A Short History of Stochastic Integration and Mathematical Finance
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On Two Dimensional Markov Processes with Branching Property^)
ON TWO DIMENSIONAL MARKOV PROCESSES WITH BRANCHING PROPERTY^) BY SHINZO WATANABE Introduction. A continuous state branching Markov process (C.B.P.) was introduced by Jirina [8] and recently Lamperti [10] determined all such processes on the half line. (A quite similar result was obtained independently by the author.) This class of Markov processes contains as a special case the diffusion processes (which we shall call Feller's diffusions) studied by Feller [2]. The main objective of the present paper is to extend Lamperti's result to multi-dimensional case. For simplicity we shall consider the case of 2-dimensions though many arguments can be carried over to the case of higher dimensions(2). In Theorem 2 below we shall characterize all C.B.P.'s in the first quadrant of a plane and construct them. Our construction is in an analytic way, by a similar construction given in Ikeda, Nagasawa and Watanabe [5], through backward equations (or in the terminology of [5] through S-equations) for a simpler case and then in the general case by a limiting procedure. A special attention will be paid to the case of diffusions. We shall show that these diffusions can be obtained as a unique solution of a stochastic equation of Ito (Theorem 3). This fact may be of some interest since the solutions of a stochastic equation with coefficients Holder continuous of exponent 1/2 (which is our case) are not known to be unique in general. Next we shall examine the behavior of sample functions near the boundaries (xj-axis or x2-axis). -
Chaotic Decompositions in Z2-Graded Quantum Stochastic Calculus
QUANTUM PROBABILITY BANACH CENTER PUBLICATIONS, VOLUME 43 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1998 CHAOTIC DECOMPOSITIONS IN Z2-GRADED QUANTUM STOCHASTIC CALCULUS TIMOTHY M. W. EYRE Department of Mathematics, University of Nottingham Nottingham, NG7 2RD, UK E-mail: [email protected] Abstract. A brief introduction to Z2-graded quantum stochastic calculus is given. By inducing a superalgebraic structure on the space of iterated integrals and using the heuristic classical relation df(Λ) = f(Λ+dΛ)−f(Λ) we provide an explicit formula for chaotic expansions of polynomials of the integrator processes of Z2-graded quantum stochastic calculus. 1. Introduction. A theory of Z2-graded quantum stochastic calculus was was in- troduced in [EH] as a generalisation of the one-dimensional Boson-Fermion unification result of quantum stochastic calculus given in [HP2]. Of particular interest in Z2-graded quantum stochastic calculus is the result that the integrators of the theory provide a time-indexed family of representations of a broad class of Lie superalgebras. The notion of a Lie superalgebra was introduced in [K] and has received considerable attention since. It is essentially a Z2-graded analogue of the notion of a Lie algebra with a bracket that is, in a certain sense, partly a commutator and partly an anticommutator. Another work on Lie superalgebras is [S] and general superalgebras are treated in [C,S]. The Lie algebra representation properties of ungraded quantum stochastic calculus enabled an explicit formula for the chaotic expansion of elements of an associated uni- versal enveloping algebra to be developed in [HPu]. -
Long Term Risk: a Martingale Approach
Long Term Risk: A Martingale Approach Likuan Qin∗ and Vadim Linetskyy Department of Industrial Engineering and Management Sciences McCormick School of Engineering and Applied Sciences Northwestern University Abstract This paper extends the long-term factorization of the pricing kernel due to Alvarez and Jermann (2005) in discrete time ergodic environments and Hansen and Scheinkman (2009) in continuous ergodic Markovian environments to general semimartingale environments, with- out assuming the Markov property. An explicit and easy to verify sufficient condition is given that guarantees convergence in Emery's semimartingale topology of the trading strate- gies that invest in T -maturity zero-coupon bonds to the long bond and convergence in total variation of T -maturity forward measures to the long forward measure. As applications, we explicitly construct long-term factorizations in generally non-Markovian Heath-Jarrow- Morton (1992) models evolving the forward curve in a suitable Hilbert space and in the non-Markovian model of social discount of rates of Brody and Hughston (2013). As a fur- ther application, we extend Hansen and Jagannathan (1991), Alvarez and Jermann (2005) and Bakshi and Chabi-Yo (2012) bounds to general semimartingale environments. When Markovian and ergodicity assumptions are added to our framework, we recover the long- term factorization of Hansen and Scheinkman (2009) and explicitly identify their distorted probability measure with the long forward measure. Finally, we give an economic interpre- tation of the recovery theorem of Ross (2013) in non-Markovian economies as a structural restriction on the pricing kernel leading to the growth optimality of the long bond and iden- tification of the physical measure with the long forward measure. -
A Stochastic Fractional Calculus with Applications to Variational Principles
fractal and fractional Article A Stochastic Fractional Calculus with Applications to Variational Principles Houssine Zine †,‡ and Delfim F. M. Torres *,‡ Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal; [email protected] * Correspondence: delfi[email protected] † This research is part of first author’s Ph.D. project, which is carried out at the University of Aveiro under the Doctoral Program in Applied Mathematics of Universities of Minho, Aveiro, and Porto (MAP). ‡ These authors contributed equally to this work. Received: 19 May 2020; Accepted: 30 July 2020; Published: 1 August 2020 Abstract: We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation. Keywords: fractional derivatives and integrals; stochastic processes; calculus of variations MSC: 26A33; 49K05; 60H10 1. Introduction A stochastic calculus of variations, which generalizes the ordinary calculus of variations to stochastic processes, was introduced in 1981 by Yasue, generalizing the Euler–Lagrange equation and giving interesting applications to quantum mechanics [1]. Recently, stochastic variational differential equations have been analyzed for modeling infectious diseases [2,3], and stochastic processes have shown to be increasingly important in optimization [4]. In 1996, fifteen years after Yasue’s pioneer work [1], the theory of the calculus of variations evolved in order to include fractional operators and better describe non-conservative systems in mechanics [5]. -
Notes on Stochastic Processes
Notes on stochastic processes Paul Keeler March 20, 2018 This work is licensed under a “CC BY-SA 3.0” license. Abstract A stochastic process is a type of mathematical object studied in mathemat- ics, particularly in probability theory, which can be used to represent some type of random evolution or change of a system. There are many types of stochastic processes with applications in various fields outside of mathematics, including the physical sciences, social sciences, finance and economics as well as engineer- ing and technology. This survey aims to give an accessible but detailed account of various stochastic processes by covering their history, various mathematical definitions, and key properties as well detailing various terminology and appli- cations of the process. An emphasis is placed on non-mathematical descriptions of key concepts, with recommendations for further reading. 1 Introduction In probability and related fields, a stochastic or random process, which is also called a random function, is a mathematical object usually defined as a collection of random variables. Historically, the random variables were indexed by some set of increasing numbers, usually viewed as time, giving the interpretation of a stochastic process representing numerical values of some random system evolv- ing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule [120, page 7][51, page 46 and 47][66, page 1]. Stochastic processes are widely used as math- ematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including physical sciences such as biology [67, 34], chemistry [156], ecology [16][104], neuroscience [102], and physics [63] as well as technology and engineering fields such as image and signal processing [53], computer science [15], information theory [43, page 71], and telecommunications [97][11][12]. -
