Continuous expansion of a filtration with a stochastic
process: the information drift
Leo Neufcourt
Michigan State University
February 26, 2020
Seminar of Financial and Actuarial Mathematics
University of Michigan
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 1 / 41
Introduction
Collaborators : Philip Protter, Dan Wang (Columbia U), Ruihua Ruan, Luyi
Shen (Ecole Polytechnique)
L. N., PhD Thesis, Columbia University (2017)
L. N. & P. Protter, Expansion of a filtration with a stochastic process: the
information drift, Submitted (2019)
L. N., R. Ruan & L. Shen, A strict local martingale study of the bitcoin
bubble, Preprint (2019)
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 2 / 41
Roadmap
1 A suggestive example from Ito
2 Semimartingales, filtrations and arbitrage
3 Expansion of a filtration with a stochastic process
4 Strict local martingales
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 3 / 41
Market model
We consider
a financial market described by a complete probability space (Ω,A,P)
(i.e. N P ∈ A)
a price process W given by a P-(standard) Brownian motion
F its natural (completed) filtration (which is here right continuous):
Ft := σ(Wu, u ≤ t)
G the filtration expanded with W1, i.e.
Gt := Ft ∨ σ(W1)
(which is here complete and right-continuous)
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 4 / 41
Suggestive example: information drift
Proposition
Let αt := W1−Wt
1−t . Then the process
Wt := Wt −∫ t
0
αudu
defines a G-Brownian motion.
1
E∫ 1
0
|W1 −Wu|1− u
du =
√2
π
∫ 1
0
du√1− u
<∞
2
E[Wt −Ws |Gs ] = E[Wt −Ws |Fs ,W1] = E[Wt −Ws |Fs ,W1 −Ws ]
= E[Wt −Ws |W1 −Ws ] =t − s
1− s(W1 −Ws)
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 5 / 41
Suggestive example: information drift II
3
E[
∫ t
s
W1 −Wu
1− udu|Gs ] =
∫ t
s
E[W1 −Wu|Gs ]
1− udu
=
∫ t
s
E[W1 −Ws |Gs ]− E[Wu −Ws |Gs ]
1− udu
=
∫ t
s
(W1 −Ws) + u−s1−s (W1 −Ws)
1− udu
= (W1 −Ws)
∫ t
s
du
1− u
=t − s
1− s(W1 −Ws)
3 And we conclude from the Levy characterization.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 6 / 41
Suggestive example: risk neutral probability
From the perspective of the filtration G, the price process W is the Ito process
dWt := dWt + αtdt
Can this drift be canceled by an equivalent change of probability? Recall that
there is no arbitrage opportunities (NFLVR) if and only if there exist a risk neutral
probability.
[Girsanov] The change of probability candidate is the stochastic exponential
dQdP
= Zt := E(
∫ t
0
αudWu) := exp(
∫ t
0
αudWu −1
2
∫ t
0
α2udu)
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 7 / 41
Suggestive example: risk neutral probability II
In general Zt := E(∫ t
0αtdWt) is a local martingale (whence supermartingale). It
is a true martingale on [0, t] if and only if E[Zt ] = 1, e.g. under Novikov’s
criterion Ee∫ t
0α2
udu <∞. Here,∫ t
0
α2udu =
∫ 1
1−t
W ′2uu2
du
where W ′1−t = W1 −Wt is a new BM.
When t < 1,∫ t
0α2udu ≤ 1
1−t∫ 1
0W 2
u du which has a finite Laplace transform
(e.g. Erdos and Kac). Thus Z is a true martingale on [0, t].
When t = 1, Ee∫ 1
0α2
udu ≤ eE∫ 1
0α2
udu = e∫ 1
0duu =∞ so Novikov criterion cannot
hold. [LIL] When t → 0, supu>tW 2
t
t2 ∼ 2t log log 1
t which is not integrable at 0.
Is it surprising ? There is an obvious G-arbitrage on [0, 1]...
