Some problems in stochastic analysis : Itô's formula for convex functions, interacting particle systems and Dyson's Brownian motion

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AUTHOR:

Nastasiya Grinberg

DEGREE:

Ph.D.

TITLE:

Some problems in stochastic analysis: Itô’s formula for convex functions, in-teracting particle systems and Dyson’s Brownian motion

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M A

E

G NS

I T A T MOLEM

U N

IV ER

SITAS WARWICEN

SIS

Some problems in stochastic analysis: Itô’s formula for

convex functions, interacting particle systems and

Dyson’s Brownian motion

by

Nastasiya Grinberg

Thesis

Submitted to the University of Warwick

for the degree of

Doctor of Philosophy

Department of Statistics

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Acknowledgments

First of all I would like to thank my supervisors Prof. Wilfrid Kendall and Dr. Jon Warren, both of whom have been infinitely patient with me all this time, and whose invaluable help allowed me to finish this exciting but tough journey. I will especially hold dear the pre-submission e-mail frenzy that I have subjected Jon to, in what seems like, in vain.

I am also grateful to all of the friends I have made in the Statistics Department and especially to my housemates, Peter and Martin, all of whom supported and encour-aged me when I most needed it.

Needless to say that I would not have been where I am now if not for my parents and my sister and their unconditional love and support. Doing most things in life would have been pointless if not for my family.

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Declarations

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Abstract

This thesis consists of two main parts: Chapter 1 is concerned with studying an ex-tension of the Itô lemma to the convex functions. We prove that the local martingale part of the decomposition of a convex function f of a continuous semimartingale can be expressed in a similar way to the classical formula with the gradient of f replaced with itssubgradient. The result itself is not new, however, our approach via Brownian perturbation is.

The second, and the largest, part of the thesis focusses on the study of a certain family of bivariate diffusionsZ(θ,µ)= (X,R)in a wedgeW ={(x,r)∈R×R+:|x| ≤r},

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Chapter 1

Itô’s formula for convex functions

of continuous semimartingales:

Brownian perturbation approach.

The first chapter of this Thesis concentrates on the study of convex functions in semi-martingale theory; it is self-contained and covers a topic different from the rest of the Thesis.

We consider an extension of the celebrated Itô’s formula for the continuous semimartingales to the class of convex functions. In particular we prove that the local martingale part of a convex function f of a d-dimensional semimartingaleX =M+A can be written in terms of an Itô stochastic integralRH(X)d M, where H(x) is some particular measurable choice of subgradient∇f(x)of f at x, andM is the martingale part of X. This result was first proved by Bouleau in[15]. Here we present a new treatment of the problem. We first prove the result forXe=X +εB,ε >0, whereB is a

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1.1 Classical Itô’s formula and its extensions

LetX be a continuousRd-valued semimartingale with decompositionXt =X0+Mt+At,

where M is a local martingale and Ais a process of finite variation, and let f be a twice continuously differentiable function. Then Itô’s lemma states that f(X)is also a continuous semimartingale and that, moreover,

f(X_t) = f(X0) +

X

i

Z t

0

∂f

∂xi

(X_s)d X_s+1 2

X

i,j

Z t

0

∂2_f

∂xi∂xj

(X_s)d〈X_s,X_s〉.

In particular, the martingale part of f(X)is given by

N_t:=X

i

Z t

0

∂f

∂x_i(Xs)d Ms=

Z t

0

∇f(X_s)d M_s ,

where _∇f(x) is the gradient of f at x (see section 1.3.1 for definition and more de-tails).

One of the most well-known, and also one of the simplest, extensions of the Itô’s formula is theTanaka’s formula(see, for example,[64, p. 169]), which states that for a standard Brownian motionBwe have

|B_t|=|B0|+

Z t

0

sgn(B_s)d B_s+L0_t ,

where sgn(x) =−1 for x ≤0 and sgn(x) =1 for x >0, and L_t is the so calledlocal timeofB at the origin, given by

L0_t =lim ε↓0

1

2ε|{s≤t:Bs∈(−ε,+ε)}|.

Note that Tanaka’s formula is also the easiest extension of Itô’s result to convex functions. Meyer proves a more general result

Theorem 1.1. (Meyer[57, p. 361]) Let X be anRd-valued semimartingale and let f be

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Moreover, for the cased=1 he derives the change of variables formula

f(Xt) = f(X0) +

Z t

0

∇−f(Xs)d Xs+

Z

R

La_tµ(d a),

where_∇₋f is the left-hand side derivative of f, andµ= 1₂∂_∂2_xf2 is viewed as a measure. The increasing processLa_t is thelocal time process of X given by

La_t(X) =lim ε↓0

1

ε Z t

0

1_[a,a+ε)(Xs)d〈Xs,Xs〉,

and which can be viewed as the time X spends at a. We refer to the above formula as theMeyer-Itô formula. The so-calledMeyer-Tanaka formulais its special case: for a continuous semimartingaleX and a function f(x) =_|x_|we have

|Xt|=|X0|+

Z t

0

sgn(Xs)d Xs+L0t .

WhenX is standard Brownian motion, we recover Tanaka’s formula.

Now consider a general convex functionf :Rd→R, not necessarily everywhere

differentiable. Every differentiable point x ∈Rd has a unique tangential hyperplane,

while at non-differentiable points there is a whole set of supporting hyperplanes. For a continuous semimartingaleX with decompositionX =M+Awe prove that the (local) martingale part of f(X)can be expressed in terms of a stochastic integral of a measur-able selection of subgradient_∇f(X)againstM. For piecewise linear 1-dimensional con-vex functions this follows from the Meyer-Tanaka formula. For example, for f(x) =_|x_| we have_∇f(x) =sgn(x). So at the origin, which is the only point where derivative is not defined, we can take the supporting line to be y =₋x. Moreover, because as we know Brownian motion spends zero time in Lebesgue-null sets, we can in fact choose

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For a continuous semimartingaleX with decompositionX =M+Awe define

kX _k_Hp=k 〈M,M〉1/2

∞ +

Z ∞

0

|dA_s_{| k}_Lp

for 1_≤p<_∞. The_Hp-space consists of all semimartingalesX such that_kX _k_Hp<∞.

