RUIN PROBABILITIES AND OPTIMAL

STOPPING FOR A DIFFUSION WITH JUMPS

Ernesto Mordecki

Universidad de la República, Montevideo, Uruguay.

Abstract. In this paper we give the ruin probabilities for a diffusion with jumps. As a related problem, we are able to give closed form solution of some optimal stopping problems for the same processes.

Keywords and Phrases: Diffusion with Jumps, Ruin Probabilities, Barrier Problems, Optimal stopping, Derivetive Pricing.

1. Introduction.

Let be given on a stochastic basis (Ω, F, 𝔽, P) a Wiener process W = (W_{t})_{t≥0}, a Poisson process N = (N_{t})_{t≥0} with intensity λ > 0, and a sequence of independent nonnegative random variables Y = (Y_{k})_{k≥1}, with identical distribution F. We will denote F ∼ exp(α) when F is exponential with parameter α > 0. Assume that the processes W, N and Y are independent, and that the filtration 𝔽 = (F_{t})_{t≥0} is the minimal filtration satisfying the usual conditions (see Jacod and Shiryaev (1987), page 2) such that the process X = (X_{t})_{t≥0} given by

X_{t} = x + σW_{t} + ∑_{k=1}^{N_{t}} Y_{k} − at,  t ≥ 0,  (1.1)

is adapted to 𝔽. Here x, σ, and a are real constants with σ and a positive. τ is a stopping time (or stopping rule) relative to 𝔽, if

τ: Ω → [0, +∞] and {τ ≤ t} ∈ F_{t} for all t ≥ 0.

Denote by M the class of all stopping times.

The first question faced in this paper is the computation of the following ruin probabilities (see Feller (1966)):

R^{+}(x) = P (∃t: X_{t} ≤ 0),  (1.2)

and

R^{−}(x) = P (∃t: X_{t} ≥ 0).  (1.3)

The exact solution to (1.2) is well known even for general process with stationary independent increments, see for instance Prabhu (1980), but the solution to (1.3) seems to be new (see anyway Skorohod (1991)).

The second problem is the following : given a Borel function g : ℝ → ℝ find a real function s : ℝ → ℝ and a stopping rule τ^{∗}, such that

s(x) = sup_{τ∈M} E(g(X_{τ})) = E(g(X_{τ^{∗}})).  (1.4)

Here s is called the cost function, and τ^{∗} (the stopping time that realizes the supremum) the optimal stopping rule.

In the present paper we give the closed form solution to the problem (1.4) when the function g is given either by

g(x) = (x − K)^{+},

or

g(x) = (K − x)^{+},

with K a real constant.

The results on optimal stopping presented are related to those in Mordecki (1997), where different functions g (used in stochastic finance) were considered.

For similar results see Mc Kean (1965), Zhang (1995), Mordecki (1996), and Shiryaev (1978) for general reference on the subject.

The paper is organized as follows. In section 2 we present the main results. The proofs, given in section 4 are based on some preliminary results, given in section 3.

2. Main Results.

2.1. Ruin Probabilities.

Theorem 1. Let X be given by (1.1) with μ = ∫_{0}^{+∞} ydF (y) and a < λμ. Then the ruin probability (1.2) is given by

R^{+}(x) = \{ \begin{matrix}e^{ηx} & if x ≥ 0, \\ 1 & if x < 0, \end{matrix}  (2.1)

for η the negative root of the equation

\frac{σ^{2}}{2} η^{2} − aη + λ ∫_{0}^{+∞}(e^{ηy} − 1)dF (y) = 0.  (2.2)

Theorem 2. Let X be given by (1.1) with F ∼ exp(α), and λ < aα. Then the ruin probability (1.3) is given by

R^{−}(x) = \{ \begin{matrix}Ae^{−p_{1}x} + Be^{−p_{2}x}, & if x < 0, \\ 1 & if x ≥ 0. \end{matrix}  (2.3)

with p_{1} and p_{2} the roots of

Q(p) = \frac{σ^{2}}{2} p^{2} + (\frac{ασ^{2}}{2} + a)p + αa − λ = 0,  (2.4)

and A, B given by

A = \frac{p_{2}(p_{1} + α)}{α(p_{2} − p_{1})}, B = \frac{p_{1}(p_{2} + α)}{α(p_{1} − p_{2})}.

