The Cameron-Martin-Girsanov (CMG) Theorem

Transcription

1 The Cameron-Martin-Girsanov (CMG) Theorem There are many versions of the CMG Theorem. In some sense, there are many CMG Theorems. The first version appeared in ] in 944. Here we present a standard version, which gives the spirit of these theorems. Introductory material Let (Ω, F, P ) be a probability space. We can introduce other probability measures on (Ω, F) and obtain other probability spaces with the same sample space Ω and the same σ-algebra of events F. One easy way to do that is to start with a random variable Θ of (Ω, F) such that Θ P -a.s. and E P Θ] =, (.) where E P ] denotes the expectation associated to the measure P. Then, we can obtain a probability measure Q on (Ω, F) by setting Q(A) = E P Θ A ] for all A F, (.2) where A is the indicator (function) of the event A. It is easy to see that Q is a probability measure on (Ω, F) and, furthermore, that Q(A) = whenever P (A) =. (.3) Recall that if two measures P and Q are related as in formula (.3), we say that Q is absolutely continuous with respect to P and denote this as Q P. (.4) In fact, the Radon-Nicodym Theorem states that the above construction of Q is the only way to obtain probability measures, which are absolutely continuous with respect to P, namely: If Q is a probability measure on (Ω, F) and Q P, then there is a random variable Θ on (Ω, F) satisfying (.), for which Q(A) = E P Θ A ] for all A F. The random variable Θ is called the Radon-Nicodym derivative of Q with respect to P. Symbolically, Θ = dq dp and it is not hard to see that Θ is unique P -a.s. Furthermore, if we also have that P Q, then dp dq = Θ. If E Q ] is the expectation associated to the measure Q of (.2), then E Q A ] = Q(A) = E P Θ A ] for all A F,

2 and hence E Q X] = E P ΘX] for any r.v. X of (Ω, F) (.5) (if E P ΘX] does not exist, then so does E Q X]). The following proposition tells us how we can express conditional expectations associated to the measure Q of (.2) in terms of conditional expectations associated to P. Proposition. Let B be a subalgebra of F. Then E Q X B] = E P ΘX B] E P Θ B] Q-a.s. (.6) Proof. For typographical convenience let us put Ψ := E P Θ B]. (.7) Of course, Ψ P -a.s., Ψ M(B) (i.e. Ψ is B-measurable), and E P Ψ] = E P EP Θ B] ] = E P Θ] =. (.8) Because of the above we can define another probability measure on (Ω, F) as Q(A) = E P Ψ A ]. (.9) Observe that, for A B we have Q(A) = E P Θ A ] = E P EP Θ A B] ] = E P A E P Θ B] ] = E P Ψ A ], that is (in view of (.9)) which also implies immediately that We continue by setting Q(A) = Q(A) for all A B, (.) E Q X] = E QX] for all X M(B). (.) Y := E P ΘX B] and Y 2 := ΨE Q X B]. (.2) Then, formula (.6) is equivalent to Y = Y 2 Q-a.s. (.3) Notice that Y, Y 2 M(B). Hence, in view of (.), to establish (.3) it suffices to show that E QY A ] = E QY 2 A ] for all A B. (.4) 2

3 Now E QY A ] = E P ΨY A ] = E P ΨEP ΘX B] A ] = E P EP ΨΘX A B] ] = E P ΨΘX A ] = E Q ΨX A ] and (with the help of (.)) ] E QY 2 A ] = E Q ΨEQ X B] A = E Q EQ ΨX A B] ] = E Q EQ ΨX A B] ] = E Q ΨX A ]. Therefore (.4) is true and the proof is completed. Why do we need to consider more than one probability measures? First of all for educational purposes, since it helps to clarify certain concepts and issues. But, also, because such situations arise in applications. For example, two persons want to divide a pie (or a property) into two pieces so that each of them feels (s)he got a fair share. However, each person may have a different measure of fairness. 2 An example A random variable X of (Ω, F) is a F-measurable function X : Ω R. Thus, X depends on Ω and F. However, X does not depend on the probability measure put on (Ω, F). It is the distribution of X which depends on the measure. For instance, if P and Q are two probability measures on (Ω, F), then the distribution functions of X with respect to P and Q respectively are F P (x) = P {X x} and F Q (x) = Q{X x}. Of course, in general F P (x) F Q (x). Thus, e.g., the phrase X is a normal random variable (i.e. X is a normally distributed random variable ) may be misleading (or, at least, equivocal), if we work with more than one probability measures, since the distribution of X depends on the measure. The same remarks apply, of course, to random vectors and stochastic processes. For example, a process which is a Brownian motion with respect to a measure P, it will probably not be a Brownian motion with respect to another measure Q. Furthermore, if X and Y are random variables which are independent with respect to P, they may not be independent with respect to another measure Q. These issues do not come up if there is only one measure, but if there are more than one measures around, then they may become very important. Let us look at a specific example (taken from 3]). Suppose we have a probability space (Ω, F, P ) and a random variable Z of this space, which 3

