text:
black student league upenn
The conference will bring together about 150 undergraduate Black Ivy League, and four-year HBCU (Historically Black Colleges and Universities) students who share diverse interests in Business, encouraging them to pursue various business-related fields, to develop professionally, and to build a supportive, valuable network within themselves. In the 1960s, things began to change. Furthermore, the BSL offers a place for students to acquaint themselves with the University and assists the students in making the Ivy-League experience more accessible and fruitful—whether through mentoring programs, socials, forums, cross-cultural events etc. Nathan discovered a passion for environmental sustainability having led initiatives like solar panel installation at his high school as a Year 12 student. Fifty-five percent of the 1.2 million Black students suspended in the U.S. live in just 13 Southern states. The BSL is built around fostering a community for Penn students of the African Diaspora. At University of Pennsylvania, the disparity between men and women on campus is far lower than the national average. Penn LPS COVID-19 Update LPS staff are not onsite, but we are still available Monday through Friday from 9 a.m. - 5 p.m. by phone and online in case you need support: (215) 898-7326 or lps@sas.upenn.edu. Since gaining acceptance into the world famous Wharton School of Business at the Ivy League's University of Pennsylvania, Tuesday, May … A haircut, a decent suit and business cards are … This university is ranked at #468 in male to female diversity nationwide. Student Clubs and Organizations. The undergraduate population is comprised of 5,472 males and 6,379 females. Visit coronavirus.upenn.edu, the University's dedicated coronavirus COVID-19 web page, for the latest updates.Penn LPS is ensuring that all students have access to the support they need. At MIT, 67.7 percent of all Black students accepted for admission decided to enroll. I left the event feeling that I could not possibly matter less. Weather dependent. While at one point it served as the active political voice on campus, it has shifted to a catalyst and facilitator of anything that may concern those who consider themselves a part of the African American Diaspora. Black students account for seven percent of Penn's undergraduate population and have benefitted the campus culture in distinct ways. The National Society of Black Engineers, University of Pennsylvania chapter was recognized as an official chapter in 1979. ... which was the forerunner for the Black Student League(BSL). Members of the Penn community may also review summaries of the results of recent survey instruments, such as the enrolled student … For more information check out our Facebook events 7 Penn ranks 1st among the Ivy League schools and 14th nationwide among doctoral/research institutions in the number of students studying abroad, according to the most recent data (Institute of International Education, 2018). Nathan Huynh | Wharton School Of Business (UPenn) Student. It will be about the resilience of black Penn Law students to rise against bigotry." Regarding the decision to remove her from teaching first-year courses, Ruger said he would expect students … This field will only be used for scheduled events, there will be no open recreation usage. The charts and links on this page provide an overview of the University of Pennsylvania by the numbers: enrollment, tuition, employment, faculty size, annual budget and more. ASA Weekend is Penn African Students Association's premier hosting of African culture and art all wrapped in a pumped, exciting and affordable weekend just for you. Enjoy this video of the neo-KKK stalking and harassing me as I try to invite a group of peaceful black students into my event at UPENN. But nowhere are Black students suspended or expelled more than in the South. The Black Student League is an organization built around fostering a familial environment that serves as a home for students of the African Diaspora at the University of Pennsylvania. The BSL offers a place for students to acquaint themselves with the University and assists students in making the Ivy-League experience more accessible and valuable through mentoring programs, socials, forums, and cross-cultural events. Resources available through the Office of student Affairs/Fraternity Sorority life Black Engineers, University of Pennsylvania added! 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2020 black student league upenn


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Welcome to Jazz in Reading
Jazz in Reading is a not-for-profit association of five 'nuts' who share a passion for jazz and seek to promote the best of live jazz across the Thames Valley on this website and to present regular concerts in the convivial surroundings of the Progress Theatre, Reading and beyond. Meet the Team
In the past year alone Jazz at Progress has played host not only to established stars of jazz like Gilad Atzmon, Henry Lowther, Tony Kofi, Orphy Robinson MBE, Steve Fishwick and Geoff Castle, representing a broad spectrum of styles from bebop to fusion, but also to the emerging talents of a new generation of musicians such as the eleven musicians who comprise Jonny Mansfield's innovative Elftet. Read reviews of past gigs
The Progress Theatre has a special atmosphere, beloved of audiences and musicians alike. 'Audiences listen here,' commented one musician who regularly plays at London's top jazz spots, while multi-instrumentalist Ben Holder said, 'It's a great venue. We had a blast. Brill audience!' The Progress also has a well-stocked bar! What's on? Is this link here a good idea? We have more info below than at Jazz at Progresss
Jazz lovers in the interval break at the Progress – photo by Zoë White
Mark Lockheart's epic six-movement orchestral jazz suite Days on Earth will form the centrepiece for the forthcoming 2019-2010 season of Jazz in Reading's promotions. A joint venture with Bracknell Jazz, this ambitious project will take place on Friday 27 September at the Wilde Theatre, South Hill Arts Centre, Bracknell as part of a major UK tour. Lockheart's star-studded ensemble will be augmented by guests and a string section, in a performance that Lockheart describes as 'the magnificent culmination of the over two billion seconds of eating, sleeping, thinking, playing and composing music'.
This is a rare opportunity to see a major new jazz work outside London and not to be missed! Link to ML blurb to be added
The Jazz at Progress programme itself gets under way on 20 September with the original sounds of Scott Willcox's Ten-Piece Big Band. Trumpeter Quentin Collins will be bringing his Sextet, including Brandon Allen and Tony Kofi on 18 October, followed by a Jazz n' Film Night on 22 November to commemorate the 80th anniversary of Blue Note Records with the Stuart Henderson Quintet and a presentation of the acclaimed documentary 'Blue Note: A Modern Jazz Story'. The ever-popular Alan Barnes and an all-star cast will round off 2019 with a 'A Jazz Christmas Carol' on 20 December.
The programme for 2020 has yet to be finalised but will definitely include visits from Andrew McCormack's Graviton and the Georgia Mancio/Kate Williams collaboration 'Finding Home'.
Lots of great jazz to look forward to and we look forward to seeing you!
Preferred dining: We have a special arrangement with The Queens Head (a short stroll from the theatre at 54 Christchurch Road), offering an excellent range of food including a full à la carte menu and specials dishes. Just show your Jazz at Progress ticket for 7% discount off food and drink on the day of the show you are attending; tickets to be presented when ordering.
email: info.queensheadreading@gmail.com
When you're in the mood to listen to live jazz, check out lots of other gigs in the area in our Jazz Gig Calendar. The calendar listings are compiled and edited by our listings editors, Paul Johnson and Jim Wade. If you're a musician, promoter or venue owner, you can submit your gig to the calendar here.
We can help you promote your jazz gigs, at a realistic and very reasonable cost. We have a growing mailing list of 860+ jazz fans in the Reading and Thames Valley area, and active Facebook and Twitter feeds, taking our reach to well over 1,800 jazz lovers in the region. Musicians, bands, venues and promoters can contact us to find out more here.
If you are looking for a jazz band for a party, wedding or other special occasion, let us know. We have extensive contacts in the jazz world and can help you find just the right band to suit your needs and budget.
Contact us to find out more here.

redpajama_set_name: RedPajamaCommonCrawl


text:
package eu.ydp.empiria.player.client.module.img.events.handlers;

import com.google.gwt.event.dom.client.TouchEndEvent;
import com.google.gwt.event.dom.client.TouchMoveEvent;
import com.google.gwt.event.dom.client.TouchStartEvent;
import com.google.gwt.user.client.ui.Widget;
import com.google.gwtmockito.GwtMockitoTestRunner;
import eu.ydp.empiria.player.client.gin.factory.TouchHandlerFactory;
import eu.ydp.empiria.player.client.module.img.events.TouchToImageEvent;
import eu.ydp.empiria.player.client.module.img.events.handlers.touch.TouchEndHandlerOnImage;
import eu.ydp.empiria.player.client.module.img.events.handlers.touch.TouchMoveHandlerOnImage;
import eu.ydp.empiria.player.client.module.img.events.handlers.touch.TouchStartHandlerOnImage;
import eu.ydp.empiria.player.client.module.img.events.handlers.touchonimage.TouchOnImageEndHandler;
import eu.ydp.empiria.player.client.module.img.events.handlers.touchonimage.TouchOnImageMoveHandler;
import eu.ydp.empiria.player.client.module.img.events.handlers.touchonimage.TouchOnImageStartHandler;
import org.junit.Test;
import org.junit.runner.RunWith;
import org.mockito.InjectMocks;
import org.mockito.Mock;

import static org.mockito.Mockito.*;

@RunWith(GwtMockitoTestRunner.class)
public class TouchHandlersOnImageInitializerTest {

    @InjectMocks
    private TouchHandlersOnImageInitializer testObj;
    @Mock
    private TouchToImageEvent touchEventsCoordinates;
    @Mock
    private Widget listenOn;
    @Mock
    private TouchHandlerFactory touchHandlerFactory;

    @Test
    public void shouldAddTouchMoveHandlerOnImage() {
        // given
        TouchOnImageMoveHandler touchOnImageMoveHandler = mock(TouchOnImageMoveHandler.class);
        TouchMoveHandlerOnImage touchMoveHandlerOnImage = mock(TouchMoveHandlerOnImage.class);
        when(touchHandlerFactory.createTouchMoveHandlerOnImage(touchOnImageMoveHandler)).thenReturn(touchMoveHandlerOnImage);

        // when
        testObj.addTouchOnImageMoveHandler(touchOnImageMoveHandler, listenOn);

        // then
        verify(listenOn).addDomHandler(touchMoveHandlerOnImage, TouchMoveEvent.getType());
    }

    @Test
    public void shouldAddTouchStartHandlerOnImage() {
        // given
        TouchOnImageStartHandler touchOnImageStartHandler = mock(TouchOnImageStartHandler.class);
        TouchStartHandlerOnImage touchStartHandlerOnImage = mock(TouchStartHandlerOnImage.class);
        when(touchHandlerFactory.createTouchStartHandlerOnImage(touchOnImageStartHandler)).thenReturn(touchStartHandlerOnImage);

        // when
        testObj.addTouchOnImageStartHandler(touchOnImageStartHandler, listenOn);

        // then
        verify(listenOn).addDomHandler(touchStartHandlerOnImage, TouchStartEvent.getType());
    }

    @Test
    public void shouldAddTouchEndHandlerOnImage() {
        // given
        TouchOnImageEndHandler touchOnImageEndHandler = mock(TouchOnImageEndHandler.class);
        TouchEndHandlerOnImage touchEndHandlerOnImage = mock(TouchEndHandlerOnImage.class);
        when(touchHandlerFactory.createTouchEndHandlerOnImage(touchOnImageEndHandler)).thenReturn(touchEndHandlerOnImage);

        // when
        testObj.addTouchOnImageEndHandler(touchOnImageEndHandler, listenOn);

