePenrose inverse is understood. 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 \((W^{2},Y^{2},Z^{2})\). The proof of Theorem5.3 is complete. But the identity \(L(x)Qx\equiv0\) precisely states that \(L\in\ker T\), yielding \(L=0\) as desired. 264276. The least-squares method minimizes the varianceof the unbiasedestimatorsof the coefficients, under the conditions of the Gauss-Markov theorem. \(Z\) Let It gives necessary and sufficient conditions for nonnegativity of certain It processes. 243, 163169 (1979), Article Hajek [28, Theorem 1.3] now implies that, for any nondecreasing convex function \(\varPhi\) on , where \(V\) is a Gaussian random variable with mean \(f(0)+m T\) and variance \(\rho^{2} T\). Part of Springer Nature. Sci. $$, $$\begin{aligned} {\mathcal {X}}&=\{\text{all linear maps ${\mathbb {R}}^{d}\to{\mathbb {S}}^{d}$}\}, \\ {\mathcal {Y}}&=\{\text{all second degree homogeneous maps ${\mathbb {R}}^{d}\to{\mathbb {R}}^{d}$}\}, \end{aligned}$$, \(\dim{\mathcal {X}}=\dim{\mathcal {Y}}=d^{2}(d+1)/2\), \(\dim(\ker T) + \dim(\mathrm{range } T) = \dim{\mathcal {X}} \), $$ (0,\ldots,0,x_{i}x_{j},0,\ldots,0)^{\top}$$, $$ \begin{pmatrix} K_{ii} & K_{ij} &K_{ik} \\ K_{ji} & K_{jj} &K_{jk} \\ K_{ki} & K_{kj} &K_{kk} \end{pmatrix} \! Now consider any stopping time \(\rho\) such that \(Z_{\rho}=0\) on \(\{\rho <\infty\}\). The other is x3 + x2 + 1. Then for any Finance 10, 177194 (2012), Maisonneuve, B.: Une mise au point sur les martingales locales continues dfinies sur un intervalle stochastique. \(\int _{0}^{t} {\boldsymbol{1}_{\{Z_{s}=0\}}}{\,\mathrm{d}} s=0\). \(0<\alpha<2\) J. Econom. 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) Geb. Polynomials . It thus remains to exhibit \(\varepsilon>0\) such that if \(\|X_{0}-\overline{x}\|<\varepsilon\) almost surely, there is a positive probability that \(Z_{u}\) hits zero before \(X_{\gamma_{u}}\) leaves \(U\), or equivalently, that \(Z_{u}=0\) for some \(u< A_{\tau(U)}\). The proof of Theorem5.3 consists of two main parts. answer key cengage advantage books introductory musicianship 8th edition 1998 chevy .. \(\widehat{b}=b\) Next, differentiating once more yields. Next, the condition \({\mathcal {G}}p_{i} \ge0\) on \(M\cap\{ p_{i}=0\}\) for \(p_{i}(x)=x_{i}\) can be written as, The feasible region of this optimization problem is the convex hull of \(\{e_{j}:j\ne i\}\), and the linear objective function achieves its minimum at one of the extreme points. Thus \(\tau _{E}<\tau\) on \(\{\tau<\infty\}\), whence this set is empty. : Hankel transforms associated to finite reflection groups. \(E\). Next, it is straightforward to verify that (i) and (ii) imply (A0)(A2), so we focus on the converse direction and assume(A0)(A2) hold. 29, 483493 (1976), Ethier, S.N., Kurtz, T.G. All of them can be alternatively expressed by Rodrigues' formula, explicit form or by the recurrence law (Abramowitz and Stegun 1972 ). Cambridge University Press, Cambridge (1994), Schmdgen, K.: The \(K\)-moment problem for compact semi-algebraic sets. Then there exist constants Math. Shop the newest collections from over 200 designers.. polynomials worksheet with answers baba yagas geese and other russian . $$, \(4 {\mathcal {G}}p(X_{t}) / h^{\top}\nabla p(X_{t}) \le2-2\delta\), \(C=\sup_{x\in U} h(x)^{\top}\nabla