truncated variation - its properties and applications in stochastic...

Truncated variation - its properties andapplications in stochastic analysis

Rafał M. Łochowski

Osaka 2015

Rafał M. Łochowski Truncated variation Osaka 2015 1 / 51

Outline1 Definition and basic properites2 The truncated variation as a measure of path irregularity3 Realized volatility estimation4 Limit theorems for segment crossings

Truncated variation - definitionFor a path f : [a;b]→ E , where E is a normed space with norm | · |, itstruncated variation is defined with the following formula

TVc(f , [a;b]) := supn

supa≤t1<t2<...<tn≤b

∑i=1

max{|f (ti+1)− f (ti)|− c,0} ,

(1)where c > 0 is the truncation parameter.

FactIf E is complete (i.e. Banach) space, then TVc(f , [a;b]) < +∞ for anyc > 0 iff f is regulated, i.e. it has right and left limits.

Truncated variation - basic propertiesLet ‖f −g‖∞ := supt∈[a;b] |f (t)−g(t)| .From the triangle inequality, for g such that ‖f −g‖∞ ≤ c/2, weimmediately get

|f (ti+1)− f (ti)|− c ≤ |g(ti+1)−g(ti)|

and from thisTVc(f , [a;b])≤ TV(g, [a;b]) , (2)

where TV(g, [a;b]) is just the (total) variation of g.

RemarkThe fact stated on the previous slide becomes clear when one recallsthat the family of regulated functions attaing values in a Banach spacecoincides with the family of functions which may be uniformlyapproximated by finite variation functions (equivalently: by stepfunctions) with an arbitrary accuracy.

Truncated variation - basic propertiesLet ‖f −g‖∞ := supt∈[a;b] |f (t)−g(t)| .From the triangle inequality, for g such that ‖f −g‖∞ ≤ c/2, weimmediately get

|f (ti+1)− f (ti)|− c ≤ |g(ti+1)−g(ti)|

and from thisTVc(f , [a;b])≤ TV(g, [a;b]) , (2)

where TV(g, [a;b]) is just the (total) variation of g.

RemarkThe fact stated on the previous slide becomes clear when one recallsthat the family of regulated functions attaing values in a Banach spacecoincides with the family of functions which may be uniformlyapproximated by finite variation functions (equivalently: by stepfunctions) with an arbitrary accuracy.

Truncated variation - variational propertyWhen f is a real function, then the bound (2) is attainable, i.e. thereexists a function f c : [a;b]→ R, such that ‖f −g‖∞ ≤ c/2 and

TVc(f , [a;b]) = TV(f c, [a;b])

Thus, we have the following variational property of TV c :

TVc(f , [a;b]) = inf‖f−g‖∞≤c/2

TV(g, [a;b]) . (3)

RemarkIt is an open question if property (3) holds in other spaces than R. Evenfor spaces E where the equality does not hold, it seems to be interestingto assess how much the left side of (3) differs from the right side.

Truncated variation - variational propertyWhen f is a real function, then the bound (2) is attainable, i.e. thereexists a function f c : [a;b]→ R, such that ‖f −g‖∞ ≤ c/2 and

TVc(f , [a;b]) = TV(f c, [a;b])

Thus, we have the following variational property of TV c :

TVc(f , [a;b]) = inf‖f−g‖∞≤c/2

TV(g, [a;b]) . (3)

RemarkIt is an open question if property (3) holds in other spaces than R. Evenfor spaces E where the equality does not hold, it seems to be interestingto assess how much the left side of (3) differs from the right side.

Truncated variation - asymptoticsFrom definition (1) or variational property (3) (for real paths) wenaturally have that

limc↓0

TVc(f , [a;b]) = TV(f , [a;b]) .

If TV(f , [a;b]) = +∞ (which is common for many important families ofstochastic processes) the rate of the divergence of TVc(f , [a;b]) to +∞

as c ↓ 0 may be viewed as the measure of irregularity of the path f

p−variationNaturally, in analysis there were many other notions of variationsintroduced. One of the most natural seems to be the p−variation,p > 0, defined as

Vp(f , [a;b]) := supn

∑i=1|f (ti+1)− f (ti)|p . (4)

For any f , if Vp(f , [a;b]) < +∞ and q > p > 0, then Vq(f , [a;b]) < +∞.

This observation makes meaningful to define the variation index

Indvar (f ) := inf{p : Vp(f , [a;b]) < +∞} ,

which may be also viewed as a measure of irregularity of the path - thegreater Indvar (f ) , the more irregular path.

RemarkIf f is a step function then Indvar (f ) = 0, otherwise usually Indvar (f )≥ 1.

∑i=1|f (ti+1)− f (ti)|p . (4)

For any f , if Vp(f , [a;b]) < +∞ and q > p > 0, then Vq(f , [a;b]) < +∞.This observation makes meaningful to define the variation index

∑i=1|f (ti+1)− f (ti)|p . (4)

For any f , if Vp(f , [a;b]) < +∞ and q > p > 0, then Vq(f , [a;b]) < +∞.This observation makes meaningful to define the variation index

The variation index vs. the asymptotics of thetruncated variationLet G ([a;b]) denote the set of real-valued, regulated functionsf : [a;b]→ R.

