Minimal Entropy and Entropic Risk Measures: A Unified Framework via Relative Entropy

Abstract

We introduce a new coherent risk measure, the minimal-entropy risk measure, which is built on the minimal-entropy $\sigma$-martingale measure—a concept inspired by the well-known minimal-entropy martingale measure used in option pricing. While the minimal-entropy martingale measure is commonly used for pricing and hedging, the minimal-entropy $\sigma$-martingale measure has not previously been studied, nor has it been analyzed as a traditional risk measure. We address this gap by clearly defining this new risk measure and examining its fundamental properties. In addition, we revisit the entropic risk measure, typically expressed through an exponential formula. We provide an alternative definition using a supremum over Kullback–Leibler divergences, making its connection to entropy clearer. We verify important properties of both risk measures, such as convexity and coherence, and extend these concepts to dynamic situations. We also illustrate their behavior in scenarios involving optimal risk transfer. Our results link entropic concepts with incomplete-market pricing and demonstrate how both risk measures share a unified entropy-based foundation.

1. Introduction

Risk measures play an essential role in both academic research and financial practice, as they provide a systematic way to assess the potential losses of a financial position. Their importance has been growing in the industry and academic research since the work of Artzner et al. (1999), which introduced an axiomatic framework for coherent risk measures. Subsequent studies, such as (Föllmer and Schied 2011, chp. 4) or Delbaen (2000), have refined and extended these ideas.

In the monetary risk-measure framework, one models a financial position using a real-valued random variable X on a probability space $(\mathsf{\Omega },\mathcal{F},\mathbf{P})$. The number $X\left(\omega \right)$ is the discounted net worth of the position if scenario $\omega$ occurs. A monetary risk measure, $\rho$, assigns a real number, $\rho \left(X\right)$, to the outcome of a random variable. This real number represents the minimal amount of capital needed to make X acceptable according to certain risk criteria. Desirable properties include monotonicity (increasing payoffs lowers the risk) and translation invariance (adding a sure amount of cash decreases the risk by the same amount), and when $\rho$ is further assumed to be convex or coherent, it reflects the benefits of diversification. We refer to Artzner et al. (1999); Delbaen (2000); Föllmer and Schied (2011) for standard references on these terms.

A preferred way to price financial claims in financial mathematics (and hence risk measurement, when viewed from a market perspective) is to determine the price as the expectation of the discounted payoff under an equivalent (local) martingale measure. However, in general markets and for certain price processes, a (local) martingale measure need not exist. Instead, from the more general perspective of no-arbitrage (or no free lunch with vanishing risk, to be more precise), one can only ensure the existence of an equivalent $\sigma$-martingale measure (E$\sigma$MM). See, for instance, Delbaen and Schachermayer (1998); Kallsen (2004); Sohns (2025) for details on $\sigma$-martingale arguments. In addition, in incomplete markets, there may be multiple such measures, and a natural question is how to select the “best” or “preferred” measure among the many.

One popular selection criterion in incomplete markets is to pick the measure that is “closest” to the real-world probability measure $\mathbf{P}$ in terms of the relative entropy (also known as the Kullback–Leibler divergence); see Delbaen et al. (2002); Frittelli (2000); Fujiwara and Miyahara (2003); Miyahara (2004). Minimizing relative entropy leads to the well-known minimal-entropy martingale measure, a construction that has proven valuable in option pricing and hedging and which is closely connected to maximizing the investor’s utility (see Section 4).

Most of the literature focuses on the local martingale setting when studying minimal-entropy measures. The corresponding minimal-entropy $\sigma$-martingale measure has not been studied, even though, in some markets, one must work with E$\sigma$MMs. We close this gap by introducing and studying the minimal-entropy $\sigma$-martingale measure:

$${\mathbf{Q}}^E=\underset{\mathbf{Q}\in {\mathcal{M}}_{\sigma }}{\mathrm{argmin}}H(\mathbf{Q}\mid \mathbf{P}),$$

where ${\mathcal{M}}_{\sigma }$ denotes the set of E$\sigma$MMs, and $H(·|\mathbf{P})$ is the relative entropy with respect to $\mathbf{P}$. The associated minimal-entropy risk measure is then

$${\rho }^E\left(X\right)={\mathbf{E}}_{{\mathbf{Q}}^E}[-X].$$

This measure is an extension of the classical minimal-entropy martingale measure (since an ELMM is a special case of an E$\sigma$MM) but is strictly more general whenever no local martingale measure exists. Notably, while the minimal-entropy martingale measure has been studied for pricing and hedging, it has not been viewed or analyzed as a traditional risk measure in the classical sense. Nor has the $\sigma$-martingale version been examined at all. In this paper, we fill this gap by proving that the minimal-entropy $\sigma$-martingale measure leads to a coherent risk measure with desirable properties.

A different risk measure and one of the most popular ones is the entropic risk measure. Because of the similar name, one might suspect similar definitions of the entropic risk measure and the minimal-entropy methods. Nevertheless, the entropic risk measure is typically introduced via the exponential formula

$$e_{\gamma }\left(X\right)={\displaystyle \frac{1}{\gamma }}\log \left(\mathbf{E}\left[e^{-\gamma X}\right]\right),$$

which does not indicate any connection to entropy.

Its name stems from its robust representation,

$$e_{\gamma }\left(X\right)=\underset{\mathbf{Q}\ll \mathbf{P}}{sup}\left\{{\mathbf{E}}_{\mathbf{Q}}[-X]-\frac{1}{\gamma }H(\mathbf{Q}\mid \mathbf{P})\right\},$$

showing that it penalizes deviations from $\mathbf{P}$ in proportion to the relative entropy. Although this measure is well understood as a convex, time-consistent risk measure, the label “entropic” might not be fully transparent when beginning from an exponential definition. Here, we show that one can equivalently start with a relative-entropy-based formulation, making the “entropic” nature more obvious. We then show that, with this alternative definition, one easily arrives at the same results and conclusions.

This measure is strictly convex (rather than linear) in the payoff, X. In contrast, the minimal-entropy risk measure only picks out a single measure—the one that eliminates arbitrage and is of the least relative entropy. As a result, it turns out to be coherent. Despite these structural differences, both measures share a fundamental entropy-based underpinning.

The foundations of entropic risk measures and minimal-entropy martingale measures (MEMM) date back over two decades, but the topic remains actively researched. Recent studies address both theory and applications. Chong et al. (2019) introduced a forward entropic risk measure based on ergodic backward-stochastic differential equations and studied its behavior at large maturities. Pichler and Schlotter (2020) analyzed entropy-based risk measures and, in particular, their convexity and dual representations. Wang et al. (2020) investigated the entropic measure transform and its applications in risk-sensitive decisions. Dhaene et al. (2015) studied MEMMs for markets that contain financial and actuarial risks. Ishikawa and Robertson (2020) examined optimal investment and pricing problems that include default risks through entropy-based techniques. Doldi et al. (2024) connected entropy martingale optimal transport to dualities in nonlinear pricing and hedging. Y. Kabanov and Sonin (2025) derived explicit characterizations of MEMMs for exponential Ornstein–Uhlenbeck volatility models. McCloud (2025) studied pricing in incomplete markets through an entropic risk measure that accounts for default risks. Finally, Marthe et al. (2025) used entropic risk measures for risk-sensitive planning within Markov decision processes.

The main contributions of this paper are as follows:

• We introduce the minimal-entropy $\sigma$-martingale measure for general semimartingale models, which is new to the literature.
• We prove that the induced minimal-entropy risk measure is coherent and extends the classical minimal-entropy martingale measure results.
• We define the entropic risk measure via its robust representation and, therefore, provide an alternative approach that highlights its connection to entropy.
• We demonstrate key properties including convexity, coherence, dynamic consistency, and optimal risk transfer for both measures, thereby revealing that minimal-entropy techniques are not only pricing tools but also valid risk measures in their own right.
• We provide some estimates, comparing the two risk measures to their real-world expectations.

The paper is organized as follows. Section 2 contains the precise definitions of both the minimal-entropy $\sigma$-martingale measure (by minimizing relative entropy under the no-arbitrage condition) and the entropic risk measure (via a relative-entropy supremum), along with existence criteria. We establish their existence and compare their properties. Section 3 focuses on the definition of monetary risk measures, convexity and coherence, proving that the minimal-entropy risk measure is coherent, while the entropic measure is convex. In Section 4, we explore duality and highlight the deeper relationship between entropy-based valuations and risk measures and, in particular, show that our definition of the entropic risk measure is equivalent to the more common definition in the literature and we elaborate that the setup works for general processes and probability measures. Section 5 provides dynamic versions, establishing time consistency. Finally, Section 6 discusses optimal risk transfer and how each of these risk measures behaves in that context. In the Appendix A, we repeat some well-known results that are used in this publication.

2. Definition and Existence

Let $(\mathsf{\Omega },\mathcal{F},\mathbf{P})$ be a probability space, $L^{\infty }$ the space of bounded random variables, and ${\mathcal{P}}_L$ the set of probability measures on $L^{\infty }\left(\mathcal{F}\right)$. Furthermore, let $\mathbb{F}={\left({\mathcal{F}}_t\right)}_{0\le t\le T}$ be a filtration with ${\mathcal{F}}_T=\mathcal{F}$, and let S be a (potentially multi-dimensional) stochastic process adapted to $\mathbb{F}$. We assume $\widehat{S}={\left({\widehat{S}}_t\right)}_{0\le t\le T}$ is the discounted price process (for details on discounting, see Sohns (2023)).

In most models, at least one probability measure exists such that the discounted process, $\widehat{S}$, is a local martingale under this probability measure. However, Delbaen and Schachermayer (1998) showed that, in an arbitrage-free market (more precisely a market that satisfies no risk with vanishing risk), you can only assume that a probability measure exists, such that $\widehat{S}$ is a $\sigma$-martingale under this probability measure. Therefore, it makes sense to study $\sigma$-martingales, equivalent $\sigma$-martingale measures, and derived risk measures.

First, let us recall the following:

Definition 1. 

A one-dimensional semimartingale, S, is called a σ-martingale if there exists a sequence of predictable sets, $D_n$, such that the following applies:

(i) 

$D_n\subset D_{n+1}$ for all n;

(ii) 

${\bigcup }_{n=1}^{\infty }{D}_n=\mathsf{\Omega }\times {\mathbb{R}}_{+}$;

(iii) 

For each $n\ge 1$, the process ${\mathbb{1}}_{D_n}•S$ is a uniformly integrable martingale.

Such a sequence, ${\left(D_n\right)}_{n\in \mathbb{N}}$, is called a σ-localizing sequence. A d-dimensional semimartingale is called a σ-martingale if each of its components is a one-dimensional σ-martingale.

These ideas, first introduced by Chou (1979) and later developed by Émery (1980); Jacod and Shiryaev (2003); Kallsen (2004), make it possible to study processes that are not martingales, even though they have no drift.

Definition 2. 

An equivalent σ-martingale measure (EσMM) is a probability measure $\mathbf{Q}$ with $\mathbf{Q}\sim \mathbf{P}$, such that $\widehat{S}$ is a σ-martingale under $\mathbf{Q}$. The set of all EσMM is denoted as ${\mathcal{M}}_{\sigma }$.

We also define a broader sets of absolutely continuous measures, $(\mathbf{Q}\ll \mathbf{P})$, under which $\widehat{S}$ is a $\sigma$-martingale:

$${\mathcal{M}}_{\sigma }^{ac}:=\left\{\mathbf{Q};\mathbf{Q}\ll \mathbf{P},\widehat{S}\mathrm{is}\mathrm{a}\sigma \text{-}\mathrm{martingale}\mathrm{under}\mathbf{Q}\right\}.$$

For more details on $\sigma$-martingales, see, for example, Goll and Kallsen (2003); Kallsen (2004); Sohns (2025).