Remark: From Theorem 2 we obtain the distribution of the supremum of the process X, when F ∼ exp(α). Denote by S = sup_{t≥0} X_{t}. Let y > x, then

P (S ≥ y) = P (∃t: x + σW_{t} + ∑_{k=1}^{N_{t}} Y_{k} − at ≥ y) =

P (∃t: y − x + σW_{t} + ∑_{k=1}^{N_{t}} Y_{k} − at ≥ 0) = R^{−}(x − y).

In the proof of Theorem 3, we use the fact that

E(S) = x − (\frac{A}{p_{1}} + \frac{B}{p_{2}} ).

For further reference (Mordecki (1977)), it is easy to compute that

E(e^{S}) = e^{x}[ − (\frac{A}{p_{1} + 1} + \frac{B}{p_{2} + 1})] < +∞.

(Observe that the result is finite because p_{i} + 1 < 0 for i = 1, 2).

2.2. Optimal Stopping.

Theorem 3. Let X be given by (1.1), and g(x) = (x − K)^{+}. Assume that F ∼ exp(α), and λ < aα. Denote

x_{0} = K + \frac{λ + \frac{α^{2}σ^{2}}{2}}{α(aα − λ)}.  (2.5)

Then, the solution to the optimal stopping problem (1.4) is

τ^{∗} = inf{t ≥ 0 : X_{t} ≥ x_{0}},

s(x) = \{ \begin{matrix}Ae^{p_{1}(x_{0}−x)} + Be^{p_{2}(x_{0}−x)}, & if x ≤ x_{0}, \\ x − K, & if x > x_{0}. \end{matrix}  (2.6)

where p_{2} < p_{1} < 0 are the roots of the equation (2.4), and the coefficients A and B are

A = \frac{p_{2}(x_{0} − K) + 1}{p_{2} − p_{1}}, B = \frac{p_{1}(x_{0} − K) + 1}{p_{1} − p_{2}}.

Theorem 4. Let X be given by (1.1) and g(x) = (K − x)^{+}. Assume that F is arbitrary. Denote by m = ∫_{0}^{+∞} ydF (y), and suppose

a < λm.  (2.8)

Let η < 0 be the unique root of the equation (2.2). Denote

x_{0} = K + \frac{1}{η} .

Then, the solution to the optimal stopping problem (1.4) is

τ^{∗} = inf{t ≥ 0 : X_{t} ≤ x_{0}},

s(x) = \{ \begin{matrix}K − x, & if x ≤ x_{0}, \\ (K − x_{0}) exp\{η(x − x_{0})\}, & if x > x_{0}. \end{matrix}  (2.9)

3. Some preliminary results.

3.1. In order to prove Theorems 1 and 2 we have to apply Itô’s formula to the process X defined in (1.1) and the functions R, defined in (2.1) and (2.3) respectively (for simplicity, we drop the superindex + and − when not neccesary). As the function R is not C^{2}(ℝ) we will apply Meyer-Itô formula (Theorem IV.51 of Protter (1992)). The second derivative of R in the sense of distributions when restricted to compacts is

μ(da) = R′′(a)da + R_{0}δ_{0}(da)

with R′′ the second derivative of R if a ≠ 0, R_{0} = R′(0^{+})−R′(0^{−}), is the difference between the lateral limits of the first derivative, and δ_{0} is the point mass at 0.

