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Superprocesses and Mckean-Vlasov Equations with Creation of Mass
Sup erpro cesses and McKean-Vlasov equations with creation of mass L. Overb eck Department of Statistics, University of California, Berkeley, 367, Evans Hall Berkeley, CA 94720, y U.S.A. Abstract Weak solutions of McKean-Vlasov equations with creation of mass are given in terms of sup erpro cesses. The solutions can b e approxi- mated by a sequence of non-interacting sup erpro cesses or by the mean- eld of multityp e sup erpro cesses with mean- eld interaction. The lat- ter approximation is asso ciated with a propagation of chaos statement for weakly interacting multityp e sup erpro cesses. Running title: Sup erpro cesses and McKean-Vlasov equations . 1 Intro duction Sup erpro cesses are useful in solving nonlinear partial di erential equation of 1+ the typ e f = f , 2 0; 1], cf. [Dy]. Wenowchange the p oint of view and showhowtheyprovide sto chastic solutions of nonlinear partial di erential Supp orted byanFellowship of the Deutsche Forschungsgemeinschaft. y On leave from the Universitat Bonn, Institut fur Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany. 1 equation of McKean-Vlasovtyp e, i.e. wewant to nd weak solutions of d d 2 X X @ @ @ + d x; + bx; : 1.1 = a x; t i t t t t t ij t @t @x @x @x i j i i=1 i;j =1 d Aweak solution = 2 C [0;T];MIR satis es s Z 2 t X X @ @ a f = f + f + d f + b f ds: s ij s t 0 i s s @x @x @x 0 i j i Equation 1.1 generalizes McKean-Vlasov equations of twodi erenttyp es. -
Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely1
The Annals ofProbability 1982. Vol. 10, No.3. 828-830 CONDITIONAL GENERALIZATIONS OF STRONG LAWS WHICH CONCLUDE THE PARTIAL SUMS CONVERGE ALMOST SURELY1 By T. P. HILL Georgia Institute of Technology Suppose that for every independent sequence of random variables satis fying some hypothesis condition H, it follows that the partial sums converge almost surely. Then it is shown that for every arbitrarily-dependent sequence of random variables, the partial sums converge almost surely on the event where the conditional distributions (given the past) satisfy precisely the same condition H. Thus many strong laws for independent sequences may be immediately generalized into conditional results for arbitrarily-dependent sequences. 1. Introduction. Ifevery sequence ofindependent random variables having property A has property B almost surely, does every arbitrarily-dependent sequence of random variables have property B almost surely on the set where the conditional distributions have property A? Not in general, but comparisons of the conditional Borel-Cantelli Lemmas, the condi tional three-series theorem, and many martingale results with their independent counter parts suggest that the answer is affirmative in fairly general situations. The purpose ofthis note is to prove Theorem 1, which states, in part, that if "property B" is "the partial sums converge," then the answer is always affIrmative, regardless of "property A." Thus many strong laws for independent sequences (even laws yet undiscovered) may be immediately generalized into conditional results for arbitrarily-dependent sequences. 2. Main Theorem. In this note, IY = (Y1 , Y2 , ••• ) is a sequence ofrandom variables on a probability triple (n, d, !?I'), Sn = Y 1 + Y 2 + .. -
On Hitting Times for Jump-Diffusion Processes with Past Dependent Local Characteristics