Definition 3. 

The relative entropy $H(\mathbf{Q}\mid \mathbf{P})$ of a probability measure, $\mathbf{Q}$, with respect to $\mathbf{P}$ is defined as

$$H(\mathbf{Q}\mid \mathbf{P})=\left\{\begin{array}{cc}{\int }_{\mathsf{\Omega }}\log \left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right)\mathrm{d}\mathbf{Q}={\mathbf{E}}_{\mathbf{Q}}\left(\log \left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right)\right)\hfill & \mathit{for}\mathbf{Q}\ll \mathbf{P},\hfill \\ \infty \hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

The relative entropy provides a notion of distance between a probability measure, $\mathbf{Q}$, and a reference probability measure, $\mathbf{P}$. Even though it can be interpreted as a distance, it is not a metric since neither the symmetry property nor the triangle inequality holds.

Beyond financial mathematics, relative entropy is a crucial concept in statistical physics and information theory, where it is referred to as the Kullback–Leibler divergence (Shunsuke 1993). It also plays a fundamental role in large deviations theory, where it underpins results such as Sanov’s theorem (Cover and Thomas 2012). A comprehensive summary of its applications can be found in Cherny and Maslov (2004).

Further illustrations of this definition are available in Dacunha-Castelle and Duflo (1986), including a demonstration of how relative entropy can be understood as a distance measure. In statistics, this interpretation is reinforced by results such as Stein’s Lemma, which can be found in (Hesse 2003, Satz 10.4).

One of the central questions in mathematical finance is how to determine the fair price of a claim, X. If the payoff at time T can be replicated with trading in the underlying asset, S, then the initial cost of the replicating strategy equals the fair price; otherwise, an arbitrage opportunity would exist.

More concretely, suppose there is an admissible self-financing strategy, $\varphi$, such that the stochastic integral $\varphi •S$ matches the terminal payoff $X_T$. Then the fair price is $\varphi •S_0$. If there exists an equivalent probability measure, $\mathbf{Q}$, under which S is a martingale, and if suitable integrability conditions hold, then $\varphi •S$ is also a martingale under $\mathbf{Q}$. In this case, the initial value equals the expected terminal value:

$$\varphi •S_0={\mathbf{E}}_{\mathbf{Q}}[\varphi •S_T]={\mathbf{E}}_{\mathbf{Q}}\left[X_T\right],$$

so the fair price is given via the expectation under $\mathbf{Q}$.

An equivalent true martingale measure may not always exist. In such cases, one often uses an equivalent local martingale measure. There are also models without any local martingale measure. However, if the market is free of arbitrage, the first fundamental theorem of asset pricing ensures the existence of at least one equivalent $\sigma$-martingale measure (see Delbaen and Schachermayer (1998)), making the discounted price process a $\sigma$-martingale.

In a complete market, this measure is unique, leading to a single fair price. In incomplete markets, there are multiple equivalent $\sigma$-martingale measures. The range of fair values then lies between

$$\underset{\mathbf{Q}}{min}{\mathbf{E}}_{\mathbf{Q}}\left[X_T\right]\mathrm{and}\underset{\mathbf{Q}}{max}{\mathbf{E}}_{\mathbf{Q}}\left[X_T\right]$$

Thus, the fair price must lie within this interval (Sohns 2023). It is, therefore, reasonable to define the minimal-entropy risk measure as an optimization problem over all equivalent $\sigma$-martingale measures.

Definition 4. 

(a) 

An equivalent σ-martingale measure ${\mathbf{Q}}^E\in {\mathcal{M}}_{\sigma }$ is called the minimal-entropy σ-martingale measure if it minimizes the relative entropy (or Kullback–Leibler divergence) $H(·\mid \mathbf{P})$ among all $\mathbf{Q}\in {\mathcal{M}}_{\sigma }$, i.e.,

$$H({\mathbf{Q}}^E\mid \mathbf{P})=inf\mathbf{Q}\in {\mathcal{M}}_{\sigma }H(\mathbf{Q}\mid \mathbf{P}).$$

The corresponding minimal-entropy risk measure ${\rho }^E$ is defined by

$${\rho }^E\left(X\right):={\mathbf{E}}_{{\mathbf{Q}}^E}[-X].$$

(b) 

For $X\in L^{\infty }$ and $\gamma >0$, the entropic risk $e_{\gamma }$ is defined as

$$e_{\gamma }\left(X\right):=\underset{\mathbf{Q}\in {\mathcal{P}}_L}{sup}\left({\mathbf{E}}_{\mathbf{Q}}[-X]-{\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})\right).$$

(1)

Remark 1. 

Equivalence of $\mathbf{Q}$ to $\mathbf{P}$ ensures that the relative entropy $H(\mathbf{Q}\mid \mathbf{P})$ remains finite since any singular part would make $\log \left(\frac{d\mathbf{Q}}{d\mathbf{P}}\right)$ infinite on a positive-$\mathbf{P}$ set. Also, if $\mathbf{Q}$ ignored events that $\mathbf{P}$ deems possible, the entropic penalty term would lose its economic meaning by excluding potentially significant outcomes.

Remark 2. 

In the above definition of $e_{\gamma }$, the parameter $\gamma >0$ can be viewed as a risk-aversion level or scaling factor, much like in exponential-utility frameworks where larger γ corresponds to greater risk aversion. Thus, $e_{\gamma }$ can be seen as an “entropic” or “exponential-penalized” valuation: the higher the γ, the stronger the penalty on deviating from the reference measure $\mathbf{P}$.

Note that the existence of these risk measures is not immediately clear. In particular, the existence of a minimal-entropy $\sigma$-martingale measure can still depend on the properties of S. For instance, if the market is not arbitrage-free (e.g., the No Free Lunch With Vanishing Risk condition fails) and S represents the price process of a tradable asset, then no equivalent $\sigma$-martingale measure can exist. Even if the market does satisfy NFLVR, one may still require additional conditions on the price processes (e.g., local boundedness or boundedness from below) to ensure that there is some equivalent $\sigma$-martingale measure carrying finite relative entropy. For further details, see Delbaen and Schachermayer (1994, 1998, 2006).

Example 1. 

• One classic example of a complete market is given via the Black–Scholes setup Black and Scholes (1973). In this model, the asset price $S_t$ typically follows

  $$\mathrm{d}{S}_t=S_t\mu \mathrm{d}t+S_t\sigma \mathrm{d}{W}_t$$

  for constants μ and $\sigma >0$, where W is a Brownian motion under the real-world probability. After discounting by the risk-free rate r, one finds a unique equivalent martingale measure, $\mathbf{Q}$, that replaces μ with r in the drift term. The market is, therefore complete, and any bounded contingent claim can be perfectly replicated. Since there is exactly one equivalent martingale measure, pricing is straightforward: every claim has a unique fair value given by its discounted expectation under $\mathbf{Q}$. In this case, the σ-martingale condition is also trivially satisfied by the unique risk-neutral measure; hence, minimizing any functional over the set of equivalent σ-martingale measures offers no ambiguity (there is only one element in that set). Consequently, the minimal-entropy measure (cf. Definition 4) coincides with the unique martingale measure, and the resulting minimal-entropy risk measure effectively matches the usual Black–Scholes pricing rule.
• By contrast, many models for asset prices are incomplete. As a specific example, consider the Heston stochastic volatility model (Heston 1993). There, one typically assumes

  $$\mathrm{d}{S}_t=S_t\mu \mathrm{d}t+S_t\sqrt{v_t}\mathrm{d}{W}_t^{\left(1\right)},\mathrm{d}{v}_t=\kappa (\theta -v_t)\mathrm{d}t+{\sigma }_{\mathrm{vol}}\sqrt{v_t}\mathrm{d}{W}_t^{\left(2\right)},$$

  where $v_t$ represents the instantaneous variance process, and $W^{\left(1\right)},W^{\left(2\right)}$ are correlated Brownian motions under the real-world probability. Unlike Black–Scholes, this model does not admit a perfect hedge for arbitrary claims, so there can be multiple local (or σ-)martingale measures equivalent to the real-world probability. In other words, the market is incomplete, so one cannot replicate every payoff. Different choices of the market price of volatility risk and jump risk (if extended) can lead to families of risk-neutral measures, all of which ensure no arbitrage but yield different pricing implications for claims.
  In such an incomplete setting, the minimal-entropy σ-martingale measure (from Definition 4) provides a systematic way to single out a preferred measure among the many. By penalizing deviations from the real-world probability through the relative entropy functional, this approach selects the measure that is “closest” (in Kullback–Leibler divergence) to the original distribution. Hence, the resulting risk measure assigns a unique fair price (or risk assessment) to each claim. For more details, see Biagini et al. (2000); Boguslavskaya and Muravey (2016); Hull and White (1987); Pham (2001); Sircar and Zariphopoulou (2004); Sohns (2022); Wiggins (1987).

Unlike the minimal-entropy $\sigma$-martingale measure, the entropic risk always exists.

Theorem 1. 

The entropic risk always exists.

Proof. 

It suffices to show that $e_{\gamma }$ is indeed a finite number.

Let $X\in L^{\infty }\left(\mathcal{F}\right)$. It is straightforward to verify the following:

$$\underset{\gamma \searrow 0}{lim}|e_{\gamma }\left(X\right)| = |{\mathbf{E}}_{\mathbf{P}}[-X]|<\infty$$

and

$$\underset{\gamma \to \infty }{lim}|e_{\gamma }\left(X\right)| = |\mathrm{ess}\sup (-X)|<\infty$$

because X is almost surely bounded.

Since $e_{\gamma }\left(X\right)$ is increasing in $\gamma$, we conclude for a fixed $\tilde{\gamma }>0$:

$$-\infty <\underset{\gamma \searrow 0}{lim}{e}_{\gamma }\left(X\right)\le e_{\tilde{\gamma }}\left(X\right)\le \underset{\gamma \to \infty }{lim}{e}_{\gamma }\left(X\right)<\infty .$$

  □

While the minimal-entropy $\sigma$-martingale measure is new in the literature, the notion of the minimal-entropy martingale measure has been extensively studied in the literature. The definition of the latter is analog to ours, but instead of equivalent $\sigma$-martingale measures, one minimizes the entropy among all equivalent martingale measures (see, for example, Grandits and Rheinländer (2002); Lee and Rheinländer (2013); Miyahara (1999); Rheinländer and Steiger (2006); Schweizer (2010)). There are numerous criteria in the literature for the existence of the minimal-entropy martingale measure. Most of these criteria impose strong conditions on the stochastic process $\widehat{S}$. A prominent example is when $\widehat{S}$ is a Lévy process (Fujiwara 2004, Theorem 3.1). Another simple result is the existence of the minimal-entropy martingale measure if $\widehat{S}$ is bounded as shown in (Frittelli 2000, Theorem 2.1). Other results depend on the specific form of the Radon–Nikodym derivative (Grandits and Rheinländer 2002) or are applicable only in discrete time settings (Föllmer and Schied 2011, Corollary 3.27).

Things are a bit easier for the minimal-entropy $\sigma$-martingale measure. We now state and prove a theorem showing that, under a mild condition—namely, the existence of at least one $\sigma$-martingale measure $\mathbf{Q}$ with finite $H(\mathbf{Q}\mid \mathbf{P})$—one confirms the existence (and uniqueness) of a minimal-entropy $\sigma$-martingale measure. This holds even if $\widehat{S}$ is not locally bounded and even if no equivalent local martingale measures exist.

