Yohann Pitrey
email: yohann.pitrey@univ-nantes.fr

Marcus Barkowsky
email: marcus.barkowsky@univ-nantes.fr

Patrick Le Callet
email: patrick.lecallet@univ-nantes.fr

Romuald Pepion
email: romuald.pepion@univ-nantes.fr

SUBJECTIVE QUALITY ASSESSMENT OF MPEG-4 SCALABLE VIDEO CODING IN A MOBILE SCENARIO

Keywords: Scalable video coding, Subjective experiments, Quality evaluation

Scalable Video Coding provides several levels of video encapsulated in a single video stream. In a transmission scenario such as broadcasting, this structure is quite advantageous as it can be used to address heterogeneous decoding targets with variable needs and requirements. However, this adaptability comes at a slight cost in coding efficiency when compared to single-layer coding. Based on subjective experiments, this cost is evaluated in this paper by comparing the new MPEG-4 Scalable Video Coding (SVC) standard with the now-established MPEG-4 AVC standard. Two scenarios are analyzed in the context of mobile transmission applications. The first scenario uses the same bitrate for SVC and AVC, leading to a slightly lower PSNR for SVC. The second scenario uses the same PSNR for SVC and AVC, leading to a slightly lower bitrate for AVC. The results of the subjective tests illustrate several interesting aspects of the relation between the performance of the two standards. First, we observe that the offset between AVC and SVC is not severe, though statistically significant in terms of user Mean Opinion Score (MOS) in such a context. Second, while adding another layer to SVC always leads to a performance loss, the impact of this loss decreases when the number of layers increases.

INTRODUCTION

Video coding has become an essential area in today's visual communications. For nearly 30 years, research has been carried out to design technics to represent the video contents with as few data as possible. One of the latest accomplishments in the field of video coding is the H.264/MPEG-4 Advanced Video Coding (AVC) standard [START_REF] Wiegand | Overview of the H.264/AVC video coding standard[END_REF]. Using spatial and temporal prediction to remove redundancy, coupled with Discrete Cosine Transform (DCT), scalar quantization and entropy coding, AVC outperforms the coding performance of previous standards in a significant way. A Network Abstraction Layer (NAL) is additionally used to ease the transmission of the video on packet-based networks. Using this new standard, high quality can be reached at relatively low bitrates to produce video streams particularly adapted for video transmission.

Nevertheless, the need for flexibility is becoming crucial in current video applications. Typically, a video service provider has to address heterogeneous display devices such as mobile phones or residential televisions, using different transmission channels such as DSL links or wireless technologies such as WiFi or GSM. Despite its high performance when compared to previous video coding standards, AVC does not provide any convenient tools to adapt a stream to different targets. Consequently, several versions of the same video content have to be encoded for each type of target. In a broadcasting context, these different versions have to be transmitted all together in order to address all types of targets. This kind of scenario called simulcast obviously leads to a waste both in bandwidth and storage space. MPEG-4 Scalable Video Coding (SVC) has been introduced by the Joint Video Team (JVT) as a response to provide flexibility for video coding and to minimize the waste of resources produced by simulcast [START_REF] Reichel | Scalable Video Coding -Joint Draft 4[END_REF]. It allows several resolutions of the same video content to be encoded as different layers of a single scalable stream. Three types of scalability are supported, to address a wide range of targets. Spatial scalability enables frame size tuning, to adapt the video contents for mobile phones as well as for higher spatial resolution display devices such as computers and high-definition televisions. Quality scalability (also referred to as fidelity scalability) provides increasing levels of quality of the stream (in terms of PSNR), essentially to adapt the bitrate needed to represent the video. Temporal scalability allows the number of frames per second to vary from one layer to another, in order to propose a balance between the motion smoothness and the amount of data to process. In addition to the three types of scalability, a new tool called inter-layer prediction is also provided in order to minimize the waste caused by the encoding of several versions of the video. Using inter-layer prediction, spatial and temporal prediction in an enhancement layer can use the information from a lower layer called the base layer to get a better prediction. The redundancy between layers is therefore reduced, improving the coding efficiency of the whole video stream.

Coding a video stream using several layers provides intermediate quality and resolution levels. The stream can be adapted more easily to the different display devices and transmission conditions. Additional value is provided by the ability to adapt the stream according to the actual resources available at a given time. This is particularly useful in mobile environments as the bandwidth and receiving conditions can vary according to environmental factors. A typical use-case would concern a user receiving a video stream on a mobile phone through a given network such as WiMax. If the receiving conditions change due to a loss of the WiMax signal (e. g. in a vehicle in motion), SVC allows to adapt the resolution of the video to the new environment. The user gets a version of the video with lower size, temporal resolution or quality, instead of having to wait for the full-size video data to reach his mobile again.

The enhanced adaptability made possible by SVC logically comes with a price in terms of coding efficiency when compared to AVC. Indeed, coding a stream using several layers re-introduces redundancy, as inter-layer prediction is not capable to fully exploit the correlations between layers. Therefore, it is to expect that to reach a given quality, the bitrate needed to encode several SVC layers is slightly higher than the bitrate needed to encode a single AVC layer. In a similar way, getting the same bitrate for SVC and AVC means a slightly lower quality for SVC.

In this paper, we evaluate the loss introduced by SVC using subjective tests in a mobile transmission context. The test setup we propose involves all three types of scalability, and uses resolutions and bitrates well-suited for lightweight applications. We evaluate the video quality using a validated methodology and controlled viewing conditions, in order to get reliable data and to build a precise analysis of the test results. This paper is organized as follows. Section 2 presents the AVC and SVC standards. Section 3 presents the encoding scenarios we used for our tests, as well as the subjective quality evaluation methodology. Section 4 analyzes the test results, while section 5 concludes the paper.

MPEG-4 SVC PERFORMANCE EVALUATION

The performance of SVC has been studied both in terms of objective and subjective quality. Rate-distortion performance analysis using the PSNR as a quality metric, such as presented in [START_REF] Schwarz | Overview of the Scalable H.264/MPEG4-AVC Extension[END_REF][START_REF] Wien | Performance Analysis of SVC[END_REF], confirm that the performance of SVC is not as good as the performance of AVC, but remains comparable if a small bitrate overhead is permitted. It is shown that with equal PSNR, SVC can save from 17% to 40% of bitrate when compared to an AVC simulcast scenario.

Objective quality metrics (and especially PSNR) do not always depict the actual perceived quality. Therefore some work has also been presented on subjective quality evaluation of the performance of SVC [5,[START_REF] Kouadio | SVC -A highly-scalable version of H.264/AVC[END_REF][START_REF] Staelens | Assessing the perceptual influence of H.264/SVC Signal-to-Noise Ratio and temporal scalability on full length movies[END_REF]. Although it involves expensive hardware and requires strict test conditions, this type of evaluation is closer to the final feeling of quality experienced by a viewer. Therefore the results of subjective tests usually allow a more detailed analysis of the impact of a particular effect on the videos.

For the verification test plan of SVC [5], subjective tests were performed on various SVC configurations, including videoconference, mobile and residential broadcasting and professional video production. This report shows that with a bitrate overhead of about 10% in favor of SVC, no significant visual quality difference could be noticed. Similar results were published by the European Broadcasting Union (EBU) [START_REF] Kouadio | SVC -A highly-scalable version of H.264/AVC[END_REF], in which the performance of SVC is compared to AVC using a subjective metric in the context of residential broadcasting (typically from Standard Definition to High Definition television). It was stated that SVC can compete with AVC, particularly when using high bitrates and a good quality for the base layer. Most of the contributions for performance analysis for SVC focus on residential TV scenarios. The mobile transmission context has been studied in [5], but the test scenario does not fully exploit the possibilities of SVC.

In this paper we report the results of subjective quality assessment tests that were performed to compare AVC simulcast and SVC in a mobile transmission context. The next section describes the subjective quality methodology we used for these tests and the encoding scenarios that were presented to the viewers.

SUBJECTIVE EXPERIMENT SETUP

In our experiment, the purpose is to compare streams encoded using SVC to their equivalent in an AVC simulcast scenario. Four commonly-used video contents are processed : Harbour, Soccer, City and Crew. They reflect a wide variety of contents, ranging from sports to documentary. The SVC video streams contain four layers with frame size, temporal frequency (fps) and bitrate designed for typical mobile transmission applications. A summary is provided for reference in Table 1.

The base layer (denoted as SVC1 in the following) is in QVGA format (320 × 240 pixels) at 15 frames per second. It is particularly suitable for devices such as mobile phones with small screen size and limited processing power. It is encoded at 100 kilobits per second ( kbps), which makes it possible to transmit over UMTS-like channels that typically have a limited transmission capacity of 384 kbps. The second layer for the SVC coding is a temporal enhancement layer (denoted as SVC2), adding another 15 frames per second to the base layer. The decoded video is thus in QVGA format at 30 fps, which makes it affordable for mobile phones with small screen size but higher processing power. The total bi-trate is equal to 250 kbps, which is suitable for HSDPA-like channels. The third layer is a dyadic spatial layer (SVC3), which doubles the frame size. This layer is thus in VGA format (640×480 pixels) at 30 fps. It is adapted for smartphonelike devices with a higher screen resolution and a medium processing power. The total bitrate is 750 kbps, which is designed for WiFi or WiMax networks. The fourth layer (SVC4) is a quality enhancement layer. Another 250 kbps is spent on enhancing the quality in terms of PSNR, leading to a total bitrate of 1000 kbps. This may be transmitted over WiMax if the network conditions are appropriate. The frame size and rate do not change from SVC3 to SVC4.

The input video sequences were generated from full-HD sequences (1920 × 1080 pixels at 60 frames per second), down-scaled using the DownConvert tool provided by the JSVM. The reference software for MPEG-4 SVC Version 8.6 was used to encode all streams [START_REF]JSVM Reference Software[END_REF]. This software is capable of encoding streams both in AVC and SVC. To encode an AVC stream, the number of layers is set to 1. In order to get a fixed bitrate, the FixedQPEncoder utility provided by the JSVM was used. This iterative tool executes the encoding with different parameters until a target bitrate value is reached. It should be stressed that the bitrates are expressed as total bitrates. This means that the sum of bitrate used to encode the full SVC stream containing four layers is equal to 1000 kbps.

From the bitstream used in condition SVC4, all the other conditions SVC3, SVC2, and SVC1 can be extracted without re-encoding. This is possible in SVC because each layer uses the lower layer as a base layer for inter-layer prediction. This is not a feature of AVC, thus two different scenarios for AVC are considered that can be useful to learn about the differences in coding efficiency between the two standards. In the first scenario, the same bitrate is allowed for AVC as for SVC. Four different encodings per video content are used, corresponding to the four conditions used for SVC. They are denoted as AVC-B1 to AVC-B4, where the letter B stands for "bitrate". Please note, that this is a different scenario for the network provider as it might no longer be possible to transmit all qualities over the same channel. The total bitrate for transmitting all four layers to the end user on the same network would be the sum of the individual bitrates, which is 2100kbit/s. In the second scenario, the same PSNR value is used for AVC as for SVC. For each decoded sequence of the four SVC layers, a PSNR value is calculated. These 16 PSNR values are then used to generate an equivalent AVC bitstream. Again, the FixedQPEncoder is used because it can handle a target PSNR value as well. These four conditions for each content are termed AVC-Q1 to AVC-Q4, where the letter Q stands for "quality".

