{'fraction_non_alphanumeric': 0.07228443449048153, 'fraction_numerical': 0.020828667413213885, 'mean_word_length': 3.5494141619969435, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 9, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Concentration of dynamic risk measures in a Brownian filtration

Motivated by liquidity risk in mathematical finance, D. Lacker introduced concentration inequalities for risk measures, i.e. upper bounds on the \emph{liquidity risk profile} of a financial loss. We derive these inequalities in the case of time-consistent dynamic risk measures when the filtration is assumed to carry a Brownian motion. The theory of backward stochastic differential equations (BSDEs) and their dual formulation plays a crucial role in our analysis. Natural by-products of concentration of risk measures are a description of the tail behavior of the financial loss and transport-type inequalities in terms of the generator of the BSDE, which in the present case can grow arbitrarily fast.


Introduction
On a fixed probability space (Ω, F , P ), we investigate concentration inequalities of the form where l is a given positive, increasing and convex function and ρ : L 2 → (−∞, +∞] is a risk measure. This type of inequalities first appeared in the work of Bobkov and Götze [3] and have recently been thoroughly studied by Lacker [19] in the context of liquidity risk in finance. Given a future uncertain loss X the inequality (1.1) yields an upper bound on the liquidity risk profile (ρ(λX)) λ≥0 of X. This (typically convex) curve captures the effect of leverage on risk. Indeed, in a frictionless market, the minimal capital requirement for (resp. the minimal price needed to hedge) a position λX should be linear in the scaling λ. This is the case for coherent risk measures. However, in an imperfect market, the minimal capital requirement (resp. superhedging price) is nonlinear in λ and the slope of the curve at a given point (e.g. λ ≥ 1) says how much an infinitesimal increase of the loss changes the minimal capital requirement. This slope is expected to be large for very illiquid claims. The question is then for which risk measures and which claims can the liquidity risk profile be controlled and bounded? see [19], where the author further gives an integrability condition under which (1.1) holds for shortfall risk measures. The goal of this note is to derive the bound (1.1) for dynamic risk measures in the Brownian setting. Given T > 0, we equip (Ω, F , P ) with the completed filtration (F t ) t∈[0,T ] of a d-dimensional Brownian motion W . As usual, we identify random variables that are equal P -a.s. and understand equalities and inequalities in this sense. A time-consistent dynamic risk measure is a family (ρ s,t ) 0≤s≤t≤T of convex increasing functionals ρ s,t : L 2 (F t ) → L 2 (F s ) such that for all 0 ≤ s ≤ t ≤ T , (A1) ρ s,t (X + η) = ρ s,t (X) + η for all X ∈ L 2 (F t ) and η ∈ L 2 (F s ) Definition 1.1. Given κ > 0, we say that the risk measure (ρ s,t ) is κ-dominated if for all X 1 , X 2 ∈ L 2 , where E κ (X 2 ) := Y 0 and the pair (Y, Z) is the unique square-integrable solution of the equation Let C 0 ([0, T ], R d ) denote the space of continuous functions from [0, T ] to R d starting from zero. A function φ : Our first main result is: Let (ρ s,t ) be a time-consistent dynamic risk measure. If 1 ρ * 0,T (1) = 0 and there is κ > 0 such that (ρ s,t ) is κ-dominated, then putting l(λ) := κ(T λ 2 + 1) for λ ≥ 0, it holds We also prove, see Theorem 2.1 below, the result for (minimal super-) solutions of backward stochastic differential equations (BSDEs) with quadratic growth. These can be seen as dynamic risk measures which are not necessarily κ-dominated. Furthermore, we prove (1.3) even for BSDEs with generators growing arbitrarily fast is z, see Corollary 2.4.
The proofs draw from convex duality results for BSDEs as developed by Drapeau et al. [11] and the (finite dimensional) Gaussian transport-entropy inequality initiated by Talagrand [28] and generalized to the framework of abstract Wiener space by Feyel and Üstünel [13]. While we can hardly overstate the importance of these inequalities, see e.g. the survey of Gozlan and Léonard [15], let us mention that Lacker [19] pointed out some interesting consequences of inequalities of the form (1.3): as mentioned above, they provide bounds on the liquidity risk profile of the financial derivative with payoff X and, on the other hand, they give new types of transportation inequalities and new descriptions of the concentration of measure phenomenon. In the same vein, a dual formulation of the inequality (1.3) leads us to a 1 ρ * 0,T denotes the convex conjugate of ρ 0,T defined as ρ * 0,T (Z) : "transport-type inequality" in terms of the penalty function (convex conjugate) of the risk measure. A particularly interesting case arises for risk measures stemming from BSDEs with superquadratic generators, as the penalty function is the integral of a subquadratic function. The next section is dedicated to the proof of Theorem 2.1, where we further derive (1.3) for minimal supersolutions of BSDEs. In the final section, we discuss some immediate mathematical consequences. In particular, deviation and transport-type inequalities.  [10], a supersolution of the BSDE with terminal condition X ∈ L 0 and generator g : Given X ∈ L 0 and a generator g define A (X) := (Y, Z) ∈ S × L : (Y, Z) fulfills (2.1) and Z dW is a supermartingale .

