Ruin probabilities with investments in random environment: smoothness

The paper deals with the ruin problem of an insurance company investing its capital reserve in a risky asset with the price dynamics given by a conditional geometric Brownian motion whose parameters depend on a Markov process describing a random variations in the economic and financial environments. We prove smoothness of the ruin probability as a function of the initial capital and obtain for it an integro-differential equation.


Introduction
Ruin models with risky investments (in other words, with stochastic interest rates) is one of the most active fields of the present day ruin theory.These models combine classical frameworks with models of asset price dynamics developed in mathematical finance.The goal of numerous studies is to provide an information about asymptotic of ruin probabilities.For this there are several approaches.One of them is based on the integro-differential equations for the ruin probabilities.An inspection of the literature reveals that in many cases these equations are derived assuming the smoothness, see, e.g., [6], [7].To our mind, the smoothness of ruin probabilities is a rather delicate property requiring a special (and rather involved) study but there are very few papers on it, see [3] and [5].
In the present note we consider the price dynamics suggested by Di Masi, Kabanov, and Runggaldier, [1], where the coefficients of a (conditional) geometric Brownian motion depend on a Markov process with finite number of states.Such a setting (sometimes referred to as stochastic volatility model, regime switching, or hidden Markov chain) reflects the random dynamics of the economic environment and seems to be adequate for a context with long term contracts typical in insurance.It was already considered in the actuarial literature, see, e.g., the papers [2], [4] where the analysis was based using implicit renewal theory and the paper [7] where a second order integro-differential equation for the ruin probability was derived using intuitive arguments.
Our main result is the theorem asserting that if the distribution of jumps has a density with two continuous and integrable derivatives then the ruin probability is two times continuously differentiable.

The model
Let (Ω, F , F = (F t ) t∈R+ , P) be a stochastic basis where are given three independent stochastic processes, W , P , and θ i generating the filtration F: 2. A compound Poisson process P = (P t ) with drift: where c is a constant, N = (N t ) is a Poisson process with intensity α > 0, and (ξ k ) is an i.i.d.sequence independent of N with the distribution F ξ = F ξ (dx).Sometimes, it is more convenient the alternative description using the notation of stochastic calculus, namely, where π = π(dt, dx) is the jump measure of P , that is a Poisson random measure R + ×R with the deterministic compensator π = π(dt, dx) = Π(dx)dt (coinciding with the mean of π) and the Lévy measure Π(dx) = αF ξ (dx).We denote by T n the consecutive jumps of the process N t = π([0, T ] × R) with the usual convention T 0 = 0.
Recall that λ jj = − k =j λ jk and λ j := −λ jj > 0 for each j.We denote τ i n the consecutive jumps of θ i with the convention τ i 0 = 0 and consider also the imbedded Markov chain ϑ i n := θ i τ i n with transition probabilities P kl = λ kl /λ k , k = l, and P kk = 0.
The conditional distribution of the length of interval τ i k+1 − τ i k with respect to the σ-algebra F τ i k is exponential with parameter λ ϑ i k .We consider the random integer-valued measure p i := n ε (τ i n ,ϑ i n ) where ε x is the unit mass at x.It can be also defined by the counting processes The compensator pi (dual predictable projection) of the random measure p i is given by the formula pi (dt, k, ) = To alleviate formulae we shall skip the superscript i (and also u) when it will not lead to an ambiguity and use for brevity the standard "dot" notation of the semimartingale stochastic calculus for integrals, with L = (L t ) standing for the (deterministic) process with L t := t.
With these conventions the Markov modulated price process S can be given as the solution of the linear integral equation In the traditional symbolical form of the Ito calculus this is written as where The solution of (homogeneous) linear equation can be represented by the formula S = E(R) where E is the stochastic exponential and R := a θ • L + σ θ • W is the relative price process, or, in a more explicit form, by the formula is the logprice process.
The process X = X u,i is defined by the formula where u > 0.
In the actuarial context the process X = X u,i is interpreted as the capital reserve of an insurance company fully invested in a risky asset whose price S i is a conditional geometric Brownian motion given the Markov process θ i describing the financial environment.
The process P describes the business activity.There are two basic models: of the non-life insurance (c < 0, F ξ ((0, ∞)) = 1) and of the annuity payments (c > 0, In the literature one can find also a mixed model where F ξ charges both half-axes. We consider here the setting where only the coefficients of the price process (i.e. the stochastic interest rate) depend on θ.The extension to the case where parameters of the business process also depends on θ is rather straightforward and has no effect on our main results.
We assume that P is not increasing (otherwise the ruin probability is zero).
Let τ u,i := inf{t > 0 : X u,i t ≤ 0} be the instant of ruin corresponding to the initial capital u and the initial regime i.
Our main result is a sufficient condition on the smoothness of survival (and ruin) probabilities as functions of the initial capital.
Theorem 1.Let ξ > 0 be a random variable with a density f which is two times continuously differentiable on (0, ∞) and such that f ′ , f ′′ ∈ L 1 (R + ).Then the functions Φ i (u) are two times continuously differentiable for u > 0.
The proof is given in Section 3.

