Datasets:
id stringlengths 15 21 | doc_id stringlengths 13 18 | text stringlengths 1 3.86k | label stringclasses 3
values | soft unknown | agreement float64 1 1 | split stringclasses 1
value | coverage float64 0 1 | op stringclasses 6
values |
|---|---|---|---|---|---|---|---|---|
1011.1329::v0:0 | 1011.1329::v0 | We consider an insurance company in the case when the premium
rate is a
bounded non-negative
random function [公式]
and the capital of the insurance company
is invested in a risky asset
whose price follows a geometric Brownian
motion with mean return [公式] and volatility
[公式] . If [公式]
we find exact the asymptotic upp... | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:1 | 1011.1329::v0 | { MJS:} primary 62P05; | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:2 | 1011.1329::v0 | 60J25; G22; G23 | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:3 | 1011.1329::v0 | Risk process; | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:4 | 1011.1329::v0 | Geometric Brownian motion; | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:5 | 1011.1329::v0 | Ruin probability | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:6 | 1011.1329::v0 | {1. }
{0} | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:7 | 1011.1329::v0 | It is well known that the analysis of activity
of an insurance company in conditions of uncertainty is of
great importance. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:8 | 1011.1329::v0 | Starting from the classical papers of
Cram\'er and Lundberg which
first considered the ruin problem
in stochastic environment,
this subject has attracted much attention. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:9 | 1011.1329::v0 | Recall that, in the classical Cram\'er--Lundberg model
satisfying the Cram\'er condition and, the positive
safety loading assumption,
the ruin probability as a function of the initial
endowment decreases exponentially (see, for example, Mikosch ). | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:10 | 1011.1329::v0 | The problem
was subsequently
extended to the case when the insurance risk
process is a general L\'evy process
(see, for example, Kl\"uppelberg et al.
for details). | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:11 | 1011.1329::v0 | More recently ruin problems have been studied
in application to an insurance company which
invests its capital in a risky asset
see, e.g., Paulsen ,
Kalshnikov and Norberg ,
Frolova et al.
and many others. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:12 | 1011.1329::v0 | It is clear that, risky investment can be dangerous:
disasters may arrive in the period when the market value of
assets is low and the company will not be able to cover losses by
selling these assets because of price fluctuations. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:13 | 1011.1329::v0 | Regulators are rather attentive to this issue and impose
stringent constraints on company portfolios. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:14 | 1011.1329::v0 | Typically, junk
bonds are prohibited and a prescribed (large) part of the portfolio
should contain non-risky assets (e.g., Treasury bonds) while in
the remaining part only risky assets with good ratings are
allowed. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:15 | 1011.1329::v0 | The common notion that investments in an asset
with stochastic
interest rate may be too risky for an insurance company can be
justified mathematically. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:16 | 1011.1329::v0 | We deal with the ruin problem for an insurance company investing
its capital in a risky asset specified by a
geometric Brownian motion | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:17 | 1011.1329::v0 | where [公式] is a standard Brownion motion and [公式] ,
[公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:18 | 1011.1329::v0 | It turns out that in this case of
{ small volatility}, i.e.
[公式] , the ruin probability is not exponential but a
power function of the initial capital with the exponent
[公式] . It will be noted that this result holds
without the requirement of positive safety loading. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:19 | 1011.1329::v0 | Also, for large volatility, i.e.
[公式] , the ruin probability equals [公式] for any
initial endowment. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:20 | 1011.1329::v0 | These results
have been obtained
under various conditions in
. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:21 | 1011.1329::v0 | Additionally, a large deviations
limiting theorems
for describing the ruin probability
was obtained by Djehiche and Nyrhinen .
Gaier et al. studied
the optimal investment problem for an insurance company. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:22 | 1011.1329::v0 | In all these papers
the premium rate was assumed to be constant. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:23 | 1011.1329::v0 | In practice
this means that the company should
obtain a premium with
the same rate continuously. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:24 | 1011.1329::v0 | We think that this condition is too restrictive
and
it significantly bounds the applicability of the above mentioned
results in practical insurance settings. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:25 | 1011.1329::v0 | The goal of this paper is to consider the ruin problem for an
insurance company for which the premium rate is specified by
a bounded non-negative random function [公式] . For the given problem,
under the condition of { small volatility},
we derive exact upper and lower bounds for the ruin
probability and in the case of e... | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:26 | 1011.1329::v0 | Particularly, we show that for the zero premium rate, i.e.