Concentration inequalities.
Further denoting by Λ + the L 2 -closure of for all x, y ∈ R d×2 n , one obtains: Assume that g is positive, decreasing in y and jointly convex. Further assume that there are a, b, c ≥ 0 such that Then, putting l(λ) := e bT cT λ 2 + a for λ ≥ 0, one has If in addition g does not depend on y, then Λ + in (2.3) can be replaced by Λ, the L 2 -closure of the set Proof. Consider the sets D := β : Ω × [0, T ] → R + : β predictable and T 0 β u du < ∞ and For q ∈ Q, denote by Q q the probability measure whose density process is given by the stochastic exponential M q := exp( q u dW u − 1 2 |q u | 2 du) and for β ∈ D, put D β s,t := exp(− t s β u du), 0 ≤ s ≤ t ≤ T and D β := D β 0,T . It follows from [11, Theorem 3.10] that the functional E admits the convex dual representation When g does not depend on y the above do not depend on β and take the form It can be checked that due to (2.2) and monotonicity of g, it holds The growth condition (2.2) on g further implies that Thus, multiplying both sides of (2.5) by D β 0,u and integrating yields for all (β, q) ∈ D × Q. Moreover, for each q ∈ Q, τ n := inf{t ≥ 0 : | t 0 q u dW u | ≥ n} ∧ T defines a sequence of stopping times such that τ n ↑ T P -a.s. Let q n := q1 [0,τ n ] . Then, the Kullback-Leibler information (or relative entropy) of Q q with respect to P is given by where the last equality follows from Girsanov theorem. Since M q is continuous, and therefore whereQ is the set of q ∈ Q such that M q = u(W T 2 −n , . . . , W kT 2 −n , . . . , W T ) for some n ∈ N and u : R d×2 n → R + bounded and Borel measurable. Hence, if X is positive, as D β ≤ 1 for all β ∈ D, and if g does not depend on y, (2.8) holds for all X ∈ L 2 . For q ∈Q and u : R d×2 n → R + a bounded Borel measurable function such that M q = u(W T 2 −n , . . . , W kT 2 −n , . . . , W T ), we denote by γ dn the canonical Gaussian measure on R d×2 n and by µ q the Borel measure on R d×2 n given by The independence of the increments of W , yields That is, H(Q q |P ) = H(µ q |γ dn ), where the right hand side denotes the Kullback-Leibler information of µ q with respect to γ dn on the measurable space (R d×2 n , B(R d×2 n )). Therefore, by Talagrand with h(x) = x 2 /2 and W 1,dn (γ dn , µ q ) denoting the Wasserstein distance of order 1 between γ dn and µ q given by where the infimum is over the set of probability measures π on R d×2 n × R d×2 n with marginals γ dn and µ q . Given n ∈ N and a 1-Lipschitz function f : It is clear that X ∈ L 2 and so, E(X) satisfies (2.8). Let q ∈Q and µ q be given by (2.9). Since the functionf (x 1 , . . . , Thus, for all λ ≥ 0 and by definition of the convex conjugate h * of h, one has so that arguing as in the computations leading to (2.10), one obtains Therefore, due to (2.8), taking the supremum above as q runs overQ yields Since h(x) = x 2 /2, it holds l(λ) = e bT cT λ 2 + a. Now, let X ∈ Λ + . There is a sequence X n of the form X n = f n (W T 2 −n , . . . , W kT 2 −n , . . . , W T ) where f n is a positive and 1-Lipschitz function on R d×2 n which converges to X in L 2 , and therefore in L 1 . As above, for all n and λ ≥ 0 one has Since E is L 1 -lower semicontinuous, see [10,Theorem 4.9], taking the limit as n goes to infinity, one has This yields (2.3).
Recall that working with positive random variables was necessary to obtain (2.8). Since this inequality holds for all X ∈ L 2 when g does not depend on y, the last claim of the theorem follows by the same arguments as above.
In case c = 0, it can be assumed w.l.o.g. that g does not depend on z, so that the functional E satisfies the representation du where due to the growth condition (2.2), the supremum can be restricted to β ∈ D satisfying 0 ≤ β ≤ b. This growth condition further ensures that g * (β) ≥ −a. Thus, for all X ∈ L 2 with X ≥ 0, one has E(λX) ≤ E[λX] + a for every λ ≥ 0.
