Truncated variation of a stochastic process - its optimality for …web.sgh.waw.pl/~rlocho/semBerlin.pdf · its value only if it is necessary for the relation kf f ck 1 1 2 c to hold

Truncated variation of a stochastic process - itsoptimality for processes with cadlag trajectories and its

limit distributions for diffusions

Rafa l Lochowski

Warsaw School of Economics

Berlin 2012

Rafa l Lochowski (Warsaw School of Economics)Truncated variation of a stochastic process - its optimality for processes with cadlag trajectories and its limit distributions for diffusionsBerlin 2012 1 / 26

Agenda

1 Definition and properties of truncated variation

2 ”Strong law of large numbers” for truncated variation of a Brownianmotion and continuous semimartingales

3 ”Central limit theorem” for truncated variation of a Brownian motion

4 ”Central limit theorem” for truncated variation of diffusions driven bya Brownian motion

5 Sketches of the proofs


Truncated variation - how it appears

Let f : [a; b]→ R be a cadlag function and let c > 0Question: what is the smallest total variation possible of a (cadlag)function from the ball

{g : ‖f − g‖∞ ≤

12 c}

?

The immediate bound frombelow reads as

infg :‖f−g‖∞≤

12 c

TV (g , [a; b]) ≥ TV c (f , [a; b]) ,

where

TV (g , [a; b]) := supn

supa≤t0<t1<...<tn≤b

n∑i=1

|g (ti )− g (ti−1)| ,

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


n∑i=1

max {|f (ti )− f (ti−1)| − c , 0} ,

and follows immediately from the inequality

|g (ti )− g (ti−1)| ≥ max {|f (ti )− f (ti−1)| − c , 0} .




{g : ‖f − g‖∞ ≤

12 c}

? The immediate bound frombelow reads as


12 c

TV (g , [a; b]) ≥ TV c (f , [a; b]) ,

where

TV (g , [a; b]) := supn


n∑i=1

|g (ti )− g (ti−1)| ,



n∑i=1

max {|f (ti )− f (ti−1)| − c , 0} ,






{g : ‖f − g‖∞ ≤

12 c}

? The immediate bound frombelow reads as


12 c

TV (g , [a; b]) ≥ TV c (f , [a; b]) ,

where

TV (g , [a; b]) := supn


n∑i=1

|g (ti )− g (ti−1)| ,



n∑i=1

max {|f (ti )− f (ti−1)| − c , 0} ,




Truncated variation - definition and another interpretation

We will call the quantity

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


n∑i=1

max {|f (ti )− f (ti−1)| − c , 0} ,

truncated variation of the function f at the level c .

Trunated variation may be interpreted not only as the lower boundobtained on the previous slide but also as the variation taking into accountonly jumps greater than c .We will also show that in fact we have the equality


12 c

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






n∑i=1

max {|f (ti )− f (ti−1)| − c , 0} ,

truncated variation of the function f at the level c .Trunated variation may be interpreted not only as the lower boundobtained on the previous slide but also as the variation taking into accountonly jumps greater than c .

We will also show that in fact we have the equality


12 c

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






n∑i=1

max {|f (ti )− f (ti−1)| − c , 0} ,

truncated variation of the function f at the level c .Trunated variation may be interpreted not only as the lower boundobtained on the previous slide but also as the variation taking into accountonly jumps greater than c .We will also show that in fact we have the equality


12 c

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


Truncated variation - optimality

Moreover, we will show that this lower bound is attainable, i.e. for somefunction f c with ‖f − f c‖∞ ≤

12 c one has


12 c

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

Remark

Since every cadlag function may be uniformly approximated with stepfunctions, total variation of which is finite, hence TV c is finite for everycadlag function.


Truncated variation - optimality

Moreover, we will show that this lower bound is attainable, i.e. for somefunction f c with ‖f − f c‖∞ ≤

12 c one has


12 c

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

Remark

Since every cadlag function may be uniformly approximated with stepfunctions, total variation of which is finite, hence TV c is finite for everycadlag function.


Optimal cadlag function from the ball{g : ‖f − g‖∞ ≤ 1

2c}

We will show also that for any c ≤ sups,u∈[a;b] |f (s)− f (u)| there exist an

unique cadlag function f c : [a; b]→ R such that ‖f c − f ‖∞ ≤12 c and for

any s ∈ (a; b]TV (f c , [a; s]) = TV c (f , [a; s]) .

The function f c appears to be a function starting from appropriate chosenstarting point f c(a) and the most ”lazy” function possible, which changesits value only if it is necessary for the relation ‖f − f c‖∞ ≤

12 c to hold.


