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Trivariate Density Revisited
Steven E. Shreve
Carnegie Mellon University
–
Conference in Honor of Ioannis Karatzas
Columbia University
June 4–8, 2012
1 / 94
Outline
Rank-based diffusion
Bang-bang control
Trivariate density
Personal reflections
2 / 94
Rank-based diffusion
E. R. Fernholz, T. Ichiba, I. Karatzas & V. Prokaj, Planar
diffusions with rank-based characteristics and perturbed Tanaka
equation, Probability Theory and Related Fields, to appear.
dX1 = g dt + σ dB1
dX2 = −h dt + ρ dB2
dX1 = −h dt + ρ dB1
dX2 = g dt + σ dB2
ρ2 + σ 2 = 1
g +h >0
B1 , B2 independent Br. motions
3 / 94
Remarks
Karatzas, et. al. examine:
I
Weak and strong existence and uniqueness in law;
I
Properties of X1 ∨ X2 and X1 ∧ X2 ;
I
The reversed dynamics of (X1 , X2 );
4 / 94
Remarks
Karatzas, et. al. examine:
I
Weak and strong existence and uniqueness in law;
I
Properties of X1 ∨ X2 and X1 ∧ X2 ;
I
The reversed dynamics of (X1 , X2 );
I
Transition probabilities (X1 , X2 ).
5 / 94
The difference process
Define
Y (t) = X1 (t) − X2 (t).
6 / 94
The difference process
Define
Y (t) = X1 (t) − X2 (t).
Then
dY = lI{Y ≤0} (g + h) dt − Il {Y >0} (g + h) dt
+lI{Y ≤0} σ dB1 − Il {Y ≤0} ρ dB2 + lI{Y >0} ρ dB1 − Il {Y >0} σ dB2
7 / 94
The difference process
Define
Y (t) = X1 (t) − X2 (t).
Then
dY = lI{Y ≤0} (g + h) dt − Il {Y >0} (g + h) dt
+lI{Y ≤0} σ dB1 − Il {Y ≤0} ρ dB2 + lI{Y >0} ρ dB1 − Il {Y >0} σ dB2
= −λ sgn Y (t) dt + dW ,
8 / 94
The difference process
Define
Y (t) = X1 (t) − X2 (t).
Then
dY = lI{Y ≤0} (g + h) dt − Il {Y >0} (g + h) dt
+lI{Y ≤0} σ dB1 − Il {Y ≤0} ρ dB2 + lI{Y >0} ρ dB1 − Il {Y >0} σ dB2
= −λ sgn Y (t) dt + dW ,
where
λ = g + h > 0,
9 / 94
The difference process
Define
Y (t) = X1 (t) − X2 (t).
Then
dY = lI{Y ≤0} (g + h) dt − Il {Y >0} (g + h) dt
+lI{Y ≤0} σ dB1 − Il {Y ≤0} ρ dB2 + lI{Y >0} ρ dB1 − Il {Y >0} σ dB2
= −λ sgn Y (t) dt + dW ,
where
λ = g + h > 0,
dW
= σ Il {Y ≤0} dB1 − Il {Y >0} dB2
|
{z
}
dW1
+ρ Il {Y >0} dB1 − lI{Y ≤0} dB2 .
|
{z
}
dW2
10 / 94
The sum process
Define
Z (t) = X1 (t) + X2 (t).
11 / 94
The sum process
Define
Z (t) = X1 (t) + X2 (t).
Then
dZ
= (g − h) dt + lI{Y ≤0} σ dB1 + Il {Y ≤0} ρ dB2
+lI{Y >0} ρ dB1 + Il {Y >0} σ dB2
12 / 94
The sum process
Define
Z (t) = X1 (t) + X2 (t).
Then
dZ
= (g − h) dt + lI{Y ≤0} σ dB1 + Il {Y ≤0} ρ dB2
+lI{Y >0} ρ dB1 + Il {Y >0} σ dB2
= ν dt + dV ,
13 / 94
The sum process
Define
Z (t) = X1 (t) + X2 (t).
Then
dZ
= (g − h) dt + lI{Y ≤0} σ dB1 + Il {Y ≤0} ρ dB2
+lI{Y >0} ρ dB1 + Il {Y >0} σ dB2
= ν dt + dV ,
where
ν = g − h,
14 / 94
The sum process
Define
Z (t) = X1 (t) + X2 (t).
