Tanmay Chakraborty
email: chakrabo@eurecom.fr

Utkarsh Trehan
email: trehan@eurecom.fr

Khawla Mallat
email: mallat@eurecom.fr

Jean-Luc Dugelay
email: dugelay@eurecom.fr

Generalizing Adversarial Explanations with Grad-CAM

Gradient-weighted Class Activation Mapping (Grad-CAM), is an example-based explanation method that provides a gradient activation heat map as an explanation for Convolution Neural Network (CNN) models. The drawback of this method is that it cannot be used to generalize CNN behaviour. In this paper, we present a novel method that extends Grad-CAM from example-based explanations to a method for explaining global model behaviour. This is achieved by introducing two new metrics, (i) Mean Observed Dissimilarity (MOD) and (ii) Variation in Dissimilarity (VID), for model generalization. These metrics are computed by comparing a Normalized Inverted Structural Similarity Index (NISSIM) metric of the Grad-CAM generated heatmap for samples from the original test set and samples from the adversarial test set. For our experiment, we study adversarial attacks on deep models such as VGG16, ResNet50, and ResNet101, and wide models such as In-ceptionNetv3 and XceptionNet using Fast Gradient Sign Method (FGSM). We then compute the metrics MOD and VID for the automatic face recognition (AFR) use case with the VGGFace2 dataset. We observe a consistent shift in the region highlighted in the Grad-CAM heatmap, reflecting its participation to the decision making, across all models under adversarial attacks. The proposed method can be used to understand adversarial attacks and explain the behaviour of black box CNN models for image analysis.

Introduction

Automatic Face Recognition (AFR) systems are extremely useful tools in today's world, used in banking, automated border control, healthcare, biometrics, and security applications. Due to the high success rate of systems based on Deep Learning (DL), they are often preferred for these tasks. Computer Vision has found its application in numerous verticals since the success of convolutional neural networks [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF]. However with the increasing use of DL models in critical applications, it has been found that DL models § Khawla is now with SAP Security Research Labs France. are subject to attacks. These attacks in the form of some calculated mathematical noise on the input image cause the model to misclassify images [23]. These adversarial attacks display specific properties, i) They are not perceptible to the human eye, ii) They are controllable, and iii) Transferability, i.e., an attack designed for one model is capable of attacking multiple models [START_REF] Demontis | Why do adversarial attacks transfer? explaining transferability of evasion and poisoning attacks[END_REF].

There is a number of literature on how to attack DL models successfully [START_REF] Alparslan | Adversarial attacks on convolutional neural networks in facial recognition domain[END_REF]. There are mainly two kinds of attacks: targeted and non-targeted attacks. Targeted attack makes a model predict a certain label for the adversarial example, while for non-targeted attacks the labels for adversarial examples are not important, as long as the model is wrong [24]. These attacks can also be subdivided into black-box attacks and white-box attacks. Black-box attacks have no information about the target model, training procedure, architecture, whereas white-box attacks know the target model, training procedure, architecture, parameters. Research has also shown adversarial attacks can also be performed across spectrum, i.e., models can be fooled with examples whose spectrum are transposed [2].

As DL models become more vulnerable to adversarial attacks, it is desirable to have defensive measure and to look into models that have adversarial robustness. In [START_REF] Xie | Feature denoising for improving adversarial robustness[END_REF], feature denoising has been proposed as a method to achieve models that are robust to a number of gradient based adversarial attacks. Adversarial attacks are often successful due to certain neurons which are more sensitive to changes than other neurons, a robust model can be achieved by identifying and controlling the sensitivity of these neurons [START_REF] Zhang | Interpreting and improving adversarial robustness of deep neural networks with neuron sensitivity[END_REF].

It is a common practice in DL to tune hyperparameters to improve model performance. The CKA-similarity algorithm was used to compare the hidden representations of broad and deep models [18]. They found that when the model capacity is large compared to the training set, a block structure emerges, which shows that the models propagate the main component of their hidden representation. A study on effect of network width on adversarial robustness suggests that for similar parameters, wider network achieves better utility but worst adversarial stability [25].

One of the classical ways to detect adversarial attack is by expanding the neural network architecture to have a binary sub-network that can classify regular examples from adversarial examples [17]. Statistical-based methods are also quite commonly found in the literature [19] [START_REF] Grosse | On the (statistical) detection of adversarial examples[END_REF] [9]. More recent methods leverage explainability of machine learning and use SHAP based signatures to detect adversarial attacks [8]. Gradient based similarity computation is also proposed under explainable methods for adversarial example detection [7].

Explainable AI concepts have been used to analyze adversarial attacks, and their defense strategies [13]. They concluded that deep networks are still vulnerable to adversarial attacks and are often not well prepared for intentional input disruption. Visual analysis based adversarial explainability is also found in the related literature [3]. They show that a model precepts adversarial examples differently. Another visual analysis method to explain why models fail observes the flow path of an adversarial example compared to a regular example and produces heatmap, and feature level map [15]. Overall visual analysis of adversarial attacks and why networks actually fail is still under explored and not generalized.

In this work, we attempt to answer three fundamental questions.

• Why do neural networks (wide vs. deep) fail under adversarial settings? • Are certain architectures better than others? • Can we find metrics that can help explain the behaviour (functioning) of CNN models when under adversarial attacks? We leverage advances in explainability to take a look at what the model sees layer by layer using Gradient-Weighted Class Activation Mapping (Grad-CAM). It is an examplebased explanation method, and it does not provide insights into global model behaviour. The other explanation methods for CNNs, such as KernelSHAP [16], also do not provide generalized results for CNNs.

To address some of the above issues, we should understand the global model behaviour. Therefore, we introduce two new metrics, (i) Mean Observed Dissimilarity (MOD) and (ii) Variation in Dissimilarity (VID), for model generalization using Grad-CAM individual heatmap comparisons. We use the Normalized Inverted Structural Similarity Index (NISSIM) metric for the Grad-CAM generated heatmap of the samples from the original test set and the samples from the adversarial test set. Thus, we extend an example-based method into a global explanatory method.

For our case study, we take five out-of-the-box models, of which three are deep models (VGG16, ResNet50, ResNet101) and two wide models (InceptionNetV3, Xcep-tionNet). We train all these models with 50 classes selected randomly from the face recognition dataset VGGFace2.

We also attack these models using the Fast Gradient Sign Method (FGSM) for different noise [10]. We generate results for both black-box attacks (attacks generated using ResNet50 and applied to all models) and a white-box attack (attacks generated for each model). For further analysis, we visualize these models layer by layer output using Grad-CAM for both with and without adversarial attacks.

As a result, we observed a global pattern displayed by all models. The shifting in the region of participation can be defined as when a model sees adversarial examples. Some parts of the input image no longer participate in the decision-making, while new parts do participate. Models are not robust to these changes. These changes are not deterministic, and given an adversarial example, there is no way to tell how it will affect the shift. This dissimilarity is why models fail when attacked, and the extent of this dissimilarity can be quantified with MOD and VID metrics. These metrics are also desirable for robust training, where the goal is to protect a model from adversarial attacks. To be noted that robust training is out of the scope of this work. The goal of this work is global explainability for CNN under an adversarial setting.

To summarize our contribution, we extend Grad-CAM, an example-based explanation method, to a global model explanation method by introducing dissimilarity metrics computed with NISSIM. We define the shifting in the region of participation for CNNs when they are under adversarial attacks. Moreover, we generalize this behaviour through our experiments across different models.

Background

We will use five established CNN models with transfer learning pre-trained on imageNet dataset for our purpose.

The adversarial examples will be generated using FGSM for both black-box and white-box attacks.

Neural networks

VGG16 VGG16 is a deep convolutional neural network with 16 layers. The authors studied the effect of deeper layers while keeping the convolution kernels of size 3x3 [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF].

ResNet50 & ResNet101 ResNets are versatile architectures that introduce residual blocks, skip connections, and shortcuts. They are considered very deep neural networks [12].

InceptionNet InceptionNet is a wide convolutional neural network with factorized convolutions and aggressive regularization. The network is wider essentially than deeper. Also, with deeper networks, there were vanishing gradients. To prevent that the authors introduced auxiliary classifiers in between. They essentially applied softmax to the outputs of two inception modules and computed an auxiliary loss over the same labels. The total loss function is a weighted sum of the auxiliary loss and the real loss [22].

XceptionNet XceptionNet is another wide network that has been developed from InceptionNet by applying depthwise separable convolutions. The number of parameters in XceptionNet is similar to InceptionNet, but the performance gain is due to more efficient use of model parameters [5].

Fast Gradient Sign Method

Creating adversarial examples requires us to add noise to the input image. FGSM is a method of generating noise in the direction of the cost function gradient concerning the data [10]. The noise is then controlled by a parameter ϵ. Given original input image x, label y, model parameter θ, and loss J. We can write adv x = x+ϵ * sign(∇ x J(θ, x, y)) this gives us the perturbations.

Grad-CAM

Grad-CAM is an example based model agnostic explanation tool for CNN that uses the gradient information of the target object and how it flows through a network to create coarse localization heatmaps for visual analysis [START_REF] Ramprasaath R Selvaraju | Grad-cam: Why did you say that?[END_REF]. The heatmap produced by Grad-CAM tells clearly for an image, which parts are under focus and considered by the CNN to come to a decision. Blue parts of the heatmap indicate no participation and red parts indicate high participation.

Metrics

In this section, we define the different metrics we introduce namely NISSIM, MOD, and VID.

Normalized Inverted Structural Similarity Index

Normalized Inverted Structural Similarity Index (NIS-SIM) metric is calculated from Structural Similarity Index (SSIM) by inverting the range and then normalizing it. SSIM is a metric that focuses on similarity, thus inverting it gives us dissimilarity. SSIM bounds to (-1, 1], where -1 means dissimilar while 1 means similar, and NISSIM bounds to (0, 1] where 0 means similar and 1 means dissimilar. Ideally we want this value as close to 0 as possible.

N ISSIM i = 1 -SSIM i 2 (1)

Mean Observed Dissimilarity

Mean Observed Dissimilarity (MOD), is the mean of the NISSIM dissimilarity over the adversarial test set for similar levels of attack. So for every adversarial set X * , calculate NISSIM value for all samples in that set, and divide by the total number of samples. This metric is bounded in between (0,1], such that 0 indicates total similarity while 1 indicates total dissimilarity. This metric is an indication on how much the model is robust under adversarial settings but still inconclusive. As models may be stable to shift, thus the need for us to look into adversarial stability with variation in dissimilarity metric.

M OD advset = 1 N N ISSIM i (2)

Variation in Dissimilarity

Variation in Dissimilarity (VID) is the variance of the NISSIM metric over the adversarial set over different levels of attack eps for a model, this shows the distribution of the attack on the model when different levels of attack are performed. This metric indicates the stability of a model under adversarial settings. Ideally, we would want the distribution to be stable for different levels of attack.

m h = 1 eps N ISSIM eps (3) V ID = (N ISSIM eps -m h ) 2 eps (4)

Case Study

We are conducting a case study on AFR systems. AFR systems use DL for many sensitive tasks such as biometrics and security monitoring. It is common for attackers to attack these sensitive systems. Since CNN models can be outwitted with adversarial examples, it is desirable to find explanations of how these models work and why they fail.

We choose three deep models, namely VGG16, ResNet50, and ResNet101, and two wide models, Incep-tionNetV3 and XceptionNet, to recognize faces from input images.

Case study pipeline

For our case study, we will use a subset of 50 randomly chosen classes from VGGFace2 dataset. VGGFace2 is a large-scale face image dataset created by scraping images from Google image search. The dataset houses 3.31 million images of 9131 classes [4].

First, we preprocess the dataset to align and crop the faces. Then, the dataset is split into 80% training, 10% testing, and 10% validation sets. The fully connected layers are trained for 60 epochs on the training set and the best weight according to the validation accuracy is stored. Once the training step is completed, the stored models are loaded and used to generate perturbations from the test set using FGSM. Then the test set is attacked with different values of ϵ from the stored perturbations and these counterexamples are stored as perturbed test sets (one test set is created for each epsilon). Then the five models are tested with these sets and the performance is noted. First, FGSM attacks were generated using the ResNet50 model and the attacks were transferred to the other models. This verified the transferability of the attacks. Then, attacks were also generated for each model and the respective models were also attacked.

Finally, Grad-CAM was used to generate heatmaps for every layer in each model and each ϵ in the perturbed test set. Subsequently, these heatmaps were analysed by computing the metrics NISSIM, MOD, and VID. We observed a consistent shifting behaviour of the region highlighted by Grad-CAM among all models that were attacked. This allowed us to generalise the behaviour of CNNs under adversarial environment.

Performance degradation

Two types of performance analysis have been performed, one without preprocessing the data and the other with preprocessing. We calculated the Peak Signal to Noise Ratio (PSNR) metric shown in Fig. 3, with the goal to To understand how much noise we introduce during the attacks. A successful attack is given in Fig. 2, to illustrate the working of adversarial attack with FGSM. The degradation of the overall accuracy of each model after an attack for different values of ϵ is given in table 1 for the set without preprocessing and table 2 for the set with preprocessing. These two tables indicate that the FGSM attacks on the model was successful, and we were able to fool the deep models. These attacks are black-box attacks, meaning that the attacks were generated using ResNet50 and all models were attacked. We can observe a pattern that wide models fail more than deep models as the ϵ increases. We also observed that preprocessing the input gave some performance benefits for all models, but the failure trend remained the same.

