239 research outputs found
A BSDE-based approach for the optimal reinsurance problem under partial information
We investigate the optimal reinsurance problem under the criterion of
maximizing the expected utility of terminal wealth when the insurance company
has restricted information on the loss process. We propose a risk model with
claim arrival intensity and claim sizes distribution affected by an
unobservable environmental stochastic factor. By filtering techniques (with
marked point process observations), we reduce the original problem to an
equivalent stochastic control problem under full information. Since the
classical Hamilton-Jacobi-Bellman approach does not apply, due to the infinite
dimensionality of the filter, we choose an alternative approach based on
Backward Stochastic Differential Equations (BSDEs). Precisely, we characterize
the value process and the optimal reinsurance strategy in terms of the unique
solution to a BSDE driven by a marked point process.Comment: 30 pages, 3 figure
Optimal excess-of-loss reinsurance for stochastic factor risk models
We study the optimal excess-of-loss reinsurance problem when both the
intensity of the claims arrival process and the claim size distribution are
influenced by an exogenous stochastic factor. We assume that the insurer's
surplus is governed by a marked point process with dual-predictable projection
affected by an environmental factor and that the insurance company can borrow
and invest money at a constant real-valued risk-free interest rate . Our
model allows for stochastic risk premia, which take into account risk
fluctuations. Using stochastic control theory based on the
Hamilton-Jacobi-Bellman equation, we analyze the optimal reinsurance strategy
under the criterion of maximizing the expected exponential utility of the
terminal wealth. A verification theorem for the value function in terms of
classical solutions of a backward partial differential equation is provided.
Finally, some numerical results are discussed
The Zakai equation of nonlinear filtering for jump-diffusion observation: existence and uniqueness
This paper is concerned with the nonlinear filtering problem for a general
Markovian partially observed system (X,Y), whose dynamics is modeled by
correlated jump-diffusions having common jump times. At any time t, the
sigma-algebra generated by the observation process Y provides all the available
information about the signal X. The central goal of stochastic filtering is to
characterize the filter which is the conditional distribution of X, given the
observed data. It has been proved in Ceci-Colaneri (2012) that the filter is
the unique probability measure-valued process satisfying a nonlinear stochastic
equation, the so-called Kushner-Stratonovich equation (KS-equation). In this
paper the aim is to describe the filter in terms of the unnormalized filter,
which is solution to a linear stochastic differential equation, called the
Zakai equation. We prove equivalence between strong uniqueness for the solution
to the Kushner Stratonovich equation and strong uniqueness for the solution to
the Zakai one and, as a consequence, we deduce pathwise uniqueness for the
solutions to the Zakai equation by applying the Filtered Martingale Problem
approach (Kurtz-Ocone (1988), Kurtz-Nappo (2011), Ceci-Colaneri (2012)). To
conclude, we discuss some particular cases.Comment: 29 page
Hedging of unit-linked life insurance contracts with unobservable mortality hazard rate via local risk-minimization
In this paper we investigate the local risk-minimization approach for a
combined financial-insurance model where there are restrictions on the
information available to the insurance company. In particular we assume that,
at any time, the insurance company may observe the number of deaths from a
specific portfolio of insured individuals but not the mortality hazard rate. We
consider a financial market driven by a general semimartingale and we aim to
hedge unit-linked life insurance contracts via the local risk-minimization
approach under partial information. The F\"ollmer-Schweizer decomposition of
the insurance claim and explicit formulas for the optimal strategy for pure
endowment and term insurance contracts are provided in terms of the projection
of the survival process on the information flow. Moreover, in a Markovian
framework, we reduce to solve a filtering problem with point process
observations.Comment: 27 page
The F\"ollmer-Schweizer decomposition under incomplete information
In this paper we study the F\"ollmer-Schweizer decomposition of a square
integrable random variable with respect to a given semimartingale
under restricted information. Thanks to the relationship between this
decomposition and that of the projection of with respect to the given
information flow, we characterize the integrand appearing in the
F\"ollmer-Schweizer decomposition under partial information in the general case
where is not necessarily adapted to the available information level. For
partially observable Markovian models where the dynamics of depends on an
unobservable stochastic factor , we show how to compute the decomposition by
means of filtering problems involving functions defined on an
infinite-dimensional space. Moreover, in the case of a partially observed
jump-diffusion model where is described by a pure jump process taking
values in a finite dimensional space, we compute explicitly the integrand in
the F\"ollmer-Schweizer decomposition by working with finite dimensional
filters.Comment: 22 page
A Benchmark Approach to Risk-Minimization under Partial Information
In this paper we study a risk-minimizing hedging problem for a semimartingale
incomplete financial market where d+1 assets are traded continuously and whose
price is expressed in units of the num\'{e}raire portfolio. According to the
so-called benchmark approach, we investigate the (benchmarked) risk-minimizing
strategy in the case where there are restrictions on the available information.
More precisely, we characterize the optimal strategy as the integrand appearing
in the Galtchouk-Kunita-Watanabe decomposition of the benchmarked claim under
partial information and provide its description in terms of the integrands in
the classical Galtchouk-Kunita-Watanabe decomposition under full information
via dual predictable projections. Finally, we apply the results in the case of
a Markovian jump-diffusion driven market model where the assets prices dynamics
depend on a stochastic factor which is not observable by investors.Comment: 31 page
GKW representation theorem and linear BSDEs under restricted information. An application to risk-minimization
In this paper we provide Galtchouk-Kunita-Watanabe representation results in
the case where there are restrictions on the available information. This allows
to prove existence and uniqueness for linear backward stochastic differential
equations driven by a general c\`adl\`ag martingale under partial information.
Furthermore, we discuss an application to risk-minimization where we extend the
results of F\"ollmer and Sondermann (1986) to the partial information framework
and we show how our result fits in the approach of Schweizer (1994).Comment: 22 page
Multitype branching processes observing particles of a given type
A multitype branching process is presented in the framework of marked trees and its structure is studied by applying the strong branching property. In particular, the Markov property and the expression for the generator are derived for the process whose components are the numbers of particles of each type. The filtering of the whole population, observing the number of particles of a given type, is discussed. Weak uniqueness for the filtering equation and a recursive structure for the linearized filtering equation are proved under a suitable assumption on the reproduction law
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