The man's fist was large though it twitched as he pounded the stamp onto the translucent page. The little money she had inside an orange handkerchief tied to her hair, coins rolling to the ground as she fled. Atrocities of our entanglement not on the bed but beside it. Using our mouths as tools for betterment, for seduction, for completion. The vertebra twists into a question mark to conform to another's. In the Patanal, the cowboys steadied the horses in the barn, the animal's labored breathing, the sigh as the coarse brush worked through the mane. The owner's daughter learning to move her hips as she practiced her samba before the steaming pot, and radio clicking, and lid drumming.
Of the men I've known, you were the most steady, reliable one near the window killing mosquitoes, gathering cool water to press to my scalp. One-sided heart I was then. Selfish one. I wanted everything. Macaws flew past in quick flock, pushing outward toward the earth's scattering filament and mystery. Though there was no purpose, though the past had nothing to do with the chase now. This grand state pumped from its own engine of blood , centuries of evolution, first as a red-eyed embryo, then reptile, then mammal, then man, pure racing, push of muscle and tendon, the tongue loose and dragging as the body made its way forward.
Each time more powerful, a new version of waking until the species grew great wings and lifted. Tina Chang Infinite and Plausible It is the smallest idea born in the interior will, that has no fury nor ignorance, no intruder but stranger, no scaffold of a plea, no mote of the hungry, no pitchfork of instinct, no ladder of pity, no carriage of lust, no wavering foot on concrete, no parish of bees, no mountains of coal, no limestone and ash, no lie poured down the stairs of a house among them, and this is the will of maker and offspring, no boot in the hallway indicating more exit than arrival, more straying than strategy, no more struggle than contained in my body now, as I wander the rooms, tearing curtains apart from their windows separating material from light.
Celestial When everything was accounted for you rummaged through my bag to find something offensive: a revolver, a notebook of misinterpreted text. I'm God's professor. His eyes two open ovens. He has a physical body and it hiccups and blesses. Tell me a story before the mudslide, tell it fast before the house falls, before it withers in the frost, before it dozes off next to the television. I couldn't tell if it was that screen or the sky spitting dust and light. Academy of American Poets Educator Newsletter. Teach This Poem. Follow Us. Find Poets.
Read Stanza. Jobs for Poets. Materials for Teachers. The Walt Whitman Award. James Laughlin Award. Ambroggio Prize. Dear Poet Project. Back Issues. For Markovian models, the martingale principle is a reformulation of the dynamic programming principle. The following result determines the dynamics of and using the martingale principle. Moreover, candidate optimal strategies for both primal and dual problems are identified.
In particular, the candidate dual optimizer follows for some and. Lemma 3. The ansatz 3. In the next section, we will start from the BSDE 3. Define 3.
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Remark 3. The form of can be obtained via the utility gradient approach; cf. Duffie and Epstein b , equation 35 , Duffie and Skiadas , theorem 2 , and Schroder and Skiadas , equation 3. The novelty of this paper is to relate the utility gradient and the minimization problem in 3. The case with zero bequest utility was considered in Schroder and Skiadas However, the utility parameter restriction equation 8 therein excludes the case.
This section verifies the identity 3. We start with the following restriction on model coefficients. Assumption 3. The processes r and are both bounded. Markovian models with unbounded market price of risk will be discussed in the next section, where more technical conditions will be imposed. Suppose that , or , and Assumption 3. Then 3. Having establish a solution to 3. Theorem 3. Then, and satisfy 3. As a direct consequence of Theorem 3. Duffie and Epstein b and Duffie and Skiadas Corollary 3.
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Let the assumptions of Theorem 3. The minimizer of the dual problem satisfies 3. Many widely used market models in the asset pricing literature come with unbounded market price of risk; e. To obtain similar results to Theorem 3. The domain E is assumed to satisfy , where is a sequence of open domains in E such that is compact and for each n. Given functions , , and , the processes in 3.
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Instead of Assumption 3. The regularity of coefficients and the nonexplosion assumption ensure that the dynamics for X is well posed; i. The assumption on the lower bound of allows for unbounded market price of risk and is readily satisfied when r is bounded from below. To present an analog of Theorem 3.
When all model coefficients are bounded, as in Assumption 3. When the market price of risk is unbounded, the last part of Assumption 3. Nevertheless, Assumption 3. Let Assumptions 3. For , 3. In particular, because , Y is bounded from above. Having constructed , and in 3. To verify their optimality, let us introduce an operator. To understand this operator, note that the solution to 3. Then, the BSDE 3. Therefore, the last two terms in the previous PDE are bounded, and is the unbounded part of the spatial operator. There exists such that i ; ii is bounded from above on E.
Its existence facilitates the proof of a verification result, leading to the following result. Suppose that , and that Assumptions 3.
