Lucy Jo's Organic Mellow Belly Low Acid Blend is a favorite among our customers looking for a low acid coffee that is easy on the stomach. We have many customers who say, "Thanks! I can drink coffee again!". Lucy Jo's Coffee Roastery is a family-run Certified Organic coffee roastery based in rural Upstate New York. We specialize in roasting high quality organic coffees. Each ingredient coffee in our blends can solidly stand as a single origin. No filler beans! We source the best quality 100% arabica organic green coffee to meet the health concerns of our customers searching for organic vs conventionally grown coffee. Our hands-on approach to small batch coffee roasting allows us to deliver the freshest roasted coffee possible with optimum flavor. We have earned a strong and growing following from our local and regional community and are excited to bring our local favorites to a larger audience. Thank you for drinking Lucy Jo's Coffee!
Froggys Freezin Fog is a water based fog machine fluid formulated to create optimum low lying fog effects for outdoor graveyard scenes. It is super thick on the front end and creates the perfect graveyard ground fog but here comes the good part; it is formulated to hang in the air to create the spooky graveyard scene as well. You get a 2 for 1 deal with the Freezin Fog that you miss out on if you use a Fast Dissipating Fluid outdoors. It is compatible with the American DJ Mister Kool Fog Machine. Freezin Fog, also known as, "Sta-Lo", "Kool Fog" or "Chiller Fog" is perfectly formulated for all water based fog machines. This fog juice was designed to be the best Halloween Fog for Outdoor Graveyard Scenes! Freezin Fog can also be used with Ice Chest Chillers, Trash Can Chillers, Kool Kore Chillers etc... Make sure to use Froggys Fast Dissipating Fog instead of Froggys Freezin Fog for Indoor Low Lying Theatrical Effects / Stage & Studio Fog that needs to dissipate before rising above the ground. You must use a Fog Chilling Device to create Low Lying Fog Effects! All Froggys Fog and Haze Fluids are made from lab-grade UV-Filtered de-ionized water and pharmaceutical grade chemicals, each approved by the FDA.
Lucy Jo's Organic Sumatra Low Acid is roasted dark to bring out the smoky earthiness Sumatran beans are known for. These naturally low acid beans are very easy drinking with good body and complexity not lost at the darker roast levels. Lucy Jo's Coffee Roastery is a family-run Certified Organic coffee roastery based in rural Upstate New York. We specialize in roasting high quality organic coffees. Each ingredient coffee in our blends can solidly stand as a single origin. No filler beans! We source the best quality 100% arabica organic green coffee to meet the health concerns of our customers searching for organic vs conventionally grown coffee. Our hands-on approach to small batch coffee roasting allows us to deliver the freshest roasted coffee possible with optimum flavor. We have earned a strong and growing following from our local and regional community and are excited to bring our local favorites to a larger audience. Thank you for drinking Lucy Jo's Coffee!
Perfect compact outdoor chair for camping trips, the beach, sporting events, outdoor concerts and more! The seat back is just 21 inches high, meeting requirements for most outdoor music festivals. This low profile chair can go anywhere, is lightweight, conveniently folds and can hold up to 250lbs. Cascade Mountain Tech offers a full one year warranty so you can purchase with confidence.
Looking for more Low Grounds similar ideas? Try to explore these searches: Firebird Long Sleeve, Short Dressing Gown, and Penn State Nittany Lions Car.
Look at latest related video about Low Grounds.
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text:
// source:

package nl.esciencecenter.ptk.vbrowser.viewers.internal;

import java.awt.BorderLayout;
import java.awt.Dimension;
import java.awt.Image;
import java.awt.Point;
import java.io.IOException;
import java.io.InputStream;
import java.util.List;
import java.util.Map;

import javax.imageio.ImageIO;
import javax.swing.JScrollPane;
import javax.swing.JViewport;
import javax.swing.SwingUtilities;

import nl.esciencecenter.ptk.data.HashMapList;
import nl.esciencecenter.ptk.data.StringList;
import nl.esciencecenter.ptk.ui.image.ImagePane;
import nl.esciencecenter.ptk.ui.image.ImagePane.ImageWaiter;
import nl.esciencecenter.ptk.vbrowser.viewers.ViewerJPanel;
import nl.esciencecenter.vbrowser.vrs.exceptions.VrsException;
import nl.esciencecenter.vbrowser.vrs.vrl.VRL;

/**
 * Implementation of an Image Viewer.<br>
 */
public class ImageViewer extends ViewerJPanel {
    private static final long serialVersionUID = 5768234709523116729L;

    /** The mimetypes I can view */
    private static String mimeTypes[] = { "image/gif", "image/jpeg", "image/bmp", "image/png" };

    private static double zoomOutFactors[] = { 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1 };

    private static double zoomInFactors[] = {
            // 100,125,150,200,300,400,500,600,800,1000%
            1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, 3, 4, 5, 6, 7, 8, 9, 10 };

    // ====================================================================
    //
    // ====================================================================
    ImagePane imagePane = null;

    // private ImageIcon icon=null;
    JScrollPane scrollPane;

    // private int offsetx;
    // private int offsety;

    int zoomIndex = 0;

    double currentZoomFactor = 1.0;

    boolean fitToScreen = false;

    private Image orgImage;

    // private JLabel imageLabel; // store image in Label Component

    @Override
    public String[] getMimeTypes() {
        return mimeTypes;
    }

    public void initGui() {
        {
            this.setSize(800, 600);

            BorderLayout thisLayout = new BorderLayout();
            this.setLayout(thisLayout);
            // this.setLayout(null); // absolute layout

            {
                this.scrollPane = new JScrollPane();
                this.scrollPane.setSize(800, 600);
                this.getContentPanel().add(scrollPane, BorderLayout.CENTER); // addToRootPane(imagePane,BorderLayout.CENTER);
                {
                    this.imagePane = new ImagePane();
                    scrollPane.setViewportView(imagePane);
                    this.imagePane.setSize(800, 600);
                    this.imagePane.setLocation(0, 0);
                }
            }

            // imagePane.setVisible(true);

            // this.imageLabel=new JLabel("ImageLabel:Loading image...");
            // this.add(this.imageLabel,BorderLayout.CENTER);
            // this.imageLabel.setLocation(0,0);
            this.setToolTipText(getViewerName());
        }

        // listeners:
        {
            new ImageViewerController(this);
        }
    }

    @Override
    public void doInitViewer() {
        initGui();
    }

    @Override
    public void doStopViewer() {
        // this.muststop=true;
        // this.imagePane.signalStop();
    }

    @Override
    public void doDisposeViewer() {
        // Help the garbage collector, images can be big:
        if (imagePane!=null){
            this.imagePane.dispose();
            this.remove(imagePane);
        }
        this.imagePane = null;
        // this.remove(imageLabel);
        // this.imageLabel=null;
    }

    @Override
    public String getViewerName() {
        // remove html color codes:
        return "ImageViewer";
    }

    @Override
    public void doStartViewer(VRL vrl, String optionalMethod) {
        doUpdate(vrl);
    }

    public void doUpdate(VRL vrl) {
        try {
            loadImage(vrl);
        } catch (Exception e) {
            notifyException("Failed to load image:" + vrl, e);
        }
    }

    public void loadImage(VRL vrl) throws Exception {
        if (vrl == null) {
            return;
        }

        notifyBusy(true);

        try {

            // load image and wait:
            // this.imagePane.loadImage(location,true);
            loadImage(vrl, false);

            // keep original image for zoom purposes;
            this.orgImage = imagePane.getImage();

            this.fitToScreen = false;

            this.zoomIndex = 0;
        } catch (Exception e) {
            throw new VrsException(e);
        } finally {
            notifyBusy(false);
        }
    }

    public void loadImage(VRL vrl, boolean wait) throws Exception {
        InputStream inps = getResourceHandler().openInputStream(vrl);

        Image image;
        image = ImageIO.read(inps);

        try {
            inps.close();
        } catch (Exception e) {
            ;
        }

        if (image == null) {
            throw new IOException("Failed to load image: Image loader returned NULL for:"
                    + vrl.toURI());
        }

        imagePane.setImage(image, wait);
    }

    /**
     * I manage my own scrollpane for panning/autoscrolling
     */
    public boolean haveOwnScrollPane() {
        return true;
    }

    /**
     * Get ScrollPane ViewPosition
     */
    public Point getViewPosition() {
        JViewport viewP = this.scrollPane.getViewport();
        return viewP.getViewPosition();
    }

    /**
     * Set ScrollPane ViewPosition
     */
    public void setViewPosition(int newx, int newy) {
        // /UIGlobal.debugPrintf(this,"moveViewPoint:"+newx+","+newy);

        JViewport viewP = this.scrollPane.getViewport();
        Dimension scrollPaneSize = scrollPane.getSize();

        // calculate maximum viewpiont size :
        Dimension maxP = scrollPane.getViewport().getView().getSize();
        maxP.width -= scrollPaneSize.width;
        maxP.height -= scrollPaneSize.height;

        if (newx > maxP.width)
            newx = maxP.width;

        if (newy > maxP.height)
            newy = maxP.height;

        viewP.setViewPosition(new Point(newx, newy));

        // this.offsetx=newx;
        // this.offsety=newy;
    }

    // ====
    // Zoom
    // ====

    private Object zoomTaskMutex = new Object();

    private Runnable zoomTask = null;

    private Thread zoomThread = null;

    public void zoomIn() {
        this.fitToScreen = false;
        if (zoomIndex < zoomInFactors.length) {
            zoomIndex++;
            doZoom();
        }
    }

    public void zoomOut() {
        this.fitToScreen = false;

        if (zoomIndex > -zoomOutFactors.length) {
            zoomIndex--;
            doZoom();
        }
    }

    private boolean doMoreZoom = false;

    /**
     * Perform Zoom: Schedule background task to do the zooming
     */
    protected void doZoom() {
        double zoomFactor = 1.0;

        final Image sourceImage = this.orgImage;
        if (sourceImage == null) {
            errorPrintf("doZoom(): NULL source image!\n");
            return;
        }

        int h = sourceImage.getHeight(null);
        int w = sourceImage.getWidth(null);

        if (fitToScreen == true) {
            double aspect = (double) w / (double) h;

            Dimension targetSize = this.scrollPane.getSize();
            double targetAspect = ((double) targetSize.width) / (double) targetSize.height;

            if (aspect > targetAspect) {
                // fit width
                w = targetSize.width;
                h = (int) (targetSize.width / aspect);
            } else {
                // fit height
                h = targetSize.height;
                w = (int) (targetSize.height * aspect);
            }
        } else {
            if (zoomIndex > 0)
                zoomFactor = zoomInFactors[zoomIndex - 1];

            if (zoomIndex < 0)
                zoomFactor = zoomOutFactors[-zoomIndex - 1];

            h = (int) (h * zoomFactor);
            w = (int) (w * zoomFactor);
        }

        final int newHeight = h;
        final int newWidth = w;

        Dimension currentSize = imagePane.getImageSize();

        // check bogus update ! (Current size already done)
        if ((currentSize.width == newWidth) && (currentSize.height == newHeight))
            return;

        synchronized (zoomTaskMutex) {
            if (zoomTask != null) {
                // check if previous zoom thread is stil active:
                if ((zoomThread != null) && (zoomThread.isAlive() == true)) {
                    // flag zoom not done !
                    this.doMoreZoom = true;
                    // IGlobal.infoPrintln(this,"Warning: Previous zoom task still running");
                    return;
                }

                // cleanup:

                this.zoomTask = null;
                this.zoomThread = null;
            }

            zoomTask = new Runnable() {
                public void run() {
                    if ((newHeight <= 0) || (newWidth <= 0)) {
                        // UIGlobal.infoPrintln(this,"*** Warning: zoomIn cancelled: image not ready");
                        return;
                    }

                    notifyBusy(true);

                    Image tempImage = sourceImage.getScaledInstance(newWidth, newHeight,
                            Image.SCALE_FAST);

                    notifyBusy(false);

                    ImageWaiter w = new ImageWaiter(tempImage);

                    try {
                        // Wait for ALL bits doesn't work anymore ???
                        w.waitForCompletion(false);
                        updateNewZoomImage(tempImage);
                    } catch (Exception e) {
                        notifyException("Failed to zoom", e);
                    }

                    //
                    // check more zoom, but only AFTER this current thread
                    // has fininshed !!!
                    //
                    checkMoreZoom();
                }
            };

            zoomThread = new Thread(zoomTask);
            zoomThread.start();
        }// synchronized(zoomTaskMutex);
    }

    private void checkMoreZoom() {
        Runnable checkZoomTask = new Runnable() {
            public void run() {

                synchronized (zoomTaskMutex) {
                    if (doMoreZoom == true) {
                        doMoreZoom = false;
                        doZoom();
                    }
                }
            }
        };

        SwingUtilities.invokeLater(checkZoomTask);
    }

    protected void updateNewZoomImage(Image img) throws Exception {
        // new image, image should already be done !
        // no checking needed

        this.imagePane.setImage(img, false);
    }

    public void resetZoom() {
        this.zoomIndex = 0;
        this.fitToScreen = false;
        this.doZoom();
    }

    public void toggleFitToScreen() {
        fitToScreen = (fitToScreen == false);
        doZoom();
    }

    public void reset() {
        this.resetZoom();
    }

    @Override
    public Map<String, List<String>> getMimeMenuMethods() {
        // Use HashMapList to keep order of menu entries: first is default(!)

