Girsanov's theorem and likelihood for random initial conditions
$begingroup$
Consider as an example the Stochastic Differential Equation
$$
text{d}Y(t) = -kappa Y(t) text{d}t + sigma text{d} B(t), qquad t geq 0
$$
where $B(t)$ is a standard Brownian motion, $kappa$ and $sigma$ are
positive constants. If $Y(0) = 0$, Girsanov's theorem tells that the
distribution of the process $Y(t)$ is absolutely continuous
w.r.t. that of a standard Brownian. If $sigma$ is known, this theorem
defines a likelihood function for an (ideal) continuously observed
path $t in [0, , T]$ with $T >0$. A maximum
likelihood estimator for $kappa$ can be consequently be derived as
described by Phillips and
Yu.
Now assume that $Y(t)$ is stationary, which implies that $Y(0)$ is
centred normal with variance $sigma^2/(2 kappa)$. Can we still then
use Girsanov's theorem to define a likelihood function? If yes, what
relations exist between this infill or continuous record
likelihood and the simply defined likelihood function arising from
partial observations $Y(t_i)$ when the instants $t_i$ tend to fill the
fixed interval $[0,,T]$?
More generally how can we cope with random initial conditions for a
continuously observed process having a state-space representation? As
a major difference with the example above, the initial state will no
longer be observed.
probability stochastic-processes stochastic-calculus
asked Dec 6 '18 at 10:42
YvesYves
596
$endgroup$
add a comment |
$begingroup$
Consider as an example the Stochastic Differential Equation
$$
text{d}Y(t) = -kappa Y(t) text{d}t + sigma text{d} B(t), qquad t geq 0
$$
where $B(t)$ is a standard Brownian motion, $kappa$ and $sigma$ are
positive constants. If $Y(0) = 0$, Girsanov's theorem tells that the
distribution of the process $Y(t)$ is absolutely continuous
w.r.t. that of a standard Brownian. If $sigma$ is known, this theorem
defines a likelihood function for an (ideal) continuously observed
path $t in [0, , T]$ with $T >0$. A maximum
likelihood estimator for $kappa$ can be consequently be derived as
described by Phillips and
Yu.
Now assume that $Y(t)$ is stationary, which implies that $Y(0)$ is
centred normal with variance $sigma^2/(2 kappa)$. Can we still then
use Girsanov's theorem to define a likelihood function? If yes, what
relations exist between this infill or continuous record
likelihood and the simply defined likelihood function arising from
partial observations $Y(t_i)$ when the instants $t_i$ tend to fill the
fixed interval $[0,,T]$?
More generally how can we cope with random initial conditions for a
continuously observed process having a state-space representation? As
a major difference with the example above, the initial state will no
longer be observed.
probability stochastic-processes stochastic-calculus
asked Dec 6 '18 at 10:42
YvesYves
596
$endgroup$
add a comment |
$begingroup$
Consider as an example the Stochastic Differential Equation
$$
text{d}Y(t) = -kappa Y(t) text{d}t + sigma text{d} B(t), qquad t geq 0
$$
where $B(t)$ is a standard Brownian motion, $kappa$ and $sigma$ are
positive constants. If $Y(0) = 0$, Girsanov's theorem tells that the
distribution of the process $Y(t)$ is absolutely continuous
w.r.t. that of a standard Brownian. If $sigma$ is known, this theorem
defines a likelihood function for an (ideal) continuously observed
path $t in [0, , T]$ with $T >0$. A maximum
likelihood estimator for $kappa$ can be consequently be derived as
described by Phillips and
Yu.
Now assume that $Y(t)$ is stationary, which implies that $Y(0)$ is
centred normal with variance $sigma^2/(2 kappa)$. Can we still then
use Girsanov's theorem to define a likelihood function? If yes, what
relations exist between this infill or continuous record
likelihood and the simply defined likelihood function arising from
partial observations $Y(t_i)$ when the instants $t_i$ tend to fill the
fixed interval $[0,,T]$?
More generally how can we cope with random initial conditions for a
continuously observed process having a state-space representation? As
a major difference with the example above, the initial state will no
longer be observed.
probability stochastic-processes stochastic-calculus
asked Dec 6 '18 at 10:42
YvesYves
596
$endgroup$
Consider as an example the Stochastic Differential Equation
$$
text{d}Y(t) = -kappa Y(t) text{d}t + sigma text{d} B(t), qquad t geq 0
$$
where $B(t)$ is a standard Brownian motion, $kappa$ and $sigma$ are
positive constants. If $Y(0) = 0$, Girsanov's theorem tells that the
distribution of the process $Y(t)$ is absolutely continuous
w.r.t. that of a standard Brownian. If $sigma$ is known, this theorem
defines a likelihood function for an (ideal) continuously observed
path $t in [0, , T]$ with $T >0$. A maximum
likelihood estimator for $kappa$ can be consequently be derived as
described by Phillips and
Yu.
Now assume that $Y(t)$ is stationary, which implies that $Y(0)$ is
centred normal with variance $sigma^2/(2 kappa)$. Can we still then
use Girsanov's theorem to define a likelihood function? If yes, what
relations exist between this infill or continuous record
likelihood and the simply defined likelihood function arising from
partial observations $Y(t_i)$ when the instants $t_i$ tend to fill the
fixed interval $[0,,T]$?
More generally how can we cope with random initial conditions for a
continuously observed process having a state-space representation? As
a major difference with the example above, the initial state will no
longer be observed.
probability stochastic-processes stochastic-calculus
probability stochastic-processes stochastic-calculus
asked Dec 6 '18 at 10:42
YvesYves
596
asked Dec 6 '18 at 10:42
YvesYves
596
edited Dec 6 '18 at 18:44
Yves
asked Dec 6 '18 at 10:42
YvesYves
596
asked Dec 6 '18 at 10:42
YvesYves
596
asked Dec 6 '18 at 10:42
YvesYves
596
596
add a comment |
add a comment |
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