Stochastic Processes and their Applications 47 ( 1993) 131-142 131 North-Holland On hitting times for jump-diffusion processes with past dependent local characteristics Manfred Schal Institut fur Angewandte Mathematik, Universitiit Bonn, Germany Received 30 March 1992 Revised 14 September 1992 It is well known how to apply a martingale argument to obtain the Laplace transform of the hitting time of zero (say) for certain processes starting at a positive level and being skip-free downwards. These processes have stationary and independent increments. In the present paper the method is extended to a more general class of processes the increments of which may depend both on time and past history. As a result a generalized Laplace transform is obtained which can be used to derive sharp bounds for the mean and the variance of the hitting time. The bounds also solve the control problem of how to minimize or maximize the expected time to reach zero. AMS 1980 Subject Classification: Primary 60G40; Secondary 60K30. hitting times * Laplace transform * expectation and variance * martingales * diffusions * compound Poisson process* random walks* M/G/1 queue* perturbation* control 1. Introduction Consider a process X,= S,- ct + uW, where Sis a compound Poisson process with Poisson parameter A which is disturbed by an independent standard Wiener process W This is the point of view of Dufresne and Gerber (1991). One can also look on X as a diffusion disturbed by jumps. This is the point of view of Ethier and Kurtz (1986, Section 4.10). Here u and A may be zero so that the jump term or the diffusion term may vanish. -
Stochastic Differential Equations
Stochastic Differential Equations Stefan Geiss November 25, 2020 2 Contents 1 Introduction 5 1.0.1 Wolfgang D¨oblin . .6 1.0.2 Kiyoshi It^o . .7 2 Stochastic processes in continuous time 9 2.1 Some definitions . .9 2.2 Two basic examples of stochastic processes . 15 2.3 Gaussian processes . 17 2.4 Brownian motion . 31 2.5 Stopping and optional times . 36 2.6 A short excursion to Markov processes . 41 3 Stochastic integration 43 3.1 Definition of the stochastic integral . 44 3.2 It^o'sformula . 64 3.3 Proof of Ito^'s formula in a simple case . 76 3.4 For extended reading . 80 3.4.1 Local time . 80 3.4.2 Three-dimensional Brownian motion is transient . 83 4 Stochastic differential equations 87 4.1 What is a stochastic differential equation? . 87 4.2 Strong Uniqueness of SDE's . 90 4.3 Existence of strong solutions of SDE's . 94 4.4 Theorems of L´evyand Girsanov . 98 4.5 Solutions of SDE's by a transformation of drift . 103 4.6 Weak solutions . 105 4.7 The Cox-Ingersoll-Ross SDE . 108 3 4 CONTENTS 4.8 The martingale representation theorem . 116 5 BSDEs 121 5.1 Introduction . 121 5.2 Setting . 122 5.3 A priori estimate . 123 Chapter 1 Introduction One goal of the lecture is to study stochastic differential equations (SDE's). So let us start with a (hopefully) motivating example: Assume that Xt is the share price of a company at time t ≥ 0 where we assume without loss of generality that X0 := 1. -
THE MEASURABILITY of HITTING TIMES 1 Introduction
Elect. Comm. in Probab. 15 (2010), 99–105 ELECTRONIC COMMUNICATIONS in PROBABILITY THE MEASURABILITY OF HITTING TIMES RICHARD F. BASS1 Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009 email: [email protected] Submitted January 18, 2010, accepted in final form March 8, 2010 AMS 2000 Subject classification: Primary: 60G07; Secondary: 60G05, 60G40 Keywords: Stopping time, hitting time, progressively measurable, optional, predictable, debut theorem, section theorem Abstract Under very general conditions the hitting time of a set by a stochastic process is a stopping time. We give a new simple proof of this fact. The section theorems for optional and predictable sets are easy corollaries of the proof. 1 Introduction A fundamental theorem in the foundations of stochastic processes is the one that says that, under very general conditions, the first time a stochastic process enters a set is a stopping time. The proof uses capacities, analytic sets, and Choquet’s capacibility theorem, and is considered hard. To the best of our knowledge, no more than a handful of books have an exposition that starts with the definition of capacity and proceeds to the hitting time theorem. (One that does is [1].) The purpose of this paper is to give a short and elementary proof of this theorem. The proof is simple enough that it could easily be included in a first year graduate course in probability. In Section 2 we give a proof of the debut theorem, from which the measurability theorem follows. As easy corollaries we obtain the section theorems for optional and predictable sets. This argument is given in Section 3. -