Then
= (g − h) dt + lI{Y ≤0} σ dB1 + Il {Y ≤0} ρ dB2
dZ
+lI{Y >0} ρ dB1 + Il {Y >0} σ dB2
= ν dt + dV ,
where
ν = g − h,
dV
= σ lI{Y ≤0} dB1 + Il {Y >0} dB2
|
{z
}
dV1
+ρ Il {Y >0} dB1 + Il {Y ≤0} dB2 .
|
{z
}
dV2
15 / 94
Summary
X1 (t) + X2 (t) = Z (t)
= X1 (0) + X2 (0) + νt + V (t),
X1 (t) − X2 (t) = Y (t)
Z
= X1 (0) + X2 (0) − λ
t
sgn Y (s) ds + W (t),
0
where V and W are correlated Brownian motions.
16 / 94
Summary
X1 (t) + X2 (t) = Z (t)
= X1 (0) + X2 (0) + νt + V (t),
X1 (t) − X2 (t) = Y (t)
Z
= X1 (0) + X2 (0) − λ
t
sgn Y (s) ds + W (t),
0
where V and W are correlated Brownian motions.
I
To determine transition probabilities for (X1 , X2 ), it suffices to
determine transition probabilites for (Z , Y ).
17 / 94
Summary
X1 (t) + X2 (t) = Z (t)
= X1 (0) + X2 (0) + νt + V (t),
X1 (t) − X2 (t) = Y (t)
Z
= X1 (0) + X2 (0) − λ
t
sgn Y (s) ds + W (t),
0
where V and W are correlated Brownian motions.
I
I
I
To determine transition probabilities for (X1 , X2 ), it suffices to
determine transition probabilites for (Z , Y ).
(Z , W ) is a Gaussian process.
Y is determined by W by
dY (t) = −λ sgn Y (t) dt + dW (t).
18 / 94
Summary
X1 (t) + X2 (t) = Z (t)
= X1 (0) + X2 (0) + νt + V (t),
X1 (t) − X2 (t) = Y (t)
Z
= X1 (0) + X2 (0) − λ
t
sgn Y (s) ds + W (t),
0
where V and W are correlated Brownian motions.
I
I
I
I
To determine transition probabilities for (X1 , X2 ), it suffices to
determine transition probabilites for (Z , Y ).
(Z , W ) is a Gaussian process.
Y is determined by W by
dY (t) = −λ sgn Y (t) dt + dW (t).
It suffices to determine the distribution of (W (t), Y (t)).
19 / 94
Bang-bang control
Minimize
Z
E
∞
e −t Xt2 dt,
0
Subject to
Z
Xt = x +
t
us ds + Wt ,
a ≤ ut ≤ b,
t ≥ 0.
0
20 / 94
Bang-bang control
Minimize
Z
E
∞
e −t Xt2 dt,
0
Subject to
Z
Xt = x +
t
us ds + Wt ,
a ≤ ut ≤ b,
t ≥ 0.
0
Beneš, Shepp & Witsenhausen (1980): Solution is ut = f (Xt ),
where
b, if x < δ,
f (x) =
a, if x ≥ δ,
√
√
and δ = ( b 2 + 2 + b)−1 − ( a2 + 2 − a)−1 .
21 / 94
Bang-bang control
Minimize
Z
E
∞
e −t Xt2 dt,
0
Subject to
Z
Xt = x +
t
us ds + Wt ,
a ≤ ut ≤ b,
t ≥ 0.
0
Beneš, Shepp & Witsenhausen (1980): Solution is ut = f (Xt ),
where
b, if x < δ,
f (x) =
a, if x ≥ δ,
√
√
and δ = ( b 2 + 2 + b)−1 − ( a2 + 2 − a)−1 .
What is the transition density for the controlled process X ?
(Beneš, et. al. computed its Laplace transform.)
22 / 94
Transition density by Girsanov
Compute the transition density for X , where
Z t
Xt = x +
f (Xs ) ds + Wt .
0
23 / 94
Transition density by Girsanov
Compute the transition density for X , where
Z t
Xt = x +
f (Xs ) ds + Wt .
0
Start with a probability measure PX under which X is a Brownian
motion. Define W by
Z t
Wt = Xt − x −
f (Xs ) ds.
0
24 / 94
Transition density by Girsanov
Compute the transition density for X , where
Z t
Xt = x +
f (Xs ) ds + Wt .
0
Start with a probability measure PX under which X is a Brownian
motion. Define W by
Z t
Wt = Xt − x −
f (Xs ) ds.
0
Change to a probability measure P under which W is a Brownian
motion. Compute the transition density for X under P.
