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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Optimal Dividend Policy of A Large Insurance Company with Solvency Constraints Zongxia Liang Department of Mathematical

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Optimal Dividend Policy of A Large Insurance Company with Solvency Constraints Zongxia Liang

Department of Mathematical Sciences Tsinghua University, Beijing 100084, China zliang@math.tsinghua.edu.cn Joint work with Jicheng Yao Hsu 100 Conference, July 5-7, 2010, Peking University

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Outline

1

Mathematical Models

2

Optimal Control Problem without solvency constraints

3

Optimal Control Problem with solvency constraints

4

Economic and financial explanation

5

8 steps to get solution

6

References

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Methods for Making Maximal Profit · · · The insurance company generally takes the following means to earn maximal profit, reduce its risk exposure and improve its security: Proportional reinsurance

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Methods for Making Maximal Profit · · · The insurance company generally takes the following means to earn maximal profit, reduce its risk exposure and improve its security: Proportional reinsurance

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Methods for Making Maximal Profit · · · The insurance company generally takes the following means to earn maximal profit, reduce its risk exposure and improve its security: Proportional reinsurance Controlling dividends payout

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Methods for Making Maximal Profit · · · The insurance company generally takes the following means to earn maximal profit, reduce its risk exposure and improve its security: Proportional reinsurance Controlling dividends payout

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Methods for Making Maximal Profit · · · The insurance company generally takes the following means to earn maximal profit, reduce its risk exposure and improve its security: Proportional reinsurance Controlling dividends payout Controlling bankrupt probability(or solvency) and so on

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Cramér-Lundberg model of cash flows

The classical model with no reinsurance, dividend pay-outs

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Cramér-Lundberg model of cash flows

The classical model with no reinsurance, dividend pay-outs

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Cramér-Lundberg model of cash flows

The classical model with no reinsurance, dividend pay-outs

The cash flow (reserve process) rt of the insurance company follows rt = r0 + pt −

Nt

  • i=1

Ui,

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Cramér-Lundberg model of cash flows

The classical model with no reinsurance, dividend pay-outs

The cash flow (reserve process) rt of the insurance company follows rt = r0 + pt −

Nt

  • i=1

Ui, where claims arrive according to a Poisson process Nt with intensity ν on (Ω, F, {Ft}t≥0, P).

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Cramér-Lundberg model of reserve process Ui denotes the size of each claim. Random variables Ui are i.i.d. and independent of the Poisson process Nt with finite first and second moments given by µ1 and µ2. p = (1 + η)νµ1 = (1 + η)νE{Ui} is the premium rate and η > 0 denotes the safety loading.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Diffusion approximation of Cramér-Lundberg model By the central limit theorem, as ν → ∞, rt

d

≈ r0 + BM(ηνµ1t, νµ2t).

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Diffusion approximation of Cramér-Lundberg model By the central limit theorem, as ν → ∞, rt

d

≈ r0 + BM(ηνµ1t, νµ2t). So we can assume that the cash flow {Rt, t ≥ 0} of insurance company is given by the following diffusion process dRt = µdt + σdWt, where the first term " µt " is the income from insureds and the second term " σWt " means the company’s risk exposure at any time t.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Making Proportional reinsurance to reduce risk The insurance company gives fraction λ(1 − a(t)) of its income to reinsurance company

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Making Proportional reinsurance to reduce risk The insurance company gives fraction λ(1 − a(t)) of its income to reinsurance company As a return, the reinsurance share with the insurance company’s risk exposure σWt by paying money (1 − a(t))σWt to insureds.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Making Proportional reinsurance to reduce risk The insurance company gives fraction λ(1 − a(t)) of its income to reinsurance company As a return, the reinsurance share with the insurance company’s risk exposure σWt by paying money (1 − a(t))σWt to insureds. The cash flow {Rt, t ≥ 0} of the insurance company then becomes dRt = (µ − (1 − a(t))λ)dt + σa(t)dWt, R0 = x. We generally assume that λ ≥ µ based on real market.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Making dividends payout for the company’s shareholders

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Making dividends payout for the company’s shareholders If Lt denotes cumulative amount of dividends paid out to the shareholders up to time t,

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Making dividends payout for the company’s shareholders If Lt denotes cumulative amount of dividends paid out to the shareholders up to time t,then the cash flow {Rt, t ≥ 0} of the company is given by dRt = (µ − (1 − a(t))λ)dt + σa(t)dWt − dLt, R0 = x, (1)