        // then
        verify(listenOn).addDomHandler(touchEndHandlerOnImage, TouchEndEvent.getType());
    }
}

redpajama_set_name: RedPajamaGithub


Left Bank Books in St. Louis, Missouri is the recipient of the Bookstore Make-Over Project sponsored by The Bookstore Training Group of Paz & Associates to celebrate their twentieth anniversary serving independent bookstores. Seventeen bookstores from across the U.S. entered the contest.
"Our goal was to select one project that would help illustrate affordable solutions to issues that many bookstores have in common," noted founder Donna Paz Kaufman. "Left Bank Books provided the perfect combination of opportunities," she added. In the store's application, owners Kris Kleindienst and Jarek Steele commented, "Frankly, it's looking really tired at this point and the layout does not work for us at all anymore." In addition to their design and merchandising expertise, Paz & Associates will contribute $2,000 for upgrades and materials, with labor provided by the store.
The design group at Paz & Associates will address display windows, the front of store experience, traffic flow throughout the space, focal point displays, signage and promotional messaging, the "Wow" factor for creating word-of-mouth marketing, and event staging. The project has an October 15 completion date. Watch for updates on this blog!


text:
Alleged drunk driver jumps Brockton Taco Bell curb
Told police he was hungry
A Brockton man was arrested for his second drunk driving offense after police say they watched him drive over a curb and sidewalk to get out of the drive-through line at Taco Bell on Belmont Street Friday night.Joshua Figueiredo, 33, of Pioneer Avenue in Brockton, was charged with his second drunk driving offense, negligent operation of a motor vehicle and driving while not in possession of a license, news partner The Enterprise reports."He must have gotten sick of waiting, and he jumped over the curbing and went onto Belmont Street," said Brockton Police Lt. William Hallisey.Figueiredo was stopped by police in a gray Toyota pickup truck around 9 p.m. Friday. Hallisey said he had a strong odor of alcohol coming from him."He stated he was hungry, and he only lived down the road and just wanted to go home," Hallisey said.According to police, Figueiredo refused field sobriety tests and a breathalyzer and asked for his lawyer.Hallisey said Figueiredo told police to either arrest him or let him go.He is scheduled to be arraigned in Brockton District Court Monday.
BROCKTON, Mass. —
A Brockton man was arrested for his second drunk driving offense after police say they watched him drive over a curb and sidewalk to get out of the drive-through line at Taco Bell on Belmont Street Friday night.
Joshua Figueiredo, 33, of Pioneer Avenue in Brockton, was charged with his second drunk driving offense, negligent operation of a motor vehicle and driving while not in possession of a license, news partner The Enterprise reports.
"He must have gotten sick of waiting, and he jumped over the curbing and went onto Belmont Street," said Brockton Police Lt. William Hallisey.
Figueiredo was stopped by police in a gray Toyota pickup truck around 9 p.m. Friday. Hallisey said he had a strong odor of alcohol coming from him.
"He stated he was hungry, and he only lived down the road and just wanted to go home," Hallisey said.
According to police, Figueiredo refused field sobriety tests and a breathalyzer and asked for his lawyer.
Hallisey said Figueiredo told police to either arrest him or let him go.
He is scheduled to be arraigned in Brockton District Court Monday.


text:
Q: How can I build programme based on nfc technology? I'm trying to write a programme, that is going to use nfc technology to transfer data between phones. The problem is, that I don't really know how to start it, what language or libraries should I choose? Or maybe there are any other technologies, that will allow me to transfer data via touching phones?

A: You can write your app following the Android-Beam guide
It can be written in Kotlin/Java Language.
Note that from Android 10, Beam feature is deprecated.
Instead you will have a new feature called Fast Share that uses Wifi Direct and other methods of pairing rather than touching.

redpajama_set_name: RedPajamaStackExchange


Run program through the queue as usual.
the total running time of the application.
30% of the total MPI time.
The command time is available in most Unix systems.
It is simple to use (no code instrumentation required).
Gives total execution time of a process and all its children in seconds.
GPROF is the GNU Project PROFiler.
•Requires recompilation of the code.
and periodic sampling of the program.
•A default gmon.out file is produced with the function call information.
name, correct directory, correct project name and so on.
We set these as environmental variables in the command line or the submission script.
The we send the job through the queue as usual.
Organizes Runs as: Applications, Experiments and Trials.
Mean and Standard Deviation Statistics given.
Function Data Window, which gives a closer look at a single function.
Hardware Counters provide Floating Point Operations (Function Data view).
Select Operation; then Apply. Derived Metric will appear as a new trial.
– Scalable all-in-one profiling package.
– Requires re-compilation of source to instrument, like Tau.


From left: Longwood Symphony members Dr. Heidi Harbison Kimberly, Dr. Mark Gebhardt, and Ramona Nee.
Ramona Nee, a lawyer from Sherborn, ran her first marathon on April 15, despite having torn a muscle in her calf about two weeks before. She ran as part of the Dana-Farber Marathon Challenge team, honoring several people in her life affected by cancer. She was elated to have crossed the finish line, but by the time she'd picked up her belongings, her muscles had stiffened in the chilly spring sunshine, and she was hobbling. So a Dana-Farber volunteer helped Nee make her way toward the Marriott Copley Place hotel, where she was to meet her husband, daughter, and in-laws.
She wanted to get one more glimpse of the finish line, so the pair walked behind the grandstand, in front of the Boston Public Library, to get to the hotel. That's where they were when the first bomb went off. Even though they were shielded from the direct blast, the force of the explosion knocked them over.
Mark Gebhardt is the chief of orthopedic surgery at Beth Israel Deaconess Medical Center. Since it was a holiday, he was at home working when his pager went off. He turned on the TV and knew immediately that his team, which treats injuries to bones and muscles, would be pressed into heavy service. Gebhardt lives within walking distance of the hospital and arrived there in minutes. The ambulances had already begun to roll in.
Upcoming classical music and opera performances in the Boston area.
Nee, Kimberly, and Gebhardt are three of the thousands who played a role in the Boston Marathon day events. They are also musicians who play in the Longwood Symphony Orchestra, an elite ensemble that draws most of its performers from Boston's medical community.
The Longwood's motto is "healing the community through music," and it's hard to think of a time when such a palliative was more acutely needed. The orchestra is dedicating its concert this Saturday to the bombing victims and it has added Elgar's elegiac "Nimrod," from the "Enigma Variations" as a special tribute. It is also offering free tickets to staff at the city's five major hospitals, creating a venue for caregivers to sort through their experiences and search for a sense of peace.
For Nee, the immediate aftermath was dominated by confusion. She'd last seen her family at Mile 25, at the designated Dana-Farber cheering section; she knew they'd been planning to walk the last mile and meet her at the hotel. She tried, unsuccessfully, to call her husband. She kept dropping her phone, and her legs were so sore that she couldn't even bend over to pick it up.
"At that point, the panic a little bit starts to set in," she said.
She would learn later that her family had turned off Boylston onto Ring Road, next to Lord & Taylor, carrying them to the other side of the street just before the first explosion occurred.
One of the eeriest scenes Nee remembers is finally entering the Marriott room where the Dana-Farber runners were to be received. Most were still on the course, so the room was filled with massage therapists and chairs for runners who would never arrive. There was an uneasy stillness. Through the windows Nee could see SWAT teams.
Fifteen minutes later, she was reunited with her family. Their joy was shadowed by the slowly dawning sense of what they had just experienced. Her 6-year-old daughter had heard someone talking about the injuries as they ran from the scene.
"Her first question to me was, 'What's a multiple amputee, Mommy?' ," Nee said.
At Brigham and Women's, Kimberly and her colleagues had less than 10 minutes to clear the emergency room and prepare for the crush of bombing victims.
"We're used to taking care of trauma," she said. "The medical part was kind of the easiest. The challenge was partly the volume of patients that arrived." The Brigham would treat 39 patients; Kimberly said 19 of them arrived in the first half-hour, many among the most critically wounded.
The first patient she treated was a young boy. "The fact that they would bring him to us instead of Children's Hospital made me wonder, what's going on out there?" He was crying and couldn't tell Kimberly his last name or his parents' names. (He was later transferred to Children's and reunited with his family.) The second patient was a woman with shrapnel in her neck.
But an unflappable sense of calm dominated. This was her job.
Meanwhile, Gebhardt was overseeing his team of orthopedists, ensuring enough resources to handle the victims at Beth Israel. Since the explosions had happened at shift change, some people stayed on to help. Others, like him, simply knew to come in.
EMTs also reacted quickly, judiciously parceling out the injured among the city's trauma centers. Environmental service teams cleaned and prepped operating rooms quickly enough to accommodate the next patient. Nutritional services workers made sure there was food and drink for patients, their families, and the exhausted medical teams.
The totality of the response, Gebhardt said, reminded him of a well-trained orchestra.
"On the surface, it looked like there were a bunch of people running in different directions. But in fact, everybody knew exactly where they needed to be. Everyone immediately assumed their roles. It was like they had rehearsed a piece.
Saturday's concert will create a space for caregivers, victims, families — united that day and ever after — to reflect and to heal.
Kimberly, a violinist, says she's always viewed music therapeutically, as a release from the anxiety and tensions of her work. It's why she's played in the Longwood since 1999. She's been thinking especially about the second suite from Ravel's ballet "Daphnis and Chloe," which is on Saturday's program. "It's not a melody you hum along to a lot," she said. "You just get caught up in the sounds. It's a good piece to listen to and process.
Nee, a violinist who also is on the Longwood board, won't be playing Saturday' but will be in the audience, reflecting on the twist of fate that brought her family so close to tragedy, and that kept them safe when others weren't.
And also, of course, about healing the community.


Newspaper veteran J.D. Creer has been named the executive editor of the Salem News, Morning Journal and East Liverpool Review.
Tammie McIntosh, publisher of the three newspapers, made the announcement Friday. In his new role, Creer will oversee editorial content for the publications serving Columbiana County.


