Optimal Stopping for a Diffusion with Jumps ^{∗}

Ernesto Mordecki Centro de Matemática. Iguá 4225, C.P. 11400. Montevideo. Uruguay. (e-mail: mordecki@cmat.edu.uy)

Abstract

In this paper we give the closed form solution of some optimal stopping problems for processes derived from a diffusion with jumps. Within the possible applications, the results can be interpreted as pricing perpetual American Options under diffusion-jump information.

Keywords and Phrases: Diffusion with jumps, Optimal stopping, American options, Derivative pricing.

JEL Classification Numbers: G12

Mathmatics Subject Classification (1991): 60G40

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 U = (U_{t})_{t≥0} or D = (D_{t})_{t≥0} given by

∗Running title: Stopping a Diffusion with Jumps

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

and

D_{t} = x + σW_{t} − ∑_{k=1}^{N_{t}} Y_{k} + at,  t ≥ 0,  (2)

are adapted to F. U stands for jumps up, and D for jumps down. 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.

Consider the following problem: given a Borel function g: R → R and a random process S = (S_{t})_{t≥0} on (Ω, F, F, P) find a real function s: R → R and a stopping rule τ^{∗}, such that

s(x) = sup_{τ∈M} E(g(S_{τ})) = E(g(S_{τ^{∗}})).  (3)

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 (3) in the following four cases. The function g is given either by

g(x) = (e^{x} − K)^{+}, or g(x) = (K − e^{x})^{+},

with K a positive constant, and the process S is either U or D.

The results presented are generalizations of Mc Kean (1965) and Mordecki (1997), for pure diffusion and pure jumps respectively, and one of the four problems (Theorem 3) was solved by Zhang (1995). The problems considered are motivated by option pricing of perpetual American Options, and this kind of models were introduced by Merton (1990). For option pricing see also Karatzas (1988) or Shiryaev et al. (1994). Other applications of the presented results, for instance to investment under uncertainty can be found in Dixit and Pindyck (1994).

The paper is organized as follows. In section 2 we present the main results. The proofs, given in section 4 are based on a general Lemma, given in section 3. In section 5 a conclusion is presented.

2 Main Results

Theorem 1 Let U be given by (1), and g(x) = (e^{x} − K)^{+}. Assume that F ∼ exp(α), α > 1 and

\frac{σ^{2}}{2} + \frac{λ}{α − 1} < a.  (4)

Denote

x_{0} = log (K \frac{λ/α − a}{σ^{2}/2 − a + λ/(α − 1)}),  (5)

Q(p) = \frac{σ^{2}}{2} p^{2} + (\frac{ασ^{2}}{2} + a)p + αa − λ.  (6)

Then, the solution to the optimal stopping problem (3) is

\begin{aligned}τ^{∗} & = & inf\{t ≥ 0: U_{t} ≥ x_{0}\}, \\ s(x) & = & \{ \begin{matrix}Ae^{p_{1}(x_{0}−x)} + Be^{p_{2}(x_{0}−x)}, & if x ≤ x_{0}, \\ e^{x} − K, & if x > x_{0}. \end{matrix} \end{aligned}  (7)

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

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

Theorem 2 Let U be given by (1) and g(x) = (K − e^{x})^{+}. Assume that F is arbitrary. Let ^{1}

a < λ ∫_{0}^{+∞} ydF (y).  (8)

Let β be the unique positive root of the equation

\frac{σ^{2}}{2} β^{2} + aβ + λ ∫_{0}^{+∞}(e^{−βy} − 1)dF (y) = 0  (9)

Denote

x_{0} = log (K \frac{β}{β + 1}).

1Conditions (8) and (11) are corrected from the published version

Then, the solution to the optimal stopping problem (3) is

τ^{∗} = inf{t ≥ 0: U_{t} ≤ x_{0}},

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

(10)

Theorem 3 Let D be given by (2) and g(x) = (e^{x} − K)^{+}. Assume that F is arbitrary. Let

\frac{σ^{2}}{2} + a + λ ∫_{0}^{+∞}(e^{−y} − 1)dF (y) < 0.  (11)

Let β > 1 be the unique positive root of the equation (8). Denote

x_{0} = log (K \frac{β}{β − 1}).

