Stochastics Calculus: Malliavin Calculus in a simplest way
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IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 3 (May. - Jun. 2013), PP 30-41 www.iosrjournals.org
www.iosrjournals.org 30 | Page
Stochastics Calculus: Malliavin Calculus in a simplest way
1Udoye, Adaobi Mmachukwu,
2. Akoh, David
, 3. Olaleye, Gabriel C.
Department of Mathematics, University of Ibadan, Ibadan.
Department of Mathematics, Federal Polytechnic, Bida
Department of Mathematics, Federal Polytechnic, Bida.
Abstract: We present the theory of Malliavin Calculus by tracing the origin of this calculus as well as giving a
simple introduction to the classical variational problem. In the work, we apply the method of integration-by-
parts technique which lies at the core of the theory of stochastic calculus of variation as provided in Malliavin Calculus. We consider the application of this calculus to the computation of Greeks, as well as discussing the
calculation of Greeks (price sensitivities) by considering a one dimensional Black-Scholes Model. The result
shows that Malliavin Calculus is an important tool which provides a simple way of calculating sensitivities of
financial derivatives to change in its underlying parameters such as Delta, Vega, Gamma, Rho and Theta.
I. Introduction The Malliavin Calculus also known as Stochastic Calculus of Variation was first introduced by Paul
Malliavin as an infinite-dimensional integration by parts technique. This calculus was designed to prove results
about the smoothness of densities of solutions of stochastic differential equations driven by Brownian motion.
Malliavin developed the notion of derivatives of Wiener functional as part of a programme for producing a
probabilistic proof of the celebrated Hörmander theorem, which states that solutions to certain stochastic
differential equations have smooth transition densities. Classical variational problems are problems that deal with selection of path from a given family of
admissible paths in order to minimize the value of some functionals. The calculus of variation originated with
attempts to solve Dido’s problem known as the isoperimetric problem. An infinite dimensional differential
calculus on the Wiener space, known as Malliavin Calculus, was initiated by Paul Malliavin (1976) with the
initial goal of giving conditions insuring that the law of a random variable has a density with respect to
Lebesgue measure, as well as estimates for this density and its derivative. Malliavin Calculus looks forward to
finding the derivative of the functions of Brownian motion which will be referred to as Malliavin derivative. We
will highlight the theory of Malliavin Calculus. In what follows, H is a real separable Hilbert space with inner
product H
.,. . Ω denotes the sample space, P denotes the probability space P.
II. The Wiener Chaos Decomposition Definition 2.1. A stochastic process W = W(h), h ϵ H defined in a complete probability space (Ω, F, P) is
called an isonormal Gaussian process if W is a centered Gaussian family such that
E (W(h)W(g)) = H
gh, for all h, g ϵ H.
Remark 2.2. The mapping h → W(h) is linear [8]. From the above, we have that
.))(()(22
)(
22
HhhWhW PL E Let G be the σ-field generated by the random variables W(h), h ϵ H ,
the main objective of this part is to find a decomposition of L2(Ω, G , P). We state some results concerning the Hermite polynomials in order to find the decomposition .
Let H xn denote the nth Hermite polynomial, then
H xn =
,!
122
22
x
n
nxn
edx
de
n
n ≥1 (1)
and H x0 = 1. These hermite polynomials are coefficients of the power expansion in t of the function
)(exp,2
2ttxxtF which can easily be seen by rewriting
22
2
1
2exp, tx
xxtF
and expanding the function around t = 0.
Stochastics Calculus: Malliavin Calculus in a simplest way
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The power expansion combines with some particular properties of F, that is
tFt
txtx
F
2exp
2
Ftxt
txtxt
F
2exp
2
and
txFt
txtxF
,
2exp,
2
provides the corresponding properties of the Hermite polynomials for n ≥ 1
xHxH nn 1
'
xHxxHxHn nnn 111
xHxH n
n
n 1
This is shown by using induction method:
To show that xHxH nn 1
'
;
Let n = 1, from
H xn =
22
22
!
1x
n
nxn
edx
de
n
we have
'
22
'
22'
1
2222
xxxx
exeedx
dexH
xHHx n 01
' 1
Let n = 2,
'
2
'
2
2
2'
22
2
2
2 22
2
1
2
1
xx
xedx
dee
dx
dexH
xx
.12
1
2
11
'2
'
2222
222
xHxxexee
xxx
Also for n = 3 we have
.12
12
6
1
6
12
2'3
'
23
3
2'
3
22
xHxxxxedx
dexH
xx
Lemma 2.3. Let X, Y be two random variables with joint Gaussian distribution such that
E (X) = E (Y) = 0 and E (X2) = E (Y2) = 1. Then for all m,n≥0, we have
m;nif
m,nif
,0
,nn!