For p ≥ 1 let V p ([a;b]) denote the subset of G ([a;b]) consisting offunctions with finite p−variation.

Next, for p ≥ 1 let Up ([a;b]) denote the subset of G ([a;b]) consistingof functions f for which

limsupc↓0

cp−1 ·TVc(f , [a;b]) < +∞.

We have U1 ([a;b]) = V 1 ([a;b]) and for any 1 < p < q we have

V p ([a;b]) ( Up ([a;b]) ( V q ([a;b]) .

The variation index vs. the asymptotics of thetruncated variationLet G ([a;b]) denote the set of real-valued, regulated functionsf : [a;b]→ R.For p ≥ 1 let V p ([a;b]) denote the subset of G ([a;b]) consisting offunctions with finite p−variation.

limsupc↓0

cp−1 ·TVc(f , [a;b]) < +∞.

V p ([a;b]) ( Up ([a;b]) ( V q ([a;b]) .

limsupc↓0

cp−1 ·TVc(f , [a;b]) < +∞.

V p ([a;b]) ( Up ([a;b]) ( V q ([a;b]) .

limsupc↓0

cp−1 ·TVc(f , [a;b]) < +∞.

V p ([a;b]) ( Up ([a;b]) ( V q ([a;b]) .

Irregularity of a typical Brownian pathIf B is a standard Brownian motion, then Indvar (B) = 2 with probability1.However, an old result of P. Levy states that V2(B, [a;b]) = +∞ withprobability 1, thus B /∈ V 2 ([0; t]) with probability 1.

On the other hand, in [LM2013] it was proven that for any t > 0,

limc→0

c ·TVc(B, [0; t]) = t, with probability 1,

thus B ∈U2 ([0; t]) with probability 1.

RemarkIn [LM2013] there was a more general fact proven: if Xt , t ≥ 0, is acontinuous semimartingale with quadratic variation < ·> then for anyt > 0,

limc→0

c ·TVc(X , [0; t]) =< X >t , with probability 1.

limc→0

Irregularity of a typical Brownian path measuredby ϕ− variationFor a continuous, increasing function ϕ : [0;+∞)→ [0;+∞) such thatϕ(0) = 0 one may define ϕ−variation as

Vϕ(f , [a;b]) := supn

∑i=1

ϕ(|f (ti+1)− f (ti)|) . (5)

An old result of S. J. Taylor [T1972] states that

ϕ0(x) = x2/ ln(max [ln(1/x) ,2])

is a function with the greatest order in the neighbourhood of 0 and suchthat for any t > 0,

Vϕ0(B, [0; t]) < +∞, with probability 1.

Irregularity of a typical Brownian path measuredby ϕ− variationFor a continuous, increasing function ϕ : [0;+∞)→ [0;+∞) such thatϕ(0) = 0 one may define ϕ−variation as

Vϕ(f , [a;b]) := supn

∑i=1

ϕ(|f (ti+1)− f (ti)|) . (5)

An old result of S. J. Taylor [T1972] states that

ϕ0(x) = x2/ ln(max [ln(1/x) ,2])

is a function with the greatest order in the neighbourhood of 0 and suchthat for any t > 0,

Vϕ0(B, [0; t]) < +∞, with probability 1.

Irregularity of a typical Brownian path measuredby ϕ− variation, cont.Moreover, for any function ϕ such that limx↓0

ϕ(x)ϕ0(x) = +∞ and t > 0 we

Vϕ(B, [0; t]) = +∞, with probability 1.

RemarkThe asymptotics of the truncated variation for functions with finiteϕ−variation is still to be investigated.

Truncated variation, p−variation and ϕ−variationnorms of a Brownian pathThe already mentioned irregularity results for the Brownian paths maybe stated using p−variation, ϕ−variation and truncated variationnorms, defined as

‖B‖p−var ,[0;T ] := (Vp(B, [0;T ]))1/p , for p ≥ 1,

‖B‖ϕ−var ,[0;T ] := inf{λ > 0 : Vϕ(B/λ, [0;T ])≤ 1} ,and

‖B‖TV ,p,[0;T ] :=

(supc>0

cp−1 ·TVc(B, [0;T ])

, for p ≥ 1.

RemarkOne always has

‖B‖TV ,p,[0;T ] ≤ ‖B‖p−var ,[0;T ].

Truncated variation, p−variation and ϕ−variationnorms of a Brownian pathThe already mentioned irregularity results for the Brownian paths maybe stated using p−variation, ϕ−variation and truncated variationnorms, defined as

‖B‖p−var ,[0;T ] := (Vp(B, [0;T ]))1/p , for p ≥ 1,

‖B‖ϕ−var ,[0;T ] := inf{λ > 0 : Vϕ(B/λ, [0;T ])≤ 1} ,and

‖B‖TV ,p,[0;T ] :=

(supc>0

cp−1 ·TVc(B, [0;T ])

, for p ≥ 1.

RemarkOne always has

‖B‖TV ,p,[0;T ] ≤ ‖B‖p−var ,[0;T ].

