On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums
|
|
- Everett Hill
- 5 years ago
- Views:
Transcription
1 J Optim Theory Appl (216) 168: DOI 1.17/s On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums Ewa Marciniak 1 Zbigniew Palmowski 2 Received: 18 December 214 / Accepted: 11 May 215 / Published online: 27 May 215 The Author(s) 215. This article is published with open access at Springerlink.com Abstract This paper concerns an optimal dividend distribution problem for an insurance company with surplus-dependent premium. In the absence of dividend payments, such a risk process is a particular case of so-called piecewise deterministic Markov processes. The control mechanism chooses the size of dividend payments. The objective consists in maximizing the sum of the expected cumulative discounted dividend payments received until the time of ruin and a penalty payment at the time of ruin, which is an increasing function of the size of the shortfall at ruin. A complete solution is presented to the corresponding stochastic control problem. We identify the associated Hamilton Jacobi Bellman equation and find necessary and sufficient conditions for optimality of a single dividend-band strategy, in terms of particular Gerber Shiu functions. A number of concrete examples are analyzed. Keywords Optimal strategy PDMP Barrier strategy Integro-differential HJB equation Gerber Shiu function Stochastic controls Mathematics Subject Classification 6G51 6G5 6K25 93E2 1 Introduction In classical collective risk theory, the surplus of an insurance company is described by the Cramér-Lundberg model. Under the assumption that the premium income per unit time is larger than the average amount claimed, the surplus in the Cramér-Lundberg B Zbigniew Palmowski zbigniew.palmowski@gmail.com 1 AGH University of Science and Technology, Kraków, Poland 2 University of Wrocław, Wrocław, Poland
2 724 J Optim Theory Appl (216) 168: model has positive first moment and has therefore the unrealistic property that it converges to infinity with probability one. In answer to this objection, De Finetti [1 introduced the dividend barrier model, in which all surpluses above a given level are transferred to a beneficiary, and raised the question of optimizing this barrier. In the mathematical finance and actuarial literature, there is a good deal of work being done on dividend barrier models and the problem of finding an optimal policy of paying dividends. Gerber and Shiu [2 and Jeanblanc and Shiryaev [3 consider the optimal dividend problem in a Brownian setting. Irbäck [4 and Zhou [5 study constant barriers.asmussen etal. [6 investigate excess-of-loss reinsurance and dividend distribution policies in a diffusion setting. Azcue and Muler [7 take a viscosity approach to investigate optimal reinsurance and dividend policies in the Cramér-Lundberg model using a Hamilton Jacobi Bellman (HJB) system of equations. Avram et al. [8,9, Kyprianou and Palmowski [11, Loeffen [12,13, Loeffen and Renaud [14 and many other authors analyze the Lévy risk processes set up from the probabilistic point of view. In this paper, we shall approach the dividend problem for a reserve-dependent risk process using the theory of piecewise deterministic Markov processes (PDMP). We also take into account the severity of ruin and therefore we consider the socalled Gerber Shiu penalty function (see, e.g., Schmidli [15 or Avram et al. [9 and references therein). For this setup, without transaction costs, we find the corresponding HJB system. We analyze the barrier strategy for which all surpluses above a given level are transferred to dividends. In particular, we find necessary and sufficient conditions for the barrier strategy to be optimal. We believe that PDMP models can better describe the situation of an insurance company, since, for example, they can invest the surplus into a bond with a fixed interest rate. Such a situation is described by a PDMP model with a linear premium (see [1). The paper is organized as follows. In Sect. 2, we introduce the basic notation and we describe the model we deal with. Section 3 is dedicated to the related one-sided and two-sided problems. In Sect. 4, we present the Verification Theorem, necessary and sufficient conditions for the barrier strategy to be optimal. In Sect. 6, wegiveall the proofs. Section 5 and 7 are devoted to some examples and concluding remarks. 2 The Model In this paper, we assume that the surplus R of an insurance company (without payment of dividends) with an initial capital x is described by the following differential equation: t N(t) R t = x + p(r s ) ds C k, (1) where p is a given deterministic positive premium function, {C k } i=1 is a sequence of i.i.d. positive random variables with d.f. F representing the claims and N is an k=1
3 J Optim Theory Appl (216) 168: independent Poisson process with intensity λ modeling the times at which the claims occur. We assume that R t a.s., EC < for a generic claim C, and the premium rate p is monotone, absolutely continuous and satisfies the following speed condition : e qt p(rt x ) dt Ax + B (2) for some constants A, B and a function r x satisfying the equation t rt x = x + p(rs x ) ds. (3) Note that r x describes a deterministic trajectory of R along which no claims appear. Remark 2.1 Constant and linear premium functions satisfy the above assumptions. For a constant premium function, we obtain the classical Cramér-Lundberg model. To approach the dividend problem, we consider the regulated risk process satisfying the following stochastic differential equation: t Xt π = x + N(t) p(xs π ) ds C k L π t, (4) where π denotes a strategy chosen from the class Π of all admissible dividend controls resulting in the cumulative amounts of dividends L π t paid up to time t. Note that ruin may be either exogeneous or endogeneous (i.e., caused by a claim or by a dividend payment). A dividend strategy is admissible, if ruin is always exogeneous or, more precisely, an admissible dividend strategy L π ={L π t, t R +} is a rightcontinuous stochastic process, adapted to the natural filtration of the risk process R that satisfies the usual conditions, and such that, at any time preceding the epoch of ruin, the dividend payment is smaller than the size of the available reserves (L π t L π t < X t π ). The object of interest is the discounted cumulative dividend paid up to the ruin time: D(π) := T π k=1 e qt dl π t, where T π := inf{t : Xt π < } is the ruin time and q is a given discount rate. Note that unless it is necessary we will write T instead of T π to simplify the notation. The objective is to maximize E x D(π), where E x is the expectation with respect to P x ( ) = P( X π = x). We will use the notation P = P and E = E. To take into account the severity of ruin, we also consider the so-called Gerber Shiu penalty function E x [e qt w(xt π )I {T < }
4 726 J Optim Theory Appl (216) 168: for some general non-positive penalty function w satisfying the integrability condition sup E [ w(y C 1 ) C 1 > y <. y Note that for q = and w = 1, we derive the ruin probability. The dividend problem consists in finding the so-called value function v given by v(x) := sup v π (x), (5) π Π where [ T v π (x) := E x e qt dl π t and the optimal strategy π Π such that + e qt w(x π T )I {T < } (6) v(x) = v π (x) for all x. 3 Preliminaries For the solution of the dividend problem, two functions, W q and G q,w, are crucial. They are related to two-sided and one-sided exit problems for R: [ E x e qτ a + I {τ + a <τ } = W q(x) W q (a), (7) G q,w (x) := E x [e qτ w(r τ )I {τ < }, (8) where x [, a, τ a + :=inf{t : R t a} and τ :=inf{t : R t < }.Fromnow on, we will assume the existence of the function W q, which follows for example from the existence of the following limit: W q (x) = lim y E x [e qτ y + I + {τ + y <τ }/E[e qτ y I {τ + y <τ }. Indeed, using the strong Markov property of R that has only negative jumps, we derive [ [ E x e qτ a + I {τ + a <τ W q (x) = lim } E a e qτ y + I {τ + y <τ } y E [e qτ y + I {τ +y <τ [ } = E x e qτ a + I {τ + a <τ } W q (a), which gives the required identity (7). For the properties of the function G q,w, we refer the reader to [18, where numerous examples are studied.