The main result of this chapter is the following

Theorem 1.2. Let f : Rd → R be convex and let X be a continuous Rd-valued

semi-martingale defined on filtered probability space(Ω,_F,_{{F }}_t_≥₀,_P)with Meyer decomposi-tion X_t=X₀+M_t+A_t. Then f(X_t)is again a continuous semimartingale; in particular, its local martingale part is given by

Z t

0

∇f(Xs)d Ms, locally inH1 ,

where_∇f(x)is some choice of subgradient of f at x, such that_∇f(X_t)is_F_t-adapted.

The above theorem was first stated and proved by Bouleau ([15, Lemma 2]). In the follow-up paper[16] he proves the conjecture stated in [15] that in fact any measurable choiceof_∇f(x)can be used. In this chapter we prove the first of the two results using an approach different to that in[15].

There are many other papers on extending the Itô’s formula by considering dif-ferent classes of functions f or stochastic processes, or both. In [72], for example, Russo and Vallois derive Itô’s formula for_C1(Rd)-functions of continuous

semimartin-gales whose time-reversals are also continuous semimartinsemimartin-gales. They also extend the formula to the case of _C1(Rd)-functions with first order derivatives being

Hölder-continuous with any parameter and the process given by a stochastic flow generated by a so-called C0(Rd,Rd)-semimartingale. In both cases the quadratic variation

pro-cess is expressed in terms of the generalised quadratic covariation propro-cess_〈f0(X),X〉t

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with a locally square integrable derivative andX a 1-dimensional Brownian motion, for which a version of Itô’s formula is derived with the finite variation part expressed again in terms of the quadratic covariation〈f0(B),B〉t. The multidimensional case (where f

belongs to the Sobolev spaceW1,2) is treated in[32]. In[51]Kendall discusses

semi-martingale decomposition ofr(B), whereris the distance function of Brownian motion on a manifold. The problem tackled in[51]is similar to ours as r fails to be differen-tiable on a set of measure zero, called the cut-locus. It is proved in[51]that r(B)is a semimartingale and its canonical decomposition is found explicitly in the sequel[23].

In the proof of our main Theorem 1.2 we need the results of Carlen and Protter

[19]and Barlow and Protter[4], which we discuss in the next section.

1.2 Convergence of semimartingales and convex functions

Paper of Barlow and Protter concerns convergence of semimartingales together with their semimartingale decomposition. We will present their results in the continuous setting, even though the original paper considered a wider class of semimartingales with jumps.

Suppose_{Xn}n≥1 is a sequence of continuous semimartingales with the decom-positionXn = X₀n+Mn+An, such that lim_n_→∞E[(Xn−X)∗] = 0. Here and in what

follows we denoteX∗=sup_t_|Xt|andX∗t =sups≤t|Xs|. Barlow and Protter prove that

under some regularity conditions imposed onMnandAnnot only that the limiting pro-cessX is again a continuous semimartingale, but that there is also convergence of the corresponding martingale and finite variation process parts of the decompositions.

Theorem 1.3. ([4, Thm. 1]) Let{Xn}n≥1 be a sequence of continuous semimartingales

in_H1with canonical decomposition Xn=X₀n+Mn+An, satisfying for some constant K

E

hZ ∞

0

|dAn_s|i≤K, (1.1a)

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and let X be a continuous process such that

lim

n→∞E[(X

n

−X)∗] =0 . (1.2)

Then X is a continuous semimartingale in _H1 and, moreover, if X = X₀+M +A is its decomposition, then

E[M∗]≤K, E[

Z ∞

0

|dA_s_|]_≤K

and

lim

n→∞kM

n

−MkH1=0 , (1.3a)

lim

n→∞E[(A

n

−A)∗] =0 . (1.3b)

In[19]Carlen and Protter prove that the assumptions of[4, Thm. 1]are sat-isfied in case when the sequence of_C2 convex functions_{fn}n≥1 of (a not necessarily continuous) semimartingaleX=M+Aconverges to a convex f, thus making the result important in our situation.

We equip the set of convex functions onRd with uniform convergence on

com-pact sets with the corresponding metric ρ, defined by ρ(f,g) = P∞_k=₁2−kρ_k(f,g), where

ρk(f,g) =

sup_|_x_|≤_k|f(x)−g(x)|

1+sup_|_x_|≤_k_|f(x)₋g(x)_| .

Theorem 1.4. ([19, Thm. 1]) Let X be a continuous Rd-valued semimartingale in H1

with X0=0and the decomposition X=M+A. For eachα >0, define

T_α=inf{t>0 :|X_t|> α}.

Let{f_n}n≥1 be a sequence of C2 convex functions on Rd and let f be a convex

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and, moreover, if its decomposition is given by f(X_t) = f(X0) +Nt+St, then for allα >0

lim

n→∞k(N

n

−N)Tα_k

H1=0 , lim

n→∞E[(S

n

−S)Tα] =_{0 ,}

where

N_tn=

Z t

0

∇f_n(X_s)d M_s

and

Sn_t =

Z t

0

∇fn(Xs)dAs+

1 2

X

i,j

Z t

0

∂2

∂x_i∂x_jf(Xs)d〈X

i s,X

j s〉

are the martingale and finite variation parts of the decomposition of fn(X)respectively.

1.3 Elements of convex analysis

In order to prove the main result of this chapter, we require some notation and re-sults from convex analysis. Proofs of the rere-sults stated in this section and more details on convex functions are given in[66]. See also[36]. We start by introducing some elementary notation.

1.3.1 Convex functions: some notation and results

Letf be any function living onRdand taking values in[−∞,+∞]. At any pointx ∈Rd

we define theone-directional derivative of f with respect to a vector y ∈Rd, if it exists,

as follows

D f(x)[y]:=lim λ↓0

f(x+λy)₋f(x)

λ .