Then, the solution to the optimal stopping problem (3) is

τ^{∗} = inf{t ≥ 0: D_{t} ≥ x_{0}},

s(x) = \{ \begin{matrix}(e^{x_{0}} − K) exp\{β(x − x_{0})\}, & if x < x_{0}, \\ e^{x} − K, & if x ≥ x_{0}. \end{matrix}

(12)

Theorem 4 Let D be given by (2), and g(x) = (K − e^{x})^{+}. Assume that F ∼ exp(α), and

\frac{λ}{α + 1} < \frac{σ^{2}}{2} + a.  (13)

Denote

x_{0} = log (K \frac{a − λ/α}{σ^{2}/2 + a − λ/(α + 1)}).  (14)

Then, the solution to the optimal stopping problem (3) is given by

\begin{aligned}τ^{∗} & = & inf\{t ≥ 0: D_{t} ≤ x_{0}\}, \\ s(x) & = & \{ \begin{matrix}K − e^{x}, & if x < x_{0}, \\ Ae^{p_{1}(x−x_{0})} + Be^{p_{2}(x−x_{0})}, & if x ≥ x_{0}. \end{matrix} \end{aligned}  (15)

where p_{1} < 0 and p_{2} < 0 are the roots of Q(p) = 0, with Q(p) given by (6) and the coefficients A and B are

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

3 Some preliminary results

First observe that both processes U and D are of the form X = (X_{t})_{t≥0} with

X_{t} = x + σW_{t} + ∑_{k=1}^{N_{t}} Z_{k} + at,  t ≥ 0,  (16)

where Z_{k} are i.i.d. random variables with support on the whole real line and a ∈ R.

In order to apply Itô’s formula to the process (s(X_{t}))_{t≥0} with s defined by (7), (10), (12) and (15) 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 (Theorem IV.51 of Protter (1992)) 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}).

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

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

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),

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

ν = ν(dx, dt) = λdtF (dx)

its compensator.

So, Itô’s formula in our case reads

s(X_{t}) − s(x) = ∫_{0}^{t}(Ls)(X_{s−})ds + M(s)_{t}  (17)

with

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

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

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

Lemma 1 Let X = (X_{t})_{t≥0} be given by (16). 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:

(i) (Ls)(x) = 0, ∀x ∈ C^{∗}.

(ii) (Ls)(x) ≤ 0, ∀x ≠ x_{0}.

(iii) 0 ≤ g(x) ≤ s(x), ∀x ∈ R.

(iv) 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| < +∞.

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

Then, under these assumptions, the pair (τ^{∗}, s) is the solution for the optimal stopping problem 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 (17) we have

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 (v) and (iii) 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 (ii)) and A_{τ^{∗}} = 0 (by condition (i)) 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

In view of the previous Lemma, the proofs of Theorems 1 to 4 reduces to the verification of conditions (i) to (v) in each case. As proofs of Theorems 1 and 4, and 2 and 3 are similar respectively, we give proofs of Theorems 1 and 2 in detail and only give a sketch of the proofs of Theorems 3 and 4.

Proof of Theorem 1. The infinitesimal generator of the process U is

(L^{U}f)(x) = \frac{1}{2}σ^{2}f′′(x) − af′(x) + λ ∫_{0}^{+∞}(f(x + y) − f(x))dF (y).

(i) For x < x_{0} and s(x) as in (7) we have

(L^{U}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})

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

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

We know Q(p_{1}) = Q(p_{2}) = 0 (see (6)).

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

is verified taking into account that

\begin{aligned}(p_{1} + α)(p_{2} + α) & = & −\frac{2λ}{σ^{2}} \\ p_{2}A + p_{1}B & = & (p_{1} + p_{2})(e^{x_{0}} − K) + e^{x_{0}} \end{aligned}  (18)

and (5).

(ii) We have to see that for x > x_{0}, L^{U}s(x) ≤ 0.