Varicose veins are called Siraja Granthi in Ayurveda. It is a condition where the blood veins get enlarged and painful. Thus the patient may find it painful to stand or sit for a couple of hours. According to Ayurveda, there might be multiple reasons to varicose vein. One of the major ones as stated before, might be long sitting or standing for hours. Another reason is the constant pressure on the lower body. Other reasons might be age or the after effects of delivery.
However, there are not much oral medications that can be suggested for varicose vein. One form of remedy that we suggest is a prescribed diet that contains garlic, ginger and onions. We ensure effective treatment with regard to the same. There are hundreds of patients who have come back to us to express their happiness in being relieved.


text:
Here's What Kim Kardarshian Planned To Do If 'KUWTK' Never Took Off
'I didn't think [the show] would go beyond season 1 or 2.'
Kim Kardashian didn't always plan on being a reality TV phenomenon, going on to own multiple clothing brands, apps, and makeup lines. If Keeping Up with the Kardashians hadn't worked out—it's currently been on the air for more than 10 years—Kim had a totally different idea of how her life would go. Click through for more details!
Photo credit: BACKGRID
"I thought I was just going to run my clothing store. I didn't think [the show] would go beyond season 1 or 2," she told Vogue India. "And then as things were happening, my mom and I were so excited."
But, she did admit the family "made so many mistakes" during the course of the show's decade-long run. Mainly, Kim said, they shouldn't have gotten "involved in anything" that came their way.
"We didn't even know what we were starting, but we knew we were in it together," the 37-year-old said. "We definitely made so many mistakes."
"At first, we would be involved in anything," Kim continued. "But it helped us work our way up to now, to launching a beauty brand completely by myself, and funding it all."
Kim's also been super busy being a mom. The makeup mogul just welcomed her third child, Chicago, via surrogate.
What do you think about Kim's backup plan if KUWTK didn't work out? Let us know in the comments!
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Kylie Jenner Launches Two Products Named After Taylor Swift Songs


Analysts predict that Alcoa Corp (NYSE:AA) will report ($0.17) earnings per share (EPS) for the current quarter, according to Zacks Investment Research. Three analysts have made estimates for Alcoa's earnings, with estimates ranging from ($0.55) to $0.08. Alcoa posted earnings of $0.77 per share in the same quarter last year, which suggests a negative year-over-year growth rate of 122.1%. The business is scheduled to issue its next quarterly earnings results after the market closes on Wednesday, April 17th.
According to Zacks, analysts expect that Alcoa will report full year earnings of $1.53 per share for the current year, with EPS estimates ranging from $0.77 to $2.61. For the next financial year, analysts expect that the firm will post earnings of $2.27 per share, with EPS estimates ranging from $1.42 to $3.37. Zacks' earnings per share calculations are an average based on a survey of research firms that follow Alcoa.
Alcoa (NYSE:AA) last announced its quarterly earnings results on Wednesday, January 16th. The industrial products company reported $0.66 earnings per share (EPS) for the quarter, topping the consensus estimate of $0.50 by $0.16. The business had revenue of $3.34 billion for the quarter, compared to the consensus estimate of $3.31 billion. Alcoa had a return on equity of 9.21% and a net margin of 1.69%. The business's revenue was up 5.4% on a year-over-year basis. During the same quarter last year, the business posted $1.04 EPS.
Several analysts recently issued reports on the company. TheStreet downgraded Alcoa from a "c-" rating to a "d+" rating in a report on Thursday, March 28th. ValuEngine raised Alcoa from a "strong sell" rating to a "sell" rating in a report on Thursday, December 20th. Zacks Investment Research raised Alcoa from a "strong sell" rating to a "hold" rating in a research note on Thursday, January 3rd. Goldman Sachs Group raised Alcoa from a "neutral" rating to a "buy" rating and set a $38.00 target price on the stock in a research note on Monday, January 14th. Finally, JPMorgan Chase & Co. cut their target price on Alcoa to $52.00 and set an "overweight" rating on the stock in a research note on Thursday, January 17th. One research analyst has rated the stock with a sell rating, five have issued a hold rating and ten have assigned a buy rating to the company. The company currently has a consensus rating of "Buy" and a consensus target price of $46.73.
A number of institutional investors have recently made changes to their positions in AA. We Are One Seven LLC bought a new stake in Alcoa during the 4th quarter valued at $35,000. Financial Architects Inc raised its stake in shares of Alcoa by 2,970.3% in the 4th quarter. Financial Architects Inc now owns 3,101 shares of the industrial products company's stock worth $82,000 after buying an additional 3,000 shares in the last quarter. IMS Capital Management bought a new stake in shares of Alcoa in the 3rd quarter worth $87,000. Cerebellum GP LLC raised its stake in shares of Alcoa by 8.1% in the 4th quarter. Cerebellum GP LLC now owns 4,798 shares of the industrial products company's stock worth $128,000 after buying an additional 361 shares in the last quarter. Finally, Point72 Hong Kong Ltd raised its stake in shares of Alcoa by 414.8% in the 3rd quarter. Point72 Hong Kong Ltd now owns 5,112 shares of the industrial products company's stock worth $207,000 after buying an additional 4,119 shares in the last quarter.
Shares of NYSE AA opened at $27.72 on Friday. The company has a market capitalization of $5.15 billion, a P/E ratio of 7.75, a P/E/G ratio of 4.49 and a beta of 1.13. Alcoa has a 12-month low of $25.01 and a 12-month high of $62.35. The company has a debt-to-equity ratio of 0.24, a quick ratio of 0.85 and a current ratio of 1.42.


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Nothing brings into focus more vividly what the rest of the world thinks about Canada than travelling abroad. Having spent the past 10 days in Ireland, I was once again reminded that when others think of Canada, they usually think good things. The problem is, beyond that, they tend not to think of us at all.
I was in Dublin as a guest speaker at the summit of the Ireland-Canada Business Association, a group fostering greater business connections, investment and trade between the two countries. There was a packed house of enthusiastic Irish business people, but I couldn't help but think how many more people would attend a summit focused on the U.S., Germany or even Australia.
Even though we're a nation nine times larger than Ireland and a member of the G7, Canada still seems like an afterthought in Dublin.
What are the economic implications of having such an invisible profile? Is it a drawback being so vanilla in a world of 101 flavours? Or can there indeed be some benefit? After all, who wants the violence or political theatre that is lifting other countries' profiles at the moment?
Certainly in terms of investment, Canada's low profile is not a benefit. Global capital managers and investors want to make money, but they need to be aware of where the opportunities are. If the only thoughts they have of Canada is moose, mountains and maple syrup, they'll overlook us.
Tourism is another enormous area of potential economic opportunity. The world's middle class of consumers, particularly in China, are looking for interesting and unique travel experiences – but they also want security and safety. Canada checks a lot of those boxes.
Finally there's the realm of culture, art and design. These overlap a bit with tourism, but taken together they really represent a broader economic driver called innovation. Countries that are innovative, creative and more risk-taking also tend to attract like-minded people.
In this very global age in which we live, the best and brightest can live anywhere. They choose to live in cities like Barcelona, Copenhagen, Austin and Melbourne for a reason – they are bursting with art and design.
I started to become despondent about Canada's feeble global profile, drowning my frustration in pints of Guinness. But then, three separate media events on three consecutive days gave me hope for my home and native land.
First, Monocle magazine. The U.K.-based magazine is the must-read on global affairs, business, culture and design. Its November issue is focused on Canada. The cover story is titled: "Canada calling: Why it's time to take a fresh look north. From Toronto to the high north, Monocle reports on a nation that's flexing mind and muscle, re-engaging diplomatically and spreading its wings." It's almost as if Monocle believes it discovered a new country no one had heard of before.
Then, The Economist magazine's Oct. 29th cover story: "Liberty Moves North: Canada's Example to the World." When the world's most respected business and political affairs publication dedicates a cover to Canada, you know attention may finally be shifting to above the 49th parallel.
Finally last week, The Lonely Planet publication awarded Canada its Top Travel Destination of 2017. That alone could do more to boost tourism to Canada than the travel agency posters of Lake Louise.
Profile matters for the economy. A strong reputation as a great place for investment, tourism, innovation, culture and design is critical in a planet crowded with competition. We've got all of the ingredients, and it's great that three major publications are finally recognizing us.
Still, Canadians cannot sit back and let business magazines and travel books do the heavy lifting for us. We need to promote ourselves more aggressively, or risk behind left behind.
This column originally appeared in The Globe and Mail on November 4, 2016.


text:
\section*{Introduction}

 Rational asset prices are the expected present value of future cash flows, properly discounted with a state price, also known as  stochastic discount factor, or Arrow price.
 In economic terms, the state price $\psi_t(\omega)$ is the price of one unit of the num\'{e}raire consumption good (or money) delivered at time $t$ when state $\omega$ prevails.
 With complete markets, the price of a financial asset is  obtained by valuing the asset's payment stream with the state price.
 State-prices determine asset prices, interest rates, and the pricing or equivalent martingale measure.

It is thus important to understand what economic theory can say about the properties of such state prices.
One might fear that the answer is: Not much, in general.
When we take a classical setup where consumption plans come from some $L^p$--space, and preferences are norm-- or Mackey--continuous on the commodity space, the famous existence theorems of general equilibrium theory return state prices in the dual space.\footnote{See the overview by \cite{MasColellZame91} for an account of infinite--dimensional general equilibrium theory. A classic in this regard is the first part of the existence proof in \cite{DuffieZame89}. }
In this generality, a state price is just a nonnegative, measurable, adapted process that satisfies some degree of integrability.
In particular, from  general equilibrium theory, we do not get continuous sample paths, or state prices that are diffusions (as one would like to have to justify a Samuelson--Black--Scholes--type model of asset markets).
One can then impose additional assumptions on preferences and endowments, of course.
\cite{DuffieZame89}, e.g., assume that endowments are It\^{o} processes, and that agents have time--additive smooth expected utility functions. From the first--order conditions of utility maximization, a representative agent argument, and It\^{o}'s lemma, Duffie and Zame obtain equilibrium state-prices that are  diffusions.

One wonders if the structure of time and uncertainty does not allow to derive more structure for state prices in complete generality.
\cite{HindyHuangKreps92} (HHK) try to develop such an approach.
They challenge the implicit assumption that preferences are  norm-- or Mackey--continuous asking when  a rational agent should consider contingent consumption plans as close.
A rational agent should treat small up or downward shifts as close, of course; but she should also react smoothly to  small shifts of (lifetime) consumption plans  over time.
After all, most agents do not care much about getting their retirement payments in twenty five years from now or twenty --five years plus one day.
Technically, one uses the weak topology for distribution functions on the time axis and some kind of $L^p$--topology for uncertainty.
HHK characterize the dual space (where prices come from).
Elements of the dual are given by state prices that are the sum of a martingale and some absolutely continuous process.
HHK's work thus shows that a suitable notion of continuity and the structure of time and uncertainty allow to derive results for state prices.
Unfortunately, there are no equilibria with prices from the HHK dual in general as these spaces are not lattices.
\cite{BankRiedel02} and \cite{Martinsda} establish existence of equilibria in bigger price spaces where the equilibrium price functional is not necessarily continuous on the whole commodity space, but only on the consumption set (the positive cone of the commodity space).

In the present paper we take the latter result as a starting point and ask: what are the positive, linear functionals that are also continuous on the consumption set?
We fully characterize the corresponding state prices, and show that under minimal continuity requirements on preferences and a suitable properness condition,\footnote{Such properness is necessary in infinite--dimensional models, see \cite{MasColellZame91} for a discussion.} equilibria with such state prices exist.