To evaluate the visual quality of the different scenarios, the Subjective Assessment Methodology for VIdeo Quality (SAMVIQ) methodology is used [START_REF]SAMVIQ Subjective Assessment Methodology for video quality[END_REF]. This methodology uses multiple stimuli assessment, which means that a viewer is al- lowed to watch each video as many times as he wants. The viewer gives a score comprised between 0 and 100 on a continuous quality scale (0 being the score of the lowest quality and 100 being the score of the highest quality). The viewer can choose the order in which he wants to view the sequences, while every sequence has to be evaluated once. As the viewer has the opportunity to revise his judgment by viewing each sequence multiple times, the accuracy of the measure is increased when compared to other subjective experiment methods.

Among the sequences presented to the viewer during a test session, two sequences are used as references. The first sequence is explicitly labeled as the high quality reference for the time of the presentation. The second sequence is hidden and randomly presented amongst other coded sequences. In this experiment, the original non-coded video stream is used as the high-quality reference. As the SAMVIQ methodology recommends not to compare two different frame formats in the same test, the QVGA and the VGA formats are handled in two separate test sessions. As a result, the two conditions 1 and 2 are processed in a first test session, while the conditions 3 and 4 are processed in a second test session. As the viewers involved in the first session were different from the viewers from the second session, it is not possible to consider the results of the two sessions as parts of the same test.

Each test session involved 15 viewers, whose age ranged from 17 to 50, with an average of 25. The screen model was a Samsung SyncMaster 1100MB reference screen. According to the ITU recommendation, the distance between the viewer and the display was equal to 6 times the height of the displayed image for the videos in QVGA format, and 4 times for the videos in VGA format. Each viewer was asked to rate 28 video sequences, during a session of approximately 30 min-Table 3. Results of the t-test and MOS comparison between the presented AVC and SVC scenarios.

QVGA Ref SVC1 SVC2 AVC-B1 AVC-B2 AVC-Q1 AVC-Q2 Ref ↑ ↑ ↑ ↑ ↑ ↑ SVC1 ↓ ↓ • ↓ • ↓ SVC2 ↓ ↑ ↑ ↓ ↑ • AVC-B1 ↓ • ↓ ↓ • ↓ AVC-B2 ↓ ↑ ↑ ↑ ↑ ↑ AVC-Q1 ↓ • ↓ • ↓ ↓ AVC-Q2 ↓ ↑ • ↑ ↓ ↑ VGA Ref SVC3 SVC4 AVC-B3 AVC-B4 AVC-Q3 AVC-Q4 Ref ↑ ↑ ↑ ↑ ↑ ↑ SVC3 ↓ ↓ ↓ ↓ • ↓ SVC4 ↓ ↑ ↓ ↓ • • AVC-B3 ↓ ↑ ↑ • ↑ • AVC-B4 ↓ ↑ ↑ • ↑ ↑ AVC-Q3 ↓ • • ↓ ↓ ↓ AVC-Q4 ↓ ↑ • • ↓ ↑ utes.
The experimental data from the presented tests is used in the next section to compare the coding performance of AVC and SVC in the presented scenarios.

EXPERIMENTAL RESULTS

Figure 1 compares the average Mean Opinion Score (MOS) of the AVC and the two SVC scenarios for the four video contents. We can observe that the scores of SVC and AVC-Q are very close for all tested conditions.

As the AVC-Q streams are designed to have the same PSNR as the corresponding SVC layers, we can state that in the context of our experiment, the PSNR is a sufficient approximation for visual quality when evaluating the same content. To reinforce this statement, Table 3 includes the results of the student t-test on the experimental data. The configurations presented to the viewer for each video content are compared in terms of statistical difference. Two configurations can be considered not statistically different when the corresponding cell in Table 3 contains a '•' symbol. On the opposite, if two configurations a and b are statistically different, it is possible to order them in terms of visual quality. If the MOS of configuration a is higher than the MOS of configuration b, the cell located at line a and column b in Table 3 contains a '↑' symbol. Otherwise (the MOS of configuration a is less than the MOS of configuration b), the cell contains a '↓' symbol. According to the t-test, we can confirm that there is no statistically significant difference between the SVC and AVC-Q scenarios.

For the first layer in QVGA at 15 frames per second, the t-test also confirms that there is no visible difference between SVC and the two AVC conditions. This result is expected as the base layer of SVC uses the same encoding algorithm as AVC.

Figure 2 displays the Mean Opinion Scores (MOS) obtained on the presented configurations for each video sequence. The results of the two test sessions (QVGA and VGA) are displayed together to ease the reading, while no relation between the two parts of the chart should be assumed. The scores obtained with the first condition are very low, typically located between 10 and 30, which corresponds to bad or low quality. The difference between AVC and SVC for this layer is very small, because the base layer of SVC is coded as a single AVC layer. Additional header information is included in the SVC version, which explains the slight difference in favor of AVC. On the second condition the scores are higher, because of the higher number of frames per second (cf. Table 1) which gives a smoother motion information. For this condition the difference between SVC and AVC is more noticeable in terms of MOS, but remains below 10 MOS points for three sequences out of four which can be considered equal. The difference is higher for CREW which is known to be a complex sequence to encode. It contains high motion and camera pan with camera light flashes causing abrupt luminance changes. The temporal enhancement between condition 1 and 2 may not allow efficient inter-layer prediction, as these light flashes are typically located on several isolated frames. So the additional frames in condition 2 may not be close in terms of content to the frames in the base layer, leading to poor inter-layer redundancy.

On the third condition (spatial enhancement from QVGA to VGA), the difference in MOS between SVC and AVC is quite constant for the four sequences, supporting the hypothesis that the lower efficiency previously observed on CREW is related to temporal scalability. The difference in terms of MOS varies around 10 points between AVC and SVC, which is a significant difference according to the t-test.

For SVC, the scores of condition 4 (VGA at 30 fps + quality enhancement) are significantly higher than the scores of condition 3. This layer adds 250 kbps of quality refinement to condition 3 (cf. Table 1), which represents 25% of the total bitrate. On the contrary for AVC, using the same bitrate constraint (750 kbps and 1000 kbps) the sequences are perceived as equivalent in a statistical sense.

When comparing AVC to SVC for the VGA format, it can be observed that AVC at 750 kbps outperforms SVC even at 1000 kbps. It has to be kept in mind though that the SVC bitstream for the VGA format already contains three layers, which allow decoding the video at intermediate quality levels. Table 4 reports the average difference in MOS between SVC and AVC for each layer. It can be observed that between conditions 1 and 2 this difference varies more than between conditions 3 and 4. This tends to show that the loss introduced by SVC is higher when the number of layers is low. Going from one single layer to two layers introduces a higher loss than going from 3 layers to 4 layers. An important parameter of this effect might be the position of the added layer in the scalability hierarchy. Adding a new layer at the bottom of the hierarchy has an impact on the whole structure of the stream, whereas adding it at the top of the hierarchy has a more limited impact. This result could help a service provider to decide whether he should add a new layer in a scalable stream configuration. In such a situation, our tests show that besides the importance of this new layer for the new target to address, its impact on the overall coding efficiency should be estimated. (VGA@30Hz+) 2.9 8.2 12.7 12.2 Table 4. Average MOS difference between SVC and the corresponding AVC-B stream.

CONCLUSION

In this paper we presented the results of subjective tests to compare the performance of MPEG-4 SVC and AVC in a mobile communication context. Two scenarios were studied for AVC, first using the same bitrate as SVC, second using the same PSNR as SVC. Using a detailed analysis of the results, we were able to confirm that the loss introduced by SVC is not severe though statistical differences are observed for all enhancement layers. The difference compared to AVC in terms of MOS is usually below 10%, which can be considered as sufficiently small. Secondly, we showed that this loss tends to stabilize when the number of layers increases. The impact of a given layer on the global coding efficiency is also related to its position in the scalable hierarchy, meaning that the more layers rely on it for prediction, the higher the impact. Finally, it was noticed that the scenario encoded with equal PSNR leads to equal visual quality, making the PSNR a good approximation of the viewer's opinion when used on each content individually. As a conclusion, our tests confirm the results previously published, while providing a precise analysis of the performance of SVC in a mobile context that fully exploits the capabilities of SVC. They show that SVC is of significant interest in mobile broadcasting, as it can address various targets with a limited cost in visual quality.

Fig. 1 .

 1 Fig. 1. Average Mean Opinion Score for the four processed sequences.

Fig. 2 .

 2 Fig.2. Mean Opinion Scores (MOS) obtained for AVC and SVC using the SAMVIQ methodology.

Table 1 .

 1 Tested layer configurations for MPEG-4 SVC.

	Layer	Frame	Frames	Bitrate	Scala-	Appli-
	Label	Size	p. sec.	(kbps)	bility	cation
	SVC1 QVGA	15	100	-	UMTS
	SVC2 QVGA	30	250	TEMP HSDPA
	SVC3	VGA	30	750	SPAT	WiMax
	SVC4	VGA	30	1000	QUAL WiMax

Table 2 .

 2 Test configurations for the AVC-B scenario.

	Layer	Frame	Frames	Bitrate
	Label	Size	p. sec.	(kbps)
	AVC-B1	QVGA	15	100
	AVC-B2	QVGA	30	250
	AVC-B3	VGA	30	750
	AVC-B4	VGA	30	1000


Yohann Pitrey

Marcus Barkowsky

Patrick Le Callet

Romuald Pépion

Evaluation of MPEG4-SVC for QoE protection in the context of transmission errors

Scalable Video Coding (SVC) provides a way to encapsulate several video layers with increasing quality and resolution in a single bitstream. Thus it is particularly adapted to address heterogeneous networks and a wide variety of decoding devices. In this paper, we evaluate the interest of SVC in a different context, which is error concealment after transmission on networks subject to packet loss. The encoded scalable video streams contain two layers with different spatial and temporal resolutions designed for mobile video communications with medium size and average to low bitrates. The main idea is to use the base layer to conceal errors in the higher layers if they are corrupted or lost. The base layer is first upscaled either spatially or temporally to reach the same resolution as the layer to conceal. Two error-concealment techniques using the base layer are then proposed for the MPEG-4 SVC standard, involving frame-level concealment and pixel-level concealment. These techniques are compared to the upscaled base layer as well as to a classical single-layer MPEG-4 AVC/H.264 error-concealment technique. The comparison is carried out through a subjective experiment, in order to evaluate the Quality-of-Experience of the proposed techniques. We study several scenarios involving various bitrates and resolutions for the base layer of the SVC streams. The results show that SVC-based error concealment can provide significantly higher visual quality than single-layer-based techniques. Moreover, we demonstrate that the resolution and bitrate of the base layer have a strong impact on the perceived quality of the concealment.

INTRODUCTION

Communication networks are subject to data-loss and corruption, due to various factors such as congestion, routing errors or hardware failures. When transmitting video data on such networks, these phenomenons create visual artifacts in the displayed video, which usually have a dramatic impact on the end-user's feeling of quality. Therefore these errors need to be detected and concealed using appropriate methods.