[公式] , the asymptotic result is
the same as in the case [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:27 | 1011.1329::v0 | Moreover, in this paper we show that in the boundary case, i.e.
[公式] ,
the company goes bankrupt
with probability [公式] for any bounded function [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:28 | 1011.1329::v0 | Indeed, an upper bound for the ruin probability
for the random function [公式] in the small volatility case
is obtained also by Ma and Sun . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:29 | 1011.1329::v0 | The paper is organised as follows. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:30 | 1011.1329::v0 | In the next section we give
the main results. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:31 | 1011.1329::v0 | In Section 3 we give the necessary results
about the tails of solutions of some linear random equation
which we apply to study the ruin problem. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:32 | 1011.1329::v0 | In Section 4 we obtain the upper bound for
the ruin probability and in Section 5 we find the corresponding
lower bound. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:33 | 1011.1329::v0 | In Section 6 we consider the
exponential premium income rate case. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:34 | 1011.1329::v0 | In Section 7 we study some ergodic
properties for an autoregressive process with random coefficient.
in Section 8 we consider
the large volatility case. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:35 | 1011.1329::v0 | {2. }
{0} | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:36 | 1011.1329::v0 | Let us consider a process [公式] of the form | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:37 | 1011.1329::v0 | where [公式] and [公式] are arbitrary constants,
[公式] is a Brownion motion, [公式] is a Poisson process with
intensity [公式] and [公式]
are i.i.d. positive
random variables with common a
distribution [公式] . Moreover, we assume that
[公式] , [公式] , [公式] are independent and
the filtration is defined as
[公式] .
Furthermore,
[公式] ... | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:38 | 1011.1329::v0 | Let [公式] (the time of ruin),
[公式] (the ruin probability). | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:39 | 1011.1329::v0 | The parameter values [公式] , [公式] , [公式] , correspond to the
Cram\'er--Lundberg
model for which the risk process is usually
written as
[公式] .
In the considered version (of non-life
insurance) the capital evolves due to a continuously incoming cash
flow with rate [公式] and outgoing random payoffs [公式] at times
forming ... | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:40 | 1011.1329::v0 | Moreover, for exponentially distributed claims the ruin
probability admits an explicit expression, see or
. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:41 | 1011.1329::v0 | We study here the case [公式] with a general random adapted
bounded function [公式] . In this case Eq.
describes the evolution of the capital of an insurance company,
which is continuously reinvested into an asset with the price
following a geometric Brownian motion . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:42 | 1011.1329::v0 | Let [公式] .
To write the upper bound for the ruin probability
we define the function : | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:43 | 1011.1329::v0 | where [公式] ,
[公式]
and
[公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:44 | 1011.1329::v0 | If [公式] and [公式] , then
[公式] ,
where
[公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:45 | 1011.1329::v0 | The proof of this theorem is given in Section 4. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:46 | 1011.1329::v0 | If [公式] and
[公式] for some [公式] ,
then there exists a constant [公式] such that
[公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:47 | 1011.1329::v0 | This result is proved in Section 5. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:48 | 1011.1329::v0 | The following theorem gives the
exact asymptotics for the exponential function [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:49 | 1011.1329::v0 | Assume that [公式] with
[公式] .
If [公式] and\\ [公式]
for some [公式] ,
then there exists a constant [公式] such that
[公式] .
Moreover, the constant [公式] is the same for
any [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:50 | 1011.1329::v0 | This result is proved in Section 6. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:51 | 1011.1329::v0 | Now we consider the large volatility case,
i.e. [公式] .