The method to derive the characterization of state prices relies on the \emph{th\'{e}orie g\'{e}n\'{e}rale} of stochastic processes as developed in \cite{DellacherieMeyer75ChpI-IV}.
It turns out that much in the same way as It\^{o}'s theory of stochastic integration is taylor--made for the Samuelson--Black--Scholes theory of asset markets, the
\emph{th\'{e}orie g\'{e}n\'{e}rale} suits our general theory for equilibrium state prices.
This is the first connection of general equilibrium theory and \emph{th\'{e}orie g\'{e}n\'{e}rale}, and we hope that more interesting results can spring from this relation in the future.

For the connoisseurs, we sketch parts of our representation theorem here.
Details concerning notation, if not obvious, are explained in Section \ref{SecModel} below.
We take a nonnegative, linear price functional on the space of all optional random measures with  total variation in $L^p$.
Fixing a stopping time $\tau$, we consider only the restriction on payment streams that pay off at time $\tau$.
This gives us a family $\left(\pi^\tau\right)_{\tau \ \mbox{\tiny stopping time}}$ of linear mappings from $L^p\left({\cal F}_\tau\right)$ to the real numbers.
With a fixed maturity, there are no issues of shifting etc., so that this mapping is norm--continuous;  Riesz' theorem gives us a random variable $z^\tau \in L^q\left({\cal F}_\tau\right)$  that represents this mapping.
So we obtain a family of random variables $\left(z^\tau\right)_{\tau \ \mbox{\tiny stopping time}}$ where every $z^\tau$  is ${\cal F}_\tau$--measurable.
We show that this large family is consistent in the sense that we have $$z^\tau=z^\sigma$$ on the event $\{\tau=\sigma\}$ for two stopping times $\sigma$ and $\tau$.
Such families are called $\mathcal{T}$--systems in the  \emph{th\'{e}orie g\'{e}n\'{e}rale}.
The question is if we can find an adapted stochastic process $\left(\psi_t\right)$ such that $\psi_\tau=z^\tau$ for all stopping times $\tau$.
In this case, one says  that $\left(\psi_t\right)$ recollects  the family $\left(z^\tau\right)_{\tau \ \mbox{\tiny stopping time}}$.
It has been shown by \cite{DellacherieLenglart80} that a $\mathcal{T}$--system can be recollected if it is left--continuous in expectation\footnote{This is not sample--path left--continuity. A $\mathcal{T}$--system $\left(w^\tau\right)_{\tau \ \mbox{\tiny stopping time}}$ is left--continuous in expectation if $\mathbb{E} w^{\tau_n} \to \mathbb{E} w^\tau$ whenever $\tau_n \uparrow \tau$ a.s.}(and this kind of continuity is necessary, in general).
Fortunately, the intertemporal topology we use here gives us even continuity in expectation.
Note that we use here the shifting property of the intertemporal topology proposed by HHK: as small shifts over time are considered as close in that topology, the price of a unit of the consumption good delivered at $\tau_n$ has to approach the price of one unit of the consumption good at $\tau$ if $\tau_n \to \tau$.
The recollecting process $(\psi_t)$ is our desired state price.
From continuity in expectation, it is  even cadlag:\footnote{This is again a classic result from the  \emph{th\'{e}orie g\'{e}n\'{e}rale}, see \cite{DellacherieMeyer75ChpI-IV}, Theorem 48.} it has right--continuous sample paths with left hand limits.
It can thus jump.
This might seem puzzling because the dual space for measures on the time axis consists of continuous functions.
And here comes the final clue: the cadlag process  $\left(\psi_t\right)$ is the optional projection of a \emph{continuous, but not necessarily adapted} stochastic process $\left(\xi_t\right)$; this is a result by \cite{Bismut78} and \cite{Emery78}.
Possible jumps in the state price density thus come from the gradual release of information under uncertainty.
To give an example, one might have $\xi_t=Z$ for an ${\cal F}_T$--measurable random variable so that the process $\xi_t$ is constant in time, but not adapted.
The optional projection is then the martingale $\psi_t=\condexp{Z}$. It is well known that in general, martingales jump when information surprises occur.

The paper is organized as follows.
The next section sets up the intertemporal model.
Section~\ref{section:compatible} contains our main theorem characterizing the state prices.
Section~\ref{section:equilibrium} establishes existence of equilibria with such prices and contains examples.

\section{Model and Notation}\label{SecModel}

We consider a stochastic pure exchange economy where a finite set $I$ of agents live in a world of uncertainty from time $0$ to time $T$.
Uncertainty is modeled by a complete probability space $(\Omega, \mathcal{F},\P)$.
Each $\omega \in \Omega$ is a state of nature which is a complete description of one possible realization of all exogenous sources of uncertainty from time $0$ to time $T$.
The sigma-field $\mathcal{F}$ is the collection of events which are distinguishable at time $T$ and $\P$ is a probability measure on $(\Omega,\mathcal{F})$.
The probability space $(\Omega,\mathcal{F},\P)$ is endowed with a filtration $\mathbb{F}=\{ \mathcal{F}(t)\colon t\in [0,T]\}$ which represents the time evolution of the agents' knowledge about the states of nature.
We assume that $\mathcal{F}(0)$ is $\P$-almost surely trivial and that $\mathbb{F}$ satisfies the usual conditions of right-continuity and completeness.
A process is said optional if it is $\mathcal{O}$-measurable where $\mathcal{O}$ is the sigma-field on $\Omega \times [0,T]$ generated by right-continuous $\mathbb{F}$-adapted processes with left-limits.

\subsection{Consumption space}

There is a single consumption good available for consumption at any time $t\in [0,T]$.
The set of positive, nondecreasing and right-continuous functions from $[0,T]$ to $\mathbb{R}_+$ is denoted by $M_+$.
We represent the consumption bundle of an agent by a process $x :(\omega,t) \mapsto x(\omega,t)$, where $x(\omega,t)$ (sometimes denoted by $x_t(\omega)$) represents the cumulative consumption from time $0$ to time $T$ and satisfies
\begin{description}
\item[(a)] for each $\omega \in \Omega$, the function $x(\omega)$ belongs to $M_+$,
\item[(b)] for each $t\in [0,T]$, the random variable $x_t$ is $\mathcal{F}(t)$-measurable and $x_T$ belongs to $L^p(\P)$
\end{description}
where $1\leq p< +\infty$.

The set of ($\P$-equivalent classes of) mappings $x : \Omega \rightarrow M_+$ such that the process $(\omega,t) \mapsto x(\omega,t)$ satisfies (a) and~(b) is denoted by $E_+$ and the linear span of $E_+$ will be denoted by $E$.
The space $E_+$ is called the consumption space and $E$ is called the commodity space.
Observe that any consumption bundle $x$ in $E_+$ is an $\mathbb{F}$-adapted process having right-continuous and bounded variation sample paths and therefore can be assimilated with an optional random measure denoted by $dx$.
If $z$ belongs to $E$ then there exist $x,y$ in $E_+$ such that $z=x-y$.
We can endow $E$ with the linear order $\geq$ defined by the cone $E_+$ in the sense that $y\geq x$ if $y-x$ belongs to $E_+$.
If $y$ belongs to $E_+$ then the order interval $[0,y]$ is defined by $[0,y]:=\{ x\in E \colon x\in E_+ \quad \mathrm{and} \quad y-x \in E_+ \}$.
The space $E$ endowed with the partial order defined by $E_+$ is a linear vector lattice (see \cite[Proposition~1]{Martinsda}).

\begin{remark}
Observe that if $x,y$ are vectors in $E$ such that $y \geq x$ then there exists $\Omega^* \in \mathcal{F}$ with $\P\Omega^*=1$ and such that for each $\omega \in \Omega^*$, the function $t \longmapsto y(\omega,t)-x(\omega,t)$ is nondecreasing with $y(\omega,0) -x(\omega,0) \geq 0$.
In particular we have for each $\omega \in \Omega^*$,
\[
y(\omega,t) \geq x(\omega,t) \quad \forall t\in [0,T].
\]
\end{remark}

\subsection{Topologies}

Since $0\leq x_t \leq x_T$ and $x_T \in L^1(\P)$ for every $x \in E_+$, the space $E$ is a subspace of $L^1(\mathcal{O},\P\otimes \kappa)$ where $\kappa = \lambda + \delta_T$ with $\lambda$ the Lebesgue measure on $\mathcal{B}$ the Borelian sigma-algebra on $[0,T]$ and $\delta_T$ the Dirac measure on $T$.
Following \cite{HindyHuang92} we consider on $E$ the restriction of the $L^p(\mathcal{O},P\otimes \kappa)$-norm, i.e., we consider the norm $\norm{\cdot}$ defined by
\[
\forall x\in E, \quad \norm{x}=\left[ \mathbb{E} \int_{[0,T]} |x(t)|^p \kappa(dt) \right] ^{\frac{1}{p}} =\left[ \mathbb{E} \int_{[0,T]} |x(t)|^p dt + \mathbb{E}|x(T)|^p \right] ^{\frac{1}{p}}.
\]
It is argued in~\cite{HindyHuang92} that this norm, called intertemporal norm, induces a topology on the set of consumption bundles that exhibits intuitive economic properties, in particular it captures the notion that consumption at adjacent dates are almost perfect substitutes except possibly at information surprises.
Usually, we refer to the topology generated by the intertemporal norm when we speak about continuity, open sets, etc.
Occasionally, we will use other topologies as well, though.
For $z \in E$ and fixed $\omega \in \Omega$, the function $z(\omega)$ can be assimilated with a signed measure $d[z(\omega)]$ on the time interval $[0,T]$.
We denote by $\normtv{z(\omega)}$ the total variation of the measure $d[z(\omega)]$ (and we will drop the $\omega$ frequently, as usual).
The expectation of the total variation of $z$ leads to the strong topology on $E$ as given by the norm
\[
\normstrong{z}:=\mathbb{E} \normtv{z}.
\]
Note that convergence in the strong topology entails convergence in the intertemporal topology.
Moreover, $E$ is a topological vector lattice when endowed with the order generated by $E_+$ and the strong topology.

If $h$ is a random variable and $\tau$ a stopping time in $\mathcal{T}$, we denote by $\delta_\tau h$ the simple random measure that delivers $h(\omega)$ units of the consumption good at time $\tau(\omega)$ and nothing elsewhere.
In particular $\delta_\tau$ is the Dirac measure on $\tau$.