So, Meyer-Itô formula gives

R(X_{t}) − R(x) = ∫_{0}^{t} R′(X_{s−})dX_{s}+

+ ∑_{0≤s≤t} (R(X_{s}) − R(X_{s−}) − R′(X_{s−})ΔX_{s})

\frac{1}{2} ∫_{ℝ} l^{X}(a, t)μ(da) =

= ∫_{0}^{t} R′(X_{s−})dX_{s}+

+ ∑_{0≤s≤t} (R(X_{s}) − R(X_{s−}) − R′(X_{s−})ΔX_{s})

\frac{1}{2} ∫_{0}^{t} R′′(X_{s−})d ⟨X, X⟩_{s} + \frac{1}{2}R_{0}l^{X}(0, t).

where l^{X}(a, t) is the local time of the process X at level a and time t. If

τ_{0} = inf{t ≥ 0: X_{t} ≤ 0},

we have l^{X}(0, τ_{0}) = 0 because X_{t} > 0 on [0, τ_{0}). Then

R(X_{τ_{0}∧t}) − R(x) = ∫_{0}^{τ_{0}∧t} R′(X_{s−})dX_{s} + \frac{1}{2} ∫_{0}^{τ_{0}∧t} R′′(X_{s−})d ⟨X, X⟩_{s}

+ ∑_{0≤s≤τ_{0}∧t} (R(X_{s}) − R(X_{s−}) − R′(X_{s−})ΔX_{s}).

Furthermore, denoting X^{c} = (X_{t}^{c})_{t≥0}, X^{d} = (X_{t}^{d})_{t≥0}, with

X_{t}^{c} = x + σW_{t} + at, and X_{t}^{d} = ∑_{i=1}^{N_{t}} Z_{i},

we have:

∫_{0}^{τ_{0}∧t}R′(X_{s−})dX_{s} + ∑_{0≤s≤τ_{0}∧t} (R(X_{s}) − R(X_{s−}) − R′(X_{s−})ΔX_{s}) =

∫_{0}^{τ_{0}∧t} R′(X_{s−})dX_{s}^{c}+
∫_{0}^{τ_{0}∧t} ∫_{ℝ}[R(X_{s−} + x) − R(X_{s−})] ∗ (μ(ω, dx, ds) − ν(dx, ds))+

∫_{0}^{τ_{0}∧t} ∫_{ℝ}[R(X_{s−} + x) − R(X_{s−})] ∗ ν(dx, ds),

where μ = μ(ω, dx, ds) is the jump measure corresponding to X^{d}, and

ν = ν(dx, dt) = λdtF (dx)

its compensator. Resuming, Itô’s formula in our case reads

R(X_{τ_{0}∧t}) − R(x) = ∫_{0}^{τ_{0}∧t}(L^{X}R)(X_{s−})ds + M(R)_{τ_{0}∧t}  (3.1)

with

(L^{X}R)(x) = \frac{1}{2}σ^{2}R′′(x) + aR′(x) + λ ∫_{ℝ}(R(x + y) − R(x))dF (x)  (3.2)

the infinitesimal generator of the process X, and the local martingale M(R) = (M(R)_{t})_{t≥0} given by

M(R)_{t} = σ ∫_{0}^{t} R′(X_{s−})dW_{s} + ∫_{0}^{t} ∫_{ℝ}(R(X_{s−} + x) − R(X_{s−})) ∗ (μ − ν). (3.3)

3.2. In order to apply Itô’s formula to the process X defined in (1.1) and the functions s defined by (2.6) and (2.9) it is enough to observe that, although s is not C^{2}(ℝ), s′′(x) is continuous for x ≠ x_{0}, and has finite lateral limits. This gives, denoting μ(da) the signed measure (when restricted to compacts) which is the second derivative of s in the generalized sense,

μ(da) = s′′(a)da.