Assume that
the distribution of
[公式] has not a finite support, i.e. [公式] for any
[公式] . If [公式] and
[公式] for some [公式] , then
[公式] for any [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:52 | 1011.1329::v0 | This theorem has been proved
by Paulsen in
for a constant premium rate, i.e. for [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:53 | 1011.1329::v0 | The key idea in the proofs of Theorem and
Theorem is based on the fact that
the function [公式] may be estimated
by the tails of solutions of some linear random equations. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:54 | 1011.1329::v0 | In the next section we study the asymptotic behaviour of those tails. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:55 | 1011.1329::v0 | {3. }
{0} | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:56 | 1011.1329::v0 | This Section contains some results from the general
renewal theory developed by Goldie for some
random equations. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:57 | 1011.1329::v0 | We consider the following two random equations | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:58 | 1011.1329::v0 | ( [公式] denoting equality of probability laws) and | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:59 | 1011.1329::v0 | where [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:60 | 1011.1329::v0 | We start with some preliminary conditions for the random variable [公式]
which are studied by Goldie (see Lemma 2.2 in ). | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:61 | 1011.1329::v0 | Let [公式] be a random variable such that, for some [公式] | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:62 | 1011.1329::v0 | and the conditional law of [公式] , given [公式] ,
be non-arithmetic. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:63 | 1011.1329::v0 | Then [公式]
and [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:64 | 1011.1329::v0 | The following result from
specifies the tail behaviour of [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:65 | 1011.1329::v0 | (Theorem 4.1 in )
Let [公式] be a random variable satisfying the conditions of
Lemma for some [公式] and [公式] be a positive random
variable for which [公式] . Then there is a unique
law for [公式] satisfying such that | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:66 | 1011.1329::v0 | where [公式]
and [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:67 | 1011.1329::v0 | Now we study the tail of [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:68 | 1011.1329::v0 | Let [公式] be a random variable satisfying the conditions of
Lemma for some [公式] . Assume also that
the distribution of [公式] is absolutely continuous with respect to
Lebesgue measure and there exists [公式] such that | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:69 | 1011.1329::v0 | and for any [公式] | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:70 | 1011.1329::v0 | where [公式] .
Then under the condition | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:71 | 1011.1329::v0 | for some [公式]
there is a unique
law for [公式] satisfying such that there exists
[公式]
and [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:72 | 1011.1329::v0 | This lemma follows directly
from Theorem 6.3 in
and Theorem 2 in . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:73 | 1011.1329::v0 | {4. }
{0} | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:74 | 1011.1329::v0 | Let [公式] be the instant of
[公式] -th jump of [公式] and let [公式] with
[公式] . We define the discrete-time process
[公式] with [公式] . Since ruin may occur only when
[公式] jumps downwards,
[公式] , where | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:75 | 1011.1329::v0 | Therefore to obtain asymptotic properties of [公式] as
[公式] we need to study the process [公式] . First of all,
we need to find a recurrence equation for this sequence. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:76 | 1011.1329::v0 | We start with resolving
of Eq. . For this we introduce the
process [公式]
which satisfies the following stochastic differential equation | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:77 | 1011.1329::v0 | The Ito formula implies that
[公式] ,
where\\ [公式] ,
[公式] and [公式] .
Moreover we can represent Eq. for
[公式] in the following way | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:78 | 1011.1329::v0 | Therefore [公式] .
From this we obtain the following random recurrence equation for [公式] | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:79 | 1011.1329::v0 | with [公式] and
[公式] . Here
[公式] and
[公式] with
[公式] .
By resolving we find the following representation
for [公式] | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:80 | 1011.1329::v0 | Moreover, taking into account here that [公式]
we obtain that [公式] ,
where | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:81 | 1011.1329::v0 | Notice that
[公式] are i.i.d. random
variables such that for [公式] | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:82 | 1011.1329::v0 | Therefore, there exists [公式] for which [公式]
and | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:83 | 1011.1329::v0 | i.e. the series
[公式]
is finite a.s. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:84 | 1011.1329::v0 | It means that the sequence [公式] have a finite limit | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:85 | 1011.1329::v0 | Taking into account that the sequence [公式] in ( )
is increasing we can estimate
[公式] as | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:86 | 1011.1329::v0 | and by ( ) we get that [公式] .