4 has the standard normal distribution (with respect to P ). That is P {Z z} = Φ(z) = z 2π e ξ2 /2 dξ, z R. (2.) For typographical convenience we write Z P N(, ). Of course, there are many random variables of (Ω, F, P ) having the standard normal distribution (e.g., due to the symmetry of Φ(z), if Z is such a variable, then so is Z). Recall that E P Z] =, V P Z] = (2.2) (where V P ] is the variance with respect to P ), and E P e µz ] = e µ2 /2 for any µ C. (2.3) Equation (2.3) can be written as E P e µz 2 µ2] =. (2.4) Thus, for a Z as above and a µ R we can introduce the probability measure Q defined by ] Q(A) = E P e µz 2 µ2 A, A F. (2.5) A question arising here is: What is the distribution of Z with respect to Q? We have ] Q{Z x} = E P e µz 2 µ2 {Z x} = e µz 2 µ2 {z x} e z2 /2 dz 2π = x 2π e (z µ)2 2 dz, i.e., with respect to Q, Z is a normal random variable with mean µ and variance ; symbolically, Z Q N(µ, ), which can be written equivalently as (Z µ) Q N(, ), i.e. Z µ has the standard normal distribution with respect to Q. To extend this example, let the random variables Z,..., Z n be independent and identically distributed with respect to P, all having the standard normal distribution. We now set Q(A) = E P e n j= µ jz j n ] 2 j= µ2 j A, A F, (2.6) where µ,... µ n R are given constants. It is not hard to see that Q is a probability measure on (Ω, F). Again, we would like to find the (joint) distribution of Z,..., Z n with respect to Q. 4

5 For a reasonable (say bounded and continuous) function g : R n R we have (recall (.5)) E Q g(z,..., Z n )] = E P e n j= µ jz j n ] 2 j= µ2 j g(z,..., Z n ) ( = g(z,..., z n )e n j= µ jz j z n 2 ) 2 j= µ z2 n2 j e dz (2π) n/2 dz n = g(z,..., z n ) e (2π) n/2 (z µ) 2 2 e (z n µ) 2 2 dz dz n. It follows that the joint probability density function of Z,..., Z n (with respect to Q) is f Q (z,..., z n ) = (z µ) 2 e 2 e (z n µ) 2 2, (2.7) (2π) n/2 which tells us that, with respect to Q, the variables Z,..., Z n are independent and furthermore Z j Q N(µ j, ), i.e. (Z j µ j ) Q N(, ), for j =,..., n. Thus, the random variable n Z j j= n µ j (2.8) is, with respect to Q, normally distributed with mean and variance n. Roughly speaking, the Cameron-Martin-Girsanov Theorem is a continuous version of the above simple example. In fact, having this example in mind, one can guess the statement of the CMG Theorem (see the remark after Theorem in the next section). 3 The Cameron-Martin-Girsanov Theorem 3. CMG Theorem in R Consider a probability space (Ω, F, P ) and an one-dimensional Brownian motion process {B t = B(t)} t (with respect to P ) with associated filtration {F t } t. Of course F t is a subalgebra of F for every t. Next, let {X t = X(t)} t be a (real-valued) stochastic process, adapted to the filtration {F t } t. We assume that for some constant T > we have j= Having X t we introduce T X 2 t dt < P -a.s. (3.) M t := e Y t, where Y t := 2 5 X 2 s ds + X s db s (3.2)