1XY
YHXH mnE
E
Proof. See the proof of lemma 1.1.1 in [8]
Stochastics Calculus: Malliavin Calculus in a simplest way
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have we
mapping oflinearity By the.9 allfor 0thatsuchLet
.,Ω,
ofsubsettotalaformevariablesrandomThe ;hW
hWh
hXeLX
PL
Lh
hW2
2
2
HEG
G
GH
Proof.
2.4Lemma
21,,...2,1 ,,0exp1
1
mmihthWtX ii
m
i
H R,E
mB hWhWXv
v
,...,B
bygiven is of Laplace that theshows (2)Equation
11E
.0Gfor 0 is,That Gset every
for zero bemust measure thezero, is transform theAs .set Borel afor
1
XX
v
G
m
GG.
B
E
R
.1,: variablesrandom by the generated ,Ω, of subspace
linear closed theasn order of choas the define we,1each For
HH
H
hhhWHPFL
n
n
2
n2.5. Definition
.,Ω,
:9 subspaces theof sum
orthogonalinfinity theinto decomposed becan ,Ω, space The .
0 nPL
PL
n
2
n
2
HG
H
G
2.6 Theorem
3 0exp
therefore,00 have we
,0, polynomial e theHermitofn combinatiolinear a as expressed becan
fact that the Using.1///such that allfor 0 that have We
.0 that show We0.n allfor toorthogonal be ,Ω,Let proof.
htWX
nhXW
nrxH
hhhWXH
XPFLX
n
r
n
n
n
2
E
E
E xH
H
H
implies(3)equationthathavewe),,,(ofsubsettotalaform,e
variablesrandomthethatstateswhichabovelemmatheBy1.normofhallforand
2W(h) PLh
t
GH
H
R
that X= 0
iable.smooth var a , and ,..., where
(4) )(),...,(
form the variablerandom all ofset
theby denote we,C and 1nLet growth). polynomial with sderivative
partial denoted (growth polynomialmost at have sderivative its of all and such that
denoted : functions abledifferenti infinitely all of Cset heconsider t We
1
1
SFhh
hWhWfF
Sf
pf
f
n
n
n
p
nn
p
H
R
RRR
Derivative Malliavin The 3.
H.
:DF mapping a is derivative thewhere
(5) .)(),...,(DF
as define is variablerandom of derivative The
1
oi
i in
n
hhWhWf
SF3.1. Definition
Stochastics Calculus: Malliavin Calculus in a simplest way
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)6(.)(,
then andLet
hFWhDF
hF
EE
H
HF 3.2 Lemma
)(),...,(
as can write
that wesuch that ,such that ,... offamily orthonomalan exists there
on W,product scalar theoflinearity the toDue .1 that Suppose .
1
11
n
n
eWeWfF
F
ehee
h
H,
HProof
.2
1exp2
is that on,distributi normal standard
theofdensity temultivaria thebe Let .Cfunction suitable afor
1
22
n
i
i
n
n
p
x
f
x
xR
.)()(
)(,
formular partsby n integratio classicalby have We
1
11
hFWeFW
xdxxxfdxxxxfhDFn n
EE
ER R
H
proof. thecompletes This
Proposition 3.3. (Nualart,2006). blediferentiaycontinousla be: RR mLet
function with bounded partial derivatives. randomais,...,Suppose 1 mFFF
andins)(Then.incomponentswithvector ,1,1 pp iF DD
.)()(1
m
i
i
i DFFFD
The proof of proposition 3.3 is similar to the proof of proposition 1.20, pp. 13
Of (Nualart, 2009). The chain rule can be extended to the case of a Lipschitz function.
Theorem 3.4 (Closability).
such that,...2,1,F and Assume 2,1
k
2 kPLF D
PLinkFFk
2,,.1
.Lin converges .2 2
1
PFD
kkt 2,1FThen D and inkFDFD tkt ,, P2L .
Proof. Let
00
,....2,1, and n
k
nn
n
knn kfIFfIF
By (1) we have n
n
k
n Linkff 2,, for all n.
