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On moment conditions for the Girsanov Theorem.

Abstract

It is well-known that the proof of the Girsanov Theorem involves the local martingale theory. Our motivation for the main result in this paper is the desire to avoid using the local martingale theory in the proof of the Girsanov Theorem, namely we will use only the martingale theory to obtain our result. Many sufficient conditions for the validity of the Girsanov Theorem have been found since the publication of the result by Girsanov in 1960. We will compare our conditions with some of these sufficient conditions.

AMS Subject Classification: 60H05.

Keywords: Girsanov theorem, martingale, local martingale, exponential processes, Schwarz's inequality.

1. Introduction

In 1960, Girsanov proved a famous result which nowadays being called the Girsanov Theorem. Let B(t) be a Brownian motion on the probability space ([OMEGA],F, P) and let {[F.sub.t]; a [less than or equal to] t [less than or equal to] b} be a filtration such that B(t) is [F.sub.t]-measurable for each t and for any s [less than or equal to] t, the random variable B(t) - B(s) is independent of the [sigma]-field [F.sub.s]. We denote by [L.sub.ad] ([OMEGA], [L.sup.2],[a, b]) the space of all stochastic processes h(t, [omega]), a [less than or equal to] t [less than or equal to] b, [omega] [member of] [omega] such that h(t) is [F.sub.t]-adapted and [[integral].sup.b.sub.a][[absolute value of h (t)].sup.2] dt < [infinity] almost surely. An exponential process [E.sub.h](t), a [less than or equal to] t [less than or equal to] b, given by h(t) [member of] [L.sub.ad] ([OMEGA],[L.sub.2], [a, b]) is a stochastic process defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the Girsanov Theorem states that if the exponential process [E.sub.h](t), 0 [less than or equal to] t [less or equal to] T given by h(t) [member of] [L.sub.ad] ([OMEGA],[L.sup.2][0, T]) is a martingale, then the process given by

W(t) = B(t) - [[integral].sup.+.sub.0] h(s) ds, 0 [less than or equal to] t [less than or equal to] T,

is a Brownian motion with respect to the probability measure Q given by dQ = [E.sub.h](T) dP. An exponential process [E.sub.h](t), 0 [less than or equal to] t [less than or equal to] T given by h(t) [member of] [L.sub.ad] ([OMEGA],[L.sup.2][0, T]) is a martingale if and only if E[[E.sub.h](T)] = 1. So the Girsanov Theorem is true if E[[E.sub.h](T)] = 1. The original proof of this theorem involves the local martingale theory. Our motivation is to present a proof involving only the martingale theory.

We denote by [L.sup.2.sub.ad] ([a, b] x [OMEGA]) the space of all stochastic processes h(t, [omega]), a [less than or equal to] t [less than or equal to] b, [omega][member of] [OMEGA] such that h(t) is [F.sub.t]-adapted and [[integral].sup.b.sub.a] E [[absolute value of h(t)].sup.2] dt < [infinity]. It is a fact that [L.sup.2.sub.ad] ([a, b] x [OMEGA]) [subset] [L.sub.ad] ([OMEGA],[L.sup.2] [a, b]). It is known that a martingale is a local martingale but not conversely. It is also known that for h(t) [member of] [L.sub.ad] ([OMEGA],[L.sup.2[a, b]), the stochastic process [[integral].sup.t.sub.a] h(t) dB(t) is a local martingale, while for h(t) [member of] [L.sup.2.sub.ad] ([a, b] x [OMEGA]), the stochastic process [[integral].sup.t.sub.a] h(t) dB(t) is a martingale. With this observation, our result is dealt for h(t) [member of] [L.sup.2.sub.ad] ([a, b] x [OMEGA]).

2. Exponential Processes

As stated above, an exponential process [E.sub.h](t), 0 [less than or equal to] t [less than or equal to] T given by h(t) [member of] [L.sub.ad] ([OMEGA],[L.sup.2][0, T]) is a martingale if and only if E[[E.sub.h](T)] = 1. Since [L.sup.2.sub.ad] ([a, b] x [OMEGA]) [subset] [L.sub.ad] ([OMEGA],[L.sup.2][a, b]), the result is obviously true for exponential processes [E.sub.h](t), 0 [less than or equal to] t [less than or equal to] T given by h [member of] [L.sup.2.sub.ad] ([a, b] x [OMEGA]). The next lemma gives another sufficient condition for the exponential process [E.sub.h](t) given by h [member of] [L.sup.2.sub.ad] ([a, b] x [OMEGA]) to be a martingale.