5 J Optim Theory Appl (216) 168: Main Results To prove the optimality of a particular strategy π among all admissible strategies Π for the dividend problem (5), we consider the following Hamilton Jacobi Bellman (HJB) system: max { Am(x) qm(x), 1 m (x) } for all x >, m(x) = w(x) for all x <, (9) where A is the full generator of R, Am(x) = p(x)m (x) + λ acting on absolutely continuous functions m such that (m(x y) m(x)) df(y), [ E m(r σi ) m(r σi ) < for any t, σ i t where {σ i } i N {} denotes the times at which the claims occur (see Davis [16 and Rolski et al. [17). In this case, m denotes the density of m. Note that any function, which is absolutely continuous and ultimately dominated by an affine function, is in the domain of the full generator A, as a consequence of the assumption that EC 1 <. Recall that, for any function m from the domain of A, the process { t } e qt m(r t ) e qs (A q) m(r s ) ds, t is a martingale. Theorem 4.1 (Verification Theorem) Let π be an admissible dividend strategy such that v π is absolutely continuous and ultimately dominated by some affine function. If (9) holds for v π then v π (x) = v(x) for all x. The proof of all theorems given here will be given in Sect. 6. Lemma 4.1 Assume that the distribution function (d.f.) F of the claim size is absolutely continuous. Then the functions W q and G q,w are continuously differentiable for all x. From now on, we assume that the claim size distribution is absolutely continuous with a density f. We will focus on the so-called barrier policy π a transferring all surpluses above a given level a to dividends.
6 728 J Optim Theory Appl (216) 168: Theorem 4.2 We have v a (x) := v πa (x) = { Wq (x) W q (a) (1 G q,w (a)) + G q,w(x), x a, x a + v a (a), x > a. (1) Moreover, v a is continuously differentiable for all x. Let H q (y) := 1 G q,w (y) W q (y). Define now a candidate for the optimal dividend barrier by } a := sup {a : H q (a) H q (x) for all x, (11) where H q () = lim x H q (x). Finally, using the above two theorems, we can give necessary and sufficient conditions for the barrier strategy to be optimal. Theorem 4.3 The value function under the barrier strategy π a is in the domain of the full generator A. The barrier policy π a is optimal and v a (x) = v(x) for all x if and only if (A q)v a (x) for all x > a. (12) Theorem 4.4 Suppose that H q (a) H q (b) for all a a b. (13) Then the barrier strategy at a is optimal, that is, v(x) = v a (x) for all x. Theorem 4.5 Suppose that f is convex and p is concave. Then the barrier strategy at a is optimal, that is, v(x) = v a (x) for all x. Theorem 4.6 Consider the problem without the penalty function (w ). Suppose that f is decreasing and p (x) q + λ, x a, where p is the density of the premium rate p. Then the barrier strategy at a is optimal, that is, v(x) = v a (x) for all x.
7 J Optim Theory Appl (216) 168: Examples In this section, we will assume that the premium function p is differentiable and the generic claim size has a density f with a rational Laplace transform. That is, there exists m N and constants {β i } i= m 1 such that the density f satisfies the following LODE: ( ) d L f (y) = dy with initial conditions f (k) () = (k =,...,m 2), where L(x) = x m + β m 1 x m 1 + +β. Note that by Theorem 4.5 if we take p concave then the barrier strategy is optimal for an exponential claim size (in this case L(x) = x + μ). From Lemma 4.1 and its proof, it follows that if the claim size distribution is absolutely continuous then W q, G q,w and v a are differentiable and satisfy and AW q (x) = qw q (x) for x, W q (x) = for x <, (14) AG q,w (x) = qg q,w (x) for x, G q,w (x) = w(x) for x <. (15) Our goal will be to find the value function v for a few examples of premium functions. The Gerber Shiu function G q,w was determined in Albrecher et al. [18. One can prove that if G q,w is differentiable then in fact G q,w C m+1 (see [17). The same holds for W q. Albrecher et al. [18 proved that G q,w satisfies the following LODE with variable coefficients of order m + 1: with the differential operator TG q,w (x) = u(x) (16) ( )( d T := L q p(x) d ) dx dx + λ λβ and the right-hand side ( ) d u(x) := λl ω(x), dx where ω(x) := x w(x z) df(z). In general, the main idea of solving the above equation is to find stable solutions s k of the fundamental system for (16) (that is, those vanishing at infinity) and then use the representation G q,w (x) = γ 1 s 1 (x) + +γ m s m (x) + Gu(x),
8 73 J Optim Theory Appl (216) 168: where G is the Green operator and the constants γ i can be computed from the initial conditions. Moreover, the form of the Green operator is found in [18, Thm.3.4. If the claim size has exponential distribution with intensity μ then we can prove that G q,w solves the following ODE: ( d 2 du 2 + ( μ + p (x) p(x) λ + q ) d p(x) du qμ ) G q,w (x) = u(x), p(x) with u(x) = λ p(x) ( d du + μ) ω(x). This allows one to find G q,w explicitly. Moreover, note that (14) is a Gerber Shiu function with zero penalty function. In contrast to the one from (7), we now have lim x W q (x) =+. This means that the optimal value function under mild conditions is a linear combination of two Gerber Shiu functions: An unstable one that vanishes on the negative half line and tends to infinity at infinity (corresponding to dividend payment, W q in our notation), and a stable one, vanishing at infinity (corresponding to the penalty payment, G q,w in our notation). From [18, we know that W q equals the unstable solution of the fundamental system for (16). One can prove that there exists a unique unstable solution (see [18for details). In the rest of this section, we will assume that the claim size has exponential distribution with intensity μ. 5.1 Linear Premium We take here p(x) = c + ɛx. By Theorem 4.5, the barrier strategy at a is optimal. In this case, and s 1 (x) = U ( q λ+q ɛ + 1, ɛ Γ(q/ɛ+1) Γ((q+λ)/(1+ɛ)) 1 ɛ x Gu(x) = ( U(x) + M() U() U(x) + 1, μx + μc ) ɛ (ɛx + c) (λ+q)/ɛ exp( μx) ( μ ) (λ+q)/ɛ ɛ (ɛx + c) (λ+q)/ɛ exp( μx μc M(v)u(v) dv M(x) ) U(v)u(v) dv, where U(u) and M(u) are Kummer functions. This gives G q,w (x) = s 1 (x) + Gu(x) for u(x) = p(x) λ ( d + μ) ω(x). Moreover, du W q (x) = C 1 M ( q λ+q ɛ + 1, ɛ + C 2 U ( q λ+q ɛ + 1, ɛ x U(v)u(v) dv ɛ ) + 1, μx + μc ) ɛ (ɛx + c) (λ+q)/ɛ exp( μx) ) (ɛx + c) (λ+q)/ɛ exp( μx), + 1, μx + μc ɛ
9 J Optim Theory Appl (216) 168: Table 1 Linear premium. Dependence of q on a μ =.3, ɛ =.2, λ =.1, c = 1 q a Table 2 Linear premium. Dependence of μ on a q =.5, ɛ =.2, λ =.1, c = 1 μ a Table 3 Linear premium. Dependence of λ on a μ =.3, q =.5, ɛ =.2, c = 1 λ a Table 4 Rational premium. Dependence of q on a μ =.3, λ =.1, c = 1 q a with C 1 and C 2 determined by the boundary conditions W q () = 1 and W q () = (λ + q)/c. Hence we can find the optimal barrier a by solving H q (a ) =. In the case of absence of the penalty function, that is, when w(x) =, we can perform some numerical analysis of the values of a. In Tables 1, 2 and 3, we present some values of a for different parameters. 5.2 Rational Premium In this subsection, we consider the rational premium with p(x) = c + 1/(1 + x).one can solve Eq. (14) and find the function W q.ifwetakew then, to get optimality of the barrier strategy using Theorem 4.6, we will assume that ɛ q + λ. Thus, in the absence of the penalty function, we can find the values of a for different parameters. In Tables 4, 5 and 6, we give some results in the case of a rational premium. Table 5 Rational premium. Dependence of μ on a q =.1, λ =.1, c = 1 μ a
10 732 J Optim Theory Appl (216) 168: Table 6 Rational premium. Dependence of λ on a q =.1, μ =.3, c = 1 λ a Note that a seems to have similar properties in both linear and rational premium examples. 6 Proofs 6.1 Proof of the Verification Theorem 4.1 The proof is based on a representation of v as the pointwise minimum of a class of controlled supersolutions of the HJB equation. We start with the observation that the value function satisfies a dynamic programming equation. Lemma 6.1 After extending v to the negative half-axis by v(x) = w(x) for x <, we have, for any stopping time τ, [ τ T v(x) = sup E x e qτ T v(xτ T π ) + e qt dl π t. π Π This follows by a straightforward adaptation of classical arguments (see, e.g., [7, pp ). We will prove that v is a supersolution of the HJB equation. Lemma 6.2 The process V π t t T := e q(t T ) v(xt T π ) + e qs dl π s (17) is a uniformly integrable (UI) supermartingale. Proof Fix arbitrary π Π, x and s, t with s < t. The process Vt π F t -measurable and is UI. Indeed, by Lemma 6.1, wehave [ t T E x [Vt π sup E x e q(t T ) v(xt T π ) + e qs dl π s = v(x). π Π is Now by integration by parts, the non-positivity of w and the non- exogeneous ruin assumption [ v(x) E x qe qs rs x ds x + e qt p(rt x ) dt (A + 1)x + B, (18) where the function r x t given in (3) satisfies (2).
