The observations of three-dimensional turbulent fluids indicate that the vorticity field of the fluid is concentrated along thin structures called vortex filaments. In his book Chorin [67] suggests probabilistic descriptions of
vortex filaments by trajectories of self-avoiding walks on a lattice. Flandoli [102] introduced a model of vortex filaments based on a three-dimensional Brownian motion. A basic problem in these models is the computation of the kynetic energy of a given configuration.
Denote by u(x) the velocity field of the fluid at point x∈ R3, and let ξ = curlube the associated vorticity field. The kynetic energy of the field will be
H= 1 2
R3|u(x)|2dx= 1 8π
R3
R3
ξ(x)ãξ(y)
|x−y| dxdy. (5.79) We will assume that the vorticity field is concentrated along a thin tube centered in a curveγ={γt,0≤t≤T}. Moreover, we will choose a random model and consider this curve as the trajectory of a three-dimensional fractional Brownian motion B = {Bt,0 ≤ t ≤T}. This can be formally expressed as
ξ(x) = Γ
R3
T 0
δ(x−y−Bs)Bãsds
ρ(dy), (5.80) where Γ is a parameter called the circuitation, andρis a probability mea- sure on R3 with compact support.
Substituting (5.80) into (5.79) we derive the following formal expression for the kynetic energy:
H=
R3
R3
Hxyρ(dx)ρ(dy), (5.81) where the so-called interaction energyHxy is given by the double integral
Hxy= Γ2 8π
3 i=1
T 0
T 0
1
|x+Bt−y−Bs|◦dBis◦dBti. (5.82) We are interested in the following problems: Is H a well defined ran- dom variable? Does it have moments of all orders and even exponential moments?
In order to give a rigorous meaning to the double integral (5.82) let us introduce the regularization of the function|ã|−1:
σn=|ã|−1∗p1/n, (5.83) where p1/n is the Gaussian kernel with variance n1. Then, the smoothed interaction energy
Hnxy=Γ2 8π
3 i=1
T 0
T 0
σn(x+Bt−y−Bs)◦dBsi
◦dBit, (5.84)
is well defined, where the integrals are path-wise Riemann-Stieltjes inte- grals. Set
Hn=
R3
R3
Hnxyρ(dx)ρ(dy). (5.85) The following result has been proved in [255]:
Theorem 5.4.1 Suppose that the measureρsatisfies
R3
R3|x−y|1−H1ρ(dx)ρ(dy)<∞. (5.86) Let Hnxy be the smoothed interaction energy defined by (5.84). Then Hn defined in (5.85) converges, for allk≥1, inLk(Ω) to a random variable H≥0 that we call the energy associated with the vorticity field (5.80).
If H = 12, fBmB is a classical three-dimensional Brownian motion. In this case condition (5.86) would be
R3
R3|x−y|−1ρ(dx)ρ(dy)<∞, which is the assumption made by Flandoli [102] and Flandoli and Gubinelli [103].
In this last paper, using Fourier approach and Itˆo’s stochastic calculus, the authors show thatEe−βH<∞for sufficiently small negativeβ.
The proof of Theorem 5.4.1 is based on the stochastic calculus of varia- tions with respect to fBm and the application of Fourier transform.
Sketch of the proof: The proof will be done in two steps:
Step 1(Fourier transform) Using 1
|z| =
R3
(2π)3e−iξ,z
|ξ|2 dξ we get
σn(x) =
R3|ξ|−2eiξ,x−|ξ|2/2n dξ.
