Stochastic Calculus and Semimartingale Model

We model a stream of cash flows as an optional stochastic process, and value the cash flows by using a continuous and strictly positive linear functional. By applying a representation theorem from the general theory of stochastic processes we are able to study this valuation principle, as well as properties of the stochastic discount factor it implies. This approach to valuation is useful in the non-presence of a financial market, as is often the case when valuing cash flows arising from insurance contracts and in the application of real options.

Before coming to the discussion of stochastic integrals, it will be useful to introduce some terminology from the general theory of stochastic processes. Some of the concepts have been defined earlier in these notes. We shall repeat the definitions for the sake of completeness.KeywordsSample PathLocal MartingalePredictable ProcessOptional SamplingMartingale Convergence TheoremThese 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.

Continuous-Time Public Good Contribution Under Uncertainty

d1. Filip Lundberg's works on risk theory were all written at a time when no general theory of stochastic processes existed, and when collective reinsurance methods, in the present day sense of the word, were entirely unknown to insurance companies. In both respects his ideas were far ahead of his time, and his works deserve to be generally recognized as pioneering works of fundamental importance.

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.

In this thesis, stochastic processes closely related to a prominent process called Brownian motion is introduced. The processes are called local Brownian motions. Two rich families are constructed, which can be used to model stochastic dynamical systems in such fields as finance, biology, and physics. Stochastic calculus with respect to local Brownian motion is developed, and strong solutions are presented for some stochastic differential equations driven by the noise generated by local Brownian motions.

This paper addresses the problem of approximating random variables in terms of sums consisting of a real constant and of a stochastic integral with respect to a given semimartingale $X$. The criterion is minimization of $\\mathbf{L}^2$–distance, or "least-squares". This problem has a straightforward and well-known solution when $X$ is a Brownian motion or, more generally, a square-integrable martingale, with respect to the underlying probability measure $P$. We address the general, semimartingale case by means of a duality approach; the adjoint variables in this duality are signed measures, absolutely continuous with respect to $P$, under which $X$ behaves like a martingale. It is shown that this duality is useful, in that the value of an appropriately formulated dual problem can be computed fairly easily; that it "has no gap" (i.e., the values of the primal and dual problems coincide); that the signed measure which is optimal for the dual problem can be easily identified whenever it exists; and that the duality is also "strong", in the sense that one can then identify the optimal stochastic integral for the primal problem. In so doing, the theory presented here both simplifies and extends the extant work on the subject. It has also natural connections and interpretations in terms of the theory of "variance-optimal" and "mean-variance efficient" portfolios in Mathematical Finance, pioneered by H. Markowitz and then greatly extended by H. Föllmer, D. Sondermann and most notably M. Schweizer.

Fractional Brownian Motions, Fractional Noises and Applications

This chapter is a self-contained introduction to the theory of stochastic integration with respect to continuous semimartingales. Apart from some simple facts on the predictable σ-field (which are included with proofs in Section 1.2), we do not need any other result from the general theory of processes. The Doob—Meyer decomposition of the square of a continuous (local) martingale is given by constructing the quadratic variation process directly. Once this is done, the stochastic integral is defined and its properties are obtained in a natural fashion, using arguments common in the theory of the Lebesgue integral, namely, the monotone class theorem and dominated convergence theorem. We first give the definition of the stochastic integral with respect to Brownian motion and then deal with the case of continuous semimartingales in stages. In the last section, we define the integral with respect to a general (r.c.l.l.) semimartingale. However, this time we state without proof some key results, including the Doob—Meyer decomposition. We show that these results, three to be precise, enable us to define the integral in the general case and derive the properties of the integral following the route adopted earlier.KeywordsBrownian MotionDominate Convergence TheoremQuadratic VariationStochastic IntegrationLocal 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.

On March 29, 1900, at the Sorbonne University in Paris, Luis Jean-Baptiste Alphonse Bachelier, at the time a 30-year-old postgraduate student, successfully defended a dissertation titled Theorie de la Speculation. In this work Bachelier singlehandedly introduced the idea that in an efficient market, the absence of unexploited profit opportunities means that the mathematical expectation of the speculator’s gain is zero, and hence, the current price of an asset is the best predictor of its future price. He showed how stock prices followed a stochastic process, developed the mathematics of Brownian motion to model such a process and came up with a formula for pricing options that is not that far from the Nobel Prize-winning solution to the same problem proposed by Fischer Black, Myron Scholes, and Robert Merton in 1973. This was all well before Einstein used essentially the same diffusion equation to model the Brownian motion in 1905, before Andrey Markov began working on continuous-time processes in 1906, and before Andrey Kolmogorov published his 1931 paper laying the foundations of the general theory of Markov processes.