11 J Optim Theory Appl (216) 168: W π s Let W π t { Π s := be the following value process: [ T π := ess sup Js π, J s π := E e qu dlu π + e qt π w(x π π Π s T π ) F s, (19) π = (π, π) ={Lu π,π, u } :π Π { }, Lu π,π L π := u, u [, s[, L π s + Lπ u s (Xπ s ), u s, where L π (x) denotes the process of cumulative dividends of the strategy π corresponding to the initial capital x. The fact that V π is a supermartingale is a direct consequence of the following P-a.s. relations: (a) Vs π = Ws π, (b) W s π E[Wt π F s, where W π is the process defined in (19). Point (b) follows by classical arguments, since the family {Jt π, π Π t } of random variables is upwards directed; see Neveu [19 and Avram et al. [9, Lem. 3.1(ii) for details. To prove (a), note that on account of the Markov property of X π, it also follows that conditional on Xs π, {X u π X s π, u s} is independent of F s. As a consequence, the following identity holds on the set {s < T π }: e qu dlu π + e qt π w(x π ) T π F s [ T π = e qs E X π s e qu dl π u + e qtπ w(x π T π ) [ T π E s = e qs v π (Xs π ) + e qu dl π u, and then we have the following representation: s + e qu dl π s J π s s T = e q(s T ) v π (Xs T π ) + e qu dl π u, which completes the proof on taking the essential supremum over the relevant family of strategies. We prove that the value function v is a solution of the HJB equation. We will denote by G the family of functions g for which M g,t I := { } e q(t T I ) g(r t TI ), t, T I := inf {t : R t / I }, (2) is a supermartinagle for any closed interval I [, [, and such that g(x) g(y) x y 1 for all x > y, g(x) w(x) for x < (21) and g is ultimately dominated by some linear function.
12 734 J Optim Theory Appl (216) 168: Lemma 6.3 We have v G. Proof Taking a strategy of not paying any dividends, by Lemma 6.2, we find that the process (2) with g = v is a supermartingale. We will prove that v(x) v(y) x y for all x > y. Let x > y. Denote by π ɛ (y) an ɛ-optimal strategy for the case X π = y. Then we take the strategy of paying x y immediately and subsequently following the strategy π ɛ (y) (note that such a strategy is admissible), so that the following holds: v(x) x y + v π ɛ (y) v(y) ɛ + x y. Since this inequality holds for any ɛ >, the stated lower bound follows. Linear domination of v by some affine function follows from (18). We now give the dual representations of the value function on a closed interval I. Assume that H I is a family of functions k for which M k,π t := e q(t τ π I ) k(x π t τ π I ) + t τ π I e qs dl π s (22) is a UI supermartingale for τ π I := inf{t : X π t / I } and k(x) v(x) for x / I. Then v(x) = min k(x) for x I. (23) k H I Indeed, let π Π, k H I and x I. Then the Optional Stopping Theorem applied to the UI Dynkin martingale yields τ π k(x) lim E x [e q(τ πi t) k(x πτ t πi t) + I t E x [ e qτ π I v(x π τ π I ) + τ π I e qs dl π (s), e qs dl π (s) where the convention exp{ } = is used. Taking the supremum over all π Π shows that k(x) v(x). Since k H I was arbitrary, it follows that inf k(x) v(x). k H I This inequality is in fact an equality since v is a member of H I by Lemma 6.2. The value function v admits a more important representation from which the Verification Theorem 4.1 follows.
13 J Optim Theory Appl (216) 168: Proposition 6.1 We have v(x) = min g(x). (24) g G Proof Since v G in view of Lemma 6.3,by(23) it suffices to prove that G H [, [. The proof of this fact is similar to the proof of the shifting lemma [9, Lem.5.5. For completeness, we give the main steps. Fix arbitrary g G, π Π and s, t with s < t. Note that M g,π is adapted and UI by the linear growth condition and arguments in the proof of Lemma 6.2 and by [9, Sec.8. Furthermore, the following (in)equalities hold true: E [ M g,π (a) t Fs T = lim E[ M g,π (b) n n t Fs T lim L π n u := n M g,π n s T where the sequence (π n ) n N of strategies is defined by π n L π n = L π and { sup{l π v : v<u,v T n}, < u < T, L π n T, u T, T n := ({t k := s + (t s) k2 } ) n, k Z {} R +, (c) = M g,π (d) s T = M s g,π, = {L π n t, t } with where the above T is calculated for the strategy π. Since s and t are arbitrary, it follows that M g,π is a supermartingale, which will complete the proof. Points (a), (c) and (d) follow from the Monotone and Dominated Convergence Theorems. To prove (b), let T i := T t i, denote M g,π n = M, L π n = L and observe that 2 n M t M s = Y i + Z i, i=1 2 n i=1 with Y i := e qt i g ( X Ti ) e qt i 1 g(x ), Ti 1 Z i := e qt i (g(x Ti ) g(x Ti ) + ΔL T i )I {ΔLTi >}. The strong Markov property of R and the definition of X π imply E[Y i F Ti 1 =e qt i 1 E [ e q(t i T i 1 ) g(x Ti ) g(x Ti 1 ) FTi 1 = e qt i 1 E XTi 1 [e qτ i g(r τi ) g(r ), (25) with τ i := T i θ Ti 1, where θ denotes the shift operator. The right-hand side of (25) is non-positive because g G. Furthermore, it follows from (21) that all the Z i are non-positive. The tower property of conditional expectation then yields E[M t M s F s. This establishes inequality (b), and the proof is complete.
14 736 J Optim Theory Appl (216) 168: Proof of the Verification Theorem 4.1 Since v π is absolutely continuous and dominated by an affine function, v π is in the domain of the full generator of R. This means that the process v π (R t TI )e t T I Avπ (Rs ) vπ (Rs ) ds is a martingale for any closed interval I [, [. By(9), it follows that v π G, which completes the proof. 6.2 Proof of Lemma 4.1 Take any x. Then fix a > such that x < a. From the definition of W q given in (7), conditioning on the first claim arrival time σ 1, we obtain W q (x) = e (λ+q)h W q (r x h ) + λ h r x t W q (r x t z) df(z) e (λ+q)t dt, (26) for h small enough, so that r x h < a. Ash, we find that W q is right-continuous at x. Moreover, rearranging terms in (26) leads to W q (r x h ) W q(x) r x h x = 1 e (λ+q)h h h rh x x W q(rh x ) h λ rh x x h h r x t W q (r x t z)df(z)e (λ+q)t dt. Letting h we conclude that W q is right-differentiable with derivative W q,+ (x) = 1 ( x ) (λ + q)w q (x) λ W q (x z) df(z). (27) p(x) Now take any x >. Equation (26) can be rewritten as h r x W q ( r x ) = t e (λ+q)h W q (x) + λ W q ( r x t z) df(z)e (λ+q)t dt, where r x is a solution to the backward equation d r t x = p( r s x)ds, r h x small enough, so that r x. We thus get left continuity and = x. Wetakeh W q (x) W q ( r x ) x r x = 1 e (λ+q)h h h x r x W q (x) h x r x λ h h r x t W q ( r x t z)df(z)e (λ+q)t dt.
