Chebyshev Inequalities for Products of Random Variables

1. Chebyshev Inequalities for Products of
Random Variables
Napat RUJEERAPAIBOON
(joint work with D. Kuhn and W. Wiesemann)
Department of Industrial Systems Engineering & Management
National University of Singapore

2. Algebra of Random Variables
Ÿ For random variables ˜ξt p1 ¤ t ¤ Tq, our goal is to compute
Pphp˜ξ1, . . . , ˜ξT q ¤ γq ? (for a given γ)
for predominant choices of the functional hp¤q.
Ÿ Varying γ € p¡V, Vq gives a complete picture of the CDF.
Stock Prices Seismic Hazards Insurances

3. Algebra of Random Variables
Ÿ For random variables ˜ξt p1 ¤ t ¤ Tq, our goal is to compute
Pphp˜ξ1, . . . , ˜ξT q ¤ γq ? (for a given γ)
for predominant choices of the functional hp¤q.
Ÿ Here, we focus on hp˜ξ1, . . . , ˜ξT q ±T
t1
˜ξt .
Stock Prices Seismic Hazards Insurances

4. Products of Random Variables
Theorem 1
For T 2,
Pp˜ξ1
˜ξ2 ¤ γq
» γ
¡V
» V
¡V
f
¢
α,
β
α
1
|α| dα dβ,
where fp¤, ¤q denotes the PDF of p˜ξ1, ˜ξ2q.
Proof. See Rohatgi, V. (1976)
Generalizable to other T ¥ 2.
Easy to derive.
But difficult to compute.

5. Geometric Brownian Motions
Theorem 2
For T 2, ˜ξt i.i.d. and log ˜ξt Npµ, σ2
q,
Pp˜ξ1
˜ξ2 ¤ γq Φ
¢
log γ ¡2µ
c
2σ
,
where Φp¤q denotes CDF of the standard normal r.v.
Proof. log ˜ξ1
˜ξ2 log ˜ξ1 log ˜ξ2 Np2µ, 2σ2
q.
Straightforward extension to other T ¥ 2.
(Log-)Normality assumption may be unjustifiable.
˜ξt ¡ 0 P-a.s. can be overly optimistic!

6. Distributional Ambiguity
Ÿ Consider an asset whose initial price is ˜S0.
˜St 1 ˜St
˜ξt 1, t 0, 1, . . .
Ÿ Total return after T periods is given by
˜ST {˜S0 ±T
t1
˜ξt , T 1, 2, . . .

7. Distributional Ambiguity
Ÿ Consider an asset whose initial price is ˜S0.
˜St 1 ˜St
˜ξt 1, t 0, 1, . . .
Ÿ Total return after T periods is given by
˜ST {˜S0 ±T
t1
˜ξt , T 1, 2, . . .
Ÿ Estimation error: distribution of t˜ξt uT
t1 is not fully known.
Ÿ Bankruptcy condition: ˜ST 0 with non-zero probability.

8. Chebyshev Ambiguity
Roy, A. (1952). Econometrica.
In analyzing economic time series, the mean µ and variance σ2
are
the only quantities that can be distilled out of the past.

9. Chebyshev Ambiguity
Roy, A. (1952). Econometrica.
In analyzing economic time series, the mean µ and variance σ2
are
the only quantities that can be distilled out of the past.
P € Ppµ, σq .
6
98
97
P :
Pp˜ξt ¥ 0q 1 dt ¤ T
EPp˜ξt q µ dt ¤ T
EPp˜ξs
˜ξt q δst σ2
µ2
ds, t ¤ T
D
GF
GE
Ÿ The ambiguity set Ppµ, σq was first studied by Chebyshev.
Ÿ Later extended by Marshall and Olkin, El Ghaoui et al. and
Vandenberghe et al., etc.

10. Chebyshev Ambiguity
Roy, A. (1952). Econometrica.
In analyzing economic time series, the mean µ and variance σ2
are
the only quantities that can be distilled out of the past.
P € Ppµ, σq .
6
98
97
P :
Pp˜ξt ¥ 0q 1 dt ¤ T
EPp˜ξt q µ dt ¤ T
EPp˜ξs
˜ξt q δst σ2
µ2
ds, t ¤ T
D
GF
GE
Ÿ The ambiguity set Ppµ, σq was first studied by Chebyshev.
Ÿ Later extended by Marshall and Olkin, El Ghaoui et al. and
Vandenberghe et al., etc.
Support information remains a challenge!

