Dynamic Risk Measures for Processes via Backward Stochastic Differential Equations Associated with Lévy Processes

In this paper, we study the dynamic risk measures for processes induced by backward stochastic differential equations driven by Teugel’s martingales associated with Lévy processes (BSDELs). The representation theorem for generators of BSDELs is provided. Furthermore, the time consistency of the coherent and convex dynamic risk measures for processes is characterized by means of the generators of BSDELs. Moreover, the coherency and convexity of dynamic risk measures for processes are characterized by the generators of BSDELs. Finally, we provide two numerical examples to illustrate the proposed dynamic risk measures.

Throughout this paper, we consider the following integral equation: where the terminal value ξ is a given F T -measurable square integrable random variable, g(·) is a given map, and H (i) t is the orthonormalized Teugel's martingale of order i associated with the Lévy process {L t , 0 ≤ t < ∞}. The above equation is called backward stochastic differential equations associated with Lévy processes (BSDELs) introduced by Bahlali et al. [1]. When Equation (1) is independent of Teugel's martingales, then Equation (1) is reduced to the following form: which is the classical backward stochastic differential equations (BSDEs) introduced by Pardoux and Peng [2] first. Pardoux and Peng [2] proved that there exists a unique adapted to illustrate the proposed dynamic risk measures. The obtained results extends the results of Briand et al. [9], Jiang [10], Penner and Réveillac [34], and Ji et al. [36]. The rest of the paper is organized as follows. In Section 2, we briefly state some preliminaries including the definitions of time-consistent dynamic convex and coherent risk measures for processes and some results on BSDELs. The definition of dynamic risk measures for processes induced by BSDELs is also provided in Section 2. In Section 3, our main results are presented, that is, the coherency and convexity of dynamic risk measures for processes are characterized by the generators of BSDELs, and the representation theorem for generators of BSDELs is provided. Section 4 contains all the proofs of the main results of this paper. We provide two numerical examples to illustrate the proposed dynamic risk measures in Section 5. Finally, conclusions are summarized.

Notations of Dynamic Risk Measures for Processes
For any positive integer n and z ∈ R n , |z| denotes its Euclidean norm. For any t ∈ [0, T], we introduce the following spaces.
• L ∞ (Ω, F t , P) is the space of random variables ξ which are F t -measurable and essentially bounded.
• H 2 T is the space of (F t )-progressively measurable processes Z : Ω × [0, T] → R such that • S 2 T is the space of (F t )-progressively measurable and càdlàg processes Y : Ω × [0, T] → R such that • R ∞ is the space of (F t )-progressively measurable and càdlàg processes ϕ : Ω × [0, T] → R such that • 2 is the space of real valued sequences (x n ) n≥0 such that • P 2 ( 2 ) is the space of predictable processes K taking values in 2 such that • E 2 is the Banach space of processes (Y, Z, K) ∈ S 2 T × H 2 T × P 2 ( 2 ) under the following norm (Y, Z, K) 2 For the convenience of the reader, we introduce the concept of related time-consistent dynamic risk measures for processes, see Cheridito et al. [22], Penner and Réveillac [34], and Ji et al. [36].
For 0 ≤ t ≤ s ≤ T, we define the projection π t,s := R ∞ → R ∞ as On a general level, a conditional risk measure ρ t is any map from R ∞ t to L ∞ (Ω, F t , P). ρ t can be described as a risk assessment at time t, which is taken into account the information available up to this time. For t ∈ [0, T], the map ρ t has the following usual axioms for all X, Y ∈ R ∞ t . (A) (Conditional cash invariance) For all m ∈ L ∞ (Ω, F t , P), is called a conditional coherent risk measure for processes, if it satisfies (A), (B), (C), and (D).
is called a conditional convex risk measure for processes, if it satisfies (A), (B), (E), and (F).
A sequence (ρ t ) t∈[0,T] is called a dynamic coherent risk measure for processes, if for each t ∈ [0, T], ρ t : R ∞ t → L ∞ (Ω, F t , P) is a conditional coherent risk measure for processes. Similarly, a sequence (ρ t ) t∈[0,T] is called a dynamic convex risk measure for processes, if for each t ∈ [0, T], ρ t : R ∞ t → L ∞ (Ω, F t , P) is a conditional convex risk measure for processes.