15 J Optim Theory Appl (216) 168: Letting h we see that W q is left-differentiable with derivative W q, (x) = 1 ( (λ + q)w q (x) λ p(x) x ) W q (x z) df(z). (28) Since F is absolutely continuous, (27) and (28) imply that W q is continuously differentiable and satisfies (14). Using the same arguments and definition (8), one can show that the function G q,w is continuously differentiable and satisfies (15). This completes the proof. 6.3 On the Value Function for the Barrier Strategy Note that for the barrier strategy until the first hitting of the barrier a, the regulated process X π a behaves like the process R. By the strong Markov property of the PDMP R t and by (7) forx [, a, wehave v a (x) = W q(x) W q (a) v [ a(a) + E x e qτ w(r τ )I {τ <τ a + }. Moreover, again using the strong Markov property, we can derive [ E x e qτ w(r τ )I {τ <τ a + } = Gq,w (x) G q,w (a) W q(x) W q (a). Hence We will prove that v a (x) = W q(x) W q (a) (v a(a) G q,w (a)) + G q,w (x). (29) v a (a) = 1, (3) from which the assertion of Theorem 4.2 immediately follows. Note that for the barrier strategy a, wehave v a (x) = x a + v a (a) for x > a. (31) Take any a > and x [, a[. From the definition of v a given in (6) and fixed a, conditioning on the first claim arrival time σ 1, we obtain r x t h v a (x) = e (λ+q)h v a (rh x ) + λ v a (rt x z) df(z) e (λ+q)t dt h + λ w(rt x z) df(z) e (λ+q)t dt, (32) r x t
16 738 J Optim Theory Appl (216) 168: where h is small enough (so that r x h, a[). Letting h we find that v a is rightcontinuous at x for all x [, a[. Moreover, rearranging terms in (32) leads to v a (rh x) v a(x) rh x x = 1 e (λ+q)h h h rh x x v a(rh x ) h λ rh x x h + h λ rh x x h h r x t h r x t v a (r x t z) df(z) e (λ+q)t dt w(r x t z) df(z) e (λ+q)t dt. Letting h we conclude that v a is right-differentiable on [, a[ with derivative satisfying x p(x)v a,+ (x) = (λ+q)v a(x) λ v a (x z) df(z) λ Now take any x, a. Equation (32) can be rewritten as x w(x z) df(z). (33) h r x v a ( r x ) = t e (λ+q)h v a (x) + λ v a ( r t x z) df(z) e (λ+q)t dt h + λ w( r t x z) df(z) e (λ+q)t dt, r x t where r x is a solution to the backward equation d r t x = p( r s x)ds, r h x = x. Wetakeh small enough, so that r x. We thus get left continuity on, a and v a (x) v a ( r h x) x r h x = 1 e (λ+q)h h h x r h x v a ( r h x ) h λ h r x t x r h x v a ( r t x z) df(z) e (λ+q)t dt h + h λ h x r h x w( r t x z) df(z) e (λ+q)t dt. h Letting h we infer that v a is left-differentiable on, a with derivative p(x)v a, (x) = (λ+q)v a(x) λ x r x t v a (x z) df(z) λ x w(x z) df(z). (34) Under the assumption that F is absolutely continuous, the function v a is differentiable on, a[. Now we will prove that it is differentiable at x = a.takex = a. Then from the definition of v a,forx = a, conditioning on the first claim arrival time, we obtain
17 J Optim Theory Appl (216) 168: h h a v a (a) = e (q+λ)h v a (a) + e λh e qt p(a) dt + λ v a (a z) df(z) e (q+λ)t dt h h t + λ w(a z) df(z) e (q+λ)t dt + λp(a) e qs e λt ds dt. (35) a Differentiating (35) with respect to h and setting h = gives a = (λ + q)v a (a) + λ v(a z) df(z) + λ a w(a z) df(z) + p(a). (36) By setting x = a in (34) and using (36), we get v a, (a) = 1. This together with (31) proves that v a has a derivative at a and (3) holds. 6.4 Proofs of Necessary and Sufficient Conditions for Optimality of the Barrier Strategy Proof of Theorem 4.3 To prove sufficiency, we need to show that v a satisfies the conditions of the Verification Theorem 4.1. From Theorem 4.2, it follows that v a is ultimately linear. Moreover, by the choice of the optimal barrier a, we know that v a (x) 1. Finally, by definition of W q and G q,w and the strong Markov property of the risk process R, it follows that are martingales. Hence e q(t T ) W q (R t T τ + a ), e q(t T ) G q,w (R t T ) e q(t T ) v a (R t T τ + a ) is a martingale. This means that v a is in the domain of the full generator of R stopped on exiting [, a and that (A q)v a (x) = forx a. To prove necessity, we assume that condition (12) is not satisfied. By the continuity of the function x (A q)v a (x), there exists an open and bounded interval J a, [ such that (A q)v a (x) > for all x J. Let π be the strategy of paying nothing if the reserve process X π takes a value in J, and following the strategy π a otherwise. If we extend v a to the negative half-axis by v a (x) = w(x) for x <, we have { E x [e qt Jv a (R TJ ), x J, v π (x) = v a (x), x / J, where T J is defined by (2). By the Optional Stopping Theorem applied to the process e qt v a (R t ), for all x J, we obtain v π (x) = E x [e qt J v a (R TJ )=v a (x) + E x [ TJ (A q)v a (R s ) ds >v a (x).
18 74 J Optim Theory Appl (216) 168: This leads to a contradiction with the optimality of the strategy π a, and the proof is complete. Proof of Theorem 4.4 In the first step, we will show that lim y x (A q)(v a v x )(y) for all x > a. (37) Let x > a. By the Dominated Convergence Theorem, we obtain lim (A q)(v a v x )(y) = p(x)(v a v x )(x) q(v a v x)(x) y x By (1), we have: + [(v a v x )(x z) (v a v x )(x) λ F(dz). i. (v a v x )(x) =. ( ) ii. (v a v x )(b) = W q (b) H q (a ) H q (x) forb [, a by the definition of a. ( ) iii. (v a v x )(u) = W q (u) H q (u) H q (x) foru [a, x by the assumption (13). iv. (v a v x )(a ), thus by iii, (v a v x )(x). v. (v a v x )(x z) (v a v x )(x) for all z by ii and iii. Thus, we have shown (37). Now assume that (12) does not hold. Then there exists x > a such that (A q)v a (x) >. By the continuity of (A q)v a, we deduce that lim y x (A q)v x (y) >, which contradicts (37). Proof of Theorem 4.5 In view of Theorem 4.3, it follows that to prove optimality of v a, we need to verify that g(x) forx > a, where Recall that g(x+a ) = p(x+a ) qv a (a ) qx+λ g(x) := Av a (x) qv a (x). (38) (v a (x+a y) v a (x+a )) f (y) dy. The desired assertion follows once the following three facts are verified: (i) g is concave on R + \{}, (ii) g(a ) = and (iii) g (a ) =. To show (i) recall that g(x) = for all x a (see the proof of Prop. 4.3). Moreover, denoting k(x, y) := v a (x + a y) v a (x + a ) and noting that 2 y 2 k(x, y), wehave 2 x 2 k(x, y) =
19 J Optim Theory Appl (216) 168: x 2 2 k(x, y) f (y) dy = k(x, y) f (y) dy y2 = y k(x, y) f (y) k(x, y) + k(x, y) f (y) dy since v a (x + a y) v x+a (a ), f (y), y k(x, ) = v a (x + a ) = 1 and f () =. Point (ii) is straightforward, and (iii) follows from the fact that g (x) = for any x < a and g is continuously differentiable. Proof of Theorem 4.6 Let g be defined by (38). Recall that by the definition of a,we have g(a ) =. Moreover, for x a,wehave x g (x) = p (x) + λ v a (y) f (x y) dy (q + λ). Note that in the case of w, v x for all x. Thus, by assumption g (x) for x and by Theorem 4.3, the strategy π a is optimal. 7 Conclusions In this paper, we solved the dividend problem with a penalty function at ruin. We found some sufficient and necessary conditions for a barrier strategy to be optimal. Unfortunately, some of them, like (12) and (13), may be difficult to verify. Moreover, we analyzed only single barrier strategies. Therefore, one can consider multi-bands strategies (see [9). It would also be interesting to consider the effect of adding fixed transaction costs that have to be paid when dividends are being paid. In the next step, it would be reasonable to examine the so-called dual model with a negative premium function and positive jumps. In such a model, the premiums are regarded as costs and claims are viewed as profits. Such a model might be appropriate for a company that specializes in inventions and discoveries (see [2). However, we leave these points for future research. Acknowledgments This work is partially supported by the National Science Centre under the grant DEC-213/9/B/ST1/1778. The second author kindly acknowledges partial support by the project RARE , a Marie Curie IRSES Fellowship within the 7th European Community Framework Programme. Open Access This article is distributed under the terms of the Creative Commons Attribution 4. International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. References 1. De Finetti, B.: Su un impostazione alternativa della teoria collettiva del rischio. Trans. XV Intern. Congress Act 2, (1957) 2. Gerber, H.U., Shiu, E.S.W.: Optimal dividends: analysis with Brownian motion. N. Am. Actuar. J. 8, 1 2 (24)