11. Ruin Probability I
Theorem 3 (Chebyshev)
If T ¡
µ
σ
¨2
2, then
sup
P€Ppµ,σq
Pp˜ST 0q 1,
where Ppµ, σq is the Chebyshev ambiguity set.
Interpretation: Companies have a finite lifespan.
Proof:
1 Define supp0
pPq
3
pξ1, . . . , ξT q € supppPq :
±T
t1 ξt 0
A
.
% It suffices to identify P s.t. supppPq supp0
pPq.

12. Ruin Probability I
Theorem 3 (Chebyshev)
If T ¡
µ
σ
¨2
2, then
sup
P€Ppµ,σq
Pp˜ST 0q 1,
where Ppµ, σq is the Chebyshev ambiguity set.
Interpretation: Companies have a finite lifespan.
Proof:
1 Define supp0
pPq
3
pξ1, . . . , ξT q € supppPq :
±T
t1 ξt 0
A
.
2 Construct a mapping T : Ppµ, σq Ñ Ppµ, σq s.t.
|supppT pPqq| |supppPq| and |supp0
pT pPqq| ¡ |supp0
pPq|.

13. Ruin Probability I
Theorem 3 (Chebyshev)
If T ¡
µ
σ
¨2
2, then
sup
P€Ppµ,σq
Pp˜ST 0q 1,
where Ppµ, σq is the Chebyshev ambiguity set.
Interpretation: Companies have a finite lifespan.
Proof:
1 Define supp0
pPq
3
pξ1, . . . , ξT q € supppPq :
±T
t1 ξt 0
A
.
2 Construct a mapping T : Ppµ, σq Ñ Ppµ, σq s.t.
|supppT pPqq| |supppPq| and |supp0
pT pPqq| ¡ |supp0
pPq|.
3 The proof completes as there is a P s.t. |supppPq| V.

14. Ruin Probability II
Theorem 4 (Chebyshev + I.I.D.)
For any T ¥ 1,
sup
P€Qpµ,σq
Pp˜ST 0q 1 ¡
¢
µ2
µ2 σ2
T
,
where Qpµ, σq
2
QT
: QT
€ Ppµ, σq
@
.
Interpretation: Nothing lasts forever.
Proof:
1 Write Pp˜ST 0q 1 ¡±T
t1 Pp˜ξt ¡ 0q 1 ¡p1 ¡Qpξ 0qqT
.
2 Derive a tight upper bound for Qpξ 0q.

15. Ruin Probability II
Theorem 4 (Chebyshev + I.I.D.)
For any T ¥ 1,
sup
P€Qpµ,σq
Pp˜ST 0q 1 ¡
¢
µ2
µ2 σ2
T
,
where Qpµ, σq
2
QT
: QT
€ Ppµ, σq
@
.
Interpretation: Nothing lasts forever.
Proof:
The Cauchy-Schwarz inequality implies
»
R
1 dQpξq¤
»
R
ξ2
dQpξq ¥
£»
R
ξ dQpξq
2
,
which in turns gives a tight upper bound on Qpξ 0q.

16. Ruin Probability II
Theorem 4 (Chebyshev + I.I.D.)
For any T ¥ 1,
sup
P€Qpµ,σq
Pp˜ST 0q 1 ¡
¢
µ2
µ2 σ2
T
,
where Qpµ, σq
2
QT
: QT
€ Ppµ, σq
@
.
Interpretation: Nothing lasts forever.
Proof:
The Cauchy-Schwarz inequality implies
p1 ¡Qpξ 0qq¤
»
R
ξ2
dQpξq ¥
£»
R
ξ dQpξq
2
,
which in turns gives a tight upper bound on Qpξ 0q.

17. Ruin Probability II
Theorem 4 (Chebyshev + I.I.D.)
For any T ¥ 1,
sup
P€Qpµ,σq
Pp˜ST 0q 1 ¡
¢
µ2
µ2 σ2
T
,
where Qpµ, σq
2
QT
: QT
€ Ppµ, σq
@
.
Interpretation: Nothing lasts forever.
Proof:
The Cauchy-Schwarz inequality implies
p1 ¡Qpξ 0qq¤pµ2
σ2
q ¥
£»
R
ξ dQpξq
2
,
which in turns gives a tight upper bound on Qpξ 0q.

