On Free Stochastic Differential Equations

0.21 MB·English

by  Vladsislav Kargin

ON FREE STOCHASTIC DIFFERENTIAL EQUATIONS
V.KARGIN
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Abstract
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The paper derives an equation for the Cauchy transform of the solution of a

n
freestochasticdifferentialequation(SDE).Thisnewequationisusedtosolve
a
J severalparticular examplesoffreeSDEs.

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1. INTRODUCTION
P
h. Free stochastic differential equations generalize classical stochastic differential equations
at tothesetting offreeprobability. Hereisanexampleofsuchanequation:
m
[ Xt = a(Xt)dt+b∗(Xt)(dZt)b(Xt)

1
Inthisequation, X isaself-adjoint operator, a(X )andb(X )areoperator-valued functions
v t t t
7 ofX ,andthedrivingnoiseZ isanoperatorprocesswithfreeincrements. Thatis,theincre-
t t
9
ments Z Z ,s > t,areassumed tobe freefrom past realizations ofZ .Theprocess Z is
6 s t t t
−
2 usuallythefreeBrownianprocess,inwhichcasetheincrementshavesemicircledistributions;
.
1 however,otherchoicesarepossible.
0
Informally, thereadermaythinkabout X asverylargerandom matricesandZ asmatri-
1 t t
1 ceswithindependent Gaussianrandomvariablesasentries. Theseentriesfollowindependent
:
v Brownian motions andweareinterested inthelawofthe eigenvalues ofX .Thefreeproba-
t
i
X bilitytheoryisaconvenient abstraction whichintendstomodelthesituation whenthesizeof
r thematricesisverylarge.
a
Thestudy offreestochastic differential equations (“freestochastic calculus”) ismoredif-
ficult than in the classical case because of non-commutativity of coefficients and noise. This
papercontributes bydeveloping anewtoolfortheanalysis oftheseequations.
The idea of free stochastic calculus was first suggested in [S]. It was later developed and
formalized in[KS],[B],[BSa],and[A],whichintroduced stochastic integration withrespect
tofreeBrownianmotionasarigorous basisforfreestochastic calculus. Theyalsoderivedan
analogoftheItoˆ formula,whichallowsustoobtainidentities likethefollowing:
a a
[W2(dW )+W (dW )W +(dW )W2] = W3 2 W dt,
t t t t t t t a − t
Z0 Z0
Date:November2010.
DepartmentofMathematics,StanfordUniversity,PaloAlto,CA94305,USA.e-mail:[email protected].
1
2 V.KARGIN
whereW denotes thefreeBrownianmotion(theWignerprocess). Ananalogous formulain
t
theclassical situation is
a a
3B2(dB ) = B3 3 B dt,
t t a − t
Z0 Z0
where B is the standard Brownian motion. Note the different coefficient before the integral
t
ontheright-hand side.
The classical Itoˆ formula is very helpful in the study of stochastic differential equations.
Unfortunately, the range of applicability of the free Itoˆ formula is smaller. This difficulty
callsforadifferent methodapplicable tothosefreeSDEs,whicharenotsolvable withtheItoˆ
formula. Onepossibility istoseekanequationfortheevolution ofthespectraldistribution of
thesolution. Suchanequation wasderivedbyBianeandSpeicherin[BSb]fortheequation
X = a(X )dt+dW . (1)
t t t
They showed that the density of the spectral probability measure of X , which we denote p
t t
andwhichgeneralizes theeigenvalue distribution ofamatrix,satisfiesthefreeFokker-Planck
equation:
∂p ∂
t
= [p (Hp +a)]. (2)
t t
∂t −∂x
HereH denotes amultipleoftheHilberttransform:
u(y)
Hu(x) := p.v. dy. (3)
x y
Z −
Still, the approach through the free Fokker-Planck equation has its own disadvantages.
First,itisapplicableonlytoequationsthathavethespecialform(1),thatis,onlytoequations
with the constant diffusion coefficient. Second, the free Fokker-Planck equation (2) is not a
bonafidepartialdifferential equationsinceitincludestheHilberttransformoperator. Forthis
reason, itissomewhatdifficulttosolvethisequation.
Thepurpose ofthispaperistoapproach thefreestochastic equations byderiving adiffer-
entialequation fortheCauchytransform ofthesolution.