Theorem 2.1. Let h [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]). Then the exponential process [E.sub.h](t), 0 [less than or equal to] t [less than or equal to] T is a martingale if

E [[integral].sup.T.sub.0] [E.sub.h][(t).sup.2] h[(t).sup.2] dt < [infinity].

Proof. Let [X.sub.t] = [[integral].sup.t.sub.0] h(s) dB(s) - 1/2 [[integral].sup.t.sub.0] h[(s).sup.2] ds. By applying Ito's formula ([5], page 103), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which gives

[E.sub.h](t) = 1 + [[integral].sup.t.sub.0] [E.sub.h](s)h(s) dB(s) for 0 [less than or equal to] t [less than or equal to] T. (2.1)

By hypothesis, if E [[integral].sup.t.sub.0] [E.sub.h][(s).sup.2]h[(s).sup.2] ds < [infinity], 0 [less than or equal to] t [less than or equal to] T, then [E.sub.h](t)h(t) [member of] [L.sup.2.sub.ad]([0, T] x [OMEGA]). Thus [[integral].sup.t.sub.0] [E.sub.h](s)h(s) dB(s) is a martingale and therefore [E.sub.h](t) is a martingale.

Theorem 2.2. If h [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]) satisfies the condition that

E [[integral].sup.t.sub.0] h[(t).sup.2][E.sub.h][(t).sup.4] dt < [infinity], (2.2)

Then

E [[integral].sup.t.sub.0] h[(t).sup.2][E.sub.h][(t).sup.2] dt < [infinity] (2.3)

and

E [[E.sub.h][(t).sup.2]] = 1 + E [[integral].sup.t.sub.0] h[(s).sup.2][E.sub.h][(s).sup.2] ds, 0 [less than or equal to] t [less than or equal to] T. (2.4)

Remark 2.3. By Theorem 2.1, it is clear that Equation 2.2 gives another sufficient condition for the exponential process [E.sub.h](t) given by h [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]) to be a martingale.

Proof. Let us write h[(t).sup.2][E.sub.h][(t).sup.2] = (h(t))(h(t)[E.sub.h][(t).sup.2]). By using Schwarz's inequality, we have

[[integral].sup.T.sub.0] h[(t).sup.2][E.sub.h][(t).sup.2] dt [less than or equal to] [([[integral].sup.t.sub.0] h[(t).sup.2])dt.sup.1/2] [([[integral].sup.t.sub.0] h[(t).sup.2[E.sub.h][(t).sup.4] dt).sup.1/2].

Taking the expectation on both sides and by Schwarz's inequality, we get

E [[integral].sup.t.sub.0] h[(t).sup.2][E.sub.h][(t).sup.2] dt [less than or equal to] E [[([[integral].sup.t.sub.0] h[(t).sup.2] dt).sup.1/2] [([[integral].sup.t.sub.0]h[(t).sup.2] [E.sub.h] [(t).sup.4] dt).sup.1/2].

[less than or equal to] [(E [[integral].sup.t.sub.0] h[(t).sup.2] dt).sup.1/2][(E [[integral].sup.t.sub.0] h[(t).sup.2] [E.sub.h] [(t).sup.4] dt).sup.1/2].

Hence Equation 2.2 implies Equation 2.3 since for h [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]), we have E [[integral].sup.t.sub.0]h[(t).sup.2] dt < [infinity].

From Equation 2.1, we have d[E.sub.h](t) = [E.sub.h](t)h(t) dB(t). By using the Ito product formula, we get

d ([E.sub.h] [(t).sup.2]) = 2 [E.sub.h] (t)[d[E.sub.h](t)] + [[d[E.sub.h](t)].sup.2]

= 2 [E.sub.h] (t)[[E.sub.h] (t)h(t)dB(t)] + [[[E.sub.h] (t)h(t)dB(t)].sup.2]

= 2 [E.sub.h] [(t).sup.2]h(t)dB(t) + [E.sub.h][(t).sup.2]h[(t).sup.2]dt.