Laws of Total Expectation and Total Variance
Laws of Total Expectation and Total Variance Definition of conditional density. Assume and arbitrary random variable X with density fX . Take an event A with P (A) > 0. Then the conditional density fXjA is defined as follows: 8 < f(x) 2 P (A) x A fXjA(x) = : 0 x2 = A Note that the support of fXjA is supported only in A. Definition of conditional expectation conditioned on an event. Z Z 1 E(h(X)jA) = h(x) fXjA(x) dx = h(x) fX (x) dx A P (A) A Example. For the random variable X with density function 8 < 1 3 64 x 0 < x < 4 f(x) = : 0 otherwise calculate E(X2 j X ≥ 1). Solution. Step 1. Z h i 4 1 1 1 x=4 255 P (X ≥ 1) = x3 dx = x4 = 1 64 256 4 x=1 256 Step 2. Z Z ( ) 1 256 4 1 E(X2 j X ≥ 1) = x2 f(x) dx = x2 x3 dx P (X ≥ 1) fx≥1g 255 1 64 Z ( ) ( )( ) h i 256 4 1 256 1 1 x=4 8192 = x5 dx = x6 = 255 1 64 255 64 6 x=1 765 Definition of conditional variance conditioned on an event. Var(XjA) = E(X2jA) − E(XjA)2 1 Example. For the previous example , calculate the conditional variance Var(XjX ≥ 1) Solution. We already calculated E(X2 j X ≥ 1). We only need to calculate E(X j X ≥ 1). Z Z Z ( ) 1 256 4 1 E(X j X ≥ 1) = x f(x) dx = x x3 dx P (X ≥ 1) fx≥1g 255 1 64 Z ( ) ( )( ) h i 256 4 1 256 1 1 x=4 4096 = x4 dx = x5 = 255 1 64 255 64 5 x=1 1275 Finally: ( ) 8192 4096 2 630784 Var(XjX ≥ 1) = E(X2jX ≥ 1) − E(XjX ≥ 1)2 = − = 765 1275 1625625 Definition of conditional expectation conditioned on a random variable. -
Probability Cheatsheet V2.0 Thinking Conditionally Law of Total Probability (LOTP)
Probability Cheatsheet v2.0 Thinking Conditionally Law of Total Probability (LOTP) Let B1;B2;B3; :::Bn be a partition of the sample space (i.e., they are Compiled by William Chen (http://wzchen.com) and Joe Blitzstein, Independence disjoint and their union is the entire sample space). with contributions from Sebastian Chiu, Yuan Jiang, Yuqi Hou, and Independent Events A and B are independent if knowing whether P (A) = P (AjB )P (B ) + P (AjB )P (B ) + ··· + P (AjB )P (B ) Jessy Hwang. Material based on Joe Blitzstein's (@stat110) lectures 1 1 2 2 n n A occurred gives no information about whether B occurred. More (http://stat110.net) and Blitzstein/Hwang's Introduction to P (A) = P (A \ B1) + P (A \ B2) + ··· + P (A \ Bn) formally, A and B (which have nonzero probability) are independent if Probability textbook (http://bit.ly/introprobability). Licensed and only if one of the following equivalent statements holds: For LOTP with extra conditioning, just add in another event C! under CC BY-NC-SA 4.0. Please share comments, suggestions, and errors at http://github.com/wzchen/probability_cheatsheet. P (A \ B) = P (A)P (B) P (AjC) = P (AjB1;C)P (B1jC) + ··· + P (AjBn;C)P (BnjC) P (AjB) = P (A) P (AjC) = P (A \ B1jC) + P (A \ B2jC) + ··· + P (A \ BnjC) P (BjA) = P (B) Last Updated September 4, 2015 Special case of LOTP with B and Bc as partition: Conditional Independence A and B are conditionally independent P (A) = P (AjB)P (B) + P (AjBc)P (Bc) given C if P (A \ BjC) = P (AjC)P (BjC). -
Lecture 19 Semimartingales