By (2) we have
kjFDFDffnnPLjtkt
Ln
j
n
k
nn
,,0!2
2
1
2
2
Hence, by Fatou Lemma,
1
22
1
.0!limlim!lim 22
nL
j
n
k
njkLn
k
n
nk
nn ffnnffnn
This gives 2,1DF , in,, kFDFD tkt PL2
.
Proposition 3.5. [8] Suppose 2,1DF is a square integrable random variable
With a decomposition given above, we have
Stochastics Calculus: Malliavin Calculus in a simplest way
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)7(.),(.1
1 tfnIFD n
n
nt
Proof. We prove the statement for a simple function (see proposition 1.2.7 of [8]).
Using a simple function, the general case results from the density of the simple function.
Let mm fIF for a symmetric function mf , then, by applying
i
n
i
ni hhWhWfDF
0
1 )(),...,(
to nnini xxxxAWAWg ,...,,...,g with )(),...,( 111 we have
m
j
n
imi
mmimAijiimit tfmIAWAWaFD1 1,...,1
11...1 .),(.)(...1)...(
Theorem 3.6. Let F be a square integrable random variable denoted by
00
.1\FE then ,Let .n
n
AnnA
n
nn fIAfIF FΒ
Proof. Assume that nn fIF such that nf is a function in nE . We also
assume that the kernel nf is of the form nBB BBn
,..., with 1 1...1 being mutually
disjoint sets of finite measure. Through the linearity of W and the properties of the
conditional expectation we have
n
i
c
iin ABWABWEBWBWEFE1
1 \)()(/)()...(\AAA
FFF
.1 ...1 ABABn nI
IV. The Divergence Operator In this section, we consider the divergence operator, defined as the adjoint of the derivative operator. If
the underlying Hilbert H is an2L -space of the form ,,2
BTL , where µ is a finite atomless
measure, we interpret the divergence operator as a stochastic integral and call it Skorohod integral because in
the Brownian motion case it coincides with the generalization of the ItÔ stochastic integral to anticipating
integrands. We first introduce the divergence operator in the framework of Gaussian isonormal process
H hhWW ),( associated with the Hilbert space H . We assume that W is defined on a complete
probability space ,, PF, and that F is generated by W. We shall consider results from[1], [8] [ and [12].
We note that the derivative operator D is a closed and unbounded operator with values in H;2 L defined
on the dense subset . of 2 L1,2D
Definition 4.1. The adjoint of the operator D denote by δ is an unbounded operator
satisfiesand;on 2HL
The domain of δ denoted as Domδ is the set of H -valued square integrable random variable
H,2 Lu where
2,1
2allfor , DE FFcuDF
H such that the c is a constant depending on u.
Let u ϵ Domδ, then δ(u) is an element of byzedcharacteri2 L
)8(,
HuDFuF EE
.foreach 2,1DF
From equation (8) above, E(δ(u)) = 0 if u ϵ Domδ and F = 1. We see SH to be the
class of smooth elementary elements of the form
n
j
jjhFu1
(9)
Stochastics Calculus: Malliavin Calculus in a simplest way
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such that jF are smooth random variables and the jh are elements of H . Applying integration-by-parts
formula, we deduce that u ϵ Domδ and
n
j j
jjjj hDFhWFu1 1
.)(H
(10)
Theorem 4.2. Let F ϵ D1,2 and u ϵ Domδ such that .;2HLFu Then Fu belongs to
Domδ and we have the equality
HuDFuFFu ,
provided the right hand side is square integrable.
Proof. Let G be any smooth random variable, we obtain
HHGDFFGDuFuDGFuG ,, EEE
., GDFuFu
H E
Lemma 4.3. Let , where:1
n
j
jj
hh SuhFDuDH
the class of smooth elementary processes of the
form ,,,1
n
j
jj hSFhFu H we have that the following commutativity relationship holds
., uDhuuD hh
H
This is true from the following:
n
j j
jjjj hDFhWFu1 1
.H
yields
n
j
jjjj
h hhDFDhWFDuD1
,,)(HH
=
n
j
n
j
jj
h
jj
h
jj hFDDhWFDhhF1 1
,)(,HH
= ., uDhu hH
Remark 4.4. Let h ϵ H and F ϵ Dh,2.Then Fh belongs to domain of δ and the following equality
is true
.FDhFWFh h
Theorem 4.5 (The Clark-Ocone formula).
