This document discusses viscosity solution methods for solving the problem of ruin in classical risk theory. It begins by introducing the problem of ruin and defining viscosity solutions. It then presents the classical risk model and derives the risk equation. The goal is to analyze the risk equation using viscosity solution methods after a change of variables. Viscosity solutions allow a broad class of collective risk problems to be studied even when the claims distribution is not smooth.

1. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Viscosity Solution Methods and the
Problem of Ruin
Khalilah Beal
University of California, Berkeley
July 15, 2015

2. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
We consider the problem of “collective ruin” in classical risk
theory, which models the risk of an insurance business.

3. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Why perturbation?
The probability that the reserve remains nonnegative, u,
satisfies an integro-di↵erential equation (IDE).
To consider solutions for large initial reserves, a gauge
parameter ✏ is introduced.
Motivated by Wentzel Kramers Brillouin (WKB) approximation
methods, the IDE is expressed in terms of the changes of
variable u✏ = 1 e w✏/✏ and u✏ = ew✏/✏.

4. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Why viscosity solutions?
Smooth solutions u✏ of the singularly perturbed IDE require
smoothness of the claims’ distribution.
Using viscosity solutions allows a broad class of collective risk
problems to be studied.
Stability of viscosity solutions implies the limit as ✏ ! 0 of w✏
satisfies a corresponding limiting equation. This equation has
simpler structure and pointwise bounds may be solutions are
determined.

5. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Viscosity Solutions
The context: solving (partial) di↵erential equations of the form
F(x, u, Du, D2
u) = 0,
where F : Rn ⇥ R ⇥ Rn ⇥ S(n).
We assume that F is proper, i.e.,
F(x, r, p, X)  F(x, s, p, Y ) whenever Y  X, r  s.

6. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
For u + c(x)u = f (x),
F(x, r, p, X) = trace(X) + c(x)r f (x).
Claim: F is proper if c is nonnegative.
Proof.
Fix x, p 2 Rn. Suppose X, Y 2 S(n) and r, s 2 R satisfy
Y  X and r  s. Then
F(x, s, p, Y ) F(x, r, p, X) = trace(X Y ) + c(x)(s r)
c(x)(s r)
0,
since X Y 0, c 0, and s r 0.

7. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Definition
Let F be proper and U ⇢ Rn. A bounded, uniformly
continuous function u is called a viscosity subsolution of
(
F(x, u, Du, D2u) = 0 in U
u = g on @U
provided
u = g on @U,
for each v 2 C1(U),
if u v has a local maximum at a point x0 2 U,
then
F(x0, u(x0), Dv(x0), D2
v(x0))  0, (1)
and
if u v has a local minimum at a point x0 2 U, then
F(x0, u(x0), Dv(x0), D2
v(x0)) 0. (2)

8. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Example: eikonal equation
For
F(x, r, p, X) = |p| c(x),
the nonlinear, first-order di↵erential equation
F = 0
is the eikonal equation.
Our BVP: (
|u0| = 1 in ( 1, 1)
u = 0 at x = ±1

9. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
What about the “usual” definition of solution?
Definition
We say u 2 C ([ 1, 1], R) is a classical solution of
(
|u0| = 1 in ( 1, 1)
u = 0 at x = ±1
(BVP)
provided
u is di↵erentiable in ( 1, 1), for each x 2 (( 1, 1),
|u0(x)| = 1, and
u(1) = u( 1) = 0.
The Mean Value Theorem implies no classical solution to this
problem exists.

10. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Claim: u(x) = 1 |x| is a viscosity solution of (BVP).
Proof.
Note that U = ( 1, 1) and @U = {±1}. Clearly, u = 0 on @U.
Fix v 2 C1(U). Suppose u v has a local max at ¯x 2 U.
WLOG, ¯x = 0. For x near 0,
u(0) v(0) u(x) v(x).
Rearranging yields
x
⇣
v(x) v(0)
x + |x|
x
⌘
0.
Now, take limits as x tends to 0.
Similar justification for the case with u v attaining a local
min at 0.

11. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Remark
Notice that
˜u(x) := |x| 1
is not a viscosity solution of the BVP.
This is reassuring, in light of the physical interpretations of the
eikonal equation.

12. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Viscosity Solutions: Moral of the Story
Viscosity solutions are useful in solving various nonlinear
problems involving di↵erentiation.
They use a maximum principle to switch between solutions
(which may not be di↵erentiable) and test functions (which are
smooth enough).

13. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Goal
Now, we formulate the risk equation which, after a change of
variables, is analyzed using viscosity solution methods.
Sometimes, formulation of the problem is the problem.

14. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
The Model
Let (⌦, U, P) be a probability space. We denote the reserve
process with initial reserve x 0 at time t 0 by
X = X(t, x, !) for ! 2 ⌦, with X(0, x, ·) = x. We suppose the
premium rate p = p(x) is deterministic, and that the random
process C = C(t) records the total, or aggregate, claims up to
time t.

15. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
The claim arrival process N
For t 0, N(t) denotes the number of claims which have
arrived during the time interval [0, t].
N = {N(t) : t 0} is the claim arrival process.
We assume N is a time homogeneous Poisson process with
intensity > 0:
P [N(t) = n] = ( t)n
n! e t
(t > 0, n = 0, 1, 2, . . .), (3)
and N(0) = 0.

16. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
The claim size process Y and distribution F
Y (n) is the size of the nth claim.
We assume Y = {Y (n) : n 2 N} is independent and identically
distributed, with E[Y ] > 0.
F{·} denotes the distribution measure and F(·) denotes the
corresponding distribution function.
We assume F(x0) = 0 if and only if x0 = 0.
Thus, our model assumes that there is no positive lower bound
on the claim sizes. Equivalently, there is almost surely a claim
of size ✏, for every ✏ > 0.

17. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
The aggregate claim amount process C
The compound Poisson process
C(t) :=
N(t)
X
n=1
Y (n) (t 0) (4)
represents the total amount of claims arriving in the time
interval (0, t]. By convention, the notation
P0
n=1 is the empty
sum and C(0) = 0. We assume the claim arrival and size
processes are independent.

18. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
The premium rate p
We assume
p is a positive-valued, non-decreasing, and smooth function.
It follows that p(x) p(0) > 0 for all x 0. Thus, our model
assumes the insurer collects a premium amount based on the
past reserves of the insurance company.

19. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
The survival probability u
We combine the foregoing processes into the reserve process of
our insurance company.
Definition
The reserve process X = X(t, x) satisfies
X(t, x) = x +
R t
0 p(X(s, x)) ds C(t). (5)
Definition
We call
u(x) := P [X(t, x) > 0 for all t > 0] (6)
the probability of ultimate survival u of the risk reserve X
satisfying (5).

20. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
The risk equation
To derive an integro-di↵erential equation satisfied by the
probability of non-ruin, we use the law of total probability.

21. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
During a time interval (0, dt] there are four possible cases:
E1 no claim occurs,
E2 one claim occurs and causes bankruptcy,
E3 one claim occurs but does not cause bankruptcy, or
E4 at least two claims occur.
By the definition of u and the law of total probability,
u(x) = P [X(t, x) > 0 for all t > 0]
=
4X
i=1
P [X(t, x) > 0 for all t > 0 | Ei ] P[Ei ].

22. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
By considering di↵erence quotients, we ultimately arrive at
p(x)u0(x) = u(x)
R x
0 u(x y) F{dy} (x 0).
This identity is called the classical risk equation, and the
boundary value problem
(
p(x)u0(x) = u(x)
R x
0 u(x y) F{dy} (x > 0)
u(1) = 1.
is the classical risk problem.

23. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
We are interested in cases for which insurer transactions involve
large sums of money. So we introduce a gauge parameter ✏ > 0
into the reserve process and premium rate:
X✏(t, ✏x) := X(t, x) for small ✏ > 0,
and define
u✏(x) := P[X✏(t, x) > 0 for all t > 0].

24. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Definition
We call
✏p✏(x)u0
✏(x) = u✏(x)
Z x/✏
0
u✏(x ✏y) F{dy} (x > 0).
the rescaled risk equation.
The corresponding boundary value problems are
(
✏p✏(x)u0
✏(x) = u✏(x)
R x/✏
0 u✏(x ✏y) F{dy} (x > 0)
u✏(1) = 1
(
✏p✏(x)u0
✏(x) = u✏(x)
R x/✏
0 u✏(x ✏y) F{dy} (x > 0)
u✏(0) = u0
for initial condition u0 2 R.

25. Viscosity
Solution
Methods and
the
Problem of
Ruin
Khalilah Beal
The problem
of ruin
Viscosity
solutions
Definition
Example
The model
The risk
equation
The rescaled
risk equation
Having defined the problem, we now express the rescaled risk
equations in terms of the changes of variable u✏ = 1 e w✏/✏
and u✏ = ew✏/✏, take limits as ✏ ! 0, and determine properties
of the limiting functions w and the sequences w✏.