Recall that the resolvent of operator X is defined as the operator-valued function of a
t
complex parameter Gt(z) := (Xt z)−1. The Cauchy transform of Xt is defined as the
−
expectation of the resolvent: gt(z) := E[(Xt z)−1]. It is useful because the knowledge
−
of the Cauchy transform is sufficient to recover all properties of the spectral probability
distribution ofX . ItturnsoutthatifX solves
t t
dX = a(X )dt+b(X )(dW )c(X ),
t t t t t
theng (z)satisfiesthefollowingequation:
t
dg
t = E(a G2)+E(b c G )E(b c G2), (4)
dt − t t t t t t t t
where we use a , b , and c to denote a(X ), b(X ), and c(X ), respectively. This is the
t t t t t t
statementofTheorem3.2below.
Equation (4)isnotausual differential equation since itinvolves expectations. Ingeneral,
these expectations are difficult to compute because the coefficients a , b , and c , and the
t t t
resolvent G are not free from each other. However, if the coefficients are polynomials, it is
t
ONFREESTOCHASTICDIFFERENTIALEQUATIONS 3
possibletoperformfurtherreductiontoadifferentialequationaswewillshowinProposition
3.4.
Aswejustsaid,theknowledgeoftheCauchytransformcanbeusedtorecoverthespectral
probabilitydistribution. Inparticular,thefreeFokker-Planckequation(2)canbederivedfrom
(4)aswillbeshowninCorollary3.6.
IncertaincasesitisnotpossibletocomputetheCauchytransformexplicitly,butitispossi-
bletodetectthebehaviorofitssingularities. Thisknowledgecanprovideuswithinformation
aboutthesupportofthespectraldistribution. Inparticular,itcanshowushowthenormofthe
solution grows.
For a simple example of this approach, let us consider the well-known case of the free
Ornstein-Uhlenbeck equation:
dX = θX dt+σdW . (5)
t t t
−
For this equation, it is easy to compute the Cauchy transform using equation (4) and re-
cover the known result that for positive θ, the spectral probability distribution of X con-
t
verges to a stationary solution, which is a semicircle distribution supported on the interval
[ σ 2/θ ,σ 2/θ ].
− | | | |
Asamoredifficultexample,consider theequation
p p
1/2 1/2
dX = θX dt+X (dW )X ,
t t t t t
whichcanbethoughtofasafreeanalogoftheequationforthe“geometricBrownianmotion”,
dx = θx dt+x dB .
t t t t
LetX = I.Equation(4)leadstothefollowingdifferential equation:
0
∂g ∂g
+z(θ 1 zg) = g(θ 1 zg),
∂t − − ∂z − − −
with the initial condition g(0,z) = (1 z)−1. The method of characteristics gives us a
−
functional equation fortheCauchytransform:
z+g 1 = e(θ 1 zg)t. (6)
− − −
θ=1/2
1.4
1.2 t=0.1
t=0.2
1 t=0.3
0.8 t=0.4
0.6
0.4
0.2
00 0.5 1 1.5 2 2.5 3 3.5 4
FIGURE 1. θ = 1/2
Whileitisdifficulttoextractanexplicitanalyticalformulaforthesolutionofthisequation,
wecaninvestigate howthesupport ofthedistribution changes withtime. Itturns outthatfor
4 V.KARGIN
θ=2
1.4
1.2
t=0.1
1
t=0.2
t=0.3
0.8
t=0.4
0.6
0.4
0.2
00 1 2 3 4 5 6 7
FIGURE 2. θ = 2
θ=−1
1.6
t=0.1
1.4
t=0.2
1.2
t=0.3
1
t=0.4
0.8
0.6
0.4
0.2
00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
FIGURE 3. θ = 1
−
θ=−1, larger times
4.5
4
t=1
3.5
3
2.5 t=0.75
2 t=0.5
1.5
t=0.25
1
0.5
00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
FIGURE 4. θ = 1,Largetimes
−
ONFREESTOCHASTICDIFFERENTIALEQUATIONS 5
θ < 0,the support of thedistribution shrinks tozero. If θ is between 0and 1, then the lower
boundaryofthesupportdecreasestozeroandtheupperboundarygrowsexponentially fastto
infinity. Ifθ > 1, then both thelowerand upper boundary ofthe support grow exponentially
fasttoinfinity.
Wecan solve equation (6)numerically and recover thedensity ofthe spectral distribution
by using Stieltjes formula. Figures 1-4 show the evolution of the density for various values
of parameters and illustrate the complexity of the behavior of the spectral distribution. For
example, Figure 2 shows that even if θ > 1, the spectral distribution does not approach
infinityimmediately. Thereisatransition periodinwhichasignificant portionofthespectral
distribution remains below λ = 1. Similarly, Figure 3 shows that for θ < 0, the distribution
does not collapse to zero immediately. Onlywhen time increases, the distribution begins the
rapidapproach tozero,asshowninFigure4.
