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## Pricing and hedging of American contingent claims in incomplete markets

### Acta Mathematicae Applicatae Sinica (1999-04-01) 15: 144-152 , April 01, 1999

This paper studies the pricing and hedging for American contingent claims in an incomplete market under mild conditions using the numeraire method to avoid changes of probability measure. When the market is incomplete, prices can not be derived by no-arbitrage arguments, since it is not possible to replicate the payoff of a given contingent claim by a controlled portfolio of the basic securities. We adopt the method of fictitious completion of [1] to provide an upper bound and a lower bound for the actual market price of the claim.

## On the spectral properties ofM-matrices and its applications

### Acta Mathematicae Applicatae Sinica (1999-10-01) 15: 418-424 , October 01, 1999

Let*A* be an*M*-matrix and*B* be a*Z*-matrix. In this paper we reveal the spectral relationship of*A* and*B* under some interesting conditions. Applying this result, we solve an open problem on splittings of an*M*-matrix and partially answer an open problem on the level diagrams for*A* and*B*.

## On Nonexistence of Algebraic Curve Solutions to Second-order Polynomial Autonomous Systems Over the Complex Field

### Acta Mathematicae Applicatae Sinica (2002-11-01) 18: 657-662 , November 01, 2002

###
*Abstract*

In this paper, by using the method of algebraic analysis, the results in our previous work are generalized. These results are of importance in the qualitative theory of polynomial autonomous systems.

## Ruin Probabilities under a Markovian Risk Model

### Acta Mathematicae Applicatae Sinica (2003-12-01) 19: 621-630 , December 01, 2003

###
*Abstract*

In this paper, a Markovian risk model is developed, in
which the occurrence of the claims is described by a point
process {*N(t)*}
_{t≥0}
with *N(t)* being the number of
jumps of a Markov chain during the interval [0,
*t*]. For the model, the
explicit form of the ruin probability Ψ(0) and the bound for the
convergence rate of the ruin probability Ψ(*u*) are given by using the generalized
renewal technique developed in this paper. Finally, we prove
that the ruin probability Ψ(*u*) is a linear combination of some
negative exponential functions in a special case when the claims
are exponentially distributed and the Markov chain has an
intensity matrix (*q*_{ij})_{i,j∈E} such that
*q*_{m} = *q*_{m1} and
*q*_{i} = *q*_{i(i+1)}, 1 ≤
*i* ≤ *m*−1.

## Boundary Value Problems for Singular Second-Order Functional Differential Equations

### Acta Mathematicae Applicatae Sinica (2002-06-01) 18: 249-254 , June 01, 2002

###
*Abstract*

Positive solutions to the boundary value problem,
$$
\left\{ {\begin{array}{*{20}l}
{{{y}\ifmmode{''}\else$''$\fi = - f{\left( {x,y{\left( {w{\left( x \right)}} \right)}} \right)},} \hfill} & {{0 < x < 1,} \hfill} \\
{{\alpha y{\left( x \right)} - \beta {y}\ifmmode{'}\else$'$\fi{\left( x \right)} = \xi {\left( x \right)},} \hfill} & {{a \leqslant x \leqslant 0,} \hfill} \\
{{\gamma y{\left( x \right)} + \delta {y}\ifmmode{'}\else$'$\fi{\left( x \right)} = \eta {\left( x \right)},} \hfill} & {{1 \leqslant x \leqslant b,} \hfill} \\
\end{array} } \right.
$$
are obtained by applying the Schauder fixed point theorem, where *w*(*x*) is a continuous function defined on [0, 1] and *f*(*x, y*) is a function defined on (0, 1)×(0, ∞), which satisfies certain restrictions and may have singularity at *y*=0. The result corrects and improves an existence theorem due to Erbe and Kong^{[1]}.