This concludes the proof.
Example 2.2. A particularly interesting example arises when g t (y, z) = |z| 2 /2. In this case, E(X) = log(E[e X ]) for every X with finite exponential moment and E is not E κ -dominated for any κ > 0. If moreover Ω = C([0, T ], R d ) is the canonical space equipped with the uniform distance and P the Wiener measure, then by [3, Theorem 1.3], the Gaussian transportation inequality of Feyel and Üstünel [13] implies the bound E(λX) ≤ λE[X]+λ 2 /2 for every λ ≥ 0 and every 1-Lipschitz function X : Ω → R.♦ Two questions emerging from Theorem 2.1 and its proof are whether the set Λ of "claims" for which the concentration inequality (2.3) holds can be enlarged, and whether Theorem 2.1 can be obtained for generators growing faster than quadratic (in z). Regarding the first question, one possibility could be to use the Kantorovich duality, (see e.g. Villani [29, Theorem 5.10]) instead of the Kantorovich-Rubinstein formula in the proof of Theorem 2.1. This, however, would destroy the interpretation of (2.3) as a bound on the liquidity risk profile. To answer the second question, one can consider a slight modification of the set Λ.
Denote by ∆W jT 2 −n := W jT 2 −n − W (j−1)T 2 −n the increment of the Brownian motion, by Λ ′ the L 2 -closure of the set Using the fact that the elements of Λ ′ have uniformly bounded Malliavin derivative, one obtains the next corollary. Refer to Nualart [23] for elements of Malliavin calculus.
Proof. Each X of the form X := f (∆W T 2 −n , ∆W 2T 2 −n , . . . , ∆W T ) for some n and f : R d×2 n → R Consider the functiong t (y, z) := b|y| + ϕ(|z|)|z|. This (deterministic) function has constant Malliavin derivative and satisfies for all y, y ′ ∈ R and z, z ′ ∈ R d . Thus, by [5, Theorem 2.2 and Remark 2.3], the BSDE with terminal condition X and generatorg admits a unique solution (Y, Z) with Y ∈ S 2 and |Z t | ≤ T e T =: K. Hence, (Y, Z) coincides with the unique solution of the BSDE with terminal condition X and the Lipschitz continuous generatorĝ t (y, z) := g t (y, z) if |z| ≤ K g t (y, Kz/|z|) if |z| > K (which is known to exist by e.g. [12]). Notice that the bound on Z does not depend on n, so thatĝ does not depend on n as well. On the one hand, g ≤g, and thus, from [10, Proposition 3.3], E(X) ≤Ẽ(X), wherẽ E(X) is the minimal supersolution of the BSDE with generatorg. Moreover,ĝ satisfies the conditions of Theorem 2.1 and Y 0 coincides withÊ(X), the minimal supersolution of the BSDE with terminal condition X and generatorĝ. Thus, whereĝ * is the convex conjugate ofĝ. Sinceĝ satisfies the growth condition (2.2) with a = 0 and c = ϕ(T e T ) + T e T θ(T e T ), as in the proof of Theorem 2.1, one has . In particular, a discretization and approximation argument already described puts us in the situation of Equation (2.7). The rest of the proof follows exactly as in the proof of Theorem 2.1. for all λ ≥ 0 and φ : C 0 ([0, T ], R d ) → R + 1-Lipschitz. If g does not depend on y, then the result also holds for φ : Proof. In view of Proposition 2.3, it suffices to show that for every φ : This in turn follows by arguments in [5] that we sketch below. The reader is referred to the proof of [5, Proposition 3.2] for details. Put t n k := kT 2 −n , with k = 0, . . . , 2 n and define the function Further denote X n : for every x, y ∈ R d×2 n . Moreover, using the fact that Brownian motion has stationary increments together with Doob maximal inequality, it follows that for every p ≥ 2 there is a constant c p depending only on p such that This shows that (X n ) n converges to φ(W ) in L p for every p > 2. Thus, (X n ) n converges to φ(W ) in L 2 and therefore φ(W ) ∈ Λ ′ .
The following is a natural example of E κ -dominated time-consistent dynamic risk measure.