Some Results on BSDELs
Let L t − = lim s t L s and ∆L t = L t − L t − . Following Nualart and Schoutens [3,4], the so-called power jumps of the Lévy process {L t , t ∈ [0, T]} are given by We denote by (H (i) ) i≥1 the Teugel's martingales, associated with the Lévy process {L t , t ∈ [0, T]}, which is a linear combination of the Y (j) , j = 1, . . . , i : 1 ] for all i ≥ 1. From Nualart and Schoutens [3], we can see that the coefficients c i,k correspond to the orthonormalization of the polynomials 1, x, x 2 , . . . with respect to the measure u(dx) = x 2 ν(dx) + σ 2 δ 0 (dx). The martingales (H (i) ) i≥1 , also called the orthonormalized ith-power-jump processes, can be chosen to be pairwise strongly orthonormal martingales, and their predictable quadratic variation processes are given by For more related results on Teugel's martingales associated with the Lévy process {L t , t ∈ [0, T]}, see Nualart and Schoutens [3,4]. For simplicity of presentation, we rewrite BSDELs (1) as where ξ ∈ L 2 (Ω, F T , P) and g(t, y, z, k) We introduce some assumptions which will be used in this paper.

Remark 2.
Following from Nualart and Schoutens [3,4], in the case of a Poisson process {N t , 0 ≤ t < ∞} with parameter λ > 0, we know that all Teugel's martingales are equal to All orthonormalized ith-power-jump processes, i ≥ 2, are equal to zero in the case of a Brownian motion {B t , 0 ≤ t < ∞}.
From Theorem 3.2 of Bahlali et al. [1], we can show the following Proposition 1 without any substantial difficulties. Therefore, we omit its proof here. Meanwhile, the following Proposition 2 is taken from Theorem 3.3 of Bahlali et al. [1]. Proposition 1. Assume that g satisfies (H1) and (H2). For i = 1, 2, let the terminal condition ξ i ∈ L 2 (Ω, F T , P), and let Y i , Z i , K i ∈ E 2 be the solution of Equation (3) corresponding to ξ = ξ 1 , ξ = ξ 2 , respectively. Then, the following estimate holds: where C is a positive constant.

Proposition 2.
For i = 1, 2, assume that g i satisfies (H1) and (H2), and let the terminal condition 3) corresponding to ξ = ξ 1 , ξ = ξ 2 , respectively. We suppose the following conditions hold: (iii) For all i ∈ N, let K (i) denote the 2 -valued stochastic process such that its i first components are equal to those of K 2 and its N\{1, 2, · · ·, i} last components are equal to those of K 1 . With this notation, we define for i ∈ N Remark 3. The third condition of comparison Theorem 2 is that we add. Without the additional condition, it does not hold in general for solutions of BSDEs associated with Lévy processes (see the counter-example in Barles et al. [42]). In the proof of Bahlali et al. [1], They actually use this condition.
In this paper, define the dynamic risk measures for processes ρ by where Y is the first component of the solution (Y t (X), Z t (X), K t (X)) of the following BSDEL: The following lemma shows the existence and uniqueness of the solution of BSDEL (6) and its proof will be postponed to Section 4. Lemma 1. Assume that g satisfies (H1) and (H2). For any X ∈ S 2 T , there exists a unique adapted solution in E 2 , denoted by (Y t,T (X), Z t,T (X), K t,T (X)), solving BSDEL (6).
Remark 4. For simplicity of the notation, we sometimes denote the solution (Y t,T (X), Z t,T (X), K t,T (X)) of BSDEL (6) by (Y t (X), Z t (X), K t (X)). Thanks to the uniqueness of the solution, for each X ∈ R ∞ , we have that Y t (X) = Y t (π t,T (X)), which is consistent with our notation ρ t (X) = ρ t (π t,T (X)). For 0 ≤ t ≤ s ≤ T, we also denote by (Y t,s (X), Z t,s (X), K t,s (X)) the solution of BSDEL (6) on [0, s] at time t. Accordingly, for all t ∈ [0, s], X ∈ R ∞ , we have that Y t,s (X) = Y t,s (π t,s (X)).