        Map<String, List<String>> mappings = new HashMapList<String, List<String>>();

        for (int i = 0; i < mimeTypes.length; i++) {
            List<String> list = new StringList(new String[] { "view:View Image" });
            mappings.put(mimeTypes[i], list);
        }

        return mappings;
    }

}

redpajama_set_name: RedPajamaGithub


A powerful motor driving the front wheels means respectable Ioniq Electric performance figures, especially from low speeds. At higher speeds wind and road noise spoil the hush, but the handling is tidy and well controlled.
The Hyundai Ioniq is available with hybrid and plug-in hybrid powertrains, as well as the all-electric model we're reviewing here. In the hybrids, smaller electric motors play only a supporting role, but in this car there's a punchy 119bhp synchronous motor with a reduction gear driving the front axle. The electricity is stored in a large lithium-ion polymer battery located underneath the rear seats and boot.
There's 295Nm of torque available from the motor, too, and in characteristic electric car fashion it's delivered from the moment you prod the accelerator. While the Ioniq Electric's 0-62mph time of 10.2 seconds isn't as quick as that of some more expensive rivals, it still feels very responsive to the accelerator. Switching to Sport mode makes a marginal difference, shaving three-tenths of a second off the 0-62mph time, but you probably won't need to switch modes very often. Top speed is 103mph, yet it's unlikely many owners will wish to explore the outer limits of performance as range will suffer greatly.
One Ioniq foible we're not so keen on is the inconsistently weighted brake pedal. It's not always easy to know how hard to push. On the upside, the car has a regenerative braking system to harness energy that would otherwise be wasted while slowing down. It practice it means you don't often have to use the brake pedal, as simply lifting off the throttle slows you down considerably.
Around town, the Ioniq rides a little more stiffly than we'd like. However, it's not uncomfortable, and the spring and damper rates do mean there's less body roll in corners than might otherwise be the case.
There steering feels a little remote and doesn't offer much feedback on what the front wheels are doing. We also noticed the steering doesn't always self-centre in a consistent fashion – returning to the straight-ahead position more positively at some times than at others.
That said, we found the Ioniq enjoyable to drive even out of town. There, the eagerness of the motor to accelerate out of corners can be fun, even though it's not an enthusiasts' car. It rides quietly, too, although wind and tyre noise become more intrusive as the speed goes up, to the extent that you'll need to turn up the radio.


text:
A Framework designed for easier developement, it's based of Love2D, but for [ComputerCraft](http://www.computercraft.info)
It also adds some other useful classes that you can use for developing awesome programs/games or whatever it is you want to create.

These are the classes included, they're displayed in the order they're loaded, you can find the mods needed for some libraries below, to make sure they work you need to install those mods( duh! ). You can find out why they're needed in the wiki.

|  Class              | Version |  Status       | Dependencies |
|---------------------|---------|---------------|--------------|
|  CLove.Cryptography | 1.0     |  Working      | None         |
|  CLove.Network      | ---     | Not completed | None         |
|  CLove.Filesystem   | 1.0     | Working       | None         |
|  CLove.Buffer       | ---     | Not completed | None         |
|  CLove.Colors       | 1.0     | Working       | None         |
|  CLove.Graphics     | 1.0     | Working       | None         |
|  CLove.Object       | ---     | Not completed | None         |
|  CLove.Event        | 1.0     | Working       | None         |
|  CLove.Audio        | ---     | Not completed | [Moarperipherals](http://www.computercraft.info/forums2/index.php?/topic/19357-cc-latest-moarperipherals/) |
|  CLove.Web          | ---     | Not completed | None         |

# Usage
To load the Framework you simply use os.loadAPI and use it's function load to load the other classes
```lua
os.loadAPI( "CLove" )
CLove:load( "classes" )
```
You can then load all the objects by doing this
```lua
CLove.Object:load( "objects" )
```

# Contact me
If you want to contact me you can either do it by sending me a message on the [ComputerCraft forums](http://www.computercraft.info/forums2/index.php?/user/4727-theoddbyte/), or you can send me a message on Skype, you can find my skype username on my ComputerCraft profile. If you're experiencing problems you need to specify which version of MC you're using, and which version of CC you're using.

[![Join the chat at https://gitter.im/TheOddByte/CLove](https://badges.gitter.im/Join%20Chat.svg)](https://gitter.im/TheOddByte/CLove?utm_source=badge&utm_medium=badge&utm_campaign=pr-badge&utm_content=badge)


When looking for nostalgic knitwear in the softest wool, with a contemporary twist, we know exactly what brand to recommend; Little Mountains is a Norwegian brand specialising in natural materials with a unique Nordic concept, featuring traditional knitting patterns as well as Scandinavian folklore in its design.
Memini has in 10 years time become the one of the best selling Norwegian designed childrenswear brands with a timeless look combined with exceptional quality. This season Memini presents a nostalgic collection inspired by Nordic fairytales and children's books.
Noa Noa Miniature is one of the brands that firmly placed Denmark on the map for kids fashion. Season after season Noa Noa Miniature delivers one collection more beautiful than the other. The quality have always been flawless, from nightwear to outerwear, Noa Noa is a brand you can rely on.
Pierrot la Lune has arrived in stores, with beautiful children's clothing for Autumn. And what you shouldn't miss is the Agatha dress in a lovely soft teal green corduroy. Timeless and just beautiful. But there's more! Read on to see the new collection from this danish brand, for children age 2 to 10 years.


" I have yet to meet a wonderful woman like Mrs. Justine. I..."
"To the family, I worked with J for a while, I had lots of..."
"What a wonderful life your Mother experienced here on the..."
To every thing there is a season, and a time to every purpose under the heaven. A time to be born and a time to die; a time to plant and a time to pluck up that which is planted. –Ecclesiates 3:1-2 KJV. Job 14:5- You have decided the length of our lives. You know how many months we will live, have set limits we cannot exceed, and we are not given a minute longer. On April 7, 2019 God looked down on His earthly garden. He saw a beautiful and precious rose named Justine Simon Henry. For over 80 years He nurtured, watered and cultivated her for others to see. He shared her beautiful and fragrant spirit with her parents, siblings, spouse, children, grandchildren and a host of other family and friends. He decided to transition her from the cares of this life to eternal reward. Visitation on Friday, April 12, 2019 at Greater Mount Pilgrim Baptist Church, 926 St. Patrick St., Donaldsonville, LA from 5:00pm to 7:00pm. Visitation on Saturday, April 13, 2019 at Virginia Baptist Church, 145 Virginia St., Belle Rose, LA from 8:00am to Religious Services at 10:00am. Burial in Church Cemetery. Justine was born on November 18, 1939 in Assumption Parish, Louisiana to the late Cullen and Louisa Simon. She was the youngest of two children. As a child and young adult, she attended Ascension Catholic School. During her career she held many positions. She worked as a nurse, nurse caregiver and as an assistant for the Ascension Catholic Church. One of her greatest joys was that of being a loving wife and mother. Justine loved the Lord and served Him faithfully. For many years she maintained her membership with the church of her youth, Virginia Baptist Church in Ewellville, Louisiana. She would occasionally visit a local ministry, Greater Mount Pilgrim Baptist Church in Donaldsonville, Louisiana. Justine met her sweetheart and love of her life, Warren Henry Sr. at his friend Homer Richard's barbershop in Donaldsonville. The couple married in 1958 and during that union 8 children were born. They believed in parenting the old time way not like today's microwave society. Genuine love and respect was the order of the day. The family would sit down for meals together 7 days a week. Justine always blessed the family with her love of cooking good, wholesome food. She lived her life in the true spirit of giving. She shared fun times and vacations with her family as well. There was no limit to her humanitarian efforts. If there was a need, she did her best to address it. To characterize her life in song, Mahalia Jackson's words resonate deeply. "If I can help somebody as I travel along…then my living shall not be in vain." Justine was a dedicated mother, aunt, grandmother and neighbor. She loved sitting on the porch and was the best neighborhood watch and Celebrity Mother that anyone could ask for. After she retired her, she decided to have a porch sweet shop that was very affectionately called, "The Sweet Spot For Kids & Son "Stef's" Barbershop" She sold the best snacks, bottles of water and had a special Friday hotdog combo that rivaled the neighborhood Sonic's offering. She would occasionally pop into the barbershop and hold conversations with family members, friends and complete strangers while they were getting their haircuts. Throughout her life she maintained a sweet and simple demeanor and embraced the love of her family and friends. She kept a special promise that she made to her husband to cook for their children and grandchildren every Sunday to carry on their family traditions. She did this faithfully until her health failed and the Good Lord called her from labor to reward. Justine was preceded in death by her parents Louisa and Cullen Simon, only sister, Lillie Mae Honoree, her husband Warren Henry Sr., and the late daughter, Tonjue Henry Christy along with several other relatives. She leaves to live an cherish her memories, 6 children – Warren Jr. (Joan), Michael, Stephon, Donald Ray (Margurite), Lillie and Dana. Grandchildren, Don Christy III (Gabriel), Tylair, Stephon Jr., Roco, Jerrell, Taylor Diamond, Ka'lynn, Keshawn, Donald Jr., Tiana, Tamia, Joshua, the late Michael Jr., Mikalyn, Lil June, 3 God children, Renee Daggs, Bianka August and Kristy Levy Boxley. She will be missed by her late sister's children, Justine Gordon (Milton), Eugene "Geno" Williams, Rev. McCullen Williams (Jacqueline) and Joyce Prince (Elijah). She also leaves behind a special sister, Mrs. Mamie Dell Terry of Houston, TX, a special adopted son, Kevin Walker, and a special adopted daughter Pamela Harvey. 13 grandchildren, 13 great-grandchildren and a host of other nieces, nephews, cousins, relatives, friends. Arrangements by Williams & Southall Funeral Home, 101 Loop 945, Donaldsonville, LA (225) 473-1900. To sign guest book or offer condolences, visit our website at www.williamsandsouthallfuneralhome.com.


Even with the advancement of technology, it seems that age isn't just a number and it may bring wisdom too after all. Just recently, a study is conducted and suggested that older generations or old folks aren't falling for tech support scams like how we'd imagine – but younger generations; their grandchildren are. Millennials or the so called tech-savvy caricatures as depicted by the media, are actually somewhat naive when it comes to technology – calls, emails or popup ads claiming to offer tech support or even prizes, but delivering none other then scams only.
The information gathered is from a collective study between Ipsos Public Affairs, a technological survey-based research group and Microsoft. A total of 1,000 adults from 12 countries were surveyed and asked about their experiences with tech-support scams, and other related issues, especially those claiming to be a Microsoft representative.
Surprisingly, the results garnered after the survey was actually encouraging. On a comprehensive scale, around 20 percent of the participants took the scams seriously, and only nine percent had actually lost money to them. However, the results in the United States turned out to be a bit worse; 33 percent of the respondents fell for scams and a whopping 21 percent saw their hard-earned cash go down the drain.
The surprising part of this is that the age breakdown is not exactly what you had in mind. It turns out that, much to the surprise to everyone, younger users were actually falling for scams more than their older counterparts. The study shows that 27 percent from the 18-24 demographic fell for scams and 13 percent of them parted with their cash. A mind-boggling 32 percent; approximately one-third of the 25-34 age group fell for a scam; 18 percent lost cash.
Moving on to their older counterparts at the age of 66 and above – where most of you would think have the highest number of fraud, given that elderly folks tend to fall for scams and tricks much easier – which is true at a point. While old folks often encounter all kinds of scams at around the same rates as their younger generation, only 11 percent of them fall for it, and shockingly only 3 percent of them actually paid for the scams.
At the very least, the good news is that the number of fraud cases recorded are still relatively small, and actually even fewer had actually paid them. Nonetheless, you're better off safe than sorry by keeping an antivirus program active on your devices, and most importantly, ignore links embedded in strange emails or on social medias. And remember, Microsoft or any big corporations will never call you first AND – there won't be a pot of gold at the end of a rainbow.