\subsection{Prices}

The weakest notion of a price is that of a nonnegative linear functional on $E$.
The algebraic dual (the space of linear functional from $E$ to $\mathbb{R}$) is denoted by $E^\star$ and $E^\star_+$ denotes the cone of nonnegative linear functionals, i.e., $\pi \in E^\star$ is nonnegative if $\pi(x) \geq 0$ for every $x\in E_+$.
If $B(T)$ denotes the space of bounded functions defined on $[0,T]$ then we let $L^q(\P,B(T))$ denote the space (up to $\P$-indistinguishability) of all $\mathcal{F}\otimes\mathcal{B}$-measurable processes $\psi: \Omega \times [0,T] \rightarrow \mathbb{R}$ such that the function
\[
\omega \rightarrow \sup_{t\in [0,T]} |\psi(\omega,t)|
\]
belongs to $L^q(\P)$ where $q\in (1,+\infty]$ is the conjugate of $p$.
There is a natural duality $\ps{\cdot}{\cdot}$ on $L^p(\P,B(T)) \times E$ defined by
\[
\ps{\psi}{z} = \mathbb{E} \int_{[0,T]} \psi(t) dz(t).
\]
The space of processes in $L^q(\P,B(T))$ that are optional is denoted by $F$ and we denote by $F_+$ the order dual cone, i.e., \[
F_+ := \{ \psi \in F \ \colon \ \ps{\psi}{x}\geq 0, \quad \forall x\in E_+ \}.
\]
The pair $\ps{F}{E}$ is a Riesz dual pair (see \cite[Proposition~1]{Martinsda}) and a process $\psi \in F$ belongs to $F_+$ if and only if $\psi(t) \geq 0$ for every $t\in [0,T]$.
To each nonnegative process $\psi \in F_+$ we can consider the nonnegative linear functional $\ps{\psi}{\cdot}$ in $E^\star_+$ defined by
\[
\forall z\in E, \quad \ps{\psi}{z} = \mathbb{E} \int \psi dz.
\]
By abuse of notations, we still denote $F_+$ (and $F$) the space of linear functionals associated to processes in $F_+$ (resp. $F$).
If a price $\pi \in E^\star_+$ is represented by an optional process $\psi \in F_+$ in the sense that $\pi=\ps{\psi}{\cdot}$, then the process $\psi$ is called a state price.
In that case the duality product $\ps{\psi}{x}$ is the value of the consumption bundle $x\in E_+$ under the price $\psi$ where $\psi(\omega,t)$ is interpreted to be the time $0$ price of one unit of consumption at time $t$ in state $\omega$, per unit of probability.

\section{Compatible Prices}\label{section:compatible}

In general, prices in $F_+$ will not be compatible  with the notion of intertemporal substitution as they might assign very different prices to consumption plans that are close in the intertemporal topology.
One might therefore aim to find prices in the topological dual $(E,\norm{\cdot})'$ of $E$.
As shown by \cite{HindyHuang92}, every linear functional $\pi \in (E,\norm{\cdot})'$ continuous for the intertemporal norm can be represented\footnote{In the sense that $\pi=\ps{\psi}{\cdot}$.} by a semimartingale $\psi$ satisfying
\[
\psi_t = A_t + M_t
\]
where $A$ is an adapted process with absolutely continuous sample path satisfying
\[
A' \in L^q(\mathcal{O},\P \otimes \kappa) \quad \mathrm{and} \quad A'_T \in L^q(\P)
\]
and $M$ is the martingale defined by
\[
M_t = \mathbb{E}[ -A'_T - A_T \vert \mathcal{F}_t], \quad \forall t\in [0,T].
\]
We denote by $K$ the space of processes $\psi$ representing linear functionals in $(E,\norm{\cdot})'$.\footnote{Observe that $K$ is a subset of $F$.}
However, there is in general no hope to obtain equilibrium prices in $K$ as it is a not a lattice.
On the other hand, all that counts for equilibrium theory are linear functionals restricted to the \emph{consumption set} $E_+$, the positive cone of the commodity space.
So we relax the requirement of continuity on the whole space and aim only for continuity on the consumption set $E_+$.
A price is called \emph{compatible} if if is continuous with respect to the intertemporal topology on $E_+$.
Denote the set of compatible prices by $H_+$.

This leads to two questions:
     \begin{itemize}
     \item What is the structure of compatible prices?
     \item Under which conditions do equilibria with compatible prices exist?
     \end{itemize}
Let us answer the first question.

\begin{theorem}\label{ThmCharCompPrices}
A nonnegative linear functional $\pi \in E^\star_+$ is a compatible price if and only if it can be written as
\[
\pi(x)=\mathbb{E} \int \psi dx, \quad \forall x \in E_+
\]
for a nonnegative, rightcontinuous processes $\psi$ with left limits that satisfies the following conditions:
  \begin{itemize}
    \item The process $\psi$ is the optional  projection\footnote{That is $\psi_\tau = \mathbb{E}[\xi_\tau \vert \mathcal{F}_\tau]$ for every stopping time $\tau \leq T$.} of a (not necessarily adapted) continuous process $\xi$ with
    \[
    \mathbb{E} \sup_{t\in[0,T]} |\xi_t| < \infty.
    \]
    \item The process
    \[
    \psi^*=\sup_{t\in[0,T]} \psi_t
    \]
    belongs to $L^q(\mathcal{F}, \P)$.
\end{itemize}
\end{theorem}

We denote by $H$ the space the rightcontinuous processes with left limits that are bounded in $L^q$ and are the optional projection of a continuous process bounded in $L^1$.
Observe that the space $H$ is a subspace of $F$ containing $K$.
If we let $H_+=H\cap F_+$ then a nonnegative linear functional $\pi$ is compatible if only if it can be represented by a process $\psi$ in $H_+$, i.e., $\pi=\ps{\psi}{\cdot}$.
We give the proof of Theorem~\ref{ThmCharCompPrices} for the case $p=1$.
Later, we indicate how to obtain the result for $p>1$.

\subsection{Sufficiency}

First, we show that every element that satisfies the conditions of the theorem  induces a compatible price.

\begin{lemma}
Let $\psi$ be a cadlag process that satisfies the assumptions in Theorem~\ref{ThmCharCompPrices}.
Then the mapping $\pi=\ps{\psi}{\cdot}$ is a compatible price.
\end{lemma}

\begin{proof}
Let $M \geq 0$ be an upper bound for $\psi$, i.e., $M \geq \sup_t \psi_t $ almost surely.
As processes $z \in E$ have integrable variation, $\ps{\psi}{\cdot}$ is well defined on $E$:
\[
\forall z\in E, \quad \left|\mathbb{E} \int \psi dz \right| \leq  M \normstrong{z} < \infty.
\]
Since $\ps{\psi}{\cdot}$ is obviously linear and nonnegative, it belongs to $E^\star_+$.

Let $\xi$ be a continuous process which optional projection $^o\xi$ coincides with $\psi$.
We first establish continuity of the functional $\ps{\psi}{\cdot}$ on the space
\[
E^k_+:=\left\{x \in E_+ : x_T \leq k \quad \mathrm{a.s.} \right\}
\]
for arbitrary $k>0$.
Let  $(x^n) \subset E^k_+$ be a sequence converging to some $x \in E^k_+$ for the intertemporal norm.
As we have
\[
\mathbb{E} \int \psi dz = \mathbb{E} \int \xi dz
\]
for all $z \in E$, it is enough to prove that
\[
\lim_n \mathbb{E} \int \xi dx^n = \mathbb{E} \int \xi dx.
\]
Suppose this is not true.
Then there is a subsequence $(y^n)$ of $(x^n)$ such that
\[
\lim_{n} d^n:= \mathbb{E} \int \xi dy^n = d \not= c :=\mathbb{E} \int \xi dx.
\]
Without loss of generality, we can assume that on a set of probability $1$, the sequence $(y^n)$ converges weakly in the sense of measures on the time axis to $x$ (see Lemma~1 in ~\cite{Martinsda} or \cite[Proposition~5]{HindyHuangKreps92}).
Then we have $\lim_n \int \xi dy^n =  \int \xi dx$ almost surely because $\xi$ is continuous.
From
\[
\left|\int \xi dy_n\right| \leq k \sup_{t \in[0,T]} |\xi_t|
\]
and the assumption on $\xi$, we get by dominated convergence that $\lim_n d^n =c$: a contradiction.

Now let $(x^n)$ be an arbitrary sequence in $E_+$ that converges to $x$.
For each $k\in \mathbb{N}$, we let $x^n_k$ and $x_k$ the optional random measures defined by
\[
dx^n_k=dx^n\wedge [\delta_0 k] \quad \mathrm{and} \quad dx_k = dx \wedge [\delta_0 k].
\]
Observe that for every $t\in [0,T]$ we have $x^n_k(t)=\min\{x^n(t),k\}$ and $x_k(t)=\min\{C(t),k\}$.
It follows immediately from dominated convergence that
\begin{equation}\label{Eqn2}
\lim_{k} \normstrong{x_k-x} = 0
\end{equation}
and
\begin{equation}\label{Eqn3}
\forall k\in \mathbb N, \quad\lim_{n} \norm{x^{n}_k-x_k} = 0.
\end{equation}
Observe that for every $(k,n)$ we have
\begin{equation}\label{eq:nk-n}
\left\vert \ps{\psi}{x^{n}_k - x^n} \right \vert \leq M \normstrong{x^{n}_k - x^n} = M \mathbb{E}[x^n - k]^+ \leq M \norm{x^n - x} + M \mathbb{E}[x_k - k]^+.
\end{equation}
For $\epsilon>0$, relation \rref{Eqn2} allow us to find $k_0$ such that
\begin{equation}\label{eq:c-ck}
\left| \ps{\psi}{x - x_k} \right| \leq M \normstrong{x - x_k} < \epsilon \quad \mathrm{and} \quad M \mathbb{E}[x_k - k]^+ \leq \varepsilon.
\end{equation}
Now fix $k=k_0$, it follows from \rref{eq:nk-n} and \rref{eq:c-ck} that for all $n \in \mathbb{N}$
\begin{eqnarray}
  \left| \ps{\psi}{x-x^n} \right| &\leq & \left| \ps{\psi}{x-x_k}\right| + \left| \ps{\psi}{x_k-x^{n}_k} \right| +  \left| \ps{\psi}{x^{n}_k-x^{n}} \right|\\
 &\leq &  2 \epsilon +\left| \ps{\psi}{x_{k}-x^{n}_k}\right| + M \norm{x^n - x}.
\end{eqnarray}
By the fact that $\pi$ is continuous on $E^k_+$ and \rref{Eqn3}, we can choose $n_0$ such that for all $n \geq n_0$
\[
\left| \ps{\psi}{x_{k}-x^{n}_k} \right| < \varepsilon \quad \mathrm{and} \quad M \norm{x^n - x} \leq \varepsilon
\]
and we finally obtain
\[
\left| \ps{\psi}{x-x^n} \right|  < 4 \epsilon
\]
for $n \geq n_0$.
This shows that $\pi$ is continuous on $E_+$.
\end{proof}

\subsection{Necessity}

The converse is the much more demanding part.
Given a compatible price $\pi \in H_+$, we have to find a \emph{density} $\psi $ that represents $\pi$.
We will frequently use the following continuity lemma that yields suitable upper bounds.

\begin{lemma}\label{LemStrongCont}
Compatible prices are continuous with respect to the strong topology.
\end{lemma}

\begin{proof}
As the strong topology is stronger than the intertemporal topology, a compatible price $\pi$ is $\normstrong{\cdot}$--continuous on $E_+$.
But the space $E$ is a topological vector lattice with respect to the strong topology.
Hence, the lattice operations are continuous with respect to the strong topology.
It follows that $\pi$ is $\normstrong{\cdot}$--continuous on the whole space $E$.
\end{proof}

