Optional Sampling Theorem Research Articles

Martingale Inequalities under G-Expectation and Their Applications

In this paper, we study the martingale inequalities under G-expectation and their applications. To this end, we introduce a new kind of random time, called G-stopping time, and then investigate the properties of a G-martingale (supermartingale) such as the optional sampling theorem and upcrossing inequalities. With the help of these properties, we can show the martingale convergence property under G-expectation.

Stopped processes and Doob's optional sampling theorem

Using the spectral measure μS of the stopping time S, we define the stopping element XS as a Daniell integral ∫XtdμS for an adapted stochastic process (Xt)t∈J that is a Daniell summable vector-valued function. This is an extension of the definition previously given for right-order-continuous sub martingales with the Doob-Meyer decomposition property. The more general definition of XS necessitates a new proof of Doob's optional sampling theorem, because the definition given earlier for sub martingales implicitly used Doob's theorem applied to martingales. We provide such a proof, thus removing the heretofore necessary assumption of the Doob-Meyer decomposition property in the result. Another advancement presented in this paper is our use of unbounded order continuity of a stochastic process, which properly characterizes the notion of continuity of sample paths almost everywhere, found in the classical theory.

Introduction to Stopping Time in Stochastic Finance Theory. Part II

Summary We start proceeding with the stopping time theory in discrete time with the help of the Mizar system [1], [4]. We prove, that the expression for two stopping times k1 and k2 not always implies a stopping time (k1 + k2) (see Theorem 6 in this paper). If you want to get a stopping time, you have to cut the function e.g. (k1 + k2) ⋂ T (see [2, p. 283 Remark 6.14]). Next we introduce the stopping time in continuous time. We are focused on the intervals [0, r] where r ∈ ℝ. We prove, that for I = [0, r] or I = [0,+∞[ the set {A ⋂ I : A ∈ Borel-Sets} is a σ-algebra of I (see Definition 6 in this paper, and more general given in [3, p.12 1.8e]). The interval I can be considered as a timeline from now to some point in the future. This set is necessary to define our next lemma. We prove the existence of the σ-algebra of the τ -past, where τ is a stopping time (see Definition 11 in this paper and [6, p.187, Definition 9.19]). If τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we can prove, that the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past (see Theorem 9 in this paper and [6, p.187 Lemma 9.21]). Suppose, that you want to use Lemma 9.21 with some events, that never occur, see as a comparison the paper [5] and the example for ST(1)={+∞} in the Summary. We don’t have the element +1 in our above-mentioned time intervals [0, r[ and [0,+1[. This is only possible if we construct a new σ-algebra on ℝ {−∞,+∞}. This construction is similar to the Borel-Sets and we call this σ-algebra extended Borel sets (see Definition 13 in this paper and [3, p. 21]). It can be proved, that {+∞} is an Element of extended Borel sets (see Theorem 21 in this paper). Now we use the interval [0,+∞] as a basis. We construct a σ-algebra on [0,+∞] similar to the book ([3, p. 12 18e]), see Definition 18 in this paper, and call it extended Borel subsets. We prove for stopping times with this given σ-algebra, that for τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we have the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past, see Theorem 25 in this paper. It is obvious, that {+∞} 2 extended Borel subsets. In general, Lemma 9.21 is important for the proof of the Optional Sampling Theorem, see 10.11 Proof of (i) in [6, p. 203].

The value of foresight

Suppose you have one unit of stock, currently worth 1, which you must sell before time T. The Optional Sampling Theorem tells us that whatever stopping time we choose to sell, the expected discounted value we get when we sell will be 1. Suppose however that we are able to see a units of time into the future, and base our stopping rule on that; we should be able to do better than expected value 1. But how much better can we do? And how would we exploit the additional information? The optimal solution to this problem will never be found, but in this paper we establish remarkably close bounds on the value of the problem, and we derive a fairly simple exercise rule that manages to extract most of the value of foresight.

Stationary distribution of the surplus process in a risk model with a continuous type investment

In this paper, we stochastically analyze the continuous time surplus process in a risk model which involves a continuous type investment. It is assumed that the investment of the surplus to other business is continuously made at a constant rate, while the surplus process stays over a given sufficient level. We obtain the stationary distribution of the surplus level and/or its moment generating function by forming martingales from the surplus process and applying the optional sampling theorem to the martingales and/or by establishing and solving an integro-differential equation for the distribution function of the surplus level.