20 742 J Optim Theory Appl (216) 168: Jeanblanc, M., Shiryaev, A.N.: Optimization of the flow of dividends. Russian Math. Surveys 5, (1995) 4. Irbäck, J.: Asymptotic theory for a risk process with a high dividend barrier. Scand. Actuarial J. 2, (23) 5. Zhou, X.: On a classical risk model with a constant dividend barrier. N. Am. Actuar. J. 9, 1 14 (25) 6. Asmussen, S., Højgaard, B., Taksar, M.: Optimal risk control and dividend distribution policies. Example of excess-of-loss reinsurance for an insurance corporation. Finance Stoch. 4, (2) 7. Azcue, P., Muler, N.: Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model. Math. Finance 15, (25) 8. Avram, F., Palmowski, Z., Pistorius, M.R.: On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17, (27) 9. Avram, F., Palmowski, Z., Pistorius, M.R.: On Gerber-Shiu functions and optimal dividend distribution for a Lévy risk-process in the presence of a penalty function. Ann. Appl. Probab. 25(4), (215) 1. Segerdahl, C.: Über einige risikotheoretische Fragestellungen. Skandinavisk Aktuartidsskrift 25, (1942) 11. Kyprianou, A., Palmowski, Z.: Distributional study of De Finetti s dividend problem for a general Lévy insurance risk process. J. Appl. Probab. 44, (27) 12. Loeffen, R.: On optimality of the barrier strategy in de Finetti s dividend problem for spectrally negative Lévy processes. Ann. Appl. Probab. 18, (28) 13. Loeffen, R.: An optimal dividends problem with transaction costs for spectrally negative Lévy processes. Insur. Math. Econ. 45, (29) 14. Loeffen, R., Renaud, J.-F.: De Finetti s optimal dividends problem with an affine penalty function at ruin. Insur. Math. Econ. 46, (29) 15. Schmidli, H.: Stochastic Control in Insurance. Springer, London (28) 16. Davis, M.H.A.: Markov Models and Optimization, Monographs on Statistics and Applied Probability. Chapman & Hall, London (1993) 17. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley, Chichester (1999) 18. Albrecher, H., Constantinescu, C., Palmowski, Z., Regensburger, G., Rosenkranz, M.: Exact and asymptotic results for insurance risk models with surplus-dependent premiums. SIAM J. Appl. Math. 73, (213) 19. Neveu, J.: Discrete Parameter Martingales. North-Holland, Amsterdam (1975) 2. Avanzi, B., Gerber, H.U.: Optimal dividends in the dual model with diffusion. Astin Bull. 38, (28)
arxiv: v1 [q-fin.pm] 23 Apr 2016
Noname manuscript No. (will be inserted by the editor) On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums arxiv:164.6892v1 [q-fin.pm] 23 Apr 216 Ewa Marciniak Zbigniew
More informationOPTIMAL DIVIDEND AND REINSURANCE UNDER THRESHOLD STRATEGY
Dynamic Systems and Applications 2 2) 93-24 OPTIAL DIVIDEND AND REINSURANCE UNDER THRESHOLD STRATEGY JINGXIAO ZHANG, SHENG LIU, AND D. KANNAN 2 Center for Applied Statistics,School of Statistics, Renmin
More informationUniversity Of Calgary Department of Mathematics and Statistics
University Of Calgary Department of Mathematics and Statistics Hawkes Seminar May 23, 2018 1 / 46 Some Problems in Insurance and Reinsurance Mohamed Badaoui Department of Electrical Engineering National
More informationAn optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density
An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density R.L. Loeffen Radon Institute for Computational and Applied Mathematics,
More informationOn the optimal dividend problem for a spectrally negative Lévy process
On the optimal dividend problem for a spectrally negative Lévy process Florin Avram Zbigniew Palmowski Martijn Pistorius Université de Pau University of Wroc law King s College London Abstract. In this
More informationON THE OPTIMAL DIVIDEND PROBLEM FOR A SPECTRALLY NEGATIVE LÉVY PROCESS. By Florin Avram Université de Pau
Submitted to the Annals of Applied Probability ON THE OPTIMAL DIVIDEND PROBLEM FOR A SPECTRALLY NEGATIVE LÉVY PROCESS By Florin Avram Université de Pau By Zbigniew Palmowski University of Wroc law and
More informationReinsurance and ruin problem: asymptotics in the case of heavy-tailed claims
Reinsurance and ruin problem: asymptotics in the case of heavy-tailed claims Serguei Foss Heriot-Watt University, Edinburgh Karlovasi, Samos 3 June, 2010 (Joint work with Tomasz Rolski and Stan Zachary
More informationPractical approaches to the estimation of the ruin probability in a risk model with additional funds
Modern Stochastics: Theory and Applications (204) 67 80 DOI: 05559/5-VMSTA8 Practical approaches to the estimation of the ruin probability in a risk model with additional funds Yuliya Mishura a Olena Ragulina
More informationStochastic Areas and Applications in Risk Theory
Stochastic Areas and Applications in Risk Theory July 16th, 214 Zhenyu Cui Department of Mathematics Brooklyn College, City University of New York Zhenyu Cui 49th Actuarial Research Conference 1 Outline
More informationThe equivalence of two tax processes
The equivalence of two ta processes Dalal Al Ghanim Ronnie Loeffen Ale Watson 6th November 218 arxiv:1811.1664v1 [math.pr] 5 Nov 218 We introduce two models of taation, the latent and natural ta processes,
More informationOptimal Dividend Strategies for Two Collaborating Insurance Companies
Optimal Dividend Strategies for Two Collaborating Insurance Companies Hansjörg Albrecher, Pablo Azcue and Nora Muler Abstract We consider a two-dimensional optimal dividend problem in the context of two
More informationThe exact asymptotics for hitting probability of a remote orthant by a multivariate Lévy process: the Cramér case
The exact asymptotics for hitting probability of a remote orthant by a multivariate Lévy process: the Cramér case Konstantin Borovkov and Zbigniew Palmowski Abstract For a multivariate Lévy process satisfying
More informationScale functions for spectrally negative Lévy processes and their appearance in economic models
Scale functions for spectrally negative Lévy processes and their appearance in economic models Andreas E. Kyprianou 1 Department of Mathematical Sciences, University of Bath 1 This is a review talk and
More informationMinimization of ruin probabilities by investment under transaction costs
Minimization of ruin probabilities by investment under transaction costs Stefan Thonhauser DSA, HEC, Université de Lausanne 13 th Scientific Day, Bonn, 3.4.214 Outline Introduction Risk models and controls
More informationOptimal stopping of a risk process when claims are covered immediately
Optimal stopping of a risk process when claims are covered immediately Bogdan Muciek Krzysztof Szajowski Abstract The optimal stopping problem for the risk process with interests rates and when claims
More informationThe optimality of a dividend barrier strategy for Lévy insurance risk processes, with a focus on the univariate Erlang mixture
The optimality of a dividend barrier strategy for Lévy insurance risk processes, with a focus on the univariate Erlang mixture by Javid Ali A thesis presented to the University of Waterloo in fulfilment
More informationarxiv: v2 [q-fin.gn] 26 Nov 2009
arxiv:96.21v2 [q-fin.gn] 26 Nov 29 De Finetti s dividend problem and impulse control for a two-dimensional insurance risk process Irmina Czarna Zbigniew Palmowski November 27, 29 Abstract. Consider two
More informationRuin probabilities in multivariate risk models with periodic common shock
Ruin probabilities in multivariate risk models with periodic common shock January 31, 2014 3 rd Workshop on Insurance Mathematics Universite Laval, Quebec, January 31, 2014 Ruin probabilities in multivariate
More informationMaximum Process Problems in Optimal Control Theory
J. Appl. Math. Stochastic Anal. Vol. 25, No., 25, (77-88) Research Report No. 423, 2, Dept. Theoret. Statist. Aarhus (2 pp) Maximum Process Problems in Optimal Control Theory GORAN PESKIR 3 Given a standard
More informationA Note On The Erlang(λ, n) Risk Process