Discussion in 'iPhone' started by kre62, Feb 28, 2013.
Some of you may recall that I broke the news about AT&T enabling FaceTime for its unlimited users a few months back. After that story went viral, AT&T shut this feature off for it's users.
Well it seems that now FaceTime over cellular has returned. It switched itself on over a week ago for me, and I've been taking my time to see if it was legit before bringing it up here. I'm not sure if this has something to do with 6.1.2, but I can confirm FaceTime is working on my iPhone5 with unlimited data, and even flipping the cell switch off and back on does not trigger the usual pop-up. My work 4S also has this enabled.
So please check your phones, AT&T unlimited users, and see if you see the same thing. If not, are you on the latest software? Report back here.
I'm on AT&T unlimited, have the latest firmware update, and when I try to enable FaceTime over cellular I still get the pop up that tells me I need wifi or an "eligible" data plan. Maybe it's rolling out only in select markets.
Give it some time probably rolling out slowly.
I placed a FT call between my 4S on 6.1.1 using AT&Ts unlimited data plan to my computer and the call went through.
Thank God this isn't an issue for me anymore now that I'm jailbroken. Sick of AT&T and their stupid restrictions.
You know before I would get excited and try it out and then dissapointed again that it wont let me turn it on.
Well it works for me just tried it, but like applejuiced my iPhone is jailbroken so it really did not matter.
Not working for me and AT&T gives me attitude over it.
AT&T Unlimited, using an iPhone 4S on iOS 6.1.2, carrier AT&T 14.0. I just tried to turn on Use Celluar Data and I still get the popup stating it requires an active Wi-Fi connection or an eligible cellular data plan. I am in what is considered the Chicago market, even though I am quite a ways west of there.
Well, looks like its still not live. Weird, both my phones are workng with no more popup. No jailbreaks.
Is this just for unlimited that is just getting it now because I've been able to make FaceTime calls over LTE for a good while now. iPhone 5, up to date, 3GB data plan.
Yes. AT&T enabled FT over LTE for non-unlimited plans a while back.
I have a 4S which doesn't have LTE and FT worked yesterday on my unlimited plan. Today, it's not working. Get the warning.
Kinda like a weather man, always breaking incorrect news, like we knew about his last incorrect forecast.
I called AT&T again. Ask if the rumors were true about AT&T settling the facetime for grandfathered in cust. I know it wasn't true but I'm going to keep getting it on record. Anyway the kept me on hold for about 15 minutes. They kept apologizing about the delay. When they finally got back to me, manager involved now. The only thing they could find was AT&T was going to allow some tiered user to use the data for facetime, what ever that means.
Does anyone know how to sign onto the different lawsuits. Every one needs to keep calling AT&T like I did. I called the tech department. Don't bother with the internet live chat. They just want to sell you a new contract and do it with a bad attitude.
I tried posting 4 things on AT&T forum.
None of them bad. They managed to delete 3 and edit one of them so much I don't even know what it says. One was deleted because they said one line was off topic, next was because the last word was "@#@&$" in place of the word "****". The gave my the whole lecture on forum rules and vulgarity. So I guess my parents are at fault for never teaching me that typing "@#@&$" was bad. The last one was a question about any law suits.
I'm following AT&T's example, I'm turning off wi-fi on my phone and only going to use cell data and keeping as many cell data hogging apps as possible running all day as long as I'm awake. Why, just because I can I think I will.
It wasn't working for me. iPhone on 6.1.2 and last night I restored cause my batt life was getting bad...again. After the restore FT has been switched on and stayed on. We'll see how long it lasts.
at&t told me yesterday that over the next few months they will be allowing some tiered contracts holders to use facetime. Big deal So now some unlimited plans will become unlimited and others will be stuck with pretend unlimited.


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text:
\section{Introduction}\label{Section:Introduction}
The relatively recent papers \cite{Yushkevich2002,Guo2008b} have shown the connection between Dynkin games and optimal switching problems where negative switching costs are allowed. In particular, \cite{Guo2008b} proved that the value of the Dynkin game is equal to the difference of the value functions for the two-mode optimal switching problem. However, there is no rigorous derivation of the dynamic programming algorithm for computing the value function in the case of signed (positive and negative) switching costs. In this paper, we resolve this issue by using a martingale approach to the optimal switching problem in discrete time.
\subsection{Literature review}
There are relatively few theoretical results on the dynamic programming method for optimal switching in discrete time. The discrete-time optimal switching problem with multiple modes was used in \cite{Carmona2008,Gassiat2012} as an approximation to the solution of a continuous time problem. The dynamic programming algorithm advocated in those papers follows from the backward induction method for solving optimal stopping problems (Chapter~\MakeUppercase{\romannumeral 1}, Section 1.1 of \cite{Peskir2006}), and requires the existence of a system of Snell envelope processes which solve the continuous-time optimal switching problem. Furthermore, their arguments for proving the existence of these processes requires strictly positive switching costs (see \cite{Martyr2014b} for related work where negative switching costs are allowed). A similar backward induction formula for the value function of an optimal switching problem with two modes and strictly positive, constant switching costs was obtained in \cite{Tanaka1990} under general non-Markovian assumptions. The paper \cite{Yushkevich2002}, which studied the optimal switching problem with two modes in a Markovian model, obtained a different type of dynamic programming equation for the value function -- one which is more in the spirit of the Wald-Bellman equations (Chapter~\MakeUppercase{\romannumeral 1}, Section 1.2 of \cite{Peskir2006}).

\subsection{Aim and results}
This paper looks at optimal switching for a finite-horizon discrete-time model which has a running reward, terminal reward and allows for negative switching costs. Our solution method is based on the martingale approach to optimal stopping problems. We provide the discrete-time analogue of the verification theorem of \cite{Djehiche2009} established for the continuous-time problem, and also justify and extend the dynamic programming method of \cite{Carmona2008,Gassiat2012} for computing the value function and optimal strategy to the case of signed switching costs. Numerical examples which utilize these results can be found in \cite{Martyr2014c}.

\section{Discrete-time optimal switching}\label{Section:Discrete-Time-Optimal-Switching}
\subsection{Definitions}
\subsubsection{Probabilistic setup.}
Let $(\Omega,\mathcal{F},\mathsf{P})$ be a complete probability space which is given. The expectation operator with respect to $\mathsf{P}$ is denoted by $\mathsf{E}$, and the indicator function of a set or event $A$ is written as $\mathbf{1}_{A}$. Let $\mathbb{T} = \lbrace 0,1,\ldots,T \rbrace$ represent a sequence of integer-valued times with $0 < T < \infty$. The probability space is equipped with a filtration $\mathbb{F} = (\mathcal{F}_{t})_{t \in \mathbb{T}}$, and it is assumed that $\mathcal{F}_{0}$ is the trivial $\sigma$-algebra, $\mathcal{F}_{0} = \lbrace \emptyset, \Omega \rbrace$, and $\mathcal{F} = \mathcal{F}_{T}$. The notation a.s. stands for ``almost-surely''. For a given $\mathbb{F}$-stopping time $\nu$, the notation $\mathcal{T}_{\nu}$ is used for the set of $\mathbb{F}$-stopping times $\tau$ such that $\nu \le \tau \le T$ \hspace{2bp} $\mathsf{P}$-a.s. Martingales, stopping times and other relevant concepts are understood to be defined with respect to the filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathsf{P})$. The usual convention to suppress the dependence on $\omega \in \Omega$ is used below.

\subsubsection{Optimal switching definitions.}
The following data for the optimal switching problem are given:
\begin{enumerate}
	\item A set of operational modes $\mathbb{I} = \left\lbrace 1,2,\ldots,m \right\rbrace$, where $2 \le m < \infty$;
	\item A reward received at time $T$ for being in mode $i \in \mathbb{I}$, which is modelled by an $\mathcal{F}_{T}$-measurable real-valued random variable $\Gamma_{i}$;
	\item A running reward received while in mode $i \in \mathbb{I}$, which is represented by a real-valued adapted process $\Psi_{i} = \left(\Psi_{i}(t)\right)_{t \in \mathbb{T}}$;
	\item A cost for switching from mode $i \in \mathbb{I}$ to $j \in \mathbb{I}$, which is modelled by a real-valued adapted process $\gamma_{i,j} = \left(\gamma_{i,j}(t)\right)_{t \in \mathbb{T}}$.
\end{enumerate}

Define a class of \emph{admissible switching controls} as follows:
\begin{definition}\label{Definition:AdmissibleSwitchingControl}
	Let $t \in \mathbb{T}$ and $i \in \mathbb{I}$ be given. An admissible switching control starting from time $t$ in mode $i$ is a sequence $\alpha = \left(\tau_{n},\iota_{n}\right)_{n \ge 0}$ with the following properties:

	\begin{enumerate}
		\item For $n \ge 0$, $\tau_{n} \in \mathcal{T}_{t}$ and satisfies $t = \tau_{0} \le \tau_{1} \le \tau_{2} \le \ldots \le T$; if $n \ge 1$ then $\mathsf{P}\left(\{\tau_{n} < T \} \cap \{\tau_{n} = \tau_{n+1}\}\right) = 0$.
		\item For $n \ge 0$, each $\iota_{n} \colon \Omega \to \mathbb{I}$ is $\mathcal{F}_{\tau_{n}}$-measurable; $\iota_{0} = i$ and $\iota_{n} \neq \iota_{n+1}$  $\mathsf{P}$-a.s. for $n \ge 0$.
	\end{enumerate}

	Let $\mathcal{A}_{t,i}$ denote the class of admissible switching controls (also called strategies) for the initial condition $(t,i) \in \mathbb{T} \times \mathbb{I}$.
\end{definition}

The switching control $\alpha = (\tau_{n},\iota_{n})_{n \ge 0}$ models the controller's decision to switch at time $\tau_{n}$, $n \ge 1$, from the active mode $\iota_{n-1}$ to another one $\iota_{n}$. The condition $\mathsf{P}\left(\{\tau_{n} < T \} \cap \{\tau_{n} = \tau_{n+1}\}\right) = 0$ for $n \ge 1$ means there is at most one switch at each time. The above definition is similar to the one given in \cite[p.~145]{Yushkevich2002}.
\begin{definition}\label{Definition:Discrete-Time-OS-ModeIndicatorEtc}
	Associated with each $\alpha = \left(\tau_{n},\iota_{n}\right)_{n \ge 0} \in \mathcal{A}_{t,i}$ are the following objects:
	\begin{itemize}
		\item a mode indicator function ${\mathbf{u} \colon \Omega \times \mathbb{T} \to \mathbb{I}}$ defined by,
		\begin{equation}\label{eq:Discrete-Time-OS-ModeIndicator}
		\mathbf{u}_{s} = \sum\limits_{n \ge 0}\iota_{n}\mathbf{1}_{\lbrace \tau_{n} \le s < \tau_{n+1}\rbrace},\hspace{1em} s \ge t.
		\end{equation}
		\item the (random) total number of switches before $T$
		\begin{equation}\label{eq:OS-Discrete-Time-RandomNumberOfSwitches}
		N(\alpha) = \sum_{n \ge 1}\mathbf{1}_{\lbrace \tau_{n} < T \rbrace}
		\end{equation}
		\item the last mode switched to before $T$
		\begin{equation}\label{eq:OS-Discrete-Time-Last-Mode-Switched-To}
		\iota_{N(\alpha)} = \sum_{n \ge 0}\iota_{n}\mathbf{1}_{\lbrace \tau_{n} < T \rbrace}\mathbf{1}_{\lbrace \tau_{n+1} = T \rbrace}, \quad \text{a.s.}
		\end{equation}
	\end{itemize}
\end{definition}

Note that the mode indicator function $\mathbf{u}$ may jump immediately from $\iota_{0}(\omega) = i$ to $\iota_{1}(\omega) \neq i$ if $\tau_{1}(\omega) = t$.

\subsection{The optimal switching problem}
Define the following performance index for switching controls with initial mode $i \in \mathbb{I}$ at time $t \in \mathbb{T}$:
\begin{equation}\label{Definition:Discrete-TimeSwitchingControlObjective}
J(\alpha;t,i) = \mathsf{E}\left[\sum_{s=t}^{T-1}\Psi_{\mathbf{u}_{s}}(s) + \Gamma_{\iota_{N(\alpha)}} - \sum\limits_{n \ge 1}\gamma_{\iota_{n-1},\iota_{n}}\left(\tau_{n}\right)\mathbf{1}_{\lbrace \tau_{n} < T \rbrace} \biggm \vert \mathcal{F}_{t}\right],\hspace{1em}\alpha \in \mathcal{A}_{t,i}
\end{equation}
The optimisation problem is to maximise the objective function $J(\alpha;t,i)$ over all admissible controls $\alpha \in \mathcal{A}_{t,i}$. The value function $V$ for the optimal switching problem is defined as a random function of the initial time and mode $(t,i)$:
\begin{equation}\label{eq:SwitchingControlValueFunction}
V(t,i) = \esssup\limits_{\alpha \in \mathcal{A}_{t,i}} J(\alpha;t,i).
\end{equation}
A switching control $\alpha^{*} \in \mathcal{A}_{t,i}$ is said to be optimal if it achieves the essential supremum in equation~\eqref{eq:SwitchingControlValueFunction}: $\mathsf{P}$-a.s.,
\begin{align*}
V(t,i) = J(\alpha^{*};t,i) \ge J(\alpha;t,i) \hspace{1 em}\forall \alpha \in \mathcal{A}_{t,i}.
\end{align*}

\begin{remark}
	Note that the analogous minimisation problem can be treated by negating the performance index~\eqref{Definition:Discrete-TimeSwitchingControlObjective} and value function~\eqref{eq:SwitchingControlValueFunction}. This follows from the relationship between the essential supremum and essential infimum: if $\Phi$ is a set of random variables on $(\Omega,\mathcal{F},\mathsf{P})$, the essential infimum of $\Phi$ with respect to $\mathsf{P}$ is defined as \cite[p.~496]{Follmer2011}:
	\[
	\essinf \Phi = \essinf_{\varphi \in \Phi} \varphi \coloneqq -\esssup_{\varphi \in \Phi}(-\varphi).
	\]
\end{remark}

\begin{remark}
	Processes or functions with super(sub)-scripts in terms of the mode indicators $\lbrace \iota_{n} \rbrace$ are interpreted in the following way:
	\begin{align*}
	Y^{\iota_{n}} & = \sum\limits_{j \in \mathbb{I}}\mathbf{1}_{\lbrace \iota_{n} = j \rbrace} Y^{j} \\
	\gamma_{\iota_{n-1},\iota_{n}}\left(\cdot\right) & = \sum\limits_{j \in \mathbb{I}}\sum\limits_{k \in \mathbb{I}}\mathbf{1}_{\lbrace \iota_{n-1} = j \rbrace}\mathbf{1}_{\lbrace \iota_{n} = k \rbrace} \gamma_{j,k}\left(\cdot\right)
	\end{align*}
	Note that the summations are finite.
\end{remark}

