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

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 highlighted ever since the seminal work of [2], which introduced an axiomatic framework for coherent risk measures. Subsequent studies, such as ([23], Chapter 4) or [12], have refined and extended these ideas.

In the monetary risk-measure framework, one models a financial position by a real-valued random variable X on a probability space $(\Omega ,\mathcal{F},\mathbf{P})$. 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 $\rho \left(X\right)$, which 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 [2,12,23] for standard references on these axioms.

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 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, [15,33,48] 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 one “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 [13,24,27,40]. Minimizing relative entropy leads to the well-known minimal entropy martingale measure, a construction which has proven valuable in option pricing and hedging and which is closely connected to maximizing the investor’s utility (see Section 4). However, most of the literature focuses on the local martingale setting. 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.

One of the most popular risk measures is the entropic risk measure. At first glance, one might suspect connections between entropic risk measures and minimal entropy methods because of the shared “entropy” label. 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 reveal 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 manifest. We show that the measure can be equivalently defined, and one arrives at the same results and conclusions quite easily.

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.

In summary, the main contributions of this paper are:

• We introduce the minimal entropy $\sigma$-martingale measure for general semimartingale models (with dividends), 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 which 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, highlighting the new $\sigma$-martingale generalization 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. 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.

2. Definition and Existence

Let $(\Omega ,\mathcal{F},\mathbf{P})$ 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 [47]).

In most models, there exists at least one probability measure such that the discounted process $\widehat{S}$ is a local martingale under this probability measure. However, [15] showed that in an arbitrage-free market (more precise 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:

Definition 1.

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

(i) 

$D_n\subset D_{n+1}$ for all n;

(ii) 

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

(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 notions, introduced by Chou [7], further explored by Émery [51], and subsequently discussed by, for example, Jacod and Shiryaev [31] and Kallsen [33], allow certain processes to fail to be true local martingales yet still satisfy a limiting or piecewise martingale property.

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 by ${\mathcal{M}}_{\sigma }$.

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

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

For more details on σ-martingales, see, for example, [28,33,48].

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

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

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 [46]. It also plays a fundamental role in large deviations theory, where it underpins results such as Sanov’s theorem [8]. A comprehensive summary of its applications can be found in [6].

Further illustrations of this definition are available in [11], 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 [30, Satz 10.4].

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.

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 σ-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 σ-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 σ-martingale measure carrying finite relative entropy. For further details, see [14,15], and [16].

As opposed to the minimal entropy σ-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:

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

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 γ, 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 σ-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 goes analogue to ours, but instead of equivalent σ-martingale measures, one minimizes the entropy among all equivalent martingale measures (see, for example, [29,38,39,43,45]). 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. For this case, Theorem 3.1 in [26] provides a proof of existence, albeit with a highly technical approach. Another simple result is the existence of the minimal entropy martingale measure if $\widehat{S}$ is bounded as shown in [24, Theorem 2.1]. Other results depend on the specific form of the Radon-Nikodym derivative, e.g., [29] or are applicable only in discrete time settings, e.g., [23, Corollary 3.27].

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

For the proof, we need some lemmas.

Lemma 1.

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

(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 which are absolutely continuous with respect to $\mathbf{P}$.

Proof. 

We have:

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

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

It is easy to see that the function $x\mapsto xlogx-x$ takes its minimum at $x=1$. Hence, we have $xlogx-x\ge -1$ 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 $\mathbf{Q}\ll \mathbf{P}$:

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

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

$$\begin{array}{cc}\hfill 0& =H(\mathbf{Q}\mid \mathbf{P})={\mathbf{E}}_{\mathbf{P}}\left[{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}log\left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right)\right]\hfill \\ \hfill & ={\mathbf{E}}_{\mathbf{P}}\left[{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}log\left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right)\right]-\underset{=0}{\underbrace{{\mathbf{E}}_{\mathbf{P}}\left[{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}-1\right]}}\hfill \\ \hfill & ={\mathbf{E}}_{\mathbf{P}}\left[\underset{\stackrel{\mathit{Equation}\left(2\right)}{\ge }0}{\underbrace{{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}log\left({\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}\right)-{\displaystyle \frac{\mathrm{d}\mathbf{Q}}{\mathrm{d}\mathbf{P}}}+1}}\right].\hfill \end{array}$$

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)

For probability measures that are not absolutely continuous with respect to $\mathbf{P}$, the statement is obvious. 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:

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

Thus, strict convexity follows.□

Lemma 2.

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

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

Then ${\mathbf{Q}}_0$ is the unique Minimal Entropy σ-Martingale Measure.

Proof. 

Define the set

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

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

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

Next, we show that Γ is convex and closed in total variation:

For any ${\mathbf{Q}}_1,{\mathbf{Q}}_2\in \Gamma$, their convex combination $\alpha {\mathbf{Q}}_1+(1-\alpha ){\mathbf{Q}}_2$, $\alpha \in [0,1]$, is also in Γ. Indeed, convex combinations preserve both the finite entropy condition (due to convexity of relative entropy) and the σ-martingale property as σ-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 Γ has a unique I-projection $\tilde{\mathbf{Q}}$ of $\mathbf{P}$, which satisfies:

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

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 $\tilde{\mathbf{Q}}\in \Gamma$, 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 σ-martingale measure. □

Lemma 3.

Let $(\Omega ,\mathcal{F},\mathbf{P})$ be a probability space, and let $\tilde{\mathbf{Q}},{\mathbf{Q}}^1,{\mathbf{Q}}^2,\cdots$ be probability measures on $(\Omega ,\mathcal{F})$ 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 σ-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

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

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 ${\mathbf{Q}}^n$ 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 directly that X is a σ-martingale under $\tilde{\mathbf{Q}}$. □

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 there exists a unique minimal entropy σ-martingale measure ${\mathbf{Q}}^E\in {\mathcal{M}}_{\sigma }$, characterized by

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

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 $\varphi \left(x\right)=xlogx$

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

and therefore the sequence $\left(\frac{\mathrm{d}{\mathbf{Q}}_n}{\mathrm{d}\mathbf{P}}\right)$ is uniformly integrable according to Theorem A3. By 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, by 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 σ-martingale with respect to all ${\mathbf{Q}}_n$ (and hence also w.r.t. all ${\mathbf{Q}}_{n_k}$). Now, we want to show it is also a σ-martingale under ${\tilde{\mathbf{Q}}}_n$. To prove this, let ${\left(D_i\right)}_i$ be a sequence of predictable sets such that ${\mathbb{1}}_{D_i}•\hat{S}$ is a martingale under a fixed ${\mathbf{Q}}_{n_k}$. For each $A\in {\mathcal{F}}_s$,

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

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 σ 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, by 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 2.

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 ${\mathcal{M}}_{\mathrm{loc}}=⌀$.