For the proof, we need some lemmas.

Lemma 1. 

Let $\mathcal{P}$ be the set of probability measures on $(\mathsf{\Omega },\mathcal{A})$. We have the following:

(a) 

$H(\mathbf{Q}\mid \mathbf{P})\ge 0$. Furthermore, $H(\mathbf{Q}\mid \mathbf{P})=0$ if and only if $\mathbf{Q}=\mathbf{P}$.

(b) 

The mapping

$$:\left\{\begin{array}{ccc}\mathcal{P}& \to & \mathbb{R}\hfill \\ \mathbf{Q}& \mapsto & H(\mathbf{Q}\mid \mathbf{P})\hfill \end{array}\right.$$

is convex and strictly convex on the set of probability measures that are absolutely continuous with respect to $\mathbf{P}$.

Proof. 

We have the following:

$$H(\mathbf{Q}\mid \mathbf{P})=\left\{\begin{array}{cc}{\mathbf{E}}_{\mathbf{P}}\left[{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\log \left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right)\right]\hfill & \mathrm{for}\mathbf{Q}\ll \mathbf{P},\hfill \\ \infty \hfill & \mathrm{else}.\hfill \end{array}\right.$$

Hence, it is reasonable to look at the function $\varphi$, which we define as a continuous extension of the function $x\mapsto x\log x$ to $[0,\infty )$. Because of $\varphi {\left(x\right)}^{″}={\displaystyle \frac{1}{x}}$ and Theorem A1, we have ${\varphi }^{″}\left(x\right)>0$ for $x>0$, which proves the strict convexity of $\varphi$.

It is easy to see that the function $x\mapsto x\log x-x$ takes its minimum at $x=1$. Hence, we have $x\log x-x\ge -1$, and thus,

$$\varphi \left(x\right)\ge x-1\mathrm{for}\mathrm{all}x\ge 0.$$

(2)

(a)

According to Equation (2), we have, for $\mathbf{Q}\ll \mathbf{P}$, the following:

$$\begin{array}{cc}\hfill H(\mathbf{Q}\mid \mathbf{P})& ={\mathbf{E}}_{\mathbf{P}}\left[{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\log \left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right)\right]\hfill \\ \hfill & \ge {\mathbf{E}}_{\mathbf{P}}\left[{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}-1\right]={\mathbf{E}}_{\mathbf{P}}\left[{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right]-1\hfill \\ \hfill & ={\mathbf{E}}_{\mathbf{Q}}\left[1\right]-1=0.\hfill \end{array}$$

Now, let $H(\mathbf{Q}\mid \mathbf{P})=0$. We must show that the equality $\mathbf{Q}=\mathbf{P}$ follows. We have the following:

$$\begin{array}{cc}\hfill 0& =H(\mathbf{Q}\mid \mathbf{P})={\mathbf{E}}_{\mathbf{P}}\left[{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\log \left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right)\right]\hfill \\ \hfill & ={\mathbf{E}}_{\mathbf{P}}\left[{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\log \left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right)\right]-\underset{=0}{\underbrace{{\mathbf{E}}_{\mathbf{P}}\left[{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}-1\right]}}\hfill \\ \hfill & ={\mathbf{E}}_{\mathbf{P}}\left[\underset{\stackrel{Equation\left(\right)}{\ge }0}{\underbrace{{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\log \left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right)-{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}+1}}\right].\hfill \end{array}$$

Hence, $\mathbf{P}$—almost everywhere, we have ${\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\log \left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right)={\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}-1$. Now, it follows that ${\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}=1\mathbf{P}$—almost surely. If $\mathbf{Q}$ is not absolutely continuous with respect to $\mathbf{P}$, the statement is obvious.

(b)

The statement is obvious for probability measures that are not absolutely continuous with respect to $\mathbf{P}$. So, we focus on the case where ${\mathbf{Q}}_1,{\mathbf{Q}}_2\ll \mathbf{P}$. For ${\lambda }_1,{\lambda }_2\ge 0$ with ${\lambda }_1+{\lambda }_2=1$, we have the following:

$$\begin{array}{cc}\hfill H({\lambda }_1{\mathbf{Q}}_1+{\lambda }_2{\mathbf{Q}}_2\mid \mathbf{P})& ={\mathbf{E}}_{\mathbf{P}}\left[\varphi \left({\lambda }_1{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_1}{\mathrm{d}\mathbf{P}}}+{\lambda }_2{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_2}{\mathrm{d}\mathbf{P}}}\right)\right]\hfill \\ \hfill & \le {\mathbf{E}}_{\mathbf{P}}\left[{\lambda }_1\varphi \left({\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_1}{\mathrm{d}\mathbf{P}}}\right)+{\lambda }_2\varphi \left({\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_2}{\mathrm{d}\mathbf{P}}}\right)\right]\hfill \\ \hfill & ={\lambda }_1{\mathbf{E}}_{\mathbf{P}}\left[\varphi \left({\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_1}{\mathrm{d}\mathbf{P}}}\right)\right]+{\lambda }_2{\mathbf{E}}_{\mathbf{P}}\left[\varphi \left({\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_2}{\mathrm{d}\mathbf{P}}}\right)\right]\hfill \\ \hfill & ={\lambda }_{1}H({\mathbf{Q}}_1\mid \mathbf{P})+{\lambda }_{2}H({\mathbf{Q}}_2\mid \mathbf{P}).\hfill \end{array}$$

Thus, strict convexity follows.  □

Lemma 2. 

Suppose there exists a measure, ${\mathbf{Q}}_0\in {\mathcal{M}}_{\sigma }^{ac}$, with

$$H({\mathbf{Q}}_0\mid \mathbf{P})<\infty \mathit{and}H({\mathbf{Q}}_0\mid \mathbf{P})\le H(\mathbf{Q}\mid \mathbf{P})\mathit{for}\mathit{all}\mathbf{Q}\in {\mathcal{M}}_{\sigma }^{ac}.$$

Then, ${\mathbf{Q}}_0$ is the unique minimal-entropy σ-martingale measure.

Proof. 

Define the set

$$\mathsf{\Gamma }:=\{\mathbf{Q}\in {\mathcal{M}}_{\sigma }^{ac};H(\mathbf{Q}\mid \mathbf{P})<\infty \}.$$

By assumption, $\mathsf{\Gamma }$ is nonempty because ${\mathbf{Q}}_0\in \mathsf{\Gamma }$. Thus, we have

$$I(\mathsf{\Gamma },\mathbf{P}):=\underset{\mathbf{Q}\in \mathsf{\Gamma }}{inf}H(\mathbf{Q}\mid \mathbf{P})<\infty .$$

Next, we show that $\mathsf{\Gamma }$ is convex and closed in total variation:

For any ${\mathbf{Q}}_1,{\mathbf{Q}}_2\in \mathsf{\Gamma }$, their convex combination $\alpha {\mathbf{Q}}_1+(1-\alpha ){\mathbf{Q}}_2$, $\alpha \in [0,1]$, is also in $\mathsf{\Gamma }$. Indeed, convex combinations preserve both the finite entropy condition (due to convexity of relative entropy) and the $\sigma$-martingale property as $\sigma$-martingales form a vector space.

The set ${\mathcal{M}}_{\sigma }^{ac}$ can be expressed through linear constraints as

$${\mathcal{M}}_{\sigma }^{ac}=\{\mathbf{Q}\ll \mathbf{P}:{\mathbf{E}}_{\mathbf{Q}}\left[f_i\right]=0\mathrm{for}\mathrm{a}\mathrm{suitable}\mathrm{family}\mathrm{of}\mathrm{bounded}\mathrm{measurable}\mathrm{functions}{f}_i\}.$$

Sets defined via linear constraints of the form $\{\mathbf{Q}\ll \mathbf{P}:{\mathbf{E}}_{\mathbf{Q}}\left[f_i\right]=0\}$ are closed in total variation topology. Intersecting with the closed set of measures having finite entropy preserves this closedness.

By applying Csiszár’s Theorem A6, we conclude that the set $\mathsf{\Gamma }$ has a unique I-projection $\tilde{\mathbf{Q}}$ of $\mathbf{P}$, which satisfies

$$H(\tilde{\mathbf{Q}}\mid \mathbf{P})=\underset{\mathbf{Q}\in \mathsf{\Gamma }}{inf}H(\mathbf{Q}\mid \mathbf{P}).$$

By hypothesis, ${\mathbf{Q}}_0$ already satisfies this minimality, so it must coincide with $\tilde{\mathbf{Q}}$. Thus, uniqueness follows directly from the strict convexity of relative entropy.

Finally, we verify that ${\mathbf{Q}}_0$ is equivalent to $\mathbf{P}$. Suppose this is not the case. Then, there exists a $\tilde{\mathbf{Q}}\in \mathsf{\Gamma }$, which is equivalent to $\mathbf{P}$, as given by hypothesis. If ${\mathbf{Q}}_0$ is not equivalent to $\mathbf{P}$, its support is strictly smaller than that of $\tilde{\mathbf{Q}}$, which contradicts minimality via Csiszár–Gibbs inequality (Theorem A7). Hence, ${\mathbf{Q}}_0\sim \mathbf{P}$.

Thus, ${\mathbf{Q}}_0\in {\mathcal{M}}_{\sigma }^{ac}$, is equivalent to $\mathbf{P}$, and minimizes entropy. Therefore, it is the unique minimal-entropy $\sigma$-martingale measure.  □

In the upcoming proof of Theorem 2 (the existence of the minimal-entropy $\sigma$-martingale measure), we construct a sequence $\left\{{\mathbf{Q}}_n\right\}$ in the class of absolutely continuous $\sigma$-martingale measures (each having finite entropy). We then take appropriate convex combinations (or a subsequence) whose Radon–Nikodým derivatives converge in $L^1\left(\mathbf{P}\right)$. A priori, it is not obvious that the limit measure ${\mathbf{Q}}_0$ remains a $\sigma$-martingale measure. The following Lemma 3 guarantees exactly this: the limiting measure ${\mathbf{Q}}_0$ is still a $\sigma$-martingale measure.

Lemma 3. 

Let $(\mathsf{\Omega },\mathcal{F},\mathbf{P})$ be a probability space, and let $\tilde{\mathbf{Q}},{\mathbf{Q}}^1,{\mathbf{Q}}^2,\dots$ be probability measures on $(\mathsf{\Omega },\mathcal{F})$ with $\tilde{\mathbf{Q}},{\mathbf{Q}}^n\sim \mathbf{P}$ for each n and

$${\displaystyle \frac{\mathrm{d}{\mathbf{Q}}^n}{\mathrm{d}\mathbf{P}}}\underset{n\to \infty }{\overset{L^1\left(\mathbf{P}\right)}{\to }}{\displaystyle \frac{\mathrm{d}\tilde{\mathbf{Q}}}{\mathrm{d}\mathbf{P}}}.$$

Suppose X is a semimartingale that is a σ-martingale under each measure ${\mathbf{Q}}^n$. Then, X is also a σ-martingale under $\tilde{\mathbf{Q}}$.

Proof. 

Since X is a $\sigma$-martingale under each measure ${\mathbf{Q}}^n$, there exists, for every n, a suitable family of predictable sets making each localized piece of X a uniformly integrable martingale under ${\mathbf{Q}}^n$. By reindexing or combining these families appropriately, we can select a single sequence ${\left(D_k\right)}_{k\in \mathbb{N}}$ of predictable sets for which ${\mathbb{1}}_{D_k}•X$ is a uniformly integrable martingale simultaneously under every measure ${\mathbf{Q}}^n$.

