Bayesian optimal investment and reinsurance with dependent financial and insurance risks

Proposition A.1 ([15, Proposition 2.2.4]).

If f ∈ C 1 ⁢ ( I ) , where I is an open interval such that x ∈ I , then f is Lipschitz near x and ∂ C ⁡ f ⁢ ( x ) = { f ′ ⁢ ( x ) } . Conversely, if f is Lipschitz near x and ∂ C ⁡ f ⁢ ( x ) reduces to a singleton { ζ } , then f is differentiable at x and f ′ ⁢ ( x ) = ζ .

Theorem A.2 ([15, Theorem 2.5.1]).

Let f be Lipschitz near x and let S be an arbitrary set of Lebesgue-measure 0 in ℝ . Moreover, the set of points at which the function f is not differentiable is denoted by Ω f . Then

Lemma A.3.

Let t ∈ [ 0 , T ] and let ( ξ , b ) ∈ 𝒰 ⁢ [ 0 , T ] be an arbitrary admissible strategy. We set

Then a possibly substochastic measure on ( Ω , 𝒢 t ) is defined by

for every t ∈ [ 0 , T ] , i.e.

The measures ℚ t ξ , b and ℙ are equivalent.

Proof.

First, we show that ( L t ξ , b ) t ≥ 0 is the Doléans–Dade exponential of the martingale ( Z t ) t ≥ 0 defined by

That is,

where

This implies the announced representation (A.2) of ( L t ξ , b ) t ≥ 0 since Ψ ^ - Ψ = ν ^ . As ( L t ξ , b ) t ≥ 0 is a non-negative local martingale, it is a supermartingale, and hence

Lemma A.4.

Let ( ξ , b ) ∈ 𝒰 ⁢ [ 0 , T ] and let L ξ , b = ( L t ξ , b ) t ∈ [ 0 , T ] be the density process given by (A.2). Then there exists a constant 0 < K 2 < ∞ such that

for all t ∈ [ 0 , T ] .

Proof.

Fix t ∈ [ 0 , T ] and ( ξ , b ) ∈ 𝒰 ⁢ [ 0 , t ] . By using [28, Theorem V.52], the unique solution of (3.3) is

Hence,

where 0 < K 2 < ∞ is independent of t ∈ [ 0 , T ] as well as ( ξ , b ) . ∎

Lemma A.5.

Suppose that ( ξ , b ) ∈ 𝒰 ⁢ [ 0 , T ] is an arbitrary strategy and h : [ 0 , T ] × Δ m → ( 0 , ∞ ) is a bounded function such that t ↦ h ⁢ ( t , p ) is absolutely continuous on [ 0 , T ] for all p ∈ Δ m and p ↦ h ⁢ ( t , p ) is continuous on Δ m for all t ∈ [ 0 , T ] . Then the function G : [ 0 , T ] × ℝ × Δ m → ℝ defined by

satisfies

where ( η t ξ , b ) t ∈ [ 0 , T ] is a martingale with respect to 𝔊 and we set ℋ ⁢ h ⁢ ( t , p ; ξ , b ) zero at those points ( t , p ) where h t does not exist.

Proof.

Let ( ξ , b ) ∈ 𝒰 ⁢ [ 0 , T ] and let h : [ 0 , T ] × Δ m → ( 0 , ∞ ) be a function satisfying the conditions stated in the lemma. Applying the product rule to

we get

and hence

By using the introduced compensated random measure Ψ ^ , the variation becomes

Substituting this into (A.5), we obtain

Therefore, by the definition of the operator ℋ given in (A.3), we have

where

with

To complete the proof, we need to show that the introduced processes are martingales with respect to 𝔊 on [ 0 , T ] . According to [13, Corollary VIII.C4], the process ( η ~ t ξ , b ) t ≥ 0 is a martingale with respect to 𝔊 if

Using the boundedness of h (by a constant K 0 ), we obtain that the expectation above is less or equal to

where, by Lemma A.4,

which yields the desired finiteness. Similarly, the martingale property of ( η ^ t ξ , b ) t ≥ 0 can be seen. Moreover, by the boundedness of h and ξ as well as Lemma A.4, it follows that

which implies the martingale property of ( η ~ t ξ , b ) t ≥ 0 . ∎

Lemma A.6.

Let α 1 ≤ ⋯ ≤ α n and β 1 ≤ ⋯ ≤ β n be real numbers and let ( p 1 , … , p n ) ∈ Δ n . Then

Proof of Theorem 4.4.