Optimal cadlag function from the ball{g : ‖f − g‖∞ ≤ 1

2c}

We will show also that for any c ≤ sups,u∈[a;b] |f (s)− f (u)| there exist an

unique cadlag function f c : [a; b]→ R such that ‖f c − f ‖∞ ≤12 c and for

any s ∈ (a; b]TV (f c , [a; s]) = TV c (f , [a; s]) .

The function f c appears to be a function starting from appropriate chosenstarting point f c(a) and the most ”lazy” function possible, which changesits value only if it is necessary for the relation ‖f − f c‖∞ ≤

12 c to hold.


The function f c and two other functionals related - UTVand DTV

Moreover, we will show also, that function f c may expressed with twoother functionals related - upward and downward truncated variations.

Upward truncated variation is defined with the following formula

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


n∑i=1

max {f (ti )− f (ti−1)− c , 0} ,

and downward truncated variation is defined with the formula

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


n∑i=1

max {f (ti−1)− f (ti )− c , 0} .


The function f c and two other functionals related - UTVand DTV

Moreover, we will show also, that function f c may expressed with twoother functionals related - upward and downward truncated variations.Upward truncated variation is defined with the following formula

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


n∑i=1

max {f (ti )− f (ti−1)− c , 0} ,

and downward truncated variation is defined with the formula

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


n∑i=1

max {f (ti−1)− f (ti )− c , 0} .


TV and two others functionals related - UTV and DTV

Between TV , UTV and DTV the following relations hold

TV c (f , [a; b]) = UTV c (f , [a; b]) + DTV c (f , [a; b]) (1)

andDTV c (f , [a; b]) = UTV c (−f , [a; b]) .

The equality (1) may be viewed as the generalisation of the Hahn-Jordandecomposition of a function with finite total variation.Moreover, it appears that the function f 0,c : [a; b]→ R defined as

f 0,c(s) = UTV c (f , [a; s])− DTV c (f , [a; s])

has increments differing from the increments of f by no more than c .






The equality (1) may be viewed as the generalisation of the Hahn-Jordandecomposition of a function with finite total variation.

Moreover, it appears that the function f 0,c : [a; b]→ R defined as








The equality (1) may be viewed as the generalisation of the Hahn-Jordandecomposition of a function with finite total variation.Moreover, it appears that the function f 0,c : [a; b]→ R defined as




The construction of the function f c

For g : [a; b]→ R we define its oscillation as

‖g‖osc = sups,u∈[a;b]

|g (s)− g (u)| .

Since for g 0,c = f − f 0,c ,∥∥g 0,c

∥∥osc≤ c , hence for some number α,

sups∈[a;b]

∣∣f (s)− f 0,c(s)− α∣∣ = sup

s∈[a;b]

∣∣g 0,c(s)− α∣∣ ≤ 1

2c (2)

and f c may be defned as

f c(s) = f 0,c(s) + α = UTV c (f , [a; s])− DTV c (f , [a; s]) + α.

Remark

When∥∥g 0,c

∥∥osc

= c , then the only number α satisfying (2) is

α = infs∈[a;b]

(g 0,c(s)) +1

2

∥∥g 0,c∥∥osc

= sups∈[a;b]

(g 0,c(s))− 1

2


.





|g (s)− g (u)| .



sups∈[a;b]

∣∣f (s)− f 0,c(s)− α∣∣ = sup

s∈[a;b]

∣∣g 0,c(s)− α∣∣ ≤ 1

2c (2)



Remark

When∥∥g 0,c

∥∥osc


α = infs∈[a;b]

(g 0,c(s)) +1

2


= sups∈[a;b]

(g 0,c(s))− 1

2


.





|g (s)− g (u)| .



sups∈[a;b]

∣∣f (s)− f 0,c(s)− α∣∣ = sup

s∈[a;b]

∣∣g 0,c(s)− α∣∣ ≤ 1

2c (2)



Remark

When∥∥g 0,c

∥∥osc


α = infs∈[a;b]

(g 0,c(s)) +1

2


= sups∈[a;b]

(g 0,c(s))− 1

2


.


Definition of the function f 0,c and construction of f c -problem with adaptation

For any stochastic process Xt , t ≥ 0, with cadlag trajectories we maydefine for every f = X (ω) the functions f 0,c and f c .