Truncated variation, p−variation and ϕ−variationnorms of a Brownian path, cont.For such defined p−variation, ϕ−variation and truncated variationnorms, we have

‖B‖p,[0;T ] =

{< +∞ if p ∈ (2;+∞),

+∞ if p ∈ [1;2];

‖B‖ϕ,[0;T ] =

{< +∞ if limx↓0

ϕ(x)ϕ0(x) < +∞,

+∞ if limx↓0ϕ(x)ϕ0(x) = +∞

‖B‖TV ,p,[0;T ] =

{< +∞ if p ∈ [2;+∞),

+∞ if p ∈ [1;2).

Truncated variation vs. p−variation norms of afractional Brownian pathSimilarly, for a fractional Brownian BH motion with the Hurst parameterH ∈ (0;1) we have

‖BH‖p,[0;T ] =

{< +∞ if p ∈ (1/H;+∞),

+∞ if p ∈ [1;1/H]

‖BH‖TV ,p,[0;T ] =

{< +∞ if p ∈ [1/H;+∞),

+∞ if p ∈ [1;1/H).

Thus, for each T > 0

BH ∈U1/H ([0;T ])\V 1/H ([0;T ]) , with probability1.

Young integrals of irregular pathsUsing truncated variation techniques it is possible to prove the following,stronger version of Young’s result from 1936 [Young, 1936]:

TheoremLet f ,g : [a;b]→ R be two functions with no common points ofdiscontinuity. If f ∈Up ([a;b]) and g ∈Uq ([a;b]) , where p > 1, q > 1,p−1 + q−1 > 1, then the Riemann Stieltjes

´ ba f (t)dg (t) exists.

Moreover, there exist a constant Cp,q, depending on p and q only, suchthat ∣∣∣∣ˆ b

af (t)dg (t)− f (a) [g (b)−g (a)]

∣∣∣∣≤ Cp,q‖f‖

p−p/qTV ,p,[a;b]‖f‖

1+p/q−posc,[a;b] ‖g‖TV ,q,[a;b],

where ‖f‖osc,[a;b] := supa≤s<t≤b |f (t)− f (s)|.

How to calculate the truncated variation?There exists an algorithm based on drawdown and drawup times. Letus define

T cU f = inf

{t ∈ (a;b] : f (t)− inf

a≤s≤tf (s) > c

Df = inf

{t ∈ (a;b] : sup

a≤s≤tf (s)− f (t) > c

If min{

T cU f ,T c

= +∞, then TVc(f , [a;b]) = 0. If not, assuming thatT c

U f < T cDf we define T c

D,−1f = a and then for k = 0,1,2, . . .

T cU,k f = inf

{t ∈ (T c

D,k−1f ;b] : f (t)− infT c

D,k−1f≤s≤tf (s) > c

T cD,k f = inf

{t ∈ (T c

U,k f ;b] : supT c

U,k f≤s≤tf (s)− f (t) > c

How to calculate the truncated variation?There exists an algorithm based on drawdown and drawup times. Letus define

T cU f = inf

{t ∈ (a;b] : f (t)− inf

a≤s≤tf (s) > c

Df = inf

{t ∈ (a;b] : sup

a≤s≤tf (s)− f (t) > c

}If min

U f ,T cDf}

= +∞, then TVc(f , [a;b]) = 0. If not, assuming thatT c

U f < T cDf we define T c

D,−1f = a and then for k = 0,1,2, . . .

T cU,k f = inf

{t ∈ (T c

D,k−1f ;b] : f (t)− infT c

D,k−1f≤s≤tf (s) > c

T cD,k f = inf

{t ∈ (T c

U,k f ;b] : supT c

U,k f≤s≤tf (s)− f (t) > c

Times T cU,k , T c

D,k , k = 0,1, . . .

TU,0c TD,0

c TU,1c

Calculation of the truncated variation, cont.Now, to calculate the truncated variation we define

mk = infs∈[TD,k−1;TU,k ]

f (s), Mk = infs∈[TU,k ;TD,k ]

f (s),

and for t such that t ∈ [TU,k ;TD,k ] we have

TVc(f , [a; t]) =k−1

∑i=0

(Mi −mi − c) +k−1

∑i=0

(Mi −mi+1− c) (6)

+ sups∈[T c

U,k f ,t]f (s)−mk − c.

Similarly, for t such that t ∈ [TD,k ;TU,k+1] we have

TVc(f , [a; t]) =k

∑i=0

(Mi −mi − c) +k−1

∑i=0

(Mi −mi+1− c) (7)

+ Mk − infs∈[T c

D,k f ,t]f (s)− c.

Stopping times T cU,k , T c

D,k , k = 0,1, . . .If Xt , t ≥ 0, is a stochastic process with cadlag (or even regulated!)trajectories, adapted to the filtration Ft , t ≥ 0, then for each trajectoryf = X(ω) the just defined times T c

U,k , T cD,k , k = 0,1, . . . are stopping

times such that

limk→+∞

T cU,k = lim

k→+∞T c

D,k = +∞, with probability 1.