25 / 94
Transition density by Girsanov
Compute the transition density for X , where
Z t
Xt = x +
f (Xs ) ds + Wt .
0
Start with a probability measure PX under which X is a Brownian
motion. Define W by
Z t
Wt = Xt − x −
f (Xs ) ds.
0
Change to a probability measure P under which W is a Brownian
motion. Compute the transition density for X under P.
"
#
dP
P{Xt ∈ B} = EX lI{Xt ∈B}
,
X
dP Ft
where
Z t
Z
1 t 2
dP = exp
f (Xs ) dXs −
f (Xs ) ds .
2 0
dPX Ft
0
26 / 94
Transition density by Girsanov (continued)
To compute P{Xt ∈ B}, we need the joint distribution of
Z t
Z t
Xt ,
f (Xs ) dXs ,
f 2 (Xs ) ds.
0
0
27 / 94
Transition density by Girsanov (continued)
To compute P{Xt ∈ B}, we need the joint distribution of
Z t
Z t
Xt ,
f (Xs ) dXs ,
f 2 (Xs ) ds.
0
0
Assume without loss of generality that δ = 0. Define
Z x
bx, if x ≤ 0,
F (x) =
f (ξ) dξ =
ax, if x ≥ 0.
0
Tanaka’s formula implies
Z t
1
F (Xt ) =
f (Xs ) dXs + (a − b)LX
t
2
0
so
Z t
1
f (Xs ) dXs = F (Xt ) − (a − b)LX
t .
2
0
28 / 94
Transition density by Girsanov (continued)
To compute P{Xt ∈ B}, we need the joint distribution of
Z t
Z t
Xt ,
f (Xs ) dXs ,
f 2 (Xs ) ds.
0
0
Assume without loss of generality that δ = 0. Define
Z x
bx, if x ≤ 0,
F (x) =
f (ξ) dξ =
ax, if x ≥ 0.
0
Tanaka’s formula implies
Z t
1
F (Xt ) =
f (Xs ) dXs + (a − b)LX
t
2
0
so
Z t
1
f (Xs ) dXs = F (Xt ) − (a − b)LX
t .
2
0
We need the joint distribution of
Z t
X
Xt , Lt ,
f 2 (Xs ) ds.
0
29 / 94
Transition density by Girsanov (continued)
Define
Z
t
Γ+ (t) =
Z0 t
Γ− (t) =
Il (0,∞) X (s) ds,
Il (−∞,0) X (s) ds = t − Γ+ (t).
0
30 / 94
Transition density by Girsanov (continued)
Define
Z
t
Γ+ (t) =
Z0 t
Γ− (t) =
Il (0,∞) X (s) ds,
Il (−∞,0) X (s) ds = t − Γ+ (t).
0
Then
Z t
f 2 (Xs ) ds = b 2 Γ− (t) + a2 Γ+ (t) = b 2 t + (a2 − b 2 )Γ+ (t).
0
31 / 94
Transition density by Girsanov (continued)
Define
Z
t
Γ+ (t) =
Z0 t
Γ− (t) =
Il (0,∞) X (s) ds,
Il (−∞,0) X (s) ds = t − Γ+ (t).
0
Then
Z t
f 2 (Xs ) ds = b 2 Γ− (t) + a2 Γ+ (t) = b 2 t + (a2 − b 2 )Γ+ (t).
0
We need the joint distribution of
Xt ,
LX
t ,
Z
Γ+ (t) =
t
Il (0,∞) (Xs ) ds,
0
where X is a Brownian motion.
32 / 94
Trivariate density
Theorem (Karatzas & Shreve (1984))
Let W be a Brownian motion, let L be its local time at zero, and
let
Z t
Γ+ (t) ,
Il (0,∞) (Ws ) ds
0
denote the occupation time of the right half-line. Then for a ≤ 0,
b ≥ 0, and 0 < t < T ,
P{WT ∈ da, LT ∈ db, Γ+ (T ) ∈ dt}
2
b
(b − a)2
b(b − a)
exp − −
= p
da db dt.
2t
2(T − t)
π t 3 (T − t)3
33 / 94
Remark
Because the occupation time of the left half-line is
Z
T
lI(−∞,0) (Wt ) dt = T − Γ+ (T ),
Γ− (T ) ,
0
we also know P{WT ∈ da, LT ∈ db, Γ− (T ) ∈ dt} for a ≤ 0, b ≥ 0,
and 0 < t < T . Applying this to −W , we obtain a formula for
P{WT ∈ da, LT ∈ db, Γ+ (T ) ∈ dt},
a ≥ 0, b ≥ 0, 0 < t < T .