So Meyer-Itô formula applies, and a minor modifications in Corollary 1 to the same formula, necessary because s is only bounded on compacts, gives

s(X_{t}) − s(x) = ∫_{0}^{t} s′(X_{s−})dX_{s} + \frac{1}{2} ∫_{0}^{t} s′′(X_{s−})d ⟨X, X⟩_{s}

+ ∑_{0≤s≤t} (s(X_{s}) − s(X_{s−}) − s′(X_{s−})ΔX_{s}).

In this case, following the notation introduced in 3.1 we have

∫_{0}^{t} s′(X_{s−})dX_{s}+ ∑_{0≤s≤t} (s(X_{s}) − s(X_{s−}) − s′(X_{s−})ΔX_{s}) =

∫_{0}^{t} s′(X_{s−})dX_{s}^{c}+
∫_{0}^{t} ∫_{ℝ}[s(X_{s−} + x) − s(X_{s−})] ∗ (μ(ω, dx, ds) − ν(dx, ds))+
∫_{0}^{t} ∫_{ℝ}[s(X_{s−} + x) − s(X_{s−})] ∗ ν(dx, ds).

So, Itô’s formula reads

s(X_{t}) − s(x) = ∫_{0}^{t}(Ls)(X_{s−})ds + M(s)_{t}.  (3.4)

with L^{X} and M(·) defined in (3.2) and (3.3).

In order to prove Theorems 3 and 4, we need the following Lemma (see Mordecki (1997)), that we include for the sake of completeness.

Lemma 3.1. Let X = (X_{t})_{t≥0} be given by (1.1). Let s and g be real Borel functions, such that s is convex with s′′ continuous for x ≠ x_{0}, for some x_{0} ∈ ℝ, and has finite lateral limits. C^{∗}, the continuation region is an open halfline of the form (−∞, x_{0}) or (x_{0}, +∞). Let

τ^{∗} = inf{t ≥ 0: X_{t} ∉ C^{∗}}.

Assume that the following five conditions hold:

(1) (Ls)(x) = 0 ∀x ∈ C^{∗}.

(2) (Ls)(x) ≤ 0 ∀x ≠ x_{0}.

(3) 0 ≤ g(x) ≤ s(x) ∀x ∈ ℝ.

(4) s(X_{τ^{∗}∧T ∧t}) ≤ Z P -a.s for all T ∈ M and for all t ∈ ℝ^{+}, with Z an integrable random variable, that is E|Z| < +∞.

(5) s(X_{τ^{∗}}) = g(X_{τ^{∗}}) P -a.s.

Then, under this assumptions, the pair (τ^{∗}, s) is the solution for the optimal stopping problem (1.4) for the function g and the process X, that is:

s(x) = sup_{τ∈M} E(g(X_{τ})) = E(g(X_{τ^{∗}})).

Proof. By (3.4)

s(X_{t}) − s(x) = ∫_{0}^{t}(Ls)(X_{s−})ds + M(s)_{t}.

We have to prove assertions (a) and (b):

(a) s(x) = Eg(X_{τ^{∗}}),

(b) s(x) ≥ Eg(X_{σ}) ∀σ ∈ M. By conditions (5) and (3) in our hypothesis this is equivalent to proving

(a’) s(x) = Es(X_{τ^{∗}}),

(b’) s(x) ≥ Es(X_{σ}) ∀σ ∈ M.

Taking into account that A_{t} = − ∫_{0}^{t}(Ls)(X_{s−})ds is increasing (by condition (2)) and A_{τ^{∗}} = 0 (by condition (1)) we have to verify

(a”) E(M_{τ^{∗}}) = 0,

(b”) E(M_{σ}) ≤ 0 ∀σ ∈ M.

As M(s) = (M(s)_{t})_{t≥0} is a local martingale with M(s)_{0} = 0, for a localizing sequence (τ_{n}) we have

E(M(s)_{t∧τ^{∗}∧τ_{n}}) = E(M(s)_{0}) = 0.