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Making dividends payout for the company’s shareholders If Lt denotes cumulative amount of dividends paid out to the shareholders up to time t,then the cash flow {Rt, t ≥ 0} of the company is given by dRt = (µ − (1 − a(t))λ)dt + σa(t)dWt − dLt, R0 = x, (1) where 1 − a(t) is called the reinsurance fraction at time t, the R0 = x means that the initial capital is x, the constants µ and λ can be regarded as the safety loadings of the insurer and reinsurer, respectively.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Optimal Control Problem for the model (1) Notations:

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Optimal Control Problem for the model (1) Notations: A policy π = {aπ(t), Lπ

t } is a pair of non-negative càdlàg

Ft-adapted processes defined on a filtered probability space (Ω, F, {Ft}t≥0, P)

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Optimal Control Problem for the model (1) Notations: A policy π = {aπ(t), Lπ

t } is a pair of non-negative càdlàg

Ft-adapted processes defined on a filtered probability space (Ω, F, {Ft}t≥0, P) A pair of Ft adapted processes π = {aπ(t), Lπ

t } is called a

admissible policy if 0 ≤ aπ(t) ≤ 1 and Lπ

t

is a nonnegative, non-decreasing, right-continuous with left limits.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Optimal Control Problem for the model (1) Notations: A policy π = {aπ(t), Lπ

t } is a pair of non-negative càdlàg

Ft-adapted processes defined on a filtered probability space (Ω, F, {Ft}t≥0, P) A pair of Ft adapted processes π = {aπ(t), Lπ

t } is called a

admissible policy if 0 ≤ aπ(t) ≤ 1 and Lπ

t

is a nonnegative, non-decreasing, right-continuous with left limits. Π denotes the whole set of admissible policies.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Optimal Control Problem for the model (1) Notations: A policy π = {aπ(t), Lπ

t } is a pair of non-negative càdlàg

Ft-adapted processes defined on a filtered probability space (Ω, F, {Ft}t≥0, P) A pair of Ft adapted processes π = {aπ(t), Lπ

t } is called a

admissible policy if 0 ≤ aπ(t) ≤ 1 and Lπ

t

is a nonnegative, non-decreasing, right-continuous with left limits. Π denotes the whole set of admissible policies. When a admissible policy π is applied, the model (1) can be rewritten as follows: dRπ

t = (µ − (1 − aπ(t))λ)dt + σaπ(t)dWt − dLπ t ,

Rπ

0 = x. (2)

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Optimal Control Problem for the model (1) General setting:

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Optimal Control Problem for the model (1) General setting: The performance function J(x, π) defined by J(x, π) = E τ π

x

e−ctdLπ

t

  • (3)
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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Optimal Control Problem for the model (1) General setting: The performance function J(x, π) defined by J(x, π) = E τ π

x

e−ctdLπ

t

  • (3)

where τ π

x = inf{t ≥ 0 : Rπ t = 0} is the time of bankruptcy,

c > 0 is a discount rate.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Optimal Control Problem for the model (1) General setting: The performance function J(x, π) defined by J(x, π) = E τ π

x

e−ctdLπ

t

  • (3)

where τ π

x = inf{t ≥ 0 : Rπ t = 0} is the time of bankruptcy,

c > 0 is a discount rate. The optimal return function V(x) defined by V(x) = sup

π∈Π

{J(x, π)}. (4)

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Optimal Control Problem for the model (1) General setting: The performance function J(x, π) defined by J(x, π) = E τ π

x

e−ctdLπ

t

  • (3)

where τ π

x = inf{t ≥ 0 : Rπ t = 0} is the time of bankruptcy,

c > 0 is a discount rate. The optimal return function V(x) defined by V(x) = sup

π∈Π

{J(x, π)}. (4) Optimal control problem for the model (1) is to find the

  • ptimal return function V(x) and the optimal policy π∗ such

that V(x) = J(x, π∗)

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Solution of optimal control problem for the model (1) does not meet safety level It well known that one can find a dividend level b0 > 0, an

  • ptimal policy π∗

b0 and an optimal return function V(x, π∗ b0) to

solve optimal control problem for the model (1), i.e., V(x) = V(x, b0) = J(x, π∗

b0)

and b0 satisfies ∞ I

{s:R

π∗ b0 (s)<b0}dL

π∗

b0

s

= 0

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Solution of optimal control problem for the model (1) does not meet safety level It well known that one can find a dividend level b0 > 0, an