The joint distribution of Parisian and hitting times of Brownian motion with application to Parisian option pricing

We study the joint law of Parisian time and hitting time of a drifted Brownian motion by using a three-state semi-Markov model, obtained through perturbation. We obtain a martingale to which we can apply the optional sampling theorem and derive the double Laplace transform. This general result is applied to address problems in option pricing. We introduce a new option related to Parisian options, being triggered when the age of an excursion exceeds a certain time or/and a barrier is hit. We obtain an explicit expression for the Laplace transform of its fair price.

An (s, k, S) fluid inventory model with exponential leadtimes and order cancellations

We consider a stochastic fluid inventory model based on a (s, k, S) policy. The content level W = {W(t): t ≥ 0} increases or decreases according to a fluid-flow rate modulated by an n-state continuous time Markov chain (CTMC). W starts at W(0) = S; whenever W(t) drops to level s, an order is placed to take the inventory back to level S, which the supplier will carry out after an exponential leadtime. However, if during the leadtime the content level reaches k, the order is suppressed. We obtain explicit formulas for the expected discounted costs. The derivations are based on the optional sampling theorem (OST) to the multidimensional martingale and on fluid flow techniques.

Optional Sampling Theorem for Deformed Submartingales

The family ${{\bf Q}}=(Q^{(n)}, {\cal F}_{n})_{n=0}^{\infty}$ of probability measures $Q^{(n)}$ defined on ${{\cal F}}_{n}$ is considered. It is called a deformation of the first kind if for all $n\in \{0,1,2,\dots\}$, $Q^{(n+1)}|_{{{\cal F}}_{n}}\ll Q^{(n)}$, and a deformation of the second kind if $Q^{(n+1)}|_{{{\cal F}}_{n}}\gg Q^{(n)}$. For finite stopping times $\tau$ measures $Q^{(\tau)}$ are introduced by the formula $Q^{(\tau)}(A)=\sum_{i=0}^\infty Q^{(i)}(A\{\tau=i\})$, where $A\in{\cal F}_{\tau}$. With the help of these measures a generalization of Doob's optional sampling theorem is formulated and proved for deformed submartingales of the first kind in the case of neighboring stopping times, and of the second kind in the case of boundedly spaced stopping times.

Valuation and Analysis of Zero-Coupon Contingent Capital Bonds

We consider the valuation and analysis of zero-coupon contingent capital bonds (CCBs) in the structural framework. Using Doob's Optional Sampling Theorem (and making virtually no assumptions on asset value dynamics, the terms of conversion or the conversion trigger) we express the value of the CCB in terms of the effective loss imposed on CCB investors at conversion and quantify the impact that contingent capital has on traditional debt and equity. We show how a variety of conversion terms can be incorporated into a single framework and describe how they can be calibrated to ensure that seniority is respected and/or equity investors are not rewarded for poor performance. We provide numerical evidence indicating that the terms of conversion can fundamentally alter the nature of the CCB, a phenomenon that is of clear interest to investors, issuers and regulators.

Constrained semi-Markov decision processes with ratio and time expected average criteria in Polish spaces

This paper deals with the ratio and time expected average criteria for constrained semi-Markov decision processes (SMDPs). The state and action spaces are Polish spaces, the rewards and costs are unbounded from above and from below, and the mean holding times are allowed to be unbounded from above. First, under general conditions we prove the existence of constrained-optimal policies for the ratio expected average criterion by developing a technique of occupation measures including the mean holding times for SMDPs, which are the generalizations of those for the standard discrete-time and continuous-time MDPs. Then, we give suitable conditions under which we establish the equivalence of the two average criteria by the optional sampling theorem, and thus we show the existence of constrained-optimal policies for the time expected average criterion. Finally, we illustrate the application of our main results with a controlled linear system, for which an exact optimal policy is obtained.

Generalization of Doob's optional sampling theorem for deformed submartingales

Generalization of Doob's optional sampling theorem for deformed submartingales

Constructing sublinear expectations on path space

We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G-expectation, and an optional sampling theorem that holds without exceptional set. Our results also shed light on the inherent limitations to constructing sublinear expectations through aggregation.

Superhedging and Dynamic Risk Measures under Volatility Uncertainty

We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. Furthermore, we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward SDE. The uniqueness of dynamic extensions of static sublinear expectations is also studied.

Doob’s optional sampling theorem in Riesz spaces

The notions of stopping times and stopped processes for continuous stochastic processes are defined and studied in the framework of Riesz spaces. This leads to a formulation and proof of Doob’s optional sampling theorem.