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 8 / 41
Roadmap
1 A suggestive example from Ito
2 Semimartingales, filtrations and arbitrage
3 Expansion of a filtration with a stochastic process
4 Strict local martingales
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 9 / 41
Market model
We consider
a financial market described by a complete probability space (Ω,A,P)
a public filtration (Ft)t∈[0,T ] satisfying the usual hypotheses
a price process (St)t∈[0,T ] given by a F-semimartingale
an expansion expansion Gt := Ft ∨Ht of F
F-semimartingale : St = S0 + Mt + At , with
M : F-local martingale, i.e. E[Mt |Fs ] = Ms , up to an increasing sequence of
stopping time (“volatility”)
A : process of finite variation, i.e.∫ T
0|dAt | <∞ (“drift”)
A portfolio is characterized by an admissible 1 strategy H and has at any time a
value Vt := V0 +∫ t
0HsdSs .
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 10 / 41
Absence of arbitrage
No Arbitrage (NA) means that there is no admissible strategy with terminal
value V such that V ≥ 0 and P(V > 0) > 0.
No Free Lunch with Vanishing Risk (NFLVR) means that no sequence of
portfolios (Hn)n with value (V n)n dominated at all times by a sequence of
admissible portfolio can converge in probability to an “arbitrage” value V
such that V ≥ 0 and P(V > 0) > 0.
Theorem (Delbaen & Schachermayer, 1994)
NFLVR holds if and only if there exist an Equivalent Local Martingale Measure
(ELMM), or risk neutral probability, i.e. a probability Q ∼ P under which S is a
local martingale, i.e. of a positive and uniformly integrable martingale Z such that
ZS is a P-local martingale
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 11 / 41
No arbitrage of the first kind
NA1 or NUPBR is a weaker form of no arbitrage.
Theorem (Kardaras, Fontana)
NA1 is equivalent to the existence of a local martingale deflator, i.e. a positive
local martingale Z such that ZS is a local martingale. Additionally, NFLVR ⇐⇒(NA and NA1).
NA1 also means that any non-trivial contingent claim ξ ≥ 0 has a strictly positive
superhedging value (minimal initial wealth for which there exist a superhedging
strategy)
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 12 / 41
Characterizing the existence of the information drift
Theorem
NA1 holds for G if and only if there exists a process α ∈ S(G) such that M is a
G-semimartingale with decomposition
M =: M +
∫ .
0
αsd [M,M]s .
α is called the information drift of M (between the filtrations F and G).
A fortiori, NFLVR for ”the insider” requires existence of the information drift; it
can be seen as a general non-degeneracy condition.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 13 / 41
Information drift of Brownian motion
Theorem
Let W be an F-Brownian motion. If there exists a process α ∈H 1(G) such that
W −∫ .
0αsds is a G-Brownian motion, then we have:
(i) αs = limt→st>s
E[Wt−Ws
t−s |Gs ]
(ii) αs = ∂∂tE[Wt |Gs ]
∣∣∣t=s
.
Conversely if there exists a process α ∈H 1(G) satisfying (i) or (ii), then and α is
the information drift of W , i.e. W −∫ .
0αsds is a G-Brownian motion.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 14 / 41
Proof.
Since W is G-adapted it is clear that (i) and (ii) are equivalent. Suppose that
M −∫ .
0αudu is a G-martingale. It follows that, for every s ≤ t,
E[Wt −Ws −
∫ t
sαudu|Gs
]= 0, hence
E[
∫ t
s
αudu|Gs]
=
∫ t
s
E[αu|Gs ]du.
By differentiating with respect to t we obtain E[αt |Gs ] = ∂∂tE[Wt |Gs ] which
establishes (ii). Conversely if (ii) holds it is also clear that
E[Wt −Ws −∫ t
s
αudu|Gs ] = E[Wt −Ws − (E[Wt |Gt ]− E[Ws |Gs ])|Gs ] = 0.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 15 / 41
Statistical arbitrage: a hard cash interpretation
We consider a market agent with initial wealth x and admissible strategies
H ∈ S : x + H ·M ≥ 0.