Due to the predictive structure of video coding techniques such as MPEG-4 AVC/H.264, [START_REF] Richardson | H.264 and MPEG-4 Video Compression[END_REF] the errors are likely to propagate along the decoded data. Detecting and predicting the impact of loss on the visual quality are thus quite difficult tasks, as well as concealing the visual artifacts using reliable data. The errors are usually concealed using spatially or temporally close areas in the decoded domain. [START_REF] Wang | Error Control and Concealment for Video Communication: A Review[END_REF][START_REF] Kung Wei-Ying | Spatial and temporal error concealment techniques for video transmission over noisy channels[END_REF] Two types of error-concealment techniques can be distinguished. Spatial techniques rely on the surrounding areas to fill lost regions by using interpolation and smoothing operations. [START_REF] Nemethova | Flexible Error Concealment for H.264 Based on Directional Interpolation[END_REF] Temporal techniques estimate the motion information of the lost parts and use the most plausible data as a basis for temporal prediction. [START_REF] Kim | Temporal Error Concealment Based on Optical Flow in the H.264/AVC Standard in Advanced Concepts for Intelligent Vision Systems[END_REF] Also, the encoding structure can be modified so that consecutive data in the bitstream corresponds to scattered areas in the decoded video. Thus, the distortions caused by loss are likely to be scattered as well, reducing the visual impact and allowing for better error concealment. [START_REF] Agrafiotis | Enhanced error concealment for video trasmission over WLANs[END_REF] Other error-concealment techniques have been introduced, based either on statistical properties of the bitstream, or on more practical considerations. For instance a study was presented where different types of gap fillings for channel switching in mobile TV were considered. [START_REF] Robitza | Acceptance of Mobile TV Channel Switching Delays[END_REF] Surprisingly, the observers preferred watching a commercial to seeing a black screen, a sender logo or a "please wait" screen. Thus, for longer outages, showing a prerecorded and prefetched commercial might be considered as an alternative to the application of error concealment strategies in particular when MPEG-4 AVC is used and no graceful degradation method exists.

Scalable Video Coding (SVC) has been introduced to provide a way to adapt video streams to variable decoding contexts. Several video layers are encoded in a single bitstream, with increasing spatio-temporal resolution and quality. In addition, inter-layer prediction is used to increase the global coding efficiency of the multi-layer stream. Thus, a layer can use the prediction and texture information from a lower layer called the base layer in order to get a better prediction. Scalable Video Coding therefore achieves higher compression rates than MPEG-4 AVC when encoding several layers jointly. A scalable video coding standard called MPEG-4 SVC was finalized in 2007, based on the MPEG-4 AVC/H.264 standard. [START_REF] Reichel | Joint Scalable Video Model JSVM-11[END_REF] It has been shown that it can allow for bitrate savings from 17% to 40% when compared to AVC, depending on the encoding configuration. [START_REF]SVC Verification Test Plan, Version 1[END_REF] SVC provides intermediate levels of video. Considering the bottom-up coding structure of SVC, the base layer can be independently decoded from the enhancement layers. Therefore, if the base layer is assumed to be unaffected by loss, the information it contains can be used in the enhancement layers in order to replace the erroneous areas. Postprocessing operations might be required as the layers are likely to have different spatial and temporal dimensions. Nonetheless, these postprocessings might produce spatial and temporal artifacts similar to classical error-concealment based on single-layer coding. Therefore, SVC represents an interesting candidate for video error-concealment. In addition, it can be easily combined with error-protection schemes. Especially, SVC is particularly adapted for unequal error-protection because of its hierarchical layer-based coding structure. The base layer is quite important for the decoding process, while it represents a limited amount of data when compared to the enhancement layers. A high-protection technique can then be applied on the base layer, while the enhancement layers can be protected in a more affordable way. [START_REF] Naghdinezhad | Unequal Error Protection for the Scalable Extension of H.264/AVC Using Genetic Algorithm[END_REF][START_REF] Bakker | Priority-based Error Protection for the Scalable Extension of H.264/SVC[END_REF] Several studies have been proposed that use SVC as a way to perform error-concealment. For example, it has been shown that the erroneous areas in the enhancement layer may be ignored by using the motion and texture information from the base layer in order to conceal the errors. [START_REF] Wang | Error prevention and concealment for scalable video coding with dual-priority transmission[END_REF] This method is quite simple and shows interesting results in terms of concealment. A more complex method was published recently using the information from the upscaled base layer to replace the erroneous areas in the enhancement layer, based on a technique called frame-hallucination. [START_REF] Ma | Frame loss error concealment for spatial scalability using hallucination[END_REF] This technique involves a training process and requires a set of available images in order to choose the best image to use for concealment. It shows interesting results when compared to the simple method based on the reconstruction of the base layer, at the price of a relatively high computational complexity. Unfortunately, in both studies the results are presented in terms of average Peak-to-Signal Ratio (PSNR) improvement, which is not representative of the human-perceived quality.

In this paper, we propose two error-concealment techniques using the base layer of SVC to conceal errors in the enhancement layer. Specifically, we evaluate the influence of the spatio-temporal characteristics of the base layer as well as the bitrate used to encode it on the quality of the concealment. These two methods are compared to a classical error-concealment technique suited for single-layer AVC, in order to evaluate the interest of SVC in such a context. The comparison is carried out through a subjective experiment, therefore the presented results reflect the visual perception of human viewers. This paper is organized as follows. Section 2 presents the process of creating the videos for the subjective experiment. Section 3 describes the setup for the subjective experiment. Section 4 presents the experimental results. Finally, section 5 concludes the paper.

DETAILS OF THE SUBJECTIVE EXPERIMENT

The process for generating the test videos is divided into four steps, from downscaling the reference sequences, encoding, transmission-error simulation to decoding and error-concealment. These four steps are described in detail in this section.

Input formats and video contents

The context of our experiment is mobile transmission. Thus we use videos in VGA format (640 × 480 pixels) at 30 frames per second (fps), which is a commonly used format for current smartphones and lightweight devices. Additionally, we use videos in QVGA format (320 × 240 pixels) at 15 and 30 fps in the SVC error-concealment scenarios as base layer videos. The input videos are generated from non-coded videos in full-HD (1920 × 1080 pixels) YUV 4:2:0 format at 60 fps using spatial and temporal downsampling. We use the tools provided by the JSVM reference software suite to perform these downscaling operations. [START_REF]JSVM Reference Software[END_REF] The source videos are first cropped to get a sequence with 4:3 aspect ratio from the 16:9 input sequence. Then spatial downscaling is performed using the Lanczos filter to get VGA (640 × 480) and QVGA (320 × 240) sequences. The cropping and downscaling steps are performed separately to ensure that the content does not appear horizontally or vertically stretched. Temporal downsampling is performed by discarding one frame out of two on each step. Thus one temporal down-scaling step is needed to get the streams at 30 fps, and two steps to get the streams at 15 fps. We use twelve different video sequences containing a wide variety of contents, ranging from documentary to sports scenes and sequences with high spatial frequency details. Figure 1 displays snapshots of these video sequences illustrating the different types of contents.

AVC and SVC encoding

After downsampling, the videos are encoded using the MPEG-4 SVC reference encoder software. This software is capable of encoding both AVC and SVC streams, thus allowing us to compare the two standards using the same coding tools. The SVC streams contain two layers. The highest layer is always in VGA format at 30 fps, with a cumulated bitrate of 600 kilobits per second (i.e.: the total bitrate for the two layers is equal to 600 kbps). The base layer is always in QVGA format. We study four different scenarios involving changes in the temporal frequency and the bitrate of the base layer. In the first scenario, the base layer is at 15 fps and is encoded at 120 kilobits per second (kbps). In the second scenario, the base layer is at 30 fps with the same bitrate of 120 kbps. In the third scenario, the base layer is at 15 fps with a higher bitrate of 200 kbps. In the fourth scenario the base layer is at 30 fps and 200 kbps. The two scenarios at 15 fps allow us to evaluate the impact of temporal upscaling on the quality, while the two scenarios at 30 fps allow us to identify the impact of the bitrate of the base layer.

One AVC stream with the same spatio-temporal resolution and bitrate as the highest layer of the SVC streams is generated for each video content. It is thus in VGA format at 30 fps, encoded at 600 kbps. We use the same coding tools as for SVC, so that the only difference between the streams is the number of layers. The size of the groups-of-pictures for SVC is equal to 16. An IDR has been inserted after 64 frames. The bitrate control is ensured by the FixedQpEncoder utility delivered with the JSVM reference software. This tool repeats the encoding process using a logarithmic search scheme to reach the desired bitrate.

In the enhancement layer of the SVC stream as well as in the AVC stream, each frame is divided into four independent slices. Each slice is shaped as a horizontal rectangle and takes a quarter of the frame. Due to the format of the videos (i.e.: VGA), losing one slice in a frame represents a spatially-significant visual loss. Additionally, it is possible to choose whether the lost slice overlaps significantly with an area of interest or not.

Bitstream extraction and loss simulation

After the encoding process, we simulate packet loss in the bitstreams to generate visual distortions in the decoded videos. We first use the bitstream extractor provided by the JSVM software suite to build a packet trace of the bitstream. This packet trace is a text file that contains one line for each slice in the bitstream. This file is processed through a loss simulator that deletes the lines corresponding to a specific area in a specified set of consecutive frames. The distorted trace-file is then merged with the video bitstream using again the bitstream extractor. The slices corresponding to the deleted lines in the trace file are discarded in the output bitstream, creating a video bitstream that contains loss. One slice out of four is lost in each distorted frame, which means that one quarter of the frame is missing during the loss interval. We manually chose to loose a slice that is part of a visually important area of the sequence. The length of the loss is set to one second out of ten seconds of video content for each sequence. The spatial and temporal locations at which the loss occurs are designed to destroy visually salient regions such as characters, high motion or fine details.

Decoding and error concealment

For the decoding process, we use a commercially available SVC decoder. Unfortunately, the latest JSVM reference decoder versions are not capable of decoding erroneous streams. Therefore it leads to unexpected crashes when missing or corrupted data occur. The decoder we use is capable of decoding both AVC and SVC streams and proposes a basic error-concealment scheme making it robust to loss. The error-concealment technique is based on a combination of a repetition of the decoder buffer and a motion vector based error concealment. Two examples of the visual artifacts produced by this technique are displayed in Figure 3.

In addition to the AVC decoded stream combined with the buffer repetition-based concealment technique, three scenarios are studied. In the first scenario, only the base layer of the SVC stream is decoded and up-scaled to get a video in VGA at 30 fps. This scenario is based on the assumption that the base layer is not affected by transmission errors. Thus, decoding and upscaling it to the full resolution provides a minimum, yet always available quality. For the two scenarios in which the base layer and the enhancement layer are both in 30 fps, the upscaling operation only consists in spatial up-sampling on each frame from QVGA to VGA. We use the Lanczos filter as an upscaler, which has shown a good efficiency while maintaining a reasonable computational complexity. [START_REF] Pitrey | Subjective Quality Evaluation of H.264 High-Definition Video Coding versus Spatial Up-Scaling and Interlacing[END_REF] For the two scenarios in which the base layer is at 15 fps, temporal up-sampling is needed to get the same resolution as the enhancement layer. To get 30 fps, we simply duplicate each frame in the upscaled version of the base layer. This solution benefits from very low computational complexity, while avoiding temporal interpolation issues which are not in the scope of this work.

The two remaining scenarios exploit the structure of the SVC stream to perform error-concealment. The first method consists in switching to the upscaled version of a frame when distorsions are detected due to transmission errors. It is referenced later on as the switch method, or frame-level concealment. In order to simulate the detection of a loss in a frame by the decoder, a frame-wise binary comparison is performed between the decoded video sequences with and without transmission errors. If a difference is detected, the decoder switches to the frame upscaled from the base layer, while it keeps the high resolution frame if no difference is detected.

The second concealment method is based on the same principle as the switch method, whereas the comparison between the frame before and after transmission is performed at the pixel level. Therefore a mask of the distorted areas of the frame is constructed and replaced by the pixels from the upscaled version. This method is referred to as the patch method, or pixel-level method. As in the previous method, the comparison is performed using a simple pixel-luminance difference. Some visual examples of the distortions produced by the different concealment methods are displayed in Figure 3.