Recently, a new concept for some stochastic process called fractional G-Brownian motion (fGBm) was developed, which generalizes the concepts of standard Brownian motion (Bm), fractional Brownian motion (fBm) and G-Brownian motion (GBm) under the framework of sublinear expectation. The fGBm can exhibit long-range dependence and feature the volatility uncertainty of financial markets simultaneously. Thus it can be a better alternative stochastic process in the financial applications. In this paper, some financial markets driven by the fGBm were considered and the corresponding arbitrage opportunities were discussed. Specifically, the financial market M that consists a risk free asset and a risky asset will admit some arbitrage opportunity if the stochastic integrals with respect to the fGBm were established in the path-wise sense, and the arbitrage opportunity can be excluded if the stochastic integrals were established in the Wick calculus sense. What is more, the arbitrage opportunity for the financial market ( M , C ) , consisting of the original market M and some contingent claim C were also investigated in the general case, and some interval of free arbitrage prices for the claim were derived by applying the generalized fractional G-Girsanov theorem and fractional G-Clark-Ocone theorem. This study generalized the well-known existing results of arbitrage theory in financial markets driven by the standard Bm, fBm and GBm.

The aims of the 2019 workshop Heat Kernels, Stochastic Processes and Functional Inequalities were: (a) to provide a forum to review recent progresses in a wide array of areas of analysis (elliptic, subelliptic and parabolic differential equations, transport, functional inequalities), geometry (Riemannian and sub-Riemannian geometries, metric measure spaces, geometric analysis and curvature), and probability (Brownian motion, Dirichlet spaces, stochastic calculus and random media) that have natural common interests, and (b) to foster, encourage and develop further interactions and cross-fertilization between these different directions of research.

It is known that any functional of Brownian motion with finite second moment can be expressed as the sum of a constant and an Ito stochastic integral. It is also known that homogeneous additive functionals of Brownian motion with finite expectations have a similar representation. This paper extends these results in several ways. It is shown that any finite functional of Brownian motion can be represented as a stochastic integral. This representation is not unique, but if the functional has a finite expectation it does have a unique representation as a constant plus a stochastic integral in which the process of indefinite integrals is a martingale. A corollary of this result is that any martingale (on a closed interval) that is measurable with respect to the increasing family of $\sigma$-fields generated by a Brownian motion is equal to a constant plus an indefinite stochastic integral. Sufficiently well-behaved Frechet-differentiable functionals have an explicit representation as a stochastic integral in which the integrand has the form of conditional expectations of the differential.

In this dissertation, we focus mainly on the further study of the new stochastic integral introduced by Ayed and Kuo in 2008. Several properties of this new stochastic integral are obtained. We first introduce the concept of near-martingale for non-adapted stochastic processes. This concept is a generalization of the martingale property for adapted stochastic processes in the It\^o theory. We prove a special case of It\^o isometry for the stochastic integral of certain instantly independent processes. We obtain some formulas for expressing a new stochastic integral in terms of It\^o integrals and Riemann integrals. Several generalized versions of It\^o's formula for the new stochastic integral obtained by Ayed and Kuo are given. We also provide some examples to illustrate the ideas.

We define a fractional Itô stochastic integral with respect to a randomly scaled fractional Brownian motion via an S -transform approach. We investigate the properties of this stochastic integral, prove an Itô formula for functions of such stochastic integrals and apply this Itô formula to the investigation of related generalized time-fractional evolution equations. We show that the constructed stochastic integrals serve as stochastic solutions for a large class of integro-differential evolution equations containing non-local operators acting on the time variable. Our results cover such special cases of randomly scaled fractional Brownian motion as gamma-grey Brownian motion, generalized grey Brownian motion, superstatistical fractional Brownian motion with random diffusion coefficient having generalized gamma distribution, generalized Fox- H processes and some other processes discussed in the literature.