Notes on Stochastic · PDF fileThe modeling of random assets in ﬁnance is based on...

Chapter 4Brownian Motion and Stochastic Calculus
The modeling of random assets in finance is based on stochastic processes,which are families (Xt)tI of random variables indexed by a time interval I. Inthis chapter we present a description of Brownian motion and a constructionof the associated It stochastic integral.
4.1 Brownian Motion
We start by recalling the definition of Brownian motion, which is a funda-mental example of a stochastic process. The underlying probability space(,F ,P) of Brownian motion can be constructed on the space = C0(R+)of continuous real-valued functions on R+ started at 0.
Definition 4.1. The standard Brownian motion is a stochastic process(Bt)tR+ such that
1. B0 = 0 almost surely,
2. The sample trajectories t 7 Bt are continuous, with probability 1.
3. For any finite sequence of times t0 < t1 < < tn, the increments
Bt1 Bt0 , Bt2 Bt1 , . . . , Btn Btn1
are mutually independent random variables.
4. For any given times 0 6 s < t, Bt Bs has the Gaussian distributionN (0, t s) with mean zero and variance t s.
In Figure 4.1 we draw three sample paths of a standard Brownian motionobtained by computer simulation using (4.2). Note that there is no point in
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computing the value of Bt as it is a random variable for all t > 0, howeverwe can generate samples of Bt, which are distributed according to the cen-tered Gaussian distribution with variance t.
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
B t
Fig. 4.1: Sample paths of a one-dimensional Brownian motion.
In particular, Property 4 above implies
IE[Bt Bs] = 0 and Var[Bt Bs] = t s, 0 6 s 6 t,
and we have
Cov(Bs, Bt) = IE[BsBt]= IE[Bs(Bt Bs)] + IE[(Bs)2]= IE[Bs] IE[Bt Bs] + IE[(Bs)2]= s, 0 6 s 6 t,
henceCov(Bs, Bt) = IE[BsBt] = min(s, t), s, t R+,
cf. also Exercise 4.1-(d).
In the sequel, the filtration (Ft)tR+ will be generated by the Brownian pathsup to time t, in other words we write
Ft = (Bs : 0 6 s 6 t), t > 0. (4.1)
Property 3 in Definition 4.1 shows that BtBs is independent of all Brownianincrements taken before time s, i.e.
(Bt Bs) (Bt1 Bt0 , Bt2 Bt1 , . . . , Btn Btn1),
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Brownian Motion and Stochastic Calculus
0 6 t0 6 t1 6 6 tn 6 s 6 t, hence Bt Bs is also independent of thewhole Brownian history up to time s, hence Bt Bs is in fact independentof Fs, s > 0.
As a consequence, Brownian motion is a continuous-time martingale, cf.also Example 2 page 2, as we have
IE[Bt | Fs] = IE[Bt Bs | Fs] + IE[Bs | Fs]= IE[Bt Bs] +Bs= Bs, 0 6 s 6 t,
because it has centered and independent increments, cf. Section 6.1.
The n-dimensional Brownian motion can be constructed as (B1t , B2t , . . . , Bnt )tR+where (B1t )tR+ , (B2t )tR+ , . . .,(Bnt )tR+ are independent copies of (Bt)tR+ .Next, we turn to simulations of 2 dimensional and 3 dimensional Brownianmotions in Figures 4.2 and 4.3. Recall that the movement of pollen particlesoriginally observed by R. Brown in 1827 was indeed 2-dimensional.
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Fig. 4.2: Two sample paths of a two-dimensional Brownian motion.
Figure 4.4 presents an illustration of the scaling property of Brownian motion.
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-2-1.5
-1-0.5
0 0.5
1 1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Fig. 4.3: Sample paths of a three-dimensional Brownian motion.
Fig. 4.4: Scaling property of Brownian motion.
4.2 Constructions of Brownian Motion
We refer to Theorem 10.28 of [Fol99] and to Chapter 1 of [RY94] for theproof of the existence of Brownian motion as a stochastic process (Bt)tR+satisfying the above Conditions 1-4. See also Problem 4.19 for a constructionbased on linear interpolation.
Brownian motion as a random walk
For convenience we will informally regard Brownian motion as a random walkover infinitesimal time intervals of length t, whose increments The animation works in Acrobat Reader on the entire pdf file.
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Bt := Bt+t Bt ' N (0, t)
over the time interval [t, t+t] will be approximated by the Bernoulli randomvariable
Bt = t (4.2)
with equal probabilities (1/2, 1/2). Figure 4.5 presents a simulation of Brow-nian motion as a random walk with t = 0.1.
Fig. 4.5: Construction of Brownian motion as a random walk.
The choice of the square root in (4.2) is in fact not fortuitous. Indeed, anychoice of (t) with a power > 1/2 would lead to explosion of the processas dt tends to zero, whereas a power (0, 1/2) would lead to a vanishingprocess.
Note that we have
IE[Bt] =12t 12
t = 0,
andVar[Bt] = IE[(Bt)2] =
12t+
12t = t.
According to this representation, the paths of Brownian motion are not dif-ferentiable, although they are continuous by Property 2, as we have
dBtdt' dt
dt= 1
dt' . (4.3)
After splitting the interval [0, T ] into N intervals
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(k 1N
T,k
NT
], k = 1, 2, . . . , N,
of length t = T/N with N large, and defining the Bernoulli randomvariable Xk as
Xk := T
with probabilities (1/2, 1/2) we have Var(Xk) = T and
Bt :=XkN
= t
is the increment of Bt over ((k 1)t, kt], and we get
BT '
0