To see this, note that for any t > 0 there exists such K < +∞ thatT c

U,K > t and T cD,K > t. Otherwise we would obtain two infinite

sequences (sk )∞

k=1 ,(Sk )∞

k=1 such that 0≤ s1 < S1 < s2 < S2 < ... < tand f (Sk )− f (sk )≥ 1

2c. But this is a contradiction, since f is aregulated function and (f (sk ))∞

k=1 ,(f (Sk ))∞

k=1 have a common limit.

Stopping times T cU,k , T c

D,k , k = 0,1, . . .If Xt , t ≥ 0, is a stochastic process with cadlag (or even regulated!)trajectories, adapted to the filtration Ft , t ≥ 0, then for each trajectoryf = X(ω) the just defined times T c

U,k , T cD,k , k = 0,1, . . . are stopping

times such that

limk→+∞

T cU,k = lim

k→+∞T c

D,k = +∞, with probability 1.

To see this, note that for any t > 0 there exists such K < +∞ thatT c

U,K > t and T cD,K > t. Otherwise we would obtain two infinite

sequences (sk )∞

k=1 ,(Sk )∞

k=1 such that 0≤ s1 < S1 < s2 < S2 < ... < tand f (Sk )− f (sk )≥ 1

2c. But this is a contradiction, since f is aregulated function and (f (sk ))∞

k=1 ,(f (Sk ))∞

k=1 have a common limit.

Calculation of the truncated variation of a processwith continuous trajectoriesWhen Xt , t ≥ 0, is a stochastic process with continuous trajectoriesthen for any t > 0, TVc(X , [0; t]) may be calculated with the stochasticsampling scheme based on the stopping times T c

U,k , T cD,k , k = 0,1, . . . .

Indeed, for continuous f = X(ω) we have XT cU,k

(ω) = f (T cU,k ) = mk + c

and XT cD,k

(ω) = f (T cD,k ) = Mk − c, hence from formulas (6) and (7) we

TVc(X , [0;T cU,k ])

=k−1

∑i=0

D,i−XT c

U,i+ c)

+k−1

∑i=0

D,i−XT c

U,i+1+ c).

Similarly,

TVc(X , [0;T cD,k ])

∑i=0

D,i−XT c

U,i+ c)

+k−1

∑i=0

D,i−XT c

U,i+1+ c).

Realized volatility estimationIf Xt , t ≥ 0, is a continuous semimartingale then the (mentionedalready) relation

c ·TVc(X , [0; t])→< X >t , with probability 1 (8)

(in the uniform convergence topology on compacts) and the statedalgorithm provides us with a tool for the realized volatility estimation.

RemarkThe rate of convergence in (8) is better than the rate of the usualalgorithms based on the calendar time or business time sampling (thiswas proved at least for some class of diffusions), and is of the sameorder as the order of tick time sampling schemes, which are specialcase of stochastic sampling schemes for realized volatility estimationintroduced by Masaaki Fukasawa [F2010].

Realized volatility estimationIf Xt , t ≥ 0, is a continuous semimartingale then the (mentionedalready) relation

c ·TVc(X , [0; t])→< X >t , with probability 1 (8)

(in the uniform convergence topology on compacts) and the statedalgorithm provides us with a tool for the realized volatility estimation.

RemarkThe rate of convergence in (8) is better than the rate of the usualalgorithms based on the calendar time or business time sampling (thiswas proved at least for some class of diffusions), and is of the sameorder as the order of tick time sampling schemes, which are specialcase of stochastic sampling schemes for realized volatility estimationintroduced by Masaaki Fukasawa [F2010].

Rate of convergence in (8)Let Xt , t ≥ 0 be the strong solution of the SDEdXt = µ(Xt)dt + σ(Xt)dBt , where B is a standard Brownian motionand µ, σ > 0 satisfy ”usual” conditions guaranteeing the existenceuniqueness of the strong solution of SDE. In [LM2013] the followingresult was proven

(c ·TVc(X , [0; t])−< X >t)⇒W<X>t/3,

where W is another standard Brownian motion, independent from Band the convergence⇒ is understood as the stable convergence in theuniform convergence topology on compacts.

Stochastic sampling schemes for realized volatilityestimationMasaaki Fukasawa considered general stochastic sampling schemesand obtained similar rates of convergence for tick sampling.

He considered a continuous semimartingale X = A + M, where Mt ,t ≥ 0, is a continuous local martingale adapted to some filtration Ft ,t ≥ 0, and At , t ≥ 0, is a finite variation process such thatAt =

´ t0 ψsd < M >s, where ψs, t ≥ 0, is locally bounded, left

continuous process adapted to Ft and a sequence of Ft−stoppingtimes 0 = τ0 < τ1 < .. . such that for every T > 0

Nτ (t) := max{k ≥ 0 : τk > t}< +∞, with probability 1.

Stochastic sampling schemes for realized volatilityestimation, cont.The statistics

RVτ (t) :=Nτ(t)−1

∑k=0

(Xτk +1−Xτk )2

is an estimator of < X >t and the rate of convergence of this statisticsdepends on the sequence τ = {τ0 < τ1 < .. .} .The best rates of convergence one obtains taking e.g. tick sampling, i.e.τc

0 = 0 and for k = 0,1, . . .