Smoothness of the survival probability
To use the theory of Markov processes we should consider not a single process θ but a family of processes θ i with initial value θ i 0 = i, i ∈ E. Automatically, other related processes also will depend on i, in many cases omitted in the notations to alleviate formulae.
The basic idea to prove the smoothness of the survival probability is to use an integral representation.Define the continuous process coinciding with X u,i on [0, T 1 ).
We consider the case of non-life insurance, i.e. with c > 0.
By virtue of the strong Markov property of (X u,i , θ i ) =j} .
On the interval [0, T 1 ) the process X u,i coincides with the continuous increasing process Y u,i and ∆X u,i = ξ 1 < 0. Putting α(dt) := αe −αt dt we get that where Clearly, where m i (dz) is the distribution of the process θ i in the Skorohod space D of the càdlàg functions with a generic point z = (z s ) and Φj (y) := EΦ j (y − |ξ 1 |).Recall that we consider the case where In this case the distribution F ξ is not involved (and this is the reason why we introduce the function Φj ).
Proof.To simplify formulae we write here "local" notations V , σ, etc. instead of V (z), σ(z), etc.Using the representation and the independence of the process (V s − V 1 ) s≥1 and the random variable Y u 1 we get that Substituting the expression for Y u 1 given by (3.1) and using abbreviated notations for the integrals where G(t, y) := G(t, e κ•Λ1 y).It follows that Let T 1 (s) := A −1 A1s and σ := √ A 1 .Then (σW t/σ 2 ) t≥0 is a Wiener process and we get that . Using the conditioning with respect to W 1 = x and taking into account that the conditional law of (W t ) t∈[0,1] in this case is that of the Brownian bridge on [0, 1] ending at the level x, that is the same as the process W s + s(x − W 1 ), s ∈ [0, 1], we get that where H(t, x, y) = G(t, e σx y) is a function taking values in [0, 1], and, hence, D and W 1/2 are independent.It follows that for any bounded Borel function g x (y) is also strictly decreasing After the change of variable we get that dy where γ s e −σ(Ds+γsv) m(s, x)ds.
Changing the variable we get that It remains to show that ρ x (.) belongs to C ∞ and its derivatives has at most exponential growth. Put , n ≥ 1. Then and, similarly, It is easily seen that Recall that Clearly, Combining this with the above bounds we easily get from (3.4) that .
It follows that there exists a constant C n > 0 such that Therefore, for some constant Cn > 0. Thus, for every x the derivative (∂ n /∂y n )Q (0) (x, B −1 x (y)) admits an integrable bound not depending on y.This implies that the function y → ρ x (y) is infinitely differentiable and Moreover, increasing in the need the constant C n we obtain the inequality Differentiating the function u → J(u, t) we have: Now we get a result on smoothness of the function Recall that c > 0 and ξ < 0. For us it is more convenient to work with the random variable ζ := −ξ > 0. The needed result follows from are measurable functions.Let ζ > 0 be a random variable with a density f two times continuously differentiable on (0, ∞) and such that f ′ , f ′′ ∈ L 1 (R + ).Then the function ] is two times continuously differentiable in u and there is a constant C depending only of κ * and σ * and such that for all t ∈ [0, 2], u > 0.
Proof.Let H : R → [0, 1] be a measurable function such that H = 0 on R − and let γ > 0 be a constant.Then the function It follows that h is two times differentiable, Let us transform our problem to the above elementary framework.Using the definitions of Y u and V we get that where the functions Φ t (y) := Φ(e κ•Λt y), m := ce −κ•Λt .Thus, To get rid of stochastic integrals in the above formula we consider on R + the strictly It follows that The change of variable s → st/A t replace the integration over the interval [0, A t ] by the integration over the interval [0, t] and the rhs of the above equality is equal to Thus, i.e. the function in parentheses above.Then h is two times continuously differentiable and there is a constant C such that for all t ∈ [0, 2] and all this implies that the function φ t (u) is two times continuously differentiable in u and the bounds (3.6) hold.
Remark 1. Minor changes in the above proof allows to get smoothness result for annuity model and for the model where the distribution F ξ charges both half-axes.

Integro-differential equations
Knowing that the survival probability is a C 2 -function, the derivation of the integrodifferential equation is easy and we get it for all variants of the model.
Due to independence, the processes P and θ have no common jumps and Σ t = Now we replace in the formula (4.2) t by τ ε h and take the expectation from both sides using the following observations: -In virtue of the strong Markov property Φ i (u) = EΦ τ ε h (X τ ε h ) (recall that Φ i (u) = 0 for u ≤ 0).
-For any ε > 0, the integrands on [0, τ ε h (ω)] are bounded from above and, therefore, the expectation of the stochastic integral over the Wiener process is zero.
-As τ ε h ≤ τ 1 , then we can replace θ s− in the integrals by i.In addition, τ ε h = h when h is small enough.
-According to the definition of dual predictable projections As a result we get that Dividing both sides of this equality by h and letting h ↓ 0 we get the result.✷ Remark 2. The obtained equation is homogeneous and holds as well for the ruin probability Ψ(u) = 1 − Φ(u).

Exponential distribution and differential equations
If F ξ (dx) = µe −x/µ dx, then the ruin probability Ψ ∈ C 2 .Note that d du

2 A − 1 r. 1 0e −WA s dM s = 1 0e
The next step is a reduction to the case of constant coefficients.The arguments are simple.The function A = σ 2 • Λ is strictly increasing.Let us denote A −1 its inverse.The law of the continuous process with independent increments (σ • W s ) s≥0 coincides with the law of the process (W As ) s≥0 .Let m(s) := ce −κ•Λs , m(r) := m(A −1 r )/σ Changing the variable we get that −WA s m(s)ds = A1 0 e −Wr m(r)dr.

2 A − 1 r
increasing function A := σ 2 • Λ with the inverse A −1 and observe that the law of the process σ • W is the same as of the process W A .Changing the variable in the integral with respect to Λ we get that e WA m • Λ t = e W m • Λ At where the function m(r) := m(A −1 r )/σ .