Finally, we take a look at the white-box attack for a particular example and see how the models perform. The results are given in the Fig. 6, 7, 8, 9, and 10. The original image given for testing is of ABD and where the model was wrong, it identified ABD as Buckley. Again, we portray that the FGSM attack was successful and the models failed.

Analysis and Discussions

For this detailed analysis, we will examine two types of heatmaps generated using Grad-CAM. First, layer by layer heatmaps for all models and epsilon values using the whitebox attack to better understand how each model behaves. And second, heatmaps for black-box attacks to understand if the behaviours are consistent and if they can be generalized.

Explaining white-box attacks

For generating the explanation heatmaps for every layer of a given model with Grad-CAM, we must first remove the last classification layer so that we can access the gradients. On the heatmap, blue means no involvement and red means the highest involvement in the decision-making. The plots are read from bottom right to top left, where bottom right is the first layer and top left is the last layer. The goal of these plots are to observe the shifting region of participation behaviour visually as the model sees them in each layer.

VGG 16 given in Fig. 6, we can observe clearly that all the attacks were successful and illustrates a clear shift of participating regions as the ϵ increases. ResNet50 given in Fig. 7, the number of layers are too many to pin point out some example, yet if observed very carefully the hidden layers as the ϵ increases, we can find a shifting in the region of participation. In ResNet101 given in Fig. 8, it seems more resilient there are some observable region shifts, but overall much less. InceptionNet v3 given in Fig. 9, seems to have learnt something different, the focus was more on forehead than face, but the overall shifting is much higher for this model, we even see focus regions getting inverted as the ϵ increases. For XceptionNet given in Fig. 10, the phenomenon is more clear, some regions get expanded, and background areas are being highlighted.

Explaining black-box attacks

For the back-box attack, we extract the raw explanation heatmap with the goal of observing more clearly the shifting region of participation phenomenon as given in Fig. 5. For each model we can observe shifting as ϵ increases. Which is why we need to quantify this shifting to generalize the phenomenon across models.

MOD and VID computation

We take the raw heatmap as computed in the blackbox attack for all adversarial groups and compute the NIS-SIM metric. This metric requires two heatmaps to compute structural dissimilarity. We use the heatmap obtained from the original image (without adversarial perturbation) as our ground truth heatmap, i.e., what the model expects to see in order to make a decision, then the second heatmap is generated from the adversarially attacked image, and we create a dataframe with all the NISSIM values for the entire test set comparing with the adversarial test set for all values of ϵ.

Next, we compute the MOD metric from this set for all models given in the table 3. The principal idea behind this table is to quantify the observed shifting for every level of ϵ value. We can see that the first column is the ground truth, all heatmaps are similar, so the MOD value is 0. The main observation to keep in mind is, as ϵ increases, the dissimilarity increases, indicating that the focus of the model is diverted when it is presented an adversarial example, this value indicates that the more the examples differ, the more likely the model will fail. We can also use this metric to explain the performance of VGG16. Since the shift was smaller, the model was less likely to fail. We also find that deep networks perform better than wide networks for similar shifts. Next, we observe the distribution of attacks with VID to understand the stability of each model across ϵ values. This distribution shows the variation of shift in the region of participation to the decision-making of a model. The goal of the distributions given in Fig. 4 is to observe the stability of each model under adversarial settings. The main idea, that is examined here, is that the lower the shift in distribution, more the model is robust to adversarial attacks. The shifting in distribution reflects distribution of the shift in the region of participation for different models. We see that the distribution for VGG16 does not shift much over the course of the attack, while the distribution for the other models is scattered and shifts to the left. This indicates that VGG16 is a stable model for this task, over the other models. The two metrics together provide valuable insights into the performance and stability of CNNs under adversarial attacks. Therefore, it is desirable to compute these metrics while we perform defence strategies such as robust training.

Shifting behaviour in the region of participation

From our extensive result analysis, we can generalize the shifting in the region of participation as seen in Fig. 1.

When adversarial examples are seen by a model, there are some parts of the input for which the participation is discarded in decision making, and some other parts start playing a role in the decision process yielding a wrong prediction. We see a shift in the focus of the model in different directions, sometimes backgrounds get highlighted, other times, participation region expands or shrinks. This phenomenon is still non-deterministic in nature, i.e. given an adversarial example it is not possible to predict how focus will be shifted. We also observe that wide networks are more susceptible to this kind of shifting under adversarial attacks. Which more concretely explains our performance degradation trend. Deeper models are much robust to this changes, for similar amount of shift, deeper models provide better performance than wider models.

Conclusions

Neural networks, the core component of a DL system, are essentially a black-box. Very little is known about why these models work, and even less about why they do not. This article attempts to understand popular deep and wide CNN architectures from the perspective of an adversarial attack. From our case study with AFR, we could conclude that neural networks fail because of a shifting behaviour in the region of participation to the decision-making, when the model sees adversarial examples, its focus changes and it now sees a different hidden representation. It appears that deep architectures are more robust to the shifting behaviour in the region of participation, even if they exhibit this behaviour, their degradation is much lower compared to wide networks. This is evident from the various metrics we computed. It is also observed that VGG16 is more stable in an adversarial environment than the other models we have studied. We introduce the NISSIM, MOD, and VID metrics to generalize adversarial behaviour of CNN by quantifying the shifting behaviour in the region of participation. Thus, we extend the example-based explanatory method to a global method for explaining model behaviour.

Figure 1 .

 1 Figure 1. The observed shifting in the region of participation highlighted by Grad-CAM

  (a) ϵ = 0.01 (b) ϵ = 0.05 (c) ϵ = 0.075 (d) ϵ = 0.1 (e) legend

Figure 4 .

 4 Figure 4. VID distribution for different ϵ, the distributions show the variation of shift in region of participation for each model.

Figure 5 .

 5 Figure 5. Black box attack example, depicting the shifting behaviour in the region of participating as ϵ increases

Figure 6 .

 6 Figure 6. Layer by layer visualization of Grad-CAM heatmaps of VGG 16, depicting shifting region of participating as ϵ increases.

Figure 7 .

 7 Figure 7. Layer by layer visualization of Grad-CAM heatmaps of ResNet50, depicting shifting in the region of participating as ϵ increases.

Figure 8 .

 8 Figure 8. Layer by layer visualization of Grad-CAM heatmaps of ResNet101, depicting shifting in the region of participating as ϵ increases.

Figure 9 .

 9 Figure 9. Layer by layer visualization of Grad-CAM heatmaps of InceptionNet v3, depicting shifting in the region of participating as ϵ increases.

Figure 10 .

 10 Figure 10. Layer by layer visualization of Grad-CAM heatmaps of XceptionNet, depicting shifting in the region of participating as ϵ increases.

Table 1 .

 1 Overall accuracy obtained with full-scale attack for data without pre-processing, depicting the success of FGSM attack

	Model	ϵ = 0.000 ϵ = 0.010 ϵ = 0.050 ϵ = 0.075 ϵ = 0.100
	VGG16	91.89%	91.55%	90.87%	89.86%	89.86%
	ResNet50	90.54%	91.89%	88.51%	78.04%	70.60%
	ResNet101	90.87%	92.22%	89.18%	75.33%	61.82%
	InceptionNet v3	88.51%	88.51%	57.09%	35.13%	25.33%
	XceptionNet	92.56%	92.56%	81.41%	69.59%	56.41%

Table 2 .

 2 Overall accuracy obtained with full-scale attack for data with pre-processing, depicting the success of FGSM attack

	Model	ϵ = 0.000 ϵ = 0.010 ϵ = 0.050 ϵ = 0.075 ϵ = 0.100
	VGG16	87.54%	85.60%	85.21%	84.82%	84.43%
	ResNet50	94.16%	94.55%	90.66%	85.21%	75.09%
	ResNet101	93.38%	93.38%	87.15%	84.82%	77.43%
	InceptionNet v3	85.21%	84.82%	66.92%	47.85%	29.96%
	XceptionNet	89.49%	90.27%	71.20%	59.14%	45.13%

Table 3 .

 3 MOD values obtained for different ϵ values, quantifying the observed shifting for every level of ϵ

	Model	ϵ = 0.000 ϵ = 0.010 ϵ = 0.050 ϵ = 0.075 ϵ = 0.100
	VGG16	0	0.098875 0.117416 0.121993 0.127394
	ResNet50	0	0.127743 0.241743 0.269364 0.292002
	ResNet101	0	0.084589 0.208446 0.245334 0.262114
	InceptionNet v3	0	0.160696 0.275883 0.296766 0.295898
	XceptionNet	0	0.162345 0.255316 0.272804	0.27981
	Mean shift	0	0.12685	0.219761 0.241252 0.251444

Acknowledgement This work has been partially supported by the European CHIST-ERA program via the French National Research Agency (ANR) within the XAIface project (grant agreement CHIST-ERA-19-XAI-011).


E Lépinette

Duc Thinh Vu
email: vu@ceremade.dauphine.fr

Limit theorems for the super-hedging prices in general models with transaction costs

Keywords: Numerical methods in finance, Super-hedging prices, Transaction costs, No-arbitrage condition Mathematics Subject Classification (2010): 49J53, 60D05, 91G20, 91G80. JEL Classification: C02, C61, G13

We propose numerical methods that provide estimations of super-hedging prices of European claims in financial market models with transaction costs. The transaction costs we consider are functions of the traded volumes and prices. Contrarily to the usual models of the literature, the transaction costs are not necessary proportional to the traded volumes, neither convex. The particular case of fixed cost is also considered. Limit theorem are established and allow to numerically compute the infimum super-hedging prices.

Introduction

Computing the super-hedging prices of a European option in presence of transaction costs is a difficult task. Indeed, the classical results of the literature focus on linear transaction costs and only dual characterizations of the super-hedging prices are formulated, see the FTAP theorems (Fundamental Theorem of Asset Pricing) by [START_REF] Guasoni | The fundamental theorem of asset pricing for continuous processes under small transaction costs[END_REF], [START_REF] Guasoni | The fundamental theorem of asset pricing under transaction costs[END_REF], [START_REF] Kabanov | Markets with transaction costs. Mathematical Theory[END_REF] among others. These results are formulated under rather strong no-arbitrage conditions (see [START_REF] Guasoni | Consistent Price Systems and Face-Lifting Pricing under Transaction Costs[END_REF], [START_REF]Consistent price systems and arbitrage opportunities of the second kind in models with transaction costs[END_REF]) and the super-hedging prices are estimated through dual characterizations based on the so-called consistent price systems, see [START_REF] Campi | A super-replication theorem in Kabanov's model for transaction costs[END_REF], [START_REF] Vallière | Hedging of American options under transaction costs[END_REF].

The interesting question is how to implement the FTAP theorem and deduce numerical estimation of the prices. Few attempts have been achieved in that direction, e.g. [START_REF] Löhne | An algorithm for calculating the set of superhedging portfolios in markets with transaction costs[END_REF] in the case of a finite probability space. The general case is difficult as we have first to identify the dual elements, i.e. the consistent price systems, which are martingales evolving in the positive duals of the solvency cones. The second step is to propose a numerical procedure to evaluate the possible super-hedging prices. There is no such a numerical method in the literature. Moreover, if the transaction costs are non linear, there is a priori no dual elements characterizing the no-arbitrage condition.

The methods we develop in this paper are based on the recent paper [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF] where the super-hedging prices are characterized for a large class of transaction cost models which are not necessary linear. Indeed, using a new approach based on a weak no-arbitrage condition, mainly the prices of the non negative claims are supposed to be non negative, we prove that the infimum prices of a European claim are solutions to a dynamic programming problem it is possible to solve backwardly, at least in discrete-time. However, in [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF], the results are merely theoretical. The authors do not provide algorithms to compute the super-hedging costs in practice. In this paper, we address this problem. To be precise, we consider financial markets with transaction costs defined by a cost process (C t ) 0≤t≤T depending on traded volumes and a process (S t ) 0≤t≤T that includes the asset prices. We shall consider the case of countably infinite t-conditional supports for S t+1 where an exact characterization of the super-hedging costs is given. The randomized procedure we propose is based on the simulation of conditionally identically distributed random variables which share the same conditional support as the price process (S t ) 0≤t≤T . We formulate a limit theorem, see Theorem 3.15, that proves the efficiency of our method.

The paper is organized as follows. In Section 2, we recall the market model studied in [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF] and the dynamic programming theorem. In Section 3, we describe the numerical scheme and the main convergence theorems. We present in Section 4 the special case of a model with one risky asset and a piecewise cost process (C t ) 0≤t≤T . In Section 5, we also give the exact solution of the super-hedging cost in the models with proportional costs and with and without fixed cost. Finally, in Section 6, we prove a limit theorem for a sequence of financial markets defined by convex cost processes.

The model

Let ξ ∈ L 0 (R d , F T ) be a contingent claim. Our goal is to characterize the set of all self-financing portfolio processes (V t ) T t=-1 such that V T = ξ. Recall that a portfolio process is by definition (see [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF]) a stochastic process (V t ) T t=-1 where V -1 ∈ Re 1 is the initial endowment expressed in cash that we may convert immediately into V 0 ∈ R d at time t = 0. By definition, we suppose that

∆V t = V t -V t-1 ∈ -G t , a.s., t = 0, • • • , T,
where, for every t ≤ T , G t is the random set of all solvent positions. Our general model is defined by a set-valued process (G t ) T t=0 adapted to the filtration (F t ) T t=0 . Precisely, we suppose that for all t ≤ T , G t is F tmeasurable in the sense of the graph Graph(G t ) = {(ω, x) : x ∈ G t (ω)} that belongs to F t × B(R d ), where B(R d ) is the Borel σ-algebra on R d and d ≥ 1 is the number of assets.