Wand Let 2,1DF is a one-dimensional Brownian motion on [0,1]. Then
T
ttt dWFDEFF0
.FE
Proof. Suppose that
0
,...2,1,n
nn nfIF (see[9] and Proposition 1.3.8 of
[12], we have
T
n
tnn
T
tt tdWtfnIEFDE0
1
10
., FF
=
T
n
tnn tdWtfInE0
1
1 ., F
Thus,
Stochastics Calculus: Malliavin Calculus in a simplest way
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T
tt FDE0
F
T
n
nn tdWtfnI0
1
1 ).(..,t0,
X
1
00 .n
nn FfIfI FE
Note. The Clark-Ocone formula and its proof can be found in [7], [8] and [10].
V. A model of a financial market The Black-Scholes Model. We consider a market consisting of a non-risky asset (Bank account) B and a risky
asset (stock) S. Let the process of the risky asset be given by
)()(exp)(20
2
tWtStS (11)
where TttWW ,0:)( is a Brownian motion defined on a complete probabilityspace
TtP t ,0, and ,, FF is a filtration generated by Brownian motion σt denote the volatility process, µ
is the mean rate of return, and are all assumed to be constant.
The price of the bond B(t) and the price of the stock S(t) satisfies the differential equations:
dttrBtdB
B(0) = 1
and
))()(()( tdWdttStdS
S(0) = 0, S(t) = St
We have that
)12()( rt
t eBtB
where r denotes the interest rate and it is a nonnegative adapted process satisfying
T
tdtr0
a.s.
Definition 5.1. Let Q be a probability measure on F, which is equivalent to P. Q is called equivalent
(Local) martingale measure (or a non-risky probability
measure) if the discounted price process TtSeSBS t
rt
ttt ,0,1 is a local martingale under Q.
We note that: A process X is sub-martingale (respectively, a super-martingale)
if and only if Xt =Mt+At (respectively, Xt =M t- At) where, M is a local martingale and A is an increasing predictable process.
Suppose dsTt T
t
2
s0 and ,0 allfor 0
a.s.
processthedefineWe.wherei
ii F
t t
ssst dsdWZ0 0
2
2
1exp
which is positive local martingale. If
12
1exp
0 0
2
T T
sss dtdW E
then the process ZT is a martingale and measure Q such that TZdP
dQ is a
probability measure, equivalent to P, such that under Q,, the process
t
stt dsWW0
~
is a Brownian motion.
In terms of the process tW~
, the price process can be expressed as
1 1
10
1
.!.,!1n n
nnnnnn
T
n
fIfntdWtfnn IXI 1-n
t0,
Stochastics Calculus: Malliavin Calculus in a simplest way
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.~
)(exp0 0
20
2
t t
sst WddsrSS
Thus, the discounted prices from a local martingale:
.2
1~exp
~
0 0
2
0
1
t t
sssttt dsWdSSBS
Definition 5.2. A derivative is contract on the risky asset that produces a payoff
H at maturity T. The payoff is an TF -measurable nonnegative random variable H.
Some fact about filtration:
Filtrations are used to model the flow of information over time.
At time t, one can decide if the event A ϵ tF has occurred or not.
Taking conditional expectation E[X TF ] of a random variable X means taking
the expectation on the basis of all information available at time t.
Proposition 5.3. (Girsanov theorem)
There exist a probability Q absolutely continuous with respect to P such that
t
st dsuWPTQ0
1 is,that has the law of Brownian motion under Q) if and
only if E(ξ1) =1 and in this case .1dP
dQ
Proof. See Proposition 4.1.2, pp. 227 of [8]
Remark 5.4. The probability P o T-1 is absolutely continuous with respect to P.
Proposition 5.5 [8] Suppose that F,G are two random variables such that F ϵ D1,2.
Let u be an H –valued random variable such that DuF = 0, H
uDF almost surely and
Gu(DuF)-1 ϵ Domδ. Then, for any differentiable function f with bounded
derivative we have
,),()()( GFHFfGFf EE
where
.)(, 1 FDGuGFH u
Proof. By chain rule, we have
.)( FDFfFfD uu
By the duality relationship, equation (8), we obtain
H
GFDuFfDGFDFfDGFf uuu 11,EEE
= .11
FDGuFfFDGuFf uu EE
We recall that in Malliavin Calculus, Integrating by part is
.00
TT
s tdWtuFuFdssuFD EEE
VI. Applications Greeks are used for risk management purposes referred to as hedging in financial mathematics. Finite
difference methods have been used to find the sensitivities of options by the use of Monte-Carlo methods, but
the speed of convergence is not so fast very close to the discontinuities. The use of Malliavin Calculus provides
a better way to calculate the greeks, both in terms of simplicity and speed of convergence. Thus, this method
provides a good solution when the payoff function is strongly discontinuous. A greek can be defined as the
derivative of a financial quantity with respect to a parameter of the model.