Thus,

[E.sub.h] [(t).sup.2] = 1 + 2 [[integral].sup.t.sub.0] [E.sub.h][(s).sup.2]h(s) dB(s) + [[integral].sup.t.sub.0] [E.sub.h][(s).sup.2]h[(s).sup.2] ds.

Taking the expectation on both sides and since [[integral].sup.t.sub.0] [E.sub.h][(s).sup.2]h(s) dB(s) is a martingale with mean zero, we get Equation 2.4.

Theorem 2.4. Suppose the exponential process [E.sub.h](t), 0 [less than or equal to] t [less than or equal to] T, given by h [member of] [L.sup.2.sub.ad]([0, T]x [OMEGA]), satisfies the condition E [[integral].sup.t.sub.0] [E.sub.h][(t).sup.4] dt < [infinity]. Then E [[integral].sup.t.sub.0] B[(t).sup.2] [E.sub.h] [(t).sup.2] dt < [infinity].

Proof. By Schwarz's inequality, we have

E [[integral].sup.t.sub.0] B[(t).sup.2] [E.sub.h] [(t).sup.2] dt [less than or equal to] E [[([[integral].sup.T.sub.0]B[(t).sup.4] dt).sup.1/2] ([[integral].sup.t.sub.0] [E.sub.h][(t).sup.4] dt).sup.1/2]]

[less than or equal to] [(E [[integral].sup.t.sub.0] B[(t).sup.4] dt).sup.1/2] [(E [[integral].sup.t.sub.0] [E.sub.h][(t).sup.4] dt).sup.1/2] < [infinity]

since E [[integral].sup.t.sub.0] B[(t).sup.4] dt = [[integral].sup.T.sub.0] E[B[(t).sup.4]] dt by Fubini's theorem and E[B[(t).sup.4]] = 3[t.sup.2].

3. Results

In this section, we present our main result, namely the Girsanov Theorem under new conditions. Throughout this section, we consider the exponential process [E.sub.h](t), 0 [less than or equal to] t [less than or equal to] T given by h [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]), unless otherwise stated. Let B(t) be a Brownian motion on the probability space ([OMEGA],F, P). Let Q be the probability measure in ([OMEGA],F) defined by dQ = [E.sub.h](T) dP, i.e.,

Q(A) = [[integral].sub.A] [E.sub.h](T) dP, A [member of] F.

Then Q and P are equivalent probability measures. To avoid confusion, from now on we denote [E.sub.P] and [E.sub.Q] to be the expectation with respect to P and Q, respectively.

The following lemma is very useful in proving our result.

Lemma 3.1. Let [theta] [member of] [L.sup.1](P) be nonnegative such that d[mu] = [theta] dP defines a probability measure. Then for any [sigma]-field G [subset] F and X [member of] [L.sup.1]([mu]), we have

[E.sub.[mu]][X| G] = [E.sub.P] [X[theta]| G]/[E.sub.P] [[theta]| G], [mu]- almost surely,

where [E.sub.[mu]] and [E.sub.P] denote the expectation with respect to [mu] and P, respectively.

Proof. See [5] page 141. _

Theorem 3.2. Consider the stochastic process W(t) = B(t) - [[integral].sup.t.sub.0] h(s) ds, 0 [less than or equal to] t [less than or equal to] T.

Suppose for h [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]) we have the following:

(a) [E.sub.P] [[integral].sup.T.sub.0] h[(t).sup.2] [E.sub.h] [(t).sup.4] dt < [infinity],

(b) [E.sub.P] [[integral].sup.T.sub.0] h[(t).sup.2]B[(t).sup.8] dt < [infinity],

(c) [E.sub.P] [([[integral].sup.t.sub.0] h[(t).sup.2] dt).sup.5] < [infinity],

(d) [E.sub.P] [[integral].sup.T.sub.0] 0 [E.sub.h][(t).sup.4] dt < [infinity].