The two concealment methods presented here are expected to have different behaviors, depending on the encoding characteristics of the layers. In the following we consider the upscaled version of the SVC base layer as a concealment method as well. This simplistic scenario might be interesting in the sense that the bitrate saved from not sending (or even encoding) the enhancement layer can be used to perform stronger error-protection on the base layer. Thus the video stream can resist more efficiently to attacks and losses, at the price of some up-scaling artifacts.

In the switch method, the frame is replaced at once by the upscaled frame. It represents an intermediate solution, as some high-quality areas might be replaced as well as distorted areas. Meanwhile, the frame keeps a globally constant (yet lower) quality, which might have an impact on the viewers opinion. The patch method takes advantage of the fact that the distortions might be localized in a particular area of the frame, which allows to keep the non-affected areas in high quality. The impact of the concealment might therefore be more limited, but discontinuities can be expected to appear at the border between the upscaled and the original areas.

In order to evaluate the performance of each of these SVC-based concealment techniques, we designed a subjective experiment to gather viewers opinions. The setup of this experiment is described in the next section.

SUBJECTIVE EXPERIMENT SETUP

The subjective experiment setup follows the recommendation ITU-R BT.500 concerning the room setup, the correct illumination and the calibration of the display. The viewing distance of 4H and other parameters which are specific for a setup using VGA resolution were taken from the VQEG Multimedia testplan. [START_REF] Hands | Multimedia Group Test Plan Draft Version 1.21, Video[END_REF] In particular, all the sequences were displayed without upscaling on a 40 inch TV-Logic LMV401 reference LCD display operating at its native resolution of 1920 × 1080 pixels at 60Hz. The VGA sequences were surrounded by a large gray border with a pixel value of Y=108 in the YCbCr color space, corresponding to 0.25 of the maximum display brightness.

For the playout, the freely available AcrVQWin program of Acreo AB Sweden was used. [START_REF] Brunnström | AcrVQWin runs subjective experiments for video quality in Windows[END_REF] It implements an Absolute Category Rating with Hidden Reference (ACR-HR) protocol according to ITU-T P.910 with sequence randomization constraints as specified in the VQEG Multimedia testplan, e.g. it is forbidden to display two conditions with the same source content. The rating scale consists of five attributes ranging from "excellent" with a MOS score of 5 to "bad" with a MOS score of 1. In the interactive graphical user interface only the attributes are shown to the observers.

In total, 29 naïve observers participated in the test. They were tested for visual acuity using a Snellen chart and for color blindness using Ishihara plates. A test session lasted for about 40 minutes and was preceded by a training session. The analysis of their votes proofed that none of them needed to be rejected by the ITU-R BT.500 screening method or the VQEG Multimedia Test plan screening method.

EXPERIMENTAL RESULTS

Figure 2 displays the average Mean Opinion Scores (MOS) and their intervals of 95% confidence for each tested configuration. Additionally, we provide the results of a student t-test performed on the data in Table 2, in order to verify the statistical significance of the results we present.

The non-coded reference gets the highest score, above 4.5 which attests the high quality of the source videos. The nondamaged SVC videos get the second highest score, with only 0.5 less on the MOS scale than the reference. The damaged-AVC gets the worst score, with a value around 1.3 which is very low when compared to the other tested configurations. The error-concealment technique used on AVC therefore shows a strong impact on the perceived quality. The cause of this large quality drop might be the flickering effect produced by the concealment technique, which is quite disturbing for the viewer. Because of the buffer repetition-based error-concealment technique used by the decoder, about half a second of video gets repeatedly displayed with new content overlayed resulting from the extrapolation of the motion concealment. As it only concerns a part of the sequence, it creates both spatial and temporal discontinuities, which are judged to be very unpleasant. The four scenarios including the SVC-based concealment techniques show the same tendency, with MOS values globally increasing when the temporal resolution and bitrate of the base layer increase. The upscaled version obtains the lowest score in each scenario. It is to notice that with equal bitrate, the version with 30 fps is preferred to the version with 15 fps. The frames in the 15 fps version get a better individual quality as the bitrate is dispatched among half the number of frames from the 30 fps version. Still, the viewers seem to prefer a fluid motion to higher quality on individual images. This is an interesting result in itself illustrating that temporal artifacts are judged as more unpleasant than spatial artifacts. This will be confirmed in the rest of this analysis. The patched and the switched versions achieve significant improvement of the visual quality when compared to the upscaled base layer in each scenario. Still, the relative behavior of the two concealment methods differs depending on the temporal frequency of the base layer.

When the temporal frequency of the base layer and the enhancement layer are identical, the patched version gets a significantly higher score than the switched version. The pixel-level concealment only affects the damaged areas in each frame, keeping the intact areas in high quality. The spatial discontinuities at the limit between the upscaled and the intact areas are preferred to the homogeneous but low-quality frame. This shows that the viewers tolerate spatial discontinuities when the global quality of the frames is higher. The impact of temporal discontinuities on the viewers' opinion is quite different. When the temporal frequency in the base layer is lower than in the enhancement layer, the patched and the switched methods get equivalent scores. This assessment is verified by the student t-test in Table 2. The quality increase with the switched version is actually quite constant from one scenario to another, so we can notice a loss in efficiency from the patched version when the temporal frequency is lower in the base layer. When upscaling the base layer from 15 to 30 fps, each frame is duplicated in order to double the number of fps. Therefore, using one frame from the base layer to conceal the errors in the enhancement layer causes spatio-temporal discontinuities in one frame out of two. This phenomenon is particularly visible when motion is high around the junction between the concealed and the non-concealed areas, such as demonstrated in Figure 3.

It is interesting to note that even the upscaled version of the lowest quality scenario (i.e.: 15 fps / 120 kbps) is preferred to the damaged AVC. This point shows that spatial and temporal consistency has a strong impact on the global perceived quality. Even a video sequence with low quality can reach a higher opinion score than a high quality video with badlyconcealed errors. Furthermore, Table 2 shows that the version which was upscaled from the base layer at 30 fps / 200 kbps is statistically equivalent in terms of visual quality to the patched and switch versions in the scenario at 15 fps / 120 kbps. This confirms that upscaling a good quality base layer can reach the same quality level as versions concealed using a lower-quality base layer. This means that transmitting a QVGA base layer at 30 fps / 200 kbps and upscaling it to VGA produces the same quality as the VGA enhancement layer encoded at 600 kbps when transmission errors call for concealment. Moreover, it can be argued that the bitrate saved from using only 200 kbps could be used to add redundancy in the bitstream and perform error protection.

The patched and switched versions in the 30 fps / 120 kbps scenario reach better or equal quality when compared to the distortions. In addition, the patched version in the 30 fps / 120 kbps scenario gets a similar score as the switched version in 30 fps / 200 kbps. The difference in bitrate between the two scenarios is indeed compensated by the superiority of the pixel-level concealment over the frame-level concealment. In this case, the spatial discontinuity produced at the border between the upscaled and the intact areas is preferred to a homogeneous quality on the whole frame. To explain this, one should consider that the quality change is more noticeable when it is applied on the whole frame than when only a limited area of the frame is affected. A temporal discontinuity is created in both cases, but its effects are more visible when the whole frame changes at once. Table 2. Comparison of the experimental results from the tested configurations through a student t-test. If the cell at row i and column j contains a '•' symbol, configurations i and j are not statistically different in terms of MOS. If the cell contains a '↑' symbol, configuration i has a significantly higher score than configuration j. If the cell contains a '↓' symbol, configuration i has a significantly lower score than configuration j. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ 2 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 3 ↓ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 4 ↓ ↑ ↑ • ↑ ↓ ↓ ↑ ↓ ↓ • ↓ ↓ ↓ 5 ↓ ↑ ↑ • ↑ ↓ ↓ ↑ ↓ ↓ • ↓ ↓ ↓ 6 ↓ ↑ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 7 ↓ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓ • ↓ 8 ↓ ↑ ↑ ↑ ↑ ↑ ↓ ↑ • • • ↓ ↓ ↓ 9 ↓ ↑ ↑ ↓ ↓ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 10 ↓ ↑ ↑ ↑ ↑ ↑ ↓ • ↑ • • ↓ ↓ ↓ 11 ↓ ↑ ↑ ↑ ↑ ↑ ↓ • ↑ • ↑ ↓ ↓ ↓ 12 ↓ ↑ ↑ • • ↑ ↓ • ↑ • ↓ ↓ ↓ ↓ 13 ↓ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓ 14 ↓ ↑ ↑ ↑ ↑ ↑ • ↑ ↑ ↑ ↑ ↑ ↓ ↓ 15 ↓ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
The presented subjective experiment answers several interesting questions. First, we showed that using SVC as an error concealment technique can produce significantly higher quality than using a classical error concealment technique on a single-layer AVC stream. It is well known that SVC introduces a loss in coding efficiency as encoding several layers of the same content produces redundancy. This loss has been evaluated in terms of visual quality by several studies and proved to produce a noticeable yet limited impact on the viewers' opinion. [START_REF]SVC Verification Test Plan, Version 1[END_REF] According to our tests, the differences between SVC and AVC are not as significant in a context such as lossy transmission, where the errors produced by data-loss have a stronger impact on the quality. Therefore, SVC can be advantageous when compared to AVC as it provides intermediate versions of the video that can be exploited for error concealment in the enhancement layers.

CONCLUSION

In this paper we studied the performances of Scalable Video Coding to perform error-concealment in terms of visual quality. The scalable streams contain two layers, among which only the highest is subject to loss. The lower layer is then upscaled to replace the lost or corrupted data. Two techniques were proposed based on this principle, working at frame and pixel level. We compared them to a classical single-layer error-concealment technique based on MPEG-4 AVC/H.264 and to a scenario where only the base layer is transmitted and upscaled. The results of our subjective experiment show that the two SVC-based error-concealment techniques perform better in terms of visual quality than the AVC-based technique. Moreover, we identified the impact of the resolution and quality of the base layer on the visual quality of the concealment.

If the two layers are at the same temporal frequency, a pixel-level technique performs well, whereas a frame-level technique is better if the two layers do not have the same temporal frequency. In addition, the bitrate used to encode the base layer is of great importance. We show that a good quality base layer combined with upsampling can perform as good as two layers using the presented concealment techniques when the base layer is in low quality. Therefore the advantage of SVC concerning error-concealment techniques can be significant, as long as the base layer is encoded with sufficient quality.

Figure 1 .

 1 Figure 1. Video contents used for the experiment. Snapshots appear as they were cropped in VGA format.

Table 1 .

 1 Video configurations used in the experiment and their associated labels.Figure2. Average Mean Opinion Scores and intervals of confidence (95%) for the tested configurations. See Table1for the correspondence between numbers and configurations.

	Format number Description / base layer format Error concealment method
	1	Reference	none
	2	Non-damaged SVC	none
	3	AVC	buffer repetition
	4	15 fps / 120 kbps	upscale
	5	15 fps / 120 kbps	patch
	6	15 fps / 120 kbps	switch
	7	30 fps / 120 kbps	upscale
	8	30 fps / 120 kbps	patch
	9	30 fps / 120 kbps	switch
	10	15 fps / 200 kbps	upscale
	11	15 fps / 200 kbps	patch
	12	15 fps / 200 kbps	switch
	13	30 fps / 200 kbps	upscale
	14	30 fps / 200 kbps	patch
	15	30 fps / 200 kbps	switch

ACKNOWLEDGEMENTS

This work was partly conducted for the SVC4QoE project funded by the French Direction Générale de la Compétitivité, de l'Industrie et des Services (DGCIS). This project is aimed at evaluating the performance of scalable video coding for the optimization of the quality of experience in a mobile video broadcasting context.

