Random surface description of capital distribution
Mykhaylo Shkolnikov
Princeton University
June 29, 2016
Outline
1
Motivation
2
Starting point: rank-based models
3
Law of large numbers
4
Central Limit Theorem
5
Some ideas from proof
Motivation
Consider an institutional investor holding an equity portfolio.
Typical portfolios include hundreds or thousands of stocks, e.g. of
S&P 500 or Russell 3000 companies.
Changes in portfolio weights triggered by characteristics of large
market, e.g. diversity or more general portfolio generating functions.
If one starts with a typical semimartingale market model, behavior of
such characteristics hard to understand due to high-dimensionality.
Our goal: exploit high-dimensionality to arrive at a tractable
infinite-dimensional model =⇒ random surface models.
1e!08
1e!06
1e!04
1e!02
Capital distribution in U.S.
1
5
10
50
100
500
1000
5000
Figure 1: Capital distribution curves: 1929–1999
cesses represented by continuous semimartingales (see, e.g., Duffie (1992) or Karatzas and Shreve
Different (1998)).
viewpoint:
remove
normalization,
combine
to a surface.
The representation
of market
weights in terms of continuous
semimartingales
is straightforward, but in order to represent the ranked market weights, it is necessary to use semimartingale
local times to capture the behavior when ranks change. The methodology for this analysis was
Towards the random surface: rank-based models
Fernholz, Karatzas ’08: pick coefficients b1 , . . . , bN , σ1 , . . . , σN
and model logarithmic market capitalizations by
dXi (t) =
N
X
k=1
1{Xi (t)=X(k) (t)} bk dt +
N
X
1{Xi (t)=X(k) (t)} σk dBi (t).
k=1
with X(1) (t) ≤ · · · ≤ X(N) (t) order statistics of (X1 (t), . . . , XN (t)),
B1 , . . . , BN are independent standard Brownian motions.
Simplest models to capture capital distribution figure: Banner,
Fernholz, Karatzas ’05, Chatterjee, Pal ’11, Ichiba, Pal, S. ’13.
Mean-field reformulation
Simple observation: pick functions b : [0, 1] → R, σ : [0, 1] → R+ such
that
b(k/N) = bk ,
σ(k/N) = σk ,
k = 1, . . . , N.
Then, rank-based model can be rewritten as
dXi (t) = b(F%(N) (t) (Xi (t))) dt + σ(F%(N) (t) (Xi (t))) dBi (t), i = 1, . . . , N
where %(N) (t) =
1
N
PN
i=1 δXi (t)
empirical measure and F%(N) (t) its
cumulative distribution function.
Suggests: large N behavior described by LLN and CLT.
Previous results for mean-field interacting diffusions
Law of large numbers. Gärtner ’88: continuous coefficients, before
Leonard ’86, Oelschläger ’84 under more assumptions.
Central limit theorems. Tanaka ’84: smooth drift, constant diffusion
coefficient, Sznitman ’85: no drift, smooth diffusion coefficient. See
also Oelschläger ’84.
Large deviations. Dawson, Gärtner ’87: continuous drift, constant
diffusion coefficient, Budhiraja, Dupuis, Fisher ’12: implicit
assumptions.
Note: none of these apply to rank-based models! In fact: CLT and
LDP hold for entirely different reasons!
Law of large numbers
Theorem (S. ’12, Jourdain, Reygner ’13). In a rank-based model
dXi (t) = b(F%(N) (t) (Xi (t))) dt + σ(F%(N) (t) (Xi (t))) dBi (t), i = 1, . . . , N
with b, σ continuous, %(N) (0) → λ it holds %(N) (·) → %(·) in
C ([0, ∞), M1 (R)) where R(t, x) := F%(t) (x) is the unique (generalized)
solution of
Rt = B(R)x + Σ(R)xx ,
and B(r ) =
Rr
0
b(a) da, Σ(r ) =
Rr
0
σ(a)2
2
R(0, ·) = Fλ (·)
da.
Porous medium equation: see e.g. Vazquez ’06.
Derivation of the porous medium equation
Pick test function f ∈ Cc∞ (R), use Itô:
N
(N)
1 X
% (t), f − %(N) (0), f = d
f (Xi (t)) − f (Xi (0))
N
i=1
N Z t
X
σ(F%(N) (Xi ))2 00
1
=
b(F%(N) (Xi )) f 0 (Xi ) +
f (Xi ) ds
N
2
i=1 0
N Z
1 X t
+
σ(F%(N) (Xi )) f 0 (Xi ) dBi
N
i=1 0
Z tD
σ(F%(N) )2 00 E
f ds + Mart.
=
%(N) , b(F%(N) )f 0 +
2
0
Porous medium equation cont.
If %(N) (·) → %(·), then expect BV part to converge to:
Z tD
σ(F% )2 00 E
%, b(F% )f 0 +
f ds.
2
0
Martingale −→ 0, so limiting equation:
Z tD
σ(F% )2 00 E
%(t), f − %(0), f =
%, b(F% )f 0 +
f ds.
2
0
Integrate by parts, recall R(t, x) = F%(t) (x):
R(t, ·), f 0 − R(0, ·), f 0 =
Z
0
So: Rt = B(R)x + Σ(R)xx .
t
B(R), f 00 + Σ(R), f 000 ds.
Towards Central Limit Theorem
LLN shows:
F%(N) (t) (x) = R(t, x) + o(1),
where o(1) is uniform on [0, T ] × R.