Letuscomparethisresultwiththeclassical analog. ByusingtheItoˆ formula, itiseasyto
showthatthesolution oftheclassical equation forthegeometric Brownianmotionis
1
x = exp (θ )t+B .
t t
{ − 2 }
Hence,withprobability1,theclassicalsolutionwilldecreaseexponentiallytozeroifθ < 1/2,
and will grow exponentially to infinity if θ > 1/2. However, the support of the solution
distribution is(0, )forallt. Thisisquiteunlikethebehavior ofthefreeSDEsolution.
∞
Note that the equation for the geometric Brownian motion can be generalized to the free
probability settinginadifferentway:
dX = θX dt+X dW +(dW )X .
t t t t t t
Thebehavior ofthe solution ofthis equations is quite different. In particular, the ratio ofthe
standarddeviationtotheexpectationis 2(e2t 1).Thisratiogrowsexponentiallyfastwith
−
t, quite unlike the previous example, where this ratio equals √t. Unfortunately, the partial
p
differential equation associated with equation is more difficult to solve and it is not clear
whetherthesolution becomesunbounded infinitetime.
Finally,letusconsider thefollowingequation:
dX = kX (dW )X ,
t t t t
and let the initial condition be X = aI. For this equation it is possible to write an explicit
0
formula for thespectral distribution ofthesolution. Aninteresting feature ofthis equation is
that the solution blows up in finite time τ = (ak) 2, by which we means that the operator
−
normofthesolution becomesinfiniteastapproaches τ.
Another interesting feature is that as time t approaches τ, the spectral distribution con-
vergestoafixeddistribution. Ifk = a 1,thenthedensityofthisdistribution is
−
√4ξ 1
f(ξ) = − ,
2πξ3
6 V.KARGIN
2
1.8
1.6
t=0.4
1.4
1.2
t=0.3
1
0.8 t=0.2
0.6
t=0.1
0.4
0.2
00 0.5 1 1.5 2 2.5 3 3.5 4
FIGURE 5. Densityfunctions fort = 0.1,0.2,0.3,0.4.
2.5
2
1.5
1
0.5
0 0 0.5 1 1.5 2 2.5 3 3.5 4
FIGURE 6. Densityfunctions fort =0.5,0.6,0.7,0.8
which is supported on the interval [1/4, ). Otherwise, it is a scaled version of this distri-
∞
bution. The behavior of the solution density for various times is illustrated in Figures 5 and
6.
SeveralspecificclassesoffreeSDEhavealreadybeeninvestigated intheliterature. Biane
andSpeicherin[BSb]andGaoin[G]studiedthefreeOrnstein-Uhlenbeckequation(5). Biane
and Speicher proved that its solution converges to a stationary process with a semicircle dis-
tribution. Gaoconsidered free Ornstein-Uhlenbeck processes withafree Levydriving noise,
andshowedthateveryself-decomposable probability measureonthereallinecanberealized
asadistribution ofsuchaprocess.
Capitaine and Donati-Matin in[CD-M]defined the freeWishart process and found that it
satisfiesthefreeSDEoftheform:
dX = λdt+ X dZ +dZ X ,
t t t t∗ t
p p
ONFREESTOCHASTICDIFFERENTIALEQUATIONS 7
where Z is the complex Wigner process. Demni in [D] studied the so-called free Jacobi
t
processes whichsatisfyequations similartothefollowing:
dX = (θI X)dt+ I X dZ X + X dZ I X .
t − − t t t t t∗ − t
p p p p
With exception of the free Ornstein-Uhlenbeck process, we study a different set of free
SDEs,andweapproachtheseequationswithadifferentpointofviewbasedonthedifferential
equations fortheCauchytransform.
ForthefreeOrnstein-Uhlebeck process, ourresultsagreewithresultsin[BSb].
The rest of the paper is organized as follows. Section 2 provides preliminary information
about free stochastic integration and Itoˆ formulas. Section 3describes main results. . In par-
ticular, Section3.1isdevoted toalocalexistence anduniqueness result. Section3.2presents
generalresultsabouttheCauchytransformofthesolutionandSection3.3providesexamples.
2. FREE STOCHASTIC INTEGRATION
2.1. ThefreeBrownianmotion. Forthebasicsoffreeprobabilitytheorywereferto[VDN]
and[NS]. Alloperators thatweconsider belong toanon-commutative W -probability space
∗
( ,E), that is, to a von Neumann operator algebra with a faithful normal trace E. We
A A
denotetheusualoperatornormby X ,andtheL2-normby X := E(X X).