Informal Introduction to Stochastic Calculus 1
MATH 425 PART V: INFORMAL INTRODUCTION TO STOCHASTIC CALCULUS G. BERKOLAIKO 1. Motivation: how to model price paths Differential equations have long been an efficient tool to model evolution of various mechanical systems. At the start of 20th century, however, science started to tackle modeling of systems driven by probabilistic effects, such as stock market (the work of Bachelier) or motion of small particles suspended in liquid (the work of Einstein and Smoluchowski). For us, the motivating questions are: Is there an equation describing the evolution of a stock price? And, perhaps more applicably, how can we simulate numerically a seemingly random path taken by the stock price? Example 1.1. We have seen in previous sections that we can model the distribution of stock prices at time t as p 2 St = S0 exp (µ − σ =2)t + σ tY ; (1.1) where Y is standard normal random variable. So perhaps, for every time t we can generate a standard normal variable, substitute it into equation (1.1) and plot the resulting St as a function of t? The result is shown in Figure 1 and it definitely does not look like a price path of a stock. The underlying reason is that we used equation (1.1) to simulate prices at different time points independently. However, if t1 is close to t2, the prices cannot be completely independent. It is the change in price that we assumed (in “Efficient Market Hypothesis") to be independent of the previous price changes. 2. Wiener process Definition 2.1. Wiener process (or Brownian motion) is a random function of t, denoted Wt = Wt(!) (where ! is the underlying random event) such that (a) Wt is a.s. -
Lecture Notes on Stochastic Calculus (Part II)
Lecture Notes on Stochastic Calculus (Part II) Fabrizio Gelsomino, Olivier L´ev^eque,EPFL July 21, 2009 Contents 1 Stochastic integrals 3 1.1 Ito's integral with respect to the standard Brownian motion . 3 1.2 Wiener's integral . 4 1.3 Ito's integral with respect to a martingale . 4 2 Ito-Doeblin's formula(s) 7 2.1 First formulations . 7 2.2 Generalizations . 7 2.3 Continuous semi-martingales . 8 2.4 Integration by parts formula . 9 2.5 Back to Fisk-Stratonoviˇc'sintegral . 10 3 Stochastic differential equations (SDE's) 11 3.1 Reminder on ordinary differential equations (ODE's) . 11 3.2 Time-homogeneous SDE's . 11 3.3 Time-inhomogeneous SDE's . 13 3.4 Weak solutions . 14 4 Change of probability measure 15 4.1 Exponential martingale . 15 4.2 Change of probability measure . 15 4.3 Martingales under P and martingales under PeT ........................ 16 4.4 Girsanov's theorem . 17 4.5 First application to SDE's . 18 4.6 Second application to SDE's . 19 4.7 A particular case: the Black-Scholes model . 20 4.8 Application : pricing of a European call option (Black-Scholes formula) . 21 1 5 Relation between SDE's and PDE's 23 5.1 Forward PDE . 23 5.2 Backward PDE . 24 5.3 Generator of a diffusion . 26 5.4 Markov property . 27 5.5 Application: option pricing and hedging . 28 6 Multidimensional processes 30 6.1 Multidimensional Ito-Doeblin's formula . 30 6.2 Multidimensional SDE's . 31 6.3 Drift vector, diffusion matrix and weak solution . -
Introduction to Stochastic Calculus
