Stochastic Areas and Applications in Risk Theory

Transcription

1 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

2 Outline Introduction Stochastic area and time change Omega risk model and Risk model with tax Conclusion and future research Zhenyu Cui 49th Actuarial Research Conference 2

3 Stochastic areas Queuing theory: the length of a queue at time t is V t (1) Overflow time τ l is the first passage time of V to a threshold. (2) The stochastic area swept by V until τ l : the cumulative waiting time experienced by all the users till the overflow of the system. Mathematical finance (1) Structural approach(merton (1974), Black and Cox (1976)): default event happens when the total value of the firm s asset first goes below the face value of its debt (2) Reduced-form approach(jarrow and Turnbull (1995)): default happens at the first jump time of a point process with an exogenously-determined intensity. (3) Stochastic area approach(yildirim (26)): default happens when the cumulative stochastic area below a threshold level exceeds an exogenous level. Zhenyu Cui 49th Actuarial Research Conference 3

4 Omega risk model Classical risk theory models the ruin event of an insurance company as the first time that there is a negative surplus(gerber and Shiu (1998)). Albrecher, Gerber and Shiu (211) introduce the Omega risk model. It distinguishes the ruin time(negative surplus) from the bankruptcy time(occupation time exceeds a level). An insurance company operates business even with a negative surplus, and it is bankrupt if the occupation time t 1 {V s<}ds of the surplus exceeds a grace period. Zhenyu Cui 49th Actuarial Research Conference 4

5 Risk model with tax Albrecher and Hipp (27) first introduced the risk model with tax, where a constant tax rate is applied to the compound Poisson risk model at profitable times(when the process coincides with running maximum). In the Levy insurance risk model with tax, Kyprianou and Zhou (29) consider surplus-dependent tax, i.e. the tax rate depends on the surplus value at the time of tax payments. Li, Tang and Zhou (213) introduce a diffusion risk model with tax and model the ruin time of the insurance company by its first exit time. Zhenyu Cui 49th Actuarial Research Conference 5

6 Given (Ω, F, F t, P) with state space J = (l, ), l <, we model the (before-tax) surplus process as a time-homogeneous diffusion V = (V t ) t [, ), which satisfies the stochastic differential equation(sde) dv t = µ(v t ) dt + σ(v t ) dw t, V = v J, (1) W is a F t -Brownian motion and µ( ) and σ( ) > are Borel functions satisfying the following conditions: there exists a constant C > such that, for all x 1, x 2 J µ(x 1 ) µ(x 2 ) + σ(x 1 ) σ(x 2 ) C x 1 x 2, µ 2 (x 1 ) + σ 2 (x 1 ) C 2 (1 + x 2 1 ), (2) Condition (2) guarantees that the SDE (1) has a unique solution that possesses the strong Markov property (see p.4, p.17, Gihman and Skorohod (1972)). Zhenyu Cui 49th Actuarial Research Conference 6

7 Denote the possible explosion time of V from its state space by ζ. Let b be a Borel measurable function such that x J, b(x), and assume the following local integrability condition b 2 ( ) σ 2 ( ) L1 loc (J). Define the function ϕ t := t b2 (V u )du, for t [, ζ]. Define { inf{u : ϕ u ζ > t}, on { t < ϕ ζ }, τ t :=, on {ϕ ζ t < }. Zhenyu Cui 49th Actuarial Research Conference 7

8 Theorem: Assume some technical assumptions, x J, b(x). (i) Define a new filtration G t = F τt, t [, ), and a new G t -adapted process X t := V τt, on { t < ϕ ζ }. Then V t = X t b2 (V s)ds = X ϕ t, P a.s., on { t < ζ}. (3) and the process X is a time-homogeneous diffusion, which solves the following SDE under P up to ϕ ζ dx t = µ(x t) b 2 (X t ) dt + σ(x t) b(x t ) db t, X = v. (4) where B t is the G t -adapted Dambis-Dubins-Schwartz Brownian motion defined in the proof. Zhenyu Cui 49th Actuarial Research Conference 8