  • ptimal policy π∗

b0 and an optimal return function V(x, π∗ b0) to

solve optimal control problem for the model (1), i.e., V(x) = V(x, b0) = J(x, π∗

b0)

and b0 satisfies ∞ I

{s:R

π∗ b0 (s)<b0}dL

π∗

b0

s

= 0 However, the b0 may be too low and it will make the company go bankrupt soon

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Solution of optimal control problem for the model (1) does not meet safety level Indeed, we proved that the b0 and π∗

b0 satisfy for any

0 < x ≤ b0 there exists ε0 > 0 such that P{τ

π∗

b0

x

≤ T} ≥ ε0 > 0, (5) where ε0 = min 4[1−Φ(

x dσ √ T )]2

exp{ 2

σ2 (λ2+δ2)T},

x √ 2πσ

T

0 t− 3

2 exp{− (x+µt)2

2σ2t }dt

  • ,

τ π

x = inf

  • t ≥ 0 : Rπ

t = 0

  • .
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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Solution of optimal control problem for the model (1) does not meet safety level Indeed, we proved that the b0 and π∗

b0 satisfy for any

0 < x ≤ b0 there exists ε0 > 0 such that P{τ

π∗

b0

x

≤ T} ≥ ε0 > 0, (5) where ε0 = min 4[1−Φ(

x dσ √ T )]2

exp{ 2

σ2 (λ2+δ2)T},

x √ 2πσ

T

0 t− 3

2 exp{− (x+µt)2

2σ2t }dt

  • ,

τ π

x = inf

  • t ≥ 0 : Rπ

t = 0

  • .

If the company’s preferred risk level is ε(≤ ε0), i.e., P[τ

π∗

b0

x

≤ T] ≤ ε, (6) then the company has to reject the policy π∗

b0 because it

does not meet safety requirement (6) by (5), and the insurance company is a business affected with a public interest,

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

The best way to the company with the model (1)

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

The best way to the company with the model (1) and insureds and policy-holders should be protected against insurer insolvencies. So the best policy π∗

b(b ≥ b0) of the

company should meet the following

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

The best way to the company with the model (1) and insureds and policy-holders should be protected against insurer insolvencies. So the best policy π∗

b(b ≥ b0) of the

company should meet the following The safety standard (6)

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

The best way to the company with the model (1) and insureds and policy-holders should be protected against insurer insolvencies. So the best policy π∗

b(b ≥ b0) of the

company should meet the following The safety standard (6) The cost for safety standard (6) being minimal

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

The best way to the company with the model (1) and insureds and policy-holders should be protected against insurer insolvencies. So the best policy π∗

b(b ≥ b0) of the

company should meet the following The safety standard (6) The cost for safety standard (6) being minimal We establish setting to solve the problems above as follows.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

General setting optimal control problem for the model (1)with solvency constraints For a given admissible policy π the performance function J(x, π) = E τ π

x

e−ctdLπ

t

  • (7)

The optimal return function V(x) = sup

b∈B

{V(x, b)} (8) where V(x, b) = supπ∈Πb{J(x, π)}, solvency constraint set B :=

  • b : P[τ πb

b

≤ T] ≤ ε, J(x, πb) = V(x, b) and πb ∈ Πb

  • ,

Πb = {π ∈ Π : ∞

0 I{s:Rπ(s)<b}dLπ s = 0} with property:

Π = Π0 and b1 > b2 ⇒ Πb1 ⊂ Πb2.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Main goal Finding value function V(x), an optimal dividend policy π∗

b∗ and

the optimal dividend level b∗ to solve the sub-optimal control problem (7) and ( 8), i.e., J(x, π∗

b∗) = V(x).

Our main results are the following

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Main Results Theorem Assume that transaction cost λ − µ > 0. Let level of risk ε ∈ (0, 1) and time horizon T be given. (i) If P[τ

π∗

b0

b0

≤ T] ≤ ε, then we find f(x) such that the value function V(x) of the company is f(x), and V(x) = V(x, b0) = J(x, π∗

b0) = V(x, 0) = f(x). The optimal

policy associated with V(x) is π∗

bo = {A∗ b0(R π∗

bo

·

), L

π∗

bo

·

}, where (R

π∗

b0

t

, L

π∗

b0

t

) is uniquely determined by the following SDE with reflection boundary:

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Main Results Theorem(continue)                dR