18. Ruin Probability II
Theorem 4 (Chebyshev + I.I.D.)
For any T ¥ 1,
sup
P€Qpµ,σq
Pp˜ST 0q 1 ¡
¢
µ2
µ2 σ2
T
,
where Qpµ, σq
2
QT
: QT
€ Ppµ, σq
@
.
Interpretation: Nothing lasts forever.
Proof:
The Cauchy-Schwarz inequality implies
p1 ¡Qpξ 0qq¤pµ2
σ2
q ¥ µ2
,
which in turns gives a tight upper bound on Qpξ 0q.

19. Ruin Probability II
Theorem 4 (Chebyshev + I.I.D.)
For any T ¥ 1,
sup
P€Qpµ,σq
Pp˜ST 0q 1 ¡
¢
µ2
µ2 σ2
T
,
where Qpµ, σq
2
QT
: QT
€ Ppµ, σq
@
.
Interpretation: Nothing lasts forever.
Proof:
1 Write Pp˜ST 0q 1 ¡±T
t1 Pp˜ξt ¡ 0q 1 ¡p1 ¡Qpξ 0qqT
.
2 Derive a tight upper bound for Qpξ 0q, i.e., σ2
{pµ2
σ2
q.
3 Combine the results from the previous steps.

20. Ruin Probability III
Remark 1 (Geometric Brownian Motion)
For any T V,
Pp˜ST 0q 0,
under the Black-Scholes economy.
Interpretation: Booming economy. If µ4
¡ µ2
σ2
, we even have
lim
TÑV
Pp˜ST ¤ ˜S0q 0.
Proof: Every ˜ξt is strictly positive with probability 1.

21. Chebyshev is a realist!
Ÿ Arguably, we have shown 3 different viewpoints of economy.
Chebyshev Chebyshev + I.I.D. Black-Scholes

22. Chebyshev is a realist!
Ÿ Arguably, we have shown 3 different viewpoints of economy.
Chebyshev Chebyshev + I.I.D. Black-Scholes
Optimist

23. Chebyshev is a realist!
Ÿ Arguably, we have shown 3 different viewpoints of economy.
Chebyshev Chebyshev + I.I.D.
Romantic
Black-Scholes
Optimist

24. Chebyshev is a realist!
Ÿ Arguably, we have shown 3 different viewpoints of economy.
Chebyshev
Realist
Chebyshev + I.I.D.
Romantic
Black-Scholes
Optimist

25. Chebyshev is a realist!
Ÿ “(Unrealistic) optimism leads people to attribute the wrong
probabilities to events.”—Coelho, M. (2010)
Chebyshev
Realist
Chebyshev + I.I.D.
Romantic
Black-Scholes
Optimist
Ÿ Chebyshev is a realist (and at most a romantic), so am I!

26. Bounds for CDFs
Chebyshev (1821–1894)
Ÿ Upper bound for the CDF.
Upγq .
sup
P€Ppµ,σq
P
¡±T
t1
˜ξt ¤ γ
©
Ÿ Lower bound for the CDF.
Lpγq .
inf
P€Ppµ,σq
P
¡±T
t1
˜ξt ¤ γ
©

27. Computing Upγq
Theorem 5
For any γ ¡ 0, Upγq can be computed from a finite SDP.
Proof:
Step 1: Formulate a moment problem representing supP.
Upγq sup
»
ξ€RT
1±T
t1 ξt ¤γpξqPpdξq
s. t.
»
ξ€RT
1Ppdξq 1
»
ξ€RT
ξPpdξq µ1
»
ξ€RT
ξξ Ppdξq σ2
I µ2
11

28. Computing Upγq
Theorem 5
For any γ ¡ 0, Upγq can be computed from a finite SDP.
Proof:
Step 2: Leverage LP duality.
Upγq inf α µβ 1 Γ, σ2
I µ2
11
s. t. α β ξ Γ, ξξ ¥ 1±T
t1 ξt ¤γpξq dξ ¥ 0
(N.B. Strong duality holds as long as µ, σ ¡ 0.)