9 (ii) Define ζ X := inf {u > : X u J}, then ζ X = ϕ ζ = ζ b2 (V s )ds, P-a.s., (iii) Let τ denote a F t stopping time of V t, t [, ζ), then τ X = ϕ τ, P-a.s., where τ X is the corresponding stopping time for X t, t [, ζ X ). Zhenyu Cui 49th Actuarial Research Conference 9

10 Notations for V Define τ x = inf{t : V t = x}, x J,. { Define the scale density s(x) := exp x 2µ(u). and the scale function is S(x) := x. s(y)dy = x. exp } σ 2 (u) du, x J { } y 2µ(u). σ 2 (u) du dy, x J. The following Sturm-Liouville ordinary differential equation 1 2 σ2 (x)g (x) + µ(x)g (x) = λg(x), λ, (5) has two independent, positive and convex solutions, respectively g +,λ ( ), and g,λ ( ). Define the auxiliary function f λ (y, z) = g,λ (y)g +,λ (z) g,λ (z)g +,λ (y), Zhenyu Cui 49th Actuarial Research Conference 1

11 Notations for X The following Sturm-Liouville ordinary differential equation 1 σ 2 (x) 2 b 2 (x) g (x) + µ(x) b 2 (x) g (x) = λg(x), λ, (6) has two independent, positive and convex solutions, respectively g +,λ ( ), and g,λ ( ). Define a pair of Laplace exponents for λ ψ ±, λ (x) = ±g, ±,λ (x) g±,λ (x), x J, (7) From properties of the solutions to the ODE (6), we have ψ, (x) = s(x), and ψ+, (x) = x s(y)dy s(x) x, (8) s(y)dy Note that ψ ±, (x) = ψ ± (x) because the diffusions V and X share the same scale density s( ) Zhenyu Cui 49th Actuarial Research Conference 11

12 Omega risk model Assume that the surplus value is modeled by a time-homogeneous diffusion V t, t [, ζ). Introduce a bankruptcy rate function ω(x) with x denoting the value of negative surplus. Note that ω(x) =, x >. Later we set ω(x) = b 2 (x), x. For x, ω(x)dt is the probability of bankruptcy within dt time units Introduce an auxiliary bankruptcy monitoring process N on the same probability space, and assume that conditional on V, N follows a Poisson process with state-dependent intensity ω(v t )1 {Vt<}, t >. Zhenyu Cui 49th Actuarial Research Conference 12

13 Define the time of bankruptcy τ ω as the first arrival time of the Poisson process N, i.e. { t } τ ω := inf t : ω(v s )1 {Vs<}ds > e 1, where e 1 is an independent exponential rv with unit rate. For λ, the Laplace transform of the bankruptcy time is [ E v [e λτω ] = P v (τ ω < e λ ) = 1 E v e ] e λ ω(v s)1 {Vs <} ds. Define the (total) exposure as E := ω(v s )1 {Vs<}ds. The probability of bankruptcy is ψ(v ) = P(τ ω < V = v ) = 1 E v [e E ], Zhenyu Cui 49th Actuarial Research Conference 13

14 Proposition For v >, if S( ) <, then E v [ e λ b 2 (V s)1 {Vs <} ds ] = 1 and if S( ) =, then For v, v E v [ e λ b 2 (V s)1 {Vs <} ds ] = s(y)dy s(y)dy [ E v e λ ] b 2 (V s)1 {Vs <} ds = g +,λ (v ) g+,λ () ψ, ψ +, λ ψ +, ψ +, λ λ () () + ψ, (), () () + ψ, (), ψ, ψ +, λ () () + ψ, (). Zhenyu Cui 49th Actuarial Research Conference 14

15 Example Assume that the surplus is modeled as the following SDE with state space J = (, ) dv t = µv 2 t dt + V t dw t, V = v J, µ (9) Consider the bankruptcy rate function: ω(x) = x 2, x < and ω(x) =, x. The probability of bankruptcy is given by µ 2 +2 µ µ 2 +2+µ e 2µv, if v >, µ > ψ(v ) = 1 1, µ <, 2µ µ 2 +2+µ e 2µv, if v, µ > Zhenyu Cui 49th Actuarial Research Conference 15