We suppose that G t (ω) is closed for every ω ∈ Ω and G t (ω)+R d + ⊆ G t (ω), for all t ≤ T . The cost value process C = (C t ) T t=0 associated to G is defined as:

C t (z) = inf{α ∈ R : αe 1 -z ∈ G t } = min{α ∈ R : αe 1 -z ∈ G t }, z ∈ R d .
We suppose that the right hand side in the definition above is non empty a.s. and -e 1 does not belong to G t a.s. where e 1 = (1, 0, • • • , 0) ∈ R d . Moreover, by assumption, C t (z)e 1 -z ∈ G t a.s. for all z ∈ R d . Note that C t (z) is the minimal amount of cash one needs to get the financial position z ∈ R d at time t. In particular, we suppose that C t (0) = 0.

If we define the liquidation value process L t (z) = -C t (-z), we get that G t = {z ∈ R d : L t (z) ≥ 0} and, as G t is supposed to be closed a.s., L t (z) is upper semicontinuous (u.s.c.) in z, see [START_REF] Lépinette | Arbitrage theory for non convex financial market models[END_REF], or equivalently C t (z) is lower semicontinuous (l.s.c.) in z. Naturally, C t (z) = C t (S t , z) depends on the available quantities and prices for the risky assets, described by an exogenous vector-valued F t -measurable random variable S t of R m + , m ≥ d, and on the quantities z ∈ R d to be traded. Here, we suppose that m ≥ d as an asset may be described by several prices and quantities offered by the market, e.g. bid and ask prices, or several pair of bid and ask prices of an order book and the associated quantities offered by the market. Some examples of models are given in [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF]. In the following, we are interested in the infimum cost one needs to super-hedge ξ, i.e. the infimum value of the initial capitals V -1 ∈ R among the portfolios (V t ) T t=-1 super-replicating ξ.

In the following, we use the notation z = (z 1 , z 2 , ..., z d ) ∈ R d and we denote z (2) = (z 2 , ..., z d ) ∈ R d-1 . Recall that the F t -measurable conditional essential supremum of a family of random variables is the smallest F t -measurable random variable that dominates the family with respect to the natural order between

[-∞, ∞]-valued random variables, i.e. X ≤ Y if P (X ≤ Y ) = 1, see [20, Section 5.3.1].

The one step hedging problem between time t -1 and t

Recall that V t-1 ≥ Gt V t by the definition of portfolio process. Then, the hedging problem V t = ξ is equivalent at time t -1 to:

L t (V t-1 -ξ) ≥ 0 ⇐⇒ V 1 t-1 ≥ ξ 1 -L t ((0, V (2)
t-1 )), ⇐⇒ V 1 t-1 ≥ ess sup F t-1 ξ 1 -L t ((0, V (2)
t-1 -ξ (2) )) , ⇐⇒ V 1 t-1 ≥ ess sup F t-1 ξ 1 + C t ((0, ξ (2) -V (2)
t-1 )) ,

⇐⇒ V 1 t-1 ≥ F ξ t-1 (V (2)
t-1 ), where

F ξ t-1 (y) = ess sup F t-1 ξ 1 + C t ((0, ξ (2) -y)) . (2.1)
By virtue of [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF]Proposition 5.7], we may suppose that

F ξ t-1 (ω, y) is jointly F t-1 × B(R d-1
)-measurable, a.s. l.s.c. (lower semi-continuous) as a function of y and convex in y if C t (s, y) is convex in y. As F t-1 is supposed to be complete, we conclude that F ξ t-1 is an F t-1 -normal integrand, see [START_REF] Rockafellar | Variational analysis[END_REF][Theorem Corollary 14.34].

The multi-step hedging problem

We denote by P t (ξ) the set of all portfolio processes starting at time t ≤ T that replicates ξ at the terminal date T :

R t (ξ) := (V s ) T s=t : V s ∈ L 0 (R d , F s ), ∆V s ∈ -G s , ∀s ≥ t + 1, V T = ξ .
The set of replicating prices of ξ at time t is

P t (ξ) := V t = (V 1 t , V (2)
t ) : (V s ) T s=t ∈ R t (ξ) .
We define the the infimum replicating cost by:

c t (ξ) := ess inf Ft {C t (V t ), V t ∈ P t (ξ)} . By Section 2.1, we know that V T -1 ∈ P T -1 (ξ) if and only if V 1 T -1 ≥ ess sup F T -1 ξ 1 + C T (0, ξ (2) -V (2)
T -1 ) a.s.

Similarly, V T -2 ∈ R T -2 (ξ) if and only if there exists V (2) T -1 ∈ L 0 (R d-1 , F T -1 ) such that V 1 T -2 ≥ ess sup F T -2 ess sup F T -1 ξ 1 + C T (0, ξ (2) -V (2)
T -1 ) + C T -1 (0, V (2)
T -1 -V (2)
T -2 ) .

By the tower property satisfied by the conditional essential supremum, we deduce that

V T -2 ∈ R T -2 (ξ) if and only if there is V (2) T -1 ∈ L 0 (R d-1 , F T -1 ) such that V 1 T -2 ≥ ess sup F T -2 ξ 1 + C T (0, ξ (2) -V (2)
T -1 ) + C T -1 (0, V (2)
T -1 -V

T -2 ) .

Recursively, we get that V t ∈ P t (ξ) if and only if, for some

V (2) s ∈ L 0 (R d-1 , F s ), s = t + 1, • • • , T -1, and with V (2) T = ξ (2) , we have V 1 t ≥ ess sup Ft ξ 1 + T s=t+1 C s (0, V (2) s -V (2)
s-1 ) .

In the following, for

u ≤ T -1, ξ u-1 ∈ L 0 (R d , F u-1 ), and ξ ∈ L 0 (R d , F T ),
we define the sets:

Π T u (ξ u-1 , ξ) := {ξ (2) u-1 } × Π T -1 s=u L 0 (R d-1 , F s ) × {ξ (2) } of all families (V (2) s ) t+1 s=u-1 such that V (2) u-1 = ξ u-1 , V (2) s ∈ L 0 (R d-1 , F s ) for all s = u, • • • , T -1 and V (2) T = ξ (2) . We set Π T u (ξ) := Π T u (0, ξ) = Π T u (ξ u-1 , ξ) when ξ (2) u-1 = 0. When u = T , we set Π T T (ξ T -1 , ξ) := {ξ (2)
T -1 }×{ξ (2) }. Therefore, the infimum replicating cost at time 0 is given by

c 0 (ξ) = ess inf F 0 V 2 ∈Π T 0 (ξ) ess sup F 0 ξ 1 + T s=0 C s (0, V 2 s -V 2 s-1 ) . For 0 ≤ t ≤ T and V t-1 ∈ L 0 (R d , F t ), we define γ ξ t (V t-1
) as:

γ ξ t (V t-1 ) := ess inf Ft V (2) ∈Π T t (V t-1 ,ξ) ess sup Ft ξ 1 + T s=t C s (0, V (2) s -V (2) s-1 ) . (2.2)
Note that γ ξ t (V t-1 ) is the infimum cost to replicate the payoff ξ when starting from the initial position (0, V

t-1 ) at time t. Observe that γ ξ t (V t-1 ) does not depend on the first component V 1 t-1 of V t . Moreover, γ ξ T (V T -1 ) = ξ 1 + C T (0, ξ (2) -V (2)
T -1 ).

As

G T + R d + ⊆ G T , we also observe that γ ξ T (V T -1 ) ≥ γ 0 T (V T -1 ) if ξ ∈ L 0 (R d + , F T ). At last, observe that c 0 (ξ) = γ ξ 0 (0).
We recall the following result from [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF]:

Proposition 2.1 (Dynamic Programming Principle). For any 0 ≤ t ≤ T -1 and V t-1 ∈ L 0 (R d , F t-1
), we have

γ ξ t (V t-1 ) = ess inf Ft Vt∈L 0 (R d ,Ft) ess sup Ft C t (0, V (2) t -V (2)
t-1 ) + γ ξ t+1 (V t ) . (2.3)
Assumption 1. The payoff ξ is hedgeable, i.e. there exists a portfolio process

(V ξ u ) T u=0 such that ξ = V ξ T .
The dynamic programming principle (2.3) allows to get γ ξ t (V t-1 ) from the cost function C t and from γ ξ t+1 . In the paper [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF], we have shown that γ ξ t is l.s.c. for any t and convex, or piecewise linear, if γ ξ t+1 satisfies the same properties.

As the term C t (0, V

t -V (2)
t-1 ) in (2.3) is F t-1 -measurable, we consider the conditional supremum

θ ξ t (V t ) := ess sup Ft γ ξ t+1 (V t )
to compute the essential supremum of (2.3). In the following, we shall use the following notations:

D ξ t (V t-1 , V t ) = C t ((0, V (2) t
-V

(2)

t-1 )) + θ ξ t (V t ), (2.4)
D ξ t (S t , V t-1 , V t ) = C t (S t , (0, V (2) t -V (2)
t-1 )) + θ ξ t (S t , V t ).

(2.5)

The second notation is used when we stress the dependence on S t . Observe that

γ ξ t (V t-1 ) = ess inf Ft Vt∈L 0 (R d ,Ft) D ξ t (V t-1 , V t ).
In order to numerically compute the minimal costs, we need to impose the finiteness of γ ξ t (V t-1 ), i.e. γ ξ t (V t-1 ) > -∞ a.s., at any time t and for all V t-1 ∈ L 0 (R d , F t-1 ). This is why, we consider the following condition, see [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF]: Definition 2.2. We say that the financial market satisfies the Absence of Early Profit condition (AEP) if, at any time t ≤ T , and for all

V t ∈ L 0 (R d , F t ), γ 0 t (V t ) > -∞ a.s..

Numerical schemes

In the following, we suppose the following assumptions on the cost process C. For any t ≤ T , the cost function C t is a lower-semi continuous Borel function defined on

R k × R d such that C t (s, 0) = 0, ∀s ∈ R k + , C t (s, x + λe 1 ) = C t (s, x) + λ, λ ∈ R, x ∈ R d , s ∈ R k + (cash invariance), C T (s, x 2 ) ≥ C T (s, x 1 ), ∀x 1 , x 2 s.t. x 2 -x 1 ∈ R d + (C T is increasing w.r.t. R d + ). Note that C T is increasing w.r.t. R d + is equivalent to G T + R d + ⊆ G T .
Moreover, for some a ≥ 0, we say that C t is a-super homogeneous if the following property holds:

C t (s, λx) ≥ λC t (s, x), ∀λ ≥ a, s ∈ R k + , x ∈ R d .

The one period model

In this section, we consider two complete sub σ-algebras F t and F t+1 such that F t ⊂ F t+1 ⊂ F and an adapted price process (S s ) s=t,t+1 satisfying the following assumption.

Assumption 2. Suppose that there is a family of F t -measurable random variables (α m t ) m≥1 such that S t+1 ∈ {α m t : m ≥ 1} a.s. and suppose that P (S t+1 = α m t |F t ) > 0 a.s. for all m ≥ 1. Moreover, we suppose that there exists continuous functions on R m , that we still denote by α m t with an abuse of notation, such that α m t = α m t (S t ). In [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF], we have shown the following: Lemma 3.1. Suppose that Assumption 2 holds. Then, for any Borel function f : R d → R, we have

ess sup Ft f (S t+1 ) = sup m≥1 f (α m t ), a.s.. Definition 3.2. The random variables {b i t+1 , i ≥ 1}, b i t+1 ∈ L 0 (R k , F t+1
), are said independent and identically distributed conditionally to F t (for short F t -i.i.d.) if, for all finite set J ⊂ N, and Borel sets B, B j , j ∈ J:

P b i t+1 ∈ B|F t = P b j t+1 ∈ B|F t , a.s. ∀i, j ≥ 1, P j∈J b j t+1 ∈ B j F t = j∈J P b j t+1 ∈ B j F t ,
θ t ∈ L 0 (R m , F t ). Let f j : R k × R m → R, j = 1, • • • , n be n ≥ 1 measurable functions such that E |f j (b 1
t+1 , θ t )||F t < ∞ a.s. (resp. f j is non negative), for all j ≤ n. Then, for any finite set J ⊂ N of cardinality n, we have:

E f k (b i t+1 , θ t )|F t = E f k (b j t+1 , θ t )|F t , a.s., i, j, k ≥ 1, E j∈J f j (b j t+1 , θ t ) F t = j∈J E f j (b j t+1 , θ t ) F t , a.s..
Proof. We prove the result by induction on n. Suppose that f j = 1 D j where

D j = B j ×A j and B j ∈ B(R k ), A j ∈ B(R m ).
Then, the claim holds by definition of the F t -i.i.d. random variables for all n ≥ 1 and the F t -measurability of θ t . By the monotone class argument, this holds for any

D 1 ∈ B(R k )⊗B(R m ) if n = 1. If n > 1,
we expand the product in the second claim and we use the induction hypothesis. Then, we repeat the arguments for D 2 ∈ B(R k ) ⊗ B(R m ) and so on. By linearity, and the induction argument after having expanding the product, we also deduce that the claim holds when

f j = n h=1 c j h 1 C j h and for any c j h ∈ R, C j h ∈ B(R k ) ⊗ B(R m ), h ≥ 1.
By standard increasing approximations, we conclude in the case where f j ≥ 0. Otherwise, we write f j = (f j ) + -(f j ) -. In particular, we get that ). The following holds: θ m t → θ t , a.s. as m → ∞. In particular, sup m θ m t = θ t a.s. Proof. We may suppose w.l.o.g. that θ t < ∞. Indeed, we may consider g(θ t ) and the sequence (g(θ m t )) m≥1 where g is a bounded strictly increasing continuous function in the contrary case. By Lemma 3.1, we get that ess sup Ft f (b 1 t+1 ) = sup m≥1 f (α m t ) = θ t a.s. For any > 0, we deduce by assumption that