τck+1 = inf

{t > tk :

∣∣∣Xt −Xτck

∣∣∣> c},

and we have stable convergence to a non-trivial limit of the processes

(RVτc (t)−< X >t) .

Stochastic sampling schemes for realized volatilityestimation, general remarks

With the tick time sampling (based on stopping times τck ) (or the

truncated variation sampling based on stopping times T cU,k , T c

D,k ,)we need to control the process very precisely (no unobserved bigfluctuations between consecutive samplings) like in the case ofthe calendar time sampling τk = k/n.

As a reward for this we obtain a better mode of the first orderconvergence (almost sure convergence instead of theconvergence in probability) and a better asymptotics of the secondorder convergence.

This is no problem as long we assume that we are able to observerounded values of the process X at any time.

The tick time sampling or the truncated variation sampling arerobust to the price rounding (market microstructure noise).

Variational properties of the tick sampling schemeThe tick time sampling scheme has also interesting property regardingthe total variation of the cadlag process X c defined as

X ct = Xτc

k, where k = max{i = 0,1,2, . . . : τ

ci ≤ t} .

This process approximates the process X with accuracy c and we have

TV(X c, [0; t]

)= c ·Nτ (t)

andRVτc (t) := c2 ·Nτc (t) = c ·TV

(X c, [0; t]

Thus, recalling the convergence results for RVτc (t) we get thatTV(X c, [0; t]

)has the same magnitude as TVc(X , [0; t]) and it

approximates uniformly X with accuracy c (which differs from theoptimal accuracy for a process with the total vatiation TVc(X , [0; t]) byfactor 2).

The truncated variation and segment crossingsThe truncated variation appears to be esspecially relevant for theinvestigation of the numbers of segment crossings. First, let us recallthe Banch Indicatrix theorem. If f : [a,b]→ R is a continuous functionthen

TV(f , [a;b]) =

Ny (f , [a;b])dy ,

whereNy (f , [a;b]) := card{x ∈ [a;b] : f (x) = y}

is called the Banach indicatrix. A generalisation of this result for thecase of regulated f is possible.Unfortunately, when TV(f , [a;b]) = +∞ this result seems to be useless.

The truncated variation and segment crossings,cont.However, when instead of considering the level crossings, oneconsiders segment crossings and instead of considering the totalvariation one considers the truncated variation then one gets thefollowing result.For any regulated f : [a;b]→ R and c > 0,

TVc(f , [a;b]) =

nyc (f , [a;b])dy . (9)

nyc (f , [a;b]) = number of times that f crosses the segment [y ;y + c].

Segment crossings - precise definitionTo be more precise, we define

nyc (f , [a,b]) = dy

c (f , [a,b]) + uyc (f , [a,b]) .

dyc (f , [a,b]) = number of downcrossings the segment [y ;y + c]..

σ0 = a, and for k = 0,1, . . .

νk = inf{t > σk : f (t) > y + c} σk+1 = inf{t > νk : f (t) < y} .

Now we definedy

c (f , [a,b]) = max{k : σk ≤ b} .

dyc (f , [a,b]) = number of downcrossings the segment [y ;y + c].

σ0 = a, and for k = 0,1, . . .

νk = inf{t > σk : f (t) < y} σk+1 = inf{t > νk : f (t) > y + c} .

uyc (f , [a,b]) = max{k : σk ≤ b} .

Segment crossings - precise definitionTo be more precise, we define

nyc (f , [a,b]) = dy

c (f , [a,b]) + uyc (f , [a,b]) .

dyc (f , [a,b]) = number of downcrossings the segment [y ;y + c]..

σ0 = a, and for k = 0,1, . . .

νk = inf{t > σk : f (t) > y + c} σk+1 = inf{t > νk : f (t) < y} .

Now we definedy

c (f , [a,b]) = max{k : σk ≤ b} .

dyc (f , [a,b]) = number of downcrossings the segment [y ;y + c].

σ0 = a, and for k = 0,1, . . .

νk = inf{t > σk : f (t) < y} σk+1 = inf{t > νk : f (t) > y + c} .

uyc (f , [a,b]) = max{k : σk ≤ b} .

Limit theorems for segment crossingsRelation (9), linking the truncated variation with the numbers ofsegment crossings, allows to obtain, for relatively broad spectrum ofstochastic processes, limit theorems for the numbers of segmentcrossings of these processes.For example, for a continuous semimartingale Xt , t ≥ 0, using thealready mentioned convergence

c ·TVc(X , [0; t])→< X >t , with probability 1

one obtains the convergenceˆR

c ·nyc (X , [0; t])dy →< X >t , with probability 1.

It is also possible to make some of the levels more important thanothers, by introducing continuous density g : R→ R, and obtainˆ

Rc ·ny

c (X , [0; t])g(y)dy →ˆ t

0g (Xs)d < X >s, with probability 1.