The two sided derivative atx in direction y exists if and only if−D f(x)[−y], given by,

−D f(x)[−y]:=lim λ↑0

f(x+λy)− f(x)

λ

is also well-defined and

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Now, if the function f is convex, then the one-directional derivative always exists and, moreover, we may write

D f(x)[y] = inf λ>0

f(x+λy)−f(x)

λ . (1.5)

FurthermoreD f(x)[y]is positively homogeneous (i.e. D f(x)[λy] =λD f(x)[y]for

λ∈(0,_∞)), convex in y withD f(x)[0] =0[66, Thm. 23.1]and

D f(x)[y]≥ −D f(x)[−y]. (1.6)

If for a convex f defined onRd and finite at some x∈Rd all directional

deriva-tives atx exist, are two-sided and finite then we have ([66, Thm. 25.2])

D f(x)[y] =_〈∇f(x),y_〉, _∀y_∈Rd ,

where

∇f(x):=

∂_f

∂x₁(x), ...,

∂f

∂x_d(x)

is thegradient of f at x = (x1, ...,xd). Note that ∂ f

∂xi(x) =D f(x)[ei], ei being thei

th

canonical basis vector ofRd.

Of course a general convex function f is not necessarily everywhere differen-tiable, a simple example being f(x) =_|x_|which is not differentiable at x =0. We can however define a set ofsubgradientsat each “troublesome” point like this.

Definition 1.5. Let f :Rd→Rbe a convex function. A subgradient∇f(x)of f at x∈Rd

is a gradient of an affine hyperplane h(x) =α+βTx, forα,β_∈_Rd_{, passing through the}

point(x,f(x))and satisfying

h(x0)≤ f(x0)

for all x06=x. We denote any subgradient at x in the direction of y ∈Rnby〈∇f(x),y〉.

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set of all subgradients atx is called thesubdifferentialoff atx, denoted∂f(x). A con-vex function with finite values issubdifferentiableeverywhere. In subsequent sections we will need the following result

Theorem 1.6. ([66, Thm. 23.2]) Let f be a convex function and x a point at which f is

finite. Then∇f(x)is a subgradient of f at x if and only if

D f(x)[y]≥ 〈∇f(x),y〉 ∀y ∈Rd\{0}. (1.7)

The above says that a subgradient at x in the direction of y will always be less or equal to the one-sided directional derivative at x with respect to y. Relation (1.7) is called the subgradient inequality and can be used as an alternative definition of a subgradient.

Finally we mention the Lipschitz continuity property of convex functions.

Theorem 1.7. ([36, Ch. 3.1, Thm. 10]) Let f be a convex function onRd and let U be

an open convex subset ofRd. Then f is continuous on U if and only if f is locally Lipschitz

on U, i.e. for all u∈U there exist constants K>0andε >0such that

|f(x)₋f(y)_{| ≤}K_kx₋y_k, _∀x,y_∈B_u(ε),

where B_u(ε)is an open ball of radiusεcentered at u and_{k · k}is the usual Euclidean norm.

1.3.2 Differential theory of convex functions

This section is devoted to studying the set D of points in the domain of f at which the supporting hyperplane is unique. In what follows we assume that f isproper, i.e. f(x) < +∞ for at least one x and f(x) > −∞ for all x. By domf we denote the effective domainof f, that is domf ={x ∈Rd : f(x)<∞}. We denote by int(domf)

the interior of domf.

Theorem 1.8. ([66, Thm. 25.2], [66, Thm. 25.4]) Let f be a proper convex function

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D f(x)[·]is a linear function onRd iff x∈ D. Moreover,Dis dense in int(dom f), and its

complement in int(dom f)is a set of measure zero.

In fact, in order for D f(x)[_·]to be linear, and hence for f to be differentiable at x, it suffices that the partial derivatives with respect to the basis vectors of_Rd _exist

atx.

So, we see that set_Dc _{of points at which a convex function} _f _{does not have a}

unique supporting hyperplane has measure zero. Consequently any process which has a probability density at each time t spends time of measure zero in_Dc_{, an important}

fact we will use in the sequel.

To prove Theorem 1.2 for a general (continuous and proper but not necessarily differentiable) convex f we will approximate it by a sequence of twice continuously differentiable convex functions fn :Rd → R, to which we know Itô’s formula can be

applied. On top of this, working with convex functions gives us an advantage of being able to deduce from the pointwise convergence of the functions something about the convergence of their corresponding gradients.

Theorem 1.9. (variation of[66, Thm. 25.7]) Let f be a convex function defined onRd

and {fn} a sequence of smooth convex functions onRd such that limn→∞fn(x) = f(x)

∀x ∈Rd. LetD ⊆int(dom f)be the set of points where f is differentiable. Then

lim

n→∞∇fn(x) =∇f(x) ∀x ∈ D. (1.8)

Proof. See proof of[66, Thm. 25.7].

This result will be used several times in Sections 5 and 6.

We conclude this section with a result concerning convergence of subgradients. Consider a sequence _{x_n_}_n_≤₁ with x_n _∈ int(domf), n_≥ 1, and x _∈ int(domf) such that lim_n_→∞xn = x. Of course in general limn→∞∇f(xn) need not exist. However

the situation when x_n = x+ε_ny for some y ∈Rd and εn →0 as n→ ∞, i.e. when

xn approaches x from a single direction y, is special. In this case it is known that

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Proposition 1.10. Let f :Rd →Rbe a convex function. For any x ∈Rd, for almost all

y∈Sd−1, where Sd−1is the unit sphere inRd,

lim

ε↓0∇f(x+εy)

exists, belongs to ∂f(x) and is unique for any selection∇f(x +εy) ∈∂f(x +εy) we may make from the subdifferential of f at x+εy for anyε >0.

Proof. First of all recall that D f(x)[y] = lim_ε_↓₀(f(x+εy)₋ f(x))/ε is a positively homogeneous function, convex in y withD f(x)[0] =0. Letg(y):=D f(x)[y]. Hence

∇g(λy) exists and is unique for allλ > 0 for almost all y _∈Rd. Fix x,y ∈Rd and

without loss of generality, by adding a suitable affine function to f, assume that

f(x) =g(y) =∇g(y) =0 .

We argue by contradiction. If theorem fails then we can find a subsequence

εn→0 and a selection∇f(x+εny)∈∂f(x+εny)such that

lim

n→∞∇f(x+εny) =h6=0 , (1.9)

and also a vectoru∈Rd with〈h,u〉>0. For suchuconsider

f(x+ε_ny+ε_nλu)₋ f(x+ε_ny)

εn

=λf(x+εny+εnλu)−f(x+εny)

εnλ

.

