Valuation of default swap with affine-type hazard rate (Q1302099)

Let \(h\) be a solution of the stochastic differential equation \(dh_t = \mu(t,h_t) dt + \sigma(t,h_t) dW_t\) where \(\mu\) and \(\sigma^2\) are both affine in the second argument. The main result of the paper is an expression for \( E[ \exp( -\int_0^t \kappa(s) h_s ds - \beta h_t)] \) for \(\beta>0\) and a strictly positive \(C^1\) function \(\kappa\). This expression is given in terms of functions satisfying a system of ordinary differential equations. For constant \(\kappa\), the above is the Laplace transform of \(( h_t, \int_0^t h_s ds)\), also obtained by \textit{B. Leblanc} [Stochastics Stochastics Rep. 57, No. 1/2, 1-35 (1996; Zbl 0891.60076)]. As an application, formulas for a default swap with hazard rate \(h\) are given.