Example 1 : maximization of logarithmic utility
Let u(x ,Y) := supE log(x + H ·M)T s.t. H ∈ S(Y) and x + H ·M ≥ 0.
Proposition (Karatzas & Pikovsky, 1993 / Ankirchner & al, 2004)
u(x ,G)− u(x ,F) = E∫ T
0
α2sd [M,M]s
Example 2 : constrained maximization of returns
vλ(x ,Y) := sup x +E(H ·M)T −λVar(H ·M)T s.t. H ∈ S(Y) and x +H ·M ≥ 0.
Proposition (2017)
vλ(x ,G)− vλ(x ,F) = E∫ T
0
α2s
4λd [M,M]s
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 16 / 41
Roadmap
1 A suggestive example from Ito
2 Semimartingales, filtrations and arbitrage
3 Expansion of a filtration with a stochastic process
4 Strict local martingales
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 17 / 41
Jacod’s condition
Definition (Jacod’s condition)
A random variable L satisfies Jacod’s condition if for a.e. t ∈ I there exists a
(non-random) σ-finite measure ηt (which can be taken invariant with time or P0)
that dominates its conditional distribution Pt , i.e.
Pt(ω, L ∈ .) ηt(.) a.s.
Proposition (Jacod)
Under Jacod’s condition we can define qLt (ω, x) := dPt(ω,.)dη(.)
∣∣∣∣σ(L)
(L)−1(x) which
satisfies qLt (., L(.)) > 0, dP× dPt-a.s.
In other words, Jacod’s condition means that the knowledge of Ft does not
impact on whether events involving L are possible or not.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 18 / 41
Initial expansions
Gt :=⋂
u>t
(Fu ∨ σ(L)
)Theorem (Jacod)
Suppose that L satisfies Jacod’s condition. Then:
(i) every F-semimartingale is a G-semimartingale, and
(ii) a continuous F-local martingale M has a decomposition
M = M +
∫ .
0
γs(ω, L(ω))d [M,M]s
where M is a G-local martingale and γ is a G-predictable process with
marginals
γt(., x) =1
qt(., x)
d [q(., x),M]td [M,M]t
.
Examples: L = W1, L = supt≤T Wt [Aksamit & Jeanblanc]
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 19 / 41
Successive initial expansions
Theorem (Kchia, Larsson & Protter)
Let M be a continuous F-local martingale and suppose that L := (L1, ..., Ln)
satisfies Jacod’s condition. Let 0 =: τ0 < τ1 < ... < τn < τn+1 := T be an
increasing sequence of (fixed) times and
Gnt :=⋂u>t
(Fu ∨ σ(Lk1t≥τk , k = 1...n)).
(i) M is a Gn-semimartingale, and
(ii) M has decomposition M =: M +∫ .
0αns d [M,M]s , where
αns (ω) :=
n∑k=1
1τk≤s≤τk+1
1
qk,ns− (ω, .)
(d [qk,n(ω, .),M]s
d [M,M]s
)(Lk(ω)).
The conditional densities qk,n(ω) are given by qk,ns := Ps (ω,(L1,...,Lk )∈dx)ηn(dx×Ωn−k )
.Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 20 / 41
Expansion with a stochastic process: how far can we go?
Gt := Ft ∨ σ(Xu, u ≤ t)
1 Static anticipative signal Xt := W1: initial expansion
2 Static anticipative signal + dynamic noise Xt := W1 + εt [Corcuera, ...]
3 Dynamic anticipative signal Xt = Wt+δ: W is not a semimartingale
(Wt+δ is ...); besides, lim sup/inf E(Wt−Ws |Gs )t−s = ±∞
4 Dynamic anticipative signal + noise Xt = Wt+δ + εt : if εt are i.i.d. centered
random variables (white noise), then for (hn)n≥1 non-decreasing with
hn →∞ and hnn → 0 we have [SLLN]
1n
∑ni=1(W
t− hin +δ
+ εt− hi
n
)a.s.−−−→
n→∞Wt+δ.