A Note On The Erlangλ, n) Risk Process Michael Bamberger and Mario V. Wüthrich Version from February 4, 2009 Abstract We consider the Erlangλ, n) risk process with i.i.d. exponentially distributed claims
More informationNecessary and sucient condition for the boundedness of the Gerber-Shiu function in dependent Sparre Andersen model
Miskolc Mathematical Notes HU e-issn 1787-2413 Vol. 15 (214), No 1, pp. 159-17 OI: 1.18514/MMN.214.757 Necessary and sucient condition for the boundedness of the Gerber-Shiu function in dependent Sparre
More informationOn Optimal Stopping Problems with Power Function of Lévy Processes
On Optimal Stopping Problems with Power Function of Lévy Processes Budhi Arta Surya Department of Mathematics University of Utrecht 31 August 2006 This talk is based on the joint paper with A.E. Kyprianou:
More informationAsymptotics of the Finite-time Ruin Probability for the Sparre Andersen Risk. Model Perturbed by an Inflated Stationary Chi-process
Asymptotics of the Finite-time Ruin Probability for the Sparre Andersen Risk Model Perturbed by an Inflated Stationary Chi-process Enkelejd Hashorva and Lanpeng Ji Abstract: In this paper we consider the
More informationRuin probabilities of the Parisian type for small claims
Ruin probabilities of the Parisian type for small claims Angelos Dassios, Shanle Wu October 6, 28 Abstract In this paper, we extend the concept of ruin in risk theory to the Parisian type of ruin. For
More informationarxiv: v2 [math.pr] 15 Jul 2015
STRIKINGLY SIMPLE IDENTITIES RELATING EXIT PROBLEMS FOR LÉVY PROCESSES UNDER CONTINUOUS AND POISSON OBSERVATIONS arxiv:157.3848v2 [math.pr] 15 Jul 215 HANSJÖRG ALBRECHER AND JEVGENIJS IVANOVS Abstract.
More informationFinite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims
Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. Finite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims Tao Jiang Abstract
More informationAnalysis of the ruin probability using Laplace transforms and Karamata Tauberian theorems
Analysis of the ruin probability using Laplace transforms and Karamata Tauberian theorems Corina Constantinescu and Enrique Thomann Abstract The classical result of Cramer-Lundberg states that if the rate
More informationOn a discrete time risk model with delayed claims and a constant dividend barrier
On a discrete time risk model with delayed claims and a constant dividend barrier Xueyuan Wu, Shuanming Li Centre for Actuarial Studies, Department of Economics The University of Melbourne, Parkville,
More informationHJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011
Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance
More informationMulti-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form
Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form Yipeng Yang * Under the supervision of Dr. Michael Taksar Department of Mathematics University of Missouri-Columbia Oct
More informationOPTIMAL DIVIDEND AND FINANCING PROBLEMS FOR A DIFFUSION MODEL WITH EXCESS-OF-LOSS REINSURANCE STRATEGY AND TERMINAL VALUE
Vol. 38 ( 218 ) No. 6 J. of Math. (PRC) OPTIMAL DIVIDEND AND FINANCING PROBLEMS FOR A DIFFUSION MODEL WITH EXCESS-OF-LOSS REINSURANCE STRATEGY AND TERMINAL VALUE LI Tong, MA Shi-xia, Han Mi (School of
More informationDeterministic Dynamic Programming
Deterministic Dynamic Programming 1 Value Function Consider the following optimal control problem in Mayer s form: V (t 0, x 0 ) = inf u U J(t 1, x(t 1 )) (1) subject to ẋ(t) = f(t, x(t), u(t)), x(t 0
More informationOptimal Execution Tracking a Benchmark
Optimal Execution Tracking a Benchmark René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Princeton, June 20, 2013 Optimal Execution
More informationErik J. Baurdoux Some excursion calculations for reflected Lévy processes
Erik J. Baurdoux Some excursion calculations for reflected Lévy processes Article (Accepted version) (Refereed) Original citation: Baurdoux, Erik J. (29) Some excursion calculations for reflected Lévy
More informationOn Finite-Time Ruin Probabilities in a Risk Model Under Quota Share Reinsurance
Applied Mathematical Sciences, Vol. 11, 217, no. 53, 269-2629 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/ams.217.7824 On Finite-Time Ruin Probabilities in a Risk Model Under Quota Share Reinsurance
More informationWorst-Case-Optimal Dynamic Reinsurance for Large Claims
Worst-Case-Optimal Dynamic Reinsurance for Large Claims by Olaf Menkens School of Mathematical Sciences Dublin City University (joint work with Ralf Korn and Mogens Steffensen) LUH-Kolloquium Versicherungs-
More informationA NEW PROOF OF THE WIENER HOPF FACTORIZATION VIA BASU S THEOREM
J. Appl. Prob. 49, 876 882 (2012 Printed in England Applied Probability Trust 2012 A NEW PROOF OF THE WIENER HOPF FACTORIZATION VIA BASU S THEOREM BRIAN FRALIX and COLIN GALLAGHER, Clemson University Abstract
More informationDistribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion
Insurance: Mathematics and Economics 28 (21) 13 2 Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion Hanspeter Schmidli Laboratory of Actuarial Mathematics,
More informationDISTRIBUTIONAL STUDY OF DE FINETTI S DIVIDEND PROBLEM FOR A GENERAL LÉVY INSURANCE RISK PROCESS
J. Appl. Prob. 44, 428 443 (27) Printed in England Applied Probability Trust 27 DISTRIBUTIONAL STUDY OF DE FINETTI S DIVIDEND PROBLEM FOR A GENERAL LÉVY INSURANCE RISK PROCESS A. E. KYPRIANOU, The University
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationarxiv: v2 [math.oc] 26 May 2015
Optimal dividend payments under a time of ruin constraint: Exponential claims arxiv:11.3793v [math.oc 6 May 15 Camilo Hernández Mauricio Junca July 3, 18 Astract We consider the classical optimal dividends
More informationPavel V. Gapeev, Neofytos Rodosthenous Perpetual American options in diffusion-type models with running maxima and drawdowns
Pavel V. Gapeev, Neofytos Rodosthenous Perpetual American options in diffusion-type models with running maxima and drawdowns Article (Accepted version) (Refereed) Original citation: Gapeev, Pavel V. and
More informationAn optimal reinsurance problem in the Cramér Lundberg model
Math Meth Oper Res DOI 1.17/s186-16-559-8 ORIGINAL ARTICLE An optimal reinsurance problem in the Cramér Lundberg model Arian Cani 1 Stefan Thonhauser 2 Received: 23 February 216 / Accepeted: 1 August 216
More informationarxiv: v1 [math.pr] 30 Mar 2014
Binomial discrete time ruin probability with Parisian delay Irmina Czarna Zbigniew Palmowski Przemys law Świ atek October 8, 2018 arxiv:1403.7761v1 [math.pr] 30 Mar 2014 Abstract. In this paper we analyze
More informationA Barrier Version of the Russian Option
A Barrier Version of the Russian Option L. A. Shepp, A. N. Shiryaev, A. Sulem Rutgers University; shepp@stat.rutgers.edu Steklov Mathematical Institute; shiryaev@mi.ras.ru INRIA- Rocquencourt; agnes.sulem@inria.fr
More informationarxiv: v1 [math.pr] 19 Aug 2017
Parisian ruin for the dual risk process in discrete-time Zbigniew Palmowski a,, Lewis Ramsden b, and Apostolos D. Papaioannou b, arxiv:1708.06785v1 [math.pr] 19 Aug 2017 a Department of Applied Mathematics
More informationON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME
ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME SAUL D. JACKA AND ALEKSANDAR MIJATOVIĆ Abstract. We develop a general approach to the Policy Improvement Algorithm (PIA) for stochastic control problems
More informationA Change of Variable Formula with Local Time-Space for Bounded Variation Lévy Processes with Application to Solving the American Put Option Problem 1
Chapter 3 A Change of Variable Formula with Local Time-Space for Bounded Variation Lévy Processes with Application to Solving the American Put Option Problem 1 Abstract We establish a change of variable
More informationRare event simulation for the ruin problem with investments via importance sampling and duality
Rare event simulation for the ruin problem with investments via importance sampling and duality Jerey Collamore University of Copenhagen Joint work with Anand Vidyashankar (GMU) and Guoqing Diao (GMU).
