General Processes: Mathematical Facts

The main purpose of this article is to propose a reasonable definition for the stochastic integration (S.I.) of progressive processes w.r.t. semimartingales. This S.I. generalizes that of predictable processes w.r.t. semimartingales as well as the stochastic Stieltjes integration. This S.I. is proposed in §1. We give also in §1 an exponential formula for semimartingales using this S.I.. The rest of this paper consists of several remarks on the theory of stochastic integration which are mostly of pedagogical interest. In §2 we propose a new construction of the S.I. of predictable processes w.r.t. local martingales. A simple proof of the integration by parts formula is given in §3. Finally, we propose in §4 a short proof of Meyer’s theorem on compensated stochastic integrals of local martingales.AMS Subject Classification60H05

Let be a standard d-dimensional Brownian motion and L a smooth functional on the space C = Cd[0, 1] of continuous functions from [0,1] to Rd, such that Denote Thenis a square integrable martingale which can be represented as a stochastic integral; consequently the random variable has the representationFor Fréchet differentiable functions L, Clark(1) gave the following formula for the integrand e:

After introducing cylindrical Gaussian random variables and Hilbert space valued Gaussian random variables, we define cylindrical Wiener process and Hilbert space valued Wiener process in a natural way. We develop the Itô stochastic integral with respect to cylindrical and Hilbert space valued Wiener processes simultaneously as they share many common features. Our construction of the Itô integral is consistent with the construction of a stochastic integral with respect to square integrable martingales, however, regarding measurability, we assume only that the integrand is adapted, following the classical ideas as presented by Liptser and Shiryayev (Statistics of Stochastic Processes. Nauka, ) and by Øksendal (Stochastic Differential Equations. Springer, New York, ). We provide detailed proofs of properties of the Itô integral and follow with the Martingale Representation Theorem. Our unified approach to stochastic integration with respect to cylindrical and Hilbert space valued Wiener processes allowed us to present the Stochastic Fubini Theorem and the Itô Formula in both cases in an almost identical way.

A general theory of stochastic integral in the abstract topological measurable space is established. The martingale measure is defined as a random set function having some martingale property. All square integrable martingale measures constitute a Hilbert space M 2. For each μ∈M 2, a real valued measure 〈μ〉 on the predictable σ-algebra ℘ is constructed. The stochastic integral of a random function \(\mathfrak{h} \in L^2 \left( {\left\langle \mu \right\rangle } \right)\) with respect to μ is defined and investigated by means of Riesz's theorem and the theory of projections. The stochastic integral operator I μis an isometry from L 2(〈μ〉) to a stable subspace of M 2, its inverse is defined as a random Radon-Nikodym derivative. Some basic formulas in stochastic calculus are obtained. The results are extended to the cases of local martingale and semimartingale measures as well.

We now return to general continuous-time financial market models in the setting of §6.1, i.e. there are d+1 primary traded assets whose price processes are given by stochastic processes S 0,... , S d , which are assumed to be adapted, right-continuous with left-limits (RCLL) and strictly positive semimartin-gales on a filtered probability space (Ω, F ℙ, F) (as usual F = (F t ) t≤T ). We assume that the market is free of arbitrage, in the sense that there exist equivalent martingale measures, but it contains non-attainable contingent claims, i.e. there are cash flows which cannot be replicated by self-financing trading strategies. In view of Theorem 6.1.5 this means that we do not have a unique equivalent martingale measures. We try to answer the obvious questions in this setting: how should we price the non-attainable contingent claims, i.e. which of the possible equivalent martingale measures should we pick for our valuation formula based on expectation, and, how can we construct hedging strategies for the nonattainable contingent claims to ‘minimise the risk? We try to answer these two questions in the general setting and then consider a prominent example of an incomplete market, a market with stochastic volatility, in more detail.KeywordsTrading StrategyStochastic VolatilityContingent ClaimMartingale MeasureIncomplete MarketThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The purpose of this paper is twofold: first, to extend the definition of the stochastic integral for processes with values in Banach spaces; and second, to define the stochastic integral as a genuine integral, with respect to a measure, that is, to provide a general integration theory for vector measures, which, when applied to stochastic processes, yields the stochastic integral along with all its properties. For the reader interested only in scalar stochastic integration, our approach should still be of interest, since it sheds new light on the stochastic integral, enlarges the class of integrable processes and presents new convergence theorems involving the stochastic integral.