As

−s(x) ≤ M(s)_{t∧τ^{∗}∧τ_{n}} ≤ s(X_{t∧τ^{∗}∧τ_{n}}) ≤ Z

(a”) follows by dominated convergence.

As M(s)_{t} ≥ −s(x), by Fatou’s Lemma the local martingale M(s) is in fact a supermartingale with EM(s)_{0} = 0 and (b”) follows.

4. Proof of the theorems.

Proof of Theorem 1. Taking into account (3.2), for x > 0

L^{X}e^{ηx} = e^{ηx}[\frac{1}{2}(ησ)^{2} − aη + λ ∫_{0}^{+∞}(e^{ηy} − 1)dF (y)] = 0.

When 0 ≤ s ≤ τ_{0} ∧ t we have X_{s−} > 0, and in consequence L^{X}(X_{s−}) = 0. Also l^{X}(0, τ_{0}) = 0, so (3.1) reads

R(X_{τ_{0}∧t}) − R(x) = M(R)_{τ_{0}∧t}

As R is a bounded function, taking expected values and limit as t goes to infinity

R(x) = E(R(X_{τ_{0}})) = E(R(X_{τ_{0}})𝕀_{{τ_{0}<∞}}) + E(R(X_{τ_{0}})𝕀_{{τ_{0}=∞}}) = P (τ_{0} < ∞).

Proof of Theorem 2. As A + B = 1, defining

R_{i}(x) = 𝕀_{{x≥0}}e^{p_{i}x} + 𝕀_{{x<0}}, i = 1, 2.

we have R = AR_{1} + BR_{2}. For x < 0:

L^{X}R_{i}(x) = p_{i}e^{p_{i}x}Q(p_{i}) + αλ\frac{p_{i}}{p_{i} + λ} = αλ\frac{p_{i}}{p_{i} + λ},

so

L^{X}R(x) = αλ\frac{Ap_{1}}{p_{1} + λ} + αλ\frac{Bp_{2}}{p_{2} + λ} = 0

by (2.5), and the proof goes as in Theorem 1.

In view of Lemma 3.1, the proofs of Theorems 3 and 4 reduces to the verification of conditions (1) to (5) in each case.

Proof of Theorem 3.

(1) For x < x_{0} and s(x) as in (2.6) we have

(L^{X}s)(x) = \frac{1}{2}σ^{2}s′′(x) − as′(x) + λ ∫_{0}^{+∞}[s(x + y) − s(x)]αe^{−αy}dy =

Ae^{p_{1}(x_{0}−x)}[\frac{1}{2}σ^{2}p_{1}^{2} + ap_{1} − λ\frac{p_{1}}{p_{1} + α}]+

Be^{p_{2}(x_{0}−x)}[\frac{1}{2}σ^{2}p_{2}^{2} + ap_{2} − λ\frac{p_{2}}{p_{2} + α}]−

λe^{−α(x_{0}−x)}[x_{0} − K + \frac{1}{α} − α\frac{A}{p_{1} + α} − α\frac{B}{p_{2} + α}] =

A\frac{p_{1}}{p_{1} + α}e^{p_{1}(x_{0}−x)}Q(p_{1}) + B \frac{p_{2}}{p_{2} + α}e^{p_{2}(x_{0}−x)}Q(p_{2})−

λe^{−α(x_{0}−x)}[x_{0} − K + \frac{1}{α} − α\frac{A}{p_{1} + α} − α\frac{B}{p_{2} + α}].

We know Q(p_{1}) = Q(p_{2}) = 0.

x_{0} − K + \frac{1}{α} − α\frac{A}{p_{1} + α} − α\frac{B}{p_{2} + α} = 0

is verified taking into account that

(p_{1} + α)(p_{2} + α) = −\frac{2λ}{σ^{2}},
p_{1}p_{2} = \frac{2(aα − λ)}{σ^{2}},
p_{1}A + p_{2}B = −1,

and (2.5).