Non-coded reference

Damaged AVC video after concealment (buffer repetition) SVC video concealed using the switch method / Upscaled SVC base layer SVC video concealed using the patch method


Bruno Bouchard
email: bouchard@ceremade.dauphine.fr

Nizar Touzi
email: touzi@cmap.polytechnique.fr

WEAK DYNAMIC PROGRAMMING PRINCIPLE FOR VISCOSITY SOLUTIONS

Keywords: Optimal control, Dynamic programming, discontinuous viscosity solutions AMS subject classifications. Primary 49L25, 60J60; secondary 49L20, 35K55

We prove a weak version of the dynamic programming principle for standard stochastic control problems and mixed control-stopping problems, which avoids the technical difficulties related to the measurable selection argument. In the Markov case, our result is tailor-made for the derivation of the dynamic programming equation in the sense of viscosity solutions.

1. Introduction. Consider the standard class of stochastic control problems in the Mayer form

V (t, x) := sup ν∈U E [f (X ν T )|X ν t = x] ,
where U is the controls set, X ν is the controlled process, f is some given function, 0 < T ≤ ∞ is a given time horizon, t ∈ [0, T ) is the time origin, and x ∈ R d is some given initial condition. This framework includes the general class of stochastic control problems under the so-called Bolza formulation, the corresponding singular versions, and optimal stopping problems.

A key-tool for the analysis of such problems is the so-called dynamic programming principle (DPP), which relates the time-t value function V (t, .) to any later time-τ value V (τ, .) for any stopping time τ ∈ [t, T ) a.s. A formal statement of the DPP is:

V (t, x) = v(t, x) := sup ν∈U E [V (τ, X ν τ )|X ν t = x] . (1.1)
In particular, this result is routinely used in the case of controlled Markov jumpdiffusions in order to derive the corresponding dynamic programming equation in the sense of viscosity solutions, see Lions [START_REF] Lions | Optimal Control of Diffusion Processes and Hamilton-Jacobi-Bellman Equations I[END_REF][START_REF] Lions | Optimal Control of Diffusion Processes and Hamilton-Jacobi-Bellman Equations, Part II: Viscosity Solutions and Uniqueness[END_REF], Fleming and Soner [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF], Touzi [START_REF] Touzi | Stochastic Control Problems, Viscosity Solutions, and Application to Finance[END_REF], for the case of controlled diffusions, and Oksendal and Sulem [START_REF] Oksendal | Applied Stochastic Control of Jump Diffusions[END_REF] for the case of Markov jump-diffusions. The statement (1.1) of the DPP is very intuitive and can be easily proved in the deterministic framework, or in discrete-time with finite probability space. However, its proof is in general not trivial, and requires on the first stage that V be measurable.

When the value function V is known to be continuous, the abstract measurability arguments are not needed and the proof of the dynamic programming principle is significantly simplified. See e.g. Fleming and Soner [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF], or Kabanov and Klueppelberg [START_REF] Kabanov | A geometric approach to portfolio optimization in models with transaction costs[END_REF] in the context of a special singular control problem in finance. Our objective is to reduce the proof to this simple context in a general situation where the value function has no a priori regularity.

The inequality "V ≤ v" is the easy one but still requires that V be measurable. Our weak formulation avoids this issue. Namely, under fairly general conditions on the controls set and the controlled process, it follows from an easy application of the tower property of conditional expectations that

V (t, x) ≤ sup ν∈U E [V * (τ, X ν τ )|X ν t = x] ,
where V * is the upper semicontinuous envelope of the function V . The proof of the converse inequality "V ≥ v" in a general probability space turns out to be difficult when the function V is not known a priori to satisfy some continuity condition. See e.g. Bertsekas and Shreve [START_REF] Bertsekas | Stochastic Optimal Control : The Discrete Time Case[END_REF], Borkar [START_REF] Borkar | Optimal Control of Diffusion Processes[END_REF], and El Karoui [START_REF] Karoui | Les Aspects probabilistes du contrôle stochastique[END_REF].

Our weak version of the DPP avoids the non-trivial measurable selection argument needed to prove the inequality V ≥ v in (1.1). Namely, in the context of a general control problem presented in Section 2, we show in Section 3 that:

V (t, x) ≥ sup ν∈U E [ϕ(τ, X ν τ )|X t = x] for every upper-semicontinuous minorant ϕ of V.
We also show that an easy consequence of this result is that

V (t, x) ≥ sup ν∈U E V * (τ ν n , X ν τ ν n )|X t = x ,
where τ ν n := τ ∧inf {s > t : |X ν s -x| > n}, and V * is the lower semicontinuous envelope of V .

This result is weaker than the classical DPP (1.1). However, in the controlled Markov jump-diffusions case, it turns out to be tailor-made for the derivation of the dynamic programming equation in the sense of viscosity solutions. Section 5 reports this derivation in the context of controlled jump diffusions.

Finally, Section 4 provides an extension of our argument in order to obtain a weak dynamic programming principle for mixed control-stopping problems.

2. The stochastic control problem. Let (Ω, F, P) be a probability space supporting a càdlàg R d -valued process Z with independent increments. Given T > 0, let F := {F t , 0 ≤ t ≤ T } be the completion of its natural filtration on [0, T ]. Note that F satisfies the usual conditions, see e.g. [START_REF] Dellacherie | Probabilité et Potentiel, Théorie du potentiel[END_REF]. We assume that F 0 is trivial and that F T = F.

For every t ≥ 0, we set F t := (F t s ) s≥0 , where F t s is the completion of σ(Z r -Z t , t ≤ r ≤ s ∨ t) by null sets of F.

We denote by T the collection of all F-stopping times. For τ 1 , τ 2 ∈ T with τ 1 ≤ τ 2 a.s., the subset T [τ1,τ2] is the collection of all τ ∈ T such that τ ∈ [τ 1 , τ 2 ] a.s. When τ 1 = 0, we simply write T τ2 . We use the notations T t [τ1,τ2] and T t τ2 to denote the corresponding sets of F t -stopping times.

Throughout the paper, the only reason for introducing the filtration F through the process Z is to guarantee the following property of the filtrations F t . remark 2.1. Notice that F t s -measurable random variables are independent of F t for all s, t ≤ T , and that F t s is the trivial degenerate σ-algebra for s ≤ t. Similarly, all F t -stopping times are independent of F t .

For τ ∈ T and a subset A of a finite dimensional space, we denote by L 0 τ (A) the collection of all F τ -measurable random variables with values in A. H 0 (A) is the collection of all F-progressively measurable processes with values in A, and H 0 rcll (A) is the subset of all processes in H 0 (A) which are right-continuous with finite left limits.

In the following, we denote by B r (z) (resp. ∂B r (z)) the open ball (resp. its boundary) of radius r > 0 and center z ∈ R , ∈ N.

Througout this note, we fix an integer d ∈ N, and we introduce the sets:

S := [0, T ] × R d andS o := (τ, ξ) : τ ∈ T T and ξ ∈ L 0 τ (R d ) .
We also denote by USC(S) (resp. LSC(S)) the collection of all upper-semicontinuous (resp. lower-semicontinuous) functions from S to R. The set of control processes is a given subset U o of H 0 (R k ), for some integer k ≥ 1, so that the controlled state process defined as the mapping:

(τ, ξ; ν) ∈ S × U o -→ X ν τ,ξ ∈ H 0 rcll (R d )for some S withS ⊂ S ⊂ S o
is well-defined and satisfies:

θ, X ν τ,ξ (θ) ∈ Sfor all(τ, ξ) ∈ S and θ ∈ T [τ,T ] .
A suitable choice of the set S in the case of jump-diffusion processes driven by Brownian motion is given in Section 5 below. Given a Borel function f : R d -→ R and (t, x) ∈ S, we introduce the reward function J : S × U -→ R:

J(t, x; ν) := E f X ν t,x (T ) (2.1)
which is well-defined for controls ν in

U := ν ∈ U o : E |f (X ν t,x (T ))| < ∞, ∀ (t, x) ∈ S . (2.2)
We say that a control ν ∈ U is t-admissible if it is F t -progressively measurable, and we denote by U t the collection of such processes. The stochastic control problem is defined by: V (t, x) := sup ν∈Ut J(t, x; ν)for(t, x) ∈ S.

(2.3) remark 2.2. The restriction to control processes that are F t -progressively measurable in the definition of V (t, •) is natural and consistent with the case where t = 0, since F 0 is assumed to be trivial, and is actually commonly used, compare with e.g. [START_REF] Yong | Stochastic Controls: Hamiltonian Systems and HJB Equations[END_REF]. It will be technically important in the following. It also seems a-priori necessary in order to ensure that Assumption A4 below makes sense, see Remark 3.2 and the proof of Proposition 5.4 below. However, we will show in Remark 5.2 below that it is not restrictive.

3. Dynamic programming for stochastic control problems. For the purpose of our weak dynamic programming principle, the following assumptions are crucial.

Assumption A For all (t, x) ∈ S and ν ∈ U t , the controlled state process satisfies:

A1 (Independence) The process X ν t,x is F t -progressively measurable. A2 (Causality) For ν ∈ U t , τ ∈ T t [t,T ] and A ∈ F t τ , if ν = ν on [t, τ ] and ν1 A = ν1 A on (τ, T ], then X ν t,x 1 A = X ν t,
ν1 [0,θ] + ν1 (θ,T ] ∈ U t .
A4 (Consistency with deterministic initial data) For all θ ∈ T t [t,T ] , we have: a. For P-a.e ω ∈ Ω, there exists νω ∈ U θ(ω) such that

E f X ν t,x (T ) |F θ (ω) ≤ J(θ(ω), X ν t,x (θ)(ω); νω ). b. For t ≤ s ≤ T , θ ∈ T t [t,s]
, ν ∈ U s , and ν := ν1 [0,θ] + ν1 (θ,T ] , we have:

E f X ν t,x (T ) |F θ (ω) = J(θ(ω), X ν t,x (
θ)(ω); ν)forP -a.e. ω ∈ Ω. remark 3.1. Assumption A2 above means that the process X ν t,x is defined (caused) by the control ν pathwise. remark 3.2. Let θ be equal to a fixed time s in A4-b. If ν is allowed to depend on F s , then the left-hand side in A4-b does not coincide with E f (X ν s,X ν t,x (s)(ω) (T )) . Hence, the above identity can not hold in this form. remark 3.3. In Section 5 below, we show that Assumption A4-a holds with equality in the jump-diffusion setting. Although we have no example of a control problem where the equality does not hold, we keep Assumption A4-a under this form because the proof only needs this requirement. remark 3.4. Assumption A3 above implies the following property of the controls set which will be needed later:

A5 (Stability under bifurcation) For ν 1 , ν 2 ∈ U t , τ ∈ T t
[t,T ] and A ∈ F t τ , we have:

ν := ν 1 1 [0,τ ] + (ν 1 1 A + ν 2 1 A c ) 1 (τ,T ] ∈ U t .
To see this, observe that τ

