On Transaction Costs in Insurance some work in progress Stefan - - PowerPoint PPT Presentation

On Transaction Costs in Insurance

some work in progress

Stefan Thonhauser

Johann Radon Institute for Computational and Applied Mathematics (RICAM)

Special Semester on Stochastics with Emphasis on Finance Concluding Workshop

Outline

Introduction Risk Model Dividends Optimization Problem General Characterization of a Solution Iterated Optimal Stopping Quasi-Variational Inequalities Examples A Good Guess Ideas - General Case

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Classical Risk Model

Let (Ω, F, P) be a probability space carrying all stochastic quantities The classical risk reserve process R = (Rt)t≥0 is described by Rt = x + ct −

Nt

• k=1

Yk Components:

◮ x ≥ 0 . . . deterministic initial capital ◮ c > 0 . . . deterministic premium rate ◮ N = (Nt)t≥0 . . . claim counting process, homogeneous Poisson

process with intensity λ > 0

◮ {Yk}k∈N . . . claim amounts, sequence of iid distributed positive

random variables with continuous distribution function FY We assume that N and Y1 are independent

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Dividends - General Strategies

Extension allows dividend payments from portfolio (De Finetti 1957)

Definition

A process L = (Lt)t≥0 is called admissible dividend strategy if

◮ predictable, c`

agl` ad (Lt− = Lt), non-decreasing

◮ no payments after ruin, Lt = Lτ L for t > τ L ◮ paying dividends can not lead to ruin, Lt+ − Lt ≤ RL

t

Lt denotes the cumulated dividends up to time t The controlled reserve process RL = (RL

t )t≥0 given by

RL

t = Rt − Lt,

VL(x) = Ex( τ L e−δt dLt), where τ L = inf{t > 0 | RL

t < 0} denotes the time of ruin of RL

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Transaction Costs - Impulse Control

Introduce fixed and proportional transaction costs: Dividend payment z is charged by kz − K with k ∈ (0, 1), K > 0 Considering optimization problems it’s enough to consider impulse controls S = {(τi, Zi)}i∈N

◮ intervention times τi and associated actions Zi ◮ 0 ≤ τi < taui+1 a.s. for all i ∈ N ◮ τi stopping time w.r.t. Ft = σ{RS

s− | s ≤ t} for t ≥ 0

◮ Zi is Fτi measurable ◮ P (limi→∞ τi ≤ T) = 0 for all T ≥ 0 ◮ a := K

k < Zi ≤ Rτi

S denotes set of admissible impulse controls Sn admissible controls with at most n interventions

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Optimization Problem

The controlled process RS = (RS

t )t≥0 for S ∈ S, is defined by

RS

t = x + ct − Nt

• n=1

Yn −

∞

• i=1

I{τi<t}Zi. τ S is its time of ruin Value of strategy S VS(x) = Ex ∞

• i=1

e−δτiu(Zi)I{τi<τ S}

• ,

with u(z) = 1

γ (kz − K)γ and γ ∈ (0, 1]

Value function of maximization problem: V (x) = sup

S∈S

VS(x)

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Properties of V

◮ V is absolutely continuous, increasing and linearly bounded ◮ Let f be continuous, increasing an linearly bounded

Value of optimal intervention: Mf (x) = sup

y∈(a,x]

{u(y) + f (x − y)}, Mf is (like f ) continuous, increasing and linearly bounded (in x) For the first characterization we need: Associated optimal stopping operator Mf (x) = sup

θ∈T

Ex

• e−δθ Mf (Rθ)
• .

Mf (x) has the same properties like f

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Iterated optimal Stopping

Define {vn}n∈N0 by

◮ v0(x) = 0 (no intervention) ◮ vn by vn(x) = Mvn−1(x) (at most n interventions) ◮ We get

vn(x) = sup

S∈Sn

VS(x) Indeed V is the limit of iterated optimal stopping

Theorem (1st Characterization)

Let {vn}n∈N be defined as above. Then the following holds lim

n→∞ vn(x) = V (x) and V (x) = MV (x).

V is the smallest fixed point of M.

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Motivation of Quasi-Variational Inequalities

At some point x ≥ 0 we can choose between

◮ intervention, it’s optimal if

MV (x) − V (x) = 0

◮ no intervention in open intervall around x

LV (x) = cV ′(x) + λ x V (x − y) dFY (y) − V (x)

• − δV (x) = 0

Therefore we expect V to fulfill QVI: max {LV , MV − V } = 0.