Then W(t) and W[(t).sup.2] - t, 0 [less than or equal to] t [less than or] T, are Q-martingales.

Proof. (I) W(t), 0 = t = T, is a Q-martingale

First note that condition (a) implies that the exponential process [E.sub.h](t) is a martingale. Also we have by Equation 2.4 that [E.sub.P] [[E.sub.h] [(t).sup.2]] < [infinity] for any 0 [less than or equal

to] t [less than or equal to] T. Now using the fact [(x + y).sup.2] [less than or equal to] 2([x.sup.2] + [y.sup.2]) and Schwarz's inequality, we get

[E.sub.P] [W[(t).sup.2]] [less than or equal to] 2[E.sub.P] [B[(t).sup.2] + [([[integral].sup.t.sub.0] h(s) ds).sup.2]]

[less than or equal to] 2 [E.sub.P] [B[(t).sup.2] + t ([[integral].sup.t.sub.0] h[(s).sup.2] ds)]

= 2 [t + t [E.sub.P] [[integral].sup.t.sub.0] h[(s).sup.2] ds] < [infinity].

Hence, [E.sub.P] ([absolute value of W(t) [E.sub.h] (T)]) [less than or equal to] [[[E.sub.P](W[(t).sup.2])].sup.1/2] [[E.sub.P] ([E.sub.h] (T)).sup.2].sup.1/2] < [infinity]. Thus we can consider the conditional expectation of W(t) [E.sub.h] (T) with respect to a [sigma]-field.

Let 0 [less than or equal to] s < t [less than or equal to] T. Since [E.sub.h](t) is a martingale, we have

[E.sub.P] [W(t) [E.sub.h] (T |[F.sub.s]] = [E.sub.P] ([E.sub.P] [W(t) [E.sub.h] (T)|[F.sub.t]]|[F.sub.s]) = [E.sub.P] (W(t)[E.sub.P] [[E.sub.h] (T)|[F.sub.t]]|[F.sub.s]) = [E.sub.P] [W(t) [E.sub.h] (t)|[F.sub.s]]. (3.1)

On the other hand, by Lemma 3.1,

[E.sub.Q] [W(t)|[F.sub.s]] = [E.sub.P] [W(t) [E.sub.h] (T)|[F.sub.s]]/[E.sub.P] [[E.sub.h] (T)|[F.sub.s]] = [E.sub.P] [W(t) [E.sub.h] (T)|[F.sub.s]]/ [E.sub.h] (s). (3.2)

It follows from Equations 3.1 and 3.2 that

[E.sub.Q][W(t)|[F.sub.s]] = [E.sub.P] [W(t) [E.sub.h] (t)|[F.sub.s]]/ [E.sub.h] (s). (3.3)

Thus if W(t) [E.sub.h] (t) is a P-martingale, then Equation 3.3 becomes

[E.sub.Q][W(t)|[F.sub.s]] = [E.sub.P] [W(t) [E.sub.h] (t)|[F.sub.s]]/[E.sub.h] (s) = W(s) [E.sub.h] (s)/[E.sub.h] (s) = W(s)

for all s [less than or equal to] t, which shows W(t), 0 [less than or equal to] t [less than or equal to] T, is a Q-martingale.

To show W(t) [E.sub.h] (t) is a P-martingale, by dW(t) = dB(t) - h(t)dt and d[E.sub.h](t) = h(t) [E.sub.h] (t) dB(t) (from Equation 2.1) and by applying the Ito product formula, we obtain

d[W(t) [E.sub.h] (t)] = [dW(t)] [E.sub.h] (t) + W(t)d[E.sub.h](t) + [dW(t)][d[E.sub.h](t)] = [1 + h(t)W(t)] [E.sub.h](t) dB(t).

Hence we have for 0 [less than or equal to] t [less than or equal to] T,

W(t) [E.sub.h] (t) = [[integral].sup.t.sub.0] [1 + h(s)W(s)] [E.sub.h] (s) dB(s) = [[integral].sup.t.sub.0] [E.sub.h](s) dB(s) + [[integral].sup.t.sub.0] h(s)W(s) [E.sub.h] (s) dB(s). (3.4)

It is clear from Equation 3.4 that if the integrals [[integral].sup.t.sub.0] [E.sub.h](s) dB(s) and [[integral].sup.t.sub.0] h(s)W(s) [E.sub.h] (s) dB(s) are P-martingales, then W(t) [E.sub.h] (t) is a P-martingale. Thus we will show that [E.sub.h](t) and h(t)W(t) [E.sub.h] (t) are in [L.sup.2.sub.ad]([0, T] x [OMEGA]).