Substituting this expression in (5.84), we obtain the following formula for the smoothed interaction energy
Hnxy = Γ2 8π
3 j=1
T 0
T 0
R3
eiξ,x+Bt−y−Bs e−|ξ|2/2n
|ξ|2
◦dBjs◦dBtj
= Γ2 8π
R3|ξ|−2eiξ,x−y−|ξ|2/2nYξ2Cdξ, (5.87) where
Yξ = T
0
eiξ,Bt◦dBt
andYξ2C=3 i=1YξiYξ
i. Integrating with respect toρyields
Hn= Γ2 8π
R3Yξ2C|ξ|−2|ρ(ξ)|2e−|ξ|2/2ndξ≥0. (5.88) From Fourier analysis and condition (5.86) we know that
R3
R3|x−y|1−H1ρ(dx)ρ(dy) =CH
R3|ρ(ξ)|2|ξ|H1−4dξ <∞. (5.89) Then, taking into account (5.89) and (5.88), in order to show the con- vergence in Lk(Ω) of Hn to a random variable H≥0 it suffices to check that
E Yξ2kC
≤Ck
1∧ |ξ|k(H1−2)
. (5.90)
Step 2 (Stochastic calculus) We will present the main arguments for the proof of the estimate (5.90) fork= 1. Relation (5.37) applied to the process ut=eiξ,Btallows us to decompose the path-wise integralYξ=T
0 eiξ,Bt◦ dBt into the sum of a divergence plus a trace term:
Yξ = T
0
eiξ,BtdBt+H T
0
iξeiξ,Btt2H−1dt. (5.91) On the other hand, applying the three dimensional version of Itˆo’s formula (5.44) we obtain
eiξ,BT= 1 + 3 j=1
T 0
iξjeiξ,BtδBtj−H T
0
t2H−1|ξ|2eiξ,Btdt. (5.92)
Multiplying both members of (5.92) by iξ|ξ|−2 and adding the result to (5.91) yields
Yξ=pξ
T 0
eiξ,BtdBt
− iξ
|ξ|2
eiξ,BT−1
:=Yξ(1)+Yξ(2),
where pξ(v) =v−|ξξ|2ξ, vis the orthogonal projection of v onξ⊥. It suffices to derive the estimate (5.90) for the term Yξ(1). Using the duality relationship (1.42) for eachj = 1,2,3 we can write
E
Yξ(1),jY(1),jξ
=E 6
eiξ,Bã, pjξDã
pjξ T
0
e−iξ,BtdBt
7
H
. (5.93)
The commutation relation D(δ(u)), hH=u, hH+δ(Du, hH) implies Dkr
T 0
e−iξ,BtdBtj
=e−iξ,Bkrδk,j+ (−iξk) T
0
1[0,t](r)e−iξ,BtdBtj. Applying the projection operators yields
pjξDr
pjξ
T 0
e−iξ,BtdBt
= e−iξ,Br
I−ξ∗ξ
|ξ|2
j,j
= e−iξ,Br
1− ξj2
|ξ|2
.
Notice that the term involving derivatives in the expectation (5.93) van- ishes. This cancellation is similar to what happens in the computation of the variance of the divergence of an adapted process, in the case of the Brownian motion. Hence,
3 j=1
E
Yξ(1),jY(1),jξ
= 2 E@
e−iξ,Bã, e−iξ,Bã A
H
= 2αH
T 0
T 0
E
eiξ,Bs−Br
|s−r|2H−2 dsdr
= 2αH
T 0
T 0
e−|s−r|
2H
2 |ξ|2 |s−r|2H−2dsdr, which behaves as |ξ|H1−2 as |ξ| tends to infinity. This completes the proof of the desired estimate fork= 1.
In the general casek≥2 the proof makes use of thelocal nondeterminism property of fBm:
Var
i
(Bti−Bsi)
≥kH
i
(ti−si)2H.
Decomposition of the interaction energy
Assume 12 < H < 23. For anyx=y,set F
Hxy= 3 i=1
T 0
t 0
1
|x+Bt−y−Bs|◦dBsi
◦dBti. (5.94)
Then HFxy exists as the limit in L2(Ω) of the sequence HFnxy defined using the approximation σn(x) of |x|−1 introduced in (5.83) and the following
decomposition holds F
Hxy = 3 i=1
T 0
t 0
1
|x−y+Bt−Br|dBirdBti.
−H2 T
0
t 0
δ0(x−y+Bt−Br)(t−r)2(2H−1)drdt.
+H(2H−1) T
0
t 0
1
|x−y+Bt−Br|(t−r)2H−2dr
dt +H
T 0
1
|x−y+BT−Br|(T−r)2H−2+ 1
|x−y+Br|r2H−1
dr.
Notice that in comparison withHxy, in the definition ofHFxy we chose to deal with the half integral over the domain
{0≤s≤t≤T},
and to simplify the notation we have omitted the constantΓ8π2. Nevertheless, it holds that Hxy = Γ8π2
F
Hxy+HFyx
, and we have proved using Fourier analysis thatHxy has moments of any order.
The following results have been proved in [255]:
1. All the terms in the above decomposition ofHFxy exists inL2(Ω) for x=y.
2. If |x−y| →0, then the terms behave as |x−y|H1−1, so they can be integrated with respect toρ(dx)ρ(dy).
3. The boundH < 23 is sharp: ForH= 23 the weighted self-intersection local time diverges.
Notes and comments
[5.1] The fractional Brownian motion was first introduced by Kolmogorov [171] and studied by Mandelbrot and Van Ness in [217], where a stochas- tic integral representation in terms of a standard Brownian motion was established.
Hurst developed in [141] a statistical analysis of the yearly water run-offs of Nile river. Suppose that x1, . . . , xnare the values ofnsuccessive yearly water run-offs. Denote by Xn = n
k=1xk the cumulative values. Then, Xk−knXn is the deviation of the cumulative valueXk corresponding tok successive years from the empirical means as calculated using data for n years. Consider the range of the amplitude of this deviation:
Rn = max
1≤k≤n
Xk− k
nXn
− min
1≤k≤n
Xk−k
nXn
and the empirical mean deviation
Sn= GH HI1
n n k=1
xk−Xn
n 2
.
Based on the records of observations of Nile flows in 622-1469, Hurst dis- covered that RSnn behaves ascnH, where H= 0.7. On the other hand, the partial sums x1+ã ã ã+xn have approximately the same distribution as nHx1, where againH is a parameter larger than 12.