(L^{U}s)(x) = \frac{σ^{2}}{2} e^{x} − ae^{x} + λ ∫_{0}^{+∞} e^{x}(e^{y} − 1)dF (y) = e^{x}(\frac{σ^{2}}{2} − a + \frac{λ}{α − 1}) < 0

in accordance with (4).

(iii) s(x) = g(x) if x > x_{0}. For x ≤ x_{0} consider the auxiliar real function

a(x) = s(x) − g(x) = Ae^{p_{1}(x_{0}−x)} + Be^{p_{2}(x_{0}−x)} − e^{x} + K

We have to see a(x) ≥ 0 for all x ≤ x_{0}. We know that a(x_{0}) = 0, a′(x_{0}) = 0 and lim_{x→−∞} a(x) = K. If a(x_{1}) = 0 with −∞ < x_{1} < x_{0} then a(x) = 0 for all x ∈ (−∞, x_{0}] by the uniqueness Theorem for Differential Equations because a(x) is the solution of a third order linear differential equation.

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

Observe now, that

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

and by Mordecki (1997) we have E(exp{sup_{0≤t≤+∞} U_{t}}) < ∞.

(v) We have

lim_{x→−∞} s(x) = lim_{x→−∞} g(x) = 0

so s(U_{τ^{∗}}) = g(U_{τ^{∗}}) = 0 on {τ^{∗} = +∞} because lim_{t→+∞} U_{t} = −∞ P − a.s., and s(x) = g(x) if x ≥ x_{0}, so s(U_{τ^{∗}}) = g(U_{τ^{∗}}) on {τ^{∗} < +∞}, concluding the proof of Theorem 1.

Proof of Theorem 2.

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

(L^{U}s)(x) = (K − e^{x_{0}})e^{−β(x−x_{0})}(\frac{σ^{2}}{2} β^{2} + aβ + λ ∫_{0}^{+∞}[e^{−βy} − 1]dF (y)) = 0

by (9).

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

(L^{U}s)(x) = e^{x}[−\frac{1}{2}σ^{2} + a + λ ∫_{0}^{+∞}(s(x + y) − s(x))e^{−x}dF (y)]

≤ e^{x}[−\frac{1}{2}σ^{2} + a + λ ∫_{0}^{+∞}(s(x_{0} + y) − s(x_{0}))e^{−x_{0}}dF (y)]

2We complete some missing arguments in the published version

≤ e^{x−x_{0}}[\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+y)−s(x))e^{−x} ≤ (s(x_{0} +y)−s(x_{0}))e^{−x_{0}} by convexity of s(log z) in z, and s′′(x_{0}−) < s′′(x_{0}+).

(iii) For x > log K, g(x) = 0. When x_{0} < x < log K the argument follows as in the proof of (iii) in Theorem 1, considering the auxiliar function a(x) = s(x) − g(x) that verifies a(x_{0}) = a′(x_{0}) = 0, a(log K) > 0, and is the solution of a second order linear differential equation.

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

0 ≤ s(U_{t}) ≤ x_{0}, for all t ≥ 0.

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

Proof of Theorem 3.

For the process D defined in (2)

(L^{D}f)(x) = \frac{1}{2}σ^{2}f′′(x) + af′(x) + λ ∫_{0}^{+∞}(f(x − y) − f(x))dF (x)

(i) If s(x) = (e^{x_{0}} − K)e^{β(x−x_{0})} for x < x_{0}

(L^{D}s)(x) = (e^{x_{0}} − K)e^{β(x−x_{0})}(\frac{σ^{2}}{2} β^{2} + aβ + λ ∫_{0}^{+∞}[e^{−βy} − 1]dF (y)) = 0

by the election of β.

(ii)^{3} In this case, for x ≥ x_{0}

(L^{D}s)(x) = e^{x}[\frac{1}{2}σ^{2} + a + λ ∫_{0}^{+∞}(s(x − y) − s(x))e^{−x}dF (y)]

≤ e^{x}[\frac{1}{2}σ^{2} + a + λ ∫_{0}^{+∞}(s(x_{0} − y) − s(x_{0}))e^{−x_{0}}dF (y)]

≤ e^{x−x_{0}}[\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−y)−s(x))e^{−x} ≤ (s(x_{0} −y)−s(x_{0}))e^{−x_{0}} by convexity of s(log z) in z, and s′′(x_{0}+) < s′′(x_{0}−).