A := T 1 A + τ 1 A c is a stopping time in T t [t,T ] (the indepen- dence of F t follows from Remark 2.1), and ν = ν 1 1 [0,τ A ] +ν 2 1 (τ A ,T ]
is the concatenation of ν 1 and ν 2 at the stopping time τ A . Given ν as constructed above, it is clear that this control can be concatenated with another control ν 3 ∈ U t by following the same argument. Iterating the above property, we therefore see that for 0 ≤ t ≤ T and τ ∈ T t [t,T ] , we have the following extension: for a finite sequence (ν 1 , . . . , ν n ) of controls in U t with ν i = ν 1 on [0, τ ], and for a partion (A i ) 1≤i≤n of Ω with A i ∈ F t τ for every i ≤ n:

ν := ν 1 1 [0,τ ] + 1 (τ,T ] n i=1 ν i 1 Ai ∈ U t .
Our main result is the following weak version of the dynamic programming principle which uses the following notation:

V * (t, x) := lim inf (t ,x )→(t,x) V (t , x ), V * (t, x) := lim sup (t ,x )→(t,x) V (t , x ),(t, x) ∈ S.
Theorem 3.5. Let Assumptions A hold true and assume that V is locally bounded. Then for every (t, x) ∈ S, and for every family of stopping times

{θ ν , ν ∈ U t } ⊂ T t [t,T ] , we have V (t, x) ≤ sup ν∈Ut E V * (θ ν , X ν t,x (θ ν )) . (3.1)
Assume further that J(.; ν) ∈ LSC(S) for every ν ∈ U o . Then, for any function ϕ : S -→ R:

ϕ ∈ USC(S) and V ≥ ϕ =⇒ V (t, x) ≥ sup ν∈U ϕ t E ϕ(θ ν , X ν t,x (θ ν )) , (3.2)
where

U ϕ t = ν ∈ U t : E ϕ(θ ν , X ν t,x (θ ν )) + < ∞ or E ϕ(θ ν , X ν t,x (θ ν )) -< ∞ .
Before proceeding to the proof of this result, we report the following consequence. Corollary 3.6. Let the conditions of Theorem 3.5 hold. For (t, x) ∈ S, let {θ ν , ν ∈ U t } ⊂ T t [t,T ] be a family of stopping times such that

X ν t,x 1 [t,θ ν ] is L ∞ -bounded for all ν ∈ U t . Then, sup ν∈Ut E V * (θ ν , X ν t,x (θ ν )) ≤ V (t, x) ≤ sup ν∈Ut E V * (θ ν , X ν t,x (θ ν )) . (3.3)
Proof. The right-hand side inequality is already provided in Theorem 3.5. Fix r > 0. It follows from standard arguments, see e.g. Lemma 3.5 in [START_REF] Reny | On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games[END_REF], that we can find a sequence of continuous functions (ϕ n ) n such that ϕ n ≤ V * ≤ V for all n ≥ 1 and such that ϕ n converges pointwise to V * on [0, T ] × B r (0). Set φ N := min n≥N ϕ n for N ≥ 1 and observe that the sequence (φ N ) N is non-decreasing and converges pointwise to V * on [0, T ] × B r (0). Applying (3.2) of Theorem 3.5 and using the monotone convergence Theorem, we then obtain:

V (t, x) ≥ lim N →∞ E φ N (θ ν , X ν t,x (θ ν )) = E V * (θ ν , X ν t,x (θ ν )) .
remark 3.7. Notice that the value function V (t, x) is defined by means of U t as the set of controls. Because of this, the lower semicontinuity of J(., ν) required in the second part of Theorem 3.5 does not imply that V is lower semicontinuous in its t-variable. See however Remark 5.3 below.

Proof. [Theorem 3.5] 1. Let ν ∈ U t be arbitrary and set θ := θ ν . The first assertion is a direct consequence of Assumption A4-a. Indeed, it implies that, for P-almost all ω ∈ Ω, there exists νω ∈ U θ(ω) such that

E f X ν t,x (T ) |F θ (ω) ≤ J(θ(ω), X ν t,x (θ)(ω); νω ) .
Since, by definition, J(θ(ω), X ν t,x (θ)(ω); νω ) ≤ V * (θ(ω), X ν t,x (θ)(ω)), it follows from the tower property of conditional expectations that

E f X ν t,x (T ) = E E f X ν t,x (T ) |F θ ≤ E V * θ, X ν t,x (θ) .
2. Let ε > 0 be given. Then there is a family (ν (s,y),ε ) (s,y)∈S ⊂ U o such that: ν (s,y),ε ∈ U s and J(s, y; ν (s,y),ε ) ≥ V (s, y) -ε,for every(s, y) ∈ S.

(3.4)

By the lower-semicontinuity of (t , x ) → J(t , x ; ν (s,y),ε ), for fixed (s, y) ∈ S, together with the upper-semicontinuity of ϕ, we may find a family (r (s,y) ) (s,y)∈S of positive scalars so that, for any (s, y) ∈ S, ϕ(s, y)-ϕ(t , x ) ≥ -ε and J(s, y; ν (s,y),ε )-J(t , x ; ν (s,y),ε ) ≤ ε for (t , x ) ∈ B(s, y; r (s,y) ), (3.5) where, for r > 0 and (s, y) ∈ S,

B(s, y; r) := {(t , x ) ∈ S : t ∈ (s -r, s], |x -y| < r} . ( 3
, r i ) i≥1 of elements of S × R, with 0 < r i ≤ r (ti,xi) for all i ≥ 1, such that S ⊂ {0} × R d ∪ (∪ i≥1 B(t i , x i ; r i )). Set A 0 := {T } × R d , C -1 := ∅,
and define the sequence

A i+1 := B(t i+1 , x i+1 ; r i+1 ) \ C i whereC i := C i-1 ∪ A i , i ≥ 0.
With this construction, it follows from (3.4), (3.5), together with the fact that V ≥ ϕ, that the countable family

(A i ) i≥0 satisfies (θ, X ν t,x (θ)) ∈ ∪ i≥0 A i P-a.s., A i ∩A j = ∅ for i = j ∈ N, and J(•; ν i,ε ) ≥ ϕ-3ε on A i for i ≥ 1, (3.7
) where ν i,ε := ν (ti,xi),ε for i ≥ 1.

We now prove (3.2). We fix

ν ∈ U t and θ ∈ T t [t,T ] . Set A n := ∪ 0≤i≤n A i , n ≥ 1. Given ν ∈ U t , we define ν ε,n s := 1 [t,θ] (s)ν s + 1 (θ,T ] (s) ν s 1 (A n ) c (θ, X ν t,x (θ)) + n i=1 1 Ai (θ, X ν t,x (θ))ν i,ε s ,fors ∈ [t, T ]. Notice that {(θ, X ν t,x (θ)) ∈ A i } ∈ F t θ
as a consequence of Assumption A1. Then, it follows from the stability under concatenation Assumption A3, Remark 3.4 that ν ε,n ∈ U t . By the definition of the neighbourhood (3.6)

, notice that θ = θ ∧ t i ≤ t i on {(θ, X ν t,x (θ)) ∈ A i }.
Then, using Assumptions A4-b, A2, and (3.7), we deduce that:

E f X ν ε,n t,x (T ) |F θ 1 A n θ, X ν t,x (θ) = E f X ν ε,n t,x (T ) |F θ 1 A0 θ, X ν t,x (θ) + n i=1 E f X ν ε,n t,x (T ) |F θ∧ti 1 Ai θ, X ν t,x (θ) = V T, X ν ε,n t,x (T ) 1 A0 θ, X ν t,x (θ) + n i=1 J(θ∧t i , X ν t,x (θ∧t i ); ν i,ε )1 Ai θ, X ν t,x (θ) ≥ n i=0 ϕ(θ, X ν t,x (θ)) -3ε 1 Ai θ, X ν t,x (θ) = ϕ(θ, X ν t,x (θ)) -3ε 1 A n θ, X ν t,x (θ) ,
which, by definition of V and the tower property of conditional expectations, implies

V (t, x) ≥ J(t, x; ν ε,n ) = E E f X ν ε,n t,x (T ) |F θ ≥ E ϕ θ, X ν t,x (θ) -3ε 1 A n θ, X ν t,x (θ) + E f X ν t,x (T ) 1 (A n ) c θ, X ν t,x (θ) .
Since f X ν t,x (T ) ∈ L 1 , it follows from the dominated convergence theorem that:

V (t, x) ≥ -3ε + lim inf n→∞ E ϕ(θ, X ν t,x (θ))1 A n θ, X ν t,x (θ) = -3ε + lim n→∞ E ϕ(θ, X ν t,x (θ)) + 1 A n θ, X ν t,x (θ) -lim n→∞ E ϕ(θ, X ν t,x (θ)) -1 A n θ, X ν t,x (θ) = -3ε + E ϕ(θ, X ν t,x (θ)) ,
where the last equality follows from the left-hand side of (3.7) and from the monotone convergence theorem, due to the fact that either E ϕ(θ, X ν t,x (θ))

+ < ∞ or E ϕ(θ, X ν t,x (θ)) -< ∞. The proof of (3.
2) is completed by the arbitrariness of ν ∈ U t and ε > 0.

remark 3.8. (Lower-semicontinuity condition I) It is clear from the above proof that it suffices to prove the lower-semicontinuity of (t, x) → J(t, x; ν) for ν in a subset Ũo of U o such that sup ν∈ Ũt J(t, x; ν) = V (t, x). Here Ũt is the subset of Ũo whose elements are F t -progressively measurable. In most applications, this allows to reduce to the case where the controls are essentially bounded or satisfy a strong integrability condition. remark 3.9. (Lower-semicontinuity condition II) In the above proof, the lowersemicontinuity assumption is only used to construct the balls B i on which J(t i , x i ; ν i,ε )-J(•; ν i,ε ) ≤ ε. Clearly, it can be alleviated, and it suffices that the lower-semicontinuity holds in time from the left, i.e. lim inf

(t ,x )→(ti,xi), t ≤ti J(t , x ; ν i,ε ) ≥ J(t i , x i ; ν i,ε ).
V (t, x) := sup ν∈Ut E T t Y ν t,x,1 (s)g s, X ν t,x (s), ν s ds + Y ν t,x,1 (T )f X ν t,x (T ) ,
where

dY ν t,x,y (s) = -Y ν t,x,y (s)k s, X ν t,x (s), ν s ds , Y ν t,x,y (t) = y > 0 .
Then, it is well known that this problem can be converted into the Mayer formulation (2.3) by augmenting the state process to (X, Y, Z), where

dZ ν t,x,y,z (s) = Y ν t,x,y (s)g s, X ν t,x (s), ν s ds , Z ν t,x,y,z (t) = z ∈ R ,
and considering the value function

V (t, x, y, z) := sup ν∈Ut E Z ν t,x,y,z (T ) + Y ν t,x,y (T )f X ν t,x (T ) = yV (t, x) + z .
In particular, V (t, x) = V (t, x, 1, 0). The first assertion of Theorem 3.5 implies

V (t, x) ≤ sup ν∈Ut E Y ν t,x,1 (θ ν )V * (θ ν , X ν t,x (θ ν )) + θ ν t Y ν t,x,1 (s)g s, X ν t,x (s), ν s ds . (3.8)
Given a upper-semicontinuous minorant ϕ of V , the function φ defined by φ(t, x, y, z) := yϕ(t, x) + z is an upper-semicontinuous minorant of V . From the second assertion of Theorem 3.5, we see that for a family {θ ν , ν

∈ U t } ⊂ T t [t,T ] , V (t, x) ≥ sup ν∈U φ t E φ θ ν , X ν t,x (θ ν ), Y ν t,x,1 (θ ν ), Z ν t,x,1,0 (θ ν ) = sup ν∈U φ t E Y ν t,x,1 (θ ν )ϕ(θ ν , X ν t,x (θ ν )) + θ ν t Y ν t,
x,1 (s)g s, X ν t,x (s), ν s ds .

