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

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)


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)
Full Text: DOI


[1] Aase, K.K., 1985. Accumulated claims and collective risk in insurance: Higher order asymptotic expansions. Scand. Actuarial. J. 65-85. · Zbl 0602.62092
[2] Chung, K.L., Williams, R.J., 1990. Introduction to Stochastic Integration. Birkhäuser, Basel. · Zbl 0725.60050
[3] Dufresne, D., 1990. The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial. J. 37-79. · Zbl 0743.62101
[4] Garrido, J., Stochastic differential equations for compounded risk reserves, Insurance: math. econom., 8, 165-173, (1989) · Zbl 0688.62056
[5] Gerber, H., Der einfluss von zins auf die ruinwahrscheinlichkeit, Mitteil. ver. schweiz. vers.math., 71, 63-70, (1971) · Zbl 0217.26804
[6] Grandell, J., 1991. Aspects of Risk Theory. Springer, Berlin. · Zbl 0717.62100
[7] Harrison, M., Ruin problems with compounding assets, J. stochastic. process. appl., 5, 67-79, (1977) · Zbl 0361.60053
[8] Karatzas, I., Shreve, S., 1991. Brownian Motion and Stochastic Calculus, Second ed. Springer, Berlin. · Zbl 0734.60060
[9] Lerche, H.R., 1986. Boundary Crossing of Brownian Motion. Springer, Berlin. · Zbl 0604.62075
[10] Norberg, R., 1995. Stochastic calculus in actuarial science. Obozrenie Prikladnoj i Promyshlennoj Matematiki 2, 823-847. (In Russian.) · Zbl 1063.65512
[11] Ø ksendal, B., 1992. Stochastic Differential Equations, third ed. Springer, Berlin.
[12] Paulsen, J., Risk theory in a stochastic environment, J. stochastic. process. appl., 21, 327-361, (1993) · Zbl 0777.62098
[13] Protter, P., 1990. Stochastic Integration and Differential Equations. Springer, Berlin. · Zbl 0694.60047
[14] Ruohonen, M., 1980. On the probability of ruin of risk processes approximated by a diffusion process. Scand. Actuarial. J. 113-120. · Zbl 0427.62075
[15] Schnieper, R., Risk processes with stochastic discounting, Mitteil. ver. schweiz. vers.math., 83, 203-218, (1983) · Zbl 0528.62088
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.