To show that ${\mathbb{1}}_{D_k}•X$ remains a uniformly integrable martingale under $\tilde{\mathbf{Q}}$, fix arbitrary times $s<t$ and an arbitrary set, $A\in {\mathcal{F}}_s$. Define the bounded random variable

$$Z={\mathbb{1}}_A\left({({\mathbb{1}}_{D_k}•X)}_t-{({\mathbb{1}}_{D_k}•X)}_s\right).$$

Since ${\mathbb{1}}_{D_k}•X$ is a martingale under each ${\mathbf{Q}}^n$, we immediately have ${\mathbf{E}}_{{\mathbf{Q}}^n}\left[Z\right]=0$ for every n. Using the convergence of the Radon–Nikodým derivatives in $L^1\left(\mathbf{P}\right)$ and boundedness of Z, we get

$${\mathbf{E}}_{\tilde{\mathbf{Q}}}\left[Z\right]={\mathbf{E}}_{\mathbf{P}}\left[Z{\displaystyle \frac{\mathrm{d}\tilde{\mathbf{Q}}}{\mathrm{d}\mathbf{P}}}\right]={\mathbf{E}}_{\mathbf{P}}\left[\underset{n\to \infty }{lim}Z{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}^n}{\mathrm{d}\mathbf{P}}}\right]=\underset{n\to \infty }{lim}{\mathbf{E}}_{\mathbf{P}}\left[Z{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}^n}{\mathrm{d}\mathbf{P}}}\right]=\underset{n\to \infty }{lim}{\mathbf{E}}_{{\mathbf{Q}}^n}\left[Z\right]=0.$$

This equality implies that the conditional expectation of ${({\mathbb{1}}_{D_k}•X)}_t$, given ${\mathcal{F}}_s$ under $\tilde{\mathbf{Q}}$ equals ${({\mathbb{1}}_{D_k}•X)}_s$, establishing the martingale property for each localized piece. Since this argument applies to each predictable set, $D_k$, it follows that X is a $\sigma$-martingale under $\tilde{\mathbf{Q}}$.   □

In a discrete-time market setting, the $\sigma$-martingale condition reduces to the usual martingale requirement on each time step. Then, the statement of the lemma is more transparent: if each ${\mathbf{Q}}^n$ makes X a martingale (so ${\mathbf{E}}_{{\mathbf{Q}}^n}\left[X_{t+1}\right|{\mathcal{F}}_t]=X_t$), and if ${\mathbf{Q}}^n\to \tilde{\mathbf{Q}}$ in $L^1$, then X remains a martingale under $\tilde{\mathbf{Q}}$. The “unified localizing sets” in discrete time simply become the entire time index set. The continuous-time version is conceptually the same but requires working with a single predictable sequence, $\left(D_k\right)$, across all ${\mathbf{Q}}^n$ for the localization.

Theorem 2. 

Suppose there exists at least one measure, ${\mathbf{Q}}_1\in {\mathcal{M}}_{\sigma }$, with finite entropy, $H({\mathbf{Q}}_1\mid \mathbf{P})<\infty$. Then, the unique minimal-entropy σ-martingale measure ${\mathbf{Q}}^E\in {\mathcal{M}}_{\sigma }$ exists.

Proof. 

We define

$${\mathcal{M}}^0=\{\mathbf{Q}\in {\mathcal{M}}_{\sigma }^{ac}:H(\mathbf{Q}\mid \mathbf{P})<\infty \}.$$

By assumption, there exists a sequence, ${\left\{{\mathbf{Q}}_n\right\}}_{n\ge 1}\in {\mathcal{M}}^0$, such that ${\left(H({\mathbf{Q}}_n\mid \mathbf{P})\right)}_{n\in \mathbb{N}}$ is decreasing and satisfies

$$H({\mathbf{Q}}_n\mid \mathbf{P})\searrow \underset{\mathbf{Q}\in {\mathcal{M}}_{\sigma }^{ac}}{inf}H(\mathbf{Q}\mid \mathbf{P}).$$

Because the sequence of the relative entropies is monotone, we obtain for the convex function $\varphi \left(x\right)=x\log x$

$$\underset{n\ge 1}{sup}{\mathbf{E}}_{\mathbf{P}}\left(\varphi \left(\frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}\right)\right)\le H({\mathbf{Q}}_1\mid \mathbf{P}),$$

and, therefore, the sequence $\left(\frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}\right)$ is uniformly integrable according to Theorem A3. Per Theorem A4, there exists a subsequence, ${\left(\frac{\mathrm{d}{\mathbf{Q}}_{n_k}}{\mathrm{d}\mathbf{P}}\right)}_{k\ge 1}$, which converges weakly in $L^1$. However, according to Mazur’s lemma (Theorem A5), there exists a sequence, ${\left(\tilde{\frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}}\right)}_{n\ge 1}$, consisting of convex combinations

$$\tilde{{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}}}=\sum _{k=n}^{N_n}{\lambda }_{n_k}{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_{n_k}}{\mathrm{d}\mathbf{P}}}\left({\lambda }_{n_k}\ge 0,\sum _k{\lambda }_{n_k}=1\right),$$

(3)

which converges in $L^1\left(\mathbf{P}\right)$ to some ${\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_0}{\mathrm{d}\mathbf{P}}}$.

The measure ${\tilde{\mathbf{Q}}}_n$ is uniquely defined if we interpret $\tilde{\frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}}$ as a Radon–Nikodym derivative. One can see directly that ${\tilde{\mathbf{Q}}}_n\ll \mathbf{P}$. By assumption, $\widehat{S}$ is a $\sigma$-martingale with respect to all ${\mathbf{Q}}_n$ (and hence also with reference to all ${\mathbf{Q}}_{n_k}$). Now, we want to show that it is also a $\sigma$-martingale under ${\tilde{\mathbf{Q}}}_n$. To prove this, let ${\left(D_i\right)}_i$ be a sequence of predictable sets, such that ${\mathbb{1}}_{D_i}•\widehat{S}$ is a martingale under a fixed ${\mathbf{Q}}_{n_k}$. For each $A\in {\mathcal{F}}_s$,

$$\begin{array}{cc}\hfill {\int }_A\left({\mathbb{1}}_{D_i}•{\widehat{S}}_t-{\mathbb{1}}_{D_i}•{\widehat{S}}_s\right)\mathrm{d}{\tilde{\mathbf{Q}}}_n& ={\int }_A\left({\mathbb{1}}_{D_i}•{\widehat{S}}_t-{\mathbb{1}}_{D_i}•{\widehat{S}}_s\right)\tilde{{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}}}\mathrm{d}\mathbf{P}\hfill \\ \hfill & ={\int }_A\left({\mathbb{1}}_{D_i}•{\widehat{S}}_t-{\mathbb{1}}_{D_i}•{\widehat{S}}_s\right)\left(\sum _{k=n}^{N_n}{\lambda }_{n_k}{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_{n_k}}{\mathrm{d}\mathbf{P}}}\right)\mathrm{d}\mathbf{P}\hfill \\ \hfill & =\sum _{k=n}^{N_n}{\lambda }_{n_k}{\int }_A\left({\mathbb{1}}_{D_i}•{\widehat{S}}_t-{\mathbb{1}}_{D_i}•{\widehat{S}}_s\right){\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_{n_k}}{\mathrm{d}\mathbf{P}}}\mathrm{d}\mathbf{P}\hfill \\ \hfill & =\sum _{k=n}^{N_n}{\lambda }_{n_k}{\int }_A\left({\mathbb{1}}_{D_i}•{\widehat{S}}_t-{\mathbb{1}}_{D_i}•{\widehat{S}}_s\right)\mathrm{d}{\mathbf{Q}}_{n_k}=0,\hfill \end{array}$$

because ${\mathbf{Q}}_{n_k}\in {\mathcal{M}}_{\sigma }^{ac}$. Thus, ${\mathbb{1}}_{D_i}•\widehat{S}$ is also a martingale under ${\tilde{\mathbf{Q}}}_n$; hence, $\widehat{S}$ is a $\sigma$ martingale under ${\tilde{\mathbf{Q}}}_n$.

Moreover, $\tilde{\frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}}$ converges in $L^1$ to $\frac{\mathrm{d}{\mathbf{Q}}_0}{\mathrm{d}\mathbf{P}}$, and so, per Lemma 3 we have ${\mathbf{Q}}_0\in {\mathcal{M}}_{\sigma }^{ac}$.

Because of the convexity of the relative entropy (Lemma 1), Equation (3), and since ${\left(H({\mathbf{Q}}_n\mid \mathbf{P})\right)}_{n\in \mathbb{N}}$ is a decreasing sequence, we have

$$\begin{array}{cc}\hfill H({\tilde{\mathbf{Q}}}_n\mid \mathbf{P})& =H\left(\sum _{k=n}^{N_n}{\lambda }_k{\mathbf{Q}}_{n_k}|\mathbf{P}\right)\hfill \\ \hfill & \le \sum _{k=n}^{N_n}{\lambda }_{k}H({\mathbf{Q}}_n\mid \mathbf{P})\le \underset{k\in \{n,\dots ,N_n\}}{max}H({\mathbf{Q}}_{n_k}\mid \mathbf{P})=H({\mathbf{Q}}_n\mid \mathbf{P}).\hfill \end{array}$$

Thus, by applying Fatou’s Lemma, we obtain

$$\begin{array}{cc}\hfill H({\mathbf{Q}}_0\mid \mathbf{P})& ={\mathbf{E}}_{\mathbf{P}}\left[\underset{n\to \infty }{lim\; inf}\tilde{{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}}}\log \left(\tilde{{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}}}\right)\right]\le \underset{n\to \infty }{lim\; inf}{\mathbf{E}}_{\mathbf{P}}\left[\tilde{{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}}}\log \left(\tilde{{\displaystyle \frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}}}\right)\right]\hfill \\ \hfill & =\underset{n\to \infty }{lim\; inf}H(\widetilde{{\mathbf{Q}}_n}\mid \mathbf{P})\le \underset{n\to \infty }{lim\; inf}H({\mathbf{Q}}_n\mid \mathbf{P}).\hfill \end{array}$$

Since $H({\mathbf{Q}}_n\mid \mathbf{P})$ is decreasing, it follows that $H({\mathbf{Q}}_0\mid \mathbf{P})$ is already minimal. Now, all conditions of Lemma 2 are satisfied, so we conclude that ${\mathbf{Q}}_0$ is equivalent to $\mathbf{P}$, proving the existence of the minimal-entropy martingale measure.

Uniqueness follows from Lemma 1 and the strict convexity of $H(·\mid \mathbf{P})$.   □

Remark 3. 

Theorem 2 shows that the “entropy-minimizing” approach extends smoothly to σ-martingale models: as soon as there is one finite-entropy σ-martingale measure, there must be a unique one of minimal entropy, even if $\widehat{S}$ is unbounded or not locally bounded. This measure can be used for entropy-based valuation, risk measurement, or other applications. The local martingale approach, in contrast, might be impossible if ${\mathcal{M}}_{\mathrm{loc}}=⌀$.

3. Convex and Coherent Risk Measures

In this section, we define the notions of monetary, convex, and coherent risk measures. We also highlight the relationships among translation invariance, monotonicity, convexity, positive homogeneity, and subadditivity.

Definition 5. 

Let $\mathcal{X}$ be the space of linear bounded, real-valued random variables.