E |f j (b i t+1 , θ t )||F t = E |f j (b 1 t+1 , θ t )||F t < ∞,
P [θ t -θ m t > |F t ] = P [θ t -max i≤m f (b i t+1 ) > |F t ] = P [θ t -f (b i t+1 ) > , ∀i ≤ m|F t ] = E m i=1 1 {θt-f (b i t+1 )> } F t , a.s.
. By Lemma 3.3, we deduce that

P [θ t -θ m t > |F t ] = P [θ t -f (b 1 t+1 ) > |F t ] m = P [ess sup Ft f (b 1 t+1 ) -f (b 1 t+1 ) > |F t ] m , a.s. We claim that P [ess sup Ft f (b 1 t+1 )-f (b 1 t+1 ) > |F t ] < 1 a.s. Indeed, assume on the contrary that P [ess sup Ft f (b 1 t+1 ) -f (b 1 t+1 ) > |F t ] = 1 on some non null set Λ t ∈ F t .
In other words, we have

E 1 {ess sup F t f (b 1 t+1 )>f (b 1 t+1 )+ } F t 1 Λt = 1 Λt .
Taking the expectation, we deduce that:

E 1 {ess sup F t f (b 1 t+1 )>f (b 1 t+1 )+ } 1 Λt = E [1 Λt ] We then deduce that 1 {ess sup F t f (b 1 t+1 )>f (b 1 t+1 )+ } 1 Λt = 1 Λt a.s. We now define θt := ess sup Ft f (b 1 t+1 )1 Ω\Λt + (ess sup Ft f (b 1 t+1 ) -)1 Λt . Observe that θt is F t - measurable and θt ≥ f (b 1 t+1 ) a.s. However, θt < ess sup Ft f (b 1 t+1
) on the non null set Λ t , in contradiction with the definition of the conditional essential supremum. Therefore,

lim m→∞ P [θ t -θ m t > |F t ] = 0, a.s.
Finally, by the dominated convergence theorem, we have

lim m→∞ P [θ t -θ m t > ] = lim m→∞ E E[1 {θt-θ m t > } |F t ] = E lim m→∞ E[1 {θt-θ m t > } |F t ] = 0.
Hence θ m t increasingly tends to θ t in probability, i.e. sup m θ m t = θ t a.s.. Assumption 3. The payoff function ξ is of the form ξ = g(S T ), where g ∈ R k + is continuous. Moreover, ξ is hedgeable, i.e. there exists a portfolio process (V ξ u ) T u=-1 such that ξ = V ξ T . We recall here two weak no-arbitrage conditions introduced in [22]: Definition 3.5. We say that the condition AIP holds at time t if the minimal cost c t (0) = γ 0 t (0) of the European zero claim ξ = 0 is 0 at time t ≤ T . We say that AIP holds if AIP holds at any time.

The following condition is more technical. Definition 3.6. We say that the condition SAIP (Strong AIP condition) holds at time t if AIP holds at time t and, for any

Z t ∈ L 0 (R d , F t ), we have D 0 t (S t , 0, Z t ) = 0 if and only if Z (2) t
= 0 a.s.. We say that SAIP holds if SAIP holds at any time.

The condition SAIP states that the minimal cost of the zero payoff is 0 at time t and this minimal cost is only attained by the zero strategy V t = 0, see [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF]. This is intuitively clear as soon as any non null transaction implies positive costs. 10

We now introduce the sequence of functions which is defined recursively as follows:

γξ T (s, v T -1 ) := ξ 1 + C T (s, (0, ξ (2) -v (2)
T -1 )), v t-1 , ξ ∈ R d , s ∈ R k , θξ t (s, v t ) := sup m γξ t+1 (α m t (s), v t ), t ≤ T -1, v t ∈ R d , Dξ t (s, v t-1 , v t ) := θξ t (s, v t ) + C t (s, (v (2)
t -v (2) t-1 )), γξ t (s, v t-1 ) := cl inf vt∈R d Dξ t (s, v t-1 , v t ) . (3.6)
Here, the notation cl(f ) designates the l.s.c. regularization of f . In this paper, we will impose later in the sequel a condition under which we have γξ

t (s, v t-1 ) := inf vt∈R d Dξ t (s, v t-1 , v t
). The introduction of the functions above is motivated by the following result proved in [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF].

Theorem 3.7. Suppose that either AIP holds and C t (s, .) is convex for fixed s or SAIP holds. Then, we have γ ξ t (S t , V t ) = γξ t (S t , V t ) a.s. and, also,

θ ξ t (S t , V t ) = θξ t (S t , V t ) a.s. and D ξ t (S t , V t-1 , V t ) = Dξ t (S t , V t-1 , V t ) for any V t-1 , V t ∈ L 0 (R d , F t ). Moreover, γξ t (s, v) is l.s.c. on R k × R d and convex in v when C t (s, .) is convex.
Recall that the family of F t -measurable random variables (α n t (S t )) n≥1 is defined in Assumption 2. We now consider an F t -i.i.d. sample of random variables {b i t+1 , i ≥ 1} that satisfies P [b 1 t+1 = α n t (S t )|F t ] > 0 a.s. for all n ≥ 1 and b 1 t+1 ∈ {α n t (S t ), n ≥ 1} a.s. Now, let us define the (random) functions Dξ T (s, x, y)

:= γξ T (s, y), Dξ t (s, x, y) := C t (s, (0, y (2) -x (2) )) + γξ t+1 (s, y), Dn T (ω, x, y) := Dξ T (s, x, y) Dn t (ω, x, y) := max i≤n Dξ t (b i t+1 (ω), x, y). (3.7)
Since γξ t+1 (s, x) is l.s.c. in s, it is Borel in s for fixed x. Then, by Lemma 3.4, we deduce that:

lim n→∞ max i≤n γξ t+1 (b i t+1 (ω), y) = sup n γξ t+1 (α n t (S t (ω)), y) = θξ t (S t (ω), y), a.s.
In particular, lim n→∞ Dn t (ω, x, y) = Dξ t (S t (ω), x, y). We now investigate the question whether inf y∈R d Dn t (ω, x, y) converge a.s.(ω) to inf y∈R d Dξ t (ω, x, y)

as n → ∞. To do so, we first recall the definition of epi-convergence, see [START_REF] Molchanov | Theory of Random Sets[END_REF]Chapter 3] or [START_REF] Rockafellar | Variational analysis[END_REF]Chapter 7]. In the following, the notation B(x, r) designates the closed ball of R d , where d ≥ 1 depends on the context, centered a point x ∈ R d and of radius r ≥ 0.

Definition 3.8. Let f n : R k → R, n ≥ 1, be a sequence of functions.

The epi-limit inferior li e f n and epi-limit superior ls e f n of (f n ) n≥1 are defined as:

li e [(f n ) n≥1 ](u) := sup k≥1 lim inf n→∞ inf v∈B(u,1/k) f n (v), ls e [(f n ) n≥1 ](u) := sup k≥1 lim sup n→∞ inf v∈B(u,1/k) f n (v).
The sequence

(f n ) n≥1 is said to be epi-convergent at point u if li e [(f n ) n≥1 ](u) = ls e [(f n ) n≥1 ](u).
We also introduce the definition of almost sure epi-convergence for random functions.

Definition 3.9. If (f n ) n≥1 , is a sequence of functions f n : Ω × R k → R such that f n is F t ⊗ B(R d
)-measurable for each n, we say that f n epi-converges to f almost surely (notation f n epi -→ f a.s.) if, for any ω outside a P -null set, and for all u:

li e [(f n (ω, •)) n≥1 ](u) = ls e [(f n (ω, •)) n≥1 ](u) = f (ω, u).
Theorem 3.10. Suppose that AIP holds and C t (s, y) is convex in y. We then have Dn t (ω, ., .) ∨ -C t (S t (ω), (0,

x (2) )) epi -→ Dξ t (S t (ω), ., .) a.s.(ω), as n → ∞.
Suppose that SAIP holds and, for any t,

C t (s, v 1 t ) ≥ C t (s, v 2 t ) if v 1 t ≥ R d + v 2 t .
li e [( Ln t (ω, •, •) n≥1 )](x, y) = ls e [( Ln t (ω, •, •) n≥1 )](x, y) = sup n Ln t (ω, x, y).
We now prove that there exists a negligible set H such that for any ω ∈ Ω\H and x, y ∈ R d × R d the following holds:

sup n Ln t (ω, x, y) = Dξ t (ω, x, y). (3.8)
By assumption on (C t ) t≥0 , we get by induction that

θ ξ t (V t ) ≥ θ 0 t (V t ) a.s. for any V t ∈ L 0 (R d , F t ). We deduce that D ξ t (V t-1 , V t ) ≥ -C t (S t (ω), (0, V (2)
t-1 )) for any for any

V t-1 , V t ∈ L 0 (R d , F t ). Indeed, under AIP, D 0 t (0, V t ) ≥ 0 a.s. hence D ξ t (V t-1 , V t ) = θ ξ t (V t ) + C t (S t , (0, V (2) t -V (2) t-1 )) ≥ θ ξ t (V t ) + C t (S t , (0, V (2)
t )) -C t (S t , (0, V (2)
t-1 )), (by subadditivity)

≥ θ 0 t (V t ) + C t (S t , (0, V (2)
t )) -C t (S t , (0, V (2)
t-1 )) ≥ D 0 t (0, V t ) -C t (S t , (0, V (2)
t-1 )) ≥ -C t (S t , (0, V (2)
t-1 )), a.s.

for any

V t-1 , V t ∈ L 0 (R d , F t ).
We now deduce that Dξ t (S t (ω), x, y) ≥ -C t S t (ω), (0, x (2) ) for every x, y a.s.(ω). Indeed, suppose on the contrary that the F t -measurable set

Γ t (ω) := (x, y) ∈ R d × R d : Dξ t (S t (ω), x, y) < -C t S t (ω), (0, x (2)
) is non-empty on the non-null set G t := {ω : Γ t (ω) = ∅}. We then deduce a measurable selection ( Vt-1 , Vt )

∈ L 0 (R d , F t ) × L 0 (R d , F t ) such that we have Dξ t (S t , Vt , Vt-1 ) < -C t (S t , (0, V (2)
t-1 )) on G t and we extend to the whole space by putting Vt-1 = 0 = Vt on the complementary set Ω \ G t . Moreover, by Theorem 5.5, we then deduce that D ξ t ( Vt , x) < -C t (S t , (0, V

t-1 ) on the non-null set G t , which is a contradiction.

Similarly, under AEP and Assumption 3, we have that [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF]. Then, by a measurable selection argument, using the fact that

D ξ t (V t-1 , V t ) ∈ R a.s. for any V t-1 , V t ∈ L 0 (R d , F t ), see
D ξ t (V t-1 , V t ) = Dξ t (S t , V t-1 , V t ) a.s.,
(s, v 1 t ) ≥ C t (s, v 2 t ) for any v 1 t , v 2 t ∈ R d such that v 1 t ≥ R d + v 2 t .
cl(f )(x) = lim inf y→x f (x) = α ∈ R : ∃(x n ) n≥1 , x n → x, lim n f (x n ) = α . Consider x 1 , x 2 ∈ R d such that x 1 ≥ R d + x 2 and a sequence (x n ) n≥1 such that x n → x 2 and f (x n ) → cl(f )(x 2 ) as n → ∞. Observe that x n + x 1 -x 2 → x 1 as n → ∞. We then have f (x n + x 1 -x 2 ) ≤ f (x n ) by our hypothesis. We deduce that cl(f )(x 1 ) ≤ lim inf n f (x n + x 1 -x 2 ) ≤ lim n f (x n ) = cl(f )(x 2 ).
Definition 3.12. We say that a set-valued mapping

K t : R k + × R d R d is

Multi-period framework

In this section, we consider the multi-period setting t = 0, • • • , T . Our goal is to determine the infimum super-hedging cost of ξ := g(S T ) = (g 1 (S T ), g (2) (S T )) at time 0, where g : R k + → R d + is a deterministic continuous function. To do so, we apply the dynamic programming principle of Proposition 2.1 to recursively compute γ ξ t (V t-1 ) for t = 0, • • • , T . Moreover, since γ ξ 0 (0) = γξ 0 (S 0 , 0) under the weak no-arbitrage condition we suppose, it is then sufficient to compute γ0 (S 0 , V 0 ) for V 0 = 0. We work under the following assumption: Assumption 4. For each t, suppose that there is a reachability set-valued mapping 1) b i 0 = S 0 for all i ∈ J 0 = N \ {0}. 2) For given t ≥ 0, we denote Ft = σ(b k u : k ∈ J u , u ≤ t) where (b k u ) k∈Ju are the random variables constructed at time t. Then, for time t + 1, and for each i ∈ J t , we generate a sequence of i.i.d. random variables α j t+1 , j ≥ 1, independent of F t such that α j t+1 ∈ L 0 (Θ, F t+1 ) for each j. Moreover, supp Ft α j t+1 = Θ. We then define for each i ∈ J t and j ≥ 1, b i,j t+1 = α j t+1 b i t . Then, J t+1 = {(i, j) : i ∈ J t , j ≥ 1}.