\subsection{Notation, conventions and assumptions}
The convention that $\sum_{s = v}^{t}\left(\cdot\right) = 0$ for any integers $t$ and $v$ with $t < v$ is used. The following terminology is referred to in later developments:
\begin{itemize}
	\item For a constant $p \ge 1$, let $L^{p}$ denote the class of random variables $Z$ satisfying $\mathsf{E}\left[\left|Z\right|^{p}\right] < \infty$.
	\item Similarly, let $\mathcal{S}^{p}$ denote the class of adapted processes $X$ satisfying $\mathsf{E}\left[\max\limits_{t \in \mathbb{T}}\left|X_{t}\right|^{p}\right] < \infty$.
\end{itemize}

\subsubsection{Assumptions.}
\begin{assumption}\label{Assumption:Optimal-Switching-Discrete-Time-Integrability}
	For each $i \in \mathbb{I}$, $\Gamma_{i} \in L^{2}$ and is $\mathcal{F}_{T}$-measurable,  $\Psi_{i} \in \mathcal{S}^{2}$, and for every $j \in \mathbb{I}$, $\gamma_{i,j} \in \mathcal{S}^{2}$.
\end{assumption}

Assumption~\ref{Assumption:Optimal-Switching-Discrete-Time-Integrability} ensures that the performance index \eqref{Definition:Discrete-TimeSwitchingControlObjective} is well defined and allows application of optimal stopping theory later. The following standard assumption on the switching costs \cite{Yushkevich2002,Guo2008b} is also imposed:
\begin{assumption}\label{assumption:SwitchingCosts}
	For every $i,j,k \in \mathbb{I}$ and $\forall t \in \mathbb{T}$: $\mathsf{P}$-a.s.,
	\begin{align}
	i. \quad & \gamma_{i,i}\left(t\right) = 0 \label{assumption:Discrete-Time-SwitchingCosts-No-Cost} \\
	ii. \quad & \gamma_{i,k}\left(t\right) < \gamma_{i,j}\left(t\right) + \gamma_{j,k}\left(t\right),\hspace{1em} \text{ if } i \neq j \text{ and } j \neq k\label{assumption:Discrete-Time-SwitchingCosts-No-Arbitrage}.
	\end{align}
\end{assumption}

Condition~\eqref{assumption:Discrete-Time-SwitchingCosts-No-Cost} says there is no additional cost for staying in the same mode. The second condition ensures that when going from one mode $i$ to another mode $k$, it is never profitable to immediately visit an intermediate mode $j$. By taking $k = i$ and using \eqref{assumption:Discrete-Time-SwitchingCosts-No-Cost}, condition~\eqref{assumption:Discrete-Time-SwitchingCosts-No-Arbitrage} also shows that it is unprofitable to switch immediately back and forth between modes.

\section{Optimal stopping and Snell envelopes}\label{Section:DiscreteSnellEnvelopes}
This section collects some results on discrete-time optimal stopping problems which are important below.

\begin{proposition}\label{Proposition:DiscreteSnellEnvelopeProperties}
	Let $U = (U_{t})_{t \in \mathbb{T}}$ be an adapted, $\mathbb{R}$-valued process that satisfies $U \in \mathcal{S}^{1}$. Then there exists an adapted, integrable $\mathbb{R}$-valued process $Z = (Z_{t})_{t \in \mathbb{T}}$ such that $Z$ is the smallest supermartingale which dominates $U$. The process $Z$ is called the Snell envelope of $U$ and it enjoys the following properties.
	\begin{enumerate}
		\item For any $t \in \mathbb{T}$, $Z_{t}$ is defined by:
		\begin{equation}
		Z_{t} = \esssup_{\tau \in \mathcal{T}_{t}}\mathsf{E}\left[U_{\tau} \vert \mathcal{F}_{t}\right].
		\end{equation}
		Moreover, $Z$ can also be defined recursively as follows: $Z_{T} \coloneqq  U_{T}$ and \\ $Z_{t} \coloneqq U_{t} \vee \mathsf{E}\left[Z_{t+1} \big \vert \mathcal{F}_{t}\right]$ for $t = T-1,\ldots,0$.
		\item For any $\theta \in \mathcal{T}$, the stopping time $\tau^{*}_{\theta} = \inf\lbrace t \ge \theta \colon Z_{t} = U_{t} \rbrace$ is optimal after $\theta$ in the sense that:
		\begin{equation}\label{eq:SnellEnvelopeAtStoppingTimes}
		Z_{\theta} = \mathsf{E}\bigl[U_{\tau^{*}_{\theta}} \big\vert \mathcal{F}_{\theta}\bigr] = \esssup_{\tau \in \mathcal{T}_{\theta}}\mathsf{E}\bigl[U_{\tau} \big\vert \mathcal{F}_{\theta}\bigr],\hspace{1em} \mathsf{P}-\text{a.s.}
		\end{equation}
		\item For any $t \in \mathbb{T}$ given and fixed, the stopped process $\left(Z_{r \wedge \tau^{*}_{t}}\right)_{t \le r \le T}$ is a martingale.
	\end{enumerate}
\end{proposition}
These results are standard and can be found in the references \cite{Neveu1975,Peskir2006,Follmer2011}.

\section{The verification theorem}\label{Section:Optimal-Switching-Discrete-Time-Verification-Theorem}
This section proposes a probabilistic solution to the optimal switching problem. The approach follows that of \cite{Djehiche2009} in continuous time, which postulates the existence of a particular system of $m$ stochastic processes and verifies (Theorem~\ref{theorem:verification}) that the components of this system solve the optimal switching problem with given initial conditions. The existence of these candidate optimal processes is proved in the following section (Theorem~\ref{theorem:ExistenceSnellEnevelope}).
\subsection{An iterative optimal stopping problem}
Suppose there exist $m$ real-valued, adapted processes $Y^{i} = \big(Y^{i}_{t}\big)_{t \in \mathbb{T}}$, $i \in \mathbb{I}$, such that $Y^{i} \in \mathcal{S}^{2}$ and
\begin{equation}\label{eq:VerificationSnellEnvelope}
Y^{i}_{t} = \esssup\limits_{\tau \in \mathcal{T}_{t}} \mathsf{E}\left[\sum_{s=t}^{\tau-1}\Psi_{i}(s) + \Gamma_{i}\mathbf{1}_{\lbrace \tau = T \rbrace} + \max\limits_{j \neq i} \left\lbrace Y^{j}_{\tau} - \gamma_{i,j}\left(\tau\right) \right\rbrace \mathbf{1}_{\lbrace \tau < T \rbrace} \biggm \vert \mathcal{F}_{t}\right].
\end{equation}

For $i \in \mathbb{I}$, define the \emph{implicit gain process} $\big(U^{i}_{t}\big)_{t \in \mathbb{T}}$ by,
\begin{equation}\label{eq:VerificationImplicitGainProcess}
U^{i}_{t} = \max\limits_{j \neq i} \left\lbrace Y^{j}_{t} - \gamma_{i,j}\left(t\right) \right\rbrace \mathbf{1}_{\lbrace t < T \rbrace} + \Gamma_{i}\mathbf{1}_{\lbrace t = T \rbrace}.
\end{equation}

Then equation~\eqref{eq:VerificationSnellEnvelope} becomes,
\begin{equation}\label{eq:VerificationSnellEnvelope2}
Y^{i}_{t} = \esssup\limits_{\tau \in \mathcal{T}_{t}}\mathsf{E}\left[\sum_{s=t}^{\tau-1}\Psi_{i}(s) + U^{i}_{\tau} \biggm \vert \mathcal{F}_{t}\right].
\end{equation}

Note that the assumptions on $Y^{i}$ and the costs guarantee that $U^{i} \in \mathcal{S}^{2}$ for every $i \in \mathbb{I}$. Recalling Proposition~\ref{Proposition:DiscreteSnellEnvelopeProperties}, $\left(Y^{i}_{t} + \sum_{s=0}^{t-1}\Psi_{i}(s)\right)_{t \in \mathbb{T}}$ can be identified as the Snell envelope of the process $\left(U^{i}_{t} + \sum_{s=0}^{t-1}\Psi_{i}(s)\right)_{t \in \mathbb{T}}$ (also see Lemma~\ref{lemma:VerificationLemma}).

\begin{theorem}[Verification Theorem]\label{theorem:verification}
	Let $i \in \mathbb{I}$ be the active mode at some fixed initial time $t \in \mathbb{T}$ and suppose $Y^{1},\ldots,Y^{m}$ as defined in equation~\eqref{eq:VerificationSnellEnvelope} are in $\mathcal{S}^{2}$. Define sequences of random times $\{\tau^{*}_{n}\}_{n \ge 0}$ and mode indicators $\{\iota^{*}_{n}\}_{n \ge 0}$ as follows:
	\begin{gather}\label{eq:OptimalStoppingStrategy}
	\begin{split}
	\tau^{*}_{0} & = t,\hspace{1pc} \iota^{*}_{0} = i \hspace{1 em} \text{and for } n \ge 1: \\
	\tau^{*}_{n} & = \inf\left\lbrace \tau^{*}_{n-1} \le s \le T \colon Y^{\iota^{*}_{n-1}}_{s} = U^{\iota^{*}_{n-1}}_{s}  \right\rbrace,\hspace{1em} \iota^{*}_{n} = \sum\limits_{j \in \mathbb{I}} j \mathbf{1}_{A^{\iota^{*}_{n-1}}_{j}}
	\end{split} \\
	\text{where } A^{\iota^{*}_{n-1}}_{j} (= A^{\iota^{*}_{n-1}}_{j}(\omega)) \text{ is the event:} \hspace{\textwidth} \nonumber \\
	A^{\iota^{*}_{n-1}}_{j} \coloneqq \left\lbrace Y^{j}_{\tau^{*}_{n}} - \gamma_{\iota^{*}_{n-1},j}\left(\tau^{*}_{n}\right) = \max\limits_{k \neq \iota^{*}_{n-1}}\left\lbrace Y^{k}_{\tau^{*}_{n}} - \gamma_{\iota^{*}_{n-1},k}\left(\tau^{*}_{n}\right) \right\rbrace \right\rbrace. \nonumber
	\end{gather}
	Then, $\alpha^{*} = \left(\tau^{*}_{n},\iota^{*}_{n}\right)_{n \ge 0} \in \mathcal{A}_{t,i}$ and satisfies
	\begin{equation}\label{eq:OptimalSwitchingProcess}
	Y^{i}_{t} = J(\alpha^{*};t,i) = \esssup\limits_{\alpha \in \mathcal{A}_{t,i}} J(\alpha;t,i)\hspace{1em} \text{a.s.}
	\end{equation}
\end{theorem}
\begin{proof}
	The proof is essentially the same as Theorem~1 of \cite{Djehiche2009}. Recall the definition of $U^{i}$ in equation~\eqref{eq:VerificationImplicitGainProcess}. At time $t$, $Y^{i}_{t}$ is given by

	\[
	Y^{i}_{t} = \esssup\limits_{\tau \in \mathcal{T}_{t}}\mathsf{E}\left[\sum_{s=t}^{\tau-1}\Psi_{i}(s) + U^{i}_{\tau} \biggm \vert \mathcal{F}_{t}\right].
	\]
	If $t = T$, then $\tau^{*}_{0} = T$, $\iota^{*}_{0} = i$ which leads to $Y^{i}_{T} = \Gamma_{i}$, and the claim follows trivially since $\Gamma_{i} = J(\alpha;T,i) = V(T,i)$ almost surely for any switching control $\alpha \in \mathcal{A}_{T,i}$.

	Suppose now that $t < T$. Lemma~\ref{lemma:AdmissibleOptimalControl} in the appendix verifies that $\alpha^{*} \in \mathcal{A}_{t,i}$. Note that the infimum in equation~\eqref{eq:OptimalStoppingStrategy} is always attained since $Y^{i}_{T} = U^{i}_{T}$ a.s. for every $i \in \mathbb{I}$. The stopping time $\tau^{*}_{1}$ in \eqref{eq:OptimalStoppingStrategy} is optimal after $t$ by Proposition~\ref{Proposition:DiscreteSnellEnvelopeProperties}. Using this together with the definition of $\iota^{*}_{1}$ gives, almost surely,
	\begin{align}\label{eq:Verification1}
	Y^{i}_{t} & = \esssup\limits_{\tau \in \mathcal{T}_{t}}\mathsf{E}\left[\sum_{s = t}^{\tau-1}\Psi_{i}(s) + U^{i}_{\tau} \biggm \vert \mathcal{F}_{t}\right] \nonumber \\
	& = \mathsf{E}\left[\sum_{s = t}^{\tau^{*}_{1} - 1}\Psi_{i}(s) + U^{i}_{\tau^{*}_{1}} \biggm \vert \mathcal{F}_{t}\right] \nonumber \\
	& = \mathsf{E}\left[\sum_{s = t}^{\tau^{*}_{1}-1}\Psi_{i}(s) + \Gamma_{i}\mathbf{1}_{\lbrace \tau^{*}_{1} = T \rbrace} + \max\limits_{j \neq i} \left\lbrace Y^{j}_{\tau^{*}_{1}} - \gamma_{i,j}\left(\tau^{*}_{1}\right)\right\rbrace \mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} \biggm \vert \mathcal{F}_{t}\right] \nonumber \\
	& = \mathsf{E}\left[\sum_{s = t}^{\tau^{*}_{1}-1}\Psi_{i}(s) + \Gamma_{i}\mathbf{1}_{\lbrace \tau^{*}_{1} = T \rbrace} + \left\lbrace Y^{\iota^{*}_{1}}_{\tau^{*}_{1}} - \gamma_{i,\iota^{*}_{1}}\left(\tau^{*}_{1}\right) \right\rbrace \mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} \biggm \vert \mathcal{F}_{t}\right]
	\end{align}