Superhedging and Dynamic Risk Measures Under Volatility Uncertainty

We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a cadlag nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. Furthermore, we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward SDE. The uniqueness of dynamic extensions of static sublinear expectations is also studied.

On Quadratic g-Evaluations/Expectations and Related Analysis

In this article we extend the notion of g-evaluation, in particular g-expectation, of Peng [8, 9] to the case where the generator g is allowed to have a quadratic growth (in the variable “z”). We show that some important properties of the g-expectations, including a representation theorem between the generator and the corresponding g-expectation—and consequently the reverse comparison theorem of quadratic BSDEs as well as the Jensen inequality—remain true in the quadratic case. Our main results also include a Doob–Meyer type decomposition, the optional sampling theorem, and the upcrossing inequality. The results of this article are important in the further development of the general quadratic nonlinear expectations (cf. [5]).

The differentiation of hypoelliptic diffusion semigroups

Basic derivative formulas are presented for hypoelliptic heat semi- groups and harmonic functions extending earlier work in the elliptic case. Fol- lowing the approach of (22), emphasis is placed on developing integration by parts formulas at the level of local martingales. Combined with the optional sampling theorem, this turns out to be an efficient way of dealing with bound- ary conditions, as well as with finite lifetime of the underlying diffusion. Our formulas require hypoellipticity of the diffusion in the sense of Malliavin cal- culus (integrability of the inverse Malliavin covariance) and are formulated in terms of the derivative flow, the Malliavin covariance and its inverse. Finally some extensions to the nonlinear setting of harmonic mappings are discussed.

Representation theorems for quadratic [formula omitted]-consistent nonlinear expectations

In this paper we extend the notion of “filtration-consistent nonlinear expectation” (or “ F -consistent nonlinear expectation”) to the case when it is allowed to be dominated by a g -expectation that may have a quadratic growth. We show that for such a nonlinear expectation many fundamental properties of a martingale can still make sense, including the Doob–Meyer type decomposition theorem and the optional sampling theorem. More importantly, we show that any quadratic F -consistent nonlinear expectation with a certain domination property must be a quadratic g -expectation as was studied in [J. Ma, S. Yao, Quadratic g -evaluations and g -martingales, 2007, preprint]. The main contribution of this paper is the finding of a domination condition to replace the one used in all the previous works (e.g., [F. Coquet, Y. Hu, J. Mémin, S. Peng, Filtration-consistent nonlinear expectations and related g -expectations, Probab. Theory Related Fields 123 (1) (2002) 1–27; S. Peng, Nonlinear expectations, nonlinear evaluations and risk measures, in: Stochastic Methods in Finance, in: Lecture Notes in Math., vol. 1856, Springer, Berlin, 2004, pp. 165–253]), which is no longer valid in the quadratic case. We also show that the representation generator must be deterministic, continuous, and actually must be of the simple form g ( z ) = μ ( 1 + | z | ) | z | , for some constant μ > 0 .

The optional sampling theorem for submartingales in the sequentially planned context

The optional sampling theorem is considered in the sequentially planned context. We prove the optional sampling theorem for direct successors and for sampling plans with a finite number of stages. Also, the theorem is studied in the general case under a uniform integrability condition; we obtain it for submartingales with a last element, and for submartingales that verify a bounded condition based on uniform integrability.

Expectation, Conditional Expectation and Martingales in Local Fields

We investigate a possible definition of expectation and conditional expectation for random variables with values in a local field such as the $p$-adic numbers. We define the expectation by analogy with the observation that for real-valued random variables in $L^2$ the expected value is the orthogonal projection onto the constants. Previous work has shown that the local field version of $L^\infty$ is the appropriate counterpart of $L^2$, and so the expected value of a local field-valued random variable is defined to be its ``projection'' in $L^\infty$ onto the constants. Unlike the real case, the resulting projection is not typically a single constant, but rather a ball in the metric on the local field. However, many properties of this expectation operation and the corresponding conditional expectation mirror those familiar from the real-valued case; for example, conditional expectation is, in a suitable sense, a contraction on $L^\infty$ and the tower property holds. We also define the corresponding notion of martingale, show that several standard examples of martingales (for example, sums or products of suitable independent random variables or ``harmonic'' functions composed with Markov chains) have local field analogues, and obtain versions of the optional sampling and martingale convergence theorems.