π∗

bo

t

= (µ − (1 − A∗

b0(R π∗

bo

t

))λ)dt + σA∗

b0(R π∗

bo

t

)dWt − dL

π∗

bo

t

, R

π∗

bo

= x, 0 ≤ R

π∗

bo

t

≤ b0, ∞

0 I {t:R

π∗ bo t

<b0}(t)dL π∗

bo

t

= 0 (9) and τ

π∗

b0

x

= inf{t : R

π∗

b0

t

= 0}. The optimal dividend level is b0. The solvency of the company is bigger than 1 − ε.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Main Results Theorem(continue) (ii) If P[τ

π∗

b0

b0

≤ T] > ε, then there is a unique b∗ > b0 satisfying P[τ

π∗

b∗

b∗

≤ T] = ε and find g(x) such that g(x) is the value function of the company, that is, g(x) = sup

b∈B

{V(x, b)} = V(x, b∗) = J(x, π∗

b∗)

(10) and b∗ ∈ B, (11) where B :=

  • b : P[τ πb

b

≤ T] ≤ ε, J(x, πb) = V(x, b) and πb ∈ Πb

  • .
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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Main Results Theorem(continue) The optimal policy associated with g(x) is π∗

b∗ = {A∗ b∗(R π∗

b∗

·

), L

π∗

b∗

·

}, where (R

π∗

b∗

·

, L

π∗

b∗

·

}) is uniquely determined by the following SDE with reflection boundary:              dR

π∗

b∗

t

= (µ − (1 − A∗

b∗(R π∗

b∗

t

))λ)dt + σA∗

b∗(R π∗

b∗

t

)dWt − dL

π∗

b∗

t

, R

π∗

b∗

= x, 0 ≤ R

π∗

b∗

t

≤ b∗, ∞

0 I {t:R

π∗ b∗ t

<b∗}(t)dL π∗

b∗

t

= 0 (12) and τ

π∗

b

x

= inf{t : R

π∗

b

t

= 0}. The optimal dividend level is b∗. The optimal dividend policy π∗

b∗ and the optimal dividend b∗

ensure that the solvency of the company is 1 − ε.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Main Results Theorem(continue) (iii) g(x, b∗) g(x, b0) ≤ 1. (13) (iv) Given risk level ε risk-based capital standard x = x(ε) to ensure the capital requirement of can cover the total given risk is determined by ϕb∗(T, x(ε)) = 1 − ε, where ϕb(T, y) satisfies    ϕb

t (t, y) = 1 2[A∗ b(y)]2σ2ϕb yy(t, y) + (λA∗ b(y) − δ)ϕb y(t, y),

ϕb(0, y) = 1, for 0 < y ≤ b, ϕb(t, 0) = 0, ϕb

y(t, b) = 0, for t > 0.

(14)

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Main Results Theorem(continue) where f(x) is defined as follows: If λ ≥ 2µ, then f(x) = f1(x, b0) = C0(b0)(eζ1x − eζ2x), x ≤ b0, f2(x, b0) = C0(b0)(eζ1b0 − eζ2b0) + x − b0, x ≥ b0. (15) If µ < λ < 2µ, then f(x) =              f3(x, b0) = x

0 X −1(y)dy, x ≤ m,

f4(x, b0) = C1(b0)

ζ1

exp (ζ1(x − m)) + C2(b0)

ζ2

exp (ζ2(x − m)), m < x < b0, f5(x, b0) = C1(b0)

ζ1

exp (ζ1(b0 − m)) + C2(b0)

ζ2

exp{ζ2(b0 − m)} +x − b0, x ≥ b0.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Main Results Theorem(continue) g(x) is defined as follows: If λ ≥ 2µ, then g(x) = f1(x, b), x ≤ b, f2(x, b), x ≥ b. (17) If µ < λ < 2µ, then g(x) =    f3(x, b), x ≤ m(b), f4(x, b), m(b) < x < b, f5(x, b), x ≥ b. (18)

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Main Results Theorem(continue) A∗(x) is defined as follows: If λ ≥ 2µ, then A∗(x) = 1 for x ≥ 0. If µ < λ < 2µ, then A∗(x) = A(x, b0) := − λ

σ2 (X −1(x))X

′(X −1(x)),

x ≤ m, 1, x > m, (19) where X −1 denotes the inverse function of X(z), and X(z) = C3(b0)z−1−c/α+C4(b0)−λ − µ α + c ln z, ∀z > 0, m(b0) = X(z1)