Introduction Conditional Expectation Martingales Brownian motion Stochastic integral Ito formula Introduction to Stochastic Calculus Rajeeva L. Karandikar Director Chennai Mathematical Institute [email protected] [email protected] Rajeeva L. Karandikar Director, Chennai Mathematical Institute Introduction to Stochastic Calculus - 1 Introduction Conditional Expectation Martingales Brownian motion Stochastic integral Ito formula A Game Consider a gambling house. A fair coin is going to be tossed twice. A gambler has a choice of betting at two games: (i) A bet of Rs 1000 on first game yields Rs 1600 if the first toss results in a Head (H) and yields Rs 100 if the first toss results in Tail (T). (ii) A bet of Rs 1000 on the second game yields Rs 1800, Rs 300, Rs 50 and Rs 1450 if the two toss outcomes are HH,HT,TH,TT respectively. Rajeeva L. Karandikar Director, Chennai Mathematical Institute Introduction to Stochastic Calculus - 2 Introduction Conditional Expectation Martingales Brownian motion Stochastic integral Ito formula Rajeeva L. Karandikar Director, Chennai Mathematical Institute Introduction to Stochastic Calculus - 3 Introduction Conditional Expectation Martingales Brownian motion Stochastic integral Ito formula A Game...... Now it can be seen that the expected return from Game 1 is Rs 850 while from Game 2 is Rs 900 and so on the whole, the gambling house is set to make money if it can get large number of players to play. The novelty of the game (designed by an expert) was lost after some time and the casino owner decided to make it more interesting: he allowed people to bet without deciding if it is for Game 1 or Game 2 and they could decide to stop or continue after first toss. -
Stochastic Calculus of Variations for Stochastic Partial Differential Equations
JOURNAL OF FUNCTIONAL ANALYSIS 79, 288-331 (1988) Stochastic Calculus of Variations for Stochastic Partial Differential Equations DANIEL OCONE* Mathematics Department, Rutgers Universiiy, New Brunswick, New Jersey 08903 Communicated by Paul Malliavin Receved December 1986 This paper develops the stochastic calculus of variations for Hilbert space-valued solutions to stochastic evolution equations whose operators satisfy a coercivity con- dition. An application is made to the solutions of a class of stochastic pde’s which includes the Zakai equation of nonlinear filtering. In particular, a Lie algebraic criterion is presented that implies that all finite-dimensional projections of the solution define random variables which admit a density. This criterion generalizes hypoelhpticity-type conditions for existence and regularity of densities for Iinite- dimensional stochastic differential equations. 0 1988 Academic Press, Inc. 1. INTRODUCTION The stochastic calculus of variation is a calculus for functionals on Wiener space based on a concept of differentiation particularly suited to the invariance properties of Wiener measure. Wiener measure on the space of continuous @-valued paths is quasi-invariant with respect to translation by a path y iff y is absolutely continuous with a square-integrable derivative. Hence, the proper definition of the derivative of a Wiener functional considers variation only in directions of such y. Using this derivative and its higher-order iterates, one can develop Wiener functional Sobolev spaces, which are analogous in most respects to Sobolev spaces of functions over R”, and these spaces provide the right context for discussing “smoothness” and regularity of Wiener functionals. In particular, functionals defined by solving stochastic differential equations driven by a Wiener process are smooth in the stochastic calculus of variations sense; they are not generally smooth in a Frechet sense, which ignores the struc- ture of Wiener measure. -
Llogl Condition for Supercritical Branching Hunt Processes
J Theor Probab (2011) 24: 170–193 DOI 10.1007/s10959-010-0322-7 L log L Condition for Supercritical Branching Hunt Processes Rong-Li Liu · Yan-Xia Ren · Renming Song Received: 17 March 2009 / Revised: 17 July 2010 / Published online: 19 October 2010 © Springer Science+Business Media, LLC 2010 Abstract In this paper we use the spine decomposition and martingale change of measure to establish a Kesten–Stigum L log L theorem for branching Hunt processes. This result is a generalization of the results in Asmussen and Hering (Z. Wahrschein- lichkeitstheor. Verw. Geb. 36:195–212, 1976) and Hering (Branching Processes, pp. 177–217, 1978) for branching diffusions. Keywords Hunt processes · Branching Hunt processes · Kesten–Stigum theorem · Martingales · Martingale change of measure Mathematics Subject Classification (2000) Primary 60J80 · 60F15 · Secondary 60J25 1 Introduction Suppose that {Zn; n ≥ 1} is a Galton–Watson process with each particle having prob- ability pn of giving birth to n offspring. Let L stand for a random variable with this := ∞ offspring distribution. Let m n=1 npn be the mean number of offspring per par- n n ticle. Then Zn/m is a non-negative martingale. Let W be the limit of Zn/m as The research of Y.