Recall that [E.sub.P] [[E.sub.h] [(t).sup.2]] < [infinity]. So [[integral].sup.t.sub.0] [E.sub.P] [[E.sub.h] [(t).sup.2]] dt < [infinity] and therefore [E.sub.h](t) [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]). Write h[(t).sup.2]W[(t).sup.2] [E.sub.h] [(t).sup.2] as (h(t)W[(t).sup.2]) (h(t) [E.sub.h] [(t).sup.2]_ and then use

Schwarz's inequality to get

[E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2]W[(t).sup.2] [E.sub.h] [(t).sup.2] dt [less than or equal to] [E.sub.P] [[([[integral].sup.t.sub.0] h[(t).sup.2]W[(t).sup.4] dt).sup.1/2] [([[integral].sup.t.sub.0] h[(t).sup.2] [E.sub.h] [(t).sup.4] dt).sup.1/2]] [less than or equal to] [[[E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2]W[(t).sup.4] dt].sup.1/2] [[[E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2] [E.sub.h] [(t).sup.4]dt].sup.1/2].

The second factor on the right hand side is finite by condition (a). For the first factor, use the inequality [(x + y).sup.4] [less than or equal to] 8([x.sup.4] + [y.sup.4]) and Schwarz's inequality to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So,

[E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2]W[(t).sup.4] dt [less than or equal to] 8 [E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2]B[(t).sup.4] dt + [T.sup.2][E.sub.P][([[integral].sup.t.sub.0] h[(t).sup.2] dt).sup.3].

By condition (c), [E.sub.P][([[integral].sup.t.sub.0] h[(t).sup.2] dt).sup.3] < [infinity]. On the other hand, by writing h[(t).sup.2]B[(t).sup.4] as h(t) (h(t)B[(t).sup.4]) and again using Schwarz's inequality, we get

[E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2]B[(t).sup.4] dt [less than or equal to] [E.sub.P] [[([[integral].sup.t.sub.0]h[(t).sup.2] dt).sup.1/2] [([[integral].sup.t.sub.0]h[(t).sup.2]B[(t).sup.8] dt).sup.1/2]].

[less than or equal to][([E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2] dt).sup.1/2] [([E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2]B[(t).sup.8] dt).sup.1/2],

which is finite by conditions (b) and (c). Hence [E.sub.P][[integral].sup.t.sub.0] h[(t).sup.2]W[(t).sup.4] dt < [infinity].

(II) W[(t).sup.2] - t, 0 [less than or equal to] t [less than or equal to] T, is a Q-martingale

Similarly as in deriving Equation 3.3, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

If [W[(t).sup.2] - t] [E.sub.h] (t), 0 [less than or equal to] t [less than or equal to] T, is a P-martingale, then W[(t).sup.2] - t, 0 [less than or equal to] t [less than or equal to] T, is a Q-martingale.

In order to show that [W[(t).sup.2] - t] [E.sub.h] (t), 0 [less than or equal to] t [less than or equal to] T, is a P-martingale, note that by the Ito product formula,

d [W[(t).sup.2] [E.sub.h] (t)] = [dW(t)]W(t) [E.sub.h] (t) + W(t) d[W(t) [E.sub.h] (t)] + [dW(t)]d[W(t) [E.sub.h] (t)] = [2 + h(t)W(t)]W(t) [E.sub.h] (t) dB(t) + [E.sub.h](t) dt.