Since all terms of the sequence are Gt-measurable, so is Wt+δ which thus
cannot be a G-semimartingale
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 21 / 41
Expansion with a stochastic process : setup
X : cadlag process
Gt := Ft ∨ σ(Xu, u ≤ t) and Gt :=⋂
u>t Gu
(πn)n≥1 : refining sequence of subdivisions of [0,T ] with |πn| →n→∞ 0
πn =: (tni )`(n)i=0 , with 0 = tn0 < ... < tn`(n) < tn`(n)+1 = T
(X n)n≥1 : sequence of cadlag (pure) jump process given by
X nt :=
∑`(n)i=0 Xttn
i1tni ≤t<tni+1
(Hn)n≥1 : non-decreasing sequence of filtration generated by X n, namely
Hnt := σ(X n
s , s ≤ t) = σ(Xtn0,Xtn1
− Xtn0, ...,Xtn
`(n)+1− Xtn
`(n))
Gn and Gn : expansions of F given by
Gnt := Ft ∨Hnt and Gnt =
⋂u>t Gnu
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 22 / 41
Convergence of filtrations in Lp
DefinitionLet p > 0.
1. A sequence of σ-algebras (Yn)n≥1 converges in Lp to a σ-algebra Y if
∀Y ∈ Lp(Y,P),E[Y |Y]Lp
−−−→n→∞
Y
2. We say that a sequence of filtrations (Yn)n≥1 converges weakly in Lp to a
filtration Y if Ynt
Lp
−−−→n→∞
Yt for every t ∈ I .
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 23 / 41
Convergence of filtrations in Lp II
Let (Gn)n≥1 be a non-decreasing sequence of filtrations and M a continuous
Gn-semimartingale with decomposition M =: Mn +∫ .
0αns d [M,M]s for every n ≥ 1
for some αn ∈ H1(Gn).
Theorem (Stability of the semimartingale property)
If Gn L2
−−−→n→∞
G and supn E∫ t
0|αn
u|d [M,M]u <∞, then M is a G-semimartingale on
[0, t].
Theorem (Convergence of information drifts)
If Gn L2
−−−→n→∞
G and supn≥1 E∫ t
0(αn
u)2d [M,M]u <∞ then M is a G-semimartingale
on [0, t] with decomposition M =: M +∫ .
0αsd [M,M]s , where α ∈ S2(G,M) and
αn L2
−−−→n→∞
α.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 24 / 41
Definition (Class Xπ)
We say that the cadlag process X is of Class Xπ, or of Class X if there is no
ambiguity, if
∀t ∈ I ,Hnt−
L2
−−−→n→1
Ht− .
Proposition
The cadlag process X is of Class X if one of the following holds:
(i) P(∆Xt 6= 0) = 0 for any fixed time t > 0.
(i’) X is continuous
(ii) H is (quasi-) left continuous.
(ii’) X is a Hunt process (e.g a Levy process)
(iii) X jumps only at totally inaccessible stopping times
(iv) π contains all the fixed times of discontinuity of X after a given rank
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 25 / 41
Expansion with a process: convergence of info. drifts
TheoremLet X be a stochastic process of Class Xπ for some sequence of subdivisions
πn := (tni )`(n)i=1 of [0,T ] and M a continuous F-local martingale. Suppose that for
every n ≥ 1 the random variable(X0,Xtn1
− Xtn0, ...,XT − Xtn
`(n)
)satisfies Jacod’s
condition and let αn be a Gn-predictable version of the process
αnt (ω) :=
∑`(n)k=0 1tnk≤t<tnk+1
1
qk,ns− (.,xk )
d [qk,n(.,xk ),M]sd [M,M]s
∣∣∣∣xk=(Xt0
,Xt1−Xt0
,··· ,Xtk−Xtk−1
)
.
1. If supn≥1E∫ T
0|αn
t |d [M,M]t <∞, M is a continuous G-semimartingale.
2. If supn E∫ T
0(αn
s )2d [M,M]s <∞, M is a continuous G-semimartingale with
decomposition M =: M +∫ .