Observe that from the definition of f 0,c it follows that the process X 0,c ,defined pathwise as X 0,c

t (ω) = f 0,c(t), where f = X (ω), is adapted to thenatural filtration of the process X , Ft = σ(Xs , 0 ≤ s ≤ t).Different observation may be done for analogously defined process X c . Toconstruct the function f c one needs to know ”future” values of theprocess X .

Remark

It is relatively easy to construct another process X c , adapted to Ft , t ≥ 0,

and such that∥∥∥X − X c

∥∥∥∞≤ 1

2 c and

TV(

X c , [0; t])≤ TV c (X , [0; t]) + c/2.



For any stochastic process Xt , t ≥ 0, with cadlag trajectories we maydefine for every f = X (ω) the functions f 0,c and f c .Observe that from the definition of f 0,c it follows that the process X 0,c ,defined pathwise as X 0,c

t (ω) = f 0,c(t), where f = X (ω), is adapted to thenatural filtration of the process X , Ft = σ(Xs , 0 ≤ s ≤ t).

Different observation may be done for analogously defined process X c . Toconstruct the function f c one needs to know ”future” values of theprocess X .

Remark



∥∥∥∞≤ 1

2 c and

TV(

X c , [0; t])≤ TV c (X , [0; t]) + c/2.





Remark



∥∥∥∞≤ 1

2 c and

TV(

X c , [0; t])≤ TV c (X , [0; t]) + c/2.





Remark



∥∥∥∞≤ 1

2 c and

TV(

X c , [0; t])≤ TV c (X , [0; t]) + c/2.


Definition of the truncated variation, upward truncatedvariation and downward tuncated variation processes

For arbitrary stochastic process Xt , t ≥ 0, with cadlag trajectories, onemay define, adapted to the natural filtration

truncated variation process

TV c (X , t) := TV c (X , [0; t]) ;

upward truncated variation process

UTV c (X , t) := UTV c (X , [0; t])

and downward truncated variation process

DTV c (X , t) := DTV c (X , [0; t]) .


Limit distributions of truncated variation processes ofBrownian motion with drift as c → 0

Let Xt = µt + Bt be a standard Brownian motion with drift µ. SinceBrownian motion has infinite total variation, hence for any t > 0

limc↓0

TV c (X , t) =∞.

It may be interesting (due to the geometric interpretation of truncatedvariation) to investigate the rate of TV c (X , t) for small cs. The answer isthe following

Fact

For any T > 0 process c · TV c (µt + Bt , t) converges almost surely in(C[0; T ],R) topology to the deterministic functionid : [0; T ]→ R, id(t) = t.


Limit distributions of truncated variation processes ofBrownian motion with drift as c → 0

Let Xt = µt + Bt be a standard Brownian motion with drift µ. SinceBrownian motion has infinite total variation, hence for any t > 0

limc↓0

TV c (X , t) =∞.

It may be interesting (due to the geometric interpretation of truncatedvariation) to investigate the rate of TV c (X , t) for small cs. The answer isthe following

Fact

For any T > 0 process c · TV c (µt + Bt , t) converges almost surely in(C[0; T ],R) topology to the deterministic functionid : [0; T ]→ R, id(t) = t.


Generalisation for continuous semimartingales

Let now Xt , t ≥ 0, be continuous semimartingale, with the decompositionXt = X0 + Mt + At , with Mt being a local martingale and At being acontinuous process with finite total variation.

Using the inequality max {|x + y | − c , 0} ≤ max {|x | − c , 0}+ |y | weobtain, that

TV c (Xt , t) ≤ TV c (Mt , t) + TV 0 (At , t)

andTV c (Mt , t) ≤ TV c (Xt , t) + TV 0 (−At , t) .

Hence we conclude easily that

limc↓0

c · TV c (X , t) = limc↓0

c · TV c (M, t)

whenever any of the above limits exist.



Let now Xt , t ≥ 0, be continuous semimartingale, with the decompositionXt = X0 + Mt + At , with Mt being a local martingale and At being acontinuous process with finite total variation.Using the inequality max {|x + y | − c , 0} ≤ max {|x | − c , 0}+ |y | weobtain, that




limc↓0


c · TV c (M, t)




Let now Xt , t ≥ 0, be continuous semimartingale, with the decompositionXt = X0 + Mt + At , with Mt being a local martingale and At being acontinuous process with finite total variation.Using the inequality max {|x + y | − c , 0} ≤ max {|x | − c , 0}+ |y | weobtain, that




limc↓0


c · TV c (M, t)



Generalisation for continuous semimartingales, cont.