(2) We have to see that for x > x_{0}, L^{X}s(x) ≤ 0:

(L^{X}s)(x) = −a + λ ∫_{0}^{+∞} ydF (y) = −a + \frac{λ}{α} < 0

by hipothesis.

(3) First, we verify that

A > 0 and B > 0.  (4.1)

We know A + B = x_{0} − K > 0. On the other side,

AB = \frac{−1}{(p_{1} − p_{2})^{2}} [p_{2}(x_{0} − K) + 1][p_{1}(x_{0} − K) + 1]

= \frac{−2}{σ^{2}(p_{1} − p_{2})^{2}(x_{0} − K)^{2}} Q(\frac{−1}{x_{0} − K}).

Now, Q(\frac{−1}{x_{0}−K}) < 0 by (2.5) and (4.1) is proved.

(4) Take Z = s(X_{τ^{∗}}) + x_{0}. Then, s(X_{τ^{∗}∧T ∧t}) ≤ Z follows from the fact that on the set {τ^{∗} < +∞} we have X_{τ^{∗}∧T ∧t} ≤ X_{τ^{∗}} and the function s is increasing. On the set {τ^{∗} = +∞} we have X_{τ^{∗}∧T ∧t} ≤ x_{0}.

Observe now, that

s(X_{τ^{∗}}) ≤ (X_{τ^{∗}} − K)^{+} ≤ sup_{0≤t≤+∞} X_{t}

and by Remark to Theorem 2 we have E({sup_{0≤t≤+∞} X_{t}}) < ∞.

(5) We have

lim_{x→−∞} s(x) = lim_{x→−∞} g(x) = 0

so s(X_{τ^{∗}}) = g(X_{τ^{∗}}) = 0 on {τ^{∗} = +∞} because lim_{t→+∞} X_{t} = −∞ P − a.s., and s(x) = g(x) if x ≥ x_{0}, so s(X_{τ^{∗}}) = g(X_{τ^{∗}}) on {τ^{∗} < +∞}.

Proof of Theorem 4.

(1) If s(x) = (K − x_{0})e^{η(x−x_{0})} for x > x_{0}

(L^{X}s)(x) = (K − x_{0})e^{η(x−x_{0})}[\frac{(ση)^{2}}{2} − aη + λ ∫_{0}^{+∞}(e^{ηy} − 1)dF (y)] = 0

by (2.2).

(2)^{1} In this case, for x < x_{0}

(L^{X}s)(x) = −as′(x) + λ ∫_{0}^{+∞}(s(x + y) − s(x))dF (y)

≤ \frac{1}{2}σ^{2}s′′(x_{0}+) − as′(x_{0}) + λ ∫_{0}^{+∞}(s(x_{0} + y) − s(x_{0}))dF (y) = 0,

because s′′(x) = 0 < s′′(x_{0}+), s′(x) = s′(x_{0}) = −1, and

s(x + y) − s(x) ≤ s(x_{0} + y) − s(x_{0})

for each y ≥ 0.

(3) For x > K, g(x) = 0. When x_{0} < x < K, we have s′′ > 0, with g′′ > 0, g(x_{0}) = s(x_{0}) and g′(x_{0}) = s′(x_{0}).

(4) is inmediate, because in this case we have

0 ≤ s(X_{t}) ≤ x_{0}.

(5) In this case we have X_{τ^{∗}} = x_{0} on the set {τ^{∗} < +∞}, and in consequence g(X_{τ^{∗}}) = s(X_{τ^{∗}}) = s(x_{0}) on {τ^{∗} < +∞}. As lim_{x→+∞} g(x) = lim_{x→+∞} s(x) = 0 the result follows.

Acknowledgements. This work was sponsored by CONICYT-BID (325/95).

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Centro de Matemática. F. de Ciencias. Iguá 4225. 11400 Montevideo, Uruguay. http://www.cmat.edu.uy/∼mordecki