Using (1.5) and homogeneity of g(y)the above is greater or equal to

λ εn

D f(x+ε_ny)[ε_nu] =λD f(x+ε_ny)[u]_≥λ〈∇f(x+ε_ny),u_〉=λ〈h,u_〉+o(1)

where the last two inequality signs come from expressions (1.7) and (1.9) respectively, ando(1)_→0 asn→ ∞. Thus we obtain

f(x+εny+εnλu)−f(x+εny)

εn

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On the other hand, since f(x) =g(y) =0, we have

f(x+εny)−f(x)

εn

= f(x+εny)

εn

=o(1). (1.11)

Hence combining (1.10) and (1.11) one obtains

f(x+ε_ny+ε_nλu)−f(x)

εn

= f(x+εny+εnλu)−f(x+εny)

εn

+ f(x+εny)−f(x)

εn

≥λ〈h,u〉+o(1).

Lettingn→ ∞, i.e.ε_n→0, the above inequality becomes

D f(x)[y+λu] =g(y+λu)_≥λ〈h,u_〉>0

⇒ g(y+λu)

λ =

g(y+λu)₋g(y)

λ ≥ 〈h,u〉>0 .

And so lettingλ→0 one obtains

〈∇g(y),u_{〉 ≥ 〈}h,u_〉>0 .

But this contradicts the assumption that_∇g(y) =0.

1.4 Piecewise linear convex functions and Meyer-Tanaka

for-mula

In this section we start our analysis of the martingale part of f(X). However, instead of treating the case of a general convex f we first prove Theorem 1.2 in a special case when f is piecewise linear. Using the Meyer-Tanaka formula, we will verify that any piecewise linear convex function of a continuous semimartingale is itself a continuous semimartingale and find the martingale part of the decomposition explicitly.

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for a detailed discussion of classical Tanaka and Itô-Tanaka formulas (ford =1). One might also find a discussion of convex functions in[65, Appendix.§3]useful.

Proposition 1.11. Let X= (X1, ...,Xd)be a continuous semimartingale living onRd, with

ithcomponent having decomposition X_ti=X₀i+M_ti+Ai_t, i∈ {1, ...,d}. Let f :Rd→Rbe a

function defined by f(x) =l1(x)∨...∨lk(x), x∈Rd, where li(x) =αi+

Pd

j=1βi jxj=αi+

βT

i x, forαi,βi∈Rd, i∈ {1, ...,k}, and x∨y :=sup{x,y}. Then f(X)is a semimartingale

with decomposition

f(X_t) = f(X₀) +

k

X

i=1

Z t

0

1_B i(Xs)β

T i d Xs+

1

2Lt , (1.12)

where B_i = {x : min_{k : sup_j_{l_j(x)} = l_k(x)} = i} and L_t is an increasing process, constant on the complement of_{t:l_i(X_t) =l_j(X_t)for any i₆= j_}. In particular the local martingale part of f(X)is given by

k

X

i=1

Z t

0

1_B i(Xs)β

T

i d Ms . (1.13)

Proof. We prove the proposition for the case when k =2 and any d _≥1 and general case follows by induction. Consider f(x) = l1(x)∨l2(x). Denote l1(Xt) = Yt and

l₂(X_t) =Z_t. Since X_t is a continuous semimartingale so are affine functionals, Y_t and Z_t, ofX_t. Let the corresponding decompositions beY =M+AandZ=N+S. Consider

f(x) =l₁(x)_∨l₂(x) = y_∨z. We can rewrite y_∨zas follows

y∨z= 1

2 |y−z|+y+z

.

Hence, using the differential notation for simplicity, we obtain

d(Yt∨Zt) =

1

2d |Yt−Zt|+Yt+Zt

= 1

2 d(|Wt|) +d Yt+d Zt

,

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above becomes

1 2

sgn(Wt)dWt+d L0t +d Yt+d Zt

,

where L0_t is the local time ofW at 0. Next

1 2

sgn(W_t)d(M_t₋N_t) +sgn(W_t)d(A_t₋S_t) +d(M_t+A_t) +d(N_t+S_t) +d L0_t=

= 1

2

sgn(W_t) +1

d M_t− sgn(W_t)−1

d N_t+

+ sgn(W_t) +1

dA_t− sgn(W_t)−1

dS_t+d L0_t

.

Now sgn(W_t) =sgn(Y_t−Z_t) =1_[Y_t_>_Z_t_]−1_[Y_t_≤_Z_t_]and so sgn(W_t) +1=21_[Y_t_>_Z_t_] and sgn(Wt)−1=−21[Yt≤Zt]. Hence we obtain

d Y_t∨Z_t

=1_[Y_t_>_Z_t_]d M_t+1_[Y_t_≤_Z_t_]d N_t+1_[Y_t_>_Z_t_]dA_t+1_[Y_t_≤_Z_t_]dS_t+ 1 2d Lt=

=1_[Y

t>Zt]d Yt+1[Yt≤Zt]d Zt+

1 2d Lt

or

Y_t_∨Z_t=Y₀_∨Z₀+

Z t

0

1_[Y

s>Zs]d Ys+ Z t

0

1_[Y

s≤Zs]d Zs+

1 2d Lt,

whereLtis a continuous increasing process, constant on the complement of{t:l1(Xt) =

l2(Xt)}. The above expression is exactly (1.12) for n=2. Noticing that x∨ y∨z =

(x_∨y)_∨z, the general case follows by induction.

Clearly the integrand in (1.13) is a measurable selection of the multivalued map∂f(x) and so Theorem 1.2 holds in the special case of convex piecewise linear functions. To illustrate this result we consider our simple example again: forf(x) =|x|

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1.5 Semimartingale decomposition of

f

(

X

e

_t

)

We are now ready to start the analysis of the general case of a convex f defined over the whole of Euclidean space _Rd_{. Let} _X _{be a continuous semimartingale in}

Rd with decomposition X = M +A and defined on some filtered probability space

(Ω,_{F_t_}_t_≥₀,_F,_P). Let(Ωe,{Fe_t}_t_≥₀,Fe,Pe)be some enlargement of this space and letB

be an(Fe_t)-standard Brownian motion independent ofX. Define theperturbed process

e

X on(Ωe,{Fe_t}_t_≥₀,Fe,Pe)by

e

X(_tε):=Xe_t:=X_t+εB_t, ε >0, t≥0 .