Example 2.5. Let g : [0, T ] × Ω × R d → R be convex and Lipschitz continuous in the third variable with Lipschitz constant κ ≥ 0. Furthermore, assume g t (0) = 0 and E[ T 0 |g t (z)| 2 dt] < ∞ for all z ∈ R d . Then, the dynamic risk measure ρ s,t (X) := Y s , where (Y, Z) is the unique solution of the BSDE is time-consistent and E κ -dominated, see [7]. ♦ Example 2.6. Random variables X ∈ Λ can be seen as claims, or financial losses whose liquidity risks is to be evaluated. For instance, consider a model with a stock following the dynamics dS t = S t (b t dt + σ t dW t ) for some b ∈ L 1 (dt) and σ j : [0, T ] → [−1, 1] measurable for all j = 1, . . . , d. Let X := F (S T ) be the payoff of a contingent claim written on S. If F (x) = log(x), i.e. X is the payoff of the so-called log-contract, see e.g. [22], then It is clear that the r.h.s. above is in Λ, as L 2 -limit of discrete-times stochastic integrals and since |σ t | ≤ 1. Thus, for every time-consistent dynamic risk measures (ρ s,t ), that is E κ -dominated for some κ > 0 there is a convex increasing function l such that ρ 0,T (λX) ≤ λE[X] + l(λ) for all λ ≥ 0. ♦ As mentioned in the introduction, there is a strong relation between BSDEs and time-consistent dynamic risk measures. On the one hand, BSDEs allow to define dynamic risk measures. This was first studied in [17,25,27] for BSDEs with Lipschitz continuous generators and extended to the quadratic growth case in [16,18] and to convex BSDEs in [10,11]. Reciprocally, under suitable additional conditions, time-consistent dynamic monetary risk measures can be seen as solutions of BSDEs. This "dynamic representation" of risk measures was initiated by [7] who introduced the concept of E κ -domination. Refer to [9,16]  Let us mention that Lacker [19] gave an integrability criterion for the concentration property of static risk measures including optimized certainty equivalent and shortfall risk measures. In the case where Ω is equipped with a metric δ, Bobkov and Ding [2] gave integral criteria on δ from which the concentration property for optimized certainty equivalent risk measures with power-type loss functions can be derived. Notice that these functionals are typically time-inconsistent, except for the entropic case.

Applications
3.1 Transport-type inequalities. Let us borrow an argument from Gozlan and Léonard [15] to show that Theorem 2.1 leads to a transport-type inequality. In what follows, we put with l as in Theorem 2.1. If g satisfies (2.12), then one has

2)
with l as in Proposition 2.3.
Proof. It follows from Theorem 2.1 and the convex dual representation of BSDEs (see [11,Theorem 3.10]) that for all X ∈ Λ, λ ≥ 0 and q ∈ L, Taking the supremum over X ∈ Λ and λ ≥ 0 yields (3.1). The proof of (3.2) is the same.
Remark 3.2. The transport-type inequality (3.1) can also be written w.r.t. subprobability measures, i.e. measures with density D β Q q for some (β, q) ∈ D × Q. In fact, the arguments of Proposition 3.1 lead to where the infimum is over probability measures on Ω × Ω with marginals Q 1 and Q 2 . Then, for g and l as in Proposition 2.3, it holds Proof. As in the proof of Proposition 3.1, it follows from Corollary (2.4) that λ( for all 1-Lipschitz function φ (i.e. satisfying (1.2)). Thus, taking the supremum over such functions φ and over λ ≥ 0 and using that W is the canonical process yields By the Kantorovich-Rubinstein formula, the above inequality implies (3.3).
In particular, when g(z) = z 2 /2, one recovers with (3.3) (admittedly with less direct arguments) the Gaussian transport-entropy inequality on C([0, 1], R d ) obtained by Feyel and Üstünel [13] and Lehec [20]. Let us give a PDE characterization of the transport-type inequality derived above. We use the notation where the infimum is over the set of probability measures π on R d × R d with marginals γ d and µ.
Hence, using P (A) ≥ 1/2, it follows this concludes the proof.

Dimension-free bounds.
For identically distributed claims, we show that the bounds obtained above become dimension-free: Proposition 3.6. Let g : Ω × [0, T ] × R × R d → [0, ∞) be a convex function (in (y, z)) increasing in y. If (X i ) i=1,...,n is a family of identically distributed random variables such that E(λX i ) ≤ E[λX i ] + l(λ) for all λ ≥ 0 and some convex increasing function l : R → [0, ∞), then E λ 1 n n i=1 X i ≤ λE[X i ] + l(λ) for all λ ≥ 0. (3.9) Proof. Since g is convex, the functional E is also convex. Thus, one has The result follows since X i are identically distributed.
The concentration inequality (3.9) is dimension-free in the sense that it does not depend on n, which implies that the bound cannot be improved by increasing the dimension. In particular, if E( 1 n n i=1 X i ) is viewed as the minimal superhedging price of the portfolio 1 n n i=1 X i , then the inequality (3.9) means that the superhedging price remains bounded from above when the portfolio increases and the bound is independent of the size of the portfolio.