0αsd [M,M]s , where α ∈ S2(G,M) and
E∫ T
0(αn
t − αt)2d [M,M]t −−−→
n→∞0.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 26 / 41
Existence of an approximating sequence of expansions
TheoremLet M be a continuous square integrable F-martingale, H another filtration and
Gt :=⋂
u>t(Fu ∨Hu). Suppose that Hn is a refining sequence of filtrations such
that Ht =∨
n≥1Hnt , and let Gnt :=
⋂u>t(Fu ∨Hn
u). Then, the following
statements are equivalent :
(i) There exists a predictable process α such that Mt := M −∫ t
0αsds defines a
continuous G-local martingale and E∫ T
0α2sds <∞.
(ii) For every n ≥ 1 there exists a predictable process αn such that
Mnt := M −
∫ t
0αnuds is a continuous Gn-local martingale and
supn≥1 E∫ T
0(αn
u)2d [M,M]u <∞.
In that case, M (resp. Mn) is a G- (resp. Gn-) square integrable martingale and
we have∫ t
0(αn
u − αu)2d [M,M]u −−−→n→∞
0.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 27 / 41
Example: A model for HF trading
W : F-Brownian motion
δ : non-negative stochastic process independent from W
ε : Markov process with stationary increments with density κt−s with respect
to Lebesgue measure
Xt := Wt+δ + εt
Gt := Ft ∨ σ(Xu, u ≤ t)
Then W has an information drift α given by
αt = lims→t
∫Xt + u −Ws
t − sκt−s(u − εs)du.
This highlights that the decay speed of the noise plays a key role in whether the
information drift exists.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 28 / 41
Example: the Bessel-3 process
Let us now consider the counterexample of the Bessel-3 process: let Z be a
Bessel-3 process and F its natural filtration; it is classical that
Wt := Zt −∫ t
0dsZs
is a F- Brownian motion.
Let G the expansion of F with Xt := inft>s Zs . The formula of Pitman shows
that W is a G-semimartingale with decomposition Wt − (2Xt −∫ t
0dsZs
) defines
a G-Brownian motion.
This implies that W is a G-semimartingale but cannot admit an information
drift since the finite variation component is singular with respect to Lebesgue
measure. We can recover this conclusion from our theorem.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 29 / 41
Example: the Bessel-3 process
Using the discretization induced by the refining family of random times
τnp := supt : Zt = pεn we have
αns =
1
Zs−∞∑p=0
1τnp<s1Zs≤(p+1)εn
1
Zs − pεn
∫ t
0
|αnt |dt = 2E
∫ t
0
ds
Zs<∞
∫ t
0
E[(αns )2]ds
∫ t
0
E[∞∑p=0
1pεn≤Zs≤(p+1)εn
pεnZ 2s (Zs − pεn)
]ds ≥∫ t
0
E[1εn≤Zs
1
Z 2s
]ds →∞
Thus supn≥1 E∫∞
0(αn
s )2ds =∞, αn cannot converge in H 2 and there cannot
exist an information drift α ∈H 2.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 30 / 41
Roadmap
1 A suggestive example from Ito
2 Semimartingales, filtrations and arbitrage
3 Expansion of a filtration with a stochastic process
4 Strict local martingales
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 31 / 41
Strict local martingales
A strict local martingale is a local martingale which is not a martingale (e.g.
Bessel-3 process Bt := 1
|W (3d)t |2
).
[Dandapani] There exists (initial) expansions G of F for which a risk neutral
probability exists but turns (some) true martingales into strict local
martingales ! This is shown for Markov stochastic volatility models under a
coupling condition on the drift and volatility of the volatility (+ abstract
conditions on the information drift)
How to detect strict local martingales?
