| 摘要(英) |
In this dissertation, we study the integral representation of positive
space-time harmonic function for Gaussian diffusion processes. A Gaussian
diffusion process $Y_{t}$ in $mathbb{R}^{d}$ is governed by
[
egin{cases}
dY_{t}=BY_{t}dt+dW_{t},\
Y_{0}=x_{0},
end{cases}
]
where $B$ is a $d imes d$ matrix, $W$ is a $d-$dimensional Brownian
motion, and $x_{0}inmathbb{R}^{d}$ is the initial value of $Y$.
$g$ is a positive space-time harmonic function for $Y_{t}$ which
satisfies
egin{align*}
(frac{partial}{partial t}+frac{1}{2} riangle+Bxcdot
abla)g=0,mbox{ }g>0mbox{ on }(0,infty) imesmathbb{R}^{d}.
end{align*}
The integral formula of $g$ is given by
egin{align*}
g(t,x) & =int_{mathbb{R}^{d}}M_{B}(t,x;z)
ho(dz),
end{align*}
where $
ho$ is a probability distribution and ${M_{B}(cdot,cdot;z);zinmathbb{R}^{d}}$
is a family of functions which is independent of $g$. To obtain such
integral representation, we consider a process associated to $g$
deduced by $X_{t}$ which is governed by
[
egin{cases}
dX_{t}=frac{
abla g(t,X_{t})}{g(t,X_{t})}dt+Bcdot X_{t}dt+dW_{t},\
X_{0}=x_{0}.
end{cases}
]
We study the limiting behavior of $X_{t}$ as $t
ightarrowinfty$.
We first obtain an interesting representation of $X_{t}$. Then the
integral formula of $g$ will follow. In Part 1, we consider the Brownian
motion, where $B=0$. In this case, we show $X_{t}$ has linear growth
with the rate given by $Y$ as $t
ightarrowinfty$. This means
egin{align*}
frac{X_{t}^{g}}{t}
ightarrow Y,mbox{ as }t
ightarrowinfty,
end{align*}
where $Y$ is a random variable. Futhermore, $X_{t}$ has remarkable
representation
egin{align*}
X_{t} & =x_{0}+tY+widehat{W}_{t},
end{align*}
where $widehat{W}_{t}$ is a Brownian motion independent of $Y$.
Using this, we obtain an integral representation for $g$, where $
ho$
(in the representation) is the disrtibution of $Y$. In Part 2, we
consider general $B$. We apply the similar approach to obtain the
growth of $X_{t}$, with different rate and a representation of $X_{t}$.
Then we can obtain the integral representation formula of $g$. We
also discuss some applications of the integral representation of space-time
harmonic functions. The first example is the integral representation
for a positive (space) harmonic functions. The second example is the
use in the calculation of the boundary crossing probability. |
| 參考文獻 |
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Finance Volume 9, 2009 - Issue 2. |
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