(a) 

A mapping, $\rho :\mathcal{X}\to \mathbb{R}\cup \{+\infty \}$, is called a monetary risk measure if it satisfies for all $X,Y\in \mathcal{X}$

• Monotonicity: if $X\le Y$ almost surely, then $\rho \left(X\right)\ge \rho \left(Y\right)$.
• Translation invariance: for all $m\in \mathbb{R}$, $\rho (X+m)=\rho \left(X\right)-m$.

(b) 

The monetary risk measure ρ is called convex if it also satisfies

• Convexity: for all $X,Y\in \mathcal{X}$ and $\lambda \in [0,1]$,

  $$\rho \left(\lambda X+(1-\lambda )Y\right)\le \lambda \rho \left(X\right)+(1-\lambda )\rho \left(Y\right).$$

(c) 

A convex risk measure, ρ, is called coherent if, in addition to monotonicity and translation invariance, it satisfies

• Positive homogeneity: for all $\alpha >0$,

  $$\rho \left(\alpha X\right)=\alpha \rho \left(X\right).$$
• Subadditivity: for all $X,Y\in \mathcal{X}$,

  $$\rho (X+Y)\le \rho \left(X\right)+\rho \left(Y\right).$$

Theorem 3. 

Let ρ be a monetary risk measure on a linear space of random variables $\mathcal{X}$.

(a) 

We say that ρ is normalized if $\rho \left(0\right)=0$. In particular, if ρ is positively homogeneous, then it follows that $\rho \left(0\right)=0$ automatically.

(b) 

Suppose ρ satisfies translation invariance and monotonicity (as in Definition 5). Then, any two of the following three properties imply the remaining third:

• Convexity;
• Positive homogeneity;
• Subadditivity.

For a detailed discussion and proof, see (Föllmer and Schied 2011, chp. 4).

Theorem 4. 

The minimal-entropy martingale measure and the entropic risk are both convex risk measures. The minimal-entropy martingale measure is also a coherent risk measure.

Proof. 

We start with the minimal-entropy martingale measure. It suffices to prove the coherence. The convexity follows from the fact that positive homogeneity and subadditivity imply convexity.

• Monotonicity. Let $X,Y\in L^{\infty }\left(\mathcal{F}\right)$ with $\mathbf{P}[X\ge Y]=1$. We have

  $${\rho }^E\left(X\right)={\mathbf{E}}_{{\mathbf{Q}}^E}[-X]\le {\mathbf{E}}_{{\mathbf{Q}}^E}[-Y]={\rho }^E\left(Y\right).$$
• Cash translability. This simply follows from

  $${\rho }^E(X+c)={\rho }^E(-X-c)={\rho }^E(-X)-c={\rho }^E\left(X\right)-c.$$
• Positive homogeneity and subadditivity. These follow directly from the linearity of the expectation.

We now address the entropic risk.

• Monotonicity. From $X\ge Y$, almost surely, ${\mathbf{E}}_{\mathbf{Q}}[-Y]\ge {\mathbf{E}}_{\mathbf{Q}}[-X]$ follows for all $\mathbf{Q}\in {\mathcal{P}}_L$, and hence

  $$\mathbf{E}[-Y]-{\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})\ge \mathbf{E}[-X]-{\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})$$

  for all $\mathbf{Q}$. Thus, $e_{\gamma }\left(X\right)\le e_{\gamma }\left(Y\right)$.
• Cash translability. With

  $$\mathbf{E}[-(X+c)]-{\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})=\mathbf{E}[-X]-{\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})-c,$$

  we obtain $e_{\gamma }(X+c)=e_{\gamma }\left(X\right)-c$.
• Convexity. We have

  $$\begin{array}{cc}\hfill & \mathbf{E}[-(\lambda X+(1-\lambda )Y)]-{\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})\hfill \\ \hfill & =\lambda \mathbf{E}[-X]+(1-\lambda )\mathbf{E}[-Y]-\lambda {\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})-(1-\lambda ){\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})\hfill \\ \hfill & =\lambda \left(\mathbf{E}[-X]-{\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})\right)+(1-\lambda )\left(\mathbf{E}[-Y]-{\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})\right).\hfill \end{array}$$
  And since, for general functions $f,g$ and $c_1,c_2\in \mathbb{R}$, we have

  $$\underset{x}{sup}(c_{1}f\left(x\right)+c_{2}g\left(x\right))\le c_1\underset{x}{sup}f\left(x\right)+c_2\underset{x}{sup}g\left(x\right),$$

  the result follows.

  □

In general, the entropic risk is not a coherent risk measure, even though it is additive for independent positions. However, there is also a coherent version of the entropic risk measure described in Föllmer and Knispel (2011), which is quite similar to the two risk measures we are examining here.

The coherence of a risk measure is particularly significant in the context of financial regulation. Regulators, such as the Basel Committee on Banking Supervision, emphasize capital requirements that are consistent with diversification benefits and penalize concentration risk (Basel Committee on Banking Supervision 2010). In particular, the subadditivity property (cf. Definition 5 and the subsequent theorem establishing that ${\rho }^E$ is coherent) ensures that merging two portfolios $\mathcal{X}$ and $\mathcal{Y}$ never results in a higher combined capital charge than the sum of their individual charges. This requirement aligns with Basel III’s core principle that holistic risk management should not be discouraged by artificially additive capital rules.

When we specialize to the minimal-entropy risk measure ${\rho }^E$ introduced in Section 2 and shown in Section 3 to be coherent, we see that it satisfies precisely these subadditivity and positive-homogeneity conditions. Concretely, if a bank adopts ${\rho }^E$ to quantify its market or counterparty exposures, then merging business units or pooling risk factors would not inflate the aggregate capital requirement beyond the sum of stand-alone capital allocations. This feature could be advantageous in an internal models approach to Basel III capital calculation, provided that the bank can justify the assumptions on market completeness or the existence of $\sigma$-martingale measures with finite entropy (see Theorem 2). Theoretically, replacing non-coherent measures (such as certain forms of value at risk) with a coherent measure like ${\rho }^E$ better captures the true diversification effect.

Nonetheless, adopting a coherent measure in practice also raises considerations:

• Model complexity. Calculating ${\rho }^E$ may require advanced numerical methods to identify or approximate the minimal-entropy $\sigma$-martingale measure.
• Data intensity. Banks must maintain high-quality market data and stress scenarios to ensure robust parameter estimation under ${\rho }^E$.
• Regulatory validation. Any internal model, including entropy-based approaches, must pass supervisory review, which entails transparency on modeling assumptions and backtesting (Basel Committee on Banking Supervision 2010).

Finally, to better understand these risk measures, we explore their relationship with the real-world probability measure. The following lemma will be helpful.

Lemma 4. 

Let $\mathbf{Q},\mathbf{R}\in {\mathcal{M}}_{\sigma }^{ac}$ be absolutely continuous σ–martingale measures, i.e., each of ${\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}},{\displaystyle \frac{\mathrm{d}\mathbf{R}}{\mathrm{d}\mathbf{P}}}$ is in $L^1\left(\mathbf{P}\right)$, and assume that $H(\mathbf{Q}\mid \mathbf{P})$ and $H(\mathbf{R}\mid \mathbf{P})$ are both finite. Then, for any bounded random variable, $X\in L^{\infty }\left(\mathcal{F}\right)$, we have

$$\left|{\mathbf{E}}_{\mathbf{Q}}\left[X\right]-{\mathbf{E}}_{\mathbf{R}}\left[X\right]\right|\le {\parallel X\parallel }_{\infty }\left(\sqrt{2H(\mathbf{Q}\mid \mathbf{P})}+\sqrt{2H(\mathbf{R}\mid \mathbf{P})}\right).$$

(4)

Proof. 

First, note that, for any probability measures $\mathbf{Q},\mathbf{R}$ (absolutely continuous w.r.t. P),

$$\begin{array}{cc}\hfill \left|{\mathbf{E}}_{\mathbf{Q}}\left[X\right]-{\mathbf{E}}_{\mathbf{R}}\left[X\right]\right|& =\left|{\mathbf{E}}_{\mathbf{P}}\left[X\frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}\right]-{\mathbf{E}}_{\mathbf{P}}\left[X\frac{\mathrm{d}\mathbf{R}}{\mathrm{d}\mathbf{P}}\right]\right|=\left|{\mathbf{E}}_{\mathbf{P}}\left[X\left(\frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}-\frac{\mathrm{d}\mathbf{R}}{\mathrm{d}\mathbf{P}}\right)\right]\right|\hfill \\ \hfill & \le {\parallel X\parallel }_{\infty }{\mathbf{E}}_{\mathbf{P}}\left[\left|\frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}-\frac{\mathrm{d}\mathbf{R}}{\mathrm{d}\mathbf{P}}\right|\right]={\parallel X\parallel }_{\infty }{\parallel \mathbf{Q}-\mathbf{R}\parallel }_{\mathrm{TV}},\hfill \end{array}$$

where ${\parallel \mathbf{Q}-\mathbf{R}\parallel }_{\mathrm{TV}}$ denotes the total variation distance:

$${\parallel \mathbf{Q}-\mathbf{R}\parallel }_{\mathrm{TV}}={\mathbf{E}}_{\mathbf{P}}\left[\left|\frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}-\frac{\mathrm{d}\mathbf{R}}{\mathrm{d}\mathbf{P}}\right|\right].$$

Next, according to the triangle inequality in total variation,

$${\parallel \mathbf{Q}-\mathbf{R}\parallel }_{\mathrm{TV}}\le {\parallel \mathbf{Q}-\mathbf{P}\parallel }_{\mathrm{TV}}+{\parallel \mathbf{P}-\mathbf{R}\parallel }_{\mathrm{TV}}.$$

Finally, we invoke the Pinsker inequality Theorem A8 and obtain

$$\begin{array}{cc}\hfill {\parallel \mathbf{Q}-\mathbf{P}\parallel }_{\mathrm{TV}}^2& \le 2H(\mathbf{Q}\mid \mathbf{P}),\hfill \\ \hfill {\parallel \mathbf{R}-\mathbf{P}\parallel }_{\mathrm{TV}}^2& \le 2H(\mathbf{R}\mid \mathbf{P}),\hfill \end{array}$$

whenever $H(\mathbf{Q}\mid \mathbf{P})$ (resp. $H(\mathbf{R}\mid \mathbf{P})$) is finite. Hence,

$${\parallel \mathbf{Q}-\mathbf{P}\parallel }_{\mathrm{TV}}\le \sqrt{2H(\mathbf{Q}\mid \mathbf{P})},{\parallel \mathbf{R}-\mathbf{P}\parallel }_{\mathrm{TV}}\le \sqrt{2H(\mathbf{R}\mid \mathbf{P})}.$$

Putting these pieces together yields

$$\begin{array}{cc}\hfill {\parallel \mathbf{Q}-\mathbf{R}\parallel }_{\mathrm{TV}}& \le {\parallel \mathbf{Q}-\mathbf{P}\parallel }_{\mathrm{TV}}+{\parallel \mathbf{P}-\mathbf{R}\parallel }_{\mathrm{TV}}\le \sqrt{2H(\mathbf{Q}\mid \mathbf{P})}+\sqrt{2H(\mathbf{R}\mid \mathbf{P})}.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill |{\mathbf{E}}_{\mathbf{Q}}\left[X\right]-{\mathbf{E}}_{\mathbf{R}}\left[X\right]|& \le {\parallel X\parallel }_{\infty }{\parallel \mathbf{Q}-\mathbf{R}\parallel }_{\mathrm{TV}}\le {\parallel X\parallel }_{\infty }\left(\sqrt{2H(\mathbf{Q}\mid \mathbf{P})}+\sqrt{2H(\mathbf{R}\mid \mathbf{P})}\right).\hfill \end{array}$$

That completes the proof.   □

Remark 4. 