## Fixed Design Nonparametric Regression with Truncated and Censored Data

### Acta Mathematicae Applicatae Sinica (2003-06-01) 19: 229-238 , June 01, 2003

###
*Abstract*

In this paper we consider a fixed design model in which the observations are subject to left truncation and right censoring. A generalized product-limit estimator for the conditional distribution at a given covariate value is proposed, and an almost sure asymptotic representation of this estimator is established. We also obtain the rate of uniform consistency, weak convergence and a modulus of continuity for this estimator. Applications include trimmed mean and quantile function estimators.

## Some greedyt-intersecting families of finite sequences

### Acta Mathematicae Applicatae Sinica (1996-10-01) 12: 377-383 , October 01, 1996

Let*n, s*_{1},*s*_{2}, ... and*s*_{n} be positive integers. Assume
$$\mathcal{M}(s_1 ,s_2 , \cdots ,s_n ) = \{ (x_1 ,x_2 , \cdots x_n )|0 \leqslant x_i \leqslant s_i ,x_i$$
is an integer for each*i*}. For
$$a = (a_1 ,a_2 , \cdots a_n ) \in \mathcal{M}(s_1 ,s_2 , \cdots ,s_n )$$
,
$$\mathcal{F} \subseteq \mathcal{M}(s_1 ,s_2 , \cdots ,s_n )$$
, and
$$A \subseteq \{ 1,2, \cdots ,n\}$$
, denote*s*_{p}(*a*)={*j*|1≤*j*≤*n*,*a*_{j}≥*p*},
$$S_p (\mathcal{F}) = \{ s_p (a)|a \in \mathcal{F}\}$$
, and
$$W_p (A) = p^{n - |A|} \prod\limits_{i \in A} {(s_i - p)}$$
.
$$\mathcal{F}$$
is called an*I*_{t}^{p}
-intersecting family if, for any a,b∈
$$\mathcal{F}$$
,*a*_{i}Λ*b*_{i}=min(*a*_{i},*b*_{i})≥*p* for at least*t* i's.
$$\mathcal{F}$$
is called a greedy*I*_{t}^{P}
-intersecting family if
$$\mathcal{F}$$
is an*I*_{t}^{p}
-intersecting family and*W*_{p}(*A*)≥*W*_{p} (*B+A*^{c}) for any*A*ε*S*_{p}(
$$\mathcal{F}$$
) and any
$$B \subseteq A$$
with |*B*|=*t*−1.

In this paper, we obtain a sharp upper bound of |
$$\mathcal{F}$$
| for greedy*I*_{t}^{p}
-intersecting families in
$$\mathcal{M}(s_1 ,s_2 , \cdots ,s_n )$$
for the case 2*p*≤*s*_{i} (1≤*i*≤*n*) and*s*_{1}>*s*_{2}>...>*s*_{n}.

## On wavelets inL 1

### Acta Mathematicae Applicatae Sinica (1994-01-01) 10: 69-74 , January 01, 1994

In this paper we give a method to characterize the smoothness of functions in*L*_{1} by an*r*-regular multiresolution analysis and its derivatives.

## PBVP for integro-differential equations of volterra type in B.S.

### Acta Mathematicae Applicatae Sinica (1991-07-01) 7: 284-288 , July 01, 1991

In this paper, we study the PBVP for integro-differential equations of Volterra type in Banach spaces. By developing monotone iterative technique for the PBVP, we get some results concerning the existence of extremal solutions, which are the limits of monotone sequences.

## Existence and uniqueness of limit cycles on a cubic kolmogorov differential system in the Predator-Prey relation

### Acta Mathematicae Applicatae Sinica (1988-08-01) 4: 245-256 , August 01, 1988

In this paper, we analyse qualitatively a cubic Kolmogorov system: $$\left\{ \begin{gathered} \frac{{dx}}{{dt}} = x[a_0 + a_1 x - a_3 x^2 - a_4 y + a_5 xy], \hfill \\ \frac{{dy}}{{dt}} = y(x - 1)(1 + by), \hfill \\ \end{gathered} \right.$$ which is one of the mathematical models in ecology describing the interaction between Predator-Prey of two populations; and give the conditions of nonexistence, existence and uniqueness of limit cycles for three different cases.