However, the results of this chapter play an important role in chapter 7, where we combine tools from the theory of Markov processes with techniques from stochastic calculus to investigate connections of Brownian motion with partial differential equations, including the probabilistic solution of the classical Dirichlet equations. problem. The construction of local times in this setting, the study of their regularity properties and the proof of the occupation density formula are very convincing illustrations of the power of stochastic calculus techniques.
Gaussian Random Variables
Theorem 1.1 Let.Xn/n1 be a sequence of real random variables such that, for each n1, Xn follows the N .mn; n2/ distribution. The random variable X follows the N .m; 2/ distribution, where mDlimmn. ii) The convergence also holds in all Lpspaces,1p<1.
Gaussian Vectors
Furthermore, PX is absolutely continuous with respect to the Lebesgue measure on E if and only if rDd, and in that case the density of X is . If r From the definition of independence of an infinite collection of fields, it suffices to prove that ifi1; : : : ;ip2Ise differ, fields.Hi Hip/are independent. A of R is the random variable PŒX2Aj.K/ given by. a) Part (ii) of the statement can be interpreted as the conditional distribution of Xknowing.K/N .pK.X/; 2/. b). On a suitable probability space.˝;F;P/ we can construct a set.Xi/i2I, indexed by the same index setI, of independentN .0; 1/random variables (see [64, ch. III] for the existence of such a set – in the sequel we only need the case if I is countable, and then an elementary construction proving only the existence of Lebesgue measure opŒ0 used; 1 possible) , and we set. Marcus and Rosen's more recent book [56] develops striking applications of the known results on Gaussian processes to Markov processes and their local time. We start by introducing the "pre-Brunian motion" (this is not a canonical terminology) which can easily be defined in terms of a Gaussian white noise on RC whose intensity is Lebesgue measure. Property (iv) of Proposition 2.3 shows that a process that has the same finite-dimensional marginal distributions as pre-Brunian motion must also be a pre-Brunian motion. It suffices to prove that for every fixed q/,X has a modification whose sampling paths are Hölder with exponent˛. According to the previous remarks, the sample paths of process XQ are Hölder with exponent˛onŒ0;. Since this holds for any finite collection1; : : : ;tkgof (strictly) positive real,F0Cis independent of .Bt;t> 0/. However,.Bt;t> 0/ D.Bt;t 0/sicB0is the point limit iBtwhen!0. Since F0C.Bt;t0/, we conclude that F0Cis is independent. itself, which gives the desired result. ii). Finally, we extend the definition of Brownian motion to the case of an arbitrary (preferably random) initial value and to an arbitrary dimension. Many previous results can be extended to d-dimensional Brownian motion with an arbitrary starting point. We will often apply this completion procedure to the canonical filtration of a random process .Xt/t0, and call the resulting filtration the completed canonical filtration of X. The reader will easily check that all results stated in Chapter 2, where we considered the canonical filtration of a Brownian motionB, remain valid if we deal with the completed canonical filtration instead. One then verifies that a process X is progressive if and only if the mapping .!;t/7!Xt.!/is measurable on˝Requipped with the-fieldP. In this way we get datA\ fT tg 2FtC DGtand dusA2GT. u t Properties of quiet times and of the related fields. 48 3 Filtrations and Martingale Theorem 3.7 Let .Xt/t0 be a progressive process with values in a measurable space .E;E/, and let T be a stopping time. In the last inequality, we need the fact that f is non-decreasing whenever.Xt/only a. We first establish continuous time analogues of classical inequalities in the discrete time setting. i) (Maximum Inequality) Let.Xt/t0 be a supermartingale with right-continuous sample paths. ii) (Doob's inequality inLp) Let.Xt/t0 be a martingale with right-continuous sample paths. EŒ1A\fT>tgXTDEŒ1A\fT>tgXt^T: By adding this equality to (3.6), we get the desired result. Then prove the claimed formula by the same method as in (c), writing EŒNUa D EŒN0 D 1and noting that the application of the optional stop. Conversely, the difference of two monotone non-decreasing continuous functions that vanish at 0 has finite variation in the sense of the previous definition. We now give a useful approximation lemma for the integral of a continuous function with respect to a function of finite variation. 