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随机微分方程 Stochastic Differential Equations M45A51

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这是一份imperial帝国理工大学 M45A51作业代写的成功案例

随机微分方程 Stochastic Differential Equations M45A51
问题 1.

$$
d q_{t}=\beta q_{t} d t+\theta \gamma(\theta) d W_{t}, q_{0}=0 .
$$
It is easy to verify that
$$
q_{t}=\theta \gamma(\theta) \int_{0}^{t} e^{-\sqrt{\theta^{2}+\beta^{2}(t-s)}} d X_{s} .
$$



证明 .

In this case the LAMN property holds with
$$
m_{T}(\theta)=e^{\theta T}, \quad \zeta(\theta)=\left(\int_{0}^{\infty} e^{-\sqrt{\theta^{2}+\beta^{2}}(t-s)} d W_{s}\right)^{2} .
$$


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M45A51 COURSE NOTES ：

The indefinite integral is defined as
$$
\int_{0}^{t} \psi(s) d W_{s}^{H}=\int_{0}^{1} \psi(t) I_{[0, t]} d W_{t}^{H}
$$
This integral has a continuous version and a Gaussian process. However,
$$
E\left(\int_{0}^{t} \psi(s) d W_{s}^{H}\right) \neq 0 .
$$
To overcome this situation, Duncan, Hu and Pasik-Duncan (2000) introduced an integral using Wick calculus for which
$$
E\left(\int_{0}^{t} f(s) d W_{s}^{H}\right)=0 .
$$








偏微分方程简介 Introduction to Partial Differential Equations MATH96018/97027/97104

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这是一份imperial帝国理工大学 MATH96018/97027/97104作业代写的成功案例

偏微分方程简介 Introduction to Partial Differential Equations MATH96018/97027/97104
问题 1.

uniformly on each compact set in the $(\zeta, \varepsilon)$ space as $y$ tends to infinity in any closed subsector of the open sector,
$$
|\arg y|<\frac{3 \pi}{5}
$$
moreover
$$
E_{3}(y ; \zeta, \varepsilon)=\frac{2}{5} y^{5 / 2}+\zeta y^{1 / 2}
$$



证明 .

and $B_{3, N}$ are polynomials in $(\zeta, \varepsilon)$.
We note that, setting
$$
\omega=\exp \left[i \frac{2}{5} \pi\right]
$$
and
$$
\mathscr{Y}{3, k}(y ; \zeta, \varepsilon)=\mathscr{Y}{3}\left(\omega^{-k} y ; \omega^{-2 k} \zeta, \omega^{-3 k} \varepsilon\right),
$$


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MATH96018/97027/97104 COURSE NOTES ：

Let us now consider the above equation in a neighborhood of $\left(0, \zeta_{0}\right)$. There exists a positive $\delta$ such that each root of the equation
$$
C_{0}(\zeta, \varepsilon)=0
$$
for some positive integer $p$, is a holomorphic function of $\varepsilon^{1 / p}$, for $0<|\varepsilon|<\delta$, that is
$$
\zeta(\varepsilon)=\zeta_{0}+\sum_{j=0}^{\infty} c_{j}\left(\varepsilon^{1 / p}\right)^{j}=g\left(\varepsilon^{1 / p}\right)
$$
This is actually a consequence of Theorem $3.2 .6$ in $[6]$, observing that $\gamma(\zeta, \varepsilon) \neq 0$ implies that the function $g$ has no polar singularity at the origin.

As a matter of fact the function $g$ is holomorphic in a full neighborhood of the origin so that $\zeta\left(\eta^{p}\right)=g(\eta)$, which is a well-defined holomorphic function of $\eta$.