As the space $\left(E, \normstrong{\cdot}\right)$ is a Banach space, the preceding lemma yields a constant $K>0$ such that
\begin{equation}\label{EqnBoundedPi}
\forall z\in E, \quad \left| \pi(z) \right| \leq K \normstrong{z}.
\end{equation}

Denote by $\mathcal{T}$ the set of all stopping times $\tau \leq T$.
A $\mathcal{T}$--system is a family $(z^\tau)_{\tau \in \mathcal{T}}$ of random variables that
satisfy (see \cite{DellacherieLenglart80})
\begin{enumerate}
  \item consistency: for $\sigma,\tau$ stopping times  $z^\tau=z^\sigma$ on the set $\{\tau=\sigma\}$,
  \item measurability: every random variable $z^\tau$ is $\cal F_\tau$--measurable.
\end{enumerate}
Fix a random variable $\tau \in \mathcal{T}$.
Define a linear mapping $Q^\tau$ from $L^1(\mathcal{F}_\tau,\P)$ into $\mathbb R$ by setting
\[
\forall Z \in L^1(\mathcal{F}_\tau,\P), \quad Q^\tau(Z)=\pi\left(\delta_\tau Z\right).
\]
Being a continuous linear mapping, it can be represented by a random variable $z^\tau \in L^\infty(\mathcal{F}_\tau,\P)$ such that
\[
\forall Z \in L^1(\mathcal{F}_\tau,\P), \quad Q^\tau(Z)=\mathbb{E}(Z  z^\tau).
\]
As $\pi$ is nonnegative we actually have $z^\tau \geq 0$ a.s.

\begin{claim}
The family $(z^\tau)_{\tau \in \mathcal{T}}$ forms a $\mathcal{T}$--system.
\end{claim}

\begin{proof}
It is sufficient to show that $z^\sigma \mathbf{1}_{\{\sigma=\tau\}}=z^\tau \mathbf{1}_{\{\sigma=\tau\}}$ almost surely.
As both $z^\sigma \mathbf{1}_{\{\sigma=\tau\}}$ and $z^\tau \mathbf{1}_{\{\sigma=\tau\}}$ are $\mathcal{F}_{\sigma \wedge \tau}$--measurable,  it is enough to show
\[
\forall Z \in L^1(\mathcal{F}_{\sigma \wedge \tau},\P),\quad \mathbb{E} z^\sigma \mathbf{1}_{\{\sigma=\tau\}} Z = \mathbb{E} z^\tau \mathbf{1}_{\{\sigma=\tau\}} Z.
\]
Take such a $Z$ in $L^1(\mathcal{F}_{\sigma \wedge \tau},\P)$.
Then
\begin{align*}
\mathbb{E}  z^\sigma  \mathbf{1}_{\{\sigma=\tau\}} Z
&= Q^\sigma\left( \mathbf{1}_{\{\sigma=\tau\} } Z\right)\\
&= \pi\left(\delta_\sigma \mathbf{1}_{\{\sigma=\tau\} } Z\right)\\
&= \pi\left(\delta_\tau \mathbf{1}_{\{\sigma=\tau\} }Z\right)\\
&= Q^\tau\left( \mathbf{1}_{\{\sigma=\tau\} } Z\right)\\
&= \mathbb{E}  z^\tau  \mathbf{1}_{\{\sigma=\tau\}} Z\,.
\end{align*}
This concludes the proof.
\end{proof}

The question is: can we find a process $(\psi_t)_{t\in [0,T]}$ such that $\psi_\tau = z^\tau$ almost surely for all stopping times $\tau \in \mathcal{T}$?
Such a question is called a \emph{problem of aggregation} in the th\'eorie g\'en\'erale of stochastic processes.
In general, aggregation is not possible without some continuity requirement (see \cite{DellacherieLenglart80}).
Therefore, we establish the following lemma.

\begin{claim}\label{LemContExp}
The $\mathcal{T}$--system $(z^\tau)_{\tau \in \mathcal{T}}$ is continuous in expectation in the sense that
\[
\lim_n \mathbb{E} z^{\tau_n}= \mathbb{E} z^\tau
\]
for all sequences of stopping times $\left(\tau_n\right)$ with $\lim_n \tau_n = \tau$.
\end{claim}

\begin{proof}
Let $(\tau_n)$ be a sequence of stopping times satisfying $\lim \tau_n = \tau$.
Then, the sequence of optional random measures $(\delta_{\tau_n})$ converge to $\delta_\tau$ in the intertemporal topology.
Continuity of the price functional $\pi$ implies
\begin{equation*}
\lim_n  \mathbb{E} z^{\tau_n} = \lim_n \pi\left(\delta_{\tau_n}\right) = \pi\left(\delta_{\tau}\right)= \mathbb{E} z^{\tau}.
\end{equation*}
\end{proof}

\begin{claim}\label{claim:uniformly integrable}
There exists a nonnegative, adapted and cadlag process $\psi \in L^\infty(\P,\mathcal{B})$ that aggregates (or recollects) $(z^\tau)_{\tau \in \mathcal{T}}$ in the sense that $\psi_\tau=z^\tau$ almost surely for every stopping time $\tau \in \mathcal{T}$.
\end{claim}

\begin{proof}
By \cite[Theorem~6]{DellacherieLenglart80}, every nonnegative $\mathcal{T}$--system can be aggregated by an optional process $\psi$.
The process $\psi$ is nonnegative because so are every $z^\tau$.
From Claim~\ref{LemContExp} the process $\psi$ is continuous in expectation.
If we prove that $\psi$ is uniformly integrable, then we can apply \cite[Theorem 48]{DM78}\footnote{The theorem is formulated for bounded processes only. The comment~50(f) in \cite{DM78} shows that it is enough to have uniform integrability.} to conclude that  $\psi$ is cadlag.
We first prove that $\psi$ is bounded in $L^1$.
Let $\tau$ be a stopping time and observe that
\[
0 \leq \mathbb{E} \psi_\tau = \mathbb{E} z^\tau = \pi(\delta_\tau).
\]
The process $\delta_\tau$ belongs to $E$ and \rref{EqnBoundedPi} yields $\pi(\delta_\tau) \leq K$ implying that $\mathbb{E} \psi_\tau < \infty$.
To establish uniform integrability, we have to show that for all $\varepsilon >0$ there exists $\delta>0$ such that for all sets $A \in \mathcal{F}$ with $\P(A) \leq \delta$ one has $\mathbb{E}\mathbf{1}_{A} \psi_\tau \leq \varepsilon$ for every stopping time $\tau$.
For $A \in \mathcal{F}$ and a stopping time $\tau$, let $c=\delta_\tau \mathbb{E}[\mathbf{1}_A \vert \mathcal{F}_\tau]$.
Since the process $c$ belongs to $E$ we have
\[
\mathbb{E} \mathbf{1}_A \psi_\tau = \mathbb{E} ( \mathbb{E}[\mathbf{1}_A\vert \mathcal{F}_\tau] \psi_\tau) = \pi(c) \leq K \normtv{c} = K \P(A).
\]
Setting $\delta=\varepsilon/K$, we obtain uniform integrability.
\end{proof}

\begin{claim}\label{claim:bounded}
The process $\psi$ is bounded.
\end{claim}

\begin{proof}
Fix $\alpha>0$ and let $\tau$ be the stopping time
\[
\tau =
\left\{
\begin{array}{ll}
\inf\left\{t \ge 0 : \psi_t \ge n\right\} & \textrm{if} \ \sup_{0 \le t \le T} \, \psi_t \geq \alpha \\
~\\
\infty & \textrm{elsewhere}
\end{array}
\right.
\]
Recall that the random variable $\psi_\tau$ is given  by $\psi_\tau(\omega) = \psi_{\tau(\omega)}(\omega) \mathbf{1}_{\{ \tau< \infty\}}$.
Consider the optional random measure $c=\delta_{\tau} \mathbf{1}_{\{ \tau < \infty\}}$. We have
\[
\pi(c)=\mathbb{E} z^{\tau} \mathbf{1}_{\{ \tau < \infty\}} =\mathbb{E} \psi_{\tau} \mathbf{1}_{\{\tau < \infty \}}.
\]
Since $\psi$ is cadlag, we get $\pi(c) = \mathbb{E} \psi_{\tau} \mathbf{1}_{\{\tau < \infty \}} \geq \alpha \P \{\tau < \infty\}.$
On the other hand, \rref{EqnBoundedPi} yields
\[
\pi(c) \leq K \normstrong{c} = K \mathbb{E} \normtv{\delta_{\tau} \mathbf{1}_{\{ \tau < \infty\}} } = K  \P \{ \tau < \infty \}.
\]
Choosing $\alpha >K$ then shows that $\P \{\tau < \infty\} =0$.
Hence the process $\psi$ is bounded, i.e., $\psi \in L^\infty(\P,B(T))$.
\end{proof}

In general, $\psi$ is not going to be continuous. However, we have the following result that goes back to \cite{Bismut78} and \cite{Emery78}.

\begin{claim}
There exists a \emph{not necessarily adapted} continuous process $\xi $ with
\[
\mathbb{E} \sup_{t \in [0,T]} | \xi_t | < \infty
\]
whose optional projection is $\psi$, that is
\[
\psi_\tau = \condexp[\mathcal{F}_\tau]{\xi_\tau}
\]
for all stopping times $\tau \in \mathcal{T}$.
\end{claim}

\begin{proof}
This is the main theorem in \cite{Bismut78} and \cite{Emery78}.
According to the notations in \cite{Bismut78} and \cite{Emery78}, we have to check the conditions that $\psi$ is regular and of class (D).
As $\psi$ is bounded, it is of class (D).
A process is regular if and only if the predictable projection of $\psi$ is equal to $\psi_-$.
This is equivalent to continuity in expectation from below (see \cite[50(d)]{DM78}).
As $\psi$ is even continuous in expectation, it is regular.
\end{proof}

\begin{claim}\label{LemRepLinfty}
For every bounded consumption plan $x\in E_+$ with $x_T \in L^\infty(\P)$ we have $\pi(x) = \ps{\psi}{x}$, i.e., \[
\pi(x) = \mathbb{E} \int \psi dx.
\]
\end{claim}

\begin{proof}
By construction, we have for every stopping time $\tau$  and $\mathcal{F}_\tau$--measurable random variable~$h$
\[
\pi(\delta_\tau h)=\mathbb{E} z^\tau h = \mathbb{E} \psi_\tau h = \mathbb{E} \xi_\tau h.
\]
Via linearity, we obtain $\pi(z)=\ps{\psi}{z}$ for every simple random measure $z$.
As simple random  measures are dense with respect to the intertemporal norm in $E_+$ and $\xi$ is continuous, we get the result for optional random measures with bounded variation in $L^\infty(\P)$ (observe that \cite{Bismut78} obtains this in his proof).
\end{proof}