(11)Rafał M. Łochowski Truncated variation Osaka 2015 30 / 51

Limit theorems for segment crossings, diffusionsSimilalry, from the already mentioned convergence(

TVc(X , [0; t])− 1c< X >t

)⇒W<X>t/3,

for Xt , t ≥ 0, being the strong solution of the SDEdXt = µ(Xt)dt + σ(Xt)dBt , where B is a standard Brownian motionand µ, σ > 0 satisfy ”usual” conditions guaranteeing the existenceuniqueness of the strong solution of SDE, and any continuous functiong : R→ R, the following convergence of the integrated numbers ofsegment crossings was obtained ([LG2014]):(ˆ

c (X , [0; t])g(y)dy− 1c

0g (Xs)d < X >s

)⇒ˆ t

0g (Xs)W<X>t

where B and W are two independent standard Brownian motions.

Limit theorems for segment down-, up- crossings,diffusionsTogether with the convergence of segment crossings, one has (obvious)first order convergence of segment down- and up- crossings (sinceuy

c (X , [0; t])−dyc (X , [0; t]) ∈ {−1,0,1}):

c ·uyc (X , [0; t])g(y)dy → 1

0g (Xs)d < X >s, with probability 1.

But in the case of the second order convergence an interestingcorrection term (the Stratonovich integral) appears:ˆ

c (X , [0; t])g(y)dy− 12 · c

0g (Xs)d < X >s

⇒ 12

0g (Xs)W<X>t

0g (Xs)◦dXs,

where B and W are two independent standard Brownian motions.Rafał M. Łochowski Truncated variation Osaka 2015 32 / 51

Segment crossings and local times of diffusionsLet Ly

t (X) be the local time of X at y ∈ R and g : R→ R be acontinuous function. Recalling the occupation times formulaˆ t

0g (Xs)d < X >s=

g (y)Lyt (X)dy .

we get ˆR

c (X , [0; t])− 1c

Lyt (X)

]g (y)dy ⇒

0g (Xs)W<X>t

On the other hand, we have the following convergence ([Kasahara,1980] for a standard Brownian motion, [LG2014, Theorem 4.5] fordiffusions)

c (X , [0; t])− 1c

Lyt (X)

]⇒WLy

t (X).

Thus we see that integrating differences nyc (X , [0; ·])− 1

c Ly (X) we getmuch faster convergence (multiplication by

√c is no needed).

0g (Xs)d < X >s=

g (y)Lyt (X)dy .

we get ˆR

c (X , [0; t])− 1c

Lyt (X)

]g (y)dy ⇒

0g (Xs)W<X>t

c (X , [0; t])− 1c

Lyt (X)

]⇒WLy

t (X).

√c is no needed).

0g (Xs)d < X >s=

g (y)Lyt (X)dy .

we get ˆR

c (X , [0; t])− 1c

Lyt (X)

]g (y)dy ⇒

0g (Xs)W<X>t

c (X , [0; t])− 1c

Lyt (X)

]⇒WLy

t (X).

√c is no needed).

”Meta-theorem”, almost sure convergenceFor a process Xt , t ≥ 0, let us denote TVc(X , ·) := TVc(X , [0; ·]) andna

c(X , ·) := nac(X , [0; ·]).

Theorem (M1)

Let Xt , t ≥ 0, be a cadlag process and assume that there exists anincreasing function ϕ : (0;+∞)→ (0;+∞) , such that limc→0+ ϕ(c) = 0,and a cadlag process ζt , t ≥ 0, with locally finite variation, such that thefollowing convergence holds

ϕ(c)TVc(X , ·)→ ζ.

Then for any continuous function f : R→ R we have the followingconvergence

nac(X , ·)f (a)da→

ˆ ·0

f (Xt−)dζt .

”Meta-theorem”, almost sure convergenceFor a process Xt , t ≥ 0, let us denote TVc(X , ·) := TVc(X , [0; ·]) andna

c(X , ·) := nac(X , [0; ·]).

Theorem (M1)

Let Xt , t ≥ 0, be a cadlag process and assume that there exists anincreasing function ϕ : (0;+∞)→ (0;+∞) , such that limc→0+ ϕ(c) = 0,and a cadlag process ζt , t ≥ 0, with locally finite variation, such that thefollowing convergence holds

ϕ(c)TVc(X , ·)→ ζ.

Then for any continuous function f : R→ R we have the followingconvergence

nac(X , ·)f (a)da→

ˆ ·0

f (Xt−)dζt .

”Meta-theorem”, stable convergence

Theorem (M2)

Let Xt , t ≥ 0, be a cadlag process and assume that there exists anincreasing function ϕ : (0;+∞)→ (0;+∞) , with limc→0+ ϕ(c) = 0, anda cadlag process ζt , t ≥ 0, with locally finite variation, such that thefollowing convergence holds

ϕ(c)TV c (X , ·)⇒ ζ

then for any continuous function f : R→ R we have the followingconvergence

nac(X , ·)f (a)da⇒

ˆ ·0

f (Xt−)dζt .

Self-similar processes

Theorem (SsP)

Let Xt , t ≥ 0, be a cadlag process such that it has stationaryincrements. Next, assume that X is a self-similar proces, that is, thereexists β ∈ (0;1) such that{

A−βXAt , t ≥ 0}

=d {Xt , t ≥ 0}

for any A > 0. Additionally we assume that for some c > 0,ETVc(X , [0,1]) < +∞ and that the tail σ-field is trivial. Then

c1/β−1TVc(X , ·)→ C · Id .