Main Results
In this section, we will state the main results of this paper. Namely, we will state the connections between the generators of BSDELs and the dynamic risk measures for processes via BSDELs. By a product, we will also give a representation theorem of the generators of BSDELs. Their proofs will be postponed to Section 4.
Before we provide the connections between the generators of BSDELs (3) and the dynamic risk measures for processes via BSDELs, we need to give a representation theorem for generators of BSDELs (3), which will be used in the later. As pointed in the Introduction, the study about representation theorems for generators is an interesting topic and is useful in financial mathematics. The following Theorem 1 is one of the main results of this paper.

Theorem 1. Assume that g satisfies (H1) and (H2). Let the terminal condition
holds true for almost every t ∈ [0, T). Furthermore, there exists a subsequence {n m } ∞ m=1 ⊂ {n} ∞ n=1 such that dP × dt-a.s., Now, we are in a position to state the connections between the generators of BSDELs and the dynamic risk measures for processes via BSDELs, which are another main results of this paper. Theorem 2. Assume that g satisfies (H1) and (H2). Denote by (Y t (X), Z t (X), K t (X)) the solution of BSDEL (6) corresponding to X ∈ R ∞ . Let ρ be defined as (5). Then, (ii) If (ρ t ) t∈[0,T] is a dynamic convex risk measure for processes, then (ρ t ) t∈[0,T] is time-consistent.
By choosing some specific generators of BSDELs, we construct dynamic risk measures for processes by means of BSDELs.
Then, ρ is a dynamic coherent risk measure.
Then, ρ is a dynamic convex risk measure. However, ρ is not a dynamic coherent risk measure.

Proofs of Main Results
In this section, we will provide the proof of Lemma 1 and all proofs of the results stated in Section 3.
Proof of Lemma 1. In order to prove the existence and uniqueness of the solution of BSDEL (6), we define a new function f X : where t ∈ [0, T], (y, z, k) ∈ R × R × 2 . It is easy to see that f X satisfies the Lipschitz condition (H2). Therefore, we only need to show that f X satisfies assumption (H1). By using assumption (H2), we obtain for all t ∈ [0, T], Notice that g satisfies assumption (H1) and X ∈ S 2 T . By taking mathematical expectation, we immediately deduce that Thus, f X satisfies assumption (H1).
In order to prove Theorem 1, we need to have two additional results. The following Proposition 3 comes from Proposition 2.2 of Jiang [33]. Proposition 4 concerning on a priori estimate for BSDELs is new and needs to be proved. Proposition 3. Let q > 1 and 1 ≤ p < q. For any (F t )-progressively measurable process where C is a positive constant and β = 2C L + 4C 2 L .
For conditional cash invariance and time-consistency of Y t (·), we have the following result. Proposition 6. Assume that g satisfies (H1) and (H2). Denote by (Y t (X), Z t (X), K t (X)) the solution of BSDEL (6) corresponding to X ∈ S 2 T . Then, we have the following statements: (i) For any m ∈ L 2 (Ω, F t , P), t ∈ [0, T], Y t X + m1 [t,T] = Y t (X) − m.
Proof. Let us prove that (i) holds. For each X ∈ S 2 T , m ∈ L 2 (Ω, F t , P), we consider the following BSDEL: Obviously, we have Thanks to uniqueness of the solution of BSDEL (6) Now we prove that (ii) holds. Suppose that g satisfies (H3). To this end, we first prove that Let us denote by (Y, Z, K) the solution of BSDEL (6) corresponding to X ∈ S 2 T . Following from the uniqueness of the solution of BSDEL (6), we obtain T]. Then, we get X ∈ S 2 T . Now, we denote by Y, Z, K the solution of BSDEL (6) corresponding to X = X. Then we have Due to the uniqueness of the solution, we have By Propositions 5 and 6(i), we get Finally, we have For conditional convexity of Y t (·), we have the following result.
Then, we have ξ n , η n ∈ L ∞ (Ω, F T , P) and ξ n → ξ, η n → η in L 2 sense. With the help of Proposition 1 and the similar argument as in (35)- (38), we deduce that (34) holds. By using assumption (H3) and Proposition 1, we also obtain that (39) holds. Thus, with the same method as in the proof of (i) ⇒ (iii), we see that g satisfies assumption (H4).
For monotonicity of Y t (·), we have the following result.
Notice that the choice of a and b is arbitrary and a ≥ b, we have that g is nonincreasing in y.
For conditional positive homogeneity of Y t (·), we have the following result.
for any ξ ∈ L 2 (Ω, F T , P), t ∈ [0, T], α ≥ 0. With the help of the same method as in the proof of (i) ⇒ (iii), we can see that g satisfies assumption (H6).