6 (notice that M t > P -a.s.). In differential notation we have Applying Itô Calculus to (3.2) yields dy t = 2 X2 t dt + X t db t, Y =. (3.3) dm t = e Y t dy t + 2 ey t X 2 t dt = M t X t db t, M =. (3.4) Of course, (3.4) can be written equivalently as an integral equation M t = + M s X s db s. (3.5) It follows that the Itô process M t (being a stochastic integral) is an F t - martingale (with respect to P ). In particular, E P M t ] = E P M ] = for all t, T ]. (3.6) Having M t, and in view of (3.6), we introduce the probability measures on (Ω, F) Q(A) = E P M T A ]; Q t (A) = E P M t A ], < t < T. (3.7) Observe that, for t T we have M t = E P M T F t ] (since M t is a martingale). Hence, as in (.), we have Q t (A) = Q(A) for all A F t. (3.8) Theorem (CMG in R ). Let (Ω, F, P ), B t, X t, F t, and T be as above. Set W t := B t X s ds, t, T ]. (3.9) Then, for any fixed T > the process W t, t T is an F t -Brownian motion on (Ω, F, Q) (i.e. with respect to Q). Proof. We will prove the theorem with the help of the Lévy characterization of Brownian motion in R : A continuous stochastic process W = {W t } in a probability space (Ω, F, Q) is an one-dimensional Brownian motion if and only if (i) W t is a martingale with respect to Ft W and (ii) Wt 2 t is a martingale with respect to Ft W. The proof of this characterization will not be given here (it can be found, e.g., in 3]). If we accept this characterization of Brownian motion, then, in order to prove Theorem we only need to check that W t of (3.9) satisfies (i) and (ii). 6

7 Let us check (i). Set K t := M t W t, (3.) where M t is given by (3.2). Then, Itô Calculus, (3.4), and (3.9) give dk t = W t dm t + M t dw t + dw t dm t ( ) = B t X s ds M t X t db t + M t (db t X t dt) + M t X t dt = ( B t ) ] X s ds M t X t + M t db t, (3.) which implies that K t = M t W t is an F t -martingale with respect to P. Now, for s t T, by (3.8) and Proposition we get E Q W t F s ] = E Qt W t F s ] = E P M t W t F s ] E P M t F s ] = M sw s M s = W s, Q-a.s. which says that W t is an F t -martingale with respect to Q. Checking of (ii), namely that W 2 t t is an F t -martingale with respect to Q, can be done in a similar way and is left as an exercise (EXERCISE ). Remark. The quantity M t of (3.2) can be viewed (at least in the case where X t is a deterministic function of t) as a continuous analog of the quantity e n j= µ jz j 2 n j= µ2 j, which appears in (2.6). Likewise, by writing B t = db s (assuming B = ) we can view the quantity W t of (3.9) as a continuous analog of the quantity displayed in (2.8). Theorem 2 (CMG in R d ). Let {B(t)} t, be a d-dimensional Brownian motion on (Ω, F, P ), with associated filtration {F t } t. Also, let {X(t)} t be a stochastic process with values in R d, adapted to the filtration {F t } t. We assume that for some constant T > we have Set T M t := e Yt, where Y t := 2 X t 2 dt < P -a.s. (3.2) X s 2 ds + X s db s (3.3) (notice that M t > P -a.s.). It is not hard to see that M t is an F t -martingale (with respect to P ) and E P M t ] = for all t, T ] (EXERCISE 2). If Q is the probability measure on (Ω, F) defined by Q(A) = E P M T A ], (3.4) 7

8 then, for any fixed T >, the process W t := B t is a d-dimensional F t -Brownian motion on (Ω, F, Q). X s ds, t, T ] (3.5) Proof. The proof is similar to the proof of Theorem and it is left as an exercise (EXERCISE 3). You may use (without proof) the Lévy characterization of Brownian motion in R d 3]: A continuous stochastic process W = {W (t)} in a probability space (Ω, F, Q) is an d-dimensional Brownian motion if and only if (i) W (t) is a martingale with respect to Ft W (meaning that each component W j (t), j =,..., d, is a martingale) and (ii) for any j, k {,..., d} we have that W j (t)w k (t) δ jk t is a martingale with respect to Ft W (here δ jk is the Kronecker delta). References ] R.H. Cameron and W.T. Martin, Transformation of Wiener integrals under translations, Annals of Mathematics, 45, , ] R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Inc. Belmont, CA, USA, ] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Second Edition, Springer, New York, 99. 4] B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Sixth Edition, Springer-Verlag, Berlin, 23. 8