K t : R k + × R d R d such that K t (s, v t-
To compute γξ 0 (S 0 , 0), we approximate γξ t (b i t , v t-1 ) by the randomization method considered in the last section that we extend to the multi-period setting.

We denote

n 1 = (n 1 u ) u=1,••• ,T a generic element in N T and, for t = 1, • • • , T , we define n t = (n t u ) u=t,••• ,T ∈ N T -t+1 . If b i t ∈ {α k t b j t-1 ; j ∈ J t-1 , k ≥ 1}, i ∈ J t ,
we set:

θn T T -1 (b i T -1 , v T -1 ) := max m≤n T T γξ T (α m T b i T -1 , v T -1 ), θn t+1 t (b i t , v t ) := max m≤n t+1 t+1 γn t+2 t+1 (α m t+1 b i t , v t ), n t+2 = (n t+1 u ) u=t+2,••• ,T , t ≤ T -1, Dn t+1 t (b i t , v t-1 , v t ) := θn t+1 t (b i t , v t ) + C t (b i t , (0, v (2)
t -v

(2)

t-1 )), t ≤ T -1, γn t+1 t (b i t , v t-1 ) := inf vt∈Kt(b i t ,v t-1 ) Dn t+1 t (b i t , v t-1 , v t ), t ≤ T -1.
Note that by assumption

γξ T (s, v T -1 ) := g 1 (s) + C T (s, (0, g 2 (s) -v (2)
T -1 )). Therefore, γξ T is l.s.c. Since K t is an upper hemicontinuous compact set-valued mapping by assumption, see [ The following theorem is our main contribution of this section. We use the convention that

n 1 → ∞, n 1 ∈ N T , if and only if n 1 i → ∞, ∀i = 1, • • • , T . Theorem 3.15.
[Limit theorem to approximate the infimum super-hedging price] Suppose that Assumption 4 holds and suppose that C t satisfies C t (s,

v 1 t ) ≥ C t (s, v 2 t ) whenever v 1 ≥ R d + v 2 t . Then: lim n 1 →∞
γn 1 0 (S 0 , 0) = γξ 0 (S 0 , 0), a.s..

Proof. By Remark 3.13, Assumption 4 implies that γξ 0 (S 0 , 0) = inf

v 1 ∈K 0 (S 0 ,0) Dξ 0 (S 0 , 0, v 1 )
where K 0 (S 0 , 0) is a compact set-valued mapping. Moreover, since γξ t+1 (., v t ) is l.s.c. hence Borel, Theorem 3.14 applies when we replace S t by each random variable b

i t ∈ α k t b j t-1 ; j ∈ J t-1 , k ≥ 1 .
Precisely, in accordance with (3.7), we shall consider: We now prove by induction that lim n 1 →∞ γn 1 0 (S 0 , 0) = γξ 0 (S 0 , 0) a.s. Observe that, at time T -1, n T =: n T ∈ N and γn

Dn t+1 t+1 t (b i t , v t-1 , v t ) = sup n≤n s t+1 γξ t+1 (α n t b i t , v t ) + C t (α n t b i t , (0, v (2)
t -v (2) t-1 )), t ≤ T -1, γn t+1 t+1 t (b i t , v t-1 ) := inf vt∈Kt(b i t ,v t-1 ) Dn t+1 t+1 t (b i t , v t-1 , v t ), t ≤ T -1, sup n t+1 t+1 γn t+1 t+1 t (b i t , v t-1 ) = γξ t (b i t , v t-1 ), t ≤ T -1,
T T -1 (b i T -1 , v T -1 ) and γn T T -1 (b i T -1 , v T -1
) coincide. So, by Theorem 3.14, we have lim

n T →∞ γn T T -1 (b i T -1 , v T -2 ) = lim n T →∞ γn T T -1 (b i T -1 , v T -2 ) = γξ T -1 (b i T -1 , v T -2 )
Now, we suppose that sup

n t+2 ∈N T -t-1 γn t+2 t+1 (b i t+1 , v t ) = γξ t+1 (b i t+1 , v t ) for any b i t+1 ∈ {α k t+1 b j t ; j ∈ J t , k ≥ 1}.
We have by definition:

Dn t+1 t (b i t , (0, v (2)
t -v

(2)

t-1 )) = θn t+1 t (b i t , v t ) + C t (b i t , (0, v (2)
t -v

(2)

t-1 )) = max m≤n t+1 t+1 γn t+2 t+1 (α m t b i t , v t ) + C t (b i t , (0, v (2)
t -v (2)
t-1 )),

n t+2 = (n t+1 u ) u=t+2,••• ,T .
Consider the directed set of all n t+1 ∈ N T -t endowed with the partial order n t+1 ≥ m t+1 if and only if n t+1 i ≥ m t+1 i for all t + 1 ≤ i ≤ T . By construction and by induction, it is easy to check that ( Dn t ) n∈N [t+1,T ] is increasing, i.e. Dn t ≥ Dm t whenever n ≥ m. Also, we may show by induction that Dn t (b i t , .) is l.s.c. for all n. By Lemma 3.16 that allows us to exchange the supremum and infimum in the following first equality, plus the induction hypothesis, we deduce that sup

n t+1 γn t+1 t (b i t , v t-1 ) = sup n t+1 inf vt∈Kt(b i t ,v t-1 ) Dn t+1 t (b i t , v t-1 , v t ) = inf vt∈Kt(b i t ,v t-1 )
sup

n t+1 Dn t+1 t (b i t , v t-1 , v t ) = inf vt∈Kt(b i t ,v t-1 )
sup

n t+1 t+1 ∈N sup n t+2 Dn t+1 t (b i t , v t-1 , v t ) = inf vt∈Kt(b i t ,v t-1 )
sup

n t+1 t+1 ∈N sup n t+2 max m≤n t+1 t+1 γn t+2 t+1 (α m t+1 b i t , v t ) + C t (b i t , (0, v (2)
t -v

(2)

t-1 )) = inf vt∈Kt(b i t ,v t-1 )
sup

n t+1 t+1 ∈N max m≤n t+1 t+1 sup n t+2 γn t+2 t+1 (α m t+1 b i t , v t ) + C t (b i t , (0, v (2)
t -v (2) t-1 )) = inf vt∈Kt(b i t ,v t-1 )
sup

n t+1 t+1 ∈N max m≤n t+1 t+1 γξ t+1 (α m t+1 b i t , v t ) + C t (b i t , (0, v (2)
t -v

(2)

t-1 )) = inf vt∈Kt(b i t ,v t-1 ) Dξ t (b i t , , v t-1 , v t ) = γξ t (b i t , v t-1 ).
To deduce the last two equalities, we use the definition of θξ t (b i t , , v t-1 , v t ) and Dξ t (b i t , , v t-1 , v t ), see (3.6) but also (3.9) in Remark 3.13. The conclusion follows by induction.

In the proof above, we have used the following lemma: Lemma 3.16 (Dini-Cartan). Consider a family of l.s.c. functions (f n ) n∈I , f n : R d → R such that for every finite set J ⊂ I, there is n 0 ∈ I with sup j∈J f j ≤ f n 0 . Consider a compact set G, then the following holds:

sup n inf x∈G f n (x) = inf x∈G sup n f n (x).
Proof. By considering an increasing homeomorphism from [-∞, +∞] onto [0, 1], we then restrict ourselves to the case sup n f n is bounded. It is clear that sup n inf x∈G f n (x) ≤ inf x∈G sup n f n (x) so that the inequality holds if the second term is -∞. For the reverse inequality, consider any a < inf x∈G sup n f n (x). For all x ∈ G, we have a < sup n f n (x). Then, there exists some k = k x such that a < f k (x). Note that the set O k := {x : a < f k (x)} is open since f k is l.s.c. By compactness argument, we deduce a finite covering of G by some O k i , j = 1, • • • , N . By our hypothesis, there exists n 0 such that a ≤

f k i (x) ≤ f n 0 (x), for all x ∈ G and i = 1, • • • , N hence we have a ≤ inf x∈G f n 0 (x) ≤ sup n inf x∈G f n (x).
Lemma 3.17. For all t , for all j ∈ J t+1 , consider b j t+1 = α k t+1 b i t where i ∈ J t and k ≥ 1. Then, b j t+1 ∈ {a n t b i t , n ≥ 1} a.s. and P [b j t+1 = a n t b i t |F t ] > 0 a.s. Moreover, {b j t+1 , j ∈ J t+1 } are F t -i.i.d. Proof. For all n ≥ 1, we have almost surely :

P b j t+1 = a n t b i t |F t = P α k t+1 b i t = a n t b i t |F t ≥ P α k t+1 = a n t |F t > 0.
The last statement follows directly from Lemma 3.3 as (α j t+1 ) j≥1 are F t -i.i.d. by assumption.

Model with one risky asset and piecewise linear costs

As we may observe in the previous section, the reachability set-valued mapping plays an important role in propagating the lower semicontinuity which, in turn, propagates the convergence property. We consider in this section a special case of convex cost functions and provide explicit expressions for the minimal super-hedging costs. In particular, under SAIP condition, we obtain an explicit expression of the reachability set K t (s, v t-1 ) when the payoff is of linear growth, i.e. ξ = (ξ 1 , ξ 2 ) ≤ R 2 + (aS T + b, c) for some a, b, c ∈ R + . We suppose the the market consists of one risk-free asset and one risky asset denoted by (S t ) 0≤t≤T . We impose the following assumption for the conditional support of the price and cost processes.

Assumption 5. The price process satisfies S t+1 ∈ {a n t S t , n ≥ 1} where the sequence (a n t ) n≥1 is deterministic and satisfies

a 1 t = min n a n t = k d t ≥ 0, a 2 t = max n a n t = k u t ∈ R + , where k d t , k u t are deterministic. The cost process C t is given by C t (S t , (x, v t )) = x + S t Ct (v 2
t-1 ) for some deterministic piecewise linear function Ct : R → R.

We recall the AEP condition in [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF] Definition 4.1. We say that the financial market satisfies the Absence of Early Profit condition (AEP) if, at any time t ≤ T , and for all

V t ∈ L 0 (R d , F t ), γ 0 t (V t ) > -∞ a.s.
. By Lemma 4.11 in [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF], AIP implies AEP if the cost function C t is either sub-additive or super-additive. Moreover, by Theorem 4.5 in [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF], AEP implies that γξ t (S t , .) > -∞ a.s. This property will be used in the proof of the following result.

= (mS T +G, K), m, G, K ∈ R, is given by γξ t (S t , v t-1 ) = G+S t h t (v 2 t-1 )
, where h t : R → R is a deterministic piecewise linear function.

Moreover, Dt (S t , v t , v t-1 ) = S t ht (v t , v t-1 ) for some deterministic piecewise linear function ht : R × R → R.

Proof. We first show by induction that, if γξ t+1 (S t+1 , v t ) = S t+1 ft+1 (v 2 t ) where ft+1 : R → R is a piecewise linear function, then γξ t (S t , v t-1 ) = S t ft (v 2 t-1 ) for some piecewise linear function ft : R → R. To do so, observe that:

θξ t (S t , v t ) = sup s∈{a n t St,n≥1} s ft+1 (v 2 t ) = max k d t S t ft+1 (v 2 t ), k u t S t ft+1 (v 2 t ) = S t max k d t ft+1 (v 2 t ), k u t ft+1 (v 2 t ) .
Since ft+1 is piecewise linear function by the hypothesis, we deduce that gt (v

2 t ) := max{k d t ft+1 (v 2 t ), k u t ft+1 (v 2 t )} is also piecewise linear by [26, Propo- sition 3.55]. Therefore, γξ t (S t , v t-1 ) = inf v 2 ∈R Dξ t (S t , v t-1 , v t ) = inf v 2 ∈R θξ t (S t , v t ) + C t (S t , v 2 t -v 2 t-1 ) = S t inf v 2 t ∈R gt (v 2 t ) + Ct (v 2 t -v 2 t-1 )
.

By [START_REF] Rockafellar | Variational analysis[END_REF]Proposition 3.55], we also deduce that gt (

v 2 t ) + Ct (v 2 t -v 2 t-1 ) is a piecewise linear function in (v 2 t , v 2 t-1
). Moreover, under AEP, we know that γξ t (S t , v t-1 ) > -∞ a.s.. Therefore, by [26, Proposition 3.55],

ft (v 2 t-1 ) := inf

v 2 ∈R gt (v 2 t ) + Ct (v 2 t -v 2 t-1 ) is a piecewise linear function in v 2 t-1 . If the payoff is ξ = (mS T + G, K), then γξ T (S T , v T -1 ) = G + S T fT (v 2 T -1 ) where fT (v 2 T 1 ) := m + CT (K -v 2 T -1
) is a piecewise linear function by assumption on C T . We then argue by induction as previously done to deduce that

γ ξ t-1 (S t-1 , v t-2 ) = G + S t-1 ft-1 (v 2 t-2 ) for some piecewise linear function ft-1 . At last, since D ξ t (S t , v t , v t-1 ) = θt (S t , v t ) + C t (S t , (0, v (2)
t -v (2)
t-1 )), the conclusion on Dt follows.

The following is our main result of this section. It states the existence of the reachability set under SAIP. Proposition 4.3. Suppose that the payoff ξ = (g 1 (S T ), g 2 (S T )) satisfies g 1 (S T ) ≤ aS T + b and g 2 (S T ) ≤ c for some a, b, c ∈ R + . We also suppose that

C t (s, v 1 ) ≥ C t (s, v 2 ) whenever v 1 ≥ R 2
+ v 2 and suppose that C t (s, .) is subadditive and 1-homogeneous.

