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Interaction between asset liability management and risk theory. (English) Zbl 0924.90012

The paper deals with the generalized Janssen model in which the asset portfolio contains \(N\) zero-coupon bonds. The rates of return are assumed to follow an Ornstein-Uhlenbeck process of the form \(dr_{t}=k(\theta-r_{t})dt+\eta dZ_{t}\), where \(Z_{t}\) is a Brownian motion and \(k,\theta,\eta\in R^{+}\). The asset \(A_{t}\), modelled by the investment in \(N\) pure-discount bonds with maturity \(T\), are determined by the stochastic differential equation \(dA_{t}=A_{t}\left(r_{t}+{\eta\lambda\over k} (1-e^{-k(T-t)})\right)dt-A_{t}{\eta\over k}\left(1-e^{-k(T-t)}\right)dZ_{t}\), where \(\lambda\) is the parameter of market risk and \(A_{T}=N\). The liability process \(B_{t}\) is modelled by a lognormal process with positive constants \(\mu_{B}, \sigma_{B}\) which is correlated with \(Z_{t}\). The assets and liabilities have no perfect match in a period \([0,T]\) if \(A_{t}\leq B_{t}\) for some \(0\leq t\leq T\). We have a final mismatching if \(A_{T} < B_{T}\). The authors study the probability of mismatching and the measure of risk at time \(t\): \(M_{t}(B_{T}-A_{T})= E[(B_{T}-A_{T})^{+}\exp\{-\int_{t}^{T}i_{u}du\}/{\mathcal F}_{t}]\), where \(i_{t}\) is a short-term interest rate; \({\mathcal F}_{t}\) is the \(\sigma\)-field of information until time \(t\); conditional expectation is taken with respect to a risk-neutral probability measure.

MSC:

91B28 Finance etc. (MSC2000)
91B30 Risk theory, insurance (MSC2010)
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