More informationGerber Shiu Risk Theory
Andreas E. Kyprianou Gerber Shiu Risk Theory Draft May 15, 213 Springer Preface These notes were developed whilst giving a graduate lecture course (Nachdiplomvorlesung) at the Forschungsinstitut für Mathematik,
More informationA Ruin Model with Compound Poisson Income and Dependence Between Claim Sizes and Claim Intervals
Acta Mathematicae Applicatae Sinica, English Series Vol. 3, No. 2 (25) 445 452 DOI:.7/s255-5-478- http://www.applmath.com.cn & www.springerlink.com Acta Mathema cae Applicatae Sinica, English Series The
More informationThreshold dividend strategies for a Markov-additive risk model
European Actuarial Journal manuscript No. will be inserted by the editor Threshold dividend strategies for a Markov-additive risk model Lothar Breuer Received: date / Accepted: date Abstract We consider
More informationA LOGARITHMIC EFFICIENT ESTIMATOR OF THE PROBABILITY OF RUIN WITH RECUPERATION FOR SPECTRALLY NEGATIVE LEVY RISK PROCESSES.
A LOGARITHMIC EFFICIENT ESTIMATOR OF THE PROBABILITY OF RUIN WITH RECUPERATION FOR SPECTRALLY NEGATIVE LEVY RISK PROCESSES Riccardo Gatto Submitted: April 204 Revised: July 204 Abstract This article provides
More informationON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER
ON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER GERARDO HERNANDEZ-DEL-VALLE arxiv:1209.2411v1 [math.pr] 10 Sep 2012 Abstract. This work deals with first hitting time densities of Ito processes whose
More informationIDLE PERIODS FOR THE FINITE G/M/1 QUEUE AND THE DEFICIT AT RUIN FOR A CASH RISK MODEL WITH CONSTANT DIVIDEND BARRIER
IDLE PERIODS FOR THE FINITE G/M/1 QUEUE AND THE DEFICIT AT RUIN FOR A CASH RISK MODEL WITH CONSTANT DIVIDEND BARRIER Andreas Löpker & David Perry December 17, 28 Abstract We consider a G/M/1 queue with
More informationBand Control of Mutual Proportional Reinsurance
Band Control of Mutual Proportional Reinsurance arxiv:1112.4458v1 [math.oc] 19 Dec 2011 John Liu College of Business, City University of Hong Kong, Hong Kong Michael Taksar Department of Mathematics, University
More informationLecture Notes on Risk Theory
Lecture Notes on Risk Theory February 2, 21 Contents 1 Introduction and basic definitions 1 2 Accumulated claims in a fixed time interval 3 3 Reinsurance 7 4 Risk processes in discrete time 1 5 The Adjustment
More informationarxiv: v5 [math.pr] 19 Jun 2015
The Annals of Applied Probability 215, Vol. 25, No. 4, 1868 1935 DOI: 1.1214/14-AAP138 c Institute of Mathematical Statistics, 215 arxiv:111.4965v5 [math.pr] 19 Jun 215 ON GERBER SHIU FUNCTIONS AND OPTIMAL
More informationExtremes and ruin of Gaussian processes
International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo. June 28-30, 2006 Extremes and ruin of Gaussian processes Jürg Hüsler Department of Math. Statistics, University
More informationRuin probabilities in a finite-horizon risk model with investment and reinsurance
Ruin probabilities in a finite-horizon risk model with investment and reinsurance R. Romera and W. Runggaldier University Carlos III de Madrid and University of Padova July 3, 2012 Abstract A finite horizon
More informationGeneralized Hypothesis Testing and Maximizing the Success Probability in Financial Markets
Generalized Hypothesis Testing and Maximizing the Success Probability in Financial Markets Tim Leung 1, Qingshuo Song 2, and Jie Yang 3 1 Columbia University, New York, USA; leung@ieor.columbia.edu 2 City
More informationInformation and Credit Risk
Information and Credit Risk M. L. Bedini Université de Bretagne Occidentale, Brest - Friedrich Schiller Universität, Jena Jena, March 2011 M. L. Bedini (Université de Bretagne Occidentale, Brest Information
More informationErnesto Mordecki 1. Lecture III. PASI - Guanajuato - June 2010
Optimal stopping for Hunt and Lévy processes Ernesto Mordecki 1 Lecture III. PASI - Guanajuato - June 2010 1Joint work with Paavo Salminen (Åbo, Finland) 1 Plan of the talk 1. Motivation: from Finance
More informationThe finite-time Gerber-Shiu penalty function for two classes of risk processes
The finite-time Gerber-Shiu penalty function for two classes of risk processes July 10, 2014 49 th Actuarial Research Conference University of California, Santa Barbara, July 13 July 16, 2014 The finite
More informationAsymptotic Irrelevance of Initial Conditions for Skorohod Reflection Mapping on the Nonnegative Orthant
Published online ahead of print March 2, 212 MATHEMATICS OF OPERATIONS RESEARCH Articles in Advance, pp. 1 12 ISSN 364-765X print) ISSN 1526-5471 online) http://dx.doi.org/1.1287/moor.112.538 212 INFORMS
More informationStochastic optimal control with rough paths
Stochastic optimal control with rough paths Paul Gassiat TU Berlin Stochastic processes and their statistics in Finance, Okinawa, October 28, 2013 Joint work with Joscha Diehl and Peter Friz Introduction
More informationOn Optimal Dividends: From Reflection to Refraction
On Optimal Dividends: From Reflection to Refraction Hans U. Gerber Ecole des hautes études commerciales Université de Lausanne CH-1015 Lausanne, Switzerland Phone: 41 1 69 3371 Fax: 41 1 69 3305 E-mail:
More informationSolving the Poisson Disorder Problem
Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, Springer-Verlag, 22, (295-32) Research Report No. 49, 2, Dept. Theoret. Statist. Aarhus Solving the Poisson Disorder Problem
More informationRichard F. Bass Krzysztof Burdzy University of Washington
ON DOMAIN MONOTONICITY OF THE NEUMANN HEAT KERNEL Richard F. Bass Krzysztof Burdzy University of Washington Abstract. Some examples are given of convex domains for which domain monotonicity of the Neumann
More informationn E(X t T n = lim X s Tn = X s
Stochastic Calculus Example sheet - Lent 15 Michael Tehranchi Problem 1. Let X be a local martingale. Prove that X is a uniformly integrable martingale if and only X is of class D. Solution 1. If If direction:
More informationThomas Knispel Leibniz Universität Hannover
Optimal long term investment under model ambiguity Optimal long term investment under model ambiguity homas Knispel Leibniz Universität Hannover knispel@stochastik.uni-hannover.de AnStAp0 Vienna, July
More informationStability of Stochastic Differential Equations
Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010
More informationAsymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables
Asymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables Jaap Geluk 1 and Qihe Tang 2 1 Department of Mathematics The Petroleum Institute P.O. Box 2533, Abu Dhabi, United Arab
More informationCharacterizations on Heavy-tailed Distributions by Means of Hazard Rate
Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 1 (23) 135 142 Characterizations on Heavy-tailed Distributions by Means of Hazard Rate Chun Su 1, Qi-he Tang 2 1 Department of Statistics
More informationOptimal Stopping and Maximal Inequalities for Poisson Processes