Lecture 19:Semimartingales 1 of 10 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 19 Semimartingales Continuous local martingales While tailor-made for the L2-theory of stochastic integration, martin- 2,c gales in M0 do not constitute a large enough class to be ultimately useful in stochastic analysis. It turns out that even the class of all mar- tingales is too small. When we restrict ourselves to processes with continuous paths, a naturally stable family turns out to be the class of so-called local martingales. Definition 19.1 (Continuous local martingales). A continuous adapted stochastic process fMtgt2[0,¥) is called a continuous local martingale if there exists a sequence ftngn2N of stopping times such that 1. t1 ≤ t2 ≤ . and tn ! ¥, a.s., and tn 2. fMt gt2[0,¥) is a uniformly integrable martingale for each n 2 N. In that case, the sequence ftngn2N is called the localizing sequence for (or is said to reduce) fMtgt2[0,¥). The set of all continuous local loc,c martingales M with M0 = 0 is denoted by M0 . Remark 19.2. 1. There is a nice theory of local martingales which are not neces- sarily continuous (RCLL), but, in these notes, we will focus solely on the continuous case. In particular, a “martingale” or a “local martingale” will always be assumed to be continuous. 2. While we will only consider local martingales with M0 = 0 in these notes, this is assumption is not standard, so we don’t put it into the definition of a local martingale. tn 3. -
Probability Cheatsheet
Probability Cheatsheet v1.1.1 Simpson's Paradox Expected Value, Linearity, and Symmetry P (A j B; C) < P (A j Bc;C) and P (A j B; Cc) < P (A j Bc;Cc) Expected Value (aka mean, expectation, or average) can be thought Compiled by William Chen (http://wzchen.com) with contributions yet still, P (A j B) > P (A j Bc) of as the \weighted average" of the possible outcomes of our random from Sebastian Chiu, Yuan Jiang, Yuqi Hou, and Jessy Hwang. variable. Mathematically, if x1; x2; x3;::: are all of the possible values Material based off of Joe Blitzstein's (@stat110) lectures Bayes' Rule and Law of Total Probability that X can take, the expected value of X can be calculated as follows: P (http://stat110.net) and Blitzstein/Hwang's Intro to Probability E(X) = xiP (X = xi) textbook (http://bit.ly/introprobability). Licensed under CC i Law of Total Probability with partitioning set B1; B2; B3; :::Bn and BY-NC-SA 4.0. Please share comments, suggestions, and errors at with extra conditioning (just add C!) Note that for any X and Y , a and b scaling coefficients and c is our http://github.com/wzchen/probability_cheatsheet. constant, the following property of Linearity of Expectation holds: P (A) = P (AjB1)P (B1) + P (AjB2)P (B2) + :::P (AjBn)P (Bn) Last Updated March 20, 2015 P (A) = P (A \ B1) + P (A \ B2) + :::P (A \ Bn) E(aX + bY + c) = aE(X) + bE(Y ) + c P (AjC) = P (AjB1; C)P (B1jC) + :::P (AjBn; C)P (BnjC) If two Random Variables have the same distribution, even when they are dependent by the property of Symmetry their expected values P (AjC) = P (A \ B jC) + P (A \ B jC) + :::P (A \ B jC) Counting 1 2 n are equal. -
Deep Optimal Stopping
Journal of Machine Learning Research 20 (2019) 1-25 Submitted 4/18; Revised 1/19; Published 4/19 Deep Optimal Stopping Sebastian Becker [email protected] Zenai AG, 8045 Zurich, Switzerland Patrick Cheridito [email protected] RiskLab, Department of Mathematics ETH Zurich, 8092 Zurich, Switzerland Arnulf Jentzen [email protected] SAM, Department of Mathematics ETH Zurich, 8092 Zurich, Switzerland Editor: Jon McAuliffe Abstract In this paper we develop a deep learning method for optimal stopping problems which directly learns the optimal stopping rule from Monte Carlo samples. As such, it is broadly applicable in situations where the underlying randomness can efficiently be simulated. We test the approach on three problems: the pricing of a Bermudan max-call option, the pricing of a callable multi barrier reverse convertible and the problem of optimally stopping a fractional Brownian motion. In all three cases it produces very accurate results in high- dimensional situations with short computing times. Keywords: optimal stopping, deep learning, Bermudan option, callable multi barrier reverse convertible, fractional Brownian motion 1. Introduction N We consider optimal stopping problems of the form supτ E g(τ; Xτ ), where X = (Xn)n=0 d is an R -valued discrete-time Markov process and the supremum is over all stopping times τ based on observations of X. Formally, this just covers situations where the stopping decision can only be made at finitely many times. But practically all relevant continuous- time stopping problems can be approximated with time-discretized versions. The Markov assumption means no loss of generality. We make it because it simplifies the presentation and many important problems already are in Markovian form. -