Since $\psi$ belongs to $L^\infty(\P,B(T))$, the random variable $\psi^\star$ defined by
\[
\psi^\star = \sup_{t\in [0,T]} \psi_t
\]
belongs to $L^\infty(\P)$.
It follows that for every consumption plan $x\in E_+$ the quantity $\ps{\psi}{x}$ is well defined as
\[
\ps{\psi}{x} = \mathbb{E} \int \psi dx \leq \mathbb{E} \psi^* x_T < \infty.
\]
We can now prove that $\pi(x)=\ps{\psi}{x}$ for every $x \in E_+$.
From Claim \ref{LemRepLinfty}, we know that $\pi(x)=\ps{\psi}{x}$ for all bounded $x$ in $L^\infty(\P,B(T))$.
Now let $x \in E_+$ be given.
Set $dx_n=dx \wedge n\delta_0$, i.e., $x_n(t) = \min\{x_t,n\}$ for every $t\in [0,T]$ .
For each $n$ the optional random measure $x_n$ is bounded and the sequence $(x_n(\omega))$ converges for the total variation norm $\normtv{.}$ to $x(\omega)$ from below for all $\omega$.
Consequently, for all nonnegative measurable functions $f$ we have
\[
\lim_n \int f dx_n =\int f dx.
\]
In particular, we have $\lim_n \int \psi dx_n = \int \psi dx$ almost surely.
By monotone convergence, we obtain
\[
\lim_n \pi(x_n)=\lim_n \mathbb{E}\int \psi dx_n = \mathbb{E}\int \psi dx =\ps{\psi}{x}
\]
and as $\pi$ is continuous with respect to the strong topology, $\pi(x)=\ps{\psi}{x}$ follows.
This concludes the proof of the theorem.

\subsection{The proof for $p>1$}

For $p>1$, the proof follows almost verbatim the above proof for $p=1$.
However, one cannot use the argument given above that establishes boundedness of the process $\psi$.
Instead, one has to use a different argument to prove that the supremum of $\psi$ is in $L^q$.
This argument is given next.

\begin{claim}
The  supremum
\[
\psi^*:=\sup_{t\in [0,T]} \psi_t
\]
satisfies
\[
\forall H\in L^p(\P), \quad \mathbb{E} \psi^* |H| \leq (K+1) \norm{H}_{L^p}.
\]
In particular the random variable $\psi^*$ belongs to $L^q$.
\end{claim}

\begin{proof}
Let $S$ be a random time (not necessarily a stopping time) and $h \in L^p_+(\P)$.
Denote by $z=\delta_S h$ and by $x=\left(z\right)^o$ its optional dual projection.
Then $x$ is an optional random measure and
\[
\mathbb{E} \psi_S h  = \mathbb{E} \int \psi dz = \mathbb{E} \int \psi dx = \pi(x)\leq K \normstrong{x}.
\]
The process $x$ is nondecreasing and $\mathcal{F}=\mathcal{F}_T$, hence
\[
\normstrong{x} = \mathbb{E} \normtv{x} = \mathbb{E} x_T = \mathbb{E} h \leq \norm{h}_{L^p}.
\]
Let $S$ be a cross--section of the set
\[
\left\{(\omega,t): \psi_t(\omega)\geq \psi^\star(\omega) - 1 \right\}.
\]
Then we have
\[
\mathbb{E} \psi^* h  \leq \mathbb{E} (\psi_S + 1) h \leq (K+1) \norm{h}_{L^p}.
\]
\end{proof}

\section{Equilibria with Compatible Prices}\label{section:equilibrium}

Each agent $i$ is characterized by a utility function $V^i : E_+ \longrightarrow \mathbb{R}$ which represents his preference relation on the space $E_+$ of consumption patterns and by a vector $e^i \in E_+$ which represents the cumulative income stream (initial endowment).
An economy is a pair
\[
\mathcal{E}=(\bm{V},\bm{e})
\]
where $\bm{V}=(V^i)_{i\in I}$ and $\bm{e}=(e^i)_{i\in I}$.
We let $e=\sum_{i\in I} e^i$ denote the aggregate endowment and if $x \in E_+$ the set $\{ y\in E_+ \ \colon \ V^i(y) > V^i(x) \}$ is denoted by $P^i(x)$.
An allocation is a vector $\bm{x}=(x^i)_{i\in I}$ where $x^i\in E_+$.
It is said feasible or attainable if $\sum_{i\in I} x^i = e$.
The set of attainable allocations is denoted by~$\mathcal{A}$.

\subsection{Equilibrium concepts}

We define hereafter the standard notion of Arrow--Debreu equilibrium.

\begin{definition}
The pair $(\psi,\bm{x})$ of a price process $\psi$ and an allocation $\bm{x}$ is called an Arrow--Debreu equilibrium if
\begin{description}
\item[(a)] the price process $\psi$ belongs to $F_+$ and $\ps{\psi}{e} >0$;
\item[(b)] the allocation $\bm{x}$ is attainable, i.e., $\bm{x}\in \mathcal{A}$; and
\item[(c)] for each agent $i$, the consumption plan $x^i$ maximizes agent i's utility over all consumption plans $y$ satisfying the budget constraint $\ps{\psi}{y} \leq \ps{\psi}{e^i}$, i.e., \[
x^i \in \mathrm{argmax} \{ V^i(y) \ \colon \ y \in E_+ \quad \textrm{and} \quad \ps{\psi}{y} \leq \ps{\psi}{e^i} \}.
\]
\end{description}
\end{definition}

A possible interpretation is that a complete set of markets open at the initial date $t=0$ for consumption good delivery at any date in any state of nature.
Markets are assumed to be competitive in the sense that agents take the price functional $\ps{\psi}{\cdot}$ as given.
Each agent can sell his initial endowment $e^i$ and buy a consumption plan $x \in E_+$ as far as he can afford it, i.e., $\ps{\psi}{x} \leq \ps{\psi}{e^i}$.
The real number $\ps{\psi}{x}$ is interpreted as the price at time $t=0$ of the consumption claim $x$, and therefore the real number $\psi(\omega,t)$ is interpreted as the time $t=0$ price (per unit of probability) of the contract that promises to deliver one unit of the unique good at time $t$ in state $\omega$.

\begin{remark}
Observe that if $(\psi,\bm{x})$ is an equilibrium then the budget constraints are binding, i.e., for each $i$, we have $\ps{\psi}{x^i}=\ps{\psi}{e^i}$.
\end{remark}

As usual in general equilibrium literature, we consider the following list of \emph{standard} assumptions.
\begin{assumption}[C]\label{hyp:C}
For each agent $i$,
\begin{description}
\item[(C.1)] the initial endowment $e^i$ belongs to $E_+$ and is not zero, i.e., $e^i>0$,
\item[(C.2)] the utility function $V^i$ is concave,
\item[(C.3)] the utility function $V^i$ is norm continuous.\footnote{Actually, it is sufficient to assume that $V^i$ is upper semi-continuous on the order interval $[0,e]$. That is, if $(x_n)_{n\in \mathbb{N}}$ is a sequence in $[0,e]$ which norm-converges to $x$ in $[0,e]$, then
\[
\limsup_{n\rightarrow \infty} V^i(x_n) \leq V^i(x).
\]}
\end{description}
\end{assumption}

We recall a well-known property of optimality for allocations.

\begin{definition}\label{def:core}
An attainable allocation $\bm{x} \in \mathcal{A}$ is said to be an {\it Edgeworth} equilibrium if there is no $0\neq \bm{\lambda} \in (\mathbb{Q}\cap [0,1])^I$ and some allocation $\bm{y}$ such that $V^i(y^i) > V^i(x^i)$ for each $i$ with $\lambda^i>0$ and satisfying $\sum_{i\in I} \lambda^i y^i = \sum_{i\in I} \lambda^i e^i$.
\end{definition}

The reader should observe that this concept is ``price free" in the sense that it is an intrinsic property of the commodity space.
It is proved in \cite{Martinsda} that every economy satisfying Assumption~C admits an Edgeworth equilibrium.
It is straightforward to check that every Arrow--Debreu equilibrium is an Edgeworth equilibrium.
The main difficulty consists in proving the converse.

\subsection{Properness of preferences}

We propose to follow the classical literature\footnote{We refer, among others, to \cite{MAS}, \cite{RZ}, \cite{YZ}, \cite{ABB87a}, \cite{ABB87b}, \cite{Zame87}, \cite{RIC}, \cite{AM}, \cite{MR},  \cite{MasColellZame91}, \cite{POD}, \cite{AndersonZame97}, \cite{AndersonZame98}, \cite{TOU98}, \cite{DF}, \cite{TOU99}, \cite{ATY01}, \cite{ShannonZame}, \cite{FLO03}, \cite{AliprantisMonteiroTourky}, \cite{AFT04} and \cite{AFT05}.} dealing with infinite dimensional commodity-price spaces by introducing the concept of proper economies.
It is a well-known fact that without some properness hypotheses on preferences, equilibrium existence may fail when the positive cone of the commodity space has empty interior.

\begin{definition}[$\tau$-properness]\label{def:tau-proper}
Let $\tau$ be a Hausdorff locally convex linear topology on $E$.
An economy $(\bm{V},\bm{e})$ is $\tau$-proper if for every Edgeworth equilibrium $\bm{x}$, for each $i$, there is a set $\widehat{P}^i(x^i)$ such that
\begin{description}
\item[(i)] the vector $x^i+e$ is a $\tau$-interior point of $\widehat{P}^i(x^i)$,
\item[(ii)] the set $\widehat{P}^i(x^i)$ is convex and satisfies the following additional convexity property
\[
\forall z\in \widehat{P}^i(x^i) \cap E_+, \quad \forall t\in (0,1), \quad tz +(1-t)x^i \in \widehat{P}^i(x^i) \cap E_+
\]
\item[(iii)] we can extend preferences in the following way
\[
\widehat{P}^i(x^i)  \cap E_+ \cap A_{x^i} \subset P^i(x^i) \subset \widehat{P}^i(x^i) \cap E_+
\]
where $A_{x^i} \subset E$ is a radial set at $x^i$.\footnote{A subset $A$ of $E$ is radial at $x\in A$ if for each $y\in E$, there exists $\bar{\alpha} \in (0,1]$ such that $(1-\alpha) x + \alpha y$ belongs to $A$ for every $\alpha \in [0,\bar{\alpha}]$.}
\end{description}
\end{definition}
We say that an economy is {\it strongly} $\tau$-proper if condition (iii) in Definition~\ref{def:tau-proper} is replaced by the following condition~(iii'):
\begin{description}
\item[(iii')] we can extend preferences in the following way
\[
\widehat{P}^i(x^i)  \cap E_+ = P^i(x^i).
\]
\end{description}
Strong $\tau$-properness was introduced by \cite{TOU99} and is used, among others, by \cite{ATY01}, \cite{AFT04} and \cite{AFT05}.
We refer to \cite{ATY} for a comparison of the different notions of properness used in the literature.
Observe that if $\mathcal{E}=(\bm{V},\bm{e})$ is an economy (satisfying the following monotonicity Assumption~M) such that for each $i$, it is possible to extend $V^i$ to a $\tau$-continuous and concave function $\widehat{V}^i : E \longrightarrow \mathbb{R}$, then the economy is $\tau$-proper.\footnote{Take $\widehat{P}^i(x) := \{ y\in E \ \colon \ \widehat{V}^i(y) > \widehat{V}^i(x) \}$.}
In other words, $\tau$-properness can be seen as a strengthening of $\tau$-continuity.
Moreover, $\tau$-properness is slightly weaker than strong $\tau$-properness.
However this slight difference is crucial in order to compare properness with the existence of smooth sub-gradients.\footnote{See also Assumption~A.7 in \cite{ShannonZame}.}
We borrow the following definition of smooth sub-gradients from \cite{BankRiedel02} (see also \cite{Martinsda}).
Recall that $K$ is the space of processes in $F$ that represent linear functionals on $E$ that are norm continuous.