	Lemma~\ref{lemma:VerificationLemma} in the appendix confirms that $Y^{\iota^{*}_{1}}$ satisfies:
	\begin{equation}\label{eq:SequentialOptimality}
	Y^{\iota^{*}_{1}}_{s} = \esssup\limits_{\tau \in \mathcal{T}_{s}}\mathsf{E}\left[\sum_{r=s}^{\tau^{*}_{1}-1}\Psi_{\iota^{*}_{1}}(r) + U^{\iota^{*}_{1}}_{\tau} \biggm \vert \mathcal{F}_{s}\right] \enskip \text{on} \enskip [\tau^{*}_{1},T].
	\end{equation}

	Use \eqref{eq:SequentialOptimality} together with the definition and optimality of $\tau^{*}_{2}$ and $\iota^{*}_{2}$ to get, almost surely,
	\begin{align}
	\mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace}Y^{\iota^{*}_{1}}_{\tau^{*}_{1}} = {} & \esssup\limits_{\tau \in \mathcal{T}_{\tau^{*}_{1}}}\mathsf{E}\left[\sum_{r=\tau^{*}_{1}}^{\tau - 1}\Psi_{\iota^{*}_{1}}(r) + U^{\iota^{*}_{1}}_{\tau} \biggm \vert \mathcal{F}_{\tau^{*}_{1}}\right]\mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} \nonumber \\
	= {} & \mathsf{E}\bigg[\sum_{r=\tau^{*}_{1}}^{\tau^{*}_{2}-1}\Psi_{\iota^{*}_{1}}(r) + \Gamma_{\iota^{*}_{1}}\mathbf{1}_{\lbrace \tau^{*}_{2} = T \rbrace} + \left\lbrace Y^{\iota^{*}_{2}}_{\tau^{*}_{2}} - \gamma_{\iota^{*}_{1},\iota^{*}_{2}}\left(\tau^{*}_{2}\right) \right\rbrace \mathbf{1}_{\lbrace \tau^{*}_{2} < T \rbrace} \biggm \vert \mathcal{F}_{\tau^{*}_{1}}\bigg]\mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace}\label{eq:Verification2}
	\end{align}

	Combining~\eqref{eq:Verification1} and~\eqref{eq:Verification2} gives the following expression for $Y^{i}_{t}$: almost surely,
	\begin{align}\label{eq:Verification3}
	Y^{i}_{t} = {} & \mathsf{E}\left[\sum_{s = t}^{\tau^{*}_{1}-1}\Psi_{i}(s) + \Gamma_{i}\mathbf{1}_{\lbrace \tau^{*}_{1} = T \rbrace} + \left\lbrace Y^{\iota^{*}_{1}}_{\tau^{*}_{1}} - \gamma_{i,\iota^{*}_{1}}\left(\tau^{*}_{1}\right) \right\rbrace \mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} \biggm \vert \mathcal{F}_{t}\right] \nonumber \\
	= {} & \mathsf{E}\left[\sum_{s = t}^{\tau^{*}_{1}-1}\Psi_{i}(s) + \mathsf{E}\left[\sum_{r=\tau^{*}_{1}}^{\tau^{*}_{2}-1}\Psi_{\iota^{*}_{1}}(r) \biggm \vert \mathcal{F}_{\tau^{*}_{1}}\right] \mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} \biggm \vert \mathcal{F}_{t}\right] \nonumber \\
	& + \mathsf{E}\left[\Gamma_{i}\mathbf{1}_{\lbrace \tau^{*}_{1} = T \rbrace} + \mathsf{E}\left[\Gamma_{\iota^{*}_{1}}\mathbf{1}_{\lbrace \tau^{*}_{2} = T \rbrace}\big \vert \mathcal{F}_{\tau^{*}_{1}}\right] \mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} \big \vert \mathcal{F}_{t}\right] \nonumber \\
	& - \mathsf{E}\left[\gamma_{i,\iota^{*}_{1}}\left(\tau^{*}_{1}\right) \mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} + \mathsf{E}\left[ \gamma_{\iota^{*}_{1},\iota^{*}_{2}}\left(\tau^{*}_{2}\right) \mathbf{1}_{\lbrace \tau^{*}_{2} < T \rbrace} \big \vert \mathcal{F}_{\tau^{*}_{1}}\right] \mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} \big \vert \mathcal{F}_{t}\right] \nonumber \\
	& + \mathsf{E}\left[\mathsf{E}\left[Y^{\iota^{*}_{2}}_{\tau^{*}_{2}}\mathbf{1}_{\lbrace \tau^{*}_{2} < T \rbrace} \big \vert \mathcal{F}_{\tau^{*}_{1}}\right] \mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} \big \vert \mathcal{F}_{t}\right]
	\end{align}

	Since $\mathbf{1}_{\lbrace \tau^{*}_{1} = T \rbrace}$, $\mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace}$ and $\gamma_{i,\iota^{*}_{1}}\left(\tau^{*}_{1}\right)$ are all $\mathcal{F}_{\tau^{*}_{1}}$-measurable, they can be brought inside the conditional expectation with respect to $\mathcal{F}_{\tau^{*}_{1}}$ in equation~\eqref{eq:Verification3}: almost surely,
	\begin{align*}
	Y^{i}_{t} = {} & \mathsf{E}\left[\mathsf{E}\left[\sum_{s = t}^{\tau^{*}_{1}-1}\Psi_{i}(s) + \sum_{r=\tau^{*}_{1}}^{\tau^{*}_{2}-1}\Psi_{\iota^{*}_{1}}(r)\mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} \biggm \vert \mathcal{F}_{\tau^{*}_{1}}\right] \biggm \vert \mathcal{F}_{t}\right] \\
	& + \mathsf{E}\left[\mathsf{E}\left[\Gamma_{i}\mathbf{1}_{\lbrace \tau^{*}_{1} = T \rbrace} + \Gamma_{\iota^{*}_{1}}\mathbf{1}_{\lbrace \tau^{*}_{2} = T \rbrace}\mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace}\big \vert \mathcal{F}_{\tau^{*}_{1}}\right] \big \vert \mathcal{F}_{t}\right] \\
	& - \mathsf{E}\left[\mathsf{E}\left[\gamma_{i,\iota^{*}_{1}}\left(\tau^{*}_{1}\right) \mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} + \gamma_{\iota^{*}_{1},\iota^{*}_{2}}\left(\tau^{*}_{2}\right) \mathbf{1}_{\lbrace \tau^{*}_{2} < T \rbrace}\mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} \big \vert \mathcal{F}_{\tau^{*}_{1}}\right] \big \vert \mathcal{F}_{t}\right] \\
	& + \mathsf{E}\left[\mathsf{E}\left[Y^{\iota^{*}_{2}}_{\tau^{*}_{2}}\mathbf{1}_{\lbrace \tau^{*}_{2} < T \rbrace}\mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} \big \vert \mathcal{F}_{\tau^{*}_{1}}\right] \big \vert \mathcal{F}_{t}\right]
	\end{align*}

	Using $\iota^{*}_{0} = i$ and the definition of the mode indicator $\mathbf{u}^{*}$ (cf. \eqref{eq:Discrete-Time-OS-ModeIndicator}) gives,
	\[
	\sum_{s = t}^{\tau^{*}_{2}-1}\Psi_{\mathbf{u}^{*}_{s}}(s) = \sum_{r = t}^{\tau^{*}_{1}-1}\Psi_{i}(r) + \sum_{r=\tau^{*}_{1}}^{\tau^{*}_{2}-1}\Psi_{\iota^{*}_{1}}(r)\mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace}.
	\]

	Lemma~\ref{lemma:AdmissibleOptimalControl} in the appendix verifies that $\mathsf{P}\left(\{\tau^{*}_{n} < T \} \cap \{\tau^{*}_{n} = \tau^{*}_{n+1}\}\right) = 0$ and ensures the above expression is well defined. Since $\tau^{*}_{1} \le \tau^{*}_{2}$ it follows that $\left\lbrace \tau^{*}_{2} < T  \right\rbrace \subset \left\lbrace \tau^{*}_{1} < T  \right\rbrace$ and therefore $\mathbf{1}_{\lbrace \tau^{*}_{2} < T \rbrace}\mathbf{1}_{\lbrace \tau^{*}_{1} < T \rbrace} = \mathbf{1}_{\lbrace \tau^{*}_{2} < T \rbrace}$ almost surely. Note that $\tau^{*}_{0} = t < T$ so $\mathbf{1}_{\lbrace \tau^{*}_{0} < T \rbrace} = 1$ and that $\mathcal{F}_{t} \subseteq \mathcal{F}_{\tau^{*}_{1}}$ since $t \le \tau^{*}_{1}$. These observations together with the tower property of conditional expectations, shows that $Y^{i}_{t}$ satisfies: almost surely,
	\begin{align}\label{eq:Verification4}
	Y^{i}_{t} = {} & \mathsf{E}\left[\mathsf{E}\left[\sum_{s = t}^{\tau^{*}_{2}-1}\Psi_{\mathbf{u}^{*}_{s}}(s) + \sum_{n = 1}^{2}\Gamma_{\iota^{*}_{n-1}}\mathbf{1}_{\lbrace \tau^{*}_{n} = T \rbrace}\mathbf{1}_{\lbrace \tau^{*}_{n-1} < T \rbrace} \biggm \vert \mathcal{F}_{\tau^{*}_{1}}\right] \biggm \vert \mathcal{F}_{t}\right] \nonumber \\
	& + \mathsf{E}\left[\mathsf{E}\left[\sum_{n = 1}^{2}-\gamma_{\iota^{*}_{n-1},\iota^{*}_{n}}\left(\tau^{*}_{n}\right)\mathbf{1}_{\lbrace \tau^{*}_{n} < T \rbrace} + Y^{\iota^{*}_{2}}_{\tau^{*}_{2}} \mathbf{1}_{\lbrace \tau^{*}_{2} < T \rbrace} \biggm \vert \mathcal{F}_{\tau^{*}_{1}}\right] \biggm \vert \mathcal{F}_{t}\right] \nonumber \\
	\begin{split}
	= {} & \mathsf{E}\left[\sum_{s = t}^{\tau^{*}_{2}-1}\Psi_{\mathbf{u}^{*}_{s}}(s) + \sum_{k = 0}^{1}\Gamma_{\iota^{*}_{k}}\mathbf{1}_{\lbrace \tau^{*}_{k} < T \rbrace}\mathbf{1}_{\lbrace \tau^{*}_{k+1} = T \rbrace} - \sum_{n = 1}^{2}\gamma_{\iota^{*}_{n-1},\iota^{*}_{n}}\left(\tau^{*}_{n}\right)\mathbf{1}_{\lbrace \tau^{*}_{n} < T \rbrace} \biggm \vert \mathcal{F}_{t}\right] \\
	& + \mathsf{E}\left[Y^{\iota^{*}_{2}}_{\tau^{*}_{2}} \mathbf{1}_{\lbrace \tau^{*}_{2} < T \rbrace} \biggm \vert \mathcal{F}_{t}\right].
	\end{split}
	\end{align}

	Let $N(\alpha^{*})$ be the total number of switches under $\alpha^{*}$ (cf. \eqref{eq:OS-Discrete-Time-RandomNumberOfSwitches}). Since $\alpha^{*}$ is admissible by Lemma~\ref{lemma:AdmissibleOptimalControl} in the appendix, for $n \ge 1$ the switching times satisfy $\tau^{*}_{n} < \tau^{*}_{n+1}$ on $\{n \le N(\alpha^{*})\}$ and $\tau^{*}_{n} = T$ on $\{n > N(\alpha^{*})\}$. Repeating the procedure of substituting for $Y^{\iota^{*}_{n}}_{\tau^{*}_{n}}$ with $ n = 2, 3,\ldots$ yields
	\begin{equation}\label{eq:Verification5}
	Y^{i}_{t} =
	\mathsf{E}\left[\sum_{s = t}^{T-1}\Psi_{\mathbf{u}^{*}_{s}}(s) + \sum_{k = 0}^{N(\alpha^{*})}\Gamma_{\iota^{*}_{k}}\mathbf{1}_{\lbrace \tau^{*}_{k} < T \rbrace}\mathbf{1}_{\lbrace \tau^{*}_{k+1} = T \rbrace} - \sum_{n = 1}^{N(\alpha^{*})}\gamma_{\iota^{*}_{n-1},\iota^{*}_{n}}\left(\tau^{*}_{n}\right)\biggm \vert \mathcal{F}_{t}\right].
	\end{equation}

	The sum of the terminal rewards collapses to a single term,
	\begin{equation}\label{eq:EquivalentTerminalCost}
	\sum_{k = 0}^{N(\alpha^{*})}\Gamma_{\iota^{*}_{k}}\mathbf{1}_{\lbrace \tau^{*}_{k} < T \rbrace}\mathbf{1}_{\lbrace \tau^{*}_{k+1} = T \rbrace} = \Gamma_{\iota^{*}_{N(\alpha^{*})}} \hspace{1em}\mathsf{P}-\text{a.s.}
	\end{equation}
	From equations~\eqref{eq:Verification5} and \eqref{eq:EquivalentTerminalCost}, one arrives at the following representation for $Y^{i}_{t}$: $\mathsf{P}$-a.s.,
	\begin{equation}\label{eq:Verification6}
	Y^{i}_{t} = \mathsf{E}\left[\sum_{s = t}^{T-1}\Psi_{\mathbf{u}^{*}_{s}}(s) + \Gamma_{\iota^{*}_{N(\alpha^{*})}} - \sum_{n \ge 1}\gamma_{\iota^{*}_{n-1},\iota^{*}_{n}}\left(\tau^{*}_{n}\right)\mathbf{1}_{\lbrace \tau^{*}_{n} < T  \rbrace} \biggm \vert \mathcal{F}_{t}\right] = J(\alpha^{*};t,i)
	\end{equation}