We establish the existence of a stochastic integral in a nuclear space setting as follows. Let , , and be nuclear spaces which satisfy the following conditions: the spaces are reflexive, complete, bornological spaces such that their strong duals also satisfy these conditions. Assume that there is a continuous bilinear mapping of into . If is an integrable, -valued predictable process and is an -valued square integrable martingale, then there exists a -valued process called the stochastic integral. The Lebesgue space of these integrable processes is studied and convergence theorems are given. Extensions to general locally convex spaces are presented.

Let χ be a family of stochastic processes on a given filtered probability space (Ω, F, (Ft)t∈T, P) with T⊆R+. Under the assumption that the set Me of equivalent martingale measures for χ is not empty, we give sufficient conditions for the existence of a unique equivalent martingale measure that minimizes the relative entropy, with respect to P, in the class of martingale measures. We then provide the characterization of the density of the minimal entropy martingale measure, which suggests the equivalence between the maximization of expected exponential utility and the minimization of the relative entropy.

We give stopped Doob inequalities for p-th moment, 0<p<∞, of stochastic convolution integrals ∫(0,t] U(t,s)φ s − dM s in a Hilbert space, where M is a Hilbert space-valued cadlag square integrable martingale, φ is an operator-valued predictable process and U(t,s) is a contraction-type evolution operator. We also generalize the previous results and try to get smaller constants. 1This research was partially supported by Sharif University of Technology.

The minimal relative entropy criterion for the selection of an equivalent martingale measure in an incomplete market seems to still hold some mystique in its financial interpretation. In this paper we work toward this interpretation by suggesting and exploring the idea of relating a martingale measure selection criterion to a CAPM-like pricing scheme. We examine this idea in the case of the minimal relative entropy criterion and we present some preliminary results. We work within a one-period financial market and show that the minimal relative entropy pricing criterion is equivalent to some CAPM-like pricing scheme where the classical beta coefficient formula has been replaced by some “entropic beta” and the market portfolio by some “appropriate” reference portfolio. Furthermore, we show that if the assets involved have returns that are jointly normal, then this “entropic beta” formula coincides with the classical beta coefficient. Additionally and for comparison reasons, we briefly illustrate that if our criterion for the choice of the martingale measure was the minimization of the variance of the Radon–Nikodym derivative, then the resulting martingale pricing and the pricing implied by the classical CAPM scheme would be the same.

In this chapter we give a brief overview of the basic mathematical concepts used for pricing financial products. Modern pricing theory is mainly based on describing the ups and downs of market prices via stochastic processes. Therefore, we start with the definition and characteristics of stochastic processes in Section 2.1. Usually the price-changes of the most basic financial instruments are modelled in terms of a so-called stochastic differential equation (SDE). The price of a financial derivative is considered to be a function of the basic financial instruments’ prices. One of the central tools in stochastic calculus which can be applied to determine the stochastic differential equation for the prices of these derivatives, Itô’s lemma, is described in Section 2.4. This concept is closely related to the stochastic integral, which is defined in Section 2.3 to be the mean square limit of some random sums, carefully put together so that the resulting limit is a (local) martingale. And in fact, martingales are one of the most important elements needed for the evaluation of financial instruments. They are defined in Section 2.5 and are used to describe the prices of financial derivatives in terms of conditional expectations. The Feynman-Kac formula provides a partial differential equation (PDE) that corresponds to such conditional expectations as described in Section 2.6. The advantage of dealing with partial differential equations is that they can be solved numerically. Also, PDE methods come naturally to applied mathematicians and physicists. This approach led Black and Scholes [BS73] to their famous equation for pricing European options. Their model will be discussed in Section 3.5. By contrast, martingales and Itô (stochastic) calculus come naturally to probabilists, and this approach led Merton [Mer74] to his option pricing formula.KeywordsStochastic ProcessStochastic Differential EquationConditional ExpectationWiener ProcessLocal MartingaleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We develop a stochastic integration theory with respect to creation, annihilation and gauge operators on the full Fock space. This is done by using a kernel representation for a large class of bounded operators on the full Fock space. It is shown that the kernels form a Banach algebra. Having established the definition of processes and stochastic integrals we go on to prove an Ito formula and use this for examining stochastic evolutions and constructing dilations of special completely positive semigroups. Explicit solutions of the corresponding stochastic differential equations are given.