\begin{definition}\label{def:U}
An economy $(\bm{V},\bm{e})$ has smooth sub-gradients in $K$ if for each $i$, for every $x \in E_+$, there exists a nonnegative optional process $\nabla V^i(x) \in K_{+}=K\cap F_+$ with
\begin{description}
\item[(U.1)] for each $j\in I$, we have $\ps{\nabla V^i(x)}{e^j} >0$,
\item[(U.2)] the vector $\nabla V^i(x)$ satisfies the subgradient property
\[
\forall y \in E_+, \quad V^i(y) - V^i(x) \leq \ps{\nabla V^i(x)}{y- x}
\]
\item[(U.3)] this subgradient is continuous in the sense that,
\[
\forall y \in E_+, \quad  \lim_{\varepsilon \downarrow 0} \ps{\nabla V^i(\varepsilon y + (1-\varepsilon) x)}{y -x} = \ps{\nabla V^i(x)}{y -x}.
\]
\end{description}
\end{definition}

\begin{remark}\label{rem:increasing}
Let $\mathcal{E}=(\bm{V},\bm{e})$ be an economy.
Preferences of agent $i$ are said increasing if $V^i(x + y) \geq V^i(x)$ for every $x$, $y$ in $E_+$; strictly increasing if $V^i(x + y) > V^i(x)$ for every $x$, $y$ in $E_+$ with $y\neq 0$.
Note that if $\mathcal{E}$ satisfies Assumption~U, then preferences of agent $i$ are increasing; they are strictly increasing if and only if $\nabla V^i(x)$ is strictly positive  for every $x \in E_+$.
\end{remark}

\begin{remark}\label{rem:desirable}
Let $(\bm{V},\bm{e})$ be an economy satisfying Assumption~U, then for each $i$, $j$ in $I$, the initial endowment $e^j$ is strongly desirable for agent $i$ in the sense that
\[
\forall x \in E_+, \quad \forall t>0, \quad V^i(x + te^j) > V^i(x).
\]
\end{remark}

\begin{remark}\label{rem:U1}
Assume that Assumption~U.2 is satisfied,
\begin{description}
\item[(a)] if preferences of agent $i$ are strictly increasing and $e^j>0$ for each $j\in J$, then Assumption~U.1 is satisfied,
\item[(b)] if preferences of agent $i$ are increasing and for each $j\in I$, there exists a strictly positive integrable adapted process $\xi^j$ such that $de^j(t)=\xi^j(t) dt$, then Assumption~U.1 is satisfied.
\end{description}
\end{remark}

In order to compare norm properness and the existence of smooth sub-gradients in $K$, we consider the following monotonicity assumption.
\begin{assumption}[M]
For every Edgeworth equilibrium $\bm{x}$, for each agent $i$, the following property is satisfied:
\[
\forall j\in I, \quad \forall t>0, \quad x^i+ t e^j + E_+ \subset P^i(x^i).
\]
\end{assumption}

\begin{remark}
From Remarks~\ref{rem:increasing}-\ref{rem:desirable}, Assumptions~C and Conditions~U.1 and U.2 imply Assumption~M.
\end{remark}

It is proved in \cite{Martinsda} that under Assumption~C.2, the existence of smooth sub-gradients in $K$ implies that the economy is weakly proper,\footnote{I.e., proper for the weak topology $\sigma(E,K)$.} in particular it is norm proper.

It was left as open question in \cite{HindyHuang92} whether a norm proper economy admits a continuous equilibrium price.
The existence results available in the literature are not general enough to be applied directly to our framework.
The topology derived from the intertemporal norm does not give rise to the mathematical properties known to be sufficient for the existence of an Arrow--Debreu equilibrium.\footnote{The topological dual space $(E,\norm{\cdot})'$ endowed with dual order defined by the cone $E^\star_+$ is not a vector lattice.}
The main contribution of this section is to prove that if an economy is proper with respect to the intertemporal norm it admits a compatible equilibrium, i.e., a price functional that is continuous on the positive cone $E_+$.

\begin{theorem}\label{th:ExistenceComp}
Under Assumptions~C and~M, if an economy is norm proper then it admits a compatible price.
\end{theorem}

\begin{proof}
Consider an economy satisfying Assumptions~C and~M and assume that it is norm proper.
From the norm properness of utility functions, there exists a family $(\psi^i)_{i\in I}$ where $\psi^i$ belongs to $K$ and supports agent $i$'s preferences.
In order to apply Proposition~2 and Theorem~2 in \cite{Martinsda}, it is sufficient to prove that the maximum of two processes in $K$ is a process in $H$.
Actually this is a consequence of the fact that $H$ is stable by taking the max.
Indeed, let $\phi$ and $\psi$ be two processes in $H$, i.e., $\phi$ and $\psi$ are nonnegative, rightcontinuous with left limits, bounded in $L^q$,  and the projection of a raw continuous process bounded in $L^1$.
We denote by $\theta$ the process defined by $\theta_t=\max\{\phi_t,\psi_t\}$.
We have to show that $\theta$ belongs to $H_+$.
It is nonnegative, rightcontinuous with left limits and bounded in $L^q$.
It remains to show that $\theta$ is the optional projection of a raw continuous process in $L^1$.
For this we can again check the conditions of the main result in \cite{Bismut78}.
To this end we have to show that $\theta$ is of class (D) and continuous in expectations.
As $\theta$ is bounded in $L^q$, it is of class (D).
Continuity in expectation is preserved by taking the max, and the proof is done.
\end{proof}

\begin{remark}
Actually, Theorem~\ref{th:ExistenceComp} is still valid if the norm properness of each utility function $V^i$ is replaced by the $\tau$-properness for any linear topology $\tau$ on $E$ such that any  linear functional $\tau$-continuous on $E$ is represented by a vector in $H$.
\end{remark}

\begin{remark}
Observe that contrary to \cite{BankRiedel02} and \cite{Martinsda}, we don't need to assume that the filtration $\mathbb{F}$ is quasi left-continuous.
This is an assumption on the way new information is revealed to the agents.
Economically, an information flow corresponds to a quasi left-continuous filtration\footnote{See \cite{HindyHuang92} for a precise definition. An information flow generated by a Brownian motion or a Poisson process is quasi left-continuous.} if information surprises (in the sense of \cite{HindyHuang92}) occur only at times which cannot be predicted.
The announcement of a policy change of the Federal reserve is an example for an information surprise which occurs at a time known in advance.
\end{remark}

\subsection{Example}

We consider {\it Hindy--Huang--Kreps preferences}, i.e. preferences given by utility functionals of the form
\[
V^i(x) = \mathbb{E} \int_{[0,T]} u^i(t,Y(x)(t)) \kappa(dt)
\]
where $u^i : [0,T] \times \mathbb{R}_+ \rightarrow \mathbb{R}$ denotes a felicity function for agent $i$, and the quantity
\[
Y(x)(t) =  \int_{[0,t]} \beta e^{-\beta(t-s)} dx(s)
\]
describes the investor's level of satisfaction obtained from his consumption up to time $t\in [0,T]$.
The constant $\beta>0$ measures how fast satisfaction decays.

We consider the linear mapping $\phi : E \rightarrow E$ defined by
\[
\forall t\in [0,T], \quad \phi(x)(t) = \int_{[0,t]} \exp\{\beta s\} dx(s).
\]
For each $x\in E$, the vector $\phi(x)$ is defined by the optional random measure $d[\phi(x)](t) = \exp\{\beta t\} dx(t)$.
The linear mapping $\phi$ is bijective and the inverse mapping $\phi^{-1}$ is given by
\[
\forall t\in [0,T], \quad \phi^{-1}(x)(t) = \int_{[0,t]} \exp\{-\beta s\} dx(s).
\]
We introduce on $E$ the following norm $\rho$:
\[
\forall x\in E, \quad \rho(x) := \norm{\phi(x)} = \mathbb{E} \int_{[0,T]} |\phi(x)(t)| \kappa(dt).
\]

It is proved in \cite[Lemma~2]{Martinsda} that the norm-topology and the $\rho$-topology coincide on $E_+$, and that the $\rho$-topological dual $(E,\rho)'$ coincides with the norm-topological dual $(E,\norm{.})'$.
In order to apply Theorem~\ref{th:ExistenceComp} it is sufficient to prove that $V^i$ is $\rho$-proper.
From \cite[Theorem]{Martinsda} this is a consequence of the following conditions: for each $i\in I$,
\begin{description}
\item[(V.1)] for each $t\in [0,T]$, the function $u^i(t,.) : \mathbb{R}_+ \rightarrow \mathbb{R}$ is continuous, strictly increasing and concave,
\item[(V.2)] for each $y\in \mathbb{R}_+$, the function $u^i(.,y) : [0,T] \rightarrow \mathbb{R}$ is $\mathcal{B}$-measurable and the function $u^i(.,0)$ belongs to $L^1(\mathcal{B},\kappa)$,
\item[(V.3)] for each $t\in [0,T]$ the right-derivative $\partial_y u^i(t,0+)$ exists and the function $\partial_y u^i(.,0+)$ belongs to $L^\infty_+(\mathcal{B},\kappa)$.
\end{description}

\section{Conclusion}

We show how the economically sensible intertemporal topology introduced by \cite{HindyHuang92} allows to derive general structural results about equilibrium state prices.
Using the  \emph{th\'{e}orie g\'{e}n\'{e}rale} of stochastic processes, we show that price functionals that are continuous on the consumption set can be represented by state prices with right-continuous sample paths that admit left limits.
Moreover, the state price is the optional projection of a process with continuous sample paths that is not necessarily adapted.

\bibliographystyle{kluwer}

redpajama_set_name: RedPajamaArXiv


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