We will use the Fact from the previous slide, Dambis and Dubins-SchwarzTheorem saying that every continuous, local martingale Mt , t ≥ 0, withM0 = 0 and infinite total variation may be represented as

Mt = B〈M,M〉t ,

where Bt is a standard Brownian motion

and the fact that the truncatedvariation does not depend on continuous and strictly increasing change oftime variable. Utilizing above facts, we obtain that

limc↓0

c ·TV c (X , t) = limc↓0

c ·TV c (M, t) = limc↓0

c ·TV c (B, 〈M,M〉t) = 〈M,M〉t .

Noticing that 〈X ,X 〉t = 〈M,M〉t , we finally obtain

limc↓0

c · TV c (X , t) = 〈X ,X 〉t .




Mt = B〈M,M〉t ,

where Bt is a standard Brownian motion and the fact that the truncatedvariation does not depend on continuous and strictly increasing change oftime variable.

Utilizing above facts, we obtain that

limc↓0



c ·TV c (B, 〈M,M〉t) = 〈M,M〉t .


limc↓0

c · TV c (X , t) = 〈X ,X 〉t .




Mt = B〈M,M〉t ,

where Bt is a standard Brownian motion and the fact that the truncatedvariation does not depend on continuous and strictly increasing change oftime variable. Utilizing above facts, we obtain that

limc↓0



c ·TV c (B, 〈M,M〉t) = 〈M,M〉t .


limc↓0

c · TV c (X , t) = 〈X ,X 〉t .




Mt = B〈M,M〉t ,

where Bt is a standard Brownian motion and the fact that the truncatedvariation does not depend on continuous and strictly increasing change oftime variable. Utilizing above facts, we obtain that

limc↓0



c ·TV c (B, 〈M,M〉t) = 〈M,M〉t .


limc↓0

c · TV c (X , t) = 〈X ,X 〉t .


”Second order” convergence for Brownian motion as c → 0

Knowing that for a standard Brownian motion

c · limc↓0

TV c (B, t) = t a. s.

we may try to investigate the (normalised) difference TV c (B, t)− tc .

It appears that

(Bt ,TV c (B, t)− t

c

)→d

(Bt ,

Bt√3

)

where the convergence →d is understood as the weak convergence in(C[0; T ],R2) topology, and B is another standard Brownian motion,independent from B.


”Second order” convergence for Brownian motion as c → 0

Knowing that for a standard Brownian motion

c · limc↓0

TV c (B, t) = t a. s.

we may try to investigate the (normalised) difference TV c (B, t)− tc .

It appears that

(Bt ,TV c (B, t)− t

c

)→d

(Bt ,

Bt√3

)

where the convergence →d is understood as the weak convergence in(C[0; T ],R2) topology, and B is another standard Brownian motion,independent from B.


”Second order” convergence for Brownian motion withdrift as c → 0

Moreover, for standard Brownian motion with drift, Wt = µt + Bt , we have

(Wt ,Tt ,Ut ,Dt)→d

(Wt ,

Bt√3,

Bt +√

3Bt

2√

3,

Bt −√

3Bt

2√

3

)(3)

where

Tt = TV c (W , t)− t

c,

Ut = UTV c (W , t)−(

1

2c+µ

2

)t,

Dt = DTV c (W , t)−(

1

2c− µ

2

)t

and the convergence →d is understood as the weak convergence in(C[0; T ],R4) topology and B is again a standard Brownian motionindependent from B.


”Second order” convergence for diffusions c → 0

Consider autonomus diffusion process defined as the (unique) strongsolution of the sde

dXt = µ (Xt) dt + σ (Xt) dBt .

For the process Xt and trunceted variation processes of Xt , under theassumption that µ, σ are lipschitz and σ is strictly positive, we haveanalogous convergence as in (3)

(Xt , Tt , Ut , Dt

)→d

(Xt ,

Bt√3,

Bt

2√

3,

Bt

2√

3

)(4)

where Tt = TV c (X , t)− 〈X 〉tc , Ut = UTV c (X , t)− 12

(〈X 〉tc + Xt

), Dt =

DTV c (X , t)− 12

(〈X 〉tc − Xt

).


The function f c - construction

In order to define the function f c we will need some other definitions. Forc > 0 we define stopping times

T cD f = inf

{s ≥ a : sup

t∈[a;s]f (t)− f (s) ≥ c

},

T cU f = inf

{s ≥ a : f (s)− inf

t∈[a;s]f (t) ≥ c

}.