16 For the surplus value modeled by V in (9), it will eventually go bankrupt with probability 1 if µ <. For µ >, we can explicitly determine the probability of bankruptcy. If µ =.5 and v = 1, then ψ(v ) = e 1 2 = If µ =.5 and v =, then ψ(v ) =.5. Zhenyu Cui 49th Actuarial Research Conference 16

17 Risk model with tax Define M t := sup s t V s. γ( ) : [v, ) [, 1) is a measurable function. Whenever the process V t coincides with its running maximum M t, the firm pays tax at rate γ(m t ). The value process after taxation satisfies du t = dv t γ(m t )dm t, t, U = v Kyprianou and Zhou (29) introduce the following function γ(u) = u u v γ(z)dz = v + u v (1 γ(z))dz, u > v. Notice that v < γ(u) u, and the representation U t = V t M t + γ(m t ). Zhenyu Cui 49th Actuarial Research Conference 17

18 For a default threshold a < v, define the time of ruin with tax as T U (a) = inf{t : U t = a}, (1) Define the Azema-Yor type stopping time as τ AY = inf{t > : V t g(m t )}, (11) for any continuous function g defined on [, ) satisfying < g(x) < x for x >. T U (a) is an Azema-Yor type stopping time with g(x) = x γ(x) + a = x v γ(z)dz + a. Zhenyu Cui 49th Actuarial Research Conference 18

19 Expected Ruin Area with Tax Proposition The expected stochastic area till the time of ruin is [ ] T U (a) E b 2 s(t) (V s )ds = 2 v S(t) S(g(t)) ( ) t b 2 ( (r)(s(r) S(g(t))) t ) s(r) σ 2 dr exp (r)s(r) S(r) S(g(r)) dr dt. g(t) v (12) Zhenyu Cui 49th Actuarial Research Conference 19

20 Example Assume that V t = σb t, σ >, γ( ) = γ [, 1), then we have g(x) = γ(x v ) + a. [ ] T U (a) E b 2 (V s )ds = { ( ) 2(v a) 2 v (2γ 1)σ (2 γ)(v a) 3(3γ 2), if 2 3 < γ < 1,, if γ 2 3. In the special case b( ) = 1, we have { (v a) 2 E[T U, if 1 (a)] = (2γ 1)σ 2 2 < γ < 1,, if γ 1 2. (13) If γ [ =.75, a =.5, σ = 1 and v = 1, then ] T E U (a) b 2 (V s )ds = 4 3, and E[T U (a)] = 1 2. Zhenyu Cui 49th Actuarial Research Conference 2

21 Conclusion and Future Research Link the stochastic area of a diffusion to the study of another diffusion. Determine the probability of bankruptcy of an Omega risk model with general bankruptcy functions. Determine the expected stochastic area until ruin in a risk model with tax. Future research (1) Extend to Omega risk model with tax (Cui (214)). (2) Extend to stochastic area of jump diffusions (Cai and Kou (211)). Zhenyu Cui 49th Actuarial Research Conference 21

22 Thank You! Q & A Zhenyu Cui 49th Actuarial Research Conference 22

23 References (partial list) Albrecher, H., and Gerber, H. and Shiu, E. (211): The optimal dividend barrier in the Gamma-Omega model, European Actuarial Journal, 1, Albrecher, H., and Hipp, C. (27): Lundberg s risk process with tax, Blatter der DGVFM, 28, Cui, Z. (213): Stochastic areas of diffusions and applications in risk theory, working paper, available at: Cui, Z. (214): Omega risk model with tax, working paper, available at: Kyprianou, A. and Zhou, X. (29): General tax structures and the Levy insurance risk model, Journal of Applied Probability, 46(4), Li, B. and Tang, Q. and Zhou, X. (213): A time-homogeneous diffusion model with tax, Journal of Applied Probability, 5(1), Zhenyu Cui 49th Actuarial Research Conference 23