Let h : [ 0 , T ] × Δ m → ( 0 , ∞ ) be a function satisfying the conditions stated in the theorem. Note that every Lipschitz function is also absolutely continuous. We set

Let us fix t ∈ [ 0 , T ] and ( ξ , b ) ∈ 𝒰 ⁢ [ t , T ] . From Lemma A.5, it follows that

where ( η t ξ , b ) t ∈ [ 0 , T ] is a martingale with respect to 𝔊 and we set ℋ ⁢ h ⁢ ( s , p s ; ξ , b ) to zero at those points s ∈ [ t , T ] where h t does not exist. Note that h is partially differentiable with respect to t almost everywhere in the sense of the Lebesgue measure according to the absolute continuity of t ↦ h ⁢ ( t , p ) for all p ∈ Δ m . The generalized HJB equation (A.4) implies

As a consequence,

due to the negativity of f. Thus, by (A.6), we get

Using the boundary condition (4.7), we obtain

Now, we take the conditional expectation in (A.7) given ( X t ξ , b , p t ) = ( x , p ) on both sides of the inequality, which yields

Taking the supremum over all investment and reinsurance strategies ( ξ , b ) ∈ 𝒰 ⁢ [ t , T ] , we obtain

To show equality, note that ( ξ s ⋆ , b s ⋆ ) given by (4.6) (with g replaced by h in A ⁢ ( s , p ) and B ⁢ ( s , p ) ) are the unique minimizers of the HJB equation (4.4). Therefore,

So, we can deduce that

This implies

Consequently,

Again, taking the conditional expectation given ( X t ξ ⋆ , b ⋆ , p t ) = ( x , p ) on both sides then yields

and the proof is complete. ∎

Proof of Lemma 4.5.

(i) The boundedness and positivity are proven by the same lines of arguments as in [7, Lemma 4.4 (a)].

(ii) This follows by conditioning.

(iii) This follows again by conditioning.

(iv) The concavity is proven in much the same way as in [7, Lemma 4.4 (c)].

(v) The Lipschitz condition is proven in much the same way as in [8, Lemma 6.1 (d)]. ∎

Proof of Theorem 4.6.

Fix t ∈ [ 0 , T ) and ( ξ , b ) ∈ 𝒰 ⁢ [ t , T ] . Let τ be the first jump time of X ξ , b after t and let t ′ ∈ ( t , T ] . It follows from Lemma 4.5 and Lemma A.5 that

where ( η t ξ , b ) t ∈ [ 0 , T ] is a martingale with respect to 𝔊 and we set ℋ ⁢ g ⁢ ( s , p s ; ξ s , b s ) to zero at those s ∈ [ t , T ] where g t ⁢ ( s , p s ) does not exist. For any ε > 0 , we can construct a strategy ( ξ ε , b ε ) ∈ 𝒰 ⁢ [ t , T ] with ( ξ s ε , b s ε ) = ( ξ s , b s ) for all s ∈ [ t , τ ∧ t ′ ] from the continuity of V such that

From the arbitrariness of ε > 0 , we conclude

Using this statement and (A.8), we obtain

where

Consequently,

By the dominated convergence theorem, we can interchange the limit and the expectation. So, we obtain by the fundamental theorem of Lebesgue calculus and 𝟙 { t ′ < τ } → 1 ℙ -a.s. for t ′ ↓ t ,

From now on, let ( ξ , b ) ∈ [ - K , K ] × [ 0 , 1 ] and let ε > 0 as well as ( ξ ¯ , b ¯ ) ∈ 𝒰 ⁢ [ t , T ] be a fixed strategy with ( ξ ¯ s , b ¯ s ) ≡ ( ξ , b ) for s ∈ [ t , t + ε ) . Then

at those points ( t , p ) where g t ⁢ ( t , p ) exists. Due to the negativity of f, we get

We show next the inequality above if g t does not exist. For this purpose, we denote by M p ⊂ [ 0 , T ] the set of points at which g p ′ ⁢ ( t ) exists for any p ∈ Δ m . On the basis of Theorem A.2, we have, for any p ∈ Δ m ,

That is, for every φ ∈ ∂ C ⁡ g p ⁢ ( t ) ⊂ [ 0 , T ] , there exist u ∈ ℕ and ( β 1 , … , β u ) ∈ Δ u such that φ = ∑ i = 1 u β i ⁢ φ i , where φ i = lim n → ∞ ⁡ g p ⁢ ( t n i ) for sequences ( t n i ) n ∈ ℕ with lim n → ∞ ⁡ t n i = t along existing g p ′ . From what has already been shown, it can be concluded that for any i = 1 , … , u ,

Thus, by the continuity of t ↦ g ⁢ ( t , p ) , p ↦ g ⁢ ( t , p ) and p ↦ J ⁢ ( p , y ) , we get for i = 1 , … , u ,

which yields

Due to the arbitrariness of φ ∈ ∂ C ⁡ g p ⁢ ( t ) and ( ξ , b ) ∈ [ - K , K ] × [ 0 , 1 ] , we obtain

Our next objective is to establish the reverse inequality. For any ε > 0 and 0 ≤ t < t ′ ≤ T , there exists a strategy ( ξ ε , t ′ , b ε , t ′ ) ∈ 𝒰 ⁢ [ t , T ] such that

By using Lemma A.5, it follows that

In the same way as before, we get

We can again interchange the limit and the infimum by the dominated convergence theorem, which yields

Thus, the same conclusion can be drawn as above, i.e.

at those points where g t ⁢ ( s , p ) exists. According to the negativity of f and the arbitrariness of ε > 0 , we get, by ε ↓ 0 ,

at those points where g t ⁢ ( s , p ) exists. In the same way as before, we obtain, in the case of non-differentiability of g with respect to t, that

Summarizing, we have equality in the previous expression. The optimality of ( ξ ⋆ , b ⋆ ) follows as in the proof of Theorem 4.4. ∎