Optimal Stopping and Maximal Inequalities for Poisson Processes D.O. Kramkov 1 E. Mordecki 2 September 10, 2002 1 Steklov Mathematical Institute, Moscow, Russia 2 Universidad de la República, Montevideo,
More informationA RISK MODEL WITH MULTI-LAYER DIVIDEND STRATEGY
A RISK MODEL WITH MULTI-LAYER DIVIDEND STRATEGY HANSJÖRG ALBRECHER AND JÜRGEN HARTINGER Abstract In recent years, various dividend payment strategies for the classical collective ris model have been studied
More informationBayesian quickest detection problems for some diffusion processes
Bayesian quickest detection problems for some diffusion processes Pavel V. Gapeev Albert N. Shiryaev We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process
More informationOn the Multi-Dimensional Controller and Stopper Games
On the Multi-Dimensional Controller and Stopper Games Joint work with Yu-Jui Huang University of Michigan, Ann Arbor June 7, 2012 Outline Introduction 1 Introduction 2 3 4 5 Consider a zero-sum controller-and-stopper
More informationWorst Case Portfolio Optimization and HJB-Systems
Worst Case Portfolio Optimization and HJB-Systems Ralf Korn and Mogens Steffensen Abstract We formulate a portfolio optimization problem as a game where the investor chooses a portfolio and his opponent,
More informationExcursions of Risk Processes with Inverse Gaussian Processes and their Applications in Insurance
Excursions of Risk Processes with Inverse Gaussian Processes and their Applications in Insurance A thesis presented for the degree of Doctor of Philosophy Shiju Liu Department of Statistics The London
More informationLecture 21 Representations of Martingales
Lecture 21: Representations of Martingales 1 of 11 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 21 Representations of Martingales Right-continuous inverses Let
More informationCoherent and convex monetary risk measures for unbounded
Coherent and convex monetary risk measures for unbounded càdlàg processes Patrick Cheridito ORFE Princeton University Princeton, NJ, USA dito@princeton.edu Freddy Delbaen Departement für Mathematik ETH
More informationConstrained Optimal Stopping Problems
University of Bath SAMBa EPSRC CDT Thesis Formulation Report For the Degree of MRes in Statistical Applied Mathematics Author: Benjamin A. Robinson Supervisor: Alexander M. G. Cox September 9, 016 Abstract
More informationPoisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation
Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation Jingyi Zhu Department of Mathematics University of Utah zhu@math.utah.edu Collaborator: Marco Avellaneda (Courant
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationOptimal dividend policies with transaction costs for a class of jump-diffusion processes
Finance & Stochastics manuscript No. will be inserted by the editor) Optimal dividend policies with transaction costs for a class of jump-diffusion processes Martin Hunting Jostein Paulsen Received: date
More informationConvergence of price and sensitivities in Carr s randomization approximation globally and near barrier
Convergence of price and sensitivities in Carr s randomization approximation globally and near barrier Sergei Levendorskĭi University of Leicester Toronto, June 23, 2010 Levendorskĭi () Convergence of
More informationMarch 16, Abstract. We study the problem of portfolio optimization under the \drawdown constraint" that the
ON PORTFOLIO OPTIMIZATION UNDER \DRAWDOWN" CONSTRAINTS JAKSA CVITANIC IOANNIS KARATZAS y March 6, 994 Abstract We study the problem of portfolio optimization under the \drawdown constraint" that the wealth
More informationOn Kusuoka Representation of Law Invariant Risk Measures
MATHEMATICS OF OPERATIONS RESEARCH Vol. 38, No. 1, February 213, pp. 142 152 ISSN 364-765X (print) ISSN 1526-5471 (online) http://dx.doi.org/1.1287/moor.112.563 213 INFORMS On Kusuoka Representation of
More informationIntroduction to self-similar growth-fragmentations
Introduction to self-similar growth-fragmentations Quan Shi CIMAT, 11-15 December, 2017 Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 1 / 34 Literature Jean Bertoin, Compensated fragmentation
More informationReflected Brownian Motion
Chapter 6 Reflected Brownian Motion Often we encounter Diffusions in regions with boundary. If the process can reach the boundary from the interior in finite time with positive probability we need to decide
More informationProf. Erhan Bayraktar (University of Michigan)
September 17, 2012 KAP 414 2:15 PM- 3:15 PM Prof. (University of Michigan) Abstract: We consider a zero-sum stochastic differential controller-and-stopper game in which the state process is a controlled
More informationONLINE APPENDIX TO: NONPARAMETRIC IDENTIFICATION OF THE MIXED HAZARD MODEL USING MARTINGALE-BASED MOMENTS
ONLINE APPENDIX TO: NONPARAMETRIC IDENTIFICATION OF THE MIXED HAZARD MODEL USING MARTINGALE-BASED MOMENTS JOHANNES RUF AND JAMES LEWIS WOLTER Appendix B. The Proofs of Theorem. and Proposition.3 The proof
More informationWITH SWITCHING ARMS EXACT SOLUTION OF THE BELLMAN EQUATION FOR A / -DISCOUNTED REWARD IN A TWO-ARMED BANDIT
lournal of Applied Mathematics and Stochastic Analysis, 12:2 (1999), 151-160. EXACT SOLUTION OF THE BELLMAN EQUATION FOR A / -DISCOUNTED REWARD IN A TWO-ARMED BANDIT WITH SWITCHING ARMS DONCHO S. DONCHEV
More informationLecture 22 Girsanov s Theorem
Lecture 22: Girsanov s Theorem of 8 Course: Theory of Probability II Term: Spring 25 Instructor: Gordan Zitkovic Lecture 22 Girsanov s Theorem An example Consider a finite Gaussian random walk X n = n
More informationPredicting the Time of the Ultimate Maximum for Brownian Motion with Drift
Proc. Math. Control Theory Finance Lisbon 27, Springer, 28, 95-112 Research Report No. 4, 27, Probab. Statist. Group Manchester 16 pp Predicting the Time of the Ultimate Maximum for Brownian Motion with
More informationPoint Process Control
Point Process Control The following note is based on Chapters I, II and VII in Brémaud s book Point Processes and Queues (1981). 1 Basic Definitions Consider some probability space (Ω, F, P). A real-valued
More informationA MODEL FOR THE LONG-TERM OPTIMAL CAPACITY LEVEL OF AN INVESTMENT PROJECT
A MODEL FOR HE LONG-ERM OPIMAL CAPACIY LEVEL OF AN INVESMEN PROJEC ARNE LØKKA AND MIHAIL ZERVOS Abstract. We consider an investment project that produces a single commodity. he project s operation yields
More informationThe Azéma-Yor Embedding in Non-Singular Diffusions
Stochastic Process. Appl. Vol. 96, No. 2, 2001, 305-312 Research Report No. 406, 1999, Dept. Theoret. Statist. Aarhus The Azéma-Yor Embedding in Non-Singular Diffusions J. L. Pedersen and G. Peskir Let
More informationSEPARABLE TERM STRUCTURES AND THE MAXIMAL DEGREE PROBLEM. 1. Introduction This paper discusses arbitrage-free separable term structure (STS) models
SEPARABLE TERM STRUCTURES AND THE MAXIMAL DEGREE PROBLEM DAMIR FILIPOVIĆ Abstract. This paper discusses separable term structure diffusion models in an arbitrage-free environment. Using general consistency
More information