List of Greeks include:
Delta: = The derivative with respect to the price of the underlying; ∆: =SV .
Stochastics Calculus: Malliavin Calculus in a simplest way
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Gamma: =Second derivative with respect to the price of the underlying;
2
2
:S
V
Vega: = Derivative with respect to the volatility; Vv:
Rho: = Derivative with respect to interest rate;rV:
Theta: = Derivative with respect to time;tV :
Greeks are useful in studying the stability of the quantity under variations of the
chosen parameters. If the price of an option is calculated using the measure Q as
xeV tTrQ E
where Ф(x) is the payoff function; the greek will be calculated under the same
simulation together with the price. The equation gives
Greek WE .xe tTrQ
where W is a random variable called Malliavin weight.
Malliavin Calculus is a special tool for calculating sensitivities of financial derivatives to change in its
underlying parameter. We now discuss the model of a financial market and the computation of the greeks.
6.1 Computation of the Greeks
We consider the Hilbert space which are constant on compact interval. We Assume that W = W(h),hϵ H
denotes an isonormal Gaussian process associated with the Hilbert space H. Let W be defined on a complete
probability space (Ω, F, P) and let F be generated by W .
Remark 6.1.1 The following observation will be important for the application of
proposition 5.5.
If u is deterministic. Then, for Gu(DuF)-1 ϵ Domδ it suffices to say that
Gu(DuF)-1 ϵ D1,2 as this implies that ., 2,11DomuDFGu
D
H
If u = DF, then the conclusion of Proposition 5.5 is written as
.2
HDF
GDFFfGFf EE
We consider an option with payoff H such that EQ(H2) < ∞. We have that
.
t
dsrQ
t HeEV
T
ts
F The price at t = 0 gives .0 HeV rTQ E Suppose
represent one of the parameters of S0, σ, r.
Let .FfH Then
.0
d
dFFfe
V QrT E (13)
From Proposition 5.5, we have
.,V0
d
dFFHFfe QrT E (14)
If f is not smooth, then (14) provides better result in combination with Monte-Carlo
Simulation than (13).
We note that the following:
In the calculation of the greeks, differentiation can be done before finding the
expectation, this will still give the same result.
The Malliavin derivative of ., TdW
dS
Tt SwtSD T
From proposition 5.5 DuF must not be zero in order to make sure that FDu
1
exist.
Let ;,0,)()(exp20
2
TttWtStS then,
Stochastics Calculus: Malliavin Calculus in a simplest way
www.iosrjournals.org 39 | Page
0
2
)()(exp2
0
S
ST TTWTS
S
In calculating the greeks, we assume that
TttWtStS ,0,)()(exp20
2
except otherwise stated.
6.2 Computation of Delta
We discuss delta of European options. Suppose H depends only on the price of the stock at maturity time T, that
is, H = Φ(ST). Let the price of the stock at time 0 be given by ,rTeE
.)(
0000
TT
rT
tT
rTT
rT
SSS
e
S
SSe
S
Se
S
V
Q
EEE
From proposition 5.5, by letting u =1, F = ST , and G = ST, we obtain
T T
TTttT
u TSdtSdtSDSD0 0
.
From the condition in the above remark, we have
T
TT
TtTT
W
T
dW
TdtSDS
0
1
0.
1
As a result,
.0
TT
QrT
WSTS
e
E
(15)
The weight is given by
weight = .0 TS
WT
6.3 Computation of Gamma
00
2
0
2
2
0
0
2
S
S
S
SSe
S
Se
S
V TTT
rTQT
rTQ
EE
.2
2
0
2
0
TT
QrT
TT
rTQ SSS
e
S
SSe
EE
We now apply the Malliavin derivative property in order to eliminate “/”. Suppose
is Lipschitz, let G = ,2
TS F = ST and u = 1.
