Risk-minimizing hedging strategies for insurance payment processes. (English) Zbl 0983.62076

Let \((\Omega,{\mathcal F},P)\) be a probability space with filtration \({\mathbf F}=(\mathcal F_t)_{0\leq t\leq T}\). Consider a financial market consisting of two assets with discounted price processes \(X=(X_t)_{0\leq t\leq T}\) and \(Y=(Y_t)_{0\leq t\leq T}\), respectively, where \(X\) is the discounted price process associated with some risky asset (a stock), and the discounted price process \(Y\) associated with the locally risk-free asset (saving account) which is assumed to be constant. A trading strategy is a process \(\varphi=(\xi,\eta)\), where \(\xi_t\) is interpreted as the amount of stocks at time \(t\), and \(\eta_t\) is the discounted deposit on the saving account at time \(t\). The pair \(\varphi_t=(\xi_t,\eta_t)\) is called the portfolio held at time \(t\). The value process associated with the strategy \(\varphi=(\xi,\eta)\) is defined by \[ V_t(\varphi)=\xi_tX_t+\eta_t,\quad 0\leq t\leq T. \] Consider an agreement between two partners, a seller (hedger) and a buyer specifying certain payments given by an \({\mathbf F}\)-adapted process \(A=(A_t)_{0\leq t\leq T}\). The simplest considered situation is where \(A\) is of the form \(A_t=-\kappa+\mathbf{1}_{\{t\geq T\}}H,\;0\leq t\leq T,\) for some constant \(\kappa\) and \(H\in L_2({\mathcal F}_T,P)\) (payments take place at time 0 and time \(T\) only). Let \(C(\varphi)=(C_t(\varphi))_{0\leq t\leq T}\) be a cost process defined by the equation \[ C_t(\varphi)=V_t(\varphi)-\int_0^t\xi_udX_u+A_t,\quad 0\leq t\leq T, \] with the initial condition \(C_0(\varphi)=V_0(\varphi)+A_0\). \(C_t(\varphi)\) comprises the hedger accumulated costs during \([0,t]\) and \(V_t(\varphi)\) may be interpreted as the value of the portfolio \(\varphi_t=(\xi_t,\eta_t)\) held at time \(t\) after the payment \(A_t\). The risk process of the strategy \(\varphi=(\xi,\eta)\) is defined by \(R_t(\varphi)= {\mathbf E}\left[(C_T(\varphi)-C_t(\varphi))^2 |{\mathcal F}_t\right]\). A strategy is said to be risk-minimizing if it minimizes \(R_t(\varphi)\) for all \(t\). Define a martingale \(V_t^{\ast}={\mathbf E}\left[A_T |{\mathcal F}_t\right],\;0\leq t\leq T\). This martingale can be uniquely decomposed by use of the Galtchouk-Kunita-Watanabe decomposition [see H. Föllmer and D. Sondermann, Contributions to Mathematical Economics, Hon. G. Debreu, 206-223 (1986; Zbl 0663.90006)] as \[ V_t^{\ast}=V_0^{\ast}+\int_0^t\xi_u^AdX_u+L_t^A, \] where \(L^A\) ia a zero-mean martingale orthogonal to \(X\) and \(\xi^A\) is a predictable process.
The author proves that there exists a unique \(0\)-admissible (\(V_T(\varphi)=0\)) risk-minimizing strategy \(\varphi=(\xi,\eta)\) for \(A\) given by \[ (\xi_t,\eta_t)=(\xi^A_t,V_t^{\ast}-A_t-\xi^A_tX_t),\quad 0\leq t\leq T. \] The presented result extends the corresponding result of Föllmer and Sondermann. The author presents some examples related to insurance. These include a general unit-linked life insurance contract driven by a Markov jump process and a claim process from non-life insurance, where the claim size distribution is affected by a traded price index.


62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
60G44 Martingales with continuous parameter
60G35 Signal detection and filtering (aspects of stochastic processes)


Zbl 0663.90006
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