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WEAK APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

创建日期：2024-05-11 17:27:00

所属分类：数学mathematics

标签： DIFFERENTIAL

WEAK APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

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WEAK APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

AND APPLICATION TO DERIVATIVE PRICING
Abstract. The authors present a new simple algorithm to approximate weakly
stochastic dierential equations in the spirit of [9][13]. They apply it to the problem
of pricingAsian options under theHeston stochastic volatilitymodel, and compare
it with other known methods. It is shown that the combination of the suggested
algorithm and quasi-Monte Carlo methods makes computations extremely fast.
1. Introduction
1.1. The Problem and its Motivation. We consider a stochastic dierential equa-
tion written in the Stratonovich form
Y(t; x) = x +
Z t
0
V0 (Y(s; x)) ds +
dX
i=1
Z t
0
Vi (Y(s; x)) dBis;
V j 2 C1b

RN;RN

;
(1)
where B =

B1; ;Bd

is a standard Brownian motion, and C1b

RN;RN

denotes
the set of RN-valued smooth functions dened over RN whose derivatives of
any order are bounded. In particular, we will use the classical notation V f (x) =PN
i=1 V
i (x)

@ f=@xi

(x) for V 2 C1b (RN;RN) and f a dierentiable function from Rn
into R: This stochastic dierential equation can be written in It o form:
Y(t; x) = x +
Z t
0
V0 (Y(s; x)) ds +
dX
i=1
Z t
0
Vi (Y(s; x)) dBis;
where
Vi0

y

= Vi0

y

+
1
2
dX
j=1
V jVij

y

:
Now, given a function f with some regularity, how can one approximate e-
ciently E

f (Y(1; x))

? It is equivalent to the following deterministic problem: if L is
the dierential operatorV0+ (1=2)
Pd
i=1 V
2
i and u is the solution of the heat equation
@u
@t
(t; x) = Lu; u (0; x) = f (x);
2000Mathematics Subject Classication. 65C30, 65C05.
Key words and phrases. Hestonmodel, numericalmethods for stochastic dierential equations, math-
ematical nance, quasi-Monte Carlo method.
This research was partially supported by the Japanese Ministry of Education, Science, Sports and
Culture, Grant-in-Aid for Scientic Research (C), 15540110, 2003.
1
2 S. NINOMIYA AND N. VICTOIR
howdoes one approximate u (1; x) (which is equal toE

f (Y(1; x))

by Feynman-Kac
theorem [7]).
This problem has had a lot of attention because of its practical importance: it
gives the evolution of the temperature in some media, and also represents price of
nancial derivatives under stochastic nancial models such as Black-Scholes [1].
Non-probabilistic methods to solve the PDE (such as nite dierence methods)
seem to only work well when L is elliptic and in low dimension. We refer to [11]
for a more detailed discussion on the subject. We will focus in this paper on
probabilistic methods.
1.2. Notation. If V is a smooth vector eld, i.e. an element of C1b

RN;RN

,
exp (V) (x) denotes the solution at time 1 of the ordinary dierential equation
dzt
dt
= V (zt) ; z0 = x:
For x 2 R, bxc denotes the integer part of x.
1.3. Probabilistic Methods.
1.3.1. Order 1. The most popular probabilistic method to approximate E
h
f

Yx1
i
is
called the Euler-Maruyama method [8]. We rst x n independent d-dimensional
random variables Z1; ;Zn such that, if X denotes a standard normal random
variables, E

p (Zk)

= E

p (X)

for all polynomial of degree less than or equal to 3.
1 Then one denes recursively the following random variables:
X(EM);n0 = x;
X(EM);n(k+1)=n = X
(EM);n
k=n +
1
n
V

X(EM);nk=n

+
1p
n
dX
i=1
Vi

X(EM);nk=n

Zik+1:
Then, one can show [8][17] that for all nice enough function f
(2)
E h f X(EM);n1 E f (Y(1; x)) C f 1n :
2 Of course, one needs an algorithm to compute E
h
f

X(EM);n1
i
: If the Zk are con-
structed from Bernoulli random variables, E
h
f

X(EM);n1
i
is a discrete sum, but one
would need to do 2nd additions, which can be rather lengthy when nd is large (one
is then forced to do someMonte-Carlo on a discrete measure). If theNi are normal
random variables, one then is forced to do use some Monte Carlo or quasi-Monte
Carlo techniques. When nd is big, quasi-Monte Carlomethod become less eective
thanMonte-Carlo, but if nd is not too high, quasi-Monte Carlo method can be very
ecient.
1Such random variable are easy to nd. One can take, for a xed i; Z ji to be d independent Bernoulli
or Gaussian random variables. A more elaborate choice of such random variables appeared in [13][16].
2Here, we have used the subdivision (k=n)k2f0; ;ng of [0; 1] : In no way taking equal time steps is
optimal. We do not want to address this problem in this paper, and we will always take subdivisions
with equal time steps.
WEAK APPROXIMATION OF SDE 3
Another method with the same rate of convergence appeared in [13], and is
called cubature on Wiener space of degree 3. It is dened with the following
recursive formula:
X(cub3);n0 = x;
X(cub3);n(k+1)=n = exp
0BBBBB@1nV0 + 1pn
dX
i=1
Zik+1Vi
1CCCCCA X(cub3);nk=n
Such algorithm can be seen as a practical application of the Wong-Zakai theo-
rem [7][18], when the Zk are normal random variables.
If Bnt = (B
n;1
t ; : : :B
n;d
t ) (n 2N) is the piecewise linear approximation of the Brown-
ian motion dened by
Bnt = (bntc + 1 nt)Bnbntc=n + (nt bntc)Bn(bntc+1)=n;
and Yn denotes the solution of the ordinary dierential equation
Ynt = x +

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