(3.9) remark 3.11. (Infinite Horizon) Infinite horizon problems can be handled similarly. Following the notations of the previous Remark 3.10, we introduce the infinite horizon stochastic control problem:

V ∞ (t, x) := sup ν∈Ut E ∞ t Y ν t,x,1 (s)g s, X ν t,x (s), ν s ds .
Then, it is immediately seen that V ∞ satisfies the weak dynamic programming principle (3.8)-(3.9).

4. Dynamic programming for mixed control-stopping problems. In this section, we provide a direct extension of the dynamic programming principle of Theorem 3.5 to the larger class of mixed control and stopping problems.

In the context of the previous section, we consider a Borel function f : R d -→ R, and we assume |f | ≤ f for some continuous function f . For (t, x) ∈ S the reward J : S × Ū × T [t,T ] -→ R:

J(t, x; ν, τ ) := E f X ν t,x (τ ) , (4.1)
which is well-defined for every control ν in

Ū := ν ∈ U o : E sup t≤s≤T f (X ν t,x (s)) < ∞ ∀ (t, x) ∈ S .
The mixed control-stopping problem is defined by:

V (t, x) := sup (ν,τ )∈ Ūt×T t [t,T ] J(t, x; ν, τ ) , (4.2)
where Ūt is the subset of elements of Ū that are F t -progressively measurable.

The key ingredient for the proof of (4.6) is the following property of the set of stopping times T T :

For all θ, τ 1 ∈ T t T and τ 2 ∈ T t [θ,T ] ,we haveτ 1 1 {τ1<θ} + τ 2 1 {τ1≥θ} ∈ T t T . (4.3)
In order to extend the result of Theorem 3.5, we shall assume that the following version of A4 holds: Assumption A4' For all (t, x) ∈ S, (ν, τ ) ∈ Ūt × T t [t,T ] and θ ∈ T t [t,T ] , we have: a. For P-a.e ω ∈ Ω, there exists (ν ω , τω ) ∈ Ūθ(ω) × T θ(ω)

[θ(ω),T ] such that

1 {τ ≥θ} (ω)E f X ν t,x (τ ) |F θ (ω) ≤ 1 {τ ≥θ} (ω)J θ(ω), X ν t,x (θ)(ω); νω , τω b. For t ≤ s ≤ T , θ ∈ T t [t,s] , (ν, τ ) ∈ Ūs × T s [s,T ]
, τ := τ 1 {τ <θ} + τ 1 {τ ≥θ} , and ν := ν1 [0,θ] + ν1 (θ,T ] , we have for P-a.e. ω ∈ Ω:

1 {τ ≥θ} (ω)E f X ν t,x (τ ) |F θ (ω) = 1 {τ ≥θ} (ω)J(θ(ω), X ν t,x (
θ)(ω); ν, τ ). Theorem 4.1. Let Assumptions A1, A2, A3 and A4' hold true. Then for every (t, x) ∈ S, and for all family of stopping times {θ ν , ν ∈ Ūt } ⊂ T t [t,T ] :

V (t, x) ≤ sup (ν,τ )∈ Ūt×T t [t,T ] E 1 {τ <θ ν } f (X ν t,x (τ )) + 1 {τ ≥θ ν } V * (θ ν , X ν t,x (θ ν )) . (4.4)
Assume further that the map (t, x) -→ J(t, x; ν, τ ) satisfies the following lowersemicontinuity property

lim inf t ↑t,x →x J(t , x ; ν, τ ) ≥ J(t, x; ν, τ )for every(t, x) ∈ S and (ν, τ ) ∈ Ū × T . (4.5)
Then, for any function ϕ ∈ USC(S) with V ≥ ϕ:

V (t, x) ≥ sup (ν,τ )∈ Ū ϕ t ×T t [t,T ] E 1 {τ <θ ν } f (X ν t,x (τ )) + 1 {τ ≥θ ν } ϕ(θ ν , X ν t,x (θ ν )) , (4.6)
where

Ūϕ t = ν ∈ Ūt : E ϕ(θ ν , X ν t,x (θ ν )) + < ∞ or E ϕ(θ ν , X ν t,x (θ ν )) -< ∞ .
For simplicity, we only provide the proof of Theorem 4.1 for optimal stopping problems, i.e. in the case where Ū is reduced to a singleton. The dynamic programming principle for mixed control-stopping problems is easily proved by combining the arguments below with those of the proof of Theorem 3.5.

Proof. (for optimal stopping problems) We omit the control ν from all notations, thus simply writing X t,x (•) and J(t, x; τ ). Inequality (4.4) follows immediately from the tower property together with Assumptions A4'-a, recall that J ≤ V * .

We next prove (4.6). Arguing as in Step 2 of the proof of Theorem 3.5, we first observe that, for every ε > 0, we can find a countable family Āi ⊂ (t i -r i , t i ]×A i ⊂ S, together with a sequence of stopping times τ i,ε in T ti [ti,T ] , i ≥ 1, satisfying Ā0 = {T } × R d and ∪ i≥0 Āi = S, Āi ∩ Āj = ∅ for i = j ∈ N,and J(•; τ i,ε ) ≥ ϕ -3ε on Āi for i ≥ 1 .

(4.7)

Set Ān := ∪ i≤n Āi , n ≥ 1. Given two stopping times θ, τ ∈ T t [t,T ] , it follows from (4.3) (and Assumption A1 in the general mixed control case) that

τ n,ε := τ 1 {τ <θ} + 1 {τ ≥θ} T 1 ( Ān ) c (θ, X t,x (θ)) + n i=1 τ i,ε 1 Āi (θ, X t,x (θ)) is a stopping time in T t [t,T ]
. We then deduce from the tower property together with Assumptions A4'-b and (4.7) that V (t, x) ≥ J(t, x; τ n,ε )

≥ E f X ν t,x (τ ) 1 {τ <θ} + 1 {τ ≥θ} (ϕ(θ, X t,x (θ)) -3ε) 1 Ān (θ, X t,x (θ)) +E 1 {τ ≥θ} f (X t,x (T ))1 ( Ān ) c (θ, X t,x (θ)) .
By sending n → ∞ and arguing as in the end of the proof of Theorem 3.5, we deduce that V (t, x) ≥ E f (X t,x (τ ))

1 {τ <θ} + 1 {τ ≥θ} ϕ(θ, X t,x (θ)) -3ε,
and the result follows from the arbitrariness of ε > 0 and τ ∈ T t [t,T ] .

Application to controlled Markov jump-diffusions.

In this section, we show how the weak DPP of Theorem 3.5 allows to derive the correponding dynamic programming equation in the sense of viscosity solutions. We refer to Crandal, Ishii and Lions [START_REF] Crandall | User's guide to viscosity solutions of second order Partial Differential Equations[END_REF] and Fleming and Soner [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF] for a presentation of the general theory of viscosity solutions.

For simplicity, we specialize the discussion to the context of controlled Markov jump-diffusions driven by a Brownian motion and a compound Poisson process. The same technology can be adapted to optimal stopping and impulse control or mixed problems, see e.g. [START_REF] Bouchard | Optimal control of trading algorithms: a general impulse control approach[END_REF].

Problem formulation and verification of Assumption A.

We shall work on the product space Ω := Ω W ×Ω N where Ω W is the set of continuous functions from [0, T ] into R d , and Ω N is the set of integer-valued measures on [0, T ] × E with

E := R m for some m ≥ 1. For ω = (ω 1 , ω 2 ) ∈ Ω, we set W (ω) = ω 1 and N (ω) = ω 2 and define F W = (F W t ) t≤T (resp. F N = (F N t ) t≤T )
as the smallest right-continuous filtration on Ω W (resp. Ω N ) such that W (resp. N ) is optional. We let P W be the Wiener measure on (Ω W , F W T ) and P N be the measure on (Ω N , F N T ) under which N is a compound Poisson measure with intensity Ñ (de, dt) = λ(de)dt, for some finite measure λ on E, endowed with its Borel tribe E. We then define the probability measure P := P W ⊗ P N on (Ω, F W T ⊗ F N T ). With this construction, W and N are independent under P. Without loss of generality, we can assume that the natural rightcontinuous filtration F = (F t ) t≤T induced by (W, N ) is complete. In the following, we shall slightly abuse notations and sometimes write N t (•) for N (•, (0, t]) for simplicity.

We let U be a closed subset of R k , k ≥ 1, and µ : S×U -→ R d and σ : S×U -→ M d be two Lipschitz continuous functions, and β : S × U × E -→ R d a measurable function, Lipschitz-continuous with linear growth in (t, x, u) uniformly in e ∈ E. Here M d denotes the set of d-dimensional square matrices.

By U o , we denote the collection of all square integrable predictable processes with values U . For every ν ∈ U o , the stochastic differential equation: dX(r) = µ (r, X(r), ν r ) dr+σ (r, X(r), ν r ) dW r + E β(r, X(r-), ν r , e)N (de, dr), t ≤ r ≤ T, (5.1) has a unique strong solution X ν τ,ξ such that X ν τ,ξ (τ ) = ξ, for any initial condition (τ, ξ) ∈ S := (τ, ξ) ∈ S o : ξ is F τ -measurable, and E |ξ| 2 < ∞ . Moreover, this solution satisfies

E sup τ ≤r≤T |X ν τ,ξ (r)| 2 < C(1 + E |ξ| 2 ), (5.2)
for some constant C which may depend on ν. remark 5.1. Clearly, less restrictive conditions could be imposed on β and N . We deliberately restrict here to this simple case, in order to avoid standard technicalities related to the definition of viscosity solutions for integro-differential operators, see e.g. [START_REF] Barles | Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited[END_REF] and the references therein.

The following remark shows that in the present case, it is not necessary to restrict the control processes ν to U t in the definition of the value function V (t, x). remark 5.2. Let Ṽ be defined by

Ṽ (t, x) := sup ν∈U E f (X ν t,x (T )) .
The difference between Ṽ (t, •) and V (t, •) comes from the fact that all controls in U are considered in the former, while we restrict to controls independent of F t in the latter. We claim that Ṽ = V , so that both problems are indeed equivalent. Clearly, Ṽ ≥ V . To see that the converse holds true, fix (t, x) ∈ [0, T ) × R d and ν ∈ U. Then, ν can be written as a measurable function of the canonical process ν((ω s ) 0≤s≤t , (ω s -ω t ) t≤s≤T ), where, for fixed (ω s ) 0≤s≤t , the map ν (ωs) 0≤s≤t : (ω s -ω t ) t≤s≤T → ν((ω s ) 0≤s≤t , (ω s -ω t ) t≤s≤T ) can be viewed as a control independent on F t . Using the independence of the increments of the Brownian motion and the compound Poisson process, and Fubini's Lemma, it thus follows that

J(t, x; ν) = E f (X ν (ωs ) 0≤s≤t t,x (T )) dP((ω s ) 0≤s≤t ) ≤ V (t, x)dP((ω s ) 0≤s≤t )
where the latter equals V (t, x). By arbitrariness of ν ∈ U, this implies that Ṽ (t, x) ≤ V (t, x). remark 5.3. By the previous remark, it follows that the value function V inherits the lower semicontinuity of the performance criterion required in the second part of Theorem 3.5, compare with Remark 3.7. This simplification is specific to the simple stochastic control problem considered in this section, and may not hold in other control problems, see e.g. [START_REF] Bouchard | Optimal control of trading algorithms: a general impulse control approach[END_REF]. Consequently, we shall deliberately ignore the lower semicontinuity of V in the subsequent analysis in order to show how to derive the dynamic programming equation in a general setting.

