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.Comment: 10 page

Goodman, Victor

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Victor Goodman

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Communications on Stochastic Analysis

Brownian super-exponents

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Communications on Stochastic Analysis Volume 1 Number 1 Article 11 4-1-2007 Brownian super-exponents Victor Goodman Follow this and additional works at: https://repository.lsu.edu/cosa  Part of the Analysis Commons, and the Other Mathematics Commons Recommended Citation Goodman, Victor (2007) "Brownian super-exponents," Communications on Stochastic Analysis: Vol. 1: No. 1, Article 11. DOI: 10.31390/cosa.1.1.11 Available at: https://repository.lsu.edu/cosa/vol1/iss1/11 BROWNIAN SUPER-EXPONENTSVICTOR GOODMANAbstract. We introduce a transform on the class of stochastic exponentialsfor d-dimensional Brownian motions. Each stochastic exponential generatesanother stochastic exponential under the transform. The new exponentialprocess is often merely a supermartingale even in cases where the originalprocess is a martingale. We determine a necessary and sufficient conditionfor the transform to be a martingale process. The condition links expectedvalues of the transformed stochastic exponential to the distribution functionof certain time-integrals.1. IntroductionIf X(t) is a d-dimensional progressively measurable process and W is a Brownianmotion under a measure P , the stochastic exponential determined by X is theprocessZX(t) = exp{∫ t0X(u) · dW(u) −12∫ t0||X(u)||2du}.The problem of checking whether ZX(t) is a true martingale is important forthe use of Girsanov’s theorem. Two well-known sufficient conditions are due toNovikov and to Kazamaki; see for example Revuz and Yor [7]. Examples wherethe process ZX(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 sufficientcondition for any stochastic exponential to form a martingale process. Their con-dition is formulated in terms of an explosion time. We consider a class of stochasticexponentials for which their condition becomes more explicit. We begin with anystochastic exponential and we describe a modification, or transform, of it whichgenerates another stochastic exponential.The transform involves a time-integral of the form∫ t0||X(u)||2ZX(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 weuse the relation to obtain bounds on the tail behavior of these distributions.2000 Mathematics Subject Classification. 60H30; 60J65.Key words and phrases. Girsanov theorem, stochastic exponential, time-integral.141Communications on Stochastic AnalysisVol. 1, No. 1 (2007) 141-149142 VICTOR GOODMANDefinition 1.1. Suppose that X(t) is a progressively measurable process suchthat for some T > 0,P{∫ T0||X(u)||2du < ∞}= 1. (1.1)If ZX(t) is the stochastic exponential generated by X(t) and y > 0, the associatedsuper-exponent process YX(t) , defined for t ≤ T , isYX,y(t) =ZX(t)y−1 + 12∫ t0||X(u)||2ZX(u)du. (1.2)Notice from Equation (1.2) that YX,y(0) = y. In addition, YX,y(t) is positive sothat the random variableexp(YX,y(t))is greater than one. We show that this random variable has a finite expectedvalue which is less than or equal to ey. This result is surprising since YX,y(t) isused as an exponent here. According to Definition 1.1, YX,y(t) itself contains anexponential factor ZX(t). For this reason, we say that the process YX,y(t) is aBrownian super-exponent.2. Transform PropertiesProposition 2.1. Suppose that a progressively measurable process X satisfies con-dition (1.1). Let YX,y(t) denote the super-exponent process in Definition 1.1. Thenfor each t ≤ T ,YX,y(t) = y +∫ t0YX,y(u)X(u) · dW(u) −12∫ t0||X(u)||2Y 2X,y(u)du. (2.1)Moreover, the processexp (YX,y(t)) (2.2)is a positive supermartingale on the interval 0 ≤ t ≤ T . In addition, the processZ̃(t) = exp (YX,y(t) − y) (2.3)is a stochastic exponential for W. This stochastic exponential is generated by thed-dimensional processYX,y(t)X(t). (2.4)Proof. It follows from the definition of ZX(t) thatdZX = ZXX · dW, d∫ t0||X||2ZXdu = ||X||2ZXdt.Direct calculation shows thatdYX,y =dZXy−1 + 12∫ t0||X||2ZXdu+ ZX d(y−1 +12∫ t0||X||2ZXdu)−1= YX,yX · dW −12||X||2Y 2X,ydt.