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Main Results Theorem(continue) ζ1 = −µ +

  • µ2 + 2σ2c

σ2 , ζ2 = −µ −

  • µ2 + 2σ2c

σ2 , b0 = 2ln |ζ2/ζ1| ζ2 − ζ1 , C0(b0) = 1 ζ1eζ1b0 − ζ2eζ2b0 , ∆ = b0 − m, z1 = z1(b0) = ζ1 − ζ2 (−ζ2 − λ/σ2)eζ1∆ + (ζ1 + λ/σ2)eζ2∆ , C1(b0) = z1 −ζ2 − (λ/σ2) ζ1 − ζ2 , C2(b0) = z1 ζ1 + (λ/σ2) ζ1 − ζ2 , C3(b0) = z1+c/α

1

λ(c + α(2µ/λ − 1)) 2(α + c)2 , α = λ2 2σ2 , C4(b0) = −(λ − µ)c (α + c)2 + (λ − µ)α (α + c)2 ln C3(b0) + (λ − µ)α (α + c)2 ln (α + c)2 (λ − µ)c .

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Economic and financial explanation For a given level of risk and time horizon, if probability of bankruptcy is less than the level of risk, the optimal control problem of (7) and (8) is the traditional (3) and (4), the company has higher solvency, so it will have good

  • reputation. The solvency constraints here do not work.

This is a trivial case.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Economic and financial explanation If probability of bankruptcy is large than the level of risk ε, the traditional optimal policy will not meet the standard of security and solvency, the company needs to find a sub-optimal policy π∗

b∗ to improve its solvency. The

sub-optimal reserve process R

π∗

b∗

t

is a diffusion process reflected at b∗, the process L

π∗

b∗

t

is the process which ensures the reflection. The sub-optimal action is to pay out everything in excess of b∗ as dividend and pay no dividend when the reserve is below b∗, and A∗(b∗, x) is the sub-optimal feedback control function. The solvency probability is 1 − ε.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Economic and financial explanation We proved that the value function is decreasing w.r.t b and the bankrupt probability is decreasing w.r.t. b, so π∗

b∗ will

reduce the company’s profit, on the other hand, in view of P[τ

π∗

b∗

b∗

≤ T] = ε, the cost of improving solvency is minimal and is g(x, b0) − g(x, b∗). Therefore the policy π∗

b∗ is the

best equilibrium action between making profit and improving solvency.

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SLIDE 55

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Economic and financial explanation We proved that the value function is decreasing w.r.t b and the bankrupt probability is decreasing w.r.t. b, so π∗

b∗ will

reduce the company’s profit, on the other hand, in view of P[τ

π∗

b∗

b∗

≤ T] = ε, the cost of improving solvency is minimal and is g(x, b0) − g(x, b∗). Therefore the policy π∗

b∗ is the

best equilibrium action between making profit and improving solvency. The risk-based capital x(ε, b∗) to ensure the capital requirement of can cover the total risk ε can be determined by numerical solution of 1 − ϕb∗(x, b∗) = ε based on (14). The risk-based capital x(ε, b∗) decreases with risk ε, i.e., x(ε, b∗) increases with solvency , so does risk-based dividend level b∗(ε).

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SLIDE 56

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Economic and financial explanation We proved that the value function is decreasing w.r.t b and the bankrupt probability is decreasing w.r.t. b, so π∗

b∗ will

reduce the company’s profit, on the other hand, in view of P[τ

π∗

b∗

b∗

≤ T] = ε, the cost of improving solvency is minimal and is g(x, b0) − g(x, b∗). Therefore the policy π∗

b∗ is the

best equilibrium action between making profit and improving solvency. The risk-based capital x(ε, b∗) to ensure the capital requirement of can cover the total risk ε can be determined by numerical solution of 1 − ϕb∗(x, b∗) = ε based on (14). The risk-based capital x(ε, b∗) decreases with risk ε, i.e., x(ε, b∗) increases with solvency , so does risk-based dividend level b∗(ε). The premium rate will increase the company’s profit.Higher risk will get higher return

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

8 steps to get solution Step 1: Prove the inequality (5) by Girsanov theorem,comparison theorem on SDE,B-D-G inequality.

slide-58
SLIDE 58

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

8 steps to get solution Step 1: Prove the inequality (5) by Girsanov theorem,comparison theorem on SDE,B-D-G inequality. Step 2: Prove Lemma 1 Assume that δ = λ − µ > 0 and define (R

