On The Randomized Schmitter Problem

Area of Science:

• Actuarial Science
• Probability Theory
• Risk Management

Background:

• The classical Schmitter problem in ruin theory traditionally assumes a deterministic initial surplus.
• Computational complexity arises in analyzing ruin probabilities with varying initial surplus levels.

Purpose of the Study:

• To revisit the Schmitter problem by introducing a randomly chosen initial surplus.
• To develop simplified analytical bounds for ruin probabilities using randomization.
• To approximate solutions for deterministic initial surplus using this new approach.

Main Methods:

• Randomization of the initial surplus level (U).
• Utilizing exponentially distributed U for computational simplification.
• Connecting the problem to m-convex ordering.
• Approximating deterministic solutions with Erlang(k)-distributed U for large k.

Main Results:

• Derived simple and sharp analytical bounds for ruin probability.
• Established a connection between the randomized Schmitter problem and m-convex ordering.
• Demonstrated the utility of randomization for computational simplification.
• Showed that Erlang(k)-distributed U approximates deterministic solutions effectively.

Conclusions:

• Randomizing the initial surplus offers significant computational advantages in ruin theory.
• The m-convex ordering provides a powerful tool for deriving bounds on ruin probabilities.
• The proposed method offers an efficient way to approximate classical ruin problems.