Conditional Expectation and Prediction Conditional Frequency Functions and Pdfs Have Properties of Ordinary Frequency and Density Functions
Conditional expectation and prediction Conditional frequency functions and pdfs have properties of ordinary frequency and density functions. Hence, associated with a conditional distribution is a conditional mean. Y and X are discrete random variables, the conditional frequency function of Y given x is pY|X(y|x). Conditional expectation of Y given X=x is Continuous case: Conditional expectation of a function: Consider a Poisson process on [0, 1] with mean λ, and let N be the # of points in [0, 1]. For p < 1, let X be the number of points in [0, p]. Find the conditional distribution and conditional mean of X given N = n. Make a guess! Consider a Poisson process on [0, 1] with mean λ, and let N be the # of points in [0, 1]. For p < 1, let X be the number of points in [0, p]. Find the conditional distribution and conditional mean of X given N = n. We first find the joint distribution: P(X = x, N = n), which is the probability of x events in [0, p] and n−x events in [p, 1]. From the assumption of a Poisson process, the counts in the two intervals are independent Poisson random variables with parameters pλ and (1−p)λ (why?), so N has Poisson marginal distribution, so the conditional frequency function of X is Binomial distribution, Conditional expectation is np. Conditional expectation of Y given X=x is a function of X, and hence also a random variable, E(Y|X). In the last example, E(X|N=n)=np, and E(X|N)=Np is a function of N, a random variable that generally has an expectation Taken w.r.t. -
Optimal Stopping Time and Its Applications to Economic Models
OPTIMAL STOPPING TIME AND ITS APPLICATIONS TO ECONOMIC MODELS SIVAKORN SANGUANMOO Abstract. This paper gives an introduction to an optimal stopping problem by using the idea of an It^odiffusion. We then prove the existence and unique- ness theorems for optimal stopping, which will help us to explicitly solve opti- mal stopping problems. We then apply our optimal stopping theorems to the economic model of stock prices introduced by Samuelson [5] and the economic model of capital goods. Both economic models show that those related agents are risk-loving. Contents 1. Introduction 1 2. Preliminaries: It^odiffusion and boundary value problems 2 2.1. It^odiffusions and their properties 2 2.2. Harmonic function and the stochastic Dirichlet problem 4 3. Existence and uniqueness theorems for optimal stopping 5 4. Applications to an economic model: Stock prices 11 5. Applications to an economic model: Capital goods 14 Acknowledgments 18 References 19 1. Introduction Since there are several outside factors in the real world, many variables in classi- cal economic models are considered as stochastic processes. For example, the prices of stocks and oil do not just increase due to inflation over time but also fluctuate due to unpredictable situations. Optimal stopping problems are questions that ask when to stop stochastic pro- cesses in order to maximize utility. For example, when should one sell an asset in order to maximize his profit? When should one stop searching for the keyword to obtain the best result? Therefore, many recent economic models often include op- timal stopping problems. For example, by using optimal stopping, Choi and Smith [2] explored the effectiveness of the search engine, and Albrecht, Anderson, and Vroman [1] discovered how the search cost affects the search for job candidates.