Thus

W[(t).sup.2] [E.sub.h] (t) = [[integral].sup.t.sub.0] [2 + h(s)W(s)] W(s) [E.sub.h] (s) dB(s) + [[integral].sup.t.sub.0] [E.sub.h](s) ds. (3.6)

Now we show the integrand in the first integral on the right of Equation 3.6 belongs to [L.sup.2.sub.ad] ([0, T] x [OMEGA]), i.e., to show the processes W(t) [E.sub.h] (t) and h(t)W[(t).sup.2] [E.sub.h] (t) are in [L.sup.2.sub.ad]([0, T] x [OMEGA]). First, by using the inequality [(x + y).sup.2] [less than or equal to] 2([x.sup.2] + [y.sup.2]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

The first expectation on the right hand side is finite by Theorem 2.4. For the second expectation, we use Schwarz's inequality to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

The second expectation in the right hand side of Inequality 3.8 is finite by condition (d). For the first expectation, by Fubini's theorem we have

[E.sub.P] [[integral].sup.t.sub.0][([[integral].sup.t.sub.0] h[(s).sup.2] ds).sup.2] dt = [[integral].sup.t.sub.0] [E.sub.P][([[integral].sup.t.sub.0] h[(s).sup.2] ds).sup.2]dt, (3.9)

which is finite by condition (c). Thus [E.sub.P] [[integral].sup.t.sub.0] [([[integral].sup.t.sub.0]h(s) ds).sup.2] [E.sub.h] [(t).sup.2] dt < [infinity] and therefore [E.sub.P] [[integral].sup.t.sub.0] W[(t).sup.2] [E.sub.h] [(t).sup.2] dt < [infinity], i.e., W(t) [E.sub.h] (t) [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]).

Next we show that h(t)W[(t).sup.2] [E.sub.h] (t) is in [L.sup.2.sub.ad] ([0, T] x [OMEGA]). By using the inequality [(x + y).sup.4] [less than or equal of] 8([x.sup.4] + [y.sup.4]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.10)

By writing h[(t).sup.2]B[(t).sup.4] [E.sub.h] [(t).sup.2] as (h(t)B[(t).sup.4]) (h(t) [E.sub.h] [(t).sup.2]) and using Schwarz's inequality,

we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

which is finite by conditions (a) and (b). On the other hand, by Schwarz's inequality,

[([[integral].sup.t.sub.0]h(s) ds).sup.4] = [[([[integral].sup.t.sub.0] (1) h(s) ds).sup.2].sup.2] [less than or equal to] [[([[integral].sup.t.sub.0] 1 ds) ([[integral].sup.t.sub.0] h[(s).sup.2] ds)].sup.2] [less than or equal to] [T.sup.2] ([[integral].sup.T.sub.0])(3.12)

Apply this to the second term in Inequality 3.10 to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By writing h[(t).sup.2][([[integral].sup.t.sub.0]h[(s).sup.2] ds).sup.2] [E.sub.h] [(t).sup.2] as [h(t)[([[integral].sup.t.sub.0] h[(s).sup.2] ds).sup.2]] (h(t) [E.sub.h] [(t).sup.2]) and using Schwarz's inequality, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is finite by conditions (a) and (c). Hence

[E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2][([[integral].sup.t.sub.0]h(s) ds).sup.4] [E.sub.h] [(t).sup.2] dt < [infinity]. (3.13)

Therefore we get from Inequality 3.10 that

[E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2]W[(t).sup.4] [E.sub.h] [(t).sup.2] dt < [infinity],

namely h(t)W[(t).sup.2] [E.sub.h] (t) [member of] [L.sup.2.sub.ad]([0, T] x [OMEGA]).

Finally we take the conditional expectation on Equation 3.6 to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.15)

Since

E {E [[[integral].sup.t.sub.s] [E.sub.h](u) du|[F.sub.s]} = E {[[integral].sup.t.sub.s] [E.sub.h](u) du} = E {[[integral].sup.t.sub.s] E [[E.sub.h] (u)|[F.sub.s]]du},

it follows that

E [[[integral].sup.t.sub.s] [E.sub.h](u) du|[F.sub.s]] = [[integral].sup.t.sub.s] E [[E.sub.h] (u)|[F.sub.s]] du = [[integral].sup.t.sub.s] [E.sub.h](s) du = [E.sub.h](s)(t - s).