We will assume that T cD f ≥ T c

U f i.e. that the first upward jump of thefunction f of size c appears before the first downward jump of the samesize c or both times are infinite, i.e. there is no upward or downward jumpof size c .Note that in the case T c

D f < T cU f we may simply consider the function −f .


The function f c - definition, cont.

Now define sequences(

T cU,k

)∞k=0

,(

T cD,k

)∞k=−1

, in the following way:

T cD,−1 = a, T c

U,0 = T cU f and for k = 0, 1, 2, ...

T cD,k = inf

s ≥ T cU,k : sup

t∈[T cU,k ;s]

f (t)− f (s) ≥ c

,

T cU,k+1 = inf

{s ≥ T c

D,k : f (s)− inft∈[T c

D,k ;s]f (t) ≥ c

}.

Remark

Note that there exists such K <∞ that T cU,K =∞ or T c

D,K =∞.Otherwise we would obtain two infinite sequences (sk)∞k=1 , (Sk)∞k=1 suchthat a ≤ s1 < S1 < s2 < S2 < ... ≤ b and f (Sk)− f (sk) ≥ 1

2 c . But this isa contradiction, since f is a cadlag function and (f (sk))∞k=1 , (f (Sk))∞k=1

have a common limit.


Some pictures

TU,0c TD,0

c TU,1c

c

c

c

-0.5

0.0

0.5

1.0

1.5

2.0

2.5


The function f c - definition, cont.

Now let us define two sequences of non-decreasing functions

mck :[T cD,k−1; T c

U,k

)→ R and Mc

k :[T cU,k ; T c

D,k

)→ R for such k that

T cD,k−1 <∞ and T c

U,k <∞ respectively, with the formulas

mck (s) = inf

t∈[T cD,k−1;s]

f (t) ,Mck (s) = sup

t∈[T cU,k ;s]

f (t) .

The function f c is now defined with the formulas

f c (s) =

inft∈[0;T c

U f ) f (t) + 12 c if s ∈

[a; T c

U,0

);

Mck (s)− 1

2 c if s ∈[T cU,k ; T c

D,k

), k = 0, 1, 2, ...;

mck+1 (s) + 1

2 c if s ∈[T cD,k ; T c

U,k+1

), k = 0, 1, 2, ....


Some pictures, cont.

-1

0

1

2


Why the function f c is optimal?

The function f c has the smallest total variation possible, since it is

monotonic on every interval of the form[T cD,k−1; T c

U,k

)or[T cU,k ; T c

D,k

),

k = 0, 1, 2, ...,K , and its variation on these intervals reads as

supt∈[T c

D,k−1;T cU,k)

f (t)− inft∈[T c

D,k−1;T cU,k)

f (t)− c

orsup

t∈[T cU,k ;T c

D,k)f (t)− inf

t∈[T cU,k ;T c

D,k)f (t)− c

respectively, thus is the smallest possible for the function from the ball{g : ‖f − g‖∞ ≤

12 c}.


Why the function f c is optimal? - cont.

There is some more accurate reasoning needed if the domain of the

function f is not just a sum of intervals of the form[T cD,k−1; T c

U,k

)and[

T cU,k ; T c

D,k

), k = 0, 1, 2, ...,K , and it is done with the minimal

decomposition of the function f c − f c(a) into a difference of twonon-decreasing functions f c

U and f cD (cf. [L2011b]).

This is possible, since its domain is a sum of the disjoint intervals, where itis monotonic, thus it has finite total variation.

The function f cU is constant on the intervals

[T cD,k−1; T c

U,k

)and the

function f cD is constant on the intervals

[T cU,k ; T c

D,k

), k = 0, 1, 2, ...,K .

From the very construction of the function f c it also follows that itbelongs to the ball

{g : ‖f − g‖∞ ≤

12 c}

and it is a cadlag function withjumps possible only in the points where also f has its jumps.


References

[L 2011a] Lochowski, R., Truncated variation, upward truncated variationand downward truncated variation of Brownian motion with drift - theircharacteristics and applications Stoch. Proc. Appl. 121, 378-393

[L 2011b] Lochowski, R., On pathwise uniform approximation of processeswith cadlag trajectories by processes with minimal total variation arXive-prints

[LM 2011] Lochowski, R., Mi los, P., On truncated variation, upwardtruncated variation and downward truncated variation for diffusionsStoch. Proc. Appl., to appear


Thank you!


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Truncated variation of a stochastic process - its optimality for …web.sgh.waw.pl/~rlocho/semBerlin.pdf · its value only if it is necessary for the relation kf f ck 1 1 2 c to hold