Fig. 4.6: Construction of Brownian motion by linear interpolation.
The following R code is used to generate Figure 4.6.
alpha=1/2;t

N. Privault
Fig. 4.7: Construction of Brownian motion by series expansions.
4.3 Wiener Stochastic Integral
In this section we construct theWiener stochastic integral of square-integrabledeterministic function with respect to Brownian motion.
Recall that the price St of risky assets has been originally modeled byBachelier as St := Bt, where is a volatility parameter. The stochasticintegral w T
0f(t)dSt =
w T0f(t)dBt
can be used to represent the value of a portfolio as a sum of profits andlosses f(t)dSt where dSt represents the stock price variation and f(t) is thequantity invested in the asset St over the short time interval [t, t+ dt].
A naive definition of the stochastic integral with respect to Brownian mo-tion would consist in letting
w T0f(t)dBt :=
w T0f(t)dBt
dtdt,
and evaluating the above integral with respect to dt. However this definitionfails because the paths of Brownian motion are not differentiable, cf. (4.3).Next we present Its construction of the stochastic integral with respect toBrownian motion. Stochastic integrals will be first constructed as integralsof simple step functions of the form
f(t) =ni=1
ai1(ti1,ti](t), t [0, T ], (4.4)
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i.e. the function f takes the value ai on the interval (ti1, ti], i = 1, 2, . . . , n,with 0 6 t0 < < tn 6 T , as illustrated in Figure 4.8.
6f
-
b rt0
a1
b rt1
a2 b rt2
b rt3 t4
a4
t
Fig. 4.8: Step function t 7 f(t).
Recall that the classical integral of f given in (4.4) is interpreted as the areaunder the curve f and computed as
w T0f(t)dt =
ni=1
ai(ti ti1).
In the next definition we adapt this construction to the setting of stochasticintegration with respect to Brownian motion. The stochastic integral (4.5)will be interpreted as the sum of profits and losses ai(Bti Bti1), i =1, 2, . . . , n, in a portfolio holding a quantity ai of a risky asset whose pricevariation is Bti Bti1 at time i = 1, 2, . . . , n.
Definition 4.2. The stochastic integral with respect to Brownian motion(Bt)t[0,T ] of the simple step function f of the form (4.4) is defined by
w T0f(t)dBt :=
ni=1
ai(Bti Bti1). (4.5)
In the next Lemma 4.3 we determine the probability distribution ofw T
0f(t)dBt
and we show that it is independent of the particular representation (4.4) cho-sen for f(t).
Lemma 4.3. Let f be a simple step function f of the form (4.4). The stochas-tic integral
w T0f(t)dBt defined in (4.5) has a centered Gaussian distribution
w T0f(t)dBt ' N
(0,
w T0|f(t)|2dt
)
with mean IE[w T
0f(t)dBt
]= 0 and variance given by the It isometry
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Var[w T
0f(t)dBt
]= IE
[(w T0f(t)dBt
)2]=
w T0|f(t)|2dt. (4.6)
Proof. Recall that if X1, X2, . . . , Xn are independent Gaussian randomvariables with prob

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