3Idem pg. 9

(iii) It is similar to (iii) in Theorem 1. Now the auxiliar function is

a(x) = (e^{x_{0}} − K)e^{β(x−x_{0})} − e^{x} + K for x ≤ x_{0}

that verifies a(x_{0}) = a′(x_{0}) = 0 and lim_{x→−∞} = K (iv) follows word by word as (iv) Theorem 1.

(v) follows as (v) in Theorem 2.

Proof of Theorem 4.

(i) is similar to (i) in Theorem 1, but the necessary relations are now (14), (18), and

p_{2}A + p_{1}B = (p_{1} + p_{2})(K − e^{x_{0}}) + e^{x_{0}}.

(ii) For x ≤ x_{0} we have

(L^{D}s)(x) = e^{x}( − \frac{1}{2}σ^{2} − a + λ ∫_{0}^{+∞}(1 − e^{−y})α exp\{−αy\}dy)

= e^{x}( − \frac{1}{2}σ^{2} − a + \frac{λ}{α + 1}) < 0

by (13).

(iii) follows as (iii) in Theorem 2.

(iv) is inmediate, because

0 ≤ s(X_{t}) ≤ K P − a.s.

(v) follows as (v) in Theorem 1.

5 Conclusion

The paper presented contains some closed form solutions to optimal stopping problems when the stopped process has jumps. As the main application is option pricing when returns are allowed to jump at poissonian times, and alternative title would have been “Pricing Perpetual American Options under Jump-Diffusion information”. The results presented are in the spirit of Shiryaev (1973) and Mc Kean (1965), though a modification in the “smooth pasting” principle was necessary to obtain the results (See Lemma 1 in section 3). The results can be used to price perpetual American Options for assets with jumps, once a reference measure in the incomplete market is choosen, as suggested by Föllmer and Scwheizer (1990).

Acknowledgements. This work was sponsored by project CONICYT-BID (325/95), and partially written at IMPA, Brazil. The author is indebted to Madani Elhidaoui who comunicated that (c) in the published proof of Theorem 2 and 3 was not correct.

References

[1] Dixit, A.K., Pindyck, R.S.: Investment under Uncertainty. Princeton, New York: Princeton University Press 1994.

[2] Föllmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. Applied Stochastic Analysis (London, 1989), Stochastic Monographs, 5, New York: Gordon and Breach 1991.

[3] Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Berlin Heidelberg New York London Paris Tokyo: Springer 1987.

[4] Karatzas, I.: On the pricing of American Options. Applied Math. Optimization 17, 37–60 1988.

[5] Mc Kean, Jr. H.P.: “Appendix: A free boundary problem for the heat equation arising from a problem in Mathematical Economics”. Industrial Management Review 6, (spring) 32–39 (1965).

[6] Merton, R.C.: Continuous Time Finance. Cambridge Oxford: Blackwell 1990.

[7] Mordecki, E.: Optimal Stopping for a Compound Poisson Process with Exponential Jumps. Publicaciones Matemáticas del Uruguay, Vol 7 (1997) pp. 55-66.

[8] Mordecki, E.: Ruin Probabilities and Optimal Stopping for a Diffusion with Jumps. (1997) Preprint.

[9] Protter, P.: Stochastic Integration and Differential Equations. Berlin Heidelberg New York London Paris Tokyo: Springer 1992.

[10] Shiryaev, A.N.: Statistical Sequential Analysis. American Math. Society. Providence 1973.

[11] Shiryaev, A.N., Kabanov, Y. M., Kramkov, D.O., Melnikov, A.V.: On the pricing of options of European and American types, II: Continuous time. Theory of Probability and Applications 39, 80–129 (1994).

[12] Zhang, X.L.: Formules quasi-explicites pour les options américaines dans un modèle de diffusion avec sauts. Mathematics and Computers in Simulation 38, 151–161 (1995).