For simplicity of notation we shall suppress the superscript(ε) wherever possible. For simplicity also but without loss of generality we can assume thatX0=Xe₀=0.

In this section we find the martingale part of f(Xe(ε))explicitly in order to take

the limit asε_→0 in the next section and hence prove Theorem 1.2. The reasoning be-hind adding a small amount of Brownian motion toX_tis as follows: we know very little about the behaviour ofXtas it is a general semimartingale. For instance, it can at some

times be trivial, i.e. constant. Hence it might spend positive amount of time in those points where f is not differentiable, that is, where it has more than one supporting hyperplane. To avoid this happening we perturbX_t by addingεB_t. Then

Lemma 1.12. Xe_t has a probability density at each t>0and, in particular, spends zero

time in any null set.

Proof. It suffices to prove that eP(Xe_t ∈ N) = 0 for any t > 0 and N ⊂ Rd with

Le b(N) =0. Then it will follow that for allt>0 the law ofXe_tunderePis absolutely

con-tinuous with respect to the Lebesgue measure, and the corresponding Radon-Nikodym derivative is the probability density ofXe_t. For any Lebesgue-null setN we have

e

P(Xe_t∈N) =E

e

P(Xe_t∈N|F_t)

,

whereFt=σ({Xs; 0≤s≤t}), and we use the tower property of conditional

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ofX_t, and hence ofFt, to obtain

EeP(X_t+εB_t∈N|F_t)

=

Z

e

P(x+εBt∈N)dµt(x),

whereµ_tis the law ofX_t. Observe thatBˆ_t :=x+εB_tis a Brownian motion started at x with_〈Bˆ_t,Bˆ_t_〉=ε2t. But we know that Brownian motion hits null-sets with probability zero. Hence, the above integral is equal to zero and the lemma is proved.

In Section 1.3.2 we have seen that_Dc_{, the set of points at which}_f _{is not}

differen-tiable, is Lebesgue-null. Consequently, by the above lemma,Xespends zero time at those

“ambiguous” points. Hence,_∇f(Xe)is almost surely everywhere defined. Moreover,

be-cause of Lemma 1.12 a particular measurable choice of_∇f(x)_∈∂f(x) at x _{∈ D}c is unimportant as it does not change the value of the stochastic integralRt

0∇f(Xes)dMes, which we will show is the martingale part of f(Xe). To do that we approximate f by a

sequence of convextwice continuously differentiablefunctions.

Let _{fn}n≥1 be a sequence of twice continuously differentiable convex func-tions on Rd converging to f with respect to ρ, i.e. such that limn→∞ρ(fn,f) =

0, where metric ρ is as defined in Section 1.2. We need to prove that stochastic integral R₀t∇f_n(Xe_s)dMe_s, the martingale part of f_n(Xe), converges in some sense to Rt

0∇f(Xes)dMes for some measurable choice of ∇f(x) ∈∂f(x), and that it is indeed the martingale part of f(Xe).

We need to note the following two inequalities

Lemma 1.13. (Adapted from[19, Lemma, p. 2])

sup

n

sup |x|≤r

|∇f_n(x)| ≤C_r<∞, ∀r>0 , (1.14)

and

sup |x|≤r

|∇f(x)| ≤Cr<∞, ∀r>0 , (1.15)

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∂f(x).

Proof. To see why inequality (1.14) is true, first notice that, since lim_n_→∞ρ(f_n,f) =0, the variation of the convex functions fnis uniformly bounded in non{|x| ≤r+1}for

anyr >0. Denote this bound byC_r. Letx_nbe such that

∇f_n(x_n) = sup |x|≤r

|∇f_n(x)|

and letu_n:=∇f_n(x_n)/|∇f_n(x_n)|. Then

|∇fn(xn)|=〈∇fn(xn),

∇f_n(x_n)

|∇f_n(x_n)|〉=〈∇fn(x),un〉=D fn(x)[un]

= inf λ>0

fn(xn+λun)−fn(xn)

λ ≤ fn(xn+un)−fn(xn). (1.16)

But, since_|x_n+u_n_{| ≤}r+1, the above is less thanC_r for allnand (1.14) follows. Now, since f_n converges to f uniformly on compact sets, we also have f_n _→ f pointwise. Therefore, for anyx,ywith_|x|,_|y|<r+1 the inequality f_n(x)−f_n(y)≤C_r,

∀n_≥1, (which follows sinceC_r bounds the variation of f_n’s) implies f(x)₋f(y)_≤C_r by virtue of taking the limitn→ ∞. So, by a calculation similar to (1.16), we have for any_∇f(x)_∈∂f(x)

|∇f(x∗)|=〈∇f(x∗),u∗〉 ≤D f(x∗)[u∗]≤ f(x∗+u∗)− f(x∗)≤Cr ,

where x∗ is such that _∇f(x∗) =sup_|_x_|≤_r_|∇f(x)| andu∗:=_∇f(x∗)/|∇f(x∗)|for any r>0.

We are now ready to prove the following

Lemma 1.14. The local martingale part of f(Xe_t)is given by the limit

lim

n→∞

Z t

0

∇fn(Xe_s)dMe_s= Z t

0

∇f(Xe_s)dMe_s (1.17)

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Proof. Since for each n ≥ 1 f_n is in C2_{, the martingale part of} _f

n(Xe) is given by R

∇fn(Xe)dM, wheree Me = M +εB. Theorem 1.4, applied to our sequence {f_n}_n_≥₁

and the semimartingaleXe, then ensures that the martingale part of the limiting process

f(Xe_t)is given by the limit of R

∇fn(Xe)dMe asntends to infinity, locally inH1. Our aim

is to prove that this limit is indeed equal toR∇f(Xe)dMe for some measurable choice of

subgradient_∇f ∈∂f.