Inequality (4) shows that two absolutely continuous σ-martingale measures $\mathbf{Q}$ and $\mathbf{R}$ with small relative entropy $H(·\mid \mathbf{P})$ yield similar expectations for any bounded payoff, X. In particular, a measure with small entropy is close to $\mathbf{P}$ in total variation distance.

Theorem 5. 

Let $X\in L^{\infty }\left(\mathcal{F}\right)$ be bounded and $\gamma >0$. Then, the entropic risk

$$e_{\gamma }\left(X\right)=\underset{\mathbf{Q}\in {\mathcal{P}}_L}{sup}\left({\mathbf{E}}_{\mathbf{Q}}[-X]-\frac{1}{\gamma }H(\mathbf{Q}\mid \mathbf{P})\right)$$

satisfies the following two-sided bound:

$${\mathbf{E}}_{\mathbf{P}}[-X]\le e_{\gamma }\left(X\right)\le {\mathbf{E}}_{\mathbf{P}}[-X]+{\displaystyle \frac{{\gamma \parallel X\parallel }_{\infty }^2}{2}}.$$

(5)

Proof. 

By taking the specific choice $\mathbf{Q}=\mathbf{P}$ in the definition of $e_{\gamma }\left(X\right)$, we see

$$e_{\gamma }\left(X\right)\ge {\mathbf{E}}_{\mathbf{P}}[-X]-{\displaystyle \frac{1}{\gamma }}H(\mathbf{P}\mid \mathbf{P})={\mathbf{E}}_{\mathbf{P}}[-X],$$

since $H(\mathbf{P}\mid \mathbf{P})=0$. This proves the left inequality in (5).

For any probability measure, $\mathbf{Q}\ll \mathbf{P}$, with finite entropy $H(\mathbf{Q}\mid \mathbf{P})$, we use the triangle bound from total variation Lemma 4 to obtain

$$\left|{\mathbf{E}}_{\mathbf{Q}}[-X]-{\mathbf{E}}_{\mathbf{P}}[-X]\right|\le {\parallel X\parallel }_{\infty }\sqrt{2H(\mathbf{Q}\mid \mathbf{P})}.$$

(6)

Hence,

$${\mathbf{E}}_{\mathbf{Q}}[-X]\le {\mathbf{E}}_{\mathbf{P}}[-X]+{\parallel X\parallel }_{\infty }\sqrt{2H(\mathbf{Q}\mid \mathbf{P})}.$$

Therefore, for any such $\mathbf{Q}$,

$$\begin{array}{cc}\hfill {\mathbf{E}}_{\mathbf{Q}}[-X]-{\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})& \le {\mathbf{E}}_{\mathbf{P}}[-X]+{\parallel X\parallel }_{\infty }\sqrt{2H(\mathbf{Q}\mid \mathbf{P})}-{\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P}).\hfill \end{array}$$

Define $h:=H(\mathbf{Q}\mid \mathbf{P})\ge 0$. Then, we want to maximize ${\parallel X\parallel }_{\infty }\sqrt{2h}-(1/\gamma )h$ over $h\ge 0$. A short calculus argument shows that the maximum occurs at

$$h^{*}={\left(\frac{{\parallel X\parallel }_{\infty }\sqrt{2}}{2/\gamma }\right)}^2={\displaystyle \frac{{\gamma }^2{\parallel X\parallel }_{\infty }^2}{2}},$$

and that the maximum value is ${\displaystyle \frac{{\gamma \parallel X\parallel }_{\infty }^2}{2}}$. Consequently,

$$\begin{array}{c}\hfill \underset{h\ge 0}{sup}\left[{\parallel X\parallel }_{\infty }\sqrt{2h}-\frac{h}{\gamma }\right]={\displaystyle \frac{{\gamma \parallel X\parallel }_{\infty }^2}{2}}.\end{array}$$

Hence, for all $\mathbf{Q}\in {\mathcal{P}}_L$ with $H(\mathbf{Q}\mid \mathbf{P})<\infty$,

$$\begin{array}{c}\hfill {\mathbf{E}}_{\mathbf{Q}}[-X]-\frac{1}{\gamma }H(\mathbf{Q}\mid \mathbf{P})\le {\mathbf{E}}_{\mathbf{P}}[-X]+{\displaystyle \frac{{\gamma \parallel X\parallel }_{\infty }^2}{2}}.\end{array}$$

Taking the supremum over all $\mathbf{Q}\in {\mathcal{P}}_L$ leads to

$$e_{\gamma }\left(X\right)=\underset{\mathbf{Q}}{sup}\left({\mathbf{E}}_{\mathbf{Q}}[-X]-\frac{1}{\gamma }H(\mathbf{Q}\mid \mathbf{P})\right)\le {\mathbf{E}}_{\mathbf{P}}[-X]+{\displaystyle \frac{{\gamma \parallel X\parallel }_{\infty }^2}{2}}.$$

This completes the proof of (5).   □

Corollary 1. 

Let ${\mathbf{Q}}^E$ be the minimal-entropy σ-martingale measure for a (bounded) price process $\widehat{S}$. Then,

$$e_{\gamma }\left(X\right)\ge {\mathbf{E}}_{{\mathbf{Q}}^E}[-X]-{\displaystyle \frac{1}{\gamma }}H\left({\mathbf{Q}}^E\mid \mathbf{P}\right)\mathit{for}\mathit{all}\mathit{bounded}X.$$

(7)

Furthermore, combining this estimate with the two-sided bound of Theorem 5, we have the following:

$$max\left\{{\mathbf{E}}_{\mathbf{P}}[-X],{\rho }^E\left(X\right)-\frac{1}{\gamma }H({\mathbf{Q}}^E\mid \mathbf{P})\right\}\le e_{\gamma }\left(X\right)\le {\mathbf{E}}_{\mathbf{P}}[-X]+{\displaystyle \frac{{\gamma \parallel X\parallel }_{\infty }^2}{2}}.$$

(8)

Proof. 

Since $e_{\gamma }\left(X\right)$ is defined as the supremum ${sup}_{\mathbf{Q}\in {\mathcal{P}}_L}\left\{{\mathbf{E}}_{\mathbf{Q}}[-X]-\frac{1}{\gamma }H(\mathbf{Q}\mid \mathbf{P})\right\},$ it clearly holds that, for any $\mathbf{Q}$ in that admissible set,

$$e_{\gamma }\left(X\right)\ge {\mathbf{E}}_{\mathbf{Q}}[-X]-\frac{1}{\gamma }H(\mathbf{Q}\mid \mathbf{P}).$$

If ${\mathbf{Q}}^E$ (the minimal-entropy $\sigma$-martingale measure) is in ${\mathcal{P}}_L$ (i.e., ${\mathbf{Q}}^E\ll \mathbf{P}$ and $H({\mathbf{Q}}^E\mid \mathbf{P})<\infty$), then simply take $\mathbf{Q}={\mathbf{Q}}^E$ in the supremum. This yields the lower bound (7).

Finally, to deduce (8), observe that $e_{\gamma }\left(X\right)\ge {\mathbf{E}}_{\mathbf{P}}[-X]$ also holds by choosing $\mathbf{Q}=\mathbf{P}$. Then, Theorem 5 provides the upper bound, so taking the maximum of these lower estimates proves (8).   □

Remark 5. 

The assumption that ${\mathbf{Q}}^E\in {\mathcal{P}}_L$ requires that the minimal-entropy measure ${\mathbf{Q}}^E$ be absolutely continuous with reference to $\mathbf{P}$ with finite relative entropy. If, for instance, the market admits no arbitrage of the first kind (NA1), and there exists a finite-entropy EσMM, one can show ${\mathbf{Q}}^E\in {\mathcal{P}}_L$. In that case, (7) provides a non-trivial lower estimate on the entropic risk regarding the minimal-entropy measure.

Remark 6. 

When the risk-aversion parameter γ tends to zero, the penalty factor $\frac{1}{\gamma }$ in the entropic risk measure

$$e_{\gamma }\left(X\right)=\underset{Q\ll P}{sup}\left\{{\mathbf{E}}_Q\left[-X\right]-\frac{1}{\gamma }H\left(Q\mid P\right)\right\}$$

becomes very large. As a result, any measure $\mathbf{Q}\ne \mathbf{P}$ with $H(\mathbf{Q}\mid \mathbf{P})>0$ gets heavily penalized in the objective, so that the supremum is forced increasingly toward $\mathbf{Q}=\mathbf{P}$.

Formally, it can be shown that

$$\underset{\gamma \to 0}{lim}{e}_{\gamma }\left(X\right)={\mathbf{E}}_{\mathbf{P}}\left[-X\right].$$

Economically, this reflects a situation of ultra-strong aversion to deviating from the reference measure $\mathbf{P}$. In the limit $\gamma \to 0$, the only measure not incurring an infinite penalty is $\mathbf{Q}=\mathbf{P}$. Thus, the entropic risk measure collapses to ${\mathbf{E}}_{\mathbf{P}}[-X]$, the plain (negative) expectation under the real-world probability $\mathbf{P}$.

4. Duality

Any convex or coherent risk measure admits a so-called robust representation. This has been discussed in several works, including Artzner et al. (1999); Föllmer and Schied (2011); Frittelli and Gianin (2002).

The usual approach to defining the entropic risk measure is as follows:

$${\displaystyle \frac{1}{\gamma }}\log \mathbf{E}\left[e^{-\gamma X}\right]$$

(9)

and its robust representation, Equation (1), is typically derived using dual representation theorems.

In this paper, we defined the entropic risk using its robust representation and now demonstrate that this definition is equivalent to the traditional one found in the literature. Since the convexity of Equation (9) has not been explicitly shown here, we present an alternative proof.

Theorem 6. 

For $\gamma >0$ and $X\in L^{\infty }\left(\mathcal{F}\right)$, we have the following:

$$\underset{\mathbf{Q}\in {\mathcal{P}}_L}{sup}\left({\mathbf{E}}_{\mathbf{Q}}(-X)-{\displaystyle \frac{1}{\gamma }}H(\mathbf{Q}\mid \mathbf{P})\right)={\displaystyle \frac{1}{\gamma }}\log {\mathbf{E}}_{\mathbf{P}}\left[e^{-\gamma X}\right].$$

Proof. 