29. Computing Upγq
Theorem 5
For any γ ¡ 0, Upγq can be computed from a finite SDP.
Proof:
Step 3: Exploit the symmetry of the objective function and target set.
Upγq inf α β pµ1q Γ, σ2
I µ2
11
s. t. α β ξ Γ, ξξ ¥ 1±T
t1 ξt ¤γpξq dξ ¥ 0

30. Computing Upγq
Theorem 5
For any γ ¡ 0, Upγq can be computed from a finite SDP.
Proof:
Step 3:
Γ
Permuted Γs
Average Γ

31. Computing Upγq
Theorem 5
For any γ ¡ 0, Upγq can be computed from a finite SDP.
Proof:
Step 3: W.l.o.g, add strucural constraints.
Upγq inf α µβ 1 Γ, σ2
I µ2
11
s. t. β β1, Γ γ1I γ211
α β ξ Γ, ξξ ¥ 1±T
t1 ξt ¤γpξq dξ ¥ 0

32. Computing Upγq
Theorem 5
For any γ ¡ 0, Upγq can be computed from a finite SDP.
Proof:
Step 4: Simplify the semi-infinite constraint.
α β ξ Γ, ξξ ¥ 1±T
t1 ξt ¤γpξq dξ ¥ 0
α βs γ2s2
γ1}ξ}2
2 ¥ 1±T
t1 ξt ¤γpξq ds ¥ 0 dξ ¥ 0 : }ξ}1 s
Put differently, we seek the worst-case ps, ξpsqq.
6
8
7
infs¥0 α βs γ2s2
γ1rinf |supsξ¥0:}ξ}1s}ξ}2
2 ¥ 0
infs¥0 α βs γ2s2
γ1rinf |supsξ¥0:}ξ}1s,
±T
t1 ξt ¤γ}ξ}2
2 ¥ 1

33. Computing Upγq
Theorem 5
For any γ ¡ 0, Upγq can be computed from a finite SDP.
Proof:
Step 4: Simplify the semi-infinite constraint.
α β ξ Γ, ξξ ¥ 1±T
t1 ξt ¤γpξq dξ ¥ 0
α βs γ2s2
γ1}ξ}2
2 ¥ 1±T
t1 ξt ¤γpξq ds ¥ 0 dξ ¥ 0 : }ξ}1 s
Put differently, we seek the worst-case ps, ξpsqq.
6
8
7
infs¥0 α βs γ2s2
γ1rT¡1
s2
|s2
s ¥ 0
infs¥0 α βs γ2s2
γ1rinf |supsξ¥0:}ξ}1s,
±T
t1 ξt ¤γ}ξ}2
2 ¥ 1

34. Computing Upγq
Theorem 5
For any γ ¡ 0, Upγq can be computed from a finite SDP.
Proof:
Step 4: Simplify the semi-infinite constraint.
α β ξ Γ, ξξ ¥ 1±T
t1 ξt ¤γpξq dξ ¥ 0
α βs γ2s2
γ1}ξ}2
2 ¥ 1±T
t1 ξt ¤γpξq ds ¥ 0 dξ ¥ 0 : }ξ}1 s
Put differently, we seek the worst-case ps, ξpsqq.
6
8
7
infs¥0 α βs γ2s2
γ1rT¡1
s2
|s2
s ¥ 0
infs¥0 α βs γ2s2
γ1rinf |s2
sξ¥0:}ξ}1s,
±T
t1 ξt ¤γ}ξ}2
2 ¥ 1

35. Computing Upγq
Theorem 5
For any γ ¡ 0, Upγq can be computed from a finite SDP.
Proof:
Step 5: 3 out of 4 cases reduce to univariate polynomial optimization.
α β ξ Γ, ξξ ¥ 1±T
t1 ξt ¤γpξq dξ ¥ 0
α βs γ2s2
γ1}ξ}2
2 ¥ 1±T
t1 ξt ¤γpξq ds ¥ 0 dξ ¥ 0 : }ξ}1 s
N.B. these are SDP-representable (Nesterov, Y. 2000).
6
8
7
infs¥0 α βs γ2s2
γ1r|s ¥ 0
infs¥0 α βs γ2s2
γ1rinf |sξ¥0:}ξ}1s,
±T
t1 ξt ¤γ}ξ}2
2 ¥ 1

36. Auxiliary Lemma
Lemma 1
The non-convex optimization problem
inf
ξ¥0
3
}ξ}2
2 : }ξ}1 s,
±T
t1 ξt ¤ γ
A
is solved by ξ pξ, ξ, . . . , ξq:
(i) ξ ξ s{T if γ ¥ ps{TqT
and (ii) ξ pT ¡1qξ s, ξξT¡1
γ o/w.
The remaining case from Step 5 reduces to a
Ÿ univariate polynomial optimization, if γ ¥ ps{TqT
.
Ÿ trivariate polynomial optimization with two ‘=’ constraints, o/w.