(2.5)BROWNIAN SUPER-EXPONENTS 143From this equation we see that YX,y − y is the sum of the Itô integral of YX,yXand the elementary integral of − 12 ||YX,yX||2. This establishes Equation (2.1).It follows immediately from Equation (2.1) that YX,y − y is the exponent of astochastic exponential. Therefore, the processexp (YX,y(t) − y)is a positive local martingale. It is well known that a positive local martingale is asupermartingale; see, for instance, Karatzas and Shreve [4]. In addition, Equation(2.3) is a direct consequence of Equation (2.1) and the definition of stochasticexponential processes. ¤Theorem 2.2. Suppose that X(t) is a deterministic function such that for someT > 0∫ T0||X(u)||2du < ∞.Let ZX(t) and YX,y denote the stochastic exponential and super-exponent processgenerated by X(t). Then for each non-negative measurable function G(u), u > 0,and t < T ,E[G(YX,y(t)) exp(Y (t) − y)]= E[G(ZX(t)y−1 − 12∫ t0||X(u)||2ZX(u)du) ;∫ t0||X(u)||2ZX(u)du <2y].(2.6)Proof. For N = 1, 2, . . . let τN be the stopping time defined byτN = inf{t ≤ T : YX,y(t) ≥ N}.It follows from Equation (2.1) thatYX,y(t ∧ τN ) − y =∫ t∧τN0YX,y(u)X(u) · dW(u) −12∫ t∧τN0||X(u)||2Y 2X,y(u)du.From this equation we see that exp(YX,y(t ∧ τN ) − y) is another stochastic expo-nential which is generated byYX,y(u)1{u<τN}X(u).Since this process is uniformly bounded in L2[0, T ], it satisfies Novikov’s condi-tion. It is well known (see Karatzas and Shreve [4]) that the associated stochasticexponential is a martingale. We apply Girsanov’s Theorem to change measureusing the Radon-Nykodym derivativeΛ(T ) = exp(YX,y(T ∧ τN ) − y).The probability measure QN is given bydQNdP= Λ(T ).Then with respect to QN the processW̃(t) = W(t) −∫ t∧τN0YX,y(u)X(u)du144 VICTOR GOODMANis a Brownian motion for t ≤ T . Since YX,y(t) is a strong solution to equation(2.1), we may consider its SDE with respect to the Brownian motion W̃:For t < τNdYX,y = YX,yX · dW −12||X||2Y 2X,ydt= YX,yX · { dW̃ + YX,yXdt} −12||X||2Y 2X,ydt= YX,yX · dW̃ +12||X||2Y 2X,ydt.(2.7)Now we have an explicit solution to the SDE in equation (2.7):YX,y(t) =Z̃X(t)y−1 − 12∫ t0||X(u)||2Z̃X(u)du. (2.8)In this equation, Z̃X(t) denotes the stochastic exponential (generated by X) withrespect to the Brownian motion W̃. Now we considerE[G(YX,y(t)) exp(YX,y(t) − y)1{t<τN}]= E[G(YX,y(t))Λ(T )1{t<τN}]= EQN [G(YX,y(t))1{t<τN}]= EQN[G(Z̃X(t)y−1 − 12∫ t0||X(u)||2Z̃X(u)du)1{t<τN}].(2.9)Here we used the identity for YX,y in Equation (2.8).Moreover, from Equation (2.8) we also havet < τN if and only if maxs≤t(Z̃X(s)y−1 − 12∫ s0||X(u)||2Z̃X(u)du)< N.This allows us to write the last expected value in Equation (2.9) asE[G(ZX(t)y−1− 12∫ t0||X(u)||2 ZX(u)du);maxs≤t(ZX(s)y−1− 12∫ s0||X(u)||2ZX(u)du)<N]since the integrand involves only the distribution of a Brownian motion for eachchoice of N . The limit of this expected value as N → ∞ isE[G(ZX(t)y−1 − 12∫ t0||X(u)||2 ZX(u)du);12∫ t0||X(u)||2ZX(u)du) < y−1].Since the limit of the first expected value in Equation (2.9) isE[G(YX,y(t)) exp(YX,y(t) − y)],the theorem is proved. ¤BROWNIAN SUPER-EXPONENTS 1453. Examples Using the TransformProposition 3.1. Suppose that X(t) is a deterministic function such that∫ t0||X(u)||2duis strictly increasing and finite for t ≤ T < ∞. Let ZX(t) and YX,y denote thestochastic exponential and super-exponent process generated by X(t). Then theprocessexp(YX,y(t))is a strict supermartingale for t ≤ T . Moreover,E[exp(YX,y(t))] = eyPr{∫ t0||X(u)||2ZX(u)du <2y}. (3.1)Proof. We apply Theorem 2.2 using the choice G(u) ≡ 1. Equation (2.6) becomesE[exp(YX,y(t) − y)] = Pr{∫ t0||X(u)||2ZX(u)du <2y},and Equation (3.1) follows. Now since each ZX(u) is a log normal random variable,the process∫ t0||X(u)||2ZX(u)du (3.2)has strictly increasing sample paths. It follows that the right hand expressionin Equation (3.1) is strictly decreasing. Therefore, exp(YX,y(t)) is a strict super-martingale. ¤Remark 3.2. Equation (3.1) provides a useful tool for investigating the distributionof a time integral given by Equation (3.2). Since each super-exponentYX,y(t) =ZX(t)y−1 + 12∫ t0||X(u)||2ZX(u)duis point-wise increasing as a function of y, it follows from the identityPr{∫ t0||X(u)||2ZX(u)du < a}= exp(−2a)E[exp(YX,2/a(t))]that the distribution function is the product of a decreasing function of a and theexplicit factor exp(−2/a) .It is not known whether exp(YX,∞(t)) has finite expectation. A finite expectedvalue would produce sharp estimates for the lower tail probability of (3.2). Weconjecture thatE[2ZX(t)∫ t0||X(u)||2ZX(u)du]= ∞.146 VICTOR GOODMANExample 3.3. In the case of d = 1 the choice X(t) ≡ σ specializes the timeintegral in (3.2) to a time integral of geometric Brownian motion:∫ t0exp(σW (u) − σ2u/2)du. (3.3)Expected values involving related time integrals appear in computational prob-lems of financial mathematics. Consequently, distribution properties of these timeintegrals 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 distributionin various integral forms, in Goodman and Kim [3] martingales techniques areused exclusively. A special case of Equation (3.1) appears in [3], Theorem 4.1:Pr{∫ t0exp(W (u) − u/2)du ≤ a}= exp(−2a)E[exp(2 exp(W (t) − t/2)a +∫ t0exp(W (u) − u/2)du)].The right hand expression for the distribution can be differentiated with respectto a. Consequently, it is shown in [3] that the density function multiplied by a2/2equals the difference between two distribution functions of time integrals of slightlydifferent geometric Brownian motions.Example 3.4. In contrast to deterministic choices for X(t), where the stochasticexponentialexp(YX,y(t))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.Consider the example of X(t) = cos(W (t)), again in the case d = 1. ThenZX(t) = exp(∫ t0cos(W (u))dW (u) −12∫ t0cos2(W (u))du)= exp(sin(W (t)) +12∫ t0[sin(W (u)) − cos2(W (u))]du)is a bounded random variable. Therefore, its super-exponent, Ycos(W ),y(t) is alsobounded. Then since the local martingaleexp(Ycos(W ),y(t))is also bounded, it is a martingale. It is of interest then to know when a super-exponent generates a martingale process.4. The Martingale ConditionTheorem 1 of Wong and Heyde [9] identifies a necessary and sufficient condi-tion for a progressively measurable process X̃ to generate a martingale stochasticexponential process. For completeness, we state their result here.BROWNIAN SUPER-EXPONENTS 147Proposition 4.1. ([9], Proposition 1) Consider a d-dimensional progressivelymeasurable process X̃(t) = ξ(W(·), t). Then there will also exist a d-dimensionalprogressively measurable processR̃(t) = ξ(W(·) +∫ ·0R̃(u)du, t)defined possibly up to an explosion time τMR , whereτMR = inf{t ∈ R+ : MR(t) =∫ t0||R̃(u)||2du = ∞}.Theorem 4.2. ([9], Theorem 1) Consider X̃(t) and R̃(t) as defined in Proposition4.1. The stochastic exponential ZX̃(T ) satisfiesP (τMR > T ) = EP [ZX̃(T )]and hence is a martingale if and only if P (τMR > T ) = 1.We apply Theorem 1 of [9] using X̃(t) = YX,y(t)X(t). That is, our generatingprocess is the one in Proposition 2.1 where the stochastic exponential process isexp(YX,y(t) − y).We first show that each generating process X implicitly defines another processX′. This allows us to identify the process R̃(t).Proposition 4.3. Suppose that a d-dimensional progressively measurable processX(t) satisfiesPr(∫ T0||X(u)||2du < ∞)= 1for some T > 0. Then there exists another progressively measurable process X′(t),so that if X̃(t) := YX,y(t)X(t)1{t≤T} in Proposition 4.1, then the process