34 / 94
Classic trivariate density
Theorem (Lévy (1948))
Let W be a Brownian motion, let MT = max0≤t≤T Wt , and let θT
be the (almost surely unique) time when W attains its maximum
on [0, T ]. Then for a ∈ R, b ≥ max{a, 0}, and 0 < t < T ,
P{WT ∈ da, MT ∈ db, θT ∈ dt}
2
b(b − a)
b
(b − a)2
= p
exp − −
da db dt.
2t
2(T − t)
π t 3 (T − t)3
The elementary proof uses the reflection principle and the Markov
property.
35 / 94
Positive part of Brownian motion
W
0
T
t
Positive part of Brownian motion
W
0
T
t
T
t
Γ+
0
Positive part of Brownian motion
W
0
T
t
T
t
Γ+
0
W+ (t) , WΓ−1 (t)
+
0
Γ+ (T ) t
38 / 94
Full decomposition
W
0
T
t
Full decomposition
W
0
T
W+ (t) , WΓ−1 (t)
+
0
Γ+ (T ) t
t
Full decomposition
W
0
T
W+ (t) , WΓ−1 (t)
+
Γ+ (T ) t
0
W− (t) , −WΓ−1 (t)
−
0
t
t
Full decomposition
W
0
T
t
T
t
W+ (t) , WΓ−1 (t)
+
Γ+ (T ) t
0
W− (t) , −WΓ−1 (t)
−
0
t
42 / 94
Local time
Local time at zero of W :
Z
Z
1 t
1 t
δ0 (Ws ) ds
= lim
lI(−,) (Ws ) ds
Lt =
↓0 4 0
2 0
Z
Z
1 t
1 t
= lim
Il (0,) (Ws ) ds = lim
Il (−,0) (Ws ) ds.
↓0 2 0
↓0 2 0
43 / 94
Tanaka’s formula
Tanaka formula:
Z
max{Wt , 0} =
t
Il (0,∞) (Ws ) dWs +
0
1
2
Z
t
δ0 (Ws ) ds
0
44 / 94
Tanaka’s formula
Tanaka formula:
Z
max{Wt , 0} =
t
Il (0,∞) (Ws ) dWs +
0
1
2
Z
t
δ0 (Ws ) ds
0
= −Bt + Lt ,
where
Z
Bt = −
t
lI(0,∞) (Ws ) dWs ,
0
45 / 94
Tanaka’s formula
Tanaka formula:
Z
max{Wt , 0} =
t
Il (0,∞) (Ws ) dWs +
0
1
2
Z
t
δ0 (Ws ) ds
0
= −Bt + Lt ,
where
Z
Bt = −
t
lI(0,∞) (Ws ) dWs ,
hBit = Γ+ (t).
0
46 / 94
Tanaka’s formula
Tanaka formula:
Z
max{Wt , 0} =
t
Il (0,∞) (Ws ) dWs +
0
1
2
Z
t
δ0 (Ws ) ds
0
= −Bt + Lt ,
where
Z
Bt = −
t
lI(0,∞) (Ws ) dWs ,
hBit = Γ+ (t).
0
Then B+ (t) , BΓ−1 (t) is a Brownian motion.
+
47 / 94
Tanaka’s formula
Tanaka formula:
Z
max{Wt , 0} =
t
Il (0,∞) (Ws ) dWs +
0
1
2
Z
t
δ0 (Ws ) ds
0
= −Bt + Lt ,
where
Z
Bt = −
t
lI(0,∞) (Ws ) dWs ,
hBit = Γ+ (t).
0
Then B+ (t) , BΓ−1 (t) is a Brownian motion.
+
Time-changed Tanaka formula:
W+ (t) = −B+ (t) + L+ (t),
where L+ (t) = LΓ−1 (t) .
+
48 / 94
Tanaka’s formula
Tanaka formula:
Z
max{Wt , 0} =
t
Il (0,∞) (Ws ) dWs +
0
1
2
Z
t
δ0 (Ws ) ds
0
= −Bt + Lt ,
where
Z
Bt = −
t
lI(0,∞) (Ws ) dWs ,
hBit = Γ+ (t).
0
Then B+ (t) , BΓ−1 (t) is a Brownian motion.
+
Time-changed Tanaka formula:
W+ (t) = −B+ (t) + L+ (t),
where L+ (t) = LΓ−1 (t) .
