By Browers Gerber Hickman

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K=1 Proof. Let a ∈ (0, ∞) and N = inf{n : Sn > a}. Then {Sn∧N } is a submartingale that is bounded from above by a + w. Thus, Sn∧N converges to an integrable random variable SN . In particular, if supn Sn ≤ a, then inf n Sn > −∞. Because this holds for all a > 0, we have that supn Sn < ∞ implies that ∞ inf n Sn > −∞. If supn Sn = ∞, then k=1 Wk = ∞. If supn Sn < ∞ and ∞ ∞ k=1 Zk = ∞, then, because inf n Sn > −∞, we also have k=1 Wk = ∞. 16. For any strategy either ruin occurs or the capital tends to inﬁnity.

Log(λ1 /λ2 ) We have either n0 = n or n0 = n + 1 . Thus, we have found V (x) for x ≤ n0 . Bibliographical Remarks The model considered in this section was introduced by de Finetti [59]. His motivation was to measure an insurance portfolio by the value of future proﬁts (as done in economics) instead of ruin probabilities (as done in actuarial science). 13. 6), and the ﬁniteness of the set {x : u(x) = 0}. Dickson and Waters [47] considered premia of size one and studied the mth moment of the discounted dividends for barrier strategies, that is, u(x) = 1Ix=b+1 for some b ∈ IIN.

For X = X ∗ equality holds, and thus f (x) = δ(x). 3 Optimal Investment and Reinsurance Let us now consider the situation where both investment and reinsurance are possible. The controlled process then fulﬁls the stochastic diﬀerential equation dXtAb = (bt θ − (θ − η) + mAt ) dt + σS bt dWtS + σI At dWtI , X0Ab = x . The corresponding Hamilton–Jacobi–Bellman equation reads sup{ 12 (A2 σI2 + b2 σS2 )f (x) + (mA + bθ − (θ − η))f (x)} = 0 . 13) A,b We are looking for an increasing, twice continuously diﬀerentiable solution with f (0) = 0 and f (∞) = 1.