R̃(t) ofthe proposition satisfiesR̃(t) =ZX′(t)y−1 − 12∫ t0||X′(u)||2ZX′(u)duX′(t)for all t < τMR . Moreover,τMR = inf{t ∈ R+ :∫ t∧T0||X′(u)||2ZX′(u)du = 2/y}.Proof. We follow the proof of Proposition 4.1. LetX̃(t) := YX,y(t)X(t)1{t≤T}.For each N = 1, 2, . . . we define a sequence of stopping times byτN = inf{t ∈ R+ :∫ t0Y 2X,y(u)||X(u)||21{u≤T}du ≥ N}.It follows from Equation (2.1) thatZX̃(t ∧ τN ) = exp(YX,y(t ∧ τN ) − y)148 VICTOR GOODMANforms a martingale. As in the proof of Theorem 2.2, we apply Girsanov’s theoremusing the Radon-Nikodym derivativeΛ(T ) = exp(YX,y(T ∧ τN ) − y)to obtain the probability measure QN wheredQN = Λ(T )dP.With respect to the measure QN , the processW̃(t) = W(t) −∫ t∧τN0YX,y(u)X(u)duis a Brownian motion. Hence, on the set {t ≤ τN ∧ T} we haveX̃(t) = ξ(W̃(·) +∫ ·0YX,y(u)X(u)du, t),that is,YX,y(t)X(t) = ξ(W̃(·) +∫ ·0YX,y(u)X(u)du, t). (4.1)Now the process YX,y(t) can also be described in terms of the Brownian motionW̃. The calculations in Equation (2.7) also apply to the stochastic case. Equation(2.8) gives an explicit formula for YX,y:YX,y(t) =Z̃X(t)y−1 − 12∫ t0||X(u)||2Z̃X(u)du. (4.2)We see that each term of Equation (4.1) is a functional of X and the Brownianmotion W̃. This demonstrates the existence of a process X so that (4.1) and (4.2)hold up to a time τN defined by the integral of YX,y(u)X(u) , using the Brownianmotion W̃.Therefore, using the identical distribution of W and the (original) measure P ,we see that there exists a progressively measurable process X′(t) so thatZX′(t)y−1 − 12∫ t0||X′(u)||2ZX′(u)duX′(t) = ξ(W(·) +∫ ·0YX′,y(u)X′(u)du, t).Here, we have abbreviated the complete expression on the right hand side using(4.2) to provide the notation. That is, YX′,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 τN increases to the stopping timeτ = inf{t ≤ T :∫ t0Y 2X′,y(u)||X′(u)||2du = ∞}.By construction, the new process X′ satisfies∫ T0||X′(u)||2du < ∞ a. s. and X′(u) = 0 for u > T.Therefore, the process YX′,y (again, defined as in (4.2)) is bounded along eachsample path up to the time where its denominator first hits zero. This defines thestopping time τMR of the Proposition. ¤BROWNIAN SUPER-EXPONENTS 149Theorem 4.4. Suppose that X(t) and X′(t) are d-dimensional processes as definedin Proposition 4.3. Then the super-exponent process YX,y(t) satisfiesE[exp(YX,y(t) − y)] = Pr{∫ t0||X′(u)||2ZX(u)du < 2/y}(4.3)for t ≤ T .Proof. From Theorem 4.2 and Proposition 4.3 we haveE[exp(YX,y(t) − y)] = Pr{τMR > t}= Pr{∫ t0Y 2X′,y(u)||X′(u)||2du < ∞}= Pr{∫ t0ZX′(u)y−1 − 12∫ u0||X′(r)||2ZX′(r)dr||X′(u)||2du < ∞}= Pr{∫ t0||X′(u)||2ZX(u)du < 2/y}.¤References1. Dufresne, D.: The integral of geometric Brownian motion; Adv. in Appl. Probab. 33 (2001)223-2412. Geman, H, and Yor, M.: Asian Options, Bessel Processes and Perpetuities; Math. Finance2 (1993) 349-3753. Goodman, V. and Kim, K.: Exponential martingales and time integrals of Brownian motionpreprint4. Karatzas, I., and Shreve, S.: Brownian Motion and Stochastic Calculus. Springer-Verlag,New York, 19915. Kim, K.: Moment Generating function of the inverse of integral of geometric BrownianMotion; Proc. Amer. Math. Soc. 132 (2004) 2753-27596. Levental, S. and Skorohod, A. V.: A necessary and sufficient condition for absence of arbi-trage with tame portfolios; Ann. Appl. Prob. 5 (1995) 906-9257. Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer-Verlag, New York, 19998. Rogers, L.C.G., and Shi, Z.: The value of an Asian option J. Appl Appl. Probab. 32 (1995)1077-10889. Wong, B. and Heyde, C.C.: On the martingale property of stochastic exponentials; J.Appl.Probab. 41 (2004) 654-66410. Yor, M.: On some exponential functionals of Brownian motion; Adv. in Appl. Probab. 24(1992) 509-531Victor Goodman: Department of Mathematics, Indiana University, Bloomington,Indiana, U.S.A.E-mail address: goodmanv@indiana.edu

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