Levy processes

Theorem (LP1)Let Xt , t ≥ 0, be a Levy process which has infinite total variation and isno monotonic on any non-degenerate interval. Moreover, assume thatEsup0≤t<T

c0D X Xt < +∞ for some c0 > 0, where we define

T cDX := inf

{t ≥ 0 : sup0≤s≤t Xs−Xt > c

}, θc

U := ET cDX

ξcU := sup0≤s<t<T c

DX (Xt −Xs− c)+ , ηcU := Eξc

If forχU (c) := θ

and for any u > 0, P(ξc

U ≤ u/χU (c))/θc

U → 1 as c→ 0+, then wehave the following convergence

χU (c)TVc(X , ·)⇒ 2 · Id .

Levy processes

T cDX := inf

}, θc

U := ET cDX

If forχU (c) := θ

χU (c)TVc(X , ·)⇒ 2 · Id .

Levy processes

T cDX := inf

}, θc

U := ET cDX

If forχU (c) := θ

χU (c)TVc(X , ·)⇒ 2 · Id .

Levy processes, cont.

T cUX := inf{t ≥ 0 : Xt − inf0≤s≤t Xs > c} , θc

D := ET cDX

ξcD := sup0≤s<t<T c

UX (Xs−Xt − c)+ , ηcD := Eξc

If forχD (c) := θ

and for any u > 0, P(ξcD ≤ u/χD (c))/θc

D→ 1 as c→ 0+, then wehave the following convergence

χD (c)TVc(X , ·)⇒ 2 · Id .

D := ET cDX

If forχD (c) := θ

χD (c)TVc(X , ·)⇒ 2 · Id .

D := ET cDX

If forχD (c) := θ

χD (c)TVc(X , ·)⇒ 2 · Id .

Consequence of Theorem M1 and Theorem SsP

Corollary (S)Let Xt , t ≥ 0, be a self-similar cadlag process as in Theorem SsP andf : R→ R be a continuous function, then

c1/β−1ˆR

nac(X , ·)f (a)da→ C

ˆ ·0

f (Xt−)dt,

where the constant C is the same as in Theorem SsP.

Consequence of Theorem M2 and Theorems LP1,LP2

Corollary (L)Let Xt , t ≥ 0, be a Levy process as in Theorem L1. If the assumptionsof Theorem L1 are satisfied then

χU (c)

nac(X , ·)f (a)da⇒ 2

ˆ ·0

f (Xt−)dt.

An analogous convergence, namely

χD (c)

nac(X , ·)f (a)da⇒ 2

ˆ ·0

f (Xt−)dt,

holds when the assumptions of Theorem L2 are satisfied.

More specific case - spectrally assymetricprocesses with ”almost” α-stable jumps

TheoremLet Xt , t ≥ 0, be a Levy process without Brownian component, with theLevy measure Π such that

Π(dx) =L(x)

(−x)1+α1x<0dx

for α ∈ (1;2) and some Borel-measurable function

L : (−∞;0)→ (0;+∞) , slowly varying at 0. Then χD(c)∼ α(α−1)Γ(2−α)

cα−1

L(−c)and

α(α−1)

Γ(2−α)

cα−1

L(−c)TVc(X , ·)⇒ 2 · Id .

A remark on local timesThe most common definition of the local times L = La

t , a ∈R, t ≥ 0, of agiven process Xt , t ≥ 0, is as the Radon-Nikodym derivative of theoccupation measure of X with respect to the Lebesgue measure in R;for every Borel measurable function f : R→ R and t > 0,ˆ t

0f (Xs)ds =

f (a)Lat da.

Notice that for any cadlag process Xt , t ≥ 0, and continuous fˆ ·0

f (Xt−)dt =

ˆ ·0

f (Xt)dt.

Thus, in view of Corollary S and Corollary L we have that the naturalcandidates for the local times La

t for a self-similar or a Levy process Xare the limits (if they exist)

C−1c1/β−1nac(X , t),

χU (c)nac(X , t) or

χD (c)nac(X , t).

A remark on local timesThe most common definition of the local times L = La

t , a ∈R, t ≥ 0, of agiven process Xt , t ≥ 0, is as the Radon-Nikodym derivative of theoccupation measure of X with respect to the Lebesgue measure in R;for every Borel measurable function f : R→ R and t > 0,ˆ t

0f (Xs)ds =

f (a)Lat da.

Notice that for any cadlag process Xt , t ≥ 0, and continuous fˆ ·0

f (Xt−)dt =

ˆ ·0

f (Xt)dt.

Thus, in view of Corollary S and Corollary L we have that the naturalcandidates for the local times La

t for a self-similar or a Levy process Xare the limits (if they exist)

C−1c1/β−1nac(X , t),

χU (c)nac(X , t) or

χD (c)nac(X , t).

Second order convergence of the truncatedvariation of strictly α-stable processesLet Xt , t ≥ 0, be strictly α-stable process with the characteristicexponent

ΨX (θ) = C0|θ|α(

1− iγ tanπα

2sgnθ

), (12)

where α ∈ (1;2) is the index, C0 > 0 is the scale parameter andγ ∈ [−1;1] is the skewness parameter. Define

A := limN→+∞

E(TV1(X , [0;N + 1])−TV1(X , [0;N])

(it is possible to prove that this limit exists) and

T ct := TVc(X , [0, t])− c1−αA · t.