Let B(r) be an open ball of radius r and B(r0) an open ball of radius r0 with r0 > r > 0, both centred at the origin. For all r,r0 > 0 define stopping times Tr :=

inf{t:X_t∈/B(r)}andTe_r0:=inf{t:Xe_t∈/B(r0)}and takeTe=T_r∧Te_r0. Assume also that

e

X_t_∧_T_e,X_t_∧_T_e ∈ H1 for all t ≥0; we know that continuous semimartingales are at least locally inH1. We consider the stopped processXe_t_∧

e

T. Note thatXt∧Te ∈B(r)⊂B(r

0₎ andXe_t_∧

e

T ∈B(r0) for all t ≥0. By Lemma 1.12 the law of Xe_t_∧

e

T has a density for all

t<Te.

Note that for proving Lemma 1.14 it would have sufficed to stopXeatTe_r0.

How-ever, in order to be consistent with localisation we will be using to prove Theorem 1.2, we useTe=T_r∧Te_r0 instead.

Notice that convergence of a continuous (local) martingale M in_Hp _is

equiv-alent to convergence of_〈M,M_〉1/2 in _Lp. So, in this case convergence in_Hp implies convergence in_Hl_{for 1}_≤_l_<_{p. In our case it is easier to prove convergence (1.17) in}

H2 and then deduce convergence in_H1. For any measurable selection_∇f _∈∂f and t>0 we have

lim

n→∞

Z t∧Te

0

∇f_n(Xe_s)− ∇f(Xe_s)

dMe_s

H2

= lim

n→∞E

h Z t∧eT

0

∇f_n(Xe_s)− ∇f(Xe_s) 2

d〈Me,Me〉_s i1/2

.

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expectation sign above as follows

Z t∧Te

0

∇f_n(Xe_s)− ∇f(Xe_s) 2

d〈Me,Me〉_s≤4C_r2₀ Z t∧Te

0

d〈M,e Me〉_s

≤4C2_r₀_〈Me,Me〉_t_∧

e

T <∞,

where the quadratic variation_〈M,e Me〉_t_∧

e

T is finite, because it is the bracket of a bounded

continuous semimartingaleXe_t_∧

e

T (see[65, Ch. IV, Thm. 1.3]). Using dominated

con-vergence theorem we can now take the limit inside the expectation sign and, since the integrand is bounded above by 4C_r2₀, we can also pull the limit inside the integral sign. We can then use almost sure convergence of ∇f_n(Xe_t) to ∇f(Xe_t) for all Xe_t ∈ D and

the fact that particular choices_∇f(Xe_t) ∈∂ f(Xe_t)for Xe_t ∈ Dc are not charged by the

integral to conclude that the limit in question is equal to

E

hZ t∧Te

0

lim

n→∞

∇f_n(Xe_s)− ∇f(Xe_s) 2

d〈M,e Me〉_s i1/2

=0 .

It follows that Rt∧Te

0 ∇fn(Xes)dMes converges to

Rt∧Te

0 ∇f(Xes)dMes in H 2_{, and}

hence in H1_{. This is true for any radiuses} _r0 _> _r _> _{0 of localisation, and so (1.17)} follows.

We also prove the following lemma concerning the semimartingale decomposi-tion of f(Xe)that we will require for the proof Theorem 1.2.

Lemma 1.15. Let Ne_t(ε) = Rt

0∇f(Xe_s(ε))dMe_s(ε) and Se(ε) be the martingale and the finite

variation parts of the decomposition of f(Xe(ε))respectively. Then for allε≤1

E

h

sup

t≤Te

|Ne

(ε) t |

i

≤K_r,r0 , (1.18a)

E

h Z Te

0

|dSe(_sε)| i

≤K_r,r0, (1.18b)

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Proof. The proof largely follows proof of Theorem 1.4 in[19]: we prove that the se-quence of continuous semimartingales_{fn(Xe)}_n_≥₁ satisfies the conditions of Theorem

1.3 of Barlow and Protter, i.e.

lim

n→∞E

h

sup

t≤eT

|f_n(Xe_t)−f(Xe_t)| i

=0 (1.19a)

sup

n≥1E

h

sup

t≤eT

|Ne_tn| i

≤K_r,r0 , (1.19b)

sup

n≥1E

hZ eT

0

|deS_sn| i

≤Kr,r0 , (1.19c)

where for eachn≥1Nen andSen are the martingale and the finite variation part of the

decomposition of f_n(Xe)respectively. Then (1.18a) and (1.18b) will follow by Theorem

1.3. The difference is only in the fact that we need to ensure that for small enoughε the constantK_r,r0 above can be taken to be independent ofε.

First of all notice that (1.19a) follows from the fact that lim_n_→∞ρ(fn,f) =0.

Next consider (1.19b). By the Burkholder-Davis-Gundy inequality we have for some constantp<∞

E

h

sup

t≤eT

|Ne_tn| i

≤pE

h

〈Nen,Nen〉

1/2

e

T

i

=pE

hZ Te

0

|∇f_n(Xe_t)|

2_d

〈Me,Me〉_t 1/2i

≤pC_r0E

h 〈Me,Me〉

1/2

e

T

i

=pC_r0E

h

(〈M,M_〉 e

T +ε

2

e

T)1/2i.

To finish we need to bound_〈Me,Me〉

e

T by some constant independent of ε. We

have_〈M,e Me〉

e

T =〈M,M〉eT+ε 2

e

T which for allε_≤1 is less than_〈M,M〉Te+Te which is in turn bounded above by_〈M,M_〉_T

r+Tr, sinceTr≥Te=Tr∧Ter0. Hence, eventually for

allε

E

h

sup

t≤Te

|Ne_tn| i

≤pC_r0E

h

(〈M,M_〉_T

r +Tr)

1/2i_,

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We now turn to (1.19c). The finite variation part of f_n(Xe_t_∧

e

T)is given by

e

Sn

t∧eT =

Z t∧Te

0

∇fn(Xe_s)dA_s+

1 2

Z t∧Te

0

X

i,j

∂2_f

n

∂x_i∂x_j(Xes)d〈Xes,Xes〉. (1.20)

WritingVar_s_≤_t(Y_s)forRt

0|d Ys|for a continuous processYt, we have

Z eT

0

|deS_sn|=Var_t_≤

e

T(eSn_t)

≤Var_t_≤ e

T

Z t

0

∇f_n(Xe_s)dA_s

+Var_t_≤ e T 1 2 Z t 0 X

i,j

∂2_f

n

∂x_i∂x_j(Xes)d〈Xes,Xes〉

.