Thus Equation 3.14 becomes

[E.sub.P] [W[(t).sup.2] [E.sub.h] (t)|[F.sub.s]] = W[(s).sup.2] [E.sub.h] (s) + [E.sub.h](s)(t - s),

which implies that for any s [less than or equal to] t,

[E.sub.P] [(W[(t).sup.2] - t) [E.sub.h] (t)|[F.sub.s]] = (W[(s).sup.2] - s) [E.sub.h] (s).

Thus [W[(t).sup.2] - t] [E.sub.h] (t), 0 [less than or equal to] t [less than or equal to] T is a P-martingale, as desired.

Now we look at the Girsanov Theorem.

Theorem 3.3. (Girsanov Theorem) Let h [member of] [L.sup.2.sub.ad]([0, T] x [OMEGA]) satisfy the conditions

(a) [E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2] [E.sub.h] [(t).sup.4] dt < [infinity],

(b) [E.sub.P] [[integral].sup.t.sub.0] h[(t).sup.2]B[(t).sup.8] dt < [infinity],

(c) [E.sub.P] [([[integral].sup.t.sub.0] h[(t).sup.2] dt).sup.5] < [infinity],

(d) [E.sub.P] [[integral].sup.t.sub.0] [E.sub.h][(t).sup.4] dt < [infinity].

Then the stochastic process

W(t) = B(t) - [[integral].sup.t.sub.0] h(s) ds, 0 [less than or equal to] t [less than or equal to] T is a Brownian motion with respect to the probability measure Q defined by dQ = [E.sub.h] (T) dP, namely Q(A) = [[integral].sub.A] [E.sub.h](T) dP for A [member of] F.

Remark 3.4. In observance of Theorem 2.2, condition (a) itself in Theorem 3.3 is sufficient for the validity of the theorem. However as can be seen in the proof of Theorem 3.2, the other conditions are needed to ensure that some of the stochastic integrals that appear in the proof are martingales.

Proof. Obviously the probability measures P and Q are equivalent. Hence Q{W(0) = 0} = 1 and W(t) is a continuous stochastic process. Let {[F.sub.t]} be the filtration given by [F.sub.t] = [sigma]{B(s); 0 [less than or equal to] s [less than or equal to] t}, 0 [less than or equal to] t [less than or equal to] T. By Theorem 3.2, W(t) and W[(t).sup.2] - t are martingales with respect to Q and [F.sub.t]. Then the Doob-Meyer decomposition of W[(t).sup.2] is given by

W[(t).sup.2] = [W[(t).sup.2] - t] + t.

So [<W>.sub.t] = t almost surely with respect to Q for each t. Hence by the Levy Characterization Theorem of Brownian motion ([5, Theorem 8.4.2, p.126]), W(t) is a Brownian motion with respect to Q.

4. Comparison of Sufficient Conditions for the Girsanov Theorem

In 1960, Girsanov [2] raised the problem of finding a sufficient condition for the Exponential process [E.sub.h](t), h [member of] [L.sub.ad] ([OMEGA],[L.sup.2][0, T]) to be a martingale. Since then many sufficient conditions have been found: Novikov [7], Kazamaki [4], Gihman and Skorohod [1], Liptser and Shiryaev [6] and Okada [9], to name a few. In this section, we compare some of these conditions for [E.sub.h](t) given by h [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]).

Consider a probability space ([OMEGA],F, P). Throughout this section, the expectation is taken with respect to P and B(t) is a Brownian motion with respect to P. Note that the problem of finding a sufficient condition for the exponential process [E.sub.h](t), 0 [less than or equal to] t [less than or] T given by h [member of] [L.sub.ad]([OMEGA],[L.sup.2][0, T]) to be a martingale is equivalent to finding sufficient conditions for the validity of the Girsanov Theorem. The following results (except (a)) are true for exponential processes [E.sub.h](t), 0 [less than or equal to] t [less than or equal to] T given by h [member of] [L.sub.ad]([OMEGA],[L.sup.2][0, T]). Since [L.sup.2.sub.ad] ([0, T] x [OMEGA]) [member of] [L.sub.ad]([OMEGA],[L.sup.2][0, T]), we only state the result for h [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]) to suit our purpose.