p(x)/4\), $$ \begin{aligned} &{\mathbb {P}}\Big[ \eta< A_{\tau(U)} \text{ and } \inf_{u\le\eta} Z_{u} = 0\Big] \\ &\ge{\mathbb {P}}\big[ \eta< A_{\tau(U)} \big] - {\mathbb {P}}\Big[ \inf_{u\le\eta } Z_{u} > 0\Big] \\ &\ge{\mathbb {P}}\big[ \eta C^{-1} < \tau(U) \big] - {\mathbb {P}}\Big[ \inf_{u\le \eta} Z_{u} > 0\Big] \\ &= {\mathbb {P}}\bigg[ \sup_{t\le\eta C^{-1}} \|X_{t} - {\overline{x}}\| < \rho \bigg] - {\mathbb {P}}\Big[ \inf_{u\le\eta} Z_{u} > 0\Big] \\ &\ge{\mathbb {P}}\bigg[ \sup_{t\le\eta C^{-1}} \|X_{t} - X_{0}\| < \rho/2 \bigg] - {\mathbb {P}} \Big[ \inf_{u\le\eta} Z_{u} > 0\Big], \end{aligned} $$, \({\mathbb {P}}[ \sup _{t\le\eta C^{-1}} \|X_{t} - X_{0}\| <\rho/2 ]>1/2\), \({\mathbb {P}}[ \inf_{u\le\eta} Z_{u} > 0]<1/3\), \(\|X_{0}-{\overline{x}}\| <\rho'\wedge(\rho/2)\), $$ 0 = \epsilon a(\epsilon x) Q x = \epsilon\big( \alpha Qx + A(x)Qx \big) + L(x)Qx. $$, \(\widehat{\mathcal {G}}p= {\mathcal {G}}p\), \(E_{0}\subseteq E\cup\bigcup_{p\in{\mathcal {P}}} U_{p}\), $$ \widehat{\mathcal {G}}p > 0\qquad \mbox{on } E_{0}\cap\{p=0\}. Am. \int_{0}^{t}\! Assume uniqueness in law holds for Write \(a(x)=\alpha+ L(x) + A(x)\), where \(\alpha=a(0)\in{\mathbb {S}}^{d}_{+}\), \(L(x)\in{\mathbb {S}}^{d}\) is linear in\(x\), and \(A(x)\in{\mathbb {S}}^{d}\) is homogeneous of degree two in\(x\). These partial sums are (finite) polynomials and are easy to compute. \(\sigma:{\mathbb {R}}^{d}\to {\mathbb {R}}^{d\times d}\) 31.1. Zhou [ 49] used one-dimensional polynomial (jump-)diffusions to build short rate models that were estimated to data using a generalized method-of-moments approach, relying crucially on the ability to compute moments efficiently. Define then \(\beta _{u}=\int _{0}^{u} \rho(Z_{v})^{1/2}{\,\mathrm{d}} B_{A_{v}}\), which is a Brownian motion because we have \(\langle\beta,\beta\rangle_{u}=\int_{0}^{u}\rho(Z_{v}){\,\mathrm{d}} A_{v}=u\). To see that \(T\) is surjective, note that \({\mathcal {Y}}\) is spanned by elements of the form, with the \(k\)th component being nonzero. We first prove an auxiliary lemma. The proof of Part(ii) involves the same ideas as used for instance in Spreij and Veerman [44, Proposition3.1]. \(A=S\varLambda S^{\top}\), we have 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. 1123, pp. $$, $$ \int_{-\infty}^{\infty}\frac{1}{y}{\boldsymbol{1}_{\{y>0\}}}L^{y}_{t}{\,\mathrm{d}} y = \int_{0}^{t} \frac {\nabla p^{\top}\widehat{a} \nabla p(X_{s})}{p(X_{s})}{\boldsymbol{1}_{\{ p(X_{s})>0\}}}{\,\mathrm{d}} s. $$, \((\nabla p^{\top}\widehat{a} \nabla p)/p\), $$ a \nabla p = h p \qquad\text{on } M. $$, \(\lambda_{i} S_{i}^{\top}\nabla p = S_{i}^{\top}a \nabla p = S_{i}^{\top}h p\), \(\lambda_{i}(S_{i}^{\top}\nabla p)^{2} = S_{i}^{\top}\nabla p S_{i}^{\top}h p\), $$ \nabla p^{\top}\widehat{a} \nabla p = \nabla p^{\top}S\varLambda^{+} S^{\top}\nabla p = \sum_{i} \lambda_{i}{\boldsymbol{1}_{\{\lambda_{i}>0\}}}(S_{i}^{\top}\nabla p)^{2} = \sum_{i} {\boldsymbol{1}_{\{\lambda_{i}>0\}}}S_{i}^{\top}\nabla p S_{i}^{\top}h p. $$, $$ \nabla p^{\top}\widehat{a} \nabla p \le|p| \sum_{i} \|S_{i}\|^{2} \|\nabla p\| \|h\|. Since \(h^{\top}\nabla p(X_{t})>0\) on \([0,\tau(U))\), the process \(A\) is strictly increasing there. is a Brownian motion. 2)Polynomials used in Electronics

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