R is deterministic ⇒ approximation cannot be good enough!
So, need the next term in the expansion!
Turns out: o(1) = O(N −1/2 ) [⇔ quantitative propagation of chaos].
So, expect
F%(N) (t) (x) = R(t, x) + N −1/2 G (t, x) + o(N −1/2 ),
where G is a Gaussian object (surface!).
Main result: Central Limit Theorem
Theorem (Kolli, S. ’16). In a rank-based model
dXi (t) = b(F%(N) (t) (Xi (t))) dt + σ(F%(N) (t) (Xi (t))) dBi (t), i = 1, . . . , N
with b, σ in C 1+ε , X1 (0), . . . , XN (0) i.i.d. with bounded density and > 2
moments, the fluctuation processes
GN (t, x) = N 1/2 (F%(N) (t) (x) − R(t, x)),
N∈N
converge in C ([0, ∞), Mfin (R)) to the mild solution of SPDE:
Central Limit Theorem cont.
Gt = (b(R)G )x +
σ(R)2
2 G xx
1/2
+ σ(R) Rx
Ẇ ,
G (0, ·) = β(Fλ (·)).
Here: Ẇ is space-time white noise and β is an independent standard
Brownian bridge.
More specifically:
Z tZ
G (t, x) =
0
σ(R(s, y )) Rx (s, y )1/2 p(s, y ; t, x) dW (s, y )
R
Z
+ β(Fλ (y )) p(0, y ; t, x) dy
R
where p(s, y ; t, x) is transition density of solution to
dX̄ (t) = b(R(t, X̄ (t))) dt + σ(R(t, X̄ (t))) dB(t).
Conclusions from theorem
G (t, x), (t, x) ∈ [0, ∞) × R is a Gaussian process.
G admits a realization which is Hölder (1/2 − δ)-continuous in x and
Hölder (1/4 − δ)-continuous in t.
Obtained refined expansion:
(t, x) 7→ F%(N) (t) (x) = R(t, x) + N −1/2 G (t, x) + o(N −1/2 )
(t, x) 7→ R(t, x) is the deterministic surface of means,
(t, x) 7→ N −1/2 G (t, x) is superimposed mean-zero random surface.
Intractable system of thousands of SDEs → 1 PDE & 1 SPDE.
Some ideas from proof
Key building block: quantitative propagation of chaos.
Consider the decoupled SDEs:
dX̄i (t) = b(R(t, X̄i (t))) dt + σ(R(t, X̄i (t))) dBi (t), 1, . . . , N
with same Brownian motions as in rank-based model.
Can show:
N
i1/p
1 X h
E sup |Xi (t) − X̄i (t)|p
≤ C N −1/2 .
N
0≤t≤T
i=1
In particular:
h
i
E sup Wp (%(N) (t), %̄(N) (t)) ≤ C N −1/2 .
0≤t≤T
Some ideas from proof cont.
Here: Wp is p-Wasserstein distance, i.e.
(N)
Wp (%
(N)
(t), %̄
N
p
1 X X(k) (t) − X̄(k) (t) .
(t)) =
N
p
k=1
Why useful? Alternative representation:
Z
(N)
(N)
F (N) (x) − F (N) (x) dx.
W1 (% (t), %̄ (t)) =
% (t)
%̄ (t)
R
So, can replace F%(N) (t) by F%̄(N) (t) in e.g.
Z
Z √
|GN (t, x)| dx =
N F%(N) (t) (x) − R(t, x) dx
R
R
=⇒ . . . (Bobkov, Ledoux ’14) tightness of GN , N ∈ N.
Some ideas from proof cont.: derivation of SPDE
Set b ≡ 0, recall McKean-Vlasov and PME equations:
Z tD
σ(F% )2 00 E
%(t), f = %(0), f +
f ds,
%,
2
0
Z t
R(t, ·), g = R(0, ·), g +
Σ(R), g 00 ds.
0
Prelimit versions:
Z tD
σ(F%(N) )2 00 E
%(N) ,
f ds + Mart.,
2
0
Z t
F%(N) (t) , g = F%(N) (0) , g +
ΣN (F%(N) ), g 00 + Mart.
%(N) (t), f = %(N) (0), f +
0
Some ideas from proof cont.: derivation of SPDE
First PME from prelimit PME, multiply by
√
N, fundamental theorem
of calculus:
GN (t), g = GN (0), g +
Z t DZ
0
1
0
E
Σ0 ((1 − r )R + rF%(N) ) dr GN , g 00 ds
√ Z t
+ N
ΣN (F%(N) ) − Σ(F%(N) ), g 00 ds + Mart.
0
Take N → ∞:
G (t), g = G (0), g +
Z
t
0
Σ (R) G , g 00 ds + Mart.
0
⇒ G generalized solution of SPDE in theorem.
Some other ingredients
Gilding ’89: gradient estimate for PME.
Krylov ’07: solution theory for linear parabolic equations with VMO
coefficients.
Krylov ’80: Itô formula for Wp1,2 functions.
Aronson ’68: heat kernel upper bound for X̄i ’s.
Bobkov, Ledoux ’14: various estimates on Wasserstein distances
between empirical measures of i.i.d. rv’s and their distribution.
del Barrio, Gine, Matran ’99: functional CLTs for empirical
measures of i.i.d. rv’s.
Rosenthal ’70: inequalities for moments of normalized binomials.
THANK YOU
FOR YOUR ATTENTION!