Second order convergence of the truncatedvariation of strictly α-stable processes, cont.

Theorem (SecondOrder)Let α ∈ (1;2) and Xt , t ≥ 0, be strictly α-stable process with thecharacteristic exponent given by formula (12), then

T c =⇒ L1 + L2,

where L1 and L2 are two independent, spectrally positive processessuch that X = L1−L2 and L1 + L2 is strictly α-stable, spectrally positiveprocess with the characteristic exponent given by formula

ΨL1+L2(θ) = C0|θ|α(

1− i tanπα

2sgnθ

Second order convergence of the truncatedvariation of strictly 1-stable processesLet Xt , t ≥ 0, be strictly 1-stable process with the characteristicexponent

ΨX (θ) = C0|θ|+ iηθ, (13)

with the scale parameter C0 > 0 and the drift η ∈ R. (The characterisitcexponent of a strictly 1-stable process is necessarily of this form.) Letus set

B = limN→+∞

E(TV1(X , [0;N + 1])−TV1(X , [0;N])−TV(Y , [0;N])

where Y = ∑N<s≤N+1 |Xs−Xs−|1|Xs−Xs−|≥1 and

T ct := TVc(X , [0, t])− 2

πC0 logc−1 · t−B · t.

Second order convergence of the truncatedvariation of strictly 1-stable processes, cont.

Theorem (SecondOrder1)Let Xt , t ≥ 0, be strictly 1-stable process with the characteristicexponent given by formula (13), then

T c =⇒M1 + M2,

where M1, M2 are two independent, spectrally positive processes suchthat X = M1−M2 and M1 + M2 is 1-stable process with thecharacteristic exponent given by formula

ΨM1+M2(θ) = C0|θ|(

1 + i2π

sgn(θ) log |θ|)− i

2(1−C)

πC0θ,

where C= Γ′(1)≈ 0.577 is the Euler-Mascheroni constant.

Second order convergences for numbers ofsegment crossingsThe immediate consequences of Theorem SecondOrder,SecondOrder1 and equality (10) is the second order convergence forthe integrated numbers of segment crossings:

if X is strictly α-stable (α ∈ (1;2)):ˆR

nyc (X , t)dy− c1−αA · t =⇒ L1 + L2

and if X is strictly 1-stable:ˆR

nyc (X , t)dy− 2

πC0 logc−1 · t−B · t =⇒M1 + M2.

Open questionsNaturally, the next step would be the investigation of the second orderconvergence for the integrated numbers of segment crossings withrespect to the measure f (y)dy :

nyc (X , t) f (y)dy .

Why it may be interesting?

1 The integral´R ny

c (X , t) f (y)dy may reveal much strongerconcentration than ny

c (X , t) for given y ∈ R.2 Investigation together with

´R ny

c (X , t) f (y)dy integrals of the form´R dy

c (X , t) f (y)dy , where dyc (X , t) is the number of

downcrossings of X from above the level y + c to the level y tilltime t, may reveal interesting correction terms.

Open questionsNaturally, the next step would be the investigation of the second orderconvergence for the integrated numbers of segment crossings withrespect to the measure f (y)dy :

nyc (X , t) f (y)dy .

Why it may be interesting?1 The integral

´R ny

c (X , t) f (y)dy may reveal much strongerconcentration than ny

c (X , t) for given y ∈ R.2 Investigation together with

´R ny

c (X , t) f (y)dy integrals of the form´R dy

c (X , t) f (y)dy , where dyc (X , t) is the number of

downcrossings of X from above the level y + c to the level y tilltime t, may reveal interesting correction terms.

Some references[F2010] M. Fukasawa, Central limit theorem for the realized volatilitybased on tick time sampling. Fin. Stoch. 14:209–233, 2010.[Kasahara, 1980] Y. Kasahara. On Levy’s downcrossing theorem. Proc.Japan Acad. Ser. A Math. Sci., 56:455–458, 1980.[LG2014] R. M. Łochowski and R. Ghomrasni. Integral and local limittheorems for numbers of level crossings for diffusions and the Skorohodproblem. Electron. J. Probab. 19(10):1–33, 2014.[LM2013] R. M. Łochowski and P. Miłos. On truncated variation, upwardtruncated variation and downward truncated variation for diffusions.Stochastic Process. Appl. 123:446-474.[LM2014] R. M. Łochowski and P. Miłos. Limit theorems for thetruncated variation and for numbers of interval crossings of Levy andself-similar processes. preprint, available from the web pagewww.akson.sgh.waw.pl\ ∼rlocho

Some references, cont.[L2014] R. M. Łochowski. On the existence of the Riemann-stieltjesintegral. Preprint arXiv:1403.5413, 2014.[T1972] S. J. Taylor. Exact asymptotic estimates of Brownian pathvariation. Duke Math. J., 39(2):219–241, 1972.[Young, 1936] L. C. Young. An inequality of the Holder type, connectedwith Stieltjes integration. Acta Math., 67(1):251–282, 1936.

Thank you!

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