π∗

b ,b

t

, L

π∗

b

t ) by the

following SDE:              dR

π∗

b ,b

t

= (µ − (1 − A∗

b(R π∗

b ,b

t

))λ)dt + σA∗

b(R π∗

b ,b

t

)dWt − dL

π∗

b

t ,

R

π∗

b ,b

= b, 0 ≤ R

π∗

b ,b

t

≤ b, ∞

0 I {t:R

π∗ b ,b t

<b}(t)dL π∗

b

t

= 0. Then limb→∞ P[τ

π∗

b

b

≤ T] = 0.

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SLIDE 59

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

8 steps to get solution Step 3: Solving HJB equation to determine the value function g(x, b)

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

8 steps to get solution Step 3: Solving HJB equation to determine the value function g(x, b) Step 4: Prove value function g(x, b) is strictly decreasing w.r.t. b

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

8 steps to get solution Step 3: Solving HJB equation to determine the value function g(x, b) Step 4: Prove value function g(x, b) is strictly decreasing w.r.t. b Step 5: Prove the probability of bankruptcy P[τ b

b ≤ T] is a

strictly decreasing function of b by Girsanov theorem,comparison theorem on SDE,B-D-G inequality and strong Markov property.

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SLIDE 62

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

8 steps to get solution Step 3: Solving HJB equation to determine the value function g(x, b) Step 4: Prove value function g(x, b) is strictly decreasing w.r.t. b Step 5: Prove the probability of bankruptcy P[τ b

b ≤ T] is a

strictly decreasing function of b by Girsanov theorem,comparison theorem on SDE,B-D-G inequality and strong Markov property. Step 6: Prove the probability of bankruptcy ψb(T, b) = P

  • τ

π∗

b

b

≤ T

  • is continuous function of b by

energy inequality approach used in PDE theory.

slide-63
SLIDE 63

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

8 steps to get solution Step 3: Solving HJB equation to determine the value function g(x, b) Step 4: Prove value function g(x, b) is strictly decreasing w.r.t. b Step 5: Prove the probability of bankruptcy P[τ b

b ≤ T] is a

strictly decreasing function of b by Girsanov theorem,comparison theorem on SDE,B-D-G inequality and strong Markov property. Step 6: Prove the probability of bankruptcy ψb(T, b) = P

  • τ

π∗

b

b

≤ T

  • is continuous function of b by

energy inequality approach used in PDE theory. Step 7: Economical analysis

slide-64
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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

8 steps to get solution Step 3: Solving HJB equation to determine the value function g(x, b) Step 4: Prove value function g(x, b) is strictly decreasing w.r.t. b Step 5: Prove the probability of bankruptcy P[τ b

b ≤ T] is a

strictly decreasing function of b by Girsanov theorem,comparison theorem on SDE,B-D-G inequality and strong Markov property. Step 6: Prove the probability of bankruptcy ψb(T, b) = P

  • τ

π∗

b

b

≤ T

  • is continuous function of b by

energy inequality approach used in PDE theory. Step 7: Economical analysis Step 8: Numerical analysis of PDE by matlab and

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

References [1] Lin He, Zongxia Liang, 2008. Optimal Financing and Dividend Control of the Insurance Company with Proportional Reinsurance Policy. Insurance: Mathematics and Economics, Vol.42, 976-983. [2] Lin He, Ping Hou and Zongxia Liang, 2008. Optimal Financing and Dividend Control of the Insurance Company with Proportional Reinsurance Policy under solvency constraints. Insurance: Mathematics and Economics, Vol.43, 474-479. [3] Lin He, Zongxia Liang, 2009. Optimal Financing and Dividend Control of the Insurance Company with Fixed and Proportional Transaction Costs.Insurance: Mathematics and Economics Vol. 44, 88-94.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

References [4] Zongxia Liang, Jicheng Yao, 2010. Nonlinear optimal stochastic control of large insurance company with insolvency probability constraints. arXiv:1005.1361 [5] Zongxia Liang, Jianping Huang, 2010. Optimal dividend and investing control of a insurance company with higher solvency

  • constraints. arXiv:1005.1360.

[6] Zongxia Liang, Jicheng Yao , 2010. Optimal dividend policy

  • f a large insurance company with positive transaction cost

under higher solvency and security. arXiv:1005.1356.

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Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints

Thank You !