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 $r$. 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 $\xi$ with respect to a given semimartingale $S$ under restricted information. Thanks to the relationship between this decomposition and that of the projection of $\xi$ 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 $\xi$ is not necessarily adapted to the available information level. For partially observable Markovian models where the dynamics of $S$ depends on an unobservable stochastic factor $X$, 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 $X$ 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