+
Conclusion: W+ is a reflected Brownian motion. It is the Brownian
motion −B+ plus the nondecreasing process L+ that grows only
when W+ is at zero.
49 / 94
Skorohod representation
Time-changed Tanaka formula:
W+ (t) = −B+ (t) + L+ (t).
Skorohod representation:
The nondecreasing process added to −B+ that grows only when
W+ = 0 is
L+ (t) = max B+ (s).
0≤s≤t
In particular,
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s).
0≤s≤t
50 / 94
W+ and B+
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s).
0≤s≤t
W+ (t) , WΓ−1 (t)
+
0
Γ+ (T ) t
W+ and B+
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s).
0≤s≤t
W+ (t) , WΓ−1 (t)
+
0
Γ+ (T ) t
0
t
−W+
W+ and B+
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s).
0≤s≤t
W+ (t) , WΓ−1 (t)
+
Γ+ (T ) t
0
B+
t
0
−W+
W+ and B+
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s).
0≤s≤t
W+ (t) , WΓ−1 (t)
+
Γ+ (T ) t
0
B+
L+ (T ) = max0≤t≤Γ+ (t) B+ (t)
t
0
−W+
54 / 94
W+ and B+ . W− and B− .
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s)
0≤s≤t
B− (t) = −W− (t) + L− (t) = −W− (t) + max B− (s).
0≤s≤t
W+
0
Γ+ (T )
T
t
W+ and B+ . W− and B− .
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s)
0≤s≤t
B− (t) = −W− (t) + L− (t) = −W− (t) + max B− (s).
0≤s≤t
W+
0
Γ+ (T )
T
t
W+ and B+ . W− and B− .
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s)
0≤s≤t
B− (t) = −W− (t) + L− (t) = −W− (t) + max B− (s).
0≤s≤t
W− run backwards
W+
0
Γ+ (T )
T
t
W+ and B+ . W− and B− .
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s)
0≤s≤t
B− (t) = −W− (t) + L− (t) = −W− (t) + max B− (s).
0≤s≤t
W− run backwards
W+
Γ+ (T )
0
0
−W+
T
t
T
t
W+ and B+ . W− and B− .
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s)
0≤s≤t
B− (t) = −W− (t) + L− (t) = −W− (t) + max B− (s).
0≤s≤t
W− run backwards
W+
Γ+ (T )
0
0
−W+
−W− run backwards
T
t
T
t
W+ and B+ . W− and B− .
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s)
0≤s≤t
B− (t) = −W− (t) + L− (t) = −W− (t) + max B− (s).
0≤s≤t
W− run backwards
W+
Γ+ (T )
0
T
t
T
t
B+
0
−W+
−W− run backwards
W+ and B+ . W− and B− .
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s)
0≤s≤t
B− (t) = −W− (t) + L− (t) = −W− (t) + max B− (s).
0≤s≤t
W− run backwards
W+
Γ+ (T )
0
B+
T
t
T
t
B− run backwards
0
−W+
−W− run backwards
W+ and B+ . W− and B− .
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s)
0≤s≤t
B− (t) = −W− (t) + L− (t) = −W− (t) + max B− (s).
0≤s≤t
W− run backwards
W+
Γ+ (T )
0
B+
T
t
T
t
B− run backwards
0
−W+
−W− run backwards
62 / 94
W+ and B+ . W− and B− .
B+ (t) = −W+ (t) + L+ (t) = −W+ (t) + max B+ (s)
0≤s≤t
B− (t) = −W− (t) + L− (t) = −W− (t) + max B− (s).
0≤s≤t
W− run backwards
W+
Γ+ (T )
0
B+
T
t
T
t
B− run backwards
0
−W+
−W− run backwards
Local time L+ (T ) = LT time has become the maximum.
Γ+ (T ) has become the time of the maximum.
63 / 94
Personal reflections
64 / 94
In the beginning....
S. Shreve, Reflected Brownian motion in the “bang-bang” control
of Browian drift, SIAM J. Control Optimization 19, 469–478,
(1981).
65 / 94
In the beginning....
S. Shreve, Reflected Brownian motion in the “bang-bang” control
of Browian drift, SIAM J. Control Optimization 19, 469–478,
(1981).
Acknowledgment in the paper
The author wishes to acknowledge the aid of V. E.
Beneš, who found an error in some preliminary work on
this subject and suggested the applicability of Tanaka’s
formula. He also wishes to thank the referee for pointing
out the uniqueness of the transition density
corresponding to the weak solution in Section 5.