Define a new probability measure, ${\mathbf{P}}_X$, as

$${\displaystyle \frac{\mathrm{d}{\mathbf{P}}_X}{\mathrm{d}\mathbf{P}}}={\displaystyle \frac{e^{-\gamma X}}{{\mathbf{E}}_{\mathbf{P}}{e}^{-\gamma X}}}.$$

Then,

$$\begin{array}{cc}\hfill H(\mathbf{Q}\mid \mathbf{P})& ={\mathbf{E}}_{\mathbf{Q}}\left[\log {\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right]\hfill \\ \hfill & ={\mathbf{E}}_{\mathbf{Q}}\log \left[{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}{\mathbf{P}}_X}}{\displaystyle \frac{\mathrm{d}{\mathbf{P}}_X}{\mathrm{d}\mathbf{P}}}\right]\hfill \\ \hfill & ={\mathbf{E}}_{\mathbf{Q}}\left[\log \left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}{\mathbf{P}}_X}}\right)+\log \left(e^{-\gamma X}\right)-\log {\mathbf{E}}_{\mathbf{P}}{e}^{-\gamma X}\right]\hfill \\ \hfill & ={\mathbf{E}}_{\mathbf{Q}}\left[\log {\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}{\mathbf{P}}_X}}\right]+{\mathbf{E}}_{\mathbf{Q}}[-\gamma X]-\log {\mathbf{E}}_{\mathbf{P}}\left[e^{-\gamma X}\right].\hfill \end{array}$$

Substituting this into the optimization problem yields

$$\begin{array}{cc}\hfill & \exp \underset{\mathbf{Q}\in {\mathcal{P}}_L}{sup}\left({\mathbf{E}}_{\mathbf{Q}}[-\gamma X]-H(\mathbf{Q}\mid \mathbf{P})\right)\hfill \\ \hfill & =\exp \left(\underset{\mathbf{Q}\in {\mathcal{P}}_L}{sup}\left({\mathbf{E}}_{\mathbf{Q}}[-\gamma X]-{\mathbf{E}}_{\mathbf{Q}}\left[\log {\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}{\mathbf{P}}_X}}\right]-{\mathbf{E}}_{\mathbf{Q}}[-\gamma X]+\log \left({\mathbf{E}}_{\mathbf{P}}\left[e^{-\gamma X}\right]\right)\right)\right)\hfill \\ \hfill & =\exp \left(\underset{\mathbf{Q}\in {\mathcal{P}}_L}{sup}\left(-{\mathbf{E}}_{\mathbf{Q}}\left[\log {\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}{\mathbf{P}}_X}}\right]+\log \left({\mathbf{E}}_{\mathbf{P}}\left[e^{-\gamma X}\right]\right)\right)\right)\hfill \\ \hfill & ={\mathbf{E}}_{\mathbf{P}}\left[e^{-\gamma X}\right]\exp \left(-\underset{\mathbf{Q}\in {\mathcal{P}}_L}{inf}{\mathbf{E}}_{\mathbf{Q}}\left[\log {\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}{\mathbf{P}}_X}}\right]\right)\hfill \\ \hfill & ={\mathbf{E}}_{\mathbf{P}}\left(e^{-\gamma X}\right),\hfill \end{array}$$

where the last equality follows from ${\mathbf{P}}_X\in {\mathcal{P}}_L$ and Lemma 1.

Taking the logarithm and dividing by $\gamma$ yields the result.   □

A detailed analysis of the dual problem for the minimal-entropy martingale measure is more complex but provides valuable insights into its economic interpretation as an equivalent local martingale measure. The dual problem, in essence, connects minimizing relative entropy to maximizing a specific utility function. Below, we elaborate briefly on this connection.

Let U be a utility function of the form

$$U\left(x\right)=-\exp (-x).$$

We aim to maximize the expected utility of the terminal value of a wealth process by selecting an optimal strategy, $\varphi$, from a set of admissible strategies, $\mathsf{\Phi }$. If the terminal value is represented as $\varphi •{\widehat{S}}_T:={\varphi }_0{\widehat{S}}_0+{\int }_0^T{\varphi }_t\mathrm{d}{\widehat{S}}_t$, then the objective becomes

$$\underset{\varphi \in \mathsf{\Phi }}{sup}\mathbf{E}\left[U(\varphi •{\widehat{S}}_T)\right]=\underset{\varphi \in \mathsf{\Phi }}{sup}\mathbf{E}\left[-\exp \left(-\varphi •{\widehat{S}}_T\right)\right].$$

Under weak conditions, Delbaen et al. (2002) show

$$\underset{\varphi \in \mathsf{\Phi }}{sup}\mathbf{E}\left[-\exp \left(-\varphi •{\widehat{S}}_T\right)\right]=-\exp \left(\underset{\mathbf{Q}\in {\mathcal{M}}_{\mathrm{loc}}}{inf}H(\mathbf{Q}\mid \mathbf{P})\right)$$

(10)

as well as

$${\displaystyle \frac{\mathrm{d}{\mathbf{Q}}^E}{\mathrm{d}\mathbf{P}}}=\exp \left(-H({\mathbf{Q}}^E\mid \mathbf{P})+{\varphi }^{*}•{\widehat{S}}_T\right),$$

where ${\varphi }^{*}$ is the strategy that maximizes the utility in Equation (10).

Further details on this duality can be found in Bellini and Frittelli (2002); Delbaen et al. (2002); Y. M. Kabanov and Stricker (2002).

One key insight from the discussion above and, in particular, Theorem 6 is that the entropic risk measure can be derived by optimizing an exponential-based functional, namely

$$e_{\gamma }\left(X\right)={\displaystyle \frac{1}{\gamma }}\log \left(\mathbf{E}\left[e^{-\gamma X}\right]\right).$$

In principle, this representation is quite general, as it only requires that the exponential moment $\mathbf{E}\left[e^{-\gamma X}\right]$ be finite. Concretely, this exponential expectation is automatically well defined for $X\in L^{\infty }(\mathsf{\Omega },\mathcal{F},\mathbf{P})$ (bounded random variables). For unbounded positions, one needs

$$\mathbf{E}\left[e^{-\gamma X}\right]<\infty ,$$

(11)

which is a standard integrability condition often seen in exponential utility frameworks (Delbaen et al. 2002; Föllmer and Schied 2011).

For many distributions commonly used in finance (e.g., lognormal, normal, or other exponential-family models), the condition (11) is satisfied for all $\gamma >0$. Thus, the entropic risk measure is well defined across a broad range of “light-tailed” or exponentially decaying distributions.

If X can exhibit extremely heavy tails (e.g., some Pareto-type or stable distributions), there may be values of $\gamma >0$ for which $\mathbf{E}\left[e^{-\gamma X}\right]=\infty$. In that case, the entropic risk measure $e_{\gamma }\left(X\right)$ may not be well defined or may be only valid for smaller $\gamma$. In practical models, one imposes truncated tails or ensures that $\gamma$ remains in a regime where $e^{-\gamma X}$ is integrable.

From the dual perspective (cf. Equation (1)), the supremum

$$\underset{\mathbf{Q}\ll \mathbf{P}}{sup}\left\{{\mathbf{E}}_{\mathbf{Q}}[-X]-\frac{1}{\gamma }H(\mathbf{Q}\mid \mathbf{P})\right\}$$

remains finite under fairly general conditions. Essentially, if X belongs to an appropriate Orlicz space or $L^p$ space for which the exponential integrals exist, then the entropic risk measure is valid, regardless of the specific distribution family (Detlefsen and Scandolo 2005; Föllmer and Schied 2011).

Hence, the underlying probability measure $\mathbf{P}$ is not required to come from a specific family (such as Gaussian or Lévy processes) for the exponential-based or entropy-based risk measure to work. The key assumption is that one can handle the necessary moment conditions (or integrability constraints) so that the exponentials are finite and the relative entropy functionals are well defined. Otherwise, if X exhibits tails so heavy that $\mathbf{E}\left[e^{-\gamma X}\right]=\infty$ for every positive $\gamma$, the approach can fail to produce a finite value.

5. Dynamic Consistency

In this section, we present dynamic versions of the entropic risk measure and the risk measure associated with the minimal-entropy martingale measure and show that they are time-consistent. Dynamic consistency is particularly important when dealing with practical financial applications. For instance, dynamic risk measures based on entropy have been successfully applied to energy markets and stochastic volatility models, as discussed in Swishchuk (2007).

Definition 6. 

(a) 

A map, ${\rho }_t:L^{\infty }\to L_t^{\infty }$, is called a dynamic risk measure.

(b) 

A dynamic risk measure is called time-consistent if, for $s\le t$,

$$\mathbf{P}[{\rho }_t\left(X\right)\ge {\rho }_t\left(Y\right)]=1\Rightarrow \mathbf{P}[{\rho }_s\left(X\right)\ge {\rho }_s\left(Y\right)]=1.$$

(c) 

The dynamic entropic risk measure $e_{\gamma ,t}$ is defined as

$$e_{\gamma ,t}\left(X\right):={\displaystyle \frac{1}{\gamma }}\log \left(\mathbf{E}\left[\exp \left(-\gamma X\right)\mid {\mathcal{F}}_t\right]\right).$$

(d) 

The dynamic minimal-entropy risk measure ${\rho }_t^E$ is defined as

$${\rho }_t^E\left(X\right):={\mathbf{E}}_{{\mathbf{Q}}^E}[-X\mid {\mathcal{F}}_t],$$

where ${\mathbf{Q}}^E$ is the minimal-entropy martingale measure.

It is possible to allow $\gamma$ to be an adapted process instead of a constant, resulting in a slightly more general definition of the dynamic entropic risk measure. However, in such cases, time consistency may not be guaranteed.

If, for $\mathbf{Q}\in {\mathcal{Q}}_t$, we define the conditional relative entropy $H_t(\mathbf{Q}\mid \mathbf{P})$ as

$$H_t(\mathbf{Q}\mid \mathbf{P})={\mathbf{E}}_{\mathbf{Q}}\left[\log {\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\mid {\mathcal{F}}_t\right].$$

The dynamic entropic risk measure can also be expressed in a robust representation involving the relative entropy

$$e_{\gamma ,t}\left(X\right)=\underset{\mathbf{Q}\in {\mathcal{Q}}_t}{sup}\left({\mathbf{E}}_{\mathbf{Q}}[-X\mid {\mathcal{F}}_t]-{\displaystyle \frac{1}{\gamma }}{H}_t(\mathbf{Q}\mid \mathbf{P})\right).$$

Further details on this duality and dynamic risk measures in general (in both continuous and discrete time) can be found in Acciaio and Penner (2011); Detlefsen and Scandolo (2005); Föllmer and Penner (2006); Penner (2007).

It is straightforward to verify that the two risk measures defined above satisfy the properties of dynamic risk measures.

Theorem 7. 

The dynamic entropic risk measure and the dynamic minimal-entropy risk measure are time-consistent.

Proof. 

We start with the entropic risk measure. We aim to show that $e_{\gamma ,s}\left(X\right)=e_{\gamma ,s}(-e_{\gamma ,t}\left(X\right))$ for $0\le s\le t$. Time consistency then follows from the intertemporal monotonicity theorem (Föllmer and Penner 2006, Proposition 4.2).

Using the tower property of conditional expectation, we have the following:

$$\begin{array}{cc}\hfill e_{\gamma ,s}\left(X\right)& ={\displaystyle \frac{1}{\gamma }}\log \left(\mathbf{E}\left[e^{-\gamma X}\mid {\mathcal{F}}_s\right]\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\gamma }}\log \left(\mathbf{E}\left[\mathbf{E}\left[e^{-\gamma X}\mid {\mathcal{F}}_t\right]\mid {\mathcal{F}}_s\right]\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\gamma }}\log \left(\mathbf{E}\left[\exp \left(-\gamma \underset{=-e_{\gamma ,t}\left(X\right)}{\underbrace{\left(-{\displaystyle \frac{1}{\gamma }}\log \left(\mathbf{E}\left[e^{-\gamma X}\mid {\mathcal{F}}_t\right]\right)\right)}}\right)\mid {\mathcal{F}}_s\right]\right)\hfill \\ \hfill & =e_{\gamma ,s}(-e_{\gamma ,t}\left(X\right)).\hfill \end{array}$$

For the minimal-entropy martingale measure, we proceed similarly:

$$\begin{array}{cc}\hfill {\rho }_s^E\left(X\right)& ={\mathbf{E}}_{{\mathbf{Q}}^E}[-X\mid {\mathcal{F}}_s]\hfill \\ \hfill & ={\mathbf{E}}_{{\mathbf{Q}}^E}\left[{\mathbf{E}}_{{\mathbf{Q}}^E}[-X\mid {\mathcal{F}}_t]\mid {\mathcal{F}}_s\right]\hfill \\ \hfill & ={\rho }_s^E(-{\rho }_t^E\left(X\right)).\hfill \end{array}$$

This proves time consistency for both measures.   □

In practice, many other risk measures are used. Therefore, it makes sense to compare the minimal-entropy risk measure with more commonly used approaches in the financial industry, such as value at risk (VaR) and conditional value at risk (CVaR, also called the expected shortfall). Below, we incorporate the standard definitions of VaR and CVaR, discuss their strengths and limitations, and highlight how they compare with the minimal-entropy framework, especially under incomplete markets and high volatility conditions.

