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Brownian Super-exponents

We introduce a transform on the class of stochastic exponentials for d-dimensional Brownian motions. Each stochastic exponential generates another stochastic exponential under the transform. The new exponential process is often merely a supermartingale even in cases where the original process is a martingale. We determine a necessary and sufficient condition for the transform to be a martingale process. The condition links expected values of the transformed stochastic exponential to the distribution function of certain time-integrals.


Introduction
If X(t) is a d-dimensional progressively measurable process and W is a Brownian motion under a measure P , the stochastic exponential determined by X is the process The problem of checking whether Z X (t) is a true martingale is important for the use of Girsanov's theorem. Two well-known sufficient conditions are due to Novikov and to Kazamaki; see for example Revuz and Yor [7]. Examples where the process Z X (t) is strictly a supermartingale appear in Goodman and Kim [3], Levental and Skorohod [6], and Wong and Heyde [9]. In their recent paper, Wong and Heyde [9] present a necessary and sufficient condition for any stochastic exponential to form a martingale process. Their condition is formulated in terms of an explosion time. We consider a class of stochastic exponentials for which their condition becomes more explicit. We begin with any stochastic exponential and we describe a modification, or transform, of it which generates another stochastic exponential.
The transform involves a time-integral of the form t 0 ||X(u)|| 2 Z X (u)du.
We derive a necessary and sufficient condition for the transform to be a martingale. Our condition is formulated in terms of the distribution of time integrals, and we use the relation to obtain bounds on the tail behavior of these distributions.
Definition 1.1. Suppose that X(t) is a progressively measurable process such that for some T > 0, If Z X (t) is the stochastic exponential generated by X(t) and y > 0, the associated super-exponent process Y X (t) , defined for t ≤ T , is Notice from Equation (1.2) that Y X,y (0) = y. In addition, Y X,y (t) is positive so that the random variable exp(Y X,y (t)) is greater than one. We show that this random variable has a finite expected value which is less than or equal to e y . This result is surprising since Y X,y (t) is used as an exponent here. According to Definition 1.1, Y X,y (t) itself contains an exponential factor Z X (t). For this reason, we say that the process Y X,y (t) is a Brownian superexponent.

Transform Properties
Proposition 2.1. Suppose that a progressively measurable process X satisfies condition (1.1). Let Y X,y (t) denote the super-exponent process in Definition 1.1. Then for each t ≤ T , is a positive supermartingale on the interval 0 ≤ t ≤ T . In addition, the processZ is a stochastic exponential for W. This stochastic exponential is generated by the d-dimensional process Y X,y (t)X(t) (2.4) Proof. It follows from the definition of Z X (t) that Direct calculation shows that From this equation we see that Y X,y − y is the sum of the Itô integral of Y X,y X and the elementary integral of − 1 2 ||Y X,y X|| 2 . This establishes Equation (2.1). It follows immediately from Equation (2.1) that Y X,y −y is the exponent of a stochastic exponential. Therefore, the process is a positive local martingale. It is well known that a positive local martingale is a supermartingale; see, for instance, Karatzas and Shreve [4]. In addition, Equation (2.3) is a direct consequence of Equation (2.1) and the definition of stochastic exponential processes.

Theorem 2.2. Suppose that X(t) is a deterministic function such that for some
Let Z X (t) and Y X,y denote the stochastic exponential and super-exponent process generated by X(t). Then for each non-negative measurable function G(u), u > 0, and t < T , Proof. For N = 1, 2, . . . let τ N be the stopping time defined by From this equation we see that exp(Y X,y (t ∧ τ N ) − y) is another stochastic exponential which is generated by Since this process is uniformly bounded in L 2 [0, T ], it satisfies Novikov's condition. It is well known (see Karatzas and Shreve [4]) that the associated stochastic exponential is a martingale. We apply Girsanov's Theorem to change measure using the Radon-Nykodym derivative Then with respect to Q N the process is a strong solution to equation (2.1), we may consider its SDE with respect to the Brownian motionW: Now we have an explicit solution to the SDE in equation (2.7): In this equation,Z X (t) denotes the stochastic exponential (generated by X) with respect to the Brownian motionW. Now we consider Here we used the identity for Y X,y in Equation (2.8).
Moreover, from Equation (2.8) we also have This allows us to write the last expected value in Equation (2.9) as since the integrand involves only the distribution of a Brownian motion for each choice of N . The limit of this expected value as N → ∞ is Since the limit of the first expected value in Equation (2.9) is the theorem is proved.