From proposition 5.5 we have
T
DST
ST
SdtSDS TT
TT
TtT
1
,1
1
0
2
TT
TTdW
dSdW
TS
0
1,
1
TTT
TT
dtS
T
WS
0
.1
T
WSS
T
WS T
TTT
T
Thus,
Stochastics Calculus: Malliavin Calculus in a simplest way
www.iosrjournals.org 40 | Page
.12
T
WSSSS T
TT
Q
TT
Q
EE
Using Malliavin property, we now eliminate the remaining “/”.
From proposition 5.5, taking .1T
W
TTSG F = ST and u = 1 gives
T
T
W
T
T
TtT
W
TTS
SdtSDS TT
111
1
0
11
TT
W
T TSS T
TT
W
TT
W TT
11
2222
T
TTT
dW
TDW
TW
02222
1,
1
T TT
TT
W
T
dt
T
dWW
0 0 2222
.1222
2
2222
T
W
TT
W
T
W
T
T
T
WW TTTTT
As a result,
.1
1222
2
T
W
TT
WS
T
WSS TT
T
QTTT
Q
EE
This implies that
.12
2
0
TT
T
QrT
WT
WS
TS
e
E (16)
6.4 Computation of Vega
T
rTQ SeV E0
T
rTQ WTSe20
2
expE
TS
T
rTQ SeE
.TWSSe TTT
QrT E
Applying proposition 5.5, let G = ST (WT – σT), F = ST and u = 1 we have
T
TW
TS
TWSdtSDTWS T
T
TTT
TtTT
1
0
11
T
W
T
W TT
T
TT dWT
DWT
W0
1,
1
T T
Tt
T WT
dt
T
dWW
0 0
TT
T WT
TWW
T
T
T
W
122
Stochastics Calculus: Malliavin Calculus in a simplest way
www.iosrjournals.org 41 | Page
.12
TT
T
QrT WT
WSe
E (17)
The above formulas still hold by means of an approximate procedure although the function and its derivative
are not Lipschitz. The important thing is that should be piecewise continuous with jump discontinuities and with a linear growth.
The formulas are applicable in the case of European call option ( (x) = (x - K)+)
and European put option ( (x) = (K - x)+), or a digital ( (x) = 1x>K).
References [1] Bally, V., Caramellio, L. & Lombardi, L. (2010). An introduction to Malliavin Calculus and its application to finace. Lambratoire
d’analysis et de Mathématiques. Appliquées, Uviversité Paris-East, Marne-la-Vallée.
[2] El-Khatilo, Y. & Hatemi, A.J. (2011). On the Price Sensitivities During Financial crisis. Proceedings of the World Congress on
Engineering Vol I. WCE 2011, July 6-8, London, U.K.
[3] Fox, C. (1950). An Introduction to Calculus of Variations. Oxford University Press.
[4] Malliavin, P. (1976). Stochastic Calculus of variations and hypoelliptic operators. In Proceedings of the International Symposium
on Stochastic differential Equations (Kryto) K.Itô edt. Wiley, New York, 195-263
[5] Malliavin, P. (1997). Stochastic Analysis. Bulletin (New Series) of American Mathematical Society, 35(1), 99-104, Jan. 1998.
[6] Malliavin, P. & Thalmaier, A. (2006). Stochastic Calculation of Variations in Mathematical Finance. Bulletin (New Series) of the
American Mathematics Society, 44(3), 487-492,July 2007.
[7] Matchie, L. (2009). Malliavin Calculus and Some Applications in Finance. African Institute for Mathematical Sciences (AIMS),
South Africa.
[8] Nualart, D. (2006). The Malliavin Calculus and Related Topics: Probability and its Applications. Springer 2nd
edition.
[9] Nualart, D.G. (2009). Lectures on Malliavin Calculus and its applications to Finance.University of Paris.
[10] Nunno, D.G. (2009). Introduction to Malliavin Calculus and Applications to Finance,Part I. Finance and Insurance, Stochastical
Analysis and Practicalmethods. Spring School, Marie Curie.
[11] Schellhorn, H. & Hedley, M. (2008). An Algorithm for the Pricing of Path-Dependent American options using Malliavin Calculus.
Proceedings of World Congress on Engineering and Computer Science (WCECS 2008). San Francisco, U.S.A.
[12] S chröter, T. (2007). Malliavin Calculus in Finance. A Special Topic Essay.St. Hugh’s College, University of Oxford.
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