The most used and well-known risk measure is value at risk (VaR). At a confidence level $\alpha \in (0,1)$, VaR typically answers the following:

“What is the smallest loss L so that X does not exceed −L with probability more than 1 − α?”

Formally, one writes

$${\mathrm{VaR}}_{\alpha }\left(X\right)=inf\left\{\ell ;\mathbf{P}[X\le -\ell ]\le 1-\alpha \right\}.$$

Due to its relative simplicity, VaR is still widely used by banks and regulatory bodies (Basel Committee on Banking Supervision 2016). However, VaR is not in general subadditive, so it fails to be coherent (Artzner et al. 1999) and does not quantify the magnitude of losses in the tail region beyond the threshold.

By comparison, the minimal-entropy risk measure ${\rho }^E$ introduced in Section 2 is coherent, thus reflecting diversification benefits more accurately. Moreover, ${\rho }^E$ depends on underlying price dynamics and the selection of a minimal-entropy $\sigma$-martingale measure, thereby embedding consistency with no-arbitrage principles in incomplete markets, while VaR typically lacks a direct link to hedging or replication arguments.

Alongside value at risk (VaR), conditional value at risk (CVaR) is likely the second most commonly used risk measure. Often referred to as the expected shortfall, CVaR at level $\alpha$ considers average tail losses above the VaR threshold. One standard formula is

$${\mathrm{CVaR}}_{\alpha }\left(X\right)={\displaystyle \frac{1}{1-\alpha }}{\int }_{1-\alpha }^1{\mathrm{VaR}}_u\left(X\right)\mathrm{d}u.$$

Basel III and the Fundamental Review of the Trading Book increasingly advocate the use of CVaR for market risk calculations, recognizing its coherent properties and superior sensitivity to extreme events (Basel Committee on Banking Supervision 2016). However, CVaR remains distribution-based, often sidelining explicit dynamic hedging or replication considerations. By contrast, ${\rho }^E$ (the minimal-entropy measure) directly incorporates no-arbitrage dynamics and penalizes large deviations from the real-world measure via the Kullback–Leibler divergence. In highly volatile or incomplete markets, ${\rho }^E$ thus captures both model risk (through entropy) and market risk (through $\sigma$-martingale constraints) more naturally than CVaR can, though at the cost of greater computational and data requirements.

Beyond VaR and CVaR, the entropic VaR (Ahmadi-Javid 2011) also introduces an entropy-based correction to the usual VaR framework, aiming for improved coherence-like properties. Meanwhile, expectile-type risk measures (Bellini and Di Bernardino 2017; Zaevski and Nedeltchev 2023) focus on generalized quantiles that can exhibit coherence under certain assumptions and capture the asymmetric tail risks. Nonetheless, these constructs typically remain tied to direct distributional assumptions, without necessarily unifying real-world dynamics and no-arbitrage arguments.

In Section 3, we see that ${\rho }^E$ fulfills the coherence axioms and, thanks to Theorem 2, is firmly rooted in a no-arbitrage framework for incomplete markets. By minimizing relative entropy with respect to $\mathbf{P}$, ${\rho }^E$ selects a risk-neutral (or $\sigma$-martingale) measure that is closest to the real-world measure, thereby unifying pricing and risk. This unification becomes particularly relevant when local martingale measures may not exist.

In contrast to the above-mentioned advantages, there are also some drawbacks. Approximating or computing the minimal-entropy $\sigma$-martingale measure can be significantly more involved than applying VaR or other distribution-based measures. In volatile environments, calibrating Radon–Nikodým derivatives so that the entropy remains finite also demands extensive data and robust modeling. Moreover, although regulators increasingly tolerate advanced internal models (see Page 13), institutions must still validate the market incompleteness assumptions and explain the chosen measure to supervisors. Hence, some market participants prefer simpler, purely distribution-based schemes like CVaR or entropic VaR because they are more transparent and easier to backtest or justify to stakeholders.

Overall, while more sophisticated than VaR and even CVaR, the minimal-entropy risk measure furnishes a coherent and market-consistent framework that directly incorporates incomplete-market structures and penalizes divergence from real-world probabilistic views. Its principal obstacles involve higher computational costs and data intensity, potentially limiting immediate adoption in practice unless sufficiently robust calibration and infrastructure are available. Nonetheless, for investors, regulators, or institutions confronting model uncertainty and market incompleteness, the minimal-entropy methodology offers a powerful alternative that more fully respects no-arbitrage and coherent-risk requirements.

6. Optimal Risk Transfer

Optimal risk transfer is a classical topic in the study of risk measures and has been studied extensively in, e.g., Barrieu and El Karoui (2005). In this section, we present a somewhat more general and extended view, highlighting in particular the property of $\gamma$-dilated families of risk measures, the associated inf-convolution approach, and how these results can be restricted to the risk-neutral setting. We then provide a brief discussion of why no non-trivial $\gamma$-dilation family can be constructed from the minimal-entropy coherent risk measure, whereas the standard entropic risk measure naturally fits into this framework. Finally, we show how restricting to (local) risk-neutral measures force the minimal-entropy measure to emerge as the unique solution to the risk-transfer problem.

Let $\rho$ be any (convex or coherent) risk measure on $L^{\infty }\left(\mathcal{F}\right)$. The fundamental question in optimal risk sharing between two entities with risk measures ${\rho }_1$ and ${\rho }_2$ is to split a position, X, into two parts $X-F$ and F. The goal is to minimize the combined risk ${\rho }_1(X-F)+{\rho }_2\left(F\right)$. This optimization problem is well expressed via the inf-convolution operator:

$$\left({\rho }_1\square {\rho }_2\right)\left(X\right):=\underset{F}{inf}\left\{{\rho }_1(X-F)+{\rho }_2\left(F\right)\right\}.$$

(12)

The optimal risk transfer is solved when the infimum in (12) is achieved via some $F^{*}$. Well-known results (see Barrieu and El Karoui (2005)) show that certain risk measures, particularly those arising from $\gamma$-dilated families, enjoy a simple solution: the optimal allocation is often a linear split of X.

Definition 7 

($\gamma$-dilated family). A family ${\left\{{\rho }_{\gamma }\right\}}_{\gamma \in I}$ of risk measures on $L^{\infty }\left(\mathcal{F}\right)$ is called γ-dilated if there exists a base risk measure ρ, such that

$${\rho }_{\gamma }\left(X\right)={\displaystyle \frac{1}{\gamma }}\rho \left(\gamma X\right)\mathit{for}\mathit{each}\gamma \in I,$$

where $I\subseteq (0,\infty )$ is some index set.

An immediate (but important) example is the entropic risk measure, which is $\gamma$-dilated by construction:

$$e_{\gamma }\left(X\right)={\displaystyle \frac{1}{\gamma }}\log \left(\mathbf{E}\left[e^{-\gamma X}\right]\right)={\displaystyle \frac{1}{\gamma }}\rho \left(\gamma X\right)$$

if one sets $\rho \left(Z\right):=\log \left(\mathbf{E}\left[e^{-Z}\right]\right)$. By contrast, a coherent risk measure with positive homogeneity cannot usually form a non-trivial $\gamma$-dilated family unless it is purely linear in $\gamma$ and thus collapses to the $\mathbf{E}$-type functional (more details below).

We restate here a known result, cf. Barrieu and El Karoui (2005), slightly paraphrased to fit our notation.

Theorem 8. 

Let ${\left\{{\rho }_{\gamma }\right\}}_{\gamma >0}$ be a γ-dilated family of convex risk measures on $L^{\infty }\left(\mathcal{F}\right)$. Then, the following statements hold:

(a) 

For any ${\gamma }_1,{\gamma }_2>0$, we have the inf-convolution identity:

$${\rho }_{{\gamma }_1+{\gamma }_2}\left(X\right)=\left({\rho }_{{\gamma }_1}\square {\rho }_{{\gamma }_2}\right)\left(X\right).$$

(13)

(b) 

The optimal allocation $F^{*}$ solving ${inf}_F\{{\rho }_{{\gamma }_1}(X-F)+{\rho }_{{\gamma }_2}\left(F\right)\}$ is linear in X, specifically

$$F^{*}={\displaystyle \frac{{\gamma }_2}{{\gamma }_1+{\gamma }_2}}X.$$

(14)

(c) 

Consequently, we also have

$${\rho }_{{\gamma }_1+{\gamma }_2}\left(X\right)={\rho }_{{\gamma }_1}(X-F^{*})+{\rho }_{{\gamma }_2}\left(F^{*}\right).$$

(d) 

Moreover, if ${\rho }^1$ and ${\rho }^2$ are two convex risk measures that each can be embedded in a single γ-dilated family (i.e., they are both instances of ${\rho }_{\gamma }$ for some base ρ), then, for any ${\gamma }_1,{\gamma }_2>0$,

$${\rho }_{{\gamma }_1}\square {\rho }_{{\gamma }_2}={\left(\rho \square \rho \right)}_{{\gamma }_1+{\gamma }_2}$$

under a suitable identification. Hence, the γ-dilation structure is preserved under inf-convolution.

Sketch of Proof 

The key insight is that, for a $\gamma$-dilated family, scaling X by $\gamma$ and adjusting the risk measure by $\frac{1}{\gamma }$ yields a consistent re-parametrization of the same base functional $\rho$. Once this is established, standard inf-convolution arguments for the exponentially (or more generally, $\rho$-) tilted function imply the linear sharing rule (14). For full details, see Barrieu and El Karoui (2005) or references therein.   □

We now remark why a coherent risk measure—in particular, the minimal entropy risk measure ${\rho }^E$—cannot generally be part of a non-trivial $\gamma$-dilated family. Recall that a coherent risk measure $\rho$ is positively homogeneous, i.e.,

$$\rho \left(\lambda X\right)=\lambda \rho \left(X\right)\mathrm{for}\mathrm{all}\lambda \ge 0.$$

Combining this with the $\gamma$-dilation requirement ${\rho }_{\gamma }\left(X\right)=\frac{1}{\gamma }\rho \left(\gamma X\right)$ quickly forces ${\rho }_{\gamma }\left(X\right)\equiv \rho \left(X\right)$ for all $\gamma$, unless $\rho \equiv 0$ or $\rho \equiv \mathrm{some}\mathrm{fixed}\mathrm{linear}\mathrm{functional}$. Indeed, for coherent $\rho$, we would have

$$\rho \left(\gamma X\right)=\gamma \rho \left(X\right),\mathrm{so}\frac{1}{\gamma }\rho \left(\gamma X\right)=\rho \left(X\right).$$

Hence, ${\rho }_{\gamma }$ cannot genuinely “depend” on $\gamma$. In short, non-trivial $\gamma$-dilated families (like the entropic class) are not coherent. This explains why the minimal-entropy risk measure, ${\rho }^E\left(X\right)={\mathbf{E}}_{{\mathbf{Q}}^E}[-X]$, cannot appear in a standard $\gamma$-dilated framework.