Therefore, to obtain the upper bound for the ruin probability
we investigate the tail behaviour of [公式] as [公式] .
To this end, first notice that we may represent [公式] in the following form | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:87 | 1011.1329::v0 | where the random variable
[公式]
has the same distribution as [公式] and is
independent of [公式] . Thus the random variable [公式]
satisfies Eq. .
We show that | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:88 | 1011.1329::v0 | where [公式] . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:89 | 1011.1329::v0 | To show ( ) we need to check the conditions
of Lemma for the random
variables [公式] and [公式] defined in ( ). | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:90 | 1011.1329::v0 | The first property in
( ) follows directly from ( ) for [公式] .
Now we show the second. | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:91 | 1011.1329::v0 | By definition of [公式] we have | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:92 | 1011.1329::v0 | Taking into account that [公式] is independent of
[公式] , the
last term in this inequality equals | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:93 | 1011.1329::v0 | i.e. [公式] .
In similar way we calculate
[公式] .
Moreover,
[公式] .
Therefore, by making use of Lemma
we get
the limiting relationship ( ) which
implies that [公式] .
Thus, to finish the proof we need to show
the inequality [公式] .
Indeed, if [公式] , then \\
[公式]
and,
therefore, in this case [公式] .
If [公式] , then, taking ... | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:94 | 1011.1329::v0 | Since by ( ) we have [公式] , therefore
by making use of ( ) and taking into account that
[公式] ( [公式] is defined in ( ))
we can estimate [公式] as | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:95 | 1011.1329::v0 | Thus, from this and ( ), we obtain that
[公式] for [公式] .
Let us consider now the case [公式] . In this case
we estimate [公式] as | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:96 | 1011.1329::v0 | We set
[公式] with [公式] .
Taking into account that
the random variables [公式] and [公式] are independent in ( ),
we obtain that | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:97 | 1011.1329::v0 | i.e. [公式] .
From this, we find | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:98 | 1011.1329::v0 | Applying this inequality to ( ), one obtains
[公式] for [公式] .
This implies Theorem . | human | {
"human": 1
} | 1 | train | 0 | none |
1011.1329::v0:99 | 1011.1329::v0 | {5. }
{0} | human | {
"human": 1
} | 1 | train | 0 | none |
End of preview. Expand in Data Studio
AI-writing-detector-dataset
面向学术论文 AI 写作检测的句级数据集：人类原文 vs DeepSeek 润色/改写，标签由构造过程确定，无需人工标注。
| 子集 | 来源 | 文档数 | train / val / test |
|---|---|---|---|
zh/ |
中文期刊论文 | 118 | 186,640 / 36,189 / 29,932 |
zh_med/ |
中文医学摘要 | 8,000 | 325,300 / 65,420 / 65,850 |
en/ |
arXiv 2007–2017 | 808 | 783,372 / 150,857 / 145,470 |
合计 1,789,030 行 / 约 35 万独立句。
构造（v3 渐进式）
v0 ─ v1(15%) ─ v2(25%) ─ v3(40%) ─ v4(50%) ─ v5(60%) ─ v6(75%) ─ v7(90%) ─ v8(100%)
polish paraphrase style compress expand style paraphrase polish
- 人类论文切句；
- 每句按固定随机序分配到一个版本，只改写一次，后续版本累积继承；
- 5 种操作：润色/改写/风格/压缩/扩写；
- 句级一对一改写，标签与文本同时产生；
- 导出 v0/v2/v4/v6/v8。
质量
| 指标 | zh | zh_med | en |
|---|---|---|---|
| 标签不一致 | 0 | 0 | 0 |
| 近拷贝率 | 0.38% | 1.1% | 0.0% |
| v8 覆盖率 | 96.4% | 98.2% | 96.0% |
字段
id / doc_id / text / label / soft / agreement / split / coverage / op
使用
from datasets import load_dataset
ds = load_dataset("johnson/AI-writing-detector-dataset", data_dir="zh")
print(ds["train"][0])
许可
Apache-2.0。原始论文版权归原作者/期刊，本数据集只提供句子级片段，不再分发 PDF。
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