Under the no-arbitrage condition SAIP, the reachability set K t (s, v t-1 ) is defined for every (s, v t-1 ) ∈ R × R and is explicitly given by:

K t (s, v t-1 ) = Bt (0, r t (s, v t-1 ) + 1)
where r t (s, v t-1 ) = sf t (v t-1 )/g t (s) and f t , g t are deterministic piecewise linear functions such that g t (s) > 0 for all s > 0.

Proof. We define ξ := (aS T + b, c) so that ξ ≤ R 2 + ξ. We show by induction that D0

t (s, v t-1 , v t ) ≤ Dξ t (s, v t-1 , v t ) ≤ Dξ t (s, v t-1 , v t )
. By the proof of [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF]Theorem 4.15], we get that

K t (s, v t-1 ) ⊆ v t : Dξ t (s, v t-1 , v t ) ≤ Dξ t (s, v t-1 , 0)
Moreover, by sub-additivity and 1-homogeneity.

D0 t (s, v t-1 , v t ) = C t (s, (0, v 2 t -v (2)
t-1 )) + θ0 t (s, v t ) ≥ -C t (s, (0, v 2 t-1 )) + D0 t (s, 0, v t ) D0 t (s, 0, v t ) ≥ |v t | D0 t (s, 0, v t /|v t |) ≥ |v t | min z∈{-1,1} D0 t (s, 0, z), ∀ |v t | ≥ 1.
We deduce that K t (S t , v t-1 ) ⊆ B(0, r t (S t , v t-1 )+1), where the radius r t (S t , v t-1 ) is given by

r t (S t , v t-1 ) := Dξ t (S t , v t-1 , 0) + C t (S t , (0, v 2 t-1 )) min z∈{-1,1} D0 t (S t , 0, z) =: S t f t (v 2 t-1 ) g t (S t )
.

Note that by Proposition 4.2, f t : R → R and g t : R → R are deterministic piecewise linear functions. Moreover, we have g t (S t ) = S t inf z∈{-1,1} a t (z) for some deterministic piecewise linear function a t . Since SAIP holds, we deduce that inf z∈{-1,1} a t (z) > 0. We then define g t (s) := s inf z∈{-1,1} a t (z) > 0 for all s > 0. The conclusion follows.

Examples

In this section, we consider two classical examples. The first one corresponds to the market with proportional transaction cost and the second one is with fixed cost. We provide the explicit expression of the reachability set-valued mapping K t for the Put option. Then, as a by-product, the minimal superhedging cost for Put option is computed.

For a sake of simplicity, we consider the binomial market model, i.e. the price process satisfies supp Ft S t+1 = k d t S t , k u t S t , where k d t , k u t ∈ R + .

Market model with proportional transaction costs

We consider a particular case of section 4 where

C t (S t , v) = v 1 + (1 + t )S t v 2 1 v 2 ≥0 + (1 -t )S t v 2 1 v 2 ≤0 . (5.15)
for some deterministic coefficient t ∈ R + . By a direct computation, see Appendix, we obtain the following Proposition 5.1. If v t-1 ∈ R 2 , the following holds:

θ0 t-1 (S t-1 , v) = -(1 -t )k d t-1 S t-1 v 2 1 v 2 ≥0 -(1 + t )k u t-1 S t-1 v 2 1 v 2 ≤0 D0 t-1 (S t-1 , 0, v) = ((1 + t-1 )S t-1 -(1 -t )k d t-1 S t-1 )v 2 1 v 2 ≥0 + ((1 -t-1 )S t-1 -(1 + t )k u t-1 S t-1 )v 2 1 v 2 ≤0
Moreover, AIP t-1 holds if and only if:

k d t-1 ≤ 1 + t-1 1 -t and k u t-1 ≥ 1 -t-1 1 + t . (5.16)
Moreover, SAIP t-1 holds if and only if the above inequalities are strict. If AIP t-1 holds, we then deduce that:

inf v 2 ∈{-1,1} D0 t-1 (S t-1 , 0, v) = S t-1 min (1 + t-1 ) -(1 -t )k d t-1 , (1 + t )k u t-1 -(1 -t-1 ) .
Proof. Recall that AIP t-1 holds if and only if D0 t-1 (S t-1 , 0, v) ≥ 0 for any v ∈ R d which is equivalent to (5.16). Moreover, suppose that SAIP t-1 holds.

If k d t-1 = 1 + t-1 1 -t , D 0 t-1 (S t-1
, 0, v) = 0 for any v 2 > 0, i.e. SAIP t-1 fails. Similarly, we get that k u t-1 > (1t-1 )/(1 + t ). At last, suppose that the inequalities in (5.16) are strict. Since S t-1 > 0 a.s., inf

v 2 ∈{-1,1} D0
t-1 (S t-1 , 0, v) > 0, a.s. so that SAIP t-1 holds by [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF]Theorem 4.15].

We apply the result above at time T and we proceed by induction, see Appendix, to deduce the following result at time T -2.

Proposition 5.2. Assume that 1 + T -1 ≤ (1 + T )k u T -1 and 1 -T -1 ≥ (1 -T )k d T -1 , we have: θ0 T -2 (S T -2 , z) = -(1 + T -1 )k d T -2 S T -2 z 2 1 z 2 ≥0 -(1 -T )k d T -1 k u T -2 S T -2 z 2 1 z 2 ≤0 , D0 T -2 (S T -2 , 0, z) = (1 + T -2 )S T -2 -(1 + T -1 )k d T -2 S T -2 ) z 2 1 z 2 ≥0 + (1 -T -2 )S T -2 -(1 -T )k d T -1 k u T -2 S T -2 z 2 1 z 2 ≤0 .
and AIP T -2 holds if and only if:

k d T -2 ≤ 1 + T -2 1 + T -1 and k u T -2 ≥ 1 -T -2 (1 -T )k d T -1 .
Moreover, SAIP T -2 holds if and only if the above inequalities are strict. Moreover, under SAIP T -2 , we have:

inf v 2 ∈{-1,1} D0 T -2 (S T -2 , 0, v) = S T -2 min ((1 + T -2 ) -(1 -T -1 )k d T -2 ), -((1 -T -2 ) -(1 + T )k d T -1 k u T -2 ) .
The assumptions of Proposition 5.2 are chosen for a sake of simplification. The computations for t < T -2 are similar. In particular, for a Put option with payoff (K -S T ) + , K > 0, we obtain a simple formula for the reachability set.

Lemma 5.3. Suppose that SAIP holds and ξ = (g(S T ), 0) where g is a continuous function bounded from above by a constant M ∈ R + . Then, there exists a reachability set K t (s, v t-1 ) = Bt (0, r t (s, v t-1 ) + 1), t ≤ T -1, closed ball of radius r t (s, v t-1 ) := λ t (s, v t-1 )/i t (s) where the functions

i t (s) := inf v 2 ∈{-1,1} D0
t (s, 0, v), λ t (s, v t-1 ) := C t (s, (0, v 2 t-1 )) + M + C t (s, (0, -v 2 t-1 )), are explicitely given by Proposition 5.1 and Proposition 5.2. In particular, we have i t (s) > 0 for all s > 0.

We illustrate the results above by a numerical example. We consider the put option payoff g(S T ) := (K -S T ) + at time T = 2. We suppose that the proportional cost coefficients 1 = 2 = 0.02. We assume that SAIP condition holds and choose k d 2 = 0.9, k u 2 = 1.1, k d 1 = 0.9, k u 1 = 1.2. The price function at time t = 0 is presented in Figure 1.

Market model with fixed cost

In this section, we consider a financial market model in presence of both proportional and fixed costs modeled by the following liquidation and cost functions:

L t (S t , v t ) := v 1 t + (v 2 t (1 -t )S t -c t ) + 1 vt>0 + (v 2 t (1 + t )S t -c t )1 vt<0 C t (S t , v t ) := -L t (S t , -v t ).
for some deterministic constant c t > 0 representing the fixed cost we need to pay to obtain a non-null position.

In this model, the cost function does not satisfy the condition property that C t (S t , λz) ≥ λC t (S t , z) for any λ ≥ 1. Then, the propagation of lower semicontinuity is not guaranteed if we only assume the SAIP condition on the market defined by this cost function. In [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF], we have introduced the horizon cost function defined as follows:

C ∞ t (s, y) := lim inf α→∞ C t (s, αy) α .
(5.17) Definition 5.4. We say that the robust no-arbitrage condition RSAIP holds at time t if the SAIP condition holds at time t for the enlarged model defined by C ∞ t . We say that RSAIP holds if it holds at any time. In [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF], we proved the following theorem analogous to Theorem 5.5: Theorem 5.5. Suppose that the condition RSAIP holds. Then, we have γ ξ t (S t , V t ) = γξ t (S t , V t ) a.s., θ ξ t (S t , V t ) = θξ t (S t , V t ) a.s. and, also, we have

D ξ t (S t , V t-1 , V t ) = Dξ t (S t , V t-1 , V t ) a.s. for any V t-1 , V t ∈ L 0 (R d , F t )
, where θξ t , Dξ t are given by (3.6). As the horizon cost function coincides with the cost function (5.15) without fixed costs, the results stated in Propositions 5.16 and 5.2 allows us to characterize the reachability set-valued mapping K t for this market. In particular, since C t ≤ C ∞ t + c t , by a straightforward computation, we deduce a simple formula of K t for the Put option: Lemma 5.6. Suppose that ξ = (g(S T ), 0) where g is a continuous function bounded from above by M ∈ R + . Then, a reachability set K t (s, v t-1 ) is explicitly given at any time t ≤ T -1 by K t (s, v t-1 ) = Bt (0, r t (s, v t-1 ) + 1), closed ball of radius r t (s, v t-1 ) := λ t (s, v t-1 )/i t (s) where

i t (s) := inf v 2 ∈{-1,1} D 0,∞ t (s, 0, v), λ t (s, v t-1 ) := C ∞ t (s, (0, v 2 t-1 )) + M + C ∞ t (s, (0, -v 2 t-1 )) + T s=t c s ,
and D 0,∞ t is given in the model without fixed cost given by Proposition 5.1 or Proposition 5.2. In particular, we have i t (s) > 0 for all s > 0.

As a numerical example, we also consider the put option payoff (K -S T ) + at time T = 2. We consider the binomial tree model as previously. In the case where the conditional support supp Ft S t is countable, we can use the randomized method established in section 3.

We use the same parameters as in Section 5.1 and we consider fixed costs c 1 = c 2 = 0.8. The price function is illustrated in Figure 3. We also visualize the ratio of put price to asset price S 0 We also compare the price of put option with and without fixed costs.

Limit theorem for convex markets

In the literature, there is few results providing limit theorems for financial market models with transaction costs, see [START_REF] Grépat | On a multi-asset version of the Kusuoka limit theorem: convergence of hedging sets[END_REF] and [START_REF] Bank | The scaling limit of superreplication prices with small transaction costs in the multivariate case[END_REF], but also [START_REF] Kusuoka | Limit theorem on option replication cost with transaction costs[END_REF] and [START_REF] Hubalek | When does convergence of asset price process imply convergence of option prices ?[END_REF] without transaction costs. In this section, we consider a sequence of markets defined by convex cost functions {C n t (S t , x), n ≥ 1} such that C n t (S t , x) ↓ C t (S t , x) as n → ∞ for some convex function C t . We associate to each C n t a dynamic programming scheme deduced by our general analysis, see [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF]:

γ ξ,n T (S T , V T -1 ) := g 1 (S T ) + C n T (S T , (0, g (2) (S T ) -V T -1 )), θ ξ,n t (S t , v t ) := ess sup Ft γ ξ,n t+1 (S t+1 , V t ), D ξ,n t (S t , V t-1 , V t ) := θ ξ,n t (S t , V t ) + C n t (S t , (0, V (2)
t -V (2) t-1 )), γ ξ,n t (S t , V t-1 ) := ess inf Ft Vt∈L 0 (R d ,Ft) D ξ,n t (S t , V t-1 , V t ).
Assumption 6. We suppose that supp Ft S t+1 = φ t (S t ) = conv{φ 1 t (S t ), ...φ J t (S t )} where φ j t : R d → R d , j ≤ J, are piecewise linear mappings in the sense of Definition 7.3.

We define γξ,n

t : R d × R d → R recursively as follows: γξ,n T (s, v T -1 ) := γ ξ,n T (s, v T -1 ), θξ,n T -1 (s, v T -1 ) := max j≤J γξ,n T (φ j T -1 (s), v t-1 ), Dξ,n t (s, v t-1 , v t ) := θξ t (s, v t ) + C n t (s, v (2)
t -v

t-1 ), γξ,n t (s, v t-1 ) := cl inf

vt∈R d Dξ,n t (s, v t-1 , v t ) .
Assumption 7. Suppose that for any t ≤ T -1, inf vt∈S d-1 (0,1) D0 t (s, 0, v t ) > 0 for all s ∈ R k + , so that there is a upper hemicontinuous reachability set-valued mapping K t (s, v t-1 ) for the super-hedging problem in the market defined by C t . Moreover, we suppose that K t is an universal reachability set in the sense that it satisfies for all n ≥ 1 and (s, v t-1 ):

γξ,n t (s, v t-1 ) = inf vt∈Kt(s,v t-1 )
Dξ,n t (s, v t-1 , v t ).