Now

Var_t_≤ e

T

Z t

0

∇f_n(Xe_s)dA_s

≤C_r0Var_t_≤

e

T

Z t

0

dA_s≤C_r0

Z t∧Tr

0

|dA_s|, (1.21)

where we have usedT_r≥T_r∧Te_r0=Teand the fact thatVar_tis an increasing process of

t. Hence, the bound on the expectation of the RHS above depends onr andr0but not onε.

Next we observe that, since f_n is convex and the bracket_〈Xe_s,Xe_s〉is an

increas-ing process of t, the second summand of (1.20), which we denote by F_tn, is also an increasing function oft. Thus,

Var_t_≤_e_T

1

2

Z t

0

X

i,j

∂2_f

n

∂x_i∂x_j(Xes)d〈Xes,Xes〉

≤Fn

e

T ≤F n Tr .

But, because fnis twice continuously differentiable, using Itô’s formula we can

also write

E[F_Tn

r] =E[fn(XeTr)−fn(Xe0)]−E hZ Tr

0

∇fn(Xe_s)dA_s i

.

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corre-sponding Lipschitz constant) we have

E[F_Tn_r]≤Kr0E[|Xe_T_r|] +C_r0E[|A_T_r|]

=K_r0_E[|X_T

r+εBTr|] +Cr0E[|ATr|] ≤K_r0(r+_E[B_T

r]) +Cr0E[|ATr|]

for allε _≤1, i.e. the bound on E[F_Tn

r]is independent of both nand ε for allε ≤1.

Combining this with (the expectation of) the bound (1.21) we obtain (1.19c). The assertion of the lemma now follows by Theorem 1.3.

1.6 Proof of Theorem 1.2

Finally we need to derive the analogous result for our original object of interest, a continuous semimartingaleX.

Proof of Theorem 1.2. We have lim_ε_↓₀Xe(ε)=Xalmost surely and, thus, for a continuous

convex f, lim_ε_→0 f(Xe(ε)) = f(X) almost surely. Note that the limit of the process

e

X(ε) as ε tends to zero lives in the enlarged probability space (Ωe,Fe,Pe), even though

the original process X is defined on (Ω,_F,_P). Crucially by Itô’s lemma f(Xe(_tε)) is a

continuous semimartingale for everyε >0. Hence, we can apply Theorem 1.3 to the sequence of semimartingales_{f(Xe(ε))}ε>₀ if we can show that the conditions (1.1a),

(1.1b) and (1.2) of the theorem are satisfied in our case.

We use the same localisation as in the proof of Lemma 1.14, i.e. we consider

e

X_t_∧ e

T foreT=T_r∧Te_r0=inf{t:X_t∈/B(r)} ∧inf{t:Xe_t∈/B(r0)}, with r0>r>0.

In view of Lemma 1.15 the only thing we need to prove in order to apply results of Theorem 1.3 is

lim ε↓0E

h

sup

t≤eT

|f(Xe_t(ε))−f(X_t)| i

=0 ,

Since by Theorem 1.7 f is Lipschitz in the ballB(r0), we have

E

h

sup

t≤Te

|f(Xe_t)− f(X_t)| i

≤Kr0E

h

sup

t≤Te

|Xe_t−X_t| i

=εKr0E

h

sup

t≤eT

|Bt|

i

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where K_r0 <∞is a Lipschitz constant depending on r0. Taking the limit ε→0 gives

the desired result.

Together with expressions (1.18a) and (1.18b) of Lemma 1.15 this ensures that the conditions of Theorem 1.3 are satisfied in our case. It follows immediately that the martingale part of f(X) is given by the limit as ε → 0 of R∇f(Xe(ε))dMe(ε), the

martingale part of f(Xe(ε)), locally in H1. All is left to prove now is that this limit is

given byR∇f(X)d M for some choice of∇f(x)∈∂f(x), i.e. that for all t>0

lim ε↓0

Z t

0

∇f(Xe(ε)

s∧Te

)dMe(ε)

s∧eT

=

Z t

0

∇f(X_s_∧_T_r)d M_s_∧_T_r (1.22)

in_H1 for allr0>r>0.

Proving the above convergence will require us to consider the limit of_∇f(Xe

(ε) t∧eT) asε tends to 0. From Proposition 1.10 we know that for all t≥0 for almost all values ofB_t the limit lim_ε_↓0∇f(Xt+εBt)exists and belongs to∂f(Xt). Denote this limit by

∇f(Xt). Also for any path of X andBfor small enoughε, i.e. eventually for allε, we

haveT_r<Te_r0. That isTe→T_rasε→0 a.s. and so

lim

ε↓0∇f(Xt∧Te+εBt∧Te) =∇f(Xt∧Tr) a.s. . (1.23)

Again we consider convergence in_H2 first, and convergence in _H1 follows. We have, using the fact that lim_ε_↓0Me_t_∧

e

T =limε↓0(Mt∧Te+εBt∧Te) =limε↓0Mt∧Te a.s.

lim ε↓0

Z t 0

∇f(Xe_s_∧

e

T)dMe_s_∧

e

T −

Z t

0

∇f(Xs∧Tr)d Ms∧Tr H2 =lim ε↓0E

Z eT

0

∇f(Xe_s)2d〈M_s,M_s〉+ Z Tr

0

∇f(Xs)2d〈Ms,Ms〉

−2

Z ∞

0

∇f(Xe_s_∧

e

T)∇f(Xs∧Tr)d〈Ms∧Te,Ms∧Tr〉 1/2

. (1.24)

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Additionally we have

Z eT

0

∇f(Xe_s)2d〈M_s,M_s〉 ≤C_r2₀〈M

e

T,MTe〉<∞

and

Z ∞

0

∇f(Xe_s_∧

e

T)∇f(Xs∧Tr)d〈Ms∧eT,Ms∧Tr〉 ≤CrCr0〈MTe,MTe〉<∞,

since_〈M_t,M_t_〉, is finite for anyt_≤Te. Appealing to the dominated and bounded

conver-gence theorems we can interchange the limit with the expectation and the integration signs respectively. Convergence (1.23) and the fact thatTe→T_r a.s. then finally yield

(1.22). Noticing that the above is true for allr0>r>0 concludes the proof.