Proposition

V is differentiable on (0, a) and fulfills the QVI a.e. .

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Abstract Strategy - QVI Control

Let g be an increasing, absolutely continuous and linearly bounded solution to the QVI Define the (admissible) impulse control Sg ∈ S, the QVI control, by τ g

1 = inf

• t > 0 | Mg(RSg

t ) = g(RSg t )

• ,

Z g

1 = argmax

• u(z) + g(RSg

τ g

1 − z) | z ∈ (a, RSg

τ g

1 ]

• and for n ≥ 2

τ g

n = inf

• t > τ g

n−1 | Mg(RSg t ) = g(RSg t )

• ,

Z g

n = argmax

• u(z) + g(RSg

τ g

n − z) | z ∈ (a, RSg

τ g

n ]

• .

{x ≥ 0 | g(x) = Mg(x)} . . . intervention region {x ≥ 0 | g(x) > Mg(x)} . . . continuation region

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Some Sort of Verification

We have

Theorem (2nd Characterization)

The strategy SV is admissible and optimal,i.e. V ≥ VS for all S ∈ S. Further we have that V is the smallest increasing, absolutely continuous and linearly bounded solution to the QVI. From the proof we get: Every solution to QVI corresponds one to one to a fixed point of M How to calculate V ?

◮ explicit solutions to Lg = 0 simplify things ◮ what to do if there is no explicit solution to QVI

there is no initial value

◮ general structure of optimal strategies ? 11 / 18

Easy Strategies

Fix some b1, b2 with

◮ 0 ≤ b1 < b2 − a ◮ below level b2 > a the

process behaves uncontrolled

◮ hitting b2 pay immediately

b2 − b1

◮ above b2 take individual

argmax such that process jumps below b2

R b2 b1 t τ x

Figure: Path - easy strategy

Advantage: below b2 explicit solutions in special cases

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Value of such a strategy b = {b1, b2} given by Vb(x) =

• f (x) u(b2−b1)

1−f (b1)

x ≤ b2, sup{x−b2≤z≤x&z>a}u(z) + f (x − z)u(b2−b1)

1−f (b1)

x > b2 where f is unique solution to Lf = 0 and f (b2) = 1 Because f (x) = Ex

• e−δθ(b2) | θ(b2) < τ
• Vb(x) = f (x) Vb(b2),

Vb(b2) = u(b2 − b1) 1 − f (b1) First idea maximize u(b2−b1)

1−f (b1) over b1, b2

Justification:

◮ if b1, b2 > 0 like smooth fit at b2, (f ′(b2) = f ′(b1)) ◮ optimality results for diffusion and L´

evy models (Ronnie’s preprint)

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Example with exponential claims

c = 2.5, α = 2, λ = 1, δ = 0.03, k = 0.99, K = 0.1, γ = 0.7

20 40 60 80 100 90 100 110 120 130

Figure: Value function

2 4 6 8 10 0.05 0.10 0.15 0.20 0.25 0.30

Figure: Difference of sup and simple

b∗ = (4.61, 4.95) is optimal by numerical verification, Vb∗ fulfills QVI It is worth to continue with the argmax

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Optimal Strategies in the classical model

The HJB equation is max{LV , 1 − V ′} = 0 General optimal strategies are of band type

◮ A = {x ∈ [0, ∞) | IDE(x) = 0 & V ′(x−) = 1}

stay at this level, pay out continuously

◮ B = {x ∈ (0, ∞) | V ′(x) = 1 & IDE(x) < 0}

pay out lump sums

◮ C = (A ∪ B)c

do nothing The sets partition R+, the areas can alternate C − A − B − C − A − B (Schmdili 2008, Albrecher and T. 2008)

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Examplary Sample Path and Value Function

t R(t) B C x1=12.96 x0=0 a=0.96 τ B x

Figure: Path of RL∗

0.96 5 10 12.96 15 x 5 10 15 20 Vx

Figure: Value function for i = 0.02 and i = 0

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Conjectures - future Program

Assume a similar band structure The following algorithm could work

◮ starting in [0, a) calculate Vb∗ ◮ find [c1, b3) where again it is optimal to do nothing, LV = 0 ◮ find [b3, c2) where it is optimal to pay out, MV = V ◮ and so on... ◮ by construction it should be a QVI control ◮ but a detailed characterization of interval boundaries is needed,

even after an succesfull construction you have to verify numerically Alternative: use discretization and iterated optimal stopping approach

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Thanks for your Attention

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