Remark 6.1. Consider the case where C, C n and S t satisfy the assumptions specified in section 4. Since C ≤ C n for all n ≥ 1 by assumption, we deduce that inf vt∈S d-1 (0,1) D0 t (s, 0, v t ) > 0 implies inf vt∈S d-1 (0,1) D0,n t (s, 0, v t ) > 0 for all n. By the proof of Proposition 4.3, it is sufficient to suppose that SAIP holds for the market defined by C. If we suppose that C t (s, v t ), C n t (s, v t ) are bounded above by |h t (s, v t )| for some continuous function h t , by the same argument as in Lemma 5.21 in [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF], we deduce that the quantities D0 t (s, v t-1 , 0) and D0,n t (s, v t-1 , 0) are bounded above by a continuous function ĥt (s, v t-1 ). Hence, an universal reachability set exits as K t (s, v t-1 ) = B(0, r t (s, v t-1 )) where

r t (s, v t-1 ) = ĥt (s, v t-1 ) + |h t (s, v t-1 )| inf vt∈S d-1 (0,1) D0 t (s, 0, v t )
.

Since r t is u.s.c., we deduce by Lemma 5.12 in [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF] that K t is upper hemicontinuous.

Theorem 6.2. Suppose that the functions φ j t : R k + → R k + , j ≤ J satisfy Assumption 6. Suppose that Assumption 7 holds. Then, for any t ≤ T -1 and for any v t-1 ∈ R d , lim n→∞ γξ,n t (s, v t-1 ) = γξ t (s, v t-1 ). Moreover, SAIP condition holds for the markets defined by C n and lim n→∞ γ ξ,n t (S t , V t ) = γ ξ t (S t , V t ) a.s. as n → ∞ for any V t ∈ L 0 (R d , F t ) and t ≤ T . 32

Proof. We first observe that γξ,n

(s, v T -1 ) = max j≤J γξ,n T (φ j T -1 (s), v t-1 ), θξ T -1 (s, v T -1 ) = max j≤J γξ T (φ j (s), v t-1
,n T -1 (s, .) = max j≤J γξ,n T (φ j (s), .) epi -→ max j∈J γξ T (φ j (s), .) = θξ T -1 (s, .), n → ∞. Since C n T -1 (s, .) ↓ C T -1 (s, .) and C T -1 (s, .
) is continuous, we deduce by the Dini theorem that the convergence is uniform on any compact subset K of R d . By [START_REF] Rockafellar | Variational analysis[END_REF]Theomrem 7.14], we deduce that C n T -1 (s, .) converges continuously to C T -1 (s, .) in the sense that C n T -1 (s, x n ) → C T -1 (s, x) whenever x n → x. We then deduce by [START_REF] Rockafellar | Variational analysis[END_REF]Theorem 7.46 ). At last, since inf vt∈S(0,1) D0 t (s, 0, v t ) > 0, SAIP holds for the market defined by C t , see [START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF]Theorem 4.16]. By Theorem 5.5, we have γξ t (S t , V t ) = γ ξ t (S t , V t ) a.s. for any V t ∈ L 0 (R d , F t ). Moreover, since D0,n t (s, 0, v t ) ≥ D0 t (s, 0, v t ), we deduce that SAIP also holds for market defined by C n t and, similarly, we have γξ,n t (S t , V t ) = γ ξ,n t (S t , V t ) a.s. for any V t ∈ L 0 (R d , F t ). The conclusion follows.

Appendix

By [START_REF] Rockafellar | Variational analysis[END_REF]Theorem 14.37], we have: Proposition 7.1. If f is an F t -normal integrand, inf y∈R d f (ω, y) is F tmeasurable and {(ω, x) ∈ Ω × R d : f (ω, x) = inf y∈R d f (ω, y)} ∈ F t ⊗ B(R d ) is a measurable closed set.

We now recall a result from [START_REF] Baptiste | Pricing without martingale measure[END_REF] which characterizes a conditional essential supremum as a pointwise supremum on a random set. Let H and F be two complete sub-σ-algebras of F T such that H ⊆ F. The conditional support of X ∈ L 0 (R d , F) with respect to H is the smallest H-graph measurable random set supp H X containing the singleton {X} a.s., see [START_REF] Baptiste | Pricing without martingale measure[END_REF]. Proposition 7.2. Let h : Ω × R k → R be a H ⊗ B(R k )-measurable function which is l.s.c. in x. Then, for all X ∈ L 0 (R k , F), ess sup H h(X) = sup x∈supp H X h(x) a.s.

Piecewise linear cost function

We recall from [START_REF] Rockafellar | Variational analysis[END_REF] the definiton of piecewise linear function: Definition 7.3. A mapping F : D → R m defined on a set D ∈ R n is piecewise linear on D if D is the union of finitely many polyhedral sets (P i ) i∈J such that, for all x ∈ P i , F (x) = A i x + B i , for some matrix A i ∈ R m×n and B i ∈ R m .

A function f : R n → R is piecewise linear if it is a real-valued piecewise linear function on its domain domf = {x : f (x) ∈ R}.

Complement to Section 4

Recall that the model is defined by one risk-free asset and one risky asset denoted by S. The cost function is given by C t (S t , v) = v 1 + S t Ct (v 2 ), (7.18) where Ct : R → R is a piecewise linear function.

By Proposition 7.2, we have: -(1

θ 0 T -
+ T )sv 2 1 v 2 ≤0 -(1 -T )sv 2 1 v 2 ≥0 = max -(1 + T )k d T -1 S T -1 v 2 1 v 2 ≤0 -(1 -T )k d T -1 S T -1 v 2 1 v 2 ≥0 , -(1 + T )k u T -1 S T -1 v 2 1 v 2 ≤0 -(1 -T )k u T -1 S T -1 v 2 1 v 2 ≥0 = -(1 -T )k d T -1 S T -1 v 2 1 v 2 ≥0 -(1 + T )k u T -1 S T -1 v 2 1 v 2 ≤0
. and

C T -1 (S T -1 , (0, v 2 -z 2 )) = (1 + T -1 )S T -1 v 2 1 v 2 -z 2 ≥0 + (1 -T -1 )S T -1 v 2 1 v 2 -z 2 ≤0 -(1 + T -1 )S T -1 z 2 1 v 2 -z 2 ≥0 + (1 -T -1 )S T -1 z 2 1 v 2 -z 2 ≤0 .
We then have:

D 0 T -1 (S T -1 , 0, v) = θ 0 T -1 (S T -1 , v) + C T -1 (S T -1 , (0, v 2 )) = ((1 + T -1 )S T -1 -(1 -T )k d T -1 S T -1 )v 2 1 v 2 ≥0 + ((1 -T -1 )S T -1 -(1 + T )k u T -1 S T -1 )v 2 1 v 2 ≤0
More generally:

D 0 T -1 (S T -1 , z, v) = θ 0 T -1 (S T -1 , v) + C T -1 (S T -1 , (0, v -z)) = (1 + T -1 )S T -1 v 2 1 v 2 -z 2 ≥0 + (1 -T -1 )S T -1 v 2 1 v 2 -z 2 ≤0 -(1 + T -1 )S T -1 z 2 1 v 2 -z 2 ≥0 + (1 -T -1 )S T -1 z 2 1 v 2 -z 2 ≤0 -(1 -T )k d T -1 S T -1 v 2 1 v 2 ≥0 -(1 + T )k u T -1 S T -1 v 2 1 v 2 ≤0 .
In the following, we assume that 1 + T -1 ≤ (1 + T )k u T -1 and, also, that

1 -T -1 ≥ (1 -T )k d T -1 .
We shall use the usual convention that inf ∅ = ∞. We get that:

γ 0 T -1 (z) = inf v∈R 2 D 0 T -1 (S T -1 , z, v) = min i=1,••• ,4 D 0,i T -1 (S T -1 , z, v),
where:

D 0,1 T -1 = inf v 2 :v 2 ≥z 2 ,v 2 ≥0 (1 + T -1 )S T -1 (v 2 -z 2 ) -(1 -T )k d T -1 S T -1 v 2 = -(1 -T )k d T -1 S T -1 z 2 1 z 2 ≤0 -(1 + T -1 )S T -1 z 2 1 z 2 ≥0 . D 0,2 T -1 = inf v 2 :v 2 ≥z 2 ,v 2 ≤0 (1 + T -1 )S T -1 (v 2 -z 2 ) -(1 + T )k u T -1 S T -1 v 2 = ∞1 z 2 >0 -(1 + T -1 )S T -1 z 2 1 z 2 ≤0 . D 0,3 T -1 = inf v 2 :v 2 ≤z 2 ,v 2 ≥0
(1

-T -1 )S T -1 (v 2 -z 2 ) -(1 -T )k d T -1 S T -1 v 2 = ∞1 z 2 <0 -(1 -T -1 )S T -1 z 2 1 z 2 ≥0 . D 0,4 T -1 = inf v 2 :v 2 ≤z 2 ,v 2 ≤0
(1

-T -1 )S T -1 (v 2 -z 2 ) -(1 + T )k u T -1 S T -1 v 2 = -(1 -T -1 )S T -1 z 2 1 z≥0 -(1 + T )k u T -1 S T -1 z 2 1 z 2 ≤0 .
We deduce that

γ 0 T -1 (S T -1 , z) = inf v∈R 2 D 0 T -1 (S T -1 , z, v) = -(1 + T -1 )S T -1 z 2 1 z 2 ≥0 -(1 -T )k d T -1 S T -1 z 2 1 z 2 ≤0 .
We now compute D 0 T -2 (S T -1 , 0, z). We have:

θ 0 T -2 (S T
+ T -1 )k d T -2 S T -2 z 2 1 z 2 ≥0 -(1 -T )k d T -1 k u T -2 S T -2 z 2 1 z 2 ≤0 +(1 + T -2 )S T -2 z 2 1 z 2 ≥0 + (1 -T -2 )S T -2 z 2 1 z 2 ≤0 = (1 + T -2 )S T -2 -(1 + T -1 )k d T -2 S T -2 ) z 2 1 z 2 ≥0 + (1 -T -2 )S T -2 -(1 -T )k d T -1 k u T -2 S T -2 z 2 1 z 2 ≤0 .
We then get the following:

Proposition 7.4. AIP holds at time T -2 if and only if the following holds:

k d T -2 ≤ 1 + T -2 1 + T -1 and k u T -2 ≥ 1 -T -2 (1 -T )k d T -1
.

Proposition 4 . 2 .

 42 Suppose that Condition AEP and Assumption 5 hold. Then the minimal hedging cost of the payoff ξ

Fig 1 :Fig 2 :

 12 Fig 1: Price of Put option

Fig 3 :

 3 Fig 3: Price of put option with fixed costs.

Fig 4 :

 4 Fig 4: Ratio price of put to asset price with fixed costs.

Fig 5 :

 5 Fig 5: Price difference between two cases

  1 (S T -1 , v) := ess sup F T -1 C T (S T , (0, -v 2 )) = sup s∈supp F T -1 S T C T (s, (0, -v 2 )) = sup s∈supp F T -1 S T -(1 + T )sv 2 1 v 2 ≤0 -(1 -T )sv 2 1 v 2 ≥0 = sup s∈[k d T -1 S T -1 ,k u T -1 S T -1 ]

  -2 , z) = ess sup F T -2 γ 0 T -1 (S T -1 , z) = sup s∈[k d T -2 S T -2 ,k u T -2 S T -2 ] γ 0 T -1 (s, z) = sup s∈[k d T -2 S T -2 ,k u T -2 S T -2 ] -(1 + T -1 )sz 2 1 z 2 ≥0 -(1 -T )k d T -1 sz 2 1 z 2 ≤0 = -(1 + T -1 )k d T -2 S T -2 z 2 1 z 2 ≥0 -(1 -T )k d T -1 k u T -2 S T -2 z 2 1 z 2 ≤0 . D 0 T -2 (S T -2 , 0, z) = θ 0 T -2 (S T -2 , z) + C T -2 (S T -2 , (0, z 2 )) = -(1

  a.s..

	Lemma 3.3. Consider a family of F t -i.i.d. random variables b i t+1 , i ≥ 1 and

  We first consider the case where AIP holds and C t (s, y) is convex in y. Let us define Lξ

Then, Dn t (ω, ., .) epi -→ Dξ t (S t (ω), ., .) a.s.. Proof. t (ω, x, y) := Dn t (ω, x, y) ∨ -C t (S t (ω), (0, x (2) )) . Observe that Ln t (ω, x, y) is l.s.c. in (x, y) as a maximum of two l.s.c. functions. As the sequence ( Ln t ) n≥1 is also non decreasing, we deduce by [26, Proposition 7.4], that for any ω:

  (2) )) as n → ∞ for any ω outside a negligible set N (x, y). Moreover, by the discussion above, we deduce a negligible set M such that for any ω ∈ Ω \ M , we have Dξ R d . Indeed, by the definition of H, we deduce that(3.8) holds for any y ∈ Q d . Now, since Dξ t (S t (ω), ., .) is convex and takes values in R, it is continuous for any ω ∈ Ω \ H. Moreover, we claim that sup n Ln Ln t (ω, x k , y k ) for any sequence x k , y k ∈ Q d such that x k → x and y k → y. Moreover, by the definition of H and the continuity of Dξ Moreover, sup n Ln t (ω, ., .) also convex as a supremum of convex functions, it is also continuous. We then deduce by continuity that (3.8) holds for any y ∈ R d . Now, we consider the second case where C t

		Indeed,
	by lower semicontinuity, we have:
	sup n	Ln t (ω, x, y) ≤ lim inf
	By Lemma 3.4, Ln

we deduce that Dξ t (S t (ω), x, y) ∈ R for any x, y, for any ω outside a negligible set. t (ω, x, y) → Dξ t (S t (ω), x, y) ∨ (-C t (S t (ω), (0, x t (S t (ω), x, y) ≥ -C t (S t (ω), (0, x (2) )) and Dξ t (S t (ω), x, y) ∈ R for any x, y. We set H := ∪ y∈Q d N (x, y) ∪ M , we claim that for any ω ∈ Ω \ H, sup n Ln t (ω, x, y) = Dξ t (S t (ω), x, y) for all x, y ∈ t (ω, x, y) < ∞ for any x, y ∈ R d and ω ∈ Ω \ H. k sup n t (S t (ω), ., .) for any ω ∈ Ω\H, we have lim inf k sup n Ln t (ω, x k , y k ) = lim inf k Dξ t (S t (ω), x k , y k ) = D ξ t (S t (ω), x, y) ∈ R. We deduce that sup n Ln t (ω, x, y) ∈ R for any x, y ∈ R d , and ω ∈ Ω \ H.