	Now let $\alpha = \left(\tau_{n},\iota_{n}\right)_{n \ge 0} \in \mathcal{A}_{t,i}$ be any admissible control. The verification theorem can be completed by showing that $J(\alpha^{*};t,i) \ge J(\alpha;t,i)$ a.s. First, note that $J(\alpha^{*};T,i) = J(\alpha;T,i)$ when $t = T$, so assume henceforth that $t < T$. Then, due to possible sub-optimality of the pair $(\tau_{1},\iota_{1})$, it is true that
	\begin{align*}
	Y^{i}_{t} & = \esssup_{\tau \in \mathcal{T}_{t}}\mathsf{E}\left[\sum_{s = t}^{\tau-1}\Psi_{i}(s) + U^{i}_{\tau} \biggm\vert \mathcal{F}_{t}\right] \\
	& \ge \mathsf{E}\left[\sum_{s = t}^{\tau_{1}-1}\Psi_{i}(s) + U^{i}_{\tau_{1}} \biggm\vert \mathcal{F}_{t}\right] \nonumber \\
	& = \mathsf{E}\left[\sum_{s = t}^{\tau_{1}-1}\Psi_{i}(s) + \Gamma_{i}\mathbf{1}_{\lbrace \tau_{1} = T \rbrace} + \max\limits_{j \neq i} \left\lbrace Y^{j}_{\tau_{1}} - \gamma_{i,j}\left(\tau_{1}\right) \right\rbrace \mathbf{1}_{\lbrace \tau_{1} < T \rbrace} \biggm\vert \mathcal{F}_{t}\right] \nonumber \\
	& \ge \mathsf{E}\left[\sum_{s = t}^{\tau_{1}-1}\Psi_{i}(s) + \Gamma_{i}\mathbf{1}_{\lbrace \tau_{1} = T \rbrace} + \left\lbrace Y^{\iota_{1}}_{\tau_{1}} - \gamma_{i,\iota_{1}}\left(\tau_{1}\right) \right\rbrace \mathbf{1}_{\lbrace \tau_{1} < T \rbrace} \biggm\vert \mathcal{F}_{t}\right].
	\end{align*}
	Repeating the arguments leading to \eqref{eq:Verification6}, replacing the equalities $(=)$ with inequalities ($\ge$) due to possible sub-optimality of $(\tau_{n},\iota_{n})$ for $n \ge 2$, eventually leads to
	\begin{align*}
	Y^{i}_{t} \ge {} & \mathsf{E}\left[\sum_{s = t}^{T-1}\Psi_{\mathbf{u}_{s}}(s) + \sum_{k = 0}^{N(\alpha) }\Gamma_{\iota_{k}}\mathbf{1}_{\lbrace \tau_{k} < T \rbrace}\mathbf{1}_{\lbrace \tau_{k+1} = T \rbrace} - \sum_{n = 1}^{N(\alpha) + 1}\gamma_{\iota_{n-1},\iota_{n}}\left(\tau_{n}\right)\mathbf{1}_{\lbrace \tau_{n} < T \rbrace}\biggm \vert \mathcal{F}_{t}\right] \\
	& + \mathsf{E}\left[Y^{\iota_{N(\alpha) + 1}}_{\tau_{N(\alpha) + 1}} \mathbf{1}_{\lbrace \tau_{N(\alpha) + 1} < T \rbrace} \biggm \vert \mathcal{F}_{t}\right] \\
	= {} & \mathsf{E}\left[\sum_{s = t}^{T-1}\Psi_{\mathbf{u}_{s}}(s) + \Gamma_{\iota_{N(\alpha)}} - \sum_{n \ge 1}\gamma_{\iota_{n-1},\iota_{n}}\left(\tau_{n}\right)\mathbf{1}_{\lbrace \tau_{n} < T\rbrace} \biggm \vert \mathcal{F}_{t}\right] \\
	= {} & J(\alpha;t,i)
	\end{align*}
	and proves that the strategy $\alpha^{*}$ is optimal.
\end{proof}
\section{Existence of the optimal processes}\label{Section:Discrete-Time-OS-Existence}
This section addresses the existence of the processes $Y^{1},\ldots,Y^{m}$ used in Theorem~\ref{theorem:verification}. The proof is a constructive one and verifies that the explicit dynamic programming scheme cited in \cite{Carmona2008,Gassiat2012} and elsewhere indeed solves the optimal switching problem.
\subsection{Backward dynamic programming}
\begin{lemma}[Backward Induction]\label{Lemma:BackwardInductionExistence}
	For each $i \in \mathbb{I}$, define the process $\tilde{Y}^{i} = \big(\tilde{Y}^{i}_{t}\big)_{t \in \mathbb{T}}$ recursively as follows:
	\begin{equation}\label{Definition:BackwardInduction}
	\begin{split}
	\tilde{Y}^{i}_{T} & = \Gamma_{i},\hspace{1 em}\text{ and for } t = T - 1,\ldots,0 : \\
	\tilde{Y}^{i}_{t} & = \max\limits_{j \neq i} \left\lbrace -\gamma_{i,j}\left(t\right) + \Psi_{j}(t) + \mathsf{E}\left[\tilde{Y}^{j}_{t+1} \vert \mathcal{F}_{t}\right] \right\rbrace \vee \left\lbrace \Psi_{i}(t) + \mathsf{E}\left[\tilde{Y}^{i}_{t+1} \vert \mathcal{F}_{t}\right]\right\rbrace.
	\end{split}
	\end{equation}
	Then $\tilde{Y}^{i}$ is $\mathbb{F}$-adapted and in $\mathcal{S}^{2}$.
\end{lemma}
\begin{proof}
	One verifies that $\tilde{Y}^{i}$ is a well-defined $\mathbb{F}$-adapted process by proceeding recursively for $t = T,\ldots,0$ using~\eqref{Definition:BackwardInduction}, noting that the conditional expectations are well-defined by the integrability conditions on the rewards and switching costs. Note that in this discrete-time setting, showing $\tilde{Y}^{i} \in \mathcal{S}^{2}$ is equivalent to showing $\tilde{Y}^{i}_{t} \in L^{2}$ for every $t \in \mathbb{T}$. Since $\tilde{Y}^{i}_{T} = \Gamma_{i} \in L^{2}$ for all $i \in \mathbb{I}$, the claim is true for $t = T$. Suppose by induction on $t = T-1,\ldots,0$ that $\tilde{Y}^{j}_{t+1} \in L^{2}$ for all $j \in \mathbb{I}$. The backward induction formula \eqref{Definition:BackwardInduction} gives:
	\begin{align}\label{eq:RecursiveIntegrability1}
	\left|\tilde{Y}^{i}_{t}\right| & = \left|\max\limits_{j \neq i} \left\lbrace -\gamma_{i,j}\left(t\right) + \Psi_{j}(t) + \mathsf{E}\left[\tilde{Y}^{j}_{t+1} \big\vert \mathcal{F}_{t}\right] \right\rbrace \vee \left\lbrace \Psi_{i}(t) + \mathsf{E}\left[\tilde{Y}^{i}_{t+1} \big\vert \mathcal{F}_{t}\right]\right\rbrace\right| \nonumber \\
	& \le \left|\max\limits_{j \neq i} \left\lbrace -\gamma_{i,j}\left(t\right) + \Psi_{j}(t) + \mathsf{E}\left[\tilde{Y}^{j}_{t+1} \big\vert \mathcal{F}_{t}\right] \right\rbrace\right| + \left|\left\lbrace \Psi_{i}(t) + \mathsf{E}\left[\tilde{Y}^{i}_{t+1} \big\vert \mathcal{F}_{t}\right]\right\rbrace\right| \nonumber \\
	& \le 2\max_{j \in I}\mathsf{E}\left[|\tilde{Y}^{j}_{t+1}|\big\vert \mathcal{F}_{t}\right] + \max\limits_{j,k \in \mathbb{I}}\max_{r \in \mathbb{T}}\left|\gamma_{j,k}(r)\right| + 2\max\limits_{j \in \mathbb{I}}\max_{r \in \mathbb{T}}\left|\Psi_{j}(r)\right|
	\end{align}
	Note that by Jensen's inequality (\cite[p.~139]{Rogers2000a}), the conditional expectation satisfies,
	\[
	\mathsf{E}\left[\left|\mathsf{E}\left[|\tilde{Y}^{j}_{t+1}|\big\vert \mathcal{F}_{t}\right]\right|^{2}\right] \le \mathsf{E}\left[|\tilde{Y}^{j}_{t+1}|^{2}\right] < +\infty,
	\]
	and is therefore also in $L^{2}$. In addition to the observation that $\mathbb{I}$ is finite and $\Psi_{j},\gamma_{j,k} \in \mathcal{S}^{2}$ for every $j,k \in \mathbb{I}$, the random variable on the right-hand side of \eqref{eq:RecursiveIntegrability1} is in $L^{2}$. Therefore $\tilde{Y}^{i}_{t} \in L^{2}$, which holds for every $i \in \mathbb{I}$ since $i$ was arbitrary. The case $t = T-1$ has already been verified so the proof by induction is complete.
\end{proof}
\subsubsection{An explicit Snell envelope system.}
	A connection between $\tilde{Y}^{i}$ in Lemma~\ref{Lemma:BackwardInductionExistence} and the Snell envelope becomes apparent upon defining a new process $\big(\hat{Y}^{i}_{t}\big)_{t \in \mathbb{T}}$ for every $i \in \mathbb{I}$ by,
	\begin{equation}\label{eq:ExistenceNewProcess}
	\hat{Y}^{i}_{t} \coloneqq \tilde{Y}^{i}_{t} + \sum_{s = 0}^{t-1} \Psi_{i}(s),
	\end{equation}
	A backward induction formula for $\hat{Y}^{i}_{t}$ is then obtained by adding the $\mathcal{F}_{t}$-measurable term $\sum_{s = 0}^{t-1} \Psi_{i}(s)$ to both sides of \eqref{Definition:BackwardInduction}:

	\begin{equation}\label{eq:ExistenceNewProcessBackwardInduction}
	\begin{split}
	\hat{Y}^{i}_{T} & = \hat{U}^{i}_{T},\hspace{1 em}\text{ and for } t = T - 1,\ldots,0 : \\
	\hat{Y}^{i}_{t} & = \hat{U}^{i}_{t} \vee \mathsf{E}\left[\hat{Y}^{i}_{t+1} \big\vert \mathcal{F}_{t}\right].
	\end{split}
	\end{equation}
	where $\big(\hat{U}^{i}_{t}\big)_{t \in \mathbb{T}}$ is the following \emph{explicit gain process}:
	\begin{align}\label{eq:ExistenceExplicitGainProcess}
	\hat{U}^{i}_{t} \coloneqq {} & \max\limits_{j \neq i} \left\lbrace -\gamma_{i,j}\left(t\right) + \sum_{s = 0}^{t-1} \left(\Psi_{i}(s) - \Psi_{j}(s)\right) + \mathsf{E}\left[\hat{Y}^{j}_{t+1} \vert \mathcal{F}_{t}\right] \right\rbrace \mathbf{1}_{\lbrace t < T \rbrace} \nonumber \\
	& + \left\lbrace \sum_{s = 0}^{T-1} \Psi_{i}(s) + \Gamma_{i} \right \rbrace\mathbf{1}_{\lbrace t = T \rbrace} \nonumber \\
	= {} & \sum_{s=0}^{t-1} \Psi_{i}(s) + \max\limits_{j \neq i} \left\lbrace -\gamma_{i,j}\left(t\right) - \sum_{s = 0}^{t-1} \Psi_{j}(s) + \mathsf{E}\left[\hat{Y}^{j}_{t+1} \vert \mathcal{F}_{t}\right] \right\rbrace \mathbf{1}_{\lbrace t < T \rbrace} \nonumber \\
	& + \Gamma_{i}\mathbf{1}_{\lbrace t = T \rbrace}.
	\end{align}
	The processes $\big(\hat{Y}^{i}_{t}\big)_{t \in \mathbb{T}}$ and $\big(\hat{U}^{i}_{t}\big)_{t \in \mathbb{T}}$ belong to $\mathcal{S}^{2}$ by integrability properties of the rewards, switching costs and as $\tilde{Y}^{i} \in \mathcal{S}^{2}$. Proposition~\ref{Proposition:DiscreteSnellEnvelopeProperties} verifies that the backward induction formula uniquely defines $\big(\hat{Y}^{i}_{t}\big)_{t \in \mathbb{T}}$ as the Snell envelope of $\big(\hat{U}^{i}_{t}\big)_{t \in \mathbb{T}}$.
\begin{theorem}[Existence]\label{theorem:ExistenceSnellEnevelope}
	Let $\big(\tilde{Y}^{i}_{t}\big)_{t \in \mathbb{T}}$, $i \in \mathbb{I}$, be the processes defined by backward induction \eqref{Definition:BackwardInduction}. Then, $\mathsf{P}$-a.s. for every $t \in \mathbb{T}$:

	\begin{equation}\label{eq:BackwardInductionSnellEnvelope}
	\tilde{Y}^{i}_{t} = \esssup\limits_{\tau \in \mathcal{T}_{t}} \mathsf{E}\left[\sum_{s=t}^{\tau-1}\Psi_{i}(s) + \Gamma_{i}\mathbf{1}_{\lbrace \tau = T \rbrace} + \max\limits_{j \neq i} \left\lbrace \tilde{Y}^{j}_{\tau} - \gamma_{i,j}\left(\tau\right) \right\rbrace \mathbf{1}_{\lbrace \tau < T \rbrace} \biggm \vert \mathcal{F}_{t}\right].
	\end{equation}
	Therefore, $\tilde{Y}^{1},\ldots,\tilde{Y}^{m}$ satisfy the verification theorem.
\end{theorem}
\begin{proof}
	For notational convenience, introduce a new process $\big(\hat{W}^{i}_{t}\big)_{t \in \mathbb{T}}$ which is defined for $t \in \mathbb{T}$ by
	\begin{equation}\label{eq:Discrete-Time-Optimal-Switching-Implicit-Gain}
	\hat{W}^{i}_{t} \coloneqq \sum_{s=0}^{t-1} \Psi_{i}(s) + \Gamma_{i}\mathbf{1}_{\lbrace t = T \rbrace} + \max\limits_{j \neq i} \left\lbrace -\gamma_{i,j}\left(t\right) - \sum_{s = 0}^{t-1} \Psi_{j}(s) + \hat{Y}^{j}_{t} \right\rbrace \mathbf{1}_{\lbrace t < T \rbrace}.
	\end{equation}
	Note that $\hat{W}^{i} \in \mathcal{S}^{2}$ by the properties of $\Gamma_{i},\Psi_{i},\gamma_{i,j}$ and $\hat{Y}^{j}$ for $i,j \in \mathbb{I}$. Equation~\eqref{eq:BackwardInductionSnellEnvelope} can be proved if
	\begin{equation}\label{eq:BackwardInductionSnellEnvelopeNewProcess}
	\forall t \in \mathbb{T}: \hat{Y}^{i}_{t} = \esssup\limits_{\tau \in \mathcal{T}_{t}} \mathsf{E}\bigl[\hat{W}^{i}_{\tau} \big \vert \mathcal{F}_{t}\bigr], \enskip \mathsf{P}-\text{a.s.}
	\end{equation}

	Indeed, if equation~\eqref{eq:BackwardInductionSnellEnvelopeNewProcess} is true then by~\eqref{eq:ExistenceNewProcess}: $\mathsf{P}-a.s.$,
	\begin{align*}
	\tilde{Y}^{i}_{t} & = \esssup\limits_{\tau \in \mathcal{T}_{t}} \mathsf{E}\bigl[\hat{W}^{i}_{\tau} \big \vert \mathcal{F}_{t}\bigr] - \sum_{s = 0}^{t-1} \Psi_{i}(s) \\
	& = \esssup\limits_{\tau \in \mathcal{T}_{t}} \mathsf{E}\left[\sum_{s = 0}^{\tau-1} \Psi_{i}(s) + \Gamma_{i}\mathbf{1}_{\lbrace \tau = T \rbrace} + \max\limits_{j \neq i} \left\lbrace \tilde{Y}^{j}_{\tau} - \gamma_{i,j}\left(\tau\right) \right\rbrace \mathbf{1}_{\lbrace \tau < T \rbrace} \biggm \vert \mathcal{F}_{t}\right] - \sum_{s = 0}^{t-1} \Psi_{i}(s) \\
	& = \esssup\limits_{\tau \in \mathcal{T}_{t}} \mathsf{E}\left[\sum_{s = t}^{\tau-1} \Psi_{i}(s) + \Gamma_{i}\mathbf{1}_{\lbrace \tau = T \rbrace} + \max\limits_{j \neq i} \left\lbrace \tilde{Y}^{j}_{\tau} - \gamma_{i,j}\left(\tau\right) \right\rbrace \mathbf{1}_{\lbrace \tau < T \rbrace} \biggm \vert \mathcal{F}_{t}\right].
	\end{align*}

	In order to prove~\eqref{eq:BackwardInductionSnellEnvelopeNewProcess}, first note by Proposition~\ref{Proposition:DiscreteSnellEnvelopeProperties} the Snell envelope of $\hat{W}^{i}$ exists and, denoting it by $Z^{i} = (Z^{i}_{t})_{t \in \mathbb{T}}$, satisfies
	\[
	Z^{i}_{t} = \esssup\limits_{\tau \in \mathcal{T}_{t}} \mathsf{E}\bigl[\hat{W}^{i}_{\tau} \big \vert \mathcal{F}_{t}\bigr]
	\]
	as well as the backward induction formula
	\begin{equation}\label{eq:Discrete-Time-Optimal-Switching-Backward-Induction-2}
	\begin{split}
	Z^{i}_{T} & = \hat{W}^{i}_{T} ,\hspace{1 em}\text{ and for } t = T - 1,\ldots,0 : \\
	Z^{i}_{t} & = \hat{W}^{i}_{t} \vee \mathsf{E}\bigl[Z^{i}_{t+1} \big\vert \mathcal{F}_{t}\bigr].
	\end{split}
	\end{equation}
	Thus establishing \eqref{eq:BackwardInductionSnellEnvelopeNewProcess} is equivalent to showing that $\hat{Y}^{i}$ is a modification of $Z^{i}$ defined in \eqref{eq:Discrete-Time-Optimal-Switching-Backward-Induction-2}. Note that by Proposition \MakeUppercase{\romannumeral 2}.36.5 of \cite{Rogers2000a} this would also mean that $\hat{Y}^{i}$ and $Z^{i}$ are indistinguishable. This shall be proved inductively.

	Note that $Z^{i}_{T} = \hat{W}^{i}_{T} = \hat{U}^{i}_{T} = \hat{Y}^{i}_{T}$ almost surely for every $i \in \mathbb{I}$. Suppose inductively for $t = T-1,\ldots,0$ that for all $i \in \mathbb{I}$, $Z^{i}_{t+1} = \hat{Y}^{i}_{t+1}$ $\mathsf{P}$-a.s. For every $i \in \mathbb{I}$ define the stopping time $\theta^{i}_{t}$ as follows
	\begin{align}\label{Definition:BackwardInductionStoppingTimes}
	\theta^{i}_{t} = \inf\left\lbrace t \le s \le T \colon \hat{Y}^{i}_{s} = \hat{U}^{i}_{s} \right\rbrace
	\end{align}
	noting that $t \le \theta^{i}_{t} \le T$ almost surely. The following lines will establish $Z^{i}_{t} = \hat{Y}^{i}_{t}$ separately on the events $\{\theta^{i}_{t} = t\}$ and $\{\theta^{i}_{t} > t\} \equiv \{\theta^{i}_{t} \ge t + 1\}$. Since $\mathsf{P}(\{\theta^{i}_{t} \ge t\}) = 1$, the previous claim would lead to $Z^{i}_{t} = \hat{Y}^{i}_{t}$ a.s. and the induction argument proves that $\hat{Y}^{i}$ is a modification of $Z^{i}$.

	\paragraph{Case 1: $Z^{i}_{t} = \hat{Y}^{i}_{t} ~on~ \{\theta^{i}_{t} = t\}$.\\}
	Since $\hat{Y}^{j}$ is the Snell envelope of $\hat{U}^{j}$ for $j \in \mathbb{I}$, it is a supermartingale and by definition of $\hat{U}^{i}$ and $\hat{W}^{i}$ this leads to $\hat{U}^{i} \le \hat{W}^{i} \le Z^{i}$ for $i \in \mathbb{I}$. Then, using \eqref{Definition:BackwardInductionStoppingTimes}, the backward induction formula and the induction hypothesis one gets
	\begin{equation}\label{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-1}
	\hat{W}^{i}_{t} \ge \hat{U}^{i}_{t} = \hat{Y}^{i}_{t} \ge \mathsf{E}\bigl[\hat{Y}^{i}_{t+1} \big\vert \mathcal{F}_{t}\bigr] = \mathsf{E}\bigl[Z^{i}_{t+1} \vert \mathcal{F}_{t}\bigr] \enskip \text{on} \enskip \{\theta^{i}_{t} = t\}.
	\end{equation}

	Using the backward induction formula \eqref{eq:Discrete-Time-Optimal-Switching-Backward-Induction-2} for $Z^{i}$ and \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-1} above also shows that
	\begin{equation}\label{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-2}
	\hat{W}^{i}_{t} = Z^{i}_{t} \enskip \text{on} \enskip \{\theta^{i}_{t} = t\}
	\end{equation}
	Using \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-2} and finiteness of $\mathbb{I}$ shows that there exists an $\mathcal{F}_{\theta^{i}_{t}}$-measurable mode $j_{*}$ (that is, $j_{*}$ is an $\mathcal{F}_{\theta^{i}_{t}}$-measurable $\mathbb{I}$-valued random variable) such that
	\begin{equation}\label{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-3}
	\begin{split}
	Z^{i}_{t} & = \sum_{s=0}^{t-1} \Psi_{i}(s) - \gamma_{i,j_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{j_{*}}(s) + \hat{Y}^{j_{*}}_{t}, \\
	j_{*} & = \argmax\limits_{j \neq i} \left\lbrace -\gamma_{i,j}\left(t\right) - \sum_{s = 0}^{t-1} \Psi_{j}(s) + \hat{Y}^{j}_{t} \right\rbrace \enskip \text{on} \enskip \{\theta^{i}_{t} = t\}.
	\end{split}
	\end{equation}
	Let us show that
	\begin{equation}\label{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-4}
	\mathbf{1}_{\{\theta^{i}_{t} = t\}}\hat{Y}^{j_{*}}_{t} = \mathbf{1}_{\{\theta^{i}_{t} = t\}}\mathsf{E}\bigl[\hat{Y}^{j_{*}}_{t+1} \big\vert \mathcal{F}_{t}\bigr] \quad \mathsf{P}-\text{a.s.}
	\end{equation}
	Since $j_{*}$ is $\mathcal{F}_{\theta^{i}_{t}}$-measurable, one has for $t \le r \le T$
	\[
	\mathsf{E}\left[\sum_{j \in \mathbb{I}}\mathbf{1}_{\{j_{*} = j\}}\hat{Y}^{j}_{r} \biggm\vert \mathcal{F}_{t}\right] = \sum_{j \in \mathbb{I}}\mathbf{1}_{\{j_{*} = j\}}\mathsf{E}\bigl[\hat{Y}^{j}_{r} \big\vert \mathcal{F}_{t}\bigr] \le \sum_{j \in \mathbb{I}}\mathbf{1}_{\{j_{*} = j\}}\hat{Y}^{j}_{t} \enskip \text{ on } \enskip \{\theta^{i}_{t} = t \}
	\]
	so that $\hat{Y}^{j_{*}} \coloneqq \sum_{j \in \mathbb{I}}\hat{Y}^{j}\mathbf{1}_{\{j_{*} = j\}}$ is a supermartingale on $[\theta^{i}_{t},T]$. Now if \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-4} is not true, by the supermartingale property of $\hat{Y}^{j_{*}}$ the event $A^{i}_{t}$ defined by
	\[
	A^{i}_{t} \coloneqq \big\{\hat{Y}^{j_{*}}_{t} > \mathsf{E}\bigl[\hat{Y}^{j_{*}}_{t+1} \big \vert \mathcal{F}_{t}\bigr]\big\} \cap \{\theta^{i}_{t} = t\}
	\]
	has positive probability. If $\mathsf{P}(A^{i}_{t}) > 0$, there exists an $\mathcal{F}_{\theta^{i}_{t}}$-measurable mode $k_{*}$ such that
	\[
	\hat{Y}^{j_{*}}_{t} = \sum_{s=0}^{t-1} \Psi_{j_{*}}(s) -\gamma_{j_{*},k_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{k_{*}}(s) + \mathsf{E}\bigl[\hat{Y}^{k_{*}}_{t+1} \vert \mathcal{F}_{t} \bigr] \enskip \text{on} \enskip A^{i}_{t}
	\]
	This leads to
	\begin{align}
	Z^{i}_{t} = {} & \sum_{s=0}^{t-1} \Psi_{i}(s) -\gamma_{i,j_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{j_{*}}(s) + \hat{Y}^{j_{*}}_{t}  \nonumber \\
	= {} & \sum_{s=0}^{t-1} \Psi_{i}(s) -\gamma_{i,j_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{j_{*}}(s) + \sum_{s=0}^{t-1} \Psi_{j_{*}}(s) \nonumber \\
	& -\gamma_{j_{*},k_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{k_{*}}(s) + \mathsf{E}\bigl[\hat{Y}^{k_{*}}_{t+1} \big\vert \mathcal{F}_{t} \bigr] \nonumber \\
	& < \sum_{s=0}^{t-1} \Psi_{i}(s) -\gamma_{i,k_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{k_{*}}(s) + \hat{Y}^{k_{*}}_{t} \label{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-5}  \enskip \text{on} \enskip A^{i}_{t}
	\end{align}
	where the inequality comes from the no-arbitrage condition~\eqref{assumption:Discrete-Time-SwitchingCosts-No-Arbitrage} and the supermartingale property of $\hat{Y}^{k_{*}}$ on $[\theta^{i}_{t},T]$. However, this contradicts the optimality of $j_{*}$ and therefore shows that \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-4} holds.