In this paper, we shall firstly illustrate why we should introduce set-valued stochastic integrals, and then we shall discuss some properties of set-valued stochastic processes and the relation between a set-valued stochastic process and its selection set. After recalling the Aumann type definition of stochastic integral, we shall introduce a new definition of Lebesgue integral of a set-valued stochastic process with respect to the time t. Finally we shall prove the presentation theorem of set-valued stochastic integral and discuss further properties that will be useful to study set-valued stochastic differential equations with their applications.

In this chapter we will consider the question as to when do all martingales adapted to a filtration \(({\mathcal F}_\centerdot )\) admit a representation as a stochastic integral with respect to a given local martingale M. This result was proved by Ito’s when the underlying filtration is the filtration generated by a multidimensional Wiener process. Ito’s had proven the integral representation property for square integrable martingales and this was extended to all martingales by Clark.

If the market is incomplete, we have several choices of equivalent martingale measures to price contingent claims .Recall that we didn't impose conditions on the preferences of the economic agents other than that they prefer more to less. So it is quite natural to specify the preferences further in order to select one of the equivalent martingale measures.The specification of the investor's attitude towards risk is done in terms of a utility function. Credit risk is increasingly being recognised as a significant area of risk and arbitrage, yet there exists relatively little research on it. In this paper we show that operational risk represents a fundamental risk to option hedging (1) and investigate it by proposing a new theoretical model. We derive an exposure indicator for the operational risk of option hedging and the resulting operational risk distribution. We obtain analytical results for various risk measures including the Value at Risk equation; this includes deriving a new analytical result for the option function of the half-normal distributions (which will be of interest to Statisticians in general). We determine an analytical solution to the price of options under operational risk (2). We conduct numerical experiments on empirical option data to validate our model and estimate the operational Value at Risk for option hedging. There exists currently a good deal of literature on hedging methods for the pricing options. It is well known that in this case a straightforward Monte Carlo simulation algorithm will be time-consuming and yield unstable results for the prices and especially the sensitivities. The knock-in/out features in the barrier option payoffs lead to slower convergence of the Monte Carlo algorithm. To address this problem the following (semi-)analytical approaches have been developed for specific models. In any case, theoretically, long-run uncertainty about demographic changes and shocks restricts the land supply and raises the land price. The consequences for housingpolicies may be significant, especially in countries (likeFrance) where policymakers attempt to develop the housingstock and to make housing more affordable for the less well-off. Now, the spatial demographic risk is local by nature: a shortfall in one place entails a surplus somewhere else; therefore, it is an insurable risk. Consequently, a future insurance market may be a tool with which to improve the way the housing market works. Empirically, we conduct the analysis for a French department, Nord. The data base used in this study is made up of individual transactions for developable land for residential purposes(19,495observations) or secondary or tertiary activities (1,667 observations) between 1989 and 2002.The results show that, during the upward trend in the land market, the price of developable land is significantly higher when the change in population in the vicinity is more volatile, and so more difficult to predict. The plot price is 7.8% (housing market) or 15.3% (office and factory market) higher when population change in the 10 nearest communes increases by one standard deviation. 2. A General Option-Pricing Formula we compare our results with those given by Merton (3).We first explain Merton's results. Merton restricted his discussion to discrete probability distributions. Let an asset have terminal value