  R d . Indeed, fix some y ∈ R d and a sequence (y k ) k≥1 in Q d such that y k → y and y k ≥ R d In the Proof of Theorem 3.10, we have used the following result: Lemma 3.11. Let f : R k → R be a function such that f that is non increasing with respect to the partial order ≥ R k + . Consider cl(f ) the lower semicontinuous regularization of f . Then, cl(f ) is non increasing w.r.t. the partial order ≥ R k + . Proof. From [26, Lemma 1.7], we have the following representation of the l.s.c. closure:

	Moreover, by the definition of H, we have sup n	γξ t+1 (b n t+1 (ω), y k ) = θξ t (S t (ω), y k )
	for any ω ∈ Ω \ H. We then deduce that sup n	γξ t+1 (b n t+1 (ω), y) = θξ t (S t (ω), y)
	for any ω ∈ Ω \ H. At last, by the definition of Dξ t and Dn t , we conclude that sup n Dn t (ω, x, y) = Dξ
	n	γξ t+1 (b n t+1 (ω), y) ≤ lim inf k	sup n	γξ t+1 (b n t+1 (ω), y k ) ≤ sup n	γξ t+1 (b n t+1 (ω), y).
	Then, we have
			θξ t (S t (ω), y) = lim inf k	θξ t (S t (ω), y k ),
		sup n	γξ t+1 (b n t+1 (ω), y) = lim inf k 14	sup n	γξ t+1 (b n t+1 (ω), y k ).

Similarly to the first case, we only need to prove sup n Dn t (ω, x, y) = Dξ t (S t (ω), x, y) for all x, y and ω outside a negligible set. By the definition of γξ t and θξ t , we can show by induction an by Lemma 3.11 that the mappings y → θξ t (s, y) and y → γξ t (s, y) are decreasing with respect to R d + . Recall the definition of N (x, y), we also denote H := ∪ y∈Q d N (x, y) ∪ M and claim that for any ω ∈ Ω \ H, sup n γξ t+1 (b n t+1 (ω), y) = θξ t (S t (ω), y), for all y ∈ + y. By lower semicontinuity and the discussion above, we have for any ω ∈ Ω \ H: θξ t (S t (ω), y) ≤ lim inf k θξ t (S t (ω), y k ) ≤ θξ t (S t (ω), y), and sup t (S t (ω), x, y) for any x, y and ω ∈ Ω \ H.

  a reachability set at time t ≤ T for the super-hedging problem if K t has compact set values and satisfies: Remark 3.[START_REF] Guasoni | The fundamental theorem of asset pricing under transaction costs[END_REF]. By[START_REF] Lépinette | Dynamic programming principle and computable prices in financial market models with transaction costs[END_REF] Theorem 4.14], under SAIP, the determining set K t (s, x) is constructed for s = S t (ω) as a closed ball B(0, r t (s, x) + 1), where r t (s, x) is an u.s.c. function. We shall see later in the model with one risky asset how to characterize K t (s, x) explicitely for every (s, x) ∈ R × R such that K t (s, x) is compact for all (s, x) and upper hemicontinuous. Moreover, By[START_REF] Aliprantis | Infinite Dimensional Analysis : A Hitchhicker's Guide, Grundlehren der Mathematischen Wissenschaften[END_REF] Lemma 17.29], the upper hemicontinuity of K implies that Theorem 3.14. Suppose that SAIP holds and C t (s,v 1 t ) ≥ C t (s, v 2 t ) for any v 1 t , v 2 t ∈ R d such that v 1 ≥ R d + v 2 .Then, we have:Moreover, for each fixedx t ∈ L 0 (R d , F t ) such that the random set K t (S t , x t ) is F t -measurable, there exists a sequence (ŷ n t+1 ) n≥1 of L 0 (R d , F t+1 ) such that ŷn t+1 ∈ arg min Kt(St,xt) ( Dn t (ω, x t , .)) a.s. and ŷn t+1 → ŷ0 t+1 ∈ L 0 (R d , F t+1 ) along a random F t+1 -measurable subsequence where ŷ0 t+1 ∈ arg min( Dξ t (S t , x t , .)). In the case where C t (s, y) is convex in y, the same conclusion holds if we replace Dn Moreover, in that case, if K t (S t , x t ) is also convex, for fixed x t ∈ L 0 (R d , F t ) such that the random set K t (S t , x t ) is F t -measurable, ŷn We prove the claim in the first case, the second case is deduced similarly using Theorem 3.10.Consider the negligible set H in the proof of Theorem 3.10 such that Dn Since K t (s, x) is compact, for any ω ∈ Ω \ N and x ∈ R d , there is a random subsequence {ŷ n k Dξ t (S t (ω), x, y), we deduce that for any ω ∈ Ω \ N :We then deduce that ŷ0 t (ω, x) ∈ arg min Kt(St(ω),x) ( Dξ t (S t (ω), x, .)) for any ω ∈ Ω\N , i.e. (3.10) holds. Using the definition of the reachability set-valued mapping K t , we conclude that ŷ0 t (ω, x) ∈ arg min( Dξ t (S t (ω), x, .)) outside a negligeable set. Recall that inf y∈Kt(St(ω),xt) Dn t (ω, x t , y) is F t+1 -measurable, see [22]. Therefore, by a measurable selection argument, we may deduce the existence of ŷn t+1 ∈ L 0 (R d , F t+1 ) such that Dn If K t (S t , x t ) is F t -measurable, consider a Castaing representation (z m t ) m≥1 of K t (S t , x t ). The generalized conditional expectation E(ŷ n t+1 |F t ) exists as ŷn t+1 ∈ K t (S t , x t ) is F t -bounded. Note that ŷn t+1 may be approximated by a sequence of F t+1 -measurable random variables in the set {z m

	lim n→∞ lim inf inf y∈Kt(St(ω),x) k→∞ Dk t (ω, x, ŷk Dn t (ω, x, y) = t (ω, x)) ≥ y∈Kt(St(ω),x) inf inf y∈Kt(St(ω),x) Dξ t (S t (ω), x, y), ∀x, y, a.s. Dξ t (S t (ω), x, y). (3.13)
	We deduce from (3.11) and (3.13) and, finally (3.12), that	(3.10)
	lim inf k→∞	Dk t (ω, x, ŷk t (ω, x)) =	inf y∈Kt(St(ω),x)	Dξ t (S t (ω), x, y) = Dξ t (S t (ω), x, ŷ0 t (ω, x))
						We
	then have:
	lim n→∞	inf y∈Kt(St(ω),x)	Dn t (ω, x, y) ≤	y∈Kt(St(ω),x) inf	Dξ
			y∈R d inf	Dξ
						y∈R d	Dξ t (S t , x t , y)|F t ,
	Theorem 3.10, we deduce by [26, Proposition 7.2] that: y∈R d ≤ inf Dξ	s.(ω) by
	γξ t (s, v t-1 ) := inf vt∈R d lim inf k→∞ Dk t (ω, x, ŷk t (ω, x)) ≥ Dξ Dξ t (s, v t-1 , v t ). t (S t (ω), x, ŷ0 t (ω, x)) The last inequality holds since inf y∈R d Dξ t (S t , x t , y) is F t -measurable. This (3.9) (3.12) implies that E(ŷ 0 t+1 |F t ) ∈ arg min( Dξ

t (S t (ω), x, y) = inf y∈Kt(St(ω),x) Dξ t (S t (ω), x, y), a.s.. Moreover, we suppose that K t (s, x) is upper hemicontinuous in (s, x), see [1, Definition 17.2]. t (ω, x, y) by Dn t (ω, x, y) ∨ -C t (S t (ω), (0, x (2) )) . t = E(ŷ n t+1 |F t ) ∈ K t (S t , x t ) a.s. and converges a.s. to ŷ0 t = E(ŷ 0 t+1 |F t ) ∈ arg min( Dξ t (S t , x t , .)). Proof. t (ω, x, y) ≤ Dξ t (ω, x, y), for all x, y and for any ω ∈ Ω \ H and n ≥ 1. t (S t (ω), x, y), ∀x, (3.11) for any ω ∈ Ω \ H. We now establish the reversed inequality. Since each Dn t is an F-normal integrand, then by [26, Theorem 13.37], we deduce that inf y∈Kt(St(ω),x) Dn t (ω, x, y) is almost surely attained at some ŷn t (ω, x). In other words, we have ŷn t (ω, x) ∈ arg min Kt(St(ω),x) ( Dn t (ω, x, .)) for any ω outside a negligible set N such that H ⊂ N . t (ω, x), k ≥ 1} of {ŷ n t (ω, x), n ≥ 1} converging to some ŷ0 t (ω, x) ∈ K t (S t (ω), x). Since Dn t (ω, ., .) epi -→ Dξ t (S t (.), ., .) a.for any ω ∈ Ω \ N . As Dξ t (S t (ω), x, ŷ0 t (ω, x)) ≥ inf y∈Kt(St(ω),x) t (ω, x t , ŷn t+1 ) = inf y∈Kt(St(ω),xt) Dn t (ω, x t , y) and ŷn t+1 ∈ K t (S t , x t ) a.s.. By [20, Lemma 2.1.2], we may suppose that ŷn t+1 ∈ K t (S t , x t ) is convergent for some random subsequence towards a F t+1 -measurable limit ŷ0 t+1 ∈ K t (S t , x t ). Moreover, by the first step, we have ŷ0 t+1 ∈ arg min Kt(St,xt) ( Dξ t (S t , x t , .)). t : m ≥ 1}. We deduce that E(ŷ n t+1 |F t ) ∈ K t (S t , x t ) if K t (S t , x t ) is convex. It is clear that E(ŷ n t+1 |F t ) converges to E(ŷ 0 t+1 |F t ) ∈ K t (S t , x t ). When the cost function is convex, Dn t (ω, x t , y) is convex. Using the Jensen inequality for conditional expectations, we get that Dξ t (S t , x t , E(ŷ 0 t+1 |F t )) ≤ E Dξ t (S t , x t , ŷ0 t+1 )|F t , ≤ E inf t (S t , x t , y). t (S t , x t , .)).

  [START_REF] Aliprantis | Infinite Dimensional Analysis : A Hitchhicker's Guide, Grundlehren der Mathematischen Wissenschaften[END_REF] ) is a compact upper hemicontinuous set-valued mapping, i.e.

	inf y∈R d	Dξ t (s, x, y) = inf y∈Kt(s,x)	Dξ

t (s, x, y), a.s.. For simplicity, we consider the model where the price process satisfies

supp Ft (S t+1 ) = {a t S t : a t ∈ Θ} , t ≤ T -1,

such that P [S t+1 = a t S t |F t ] > 0 a.s. for all a t ∈ Θ, where Θ = {a n t , n ≥ 1} is a deterministic sequence of positive numbers. Consider a sequence of random variables {b i t , i ∈ J t , t = 0, • • • , T } in R k×T generated by the following procedure:

  ).Under Assumption 6 holds, the mapping (s, v t-1 ) → (φ j (s), v t-1 ) is piecewise linear in the sense of Definition 7.3. Since, γξ,n

		T	is convex, we deduce by
	[26, Exercies 2.20] that γξ,n T (φ j (.), .) is jointly convex. Moreover, since we
	have lim n→∞	γξ,n

T (φ j (s), .)

epi

= γξ

T (φ j (s), .), for any j ≤ J, we deduce by

[START_REF] Rockafellar | Variational analysis[END_REF] Proposition 7

.48] that: θξ

  As in the case t = T -1, we deduce by induction that lim n→∞ Dξ,n t (s, v t-1 , .)

			] that
		Dξ,n
	Suppose that lim n→∞	Dξ,n
			and takes
	real values, it is also continuous. We deduce by [26, Proposition 7.4] that
	lim n→∞	γξ,n t+1 (s, .)

T -1 (s, v T -2 , .) epi -→ Dξ,n T -1 (s, v T -2 , .), n → ∞. t+1 (s, v t , .) epi = Dξ,n t+1 (s, v t , .) and, by induction, let us show that lim n→∞ Dξ,n t (s, v t-1 , .) epi = Dξ,n t (s, v t-1 , .). Since K t+1 (s, .) is compact, we deduce that γξ,n t+1 (s, .) ↓ γξ t+1 (s, .). Since γξ t+1 (s, .) is convex epi = γξ t+1 (s, .). epi = Dξ t (s, v t-1 , .