	Using \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-3} together with \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-4} and the definition of $\hat{Y}^{i}$ yields:
	\[
	Z^{i}_{t} = \sum_{s=0}^{t-1} \Psi_{i}(s) -\gamma_{i,j_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{j_{*}}(s) + \mathsf{E}\bigl[\hat{Y}^{j_{*}}_{t+1} \big\vert \mathcal{F}_{t}\bigr] \le \hat{Y}^{i}_{t} \enskip \text{on} \enskip \{\theta^{i}_{t} = t\}
	\]
	and using this with \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-1} and \eqref {eq:Proof-BackwardInductionSnellEnvelopeNewProcess-2} gives:
	\begin{equation}\label{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-6}
	\hat{W}^{i}_{t} = \hat{U}^{i}_{t} = \hat{Y}^{i}_{t} = Z^{i}_{t} \enskip \text{on} \enskip \{\theta^{i}_{t} = t\}.
	\end{equation}

	\paragraph{Case 2: $Z^{i}_{t} = \hat{Y}^{i}_{t} ~on~ \{\theta^{i}_{t} \ge t + 1\}$.\\}
	Note that $\{\theta^{i}_{t} \ge t + 1\} \equiv \{\theta^{i}_{t} > t\}$ and is therefore $\mathcal{F}_{t}$-measurable. The properties of the Snell envelopes $\hat{Y}^{i}$ and $Z^{i}$ together with the induction hypothesis then give
	\begin{equation}\label{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-7}
	\hat{Y}^{i}_{t} = \mathsf{E}\bigl[\hat{Y}^{i}_{t+1} \big \vert \mathcal{F}_{t}\bigr] = \mathsf{E}\bigl[Z^{i}_{t+1} \big \vert \mathcal{F}_{t}\bigr] \le Z^{i}_{t} \enskip \text{on} \enskip \{\theta^{i}_{t} \ge t + 1\}.
	\end{equation}
	Suppose that $Z^{i}$ is a strict supermartingale on $\{\theta^{i}_{t} \ge t + 1\}$ with positive probability. Then the $\mathcal{F}_{t}$-measurable event $B^{i}_{t}$ defined by
	\[
	B^{i}_{t} \coloneqq \left\{\mathsf{E}\left[Z^{i}_{t+1} \big \vert \mathcal{F}_{t}\right] < Z^{i}_{t} \right\} \cap \{\theta^{i}_{t} \ge t + 1\}
	\]
	has positive probability, and implies the existence of an $\mathcal{F}_{t}$-measurable mode $j_{*}$ such that
	\begin{equation}\label{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-8}
	\begin{split}
	\hat{Y}^{i}_{t} < Z^{i}_{t} & = \sum_{s=0}^{t-1} \Psi_{i}(s) - \gamma_{i,j_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{j_{*}}(s) + \hat{Y}^{j_{*}}_{t}, \\
	j_{*} & = \argmax\limits_{j \neq i} \left\lbrace -\gamma_{i,j}\left(t\right) - \sum_{s = 0}^{t-1} \Psi_{j}(s) + \hat{Y}^{j}_{t} \right\rbrace \enskip \text{on} \enskip B^{i}_{t}.
	\end{split}
	\end{equation}
	But by definition of $\hat{Y}^{i}$, it is true that
	\[
	\hat{Y}^{i}_{t} \ge \sum_{s=0}^{t-1} \Psi_{i}(s) - \gamma_{i,j_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{j_{*}}(s) + \mathsf{E}\bigl[\hat{Y}^{j_{*}}_{t+1} \big\vert \mathcal{F}_{t}\bigr] \enskip \text{on} \enskip B^{i}_{t}
	\]
	and using this in \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-8} shows
	\[
	\mathsf{E}\bigl[\hat{Y}^{j_{*}}_{t+1} \big\vert \mathcal{F}_{t}\bigr] < \hat{Y}^{j_{*}}_{t} \enskip \text{on} \enskip B^{i}_{t}.
	\]
	Using the same arguments leading up to \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-5}, one can show that there exists an $\mathcal{F}_{t}$-measurable mode $k_{*}$ such that
	\begin{align*}
	Z^{i}_{t} = {} & \sum_{s=0}^{t-1} \Psi_{i}(s) -\gamma_{i,j_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{j_{*}}(s) + \hat{Y}^{j_{*}}_{t}  \\
	= {} & \sum_{s=0}^{t-1} \Psi_{i}(s) -\gamma_{i,j_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{j_{*}}(s) + \sum_{s=0}^{t-1} \Psi_{j_{*}}(s) \nonumber \\
	& -\gamma_{j_{*},k_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{k_{*}}(s) + \mathsf{E}\bigl[\hat{Y}^{k_{*}}_{t+1} \vert \mathcal{F}_{t} \bigr] \nonumber \\
	& < \sum_{s=0}^{t-1} \Psi_{i}(s) -\gamma_{i,k_{*}}(t) - \sum_{s = 0}^{t-1} \Psi_{k_{*}}(s) + \hat{Y}^{k_{*}}_{t}  \enskip \text{on} \enskip B^{i}_{t}
	\end{align*}
	which contradicts the optimality of $j_{*}$. Thus $\mathsf{P}\big(B^{i}_{t}\big) = 0$ and this shows
	\begin{equation}\label{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-9}
	\mathbf{1}_{\{\theta^{i}_{t} \ge t + 1\}}\mathsf{E}\bigl[Z^{i}_{t+1} \big \vert \mathcal{F}_{t}\bigr] = Z^{i}_{t}\mathbf{1}_{\{\theta^{i}_{t} \ge t + 1\}},\enskip \mathsf{P}-\text{a.s.}
	\end{equation}
	Finally, putting \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-9}, \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-7} and \eqref{eq:Proof-BackwardInductionSnellEnvelopeNewProcess-6} together gives
	\[
	\hat{Y}^{i}_{t} = Z^{i}_{t},\quad \mathsf{P}-\text{a.s.}
	\]
	Since the case $t = T-1$ is true and $i \in \mathbb{I}$ was arbitrary, the proof by induction is complete. Therefore, for every $i \in \mathbb{I}$,
	\[
	\forall t \in \mathbb{T}: \hat{Y}^{i}_{t} = Z^{i}_{t} \quad \mathsf{P}-\text{a.s.}
	\]
	which means $\hat{Y}^{i}$ is a modification of (and therefore indistinguishable from) $Z^{i}$, whence~\eqref{eq:BackwardInductionSnellEnvelopeNewProcess} follows.
\end{proof}
\begin{remark}
	By equation~\eqref{eq:BackwardInductionSnellEnvelopeNewProcess} and Proposition~\ref{Proposition:DiscreteSnellEnvelopeProperties}, the arguments used in the previous proof show that $\hat{Y}^{i},~i \in \mathbb{I}$, satisfies:
	\[
	\begin{split}
	\tilde{Y}^{i}_{T} & = \Gamma_{i},\hspace{1 em}\text{ and for } t = T - 1,\ldots,0 : \\
	\tilde{Y}^{i}_{t} & = \max\limits_{j \neq i} \left\lbrace -\gamma_{i,j}\left(t\right) + \tilde{Y}^{j}_{t} \right\rbrace \vee \left\lbrace \Psi_{i}(t) + \mathsf{E}\bigl[\tilde{Y}^{i}_{t+1} \big\vert \mathcal{F}_{t}\bigr]\right\rbrace.
	\end{split}
	\]
	which is the (non-Markovian) analogue to the implicit backward induction formula of \cite{Gassiat2012}. However, the steps carried out here are in the reverse direction to \cite[pp.~2037]{Gassiat2012}, where the explicit backward induction formula was derived from the implicit one. The latter was obtained from an approximation to a continuous-time optimal switching problem with non-negative switching costs.
\end{remark}

\section{Conclusion}
In this paper we used a probabilistic approach to solve a finite-horizon discrete-time optimal switching problem for a model with a running reward, terminal reward and signed switching costs. The approach, which works without Markovian assumptions, reduced the switching problem to iterated optimal stopping problems defined in terms of (coupled) Snell envelopes, just as in the verification theorem of \cite{Djehiche2009} in the continuous-time case. We were able to define the Snell envelopes by an explicit backward induction scheme, thereby extending the numerical methods of \cite{Carmona2008,Gassiat2012} to problems with negative switching costs.

redpajama_set_name: RedPajamaArXiv


[2014.08.27] AKB48 - 37th Single "Kokoro no Placard"
Re: [2014.08.27] AKB48 - 37th Single "Kokoro no Placard"
3rd round handshake sales, as romanised by pinkmonster@onehallyu, thanks!
Handshake Results up to Round 3 (it only includes members that have sold out at least one slot or are close to doing so).
On July 21, it was announced that AKB48's Team 8, which is comprised of members hailing from all 47 prefectures of Japan, will record their first original track entitled "47 no Suteki na Machi e." The track is for their upcoming 37th single "Kokoro no Placard," which is set to be released on August 27.
Each of the members will sing their home-prefecture name in the interlude of the track. Hyogo-born member Yamada Nanami (15) stated, "I would like to travel to many places so that many people can hear the song."
Was I the only one here looking for TGSK in that?


text:
The Sun Times is a local newspaper which services the Bruce-Grey-Owen Sound area in the Canadian province of Ontario. Its headquarters are in Owen Sound.  The Times newspaper was founded in 1853 and The Sun newspaper was founded in 1893; they amalgamated in 1918. Daily editions of the amalgamated paper started in 1922.

The newspaper is owned and operated by Postmedia.

The current editor is Doug Edgar, a former reporter for the paper.

See also
List of newspapers in Canada

External links

Postmedia Network publications
Mass media in Owen Sound
Daily newspapers published in Ontario
Publications established in 1918
1918 establishments in Ontario

redpajama_set_name: RedPajamaWikipedia


text:
Франц Сенн (* 19. Березень 1831 в Унтерлангенфельді; † 31 Січень 1884 в Нойштифт-ім-Штубайталі ) — австрійський священик і альпініст. У 1869 році він був одним із засновників Німецького альпійського клубу, також відомий як священик льодовика.

Життя
Франц Зенн, який походив із фермерської родини в Ецталі, навчався після відвідування єзуїтської середньої школи в Інсбруку (до 1851) спочатку чотири семестри філософії в Інсбруку та Мюнхені, потім теології в семінарії в Бріксені. У 1856 році висвячений на священика . Після багатьох років роботи кооператора в Замсі, Серфаусі та Ландеку, його місця роботи були з 1860 року до Венту в Ецталі, оточеному льодовиками, які він розробив для туризму. Разом з гірським провідником Кіпріаном Гранбіхлером та іншими він здійснив декілька перших сходжень, таких як Фінайльшпітце, Вернагтшпіце та Кройцшпіце. З 1872 по 1881 рік Сенн працював у Наудерсі, а потім у Нойштифті в Штубайталі, де й помер 31 року. Січень 1884 р. помер і там, де досі пам'ятний камінь позначає місце його могили.

Коли в 1860 році Франц Сенн прийняв виборство Святого Якоба у Венті як викан і тимчасовий, він незабаром зрозумів, що рівень життя населення можна підвищити лише за допомогою розвитку туризму . Його ідея була прокласти шляхи, підйоми та побудувати укриття в той час, коли майже не було карт цієї великої льодовикової території. Щоб зробити гірський світ Вент більш відомим, Сенн замовив панораму з вершини Кройцшпітце, спочатку в 1868 році від пейзажиста Карла Бріцці, а наступного року — від Карла Йордана та Георга Енгельгардта завдяки тому, що Сенн відчував незадовільний результат.У 1869 році Сенн разом із однодумцями заснував у Мюнхені Німецький альпійський клуб; Це також сталося через те, що люди були незадоволені більш науковою спрямованістю Австрійського альпійського клубу, заснованого у Відні в 1862 році. У 1883 році його хвороба на туберкульоз загострилася, тому він відправився в Меран, щоб одужати. Однак він так і не одужав і помер 31 березня. Січень 1884 р.

Сенна іноді вважають засновником туризму в Тіролі. Будинок, де він народився, де помер 19 У березні 1831 року в Унтерленгенфельді народився, його все ще можна побачити ззовні в Ецталі сьогодні. Франц Зенн також був членом KDStV Aenania Munich у CV.

Відзнаки
У Ленгенфельді, Венті, Наудерсі та Нойштіфті в Штубайталі є меморіали та меморіальні дошки для Зенна. Сеннкогель (3398 м u. A.) на гребені Ецтальських Альп і Франц-Зенн-Хютте в Обербергталі були названі його іменем. Франц-Зенн-Вег у Венті також названий на його честь.

Літератури

 Йозеф Муссхаузер: Священик льодовика. Роман Франца Сенна. Видавництво Розенхайма Альфред Форг, Розенхайм 1962
 Луї Обервальдер : пастор Франц Сенн, забутий вихователь дорослих на його 100-річчя. річниця смерті . Інсбрук. - 1984
 Луї Обервальдер, Ніко Майяндер, Ганс Гайд, Пітер Хасслахер і Франц Флірі: Франц Сенн. Піонер альпінізму та засновник Альпійського клубу. Тіролія, Інсбрук та Відень 2004 р., ISBN 3-7022-2629-X
 Екарт Ролофф : Гори - Божий дар для фермерів. (Підзаголовок: Син фермера і священик Франц Зенн рано знав, який прибуток може принести туризм. Так він став засновником Німецького альпійського клубу.) У: Landwirtschaftliche Magazin Rheinland, випуск 30/2010, с. 64-65.

 Wolfgang G. Schöpf: Франц Сен.

Веб-Посилання

 Personenmappe zu Franz Senn (I) (PDF) im Historischen Alpenarchiv der Alpenvereine in Deutschland, Österreich und Südtirol (temporär offline)
 Personenmappe zu Franz Senn (II) (PDF) im Historischen Alpenarchiv der Alpenvereine in Deutschland, Österreich und Südtirol (temporär offline)

Предмети

Померли 1884
Народились 1831


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