Example. As was mentioned before, in[16]Bouleau has proved thatanymeasurable choice of subgradient _∇f(Xt) works for the stochastic integral of Theorem 1.2. A

function

∇ef(x) =lim

θ↓0E[∇f(x+θN)], (1.25) whereN is a standardd-dimensional Gaussian random variable, is a particular exam-ple._∇ef(x)can be regarded as a sort of an average of (sub)gradients within the vicinity ofx. To verify that it does indeed define a subgradient of f at each x _∈Rd we check

the subgradient inequality (1.7) of Theorem 1.6. For any y∈Rd\{0}we have

〈∇ef(x),y_〉=_〈lim

θ↓0E[∇f(x+θN)],y〉=limθ↓0E[〈∇f(x+θN),y〉]. (1.26)

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〈∇f(x+θN),y_{〉 ≤}D(x+θN)[y] = inf λ>0

f(x+θN+λy)− f(x+θN)

λ ≤ f(x+θN+ y)₋f(x+θN)_≤K_|y_|

for some Lipshitz constantK < _∞depending on x andN. Appealing to the bounded convergence theorem now allows us to take the limit inside the expectation in equation (1.26) above

〈∇ef(x),y_〉=E[〈lim

θ↓0∇f(x+θN),y〉]. (1.27) But by Proposition 1.10 lim_θ_↓₀_∇f(x +θN) exists, is unique and belongs to

∂f(x)for almost allN. Denote this limit by_∇∗f(x). Then (1.27) is equal to

E[〈∇∗f(x),y〉]≤E[D f(x)[y]] =D f(x)[y].

Hence we have〈∇ef(x),y〉 ≤D f(x)[y]for any y ∈Rd\{0}for all x, and so∇ef(x)

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Chapter 2

Elements of representation theory.

Starting with the present chapter the thesis is taking up a study of a different subject to that of Chapter 1. Chapters 3, 4 and 5 are concerned with a certain family of bivariate Markov chains, diffusions and repelling particle systems, respectively.

In this chapter we outline some basic results from representation theory which we will employ in later chapters. In particular we will describe finite-dimensional repre-sentations of the universal enveloping algebraU(gl(n))and its quantisationUq(gl(n)),

treating separately the case ofU(sl(2))andU_q(sl(2)), and explain how they are related to combinatorics. We start with some basic definitions and notation.

2.1 Basics

Given a groupG, Representation theory is concerned with finding all finite- and infinite-dimensional vector spaces on which elements of the group act as automorphisms and describing these actions. Thus homomorphisms, structure preserving maps between two algebraic objects, play a fundamental role in Representation theory. Agroup ho-momorphism between two groups (G,_·) and (H,_×) is a map ϕ : G → H such that

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anautomorphismis an endomorphism which is a bijection. Now

Definition 2.1. (Group representation) A representation of a group G on a complex vector

space V is a homomorphismρ

ρ:G_→Aut(V),

where Aut(V)is a family of automorphisms of V .

In other words, a representation of G on V is a mapρsuch that for all g ∈G the mapρ(g):V _→V is linear and invertible and such that for allg,g0_∈G we have

ρ(g g0)(v) =ρ(g)ρ(g0)(v), v ∈ V. We say that V is a G-module. The dimension of V is called thedimension of the representation. It is customary (if slightly inaccurate) to call the vector spaceV itself a representation of G and write g vas a shorthand for

ρ(g)(v). IfV is endowed with an inner product_〈·,_·〉then it is aunitary representation if the group acts on members ofV unitarily, i.e. for anyg∈G andv,u∈V

〈g v,gu〉=〈v,u〉.

Next we define thedirect sumand thetensor productof two vector spacesV and W. The direct sum V ⊕W is a Cartesian product V ×W with element-wise addition

(v1,w1) + (v2,w2) = (v1+v2,w1+w2) for v1,v2 ∈ V, w1,w2 ∈W. If V and W are inner product spaces, so is its direct sum via_〈(v1,w1),(v2,w2)〉=〈v1,v2〉+〈w1,w2〉. Of course we have dim(V ⊕W) =dim(V) +dim(W).

Suppose now V and W are finite-dimensional; to construct a direct product V⊗W we associate to each pair(u,v)an elementu⊗vvia a bi-linear mapφ:V×W →

V _⊗W. The map φ is unique up to an isomorphism. Consider a space of all linear combinationsF :=_{P

nαnun⊗vn,(un,vn)∈U×V,αn∈C}and a subspaceSgenerated

by all the elements ofF of the form

u⊗w+v⊗w−(u+v)⊗w, u⊗w+u⊗v−u⊗(w+v),

(αu)⊗v−α(u⊗v), u⊗(αv)−α(u⊗v)

Some problems in stochastic analysis : Itô's formula for convex functions, interacting particle systems and Dyson's Brownian motion

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AUTHOR:

DEGREE:

TITLE:

DATE OF DEPOSIT:

AUTHOR'S SIGNATURE:

Some problems in stochastic analysis: Itô’s formula for

convex functions, interacting particle systems and

Dyson’s Brownian motion

Nastasiya Grinberg

Thesis

Doctor of Philosophy

Department of Statistics

Contents

Acknowledgments

Declarations

Abstract

Chapter 1

Itô’s formula for convex functions

of continuous semimartingales:

Brownian perturbation approach.

1.1

Classical Itô’s formula and its extensions

1.2

Convergence of semimartingales and convex functions

1.3

Elements of convex analysis

1.3.1

Convex functions: some notation and results

1.3.2

Differential theory of convex functions

1.4

Piecewise linear convex functions and Meyer-Tanaka

for-mula

1.5

Semimartingale decomposition of

f

(

X

e

)

1.6

Proof of Theorem 1.2

Chapter 2

Elements of representation theory.

2.1

Basics