Let f : R d -→ R be a lower semicontinuous function with linear growth, and define the performance criterion J by (2.1). Then, it follows that U = U o and, from (5.2) and the almost sure continuity of (t, x) → X ν t,x (T ), that J(., ν) is lower semicontinuous, as required in the second part of Theorem 3.5.

The value function V is defined by (2.3). Various types of conditions can be formulated in order to guarantee that V is locally bounded. For instance, if f is bounded from above, this condition is trivially satisfied. Alternatively, one may restrict the set U to be bounded, so that the linear growth of f implies corresponding bounds for V . We do not want to impose such a constraint because we would like to highlight the fact that our methodology applies to general singular control problems. We then leave this issue as a condition which is to be checked by specific arguments to the case in hand.

Proposition 5.4. In the above controlled diffusion context, assume further that V is locally bounded. Then, the value function V satisfies the weak dynamic programming principle (3.1)-(3.2).

Proof. Conditions A1, A2 and A3 from Assumption A are obviously satisfied in the present context. It remains to check that A4 holds true. For ω ∈ Ω and r ≥ 0, we denote ω r

• := ω .∧r and T r (ω

)(•) := ω •∨r -ω r so that ω • = ω r • + T r (ω)(•). Fix (t, x) ∈ S, ν ∈ U t , θ ∈ T t [t,T ]
, and observe that, by the flow property,

E f X ν t,x (T ) |F θ (ω) = f X ν(ω θ(ω) +T θ(ω) (ω)) θ(ω),X ν t,x (θ)(ω) (T )(T θ(ω) (ω)) dP(T θ(ω) (ω)) = f X ν(ω θ(ω) +T θ(ω) (ω)) θ(ω),X ν t,x (θ)(ω) (T )(T θ(ω) (ω)) dP(ω) = J(θ(ω), X ν t,x (θ)(ω); νω )
where, νω (ω) := ν(ω θ(ω) + T θ(ω) (ω)) is an element of U θ(ω) . This already proves A4-a.

As for A4-b, note that if ν := ν1 [0,θ] + ν1 (θ,T ] with ν ∈ U s and θ ∈ T t [t,s] , then the same computations imply

E f X ν t,x (T ) |F θ (ω) = f X ν(ω θ(ω) +T θ(ω) (ω)) θ(ω),X ν t,x (θ)(ω) (T )(T θ(ω) (ω)) dP(ω),
where we used the flow property together with the fact that X ν t,x = X ν t,x on [t, θ] and that the dynamics of X ν t,x depends only on ν after θ. Now observe that ν is independent of F s and therefore on ω θ(ω) since θ ≤ s P -a.s. It follows that

E f X ν t,x (T ) |F θ (ω) = f X ν(Ts(ω)) θ(ω),X ν t,x (θ)(ω) (T )(T θ(ω) (ω)) dP(ω) = J(θ(ω), X ν t,x (
θ)(ω); ν) . remark 5.5. It can be similarly proved that A4' holds true, in the context of mixed control-stopping problems.

PDE derivation.

We can now show how our weak formulation of the dynamic programming principle allows to characterize the value function as a discontinuous viscosity solution of a suitable Hamilton-Jacobi-Bellman equation.

Let C 0 denote the set of continuous maps on [0, T ]×R d endowed with the topology of uniform convergence on compact sets. To (t, x, p, A, ϕ) ∈ [0, T ]×R d ×R d ×M d ×C 0 , we associate the Hamiltonian of the control problem:

H(t, x, p, A, ϕ) := inf u∈U H u (t, x, p, A, ϕ),
where, for u ∈ U ,

H u (t, x, p, A, ϕ) := -µ(t, x, u), p - 1 2 Tr [(σσ )(t, x, u)A] - E (ϕ(t, x + β(t, x, u, e)) -ϕ(t, x)) λ(de),
and σ is the transpose of the matrix σ.

Notice that the operator H is upper-semicontinuous, as an infimum over a family of continuous maps (note that β is locally bounded uniformly with respect to its last argument and that λ is finite, by assumption). However, since the set U may be unbounded, it may fail to be continuous. We therefore introduce the corresponding lower-semicontinuous envelope:

H * (z) := lim inf z →z H(z )forz = (t, x, p, A, ϕ) ∈ S × R d × M d × C 0 . Corollary 5.6. Assume that V is locally bounded. Then: (i) V * is a viscosity subsolution of -∂ t V * + H * (., DV * , D 2 V * , V * ) ≤ 0on[0, T ) × R d . (ii) V * is a viscosity supersolution of -∂ t V * + H(., DV * , D 2 V * , V * ) ≥ 0on[0, T ) × R d .
Proof. 1. We start with the supersolution property. Assume to the contrary that there is (t

0 , x 0 ) ∈ [0, T ) × R d together with a smooth function ϕ : [0, T ) × R d -→ R satisfying 0 = (V * -ϕ)(t 0 , x 0 ) < (V * -ϕ)(t, x)for all(t, x) ∈ [0, T ) × R d , (t, x) = (t 0 , x 0 ), such that -∂ t ϕ + H(., Dϕ, D 2 ϕ, ϕ) (t 0 , x 0 ) < 0.
(5.3)

For ε > 0, let φ be defined by

φ(t, x) := ϕ(t, x) -ε(|t -t 0 | 2 + |x -x 0 | 4 ),
and note that φ converges uniformly on compact sets to ϕ as ε → 0. Since H is upper-semicontinuous and (φ, ∂ t φ, Dφ, D 2 φ)(t 0 , x 0 ) = (ϕ, ∂ t ϕ, Dϕ, D 2 ϕ)(t 0 , x 0 ), we can choose ε > 0 small enough so that there exist u ∈ U and r > 0, with t 0 + r < T , satisfying

-∂ t φ + H u (., Dφ, D 2 φ, φ) (t, x) < 0for all(t, x) ∈ B r (t 0 , x 0 ), (5.4)
where we recall that B r (t 0 , x 0 ) denotes the ball of radius r and center (t 0 , x 0 ). Let (t n , x n ) n be a sequence in B r (t 0 , x 0 ) such that (t n , x n , V (t n , x n )) → (t 0 , x 0 , V * (t 0 , x 0 )), and let X n • := X u tn,xn (•) denote the solution of (5.1) with constant control ν = u and initial condition X n tn = x n , and consider the stopping time θ n := inf {s ≥ t n : (s, X n s ) / ∈ B r (t 0 , x 0 )} .

Note that θ n < T since t 0 + r < T . Applying Itô's formula to φ(•, X n ), and using (5.4) and (5.2), we see that φ(t n , x n ) = E φ(θ n , X n θn ) -θn tn ∂ t φ -H u (., Dφ, D 2 φ, φ) (s, X n s )ds ≤ E φ(θ n , X n θn ) .

Now observe that ϕ ≥ φ + η on ([0, T ] × R d ) \ B r (t 0 , x 0 ) for some η > 0. Hence, the above inequality implies that φ(t n , x n ) ≤ E ϕ(θ n , X n θn ) -η. Since (φ -V )(t n , x n ) → 0, we can then find n large enough so that V (t n , x n ) ≤ E ϕ(θ n , X n θn ) -η/2for sufficiently largen ≥ 1.

On the other hand, it follows from (3.2) that:

V (t n , x n ) ≥ sup ν∈Ut n E ϕ(θ n , X ν tn,xn (θ n )) ≥ E ϕ(θ n , X n θn ) ,
which is the required contradiction.

2. We now prove the subsolution property. Assume to the contrary that there is (t 0 , x 0 ) ∈ [0, T ) × R d together with a smooth function ϕ : [0, T ) × R d -→ R satisfying 0 = (V * -ϕ)(t 0 , x 0 ) > (V * -ϕ)(t, x)for all(t, x) ∈ [0, T ) × R d , (t, x) = (t 0 , x 0 ), such that -∂ t ϕ + H * (., Dϕ, D 2 ϕ, ϕ) (t 0 , x 0 ) > 0.

(5.5)

For ε > 0, let φ be defined by φ(t, x) := ϕ(t, x) + ε(|t -t 0 | 2 + |x -x 0 | 4 ), and note that φ converges uniformly on compact sets to ϕ as ε → 0. By the lowersemicontinuity of H * , we can then find ε, r > 0 such that t 0 + r < T and -∂ t φ + H u (., Dφ, D 2 φ, φ) (t, x) > 0for everyu ∈ U and (t, x) ∈ B r (t 0 , x 0 ). (5.6) Since (t 0 , x 0 ) is a strict maximizer of the difference V * -φ, it follows that sup ([0,T ]×R d )\Br(t0,x0) (V * -φ) ≤ -2η for some η > 0 .

(5.7)

Let (t n , x n ) n be a sequence in B r (t 0 , x 0 ) such that (t n , x n , V (t n , x n )) → (t 0 , x 0 , V * (t 0 , x 0 )).

For an arbitrary control ν n ∈ U tn , let X n := X ν n tn,xn denote the solution of (5.1) with initial condition X n tn = x n , and set θ n := inf {s ≥ t n : (s, X n s ) / ∈ B r (t 0 , x 0 )} .

Notice that θ n < T as a consequence of the fact that t 0 + r < T . We may assume without loss of generality that

|(V -φ)(t n , x n )| ≤ ηfor alln ≥ 1. (5.8)
Applying Itô's formula to φ(•, X n ) and using (5.6) leads to φ(t n , x n ) = E φ(θ n , X n θn ) -θn tn ∂ t φ -H ν n (., Dφ, D 2 φ, φ) (s, X n s )ds ≥ E φ(θ n , X n θn ) .

In view of (5.7), the above inequality implies that φ(t n , x n ) ≥ E V * (θ n , X n θn ) + 2η, which implies by (5.8) that:

V (t n , x n ) ≥ E V * (θ n , X n θn ) + ηforn ≥ 1.
Since ν n ∈ U tn is arbitrary, this contradicts (3.1) for n ≥ 1 fixed.

remark 3 .

 3 [START_REF] Lions | Optimal Control of Diffusion Processes and Hamilton-Jacobi-Bellman Equations I[END_REF]. (The Bolza and Lagrange formulations) Consider the stochastic control problem under the so-called Lagrange formulation:

  .6) Note that we do not use here balls of the usual form B r (s, y) and consider the topoly induced by half-closed intervals on [0, T ]. The fact that t ≤ s for (t , x ) ∈ B(s, y; r) will play an important role when appealing to Assumption A4-b in Step 3 below. Clearly, {B(s, y; r) : (s, y) ∈ S, 0 < r ≤ r (s,y) } forms an open covering of (0, T ] × R d . It then follows from the Lindelöf covering Theorem, see e.g.[START_REF] Dugundj | Topology, Allyn and Bacon series in Advanced Mathematics[END_REF] Theorem 6.3 Chap. VIII, that we can find a countable sequence (t i , x i

Acknowledgments. The authors are grateful to Nicole El Karoui and Marcel Nutz for fruitful comments. This research is part of the Chair Financial Risks of the Risk Foundation sponsored by Société Générale, the Chair Derivatives of the Future sponsored by the Fédération Bancaire Française, the Chair Finance and Sustainable Development sponsored by EDF and Calyon, and the Chair Les particuliers face au risque sponsored by Groupama.