2 ∗
k k k k
The spectral probability distribution of a self-adjoint operator X is a probability
p∈ A
measureµonRsuchthat
E Xk = xkµ(dx).
R
(cid:16) (cid:17) Z
ItsCauchytransform isthefunction
µ(dx)
g (z) = .
X
R x z
Z −
ItcanbedefineddirectlyintermsofoperatorX astheexpectationoftheresolvent: g (z) =
X
E[GX (z)],whereGX (z) := (X z)−1.Theprobability measureµcanberecovered from
−
itsCauchytransform bytheStieltjesinversion formula:
1
µ(B)= lim Img(x+iε)dx, (7)
π ε↓0 ZB
provided thatB isBorelandµ(∂B) = 0.
This fact is the starting point of our approach, since we will study the evolution of the
Cauchytransform asatooltoinvestigate theevolution ofthecorresponding probability mea-
sure.
The most important concept in free probability theory is that of free independence. Let
A denote an arbitrary element of algebra . The sub-algebras ,..., of algebra
i i 1 n
A A A A
(and operators that generate them) are said to be freely independent or free, if the following
condition holds:
E(A ...A ) = 0,
i(1) i(m)
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provided thatE(A ) = 0and i(s+1) = i(s)foreverys. Twoparticular consequences of
i(s)
6
thisdefinitionisthat(i)E(AB) = E(A)E(B)ifAandB arefree,and(ii)
E(AX AX ) = E(A2)E(X )E(X ), (8)
1 2 1 2
ifAisfreefromX andX andE(A) =0.
1 2
ThefreeBrownianmotion,ortheWignerprocess,isafamilyofoperatorsW ,wheret 0,
t
≥
that satisfies the following properties: (1) W = 0; (2) the increments of W are free in the
0 t
sense of Voiculescu, i.e., if t > s, then W W is free from the subalgebra which is
t s s
− W
generated by all W with τ s , and (3) the spectral distribution of W W is semicircle
τ t s
≤ −
withzeroexpectation andvariancet s.
−
The choice of W is not unique, and in the rest of the paper we assume that a particular
t
realization ofW isfixed.
t
2.2. Free stochastic integral. Itoˆ-style free stochastic integration with respect to the free
Brownian motion was defined and studied in [KS] and [BSa]. Their results show that under
certainassumptions ontheoperator coefficientsa andb ,itispossibletodefinetheintegral
t t
1
I = a (dW )b ,
t t t
Z0
whereW isthefreeBrownianmotion.
t
Let us briefly recall the construction of the integral. For details of the construction, the
readerisadvisedtoseeDefinition2.2.1andSection3in[BSa]andSection3andTheorem14
in[A]. Suppose thata andb arefunctions ofW ,τ t.Thatis,leta andb belong tothe
t t τ t t
≤
sub-algebra . Assume also that max a , b C for all t [0,1] and that t a
t t t t
W {k k k k} ≤ ∈ →
and t b are continuous mappings in the operator norm. Let t ,...,t and τ ,...,τ be
t 0 n 1 n
→
realnumberssuchthat
0 = t t ... t = 1,
0 1 n
≤ ≤ ≤
and
0 τ t .
k k 1
≤ ≤ −
Wedenotethesetoft ,...,t andτ ,...,τ as∆. Let
0 n 1 n
d(∆) = max (t τ ).
k k
1 k n −
≤ ≤
Considerthesum
n
I(∆) = a (W W )b
τi ti − ti−1 τi
i=1
X
It turns out that as d(∆) 0, the sums I(∆) converge in operator norm and the limit
→
does not depend on the choice of t and τ . The limit is called the free stochastic integral
i i
1
and denoted as a (dW )b . An important point in the proof of convergence is that the
0 t t t
convergence of sums in the operator norm depends on a free analogue of the Burkholder-
R
Gundymartingaleinequalities.