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 32 / 41
The bitcoin bubble
2013 2014 2015 2016 2017 2018 2019
0
2500
5000
7500
10000
12500
15000
17500
20000
Total capitalization > $100bn
adaaeaionantb10b20b40batbchbnbbsvbtcbtcpbtgbtknbtmcennzctxccvcdaidash
dcrdgbdogedrgndxyelfengeosetcethethosfungasgnogntgoldgusdicnicxkcsknc
liborusdloomlrclskltcmaidmanamtlnasneoomgpaxpaypivxpolypowrpptqashreprhocsalt
sntsp500srntrxtusdusdcusdtvenverivtcwaveswtcxemxlmxmrxrpxvgzeczilzrx
Price history of the main cryptocurrencies (www.coinmetrics.io)
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 33 / 41
Characterization of strict local martingales
Suppose that under a risk-neutral probability the local martingale S follows a
diffusion
dSt = σ(St)StdWt ,
where σ > 0 on (0,∞).
Theorem (Delbaen & Shirakawa, 1997)
S is a strict local martingale if and only if∫ T
01
xσ(x)2 <∞.
A natural candidate to estimate x 7→ σ(x) : the Florens-Zmirou estimator, where
Xt := log(St).
SFZn (x) :=
∑ni=1 1|Sti
−x|<hnn(Xti+1 − Xti )2∑n
i=1 1|Sti−x|<hn
.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 34 / 41
Bitcoin and microstructure noise
Bitcoin price paths exhibit typical patterns of microstructure noise
0 200000 400000 600000 800000 1000000number of points
800000
1000000
1200000
1400000
1600000
1800000
QV (y
r2 )
Quadratic variation of log-prices between 01/2014 and 12/2016 according to discretization size
QVy = 0.584 + 1.21e6QV / 2ny = 0.898
0
10
20
30
40
50
[ 2]
Model : Yti := Xti + εti [Zhang, Mykland & Aıt-Sahalia, 2004]
E[ε2] = limn→∞QVn(Y )
2n a.s.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 35 / 41
Subsampling against microstructure noise
When microstructure noise is present its variance dominates the FZ
estimator: limn→∞SFZn (x)2n = E[ε2]
This can be corrected using a subsampled FZ estimator :
SFZ ,subn (x) :=
∑[n/λ]i=1 1|St[λi ]−x|<hn n[(Xt[λ(i+1)]
−Xt[λi ])2− 1
λ
∑[λ(i+1)]−1
j=[λi ](Xtj+1
−Xtj)2]∑n
i=1 1|Sti−x|<hn.
SFZ ,subn (x) converges at the same rate as the FZ estimator in absence of
noise, i.e. if nh2n →∞ then SFZ ,sub
n (x)P−−−→
n→∞σ2(x) + o(E[ε2]) [L.N., Ruan &
Shen].
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 36 / 41
A bubble detection procedure
Following [Jarrow, Kchia & Protter]
At every time t:
Step 1 Compute SFZ ,sub to estimate x 7→ σ2(x).
Step 2 Estimate the right-tail asymptotics x 7→ σ2(x) via a regression
(parametric or RKHS) and decide whether Bt is 0 or 1
Step 3 Smoothen the bubble estimator t 7→ Bt with a Hidden Markov
Chain model
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 37 / 41
Results: volatility estimate
Birth:6 8 10 12
price0.0
0.2
0.4
0.6
0.8
1.0
1.2
vola
tility
1e9
6 8 10 12 6 8 10 12 6 8 10 12 6 8 10 120
50
100
150
200
Death:
price0
1
2
3
4
5
6
vola
tility
1e10
100 2000
1
2
3
4
5
6
1e10
100 200 100 200 100 200 100 2000
25
50
75
100
125
150
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 38 / 41
Results: bubbles
2012 2013 2014 2015 2016 2017 20180
2500
5000
7500
10000
12500
15000
17500
20000
price
(USD
)
Bubbles in bitcoin prices
price1bubble
log-price
2
4
6
8
10
log-
price
(USD
)
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 39 / 41
Conclusion
There are various situations in which asset prices can differ significantly from
martingales
This has major implications in term of arbitrage opportunities
Expansions of filtrations can change deeply the properties of semimartingales
These new expansion models can be applied to find new statistics on
stochastic processes which can help identify the information structure of the
market.
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 40 / 41
Thank you!
Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 41 / 41