Numerical Illustrations
In this section, we will provide two numerical examples to illustrate the proposed dynamic risk measures. Example 1. We suppose that the generator of the BSDEL (6) is independent of (y, k) and is given by For any X ∈ S 2 T and σ ∈ (0, 1], we consider the following equation: The solution of (44) is Let ρ t (X) = −σ 2 t + σB t , ∀t ∈ [0, T], X ∈ R ∞ .
By Theorem 2, we obtain that ρ is a dynamic convex risk measure. In the following, we will present some numerical illustrations for this example. Set T = 5. The curves of ρ t (X) as a function of t (for σ = 0.1, 0.25, 0.5, 0.75) and as a function of σ (for t = 1, 2, 3, 4) are plotted in Figures 1 and 2, respectively. From Figures 1 and 2, it is interesting to note that the dynamic risk measures ρ tend to decline on the whole, which is consistent with our intuitive understanding: in securities trading, when the stock price drops, the loss of investors increases and the corresponding cost risk also increases, as a result, the absolute value of the dynamic risk measure becomes larger. Furthermore, we find that the values of the dynamic risk measures ρ appear positive on some time interval, which can be interpreted by the effect of the large disturbance of Brownian motion at some point. We mention that the fluctuations of the dynamic risk measures ρ become more stable in Figure 1 when σ becomes smaller, and the downward trends of the dynamic risk measures ρ become more obvious in Figure 2 when t becomes bigger. That is because of choosing to invest in low-risk assets and increasing of investment risk, respectively.
In a financial market, some investors may venture among certain European-type contingent claims, some bonds, some stocks, and so on. Depending on an investor's appetite for risk, he/she may choose a curve in Figure 1 as their investment target, or choose a reasonable time of trading based on the impact of level of risk appetite in Figure 2. For example, in order to get more returns, a risk-lover may choose a curve of σ = 0.75 in Figure 1 as his/her investment target in high-risk assets such as certain European-type contingent claims and some stocks. On the contrary, a risk-averse investor may choose a curve of σ = 0.1 in Figure 1 as his/her investment target.
From Theorem 2, we obtain that ρ is a dynamic convex risk measure. In the following, we will also present some numerical illustrations for the example. Set T = 5. The curves of ρ t (X) as a function of t (for σ = 0.1, 0.25, 0.5, 0.75) and as a function of σ (for t = 1, 2, 3, 4) are plotted in Figures 3 and 4, respectively. It is interesting to note that, comparing with Figure 3, the downward trends of the dynamic risk measures ρ are clearly obvious in Figure 4. We mention that the fluctuations of the dynamic risk measures ρ become more stable in Figure 3 when σ becomes smaller, and there are no significant differences among the dynamic risk measures ρ in Figure 4 when t changes. Further, comparing Figures 3 and 4 with Figures 1 and 2, although the changing trends of the corresponding figures are similar, the fluctuation range of the former is smaller. This is because the solution of the current example is less affected by the diffusion term, which leads to a slower evolution speed than that of Example 1.
In a financial market, depending on investors' appetite for risk, they may choose different investments in Figure 3. Figure 4 suggests that there may not be much difference for investors who choose a reasonable time of trading. Thus, the risk lovers, taking more risks, may choose the time t = 4 of trading to get more returns.

Conclusions
In this paper, we study the dynamic risk measures for processes induced by backward stochastic differential equations driven by Teugel's martingales associated with Lévy processes. The representation theorem for generators of BSDELs is provided. Furthermore, the time-consistency of the coherent and convex dynamic risk measures for processes is characterized by means of the generators of BSDELs. Moreover, the coherency and convexity of dynamic risk measures for processes are characterized by the generators of BSDELs. Finally, we provide two numerical examples to illustrate the proposed dynamic risk measures.