ONFREESTOCHASTICDIFFERENTIALEQUATIONS 9
Averyusefultoolinthestudyofstochastic integralsistheItoˆ formula. Afreeprobability
analogueoftheItoˆ formulawasdevelopedin[BSa]. Intermsofformalrules,itcanbewritten
asfollows:
a dt b dt = a dt b dW c = a dW b c dt = 0,
t t t t t t t t t t
· · ·
a dW b c dW d = E(b c )a d dt. (9)
t t t t t t t t t t
·
Notethattheruleinthesecondlineissignificantly different fromtheclassical case. Interms
offreestochastic integrals, thesecondrulecanbewrittenasfollows:
1 1 1 t
a dW b c dW d = ( a dW b )c dW d
t t t t t t τ τ τ t t t
·
Z0 Z0 Z0 Z0
1 t
+ a dW b ( c dW d )
t t t τ τ τ
Z0 Z0
1
+ E(b c )a d dt.
t t t t
Z0
(Compare Theorem 4.1.2. in [BSa] or Proposition 8 and Corollary 10 in [A].) Here is an
illustration (aparticular caseofProposition 4.3.2in[BSa]):
LetW bethefreeBrownianmotionanddefine
t
∂(Wn) := Wn 1dW +Wn 2(dW )W +...+(dW )Wn 1.
t t − t t − t t t t −
Then,
α
(W )n = d(Wn)
a t
Z0
α α
= ∂(Wn)+ Wn k l 2(dW )Wk(dW )Wl,
t t − −− t t t t
Z0 Z0 0 k+l n 2
≤ X≤ −
anditfollowsthat
a ⌊n/2⌋−1 a
∂(Wn) = (W )n (n 2k 1)C Wn 2k 2tkdt, (10)
t a − − − k t − −
Z0 k=0 Z0
X
whereC aretheCatalannumbers,
k
1 2k
C := = E[(W )2k].
k 1
k+1 k
(cid:18) (cid:19)
AnanalogousformulafortheclassicalItoˆ integralwithrespecttotheBrownianmotionB
t
isquitedifferent:
a n(n 1) a
nBtn−1(dBt) = Ban− 2− Btn−2dt.
Z0 Z0
Below,wewillusethefreeItoˆ formulainordertocomputethemomentsofvariableX .
t
10 V.KARGIN
3. FREE STOCHASTIC DIFFERENTIAL EQUATIONS
3.1. Existenceanduniqueness. Afreestochastic differential equation (freeSDE)
dX = a(X )dt+b(X )(dW )c(X ) (11)
t t t t t
isaconvenient shortcut notation forthefollowingintegralequation:
t t
X = X + a(X )dτ + b(X )dW c(X ). (12)
t 0 τ τ τ τ
Z0 Z0
Weconsideronlyequationswiththecoefficientsthatdonotdependexplicitlyontime,andwe
willalwaysassumethata(X ),b(X ),andc(X )arelocallyoperator Lipschitzfunctions. (A
t t t
functionf : R Ciscalledlocallyoperator-Lipschitz, ifitisalocallybounded,measurable
→
function, andifforallA > 0,thereisaconstant K > 0,suchthat
A
f(X) f(Y) K X Y
A
k − k ≤ k − k
forallself-adjointoperatorsX andY withthenormlessthanA.Forexample,allpolynomials
arelocallyoperator-Lipschitz.)
Equations(11)and(12)areparticular casesofthefollowingmoregeneralequations:
m
dX = a(X )dt+ b (X )(dW )c (X ) (13)
t t i t t i t
i=1
X
and
t m t
X = X + a(X )dτ + b (X )dW c (X ). (14)
t 0 τ i τ τ i τ
Z0 i=1Z0
X
OurresultsfortheCauchytransformofX canbeextendedtothismoregeneralsettingat
t
theexpenseofmorecumbersomenotation.
Theexistenceofthesolutionofequation(14)mayfailforlargetifthenormofthesolution
approachesinfinityinfinitetime. However,forsufficientlysmallt > 0wehavethefollowing
local existence result. (See Theorem 3.1 in [BSb] for a sufficient condition of the global
existence in a simpler class of free SDEs, and Theorem 5.2.1 in [O] for an existence and
uniqueness resultinthecaseofclassical SDEs).
Theorem 3.1. Suppose that a , b , and c are locally operator Lipshitz functions and X is
i i i
bounded inoperator norm. Then, there exist t > 0andafamilyofoperators X defined for
0 t
allt [0,t )andbounded inoperatornorm,suchthatX = X,andX isauniquesolution
0 0 t
∈
of(14)fort < t .
0
Proof: TheproofproceedsbyPicard’smethodofsuccessiveapproximations. Wewillgive
(0)
theproveforthecasem = 1.Thegeneralcaseissimilar. DefineX =X,and
t
t t
X(N+1) = X + a(X(N))dτ + b(X(N))dW c(X(N)). (15)
t τ τ τ τ
Z0 Z0
Weaimtoshowthatthisprocessconvergesforallsufficientlysmallt. Forthis,itisenoughto
showthatforasufficiently smallt > 0 andallt < t andN 1,thefollowing twoclaims
0 0
≥