-X. Ren was supported by NSFC (Grant No. 10871103 and 10971003). R.-L. Liu · Y.-X. Ren LMAM School of Mathematical Sciences, Peking University, Beijing, 100871, P.R. China R.-L. Liu e-mail: [email protected] Y.-X. Ren e-mail: [email protected] R. Song () Department of Mathematics, The University of Illinois, Urbana, IL 61801, USA e-mail: [email protected] J Theor Probab (2011) 24: 170–193 171 n →∞. -
Strong Law of Large Numbers for Supercritical Superprocesses Under Second Moment Condition
Front. Math. China 2015, 10(4): 807–838 DOI 10.1007/s11464-015-0482-y Strong law of large numbers for supercritical superprocesses under second moment condition Zhen-Qing CHEN1, Yan-Xia REN2, Renming SONG3, Rui ZHANG4 1 Department of Mathematics, University of Washington, Seattle, WA 98195, USA 2 LMAM School of Mathematical Sciences & Center for Statistical Science, Peking University, Beijing 100871, China 3 Department of Mathematics, University of Illinois, Urbana, IL 61801, USA 4 LMAM School of Mathematical Sciences, Peking University, Beijing 100871, China c Higher Education Press and Springer-Verlag Berlin Heidelberg 2015 Abstract Consider a supercritical superprocess X = {Xt,t 0} on a locally compact separable metric space (E,m). Suppose that the spatial motion of X is a Hunt process satisfying certain conditions and that the branching mechanism is of the form ψ(x, λ)=−a(x)λ + b(x)λ2 + (e−λy − 1+λy)n(x, dy),x∈ E, λ > 0, (0,+∞) ∈ B ∈ B+ ∞ where a b(E),b b (E), and n is a kernel from E to (0, + ) +∞ 2 ∞ P satisfying supx∈E 0 y n(x, dy) < + . Put Ttf(x)= δx f,Xt . Suppose that the semigroup {Tt; t 0} is compact. Let λ0 be the eigenvalue of the (possibly non-symmetric) generator L of {Tt} that has the largest real part among all the eigenvalues of L, which is known to be real-valued. Let φ0 and φ0 be the eigenfunctions of L and L (the dual of L) associated with λ0, respectively. Assume λ0 > 0. Under some conditions on the spatial motion and the φ0-transform of the semigroup {Tt}, we prove that for a large class of suitable functions f, −λ0t lim e f,Xt = W∞ φ0(y)f(y)m(dy), Pμ-a.s., t→+∞ E for any finite initial measure μ on E with compact support, where W∞ is −λ t the martingale limit defined by W∞ := limt→+∞ e 0 φ0,Xt. -
Classical Limit Theorems for Measure-Valued Markov Processes
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector JOURNAL OF MULTIVARIA’IX ANALYSIS 9,234-247 (1979) Classical Limit Theorems for Measure-Valued Markov Processes ALANF.KhRR The Johns Hopkins University Communicated by A. Dwretzky Laws of large numbers, central liiit theorems, and laws of the iterated logarithm are obtained for discrete and continuous time Markov processes whose state space is a set of measures. These results apply to each measure- valued stochastic process itself and not simply to its real-valued functionals. 1. INTRODUCTION In this note we derive classicallimit theorems for Markov chains and Markov processestaking values in a set of measures.The results obtained are strong laws of large numbers, central limit theorems, and laws of the iterated logarithm for the processesthemselves, rather than only for functionals thereof. Of course, the latter classof results follows from those presented here. Our notation and terminology are those of [3]. We denote by E a LCCB space, by 8 the Bore1 u-algebra on E, by C, the Banach space of continuous real-valued functions on E vanishing at infinity and by C, the family of those functions in C, with compact support. Denote by K+ the family of finite measureson B and let K = K+ - K+ . Then, cf. [3, p. 91, K = C$; .f shall denote the Bore1 u-algebra on K generated by the weak* topology induced on K by C’s. The notation m(f) = Jf d m is used frequently below. Stochastic processeswith state spaceK+ we think of as measure-valued.