These facts lead to the conclusion that one cannot assume thatx1, . . . , xn
are values of a sequence of independent and identically distributed random variables. Some alternative models are required in order to explain the empirical facts. One possibility is to assume thatx1, . . . , xnare values of the increments of a fractional Brownian motion. Motivated by these empirical observations, Mandelbrot has given the name of Hurst parameter to the parameter H of fBm.
The fact that for H > 12 fBm is not a semimartingale has been first proved by [198] (see also Example 4.9.2 in Liptser and Shiryaev [201]).
Rogers in [296] has established this result for anyH =12.
[5.2] Different approaches have been used in the literature in order to define stochastic integrals with respect to fBm. Lin [198] and Dai and Heyde [73] have defined a stochastic integral T
0 φsdBs as limit in L2 of Riemann sums in the caseH > 12. This integral does not satisfy the prop- erty E(T
0 φsdBs) = 0 and it gives rise to change of variable formulae of Stratonovich type. A new type of integral with zero mean defined by means of Wick products was introduced by Duncan, Hu and Pasik-Duncan in [86], assuming H > 12. This integral turns out to coincide with the divergence operator (see also Hu and ỉksendal [140]).
A construction of stochastic integrals with respect to fBm with parame- ter H ∈ (0,1) by a regularization technique was developed by Carmona and Coutin in [58]. The integral is defined as the limit of approximating integrals with respect to semimartingales obtained by smoothing the singu- larity of the kernelKH(t, s). The techniques of Malliavin Calculus are used in order to establish the existence of the integrals. The ideas of Carmona and Coutin were further developed by Al`os, Mazet and Nualart in the case 0< H < 12 in [8].
The interpretation of the divergence operator as a stochastic integral was introduced by Decreusefont and ¨Ust¨unel in [78]. A stochastic calculus for the divergence process has been developed by Al`os, Mazet and Nualart in [9], among others.
A basic reference for the stochastic calculus with respect to the fBM is the recent monograph by Hu [139]. An Itˆo’s formula for H ∈(0,1) in the framework of white noise analysis has been established by Bender in [22].
We refer to [124] and [123] for the stochastic calculus with respect to fBM based on symmetric integrals.
The results on the stochastic calculus with respect to the fBm are based on the papers [11] (caseH > 12), [7] and [66] (case H < 12).
[5.3] In [202], Lyons considered deterministic integral equations of the form
xt=x0+ t
0
σ(xs)dgs,
0≤t≤T, where theg: [0, T]→Rdis a continuous functions with bounded p-variation for somep∈[1,2). This equation has a unique solution in the space of continuous functions of bounded p-variation if each component of g has a H¨older continuous derivative of orderα > p−1. Taking into account that fBm of Hurst parameter H has locally bounded p-variation paths for p >1/H, the result proved in [202] can be applied to Equation (5.78) in the case σ(s, x) =σ(x), and b(s, x) = 0, provided the coefficient σhas a H¨older continuous derivative of orderα > H1 −1.
The rough path analysis developed by Lyons in the [203] is a powerful technique based on the notion ofp-variation which permits to handle dif- ferential equations driven by irregular functions (see also the monograph [204] by Lyons and Qian). In [70] Coutin and Qian have established the existence of strong solutions and a Wong-Zakai type approximation limit for stochastic differential equations driven by a fractional Brownian motion with parameterH > 14 using the approach of rough path analysis.
In [299] Ruzmaikina establishes an existence and uniqueness theorem for ordinary differential equations driven by a H¨older continuous function using H¨older norms.
The generalized Stieltjes integral defined in (5.64), based on the frac- tional integration by parts formula, was introduced by Z¨ahle in [355]. In this paper, the author develops an approach to stochastic calculus based on the fractional calculus. As an application, in [356] the existence and uniqueness of solutions is proved for differential equations driven by a frac- tional Brownian motion with parameterH > 12, in a small random interval, provided the diffusion coefficient is a contraction in the spaceW2,β∞, where
1
2 < β < H. HereW2,β∞denotes the Besov-type space of bounded measur- able functionsf : [0, T]→Rsuch that
T 0
T 0
(f(t)−f(s))2
|t−s|2β+1 dsdt <∞.
In [254] Nualart and Rascanu have established the existence and uniqueness of solution for Equation (5.78) using an a priori estimate based on the fractional integration by parts formula, following the approach of Z¨ahle.
[5.4] The results of this section have been proved by Nualart, Rovira and Tindel in [255].
6
Malliavin Calculus in finance
In this chapter we review some applications of Malliavin Calculus to mathe- matical finance. First we discuss a probabilistic method for numerical com- putations of price sensitivities (Greeks) based on the integration by parts formula. Then, we discuss the use of Clark-Ocone formula to find hedging portfolios in the Black-Scholes model. Finally, the last section deals with the computation of additional expected utility for insider traders.