37. Computing Upγq
Theorem 5
For any γ ¡ 0, Upγq can be computed from a finite SDP.
Proof:
Step 5: All cases reduce to univariate polynomial optimization.
α β ξ Γ, ξξ ¥ 1±T
t1 ξt ¤γpξq dξ ¥ 0
α βs γ2s2
γ1}ξ}2
2 ¥ 1±T
t1 ξt ¤γpξq ds ¥ 0 dξ ¥ 0 : }ξ}1 s
N.B. these are SDP-representable (Nesterov, Y. 2000).
6
8
7
infs¥0 α βs γ2s2
γ1r|s ¥ 0
infs¥0 α βs γ2s2
γ1r|s ¥ 1

38. Computing Lpγq
Theorem 6
For any γ ¡ 0, Lpγq can be computed from a finite SDP.
Proof:
Lpγq inf
P€Ppµ,σq
P
¡±T
t1
˜ξt ¤ γ
©
1 Formulate Lpγq as a semi-infinite linear program and dualize.
2 Add structural constraints on the dual variables.
3 Reduce dual constraints to univariate polynomial optimization.

39. Application to Portfolio Selection
Corollary 1
Let ν and Σ denote the mean vector and the covariance matrix of the
asset return distribution. Any portfolio w solving
max
w€W,γ
γ
s.t. P
¡±T
t1
˜ξt ¤ γ
©
¤ dP € Ppw ν,
c
w Σwq
is mean-variance efficient.
Interpretation: Find a portfolio w whose total return falls below γ
with probability at most À 15%.
Ÿ Generalize exact single-period results (El Ghaoui et al. 2003).
Ÿ Generalize approx multi-period results (R. et al. 2016).
Markowitz, H. (1952)

40. Conclusions
Ÿ Chebyshev ambiguity “companies have a finite lifespan”.
Ÿ Chebyshev bounds Lpγq and Upγq can be computed from SDPs.
Ÿ Proof techniques from this talk lay the groundwork for:
Ÿ Pairwise correlations among tξt uT
t1.
Ÿ Imprecise variances.
Ÿ Other functionals hp¤q e.g. sums, maxima and minima.

41. References
Ÿ El Ghaoui, L., Oks, M. and Oustry, F.
Worst-case value-at-risk and robust portfolio optimization.
Operations Research, 2003.
Ÿ Marshall, A. and Olkin, I.
Multivariate Chebyshev Inequalities.
Annals of Mathematical Statistics, 1960.
Ÿ Rujeerapaiboon, N., Kuhn, D. and Wiesemann, W.
Chebyshev Inequalities for Products of Random Variables.
Mathematics of Operations Research, 2018.
Ÿ Vandenberghe, L., Boyd, S. and Comanor, K.
Generalized Chebyshev bounds via semidefinite programming.
SIAM Review, 2007.
Appreciate the artworks from tRoundicons, Smashicons, Dimitry
Miroliubov, Freepiku@flaticon.com
napat.rujeerapaiboon@nus.edu.sg

42. Proof of Theorem 1
We let gp¤q denote the PDF of ˜ξ1
˜ξ2 and write
Pp˜ξ1
˜ξ2 ¤ γq
» V
0
» γ{α
¡V
fpα, βqdβ dα
» 0
¡V
» V
γ{α
fpα, βqdβ dα.
Taking a derivative w.r.t. γ and applying Fundamental Theorem of
Calculus yields
gpγq
» V
0
fpα, γ{αq1
α
dα
» 0
¡V
¡fpα, γ{αq1
α
dα
» V
¡V
fpα, γ{αq 1
|α| dα.
Finally, we can recover the CDF of ˜ξ1
˜ξ2 from the PDF gp¤q.