Then the statements of Theorem 3. The proof of Theorem 3. Therefore, the following uniqueness results for 3. The optimality of has been verified in Xing , theorem 2. First, Xing restricts strategies to a permissible class that is smaller than the current admissible class. It is the duality inequality 2.
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Second, Xing , assumption 2. This integrability condition translates to model parameter restrictions; see Xing , proposition 3. Rather than forcing to satisfy this integrability condition, which is a sufficient condition for the existence of Epstein—Zin utility, we show that the class D Epstein—Zin utility exists for ; hence, belongs to , which abstractly envelops all Epstein—Zin utilities and, in particular, contains those satisfying the integrability condition.
As a result, the aforementioned model parameter restrictions for the Heston model and the Kim—Omberg model can be removed in the following examples. Example 3. These parameter restrictions ensure the existence of a strictly positive process X. The inverse Heston model studied in Chacko and Viceira : , ,. When and , the previous model is not an affine model, but is in the class of essentially affine models proposed by Duffee The following result specifies Assumptions 3.
Proposition 3. Assume and the following conditions: i ; ii Either or ; iii and. Feller's test ensures the existence of a strictly positive process X satisfying the previous stochastic differential equation SDE. Using X to model the variance process has received a large amount of empirical support; see the survey in Carr and Sun , p.
Given , , consider the following asset dynamics: Denote. The following result provides sufficient conditions for Assumptions 3. Assume , and the following conditions: i ; ii Either or ; iii and. The following result from Xing , proposition 3. Assume and either of the following parameter restrictions hold: i and ; ii. Note that. The proof is split into two cases. We will first prove 4. On the other hand, the class D property of for and the boundedness of for ensure the integrability of ; hence, the class D property of the process.
Due to , we have The class D property of both upper and lower bounds implies the class D property of. As and are finite, both are martingales. It follows that is a local martingale. Therefore, there exists a local martingale L such that where is an increasing process due to 2. As a result, is a local supermartingale. On the other hand, we have seen that is of class D. Moreover, is of class D , thanks to the boundedness of and class D property of. Hence, the local supermartingale is a supermartingale. Hence, 4. To show that the inequality in 4.
To this end, take. Lemma A. It then follows from 2. Therefore, 4. In this case,. We show for any first. To this end, for and , define an increasing process. Equation 2. Because , we have Therefore, is of class D , and a similar argument as in the previous case confirms 4. To show the inequality in 4. Moreover, due to , 4. Let the filtration be generated by some Brownian motion B. Solving 2. The previous BSDE translates to 4. Define and. To treat the terminal condition , we consider an approximated terminal condition with and its associated solution.
Proceed as in the proof of Schroder and Skiadas , theorem A2 , is constructed as. Coming back to , the statement in i is confirmed. Our assumption on D implies the integrability of and. Moreover, because , we have ; therefore, the generator of 4. Then the statement in ii is confirmed following the proof of Xing , proposition 2. The statement for the primal problem is proved in Xing , see the argument leading to equation 2.
In particular, because all investment opportunities are driven by W , it suffices to consider the martingale part of as a stochastic integral with respect to W. Let us outline the argument for the primal problem. Parameterize c by and suppose that for some processes and. Calculation shows 4.
For the dual problem, suppose that for some processes and.
It then remains to obtain the minimizer for. Plugging , back into and back into , we obtain that both and are the same as in 3. Therefore, both and are solution to 3. Because is bounded, implies that is bounded as well. Therefore, defines a probability measure equivalent to ; hence, 3. This inequality implies that 4. On the other hand, Assumption 3. We denote and. Due to the exponential term in y , we introduce a truncated version of 4. Note that and. We split the following discussion into two cases. Case , or : In this case ; hence, the second term in is positive, and therefore for all n.
Comparison theorem for quadratic BSDE cf. Kobylanski, , theorem 2. As a result, for all n. For any , , therefore, is a solution to 4. Case , or : In this case , hence the second term in is negative. As a result, 4. Consider the BSDE which has the solution and. Then the comparison theorem for quadratic BSDE yields that , for all t and n , where.
As a result, implies that for all n. For any , ; therefore, is a solution to 4. Finally, we will show in both cases. Combining the previous inequality with 4. As is bounded; hence, it belongs to. It then follows from Kazamaki , theorem 3. For any solution of 3. It is clear that and. We will prove are of class D , and 4.
Therefore, and. Take and denote.
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We have from and that Combining this identity with 2. For the exponential local martingale Q , note that and. The boundedness of and imply as well. A similar argument yields. It then follows from Kazamaki , theorem 2. Coming back to 4. To verify 4. Taking a localizing sequence , we obtain on. Sending , the monotone convergence theorem and the class D property of yield The class D property of ensures a.
Subtracting it from both sides of the previous equation, 4. Therefore, the class D property of Q implies the same property of and. The discussion after 4. A similar localization argument as the previous step confirms 4. Remark 4. A careful examination reveals that the previous proof only requires to be bounded from above and Q to be a martingale.