Examples Using the Transform
is strictly increasing and finite for t ≤ T < ∞. Let Z X (t) and Y X,y denote the stochastic exponential and super-exponent process generated by X(t). Then the process exp(Y X,y (t)) is a strict supermartingale for t ≤ T . Moreover, Proof. We apply Theorem 2.2 using the choice G(u) ≡ 1. Equation (2.6) becomes and Equation (3.1) follows. Now since each Z X (u) is a log normal random variable, the process has strictly increasing sample paths. It follows that the right hand expression in Equation (3.1) is strictly decreasing. Therefore, exp(Y X,y (t)) is a strict supermartingale.

Remark 3.2. Equation (3.1) provides a useful tool for investigating the distribution of a time integral given by Equation (3.2). Since each super-exponent
Y X,y (t) = Z X (t) is point-wise increasing as a function of y, it follows from the identity that the distribution function is the product of a decreasing function of a and the explicit factor exp(−2/a) . It is not known whether exp(Y X,∞ (t)) has finite expectation. A finite expected value would produce sharp estimates for the lower tail probability of (3.2). We conjecture that Expected values involving related time integrals appear in computational problems of financial mathematics. Consequently, distribution properties of these time integrals have been studied by many authors; see Dufresne [1], Geman and Yor [2], Rogers and Shi [8], and Goodman and Kim [3].
Although most works have used analytic techniques to express the distribution in various integral forms, in Goodman and Kim [3] martingales techniques are used exclusively. A special case of Equation (3.1) appears in [3], Theorem 4.1: The right hand expression for the distribution can be differentiated with respect to a. Consequently, it is shown in [3] that the density function multiplied by a 2 /2 equals the difference between two distribution functions of time integrals of slightly different geometric Brownian motions.

Example 3.4.
In contrast to deterministic choices for X(t), where the stochastic exponential is never a martingale, stochastic choices for X may produce martingales. Of course, the introduction of a stopping time, as we have seen in the proof of Theorem 2.2, may produce a martingale. In other cases, stopping times are not required. is a bounded random variable. Therefore, its super-exponent, Y cos(W ),y (t) is also bounded. Then since the local martingale exp(Y cos(W ),y (t)) is also bounded, it is a martingale. It is of interest then to know when a superexponent generates a martingale process.

The Martingale Condition
Theorem 1 of Wong and Heyde [9] identifies a necessary and sufficient condition for a progressively measurable processX to generate a martingale stochastic exponential process. For completeness, we state their result here. and hence is a martingale if and only if P (τ MR > T ) = 1.

Proposition 4.1. ([9], Proposition 1) Consider a d-dimensional progressively measurable processX(t) = ξ(W(·), t). Then there will also exist a d-dimensional progressively measurable process
We apply Theorem 1 of [9] usingX(t) = Y X,y (t)X(t). That is, our generating process is the one in Proposition 2.1 where the stochastic exponential process is exp(Y X,y (t) − y).
We first show that each generating process X implicitly defines another process X ′ . This allows us to identify the processR(t).

Proposition 4.3. Suppose that a d-dimensional progressively measurable process
for some T > 0. Then there exists another progressively measurable process X ′ (t), so that ifX(t) := Y X,y (t)X(t)1 {t≤T } in Proposition 4.1, then the processR(t) of the proposition satisfies for all t < τ MR . Moreover, Proof. We follow the proof of Proposition 4.1. Let For each N = 1, 2, . . . we define a sequence of stopping times by It follows from Equation (2.1) that forms a martingale. As in the proof of Theorem 2.2, we apply Girsanov's theorem using the Radon-Nikodym derivative to obtain the probability measure Q N where With respect to the measure Q N , the process is a Brownian motion. Hence, on the set {t ≤ τ N ∧ T } we havẽ That is, Now the process Y X,y (t) can also be described in terms of the Brownian motioñ W. The calculations in Equation (2.7) also apply to the stochastic case. Equation (2.8) gives an explicit formula for Y X,y : We see that each term of Equation (4.1) is a functional of X and the Brownian motionW. This demonstrates the existence of a process X so that (4.1) and (4.2) hold up to a time τ N defined by the integral of Y X,y (u)X(u) , using the Brownian motionW.
Therefore, using the identical distribution of W and the (original) measure P , we see that there exists a progressively measurable process X ′ (t) so that Here, we have abbreviated the complete expression on the right hand side using (4.2) to provide the notation. That is, Y X ′ ,y denotes the expression in Equation (4.2) but in the original Brownian motion and X is replaced by the process X ′ .
As N → ∞ the stopping time τ N increases to the stopping time τ = inf{t ≤ T : t 0 Y 2 X ′ ,y (u)||X ′ (u)|| 2 du = ∞} By construction, the new process X ′ satisfies T 0 ||X ′ (u)|| 2 du < ∞ a. s. and X ′ (u) = 0 for u > T Therefore, the process Y X ′ ,y (again, defined as in (4.2)) is bounded along each sample path up to the time where its denominator first hits zero. This defines the stopping time τ MR of the Proposition.