Ruin problems with assets and liabilities of diffusion type. (English) Zbl 0962.60075

Norberg, Ragnar

Stochastic Processes Appl. 81, No. 2, 255-269 (1999).

The author studies the properties of ruin probabilities for a risky business when its cash balance process \(X\) is a Markov diffusion with drift, i.e. satisfies an Itô s.d.e. \(dX_{t}=\mu (t,X_{t}) dt+\sigma (t,X_{t}) dW_{t}\), \(W\) being a standard Brownian motion, with coefficients dependent only on time and the current level of cash balances. The analyzed ruin functions are of two types: the probability of ruin at a certain future date (transition probability) and the probability of ruin at any time prior to a given future date (crossing probability), both conditioned on the present cash balance level. Accordingly, there exist two types of ruin event martingales, which are the above named probabilities conditioned on the presently available information. Using Itô’s formula and the properties of martingales, the author derives partial differential equations for both ruin function types. These p.d.e. happen to be the same but with different initial and boundary conditions for the transition and crossing cases.
Solutions to the derived p.d.e. are obtained in a closed form when the cash balance process is time-homogeneous (coefficients \(\mu \) and \(\sigma \) are functions of \(X\) only) and the time horizon of possible ruin is set to infinity. Important special cases are, in particular, the ones of constant coefficients in the Itô equations for the net claims and the rate of return processes (the latter are the two variables determining the cash balances). Under these assumptions, the author derives a superexponential upper bound for the crossing probability in the infinite ruin time case.

Reviewer: A.Derviz (Praha)

MSC:

60J60	Diffusion processes
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60G35	Signal detection and filtering (aspects of stochastic processes)

Keywords:

actuarial risk theory; finance; martingales; functionals of Brownian motions

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