Theorem 4.1. For h [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]), we get E[[E.sub.h](T)] = 1 under any one of the following:

(a) E [[integral].sup.t.sub.0] h[(t).sup.2] [E.sub.h] [(t).sup.4] dt < [infinity].

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(d) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for some [delta] > 0.

(e) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for [delta] > 0, 0 [less than or equal to] t [less than or equal to] T and some [delta] > 0.

(f) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for some [delta] > 0.

(g) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for some [epsilon] > 0 and constant C.

Proof.

(a) A result of Theorems 2.2 and 2.1.

(b) See [8].

(c) See [4].

(d) See [1, 3].

(e) See [3].

(f) See [6].

(g) Fix [lambda] > 1 and choose a finite partition {[t.sub.j]} of [0,T] such that each of [lambda][t.sub.1], [lambda]([t.sub.2] - [t.sub.1]), ..., [lambda]([t.sub.T] - [t.sub.n-1]) is less than [epsilon]. Then by Jensen's inequality ([10]), for each j we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore the conclusion follows from (f).

Theorem 4.2. From Theorem 4.1, we have the following implications.

1. (b) [??] (c).

2. (b) [??] (a) if E[h[(t).sup.4]] < [infinity].

3. (d) [??] (b).

4. (d) [??] (c).

5. (d) [??] (e).

6. (d) [??] (f).

7. (f) [??] (b).

8. (f) [??] (c).

9. (g) [??] (e).

Proof.

1. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have by Schwarz's inequality that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. First we show that E [[E.sub.h] [(t).sup.8]] < [infinity]. Note that if h [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]), then 16h [member of] [L.sup.2.sub.ad] ([0, T] x [OMEGA]). So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Apply Holder's inequality with p = 16/15 and q = 16 to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We thus have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now if E [h[(t).sup.4]] < [infinity], we have

E [h[(t).sup.2] [E.sub.h] [(t).sup.4]] [less than or equal to] [(E [h[(t).sup.4].sup.1/2] [(E [[epsilon].sub.h][(t).sup.8]).sup.1/2] < [infinity].

Thus E [[integral].sup.t.sub.0] h[(t).sup.2] [E.sub.h] [(t).sup.4] dt = [[integral].sup.t.sub.0] E [h[(t).sup.2] [E.sub.h] [(t).sup.4]] dt < [infinity].

3. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the implication follows.

4. Since (d) implies (b) and (b) implies (c), the implication follows.

5. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for t + [alpha] [less than or equal to] T, the implication follows.

6. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the implication follows.

7. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the implication follows.

8. Since (f) implies (b) and (b) implies (c), the implication follows.

9. See the proof of Theorem 4.1 (g).

References

[1] I.I. Gihman and A.V. Skorohod. Stochastic Differential Equations, Springer-Verlag, Berlin, 1972.

[2] I.V. Girsanov. On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures, Theory of Probability and its Applications, 5(3):285-301, 1960.

[3] G. Kallianpur. Stochastic Filtering Theory, Applications of Mathematics, No. 13, Springer-Verlag, NewYork, 1980.

[4] N. Kazamaki. Continuous Exponential Martingales and BMO, Springer-Verlag, Lecture Notes in Mathematics No. 1579, Berlin, 1994.

[5] H.-H. Kuo. Introduction to Stochastic Integration, Springer, New York, 2005.

[6] R.S. Liptser and A.N. Shiryaev. Statistics of Random Processes I: General Theory, (2nd, Revised and Expanded), Springer, Berlin, 2001.

[7] A. A. Novikov. On Moment Inequalitites for Stochastic Integrals, Theory of Probability and its Applications, 16(3):538-541, 1971.

[8] A. A. Novikov. On An Identity for Stochastic Integrals, Theory of Probability and its Applications, 17:717-720, 1972.

[9] T. Okada. A Criterion for Uniform Integrability of Exponential Martingales, Tohoku Mathematical Journal, 34:495-498, 1982.

[10] J. Yeh, Lectures on Real Analysis, World Scientific Publishing Co. Pte. Ltd., Singapore, 2000.

See Keong Lee

Department of Mathematics, Louisiana State University,

Baton Rouge, LA 70803, USA

E-mail: sklin_04@math.lsu.edu
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