66 / 94
In the beginning....
S. Shreve, Reflected Brownian motion in the “bang-bang” control
of Browian drift, SIAM J. Control Optimization 19, 469–478,
(1981).
Acknowledgment in the paper
The author wishes to acknowledge the aid of V. E.
Beneš, who found an error in some preliminary work on
this subject and suggested the applicability of Tanaka’s
formula. He also wishes to thank the referee for pointing
out the uniqueness of the transition density
corresponding to the weak solution in Section 5.
I
Work on stochastic control (monotone follower, bounded
variation follower, finite-fuel, ....)
67 / 94
In the beginning....
S. Shreve, Reflected Brownian motion in the “bang-bang” control
of Browian drift, SIAM J. Control Optimization 19, 469–478,
(1981).
Acknowledgment in the paper
The author wishes to acknowledge the aid of V. E.
Beneš, who found an error in some preliminary work on
this subject and suggested the applicability of Tanaka’s
formula. He also wishes to thank the referee for pointing
out the uniqueness of the transition density
corresponding to the weak solution in Section 5.
I
I
Work on stochastic control (monotone follower, bounded
variation follower, finite-fuel, ....)
Work on optimal investment, consumption and duality with
John Lehoczky, Suresh Sethi, Gan-Lin Xu, Jaksa Cvitanič ....
68 / 94
....and then THE BOOK,
69 / 94
....and then THE BOOK,
70 / 94
and mathematical finance took off.
71 / 94
and mathematical finance took off.
72 / 94
Ten things I learned from Ioannis Karatzas
73 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
74 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
75 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day,
76 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
77 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
78 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata
79 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata (From the Greek ληµµατ α; not commonly used in
West Virginia dialect.)
80 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata (From the Greek ληµµατ α; not commonly used in
West Virginia dialect.)
7. Coördinate
81 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata (From the Greek ληµµατ α; not commonly used in
West Virginia dialect.)
7. Coördinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)
82 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata (From the Greek ληµµατ α; not commonly used in
West Virginia dialect.)
7. Coördinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)
I
Scholarship
83 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata (From the Greek ληµµατ α; not commonly used in
West Virginia dialect.)
7. Coördinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)
I
Scholarship
6. Appreciate the work of others.
84 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata (From the Greek ληµµατ α; not commonly used in
West Virginia dialect.)
7. Coördinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)
I
Scholarship
6. Appreciate the work of others.
5. Revise, revise, revise.
85 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata (From the Greek ληµµατ α; not commonly used in
West Virginia dialect.)
7. Coördinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)
I
Scholarship
6. Appreciate the work of others.
5. Revise, revise, revise.
I
Advising
86 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata (From the Greek ληµµατ α; not commonly used in
West Virginia dialect.)
7. Coördinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)
I
Scholarship
6. Appreciate the work of others.
5. Revise, revise, revise.
I
Advising
4. Choose excellent students.
87 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata (From the Greek ληµµατ α; not commonly used in
West Virginia dialect.)
7. Coördinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)
I
Scholarship
6. Appreciate the work of others.
5. Revise, revise, revise.
I
Advising
4. Choose excellent students.
3. Teach students to appreciate the work of others and to revise,
revise, revise.
88 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata (From the Greek ληµµατ α; not commonly used in
West Virginia dialect.)
7. Coördinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)
I
Scholarship
6. Appreciate the work of others.
5. Revise, revise, revise.
I
Advising
4. Choose excellent students.
3. Teach students to appreciate the work of others and to revise,
revise, revise.
I
Administration
89 / 94
Ten things I learned from Ioannis Karatzas
I
Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I
Spelling
8. Lemmata (From the Greek ληµµατ α; not commonly used in
West Virginia dialect.)
7. Coördinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)
I
Scholarship
6. Appreciate the work of others.
5. Revise, revise, revise.
I
Advising
4. Choose excellent students.
3. Teach students to appreciate the work of others and to revise,
revise, revise.
I
Administration
2. Avoid it.
90 / 94
Number one
91 / 94
Number one
1. A professional colleague who is also a friend is more precious
than Euros/Drachmae.
92 / 94
Number one
1. A professional colleague who is also a friend is more precious
than Euros/Drachmae.
Thank you, Yannis,
93 / 94
Number one
1. A professional colleague who is also a friend is more precious
than Euros/Drachmae.
Thank you, Yannis,
for all you have done
I
for your students,
I
for your colleagues,
I
for science,
I
and for me personally.
94 / 94