74 4 Continuous Semimartingales that the initial value of a finite variational process is0 will be convenient for certain uniqueness theorems. Proof By the observations preceding the statement of Theorem 4.2, we know that the sample paths of HA are finite variation functions. A local continuous martingale M such that there exists a random variable Z2 L1 mejMtj Z for every t 0 (in particular a bounded local continuous martingale) is a uniformly integrable martingale. iii). The desired result is an immediate consequence of (ii) since MTn is a continuous local martingale and MTnj n. From Theorem 4.7(ii) we get that this continuous local martingale is a uniformly integrable martingale. Again from Theorem 4.7(ii), this continuous local martingale is a uniformly integrable martingale, so for every. If is a finite union of intervals, it follows from (4.7) and another application of the Cauchy-Schwarz inequality. Since every nonnegative Borel function is a monotonically increasing limit of simple functions with bounded support, an application of the monotone convergence theorem completes the proof. This result was used in the construction of the quadratic variation of the federal local martingale.). Show that the conclusion of the previous question still holds if we just assume that Mi is a federated local martingale with M0D0. We then note that the Miara federal martingales are orthogonal and that their corresponding quadratic variations are given by . the orthogonality of the Mias martingales and the formula of the last representation can be easily verified, for example by using the approximations hM;Ni). 0 HsKsdhM;Nis: (5.5) The following property of "associability" of stochastic integrals, which is analogous to property (4.1) for integrals with respect to finite variational processes, is very useful. To complete the proof of case pD1 of the theorem, it suffices to check that. When XDM is a connected local martingale, we know from the definition of quadratic variation that M2 hM;Mi is a connected local martingale. We begin with a striking characterization of the real Brownian motion as a unique locally continuous martingale M such that hM;Mit D t. In particular, a local continuous martingale M is an an.Ft/-Brownian motion if and only if hM;Mit D t, for every t 0, or equivalently if and only if M2t t is a local continuous martingale. Finally, X is custom and has independent steps with respect to the filtering. Ft/so X is ad-dimensional .Ft/-. By Theorem 3.7, the process.ˇr/r0 is adapted with respect to the filtering.Gr/defined byGr DFr for everyr0, andG1 D F1. To complete the validation, setqDp=22.0; 1/and integrate each side of the last limit with respect to the measure xq1dx. Proposition 4.7(ii) then shows that the local continuous martingaleM, which is dominated by the variable M1, is uniformly integrable. The following is the presentation formula of the theorem and its uniqueness is also simple. ut Consequences Let us state two important consequences of the representation theorem. At the outset of the argument, we can assume that the sample paths M.n/for each are continuous. We get that MDQ is a federal local martingale under P and thus MQ is a federal local martingale under Q. a). To see this, it suffices to consider the case where X D M is a federal local martingale (under P). The equality between the two ends of the last screen is the Cameron-Martin formula. Explain the definition of the stochastic integrals that appear in the definition of ˇt, then show that the process .ˇt/t0 is a .Ft/-Brownian motion starting at 0. From the previous formulas we see that the finite-dimensional marginals of the process X are completely determined by the semigroup.Qt/t0 and the law of X0 (initial distribution). A major motivation for the introduction of the resolvent is the fact that it allows one to construct certain supermartingales related to a Markov process. Then D.L/ is a linear subspace of C0.E/ and LWD.L/!C0.E/ is a linear operator called the generator of the semigroup .Qt/t0. Note In general, it is very difficult to determine the exact domain of the generator. QXt.!/ are cadlàg asE-valued mappings, and not only asE-valued mappings (we already know that for each fixedt0,XQt.!/DXt.!/a.s. inEwith probability one, but this does not mean that the sample paths and their left boundaries remain inE). In addition, we know that the sample paths for.Yt/t0 are càdlàg (remember thath./D0of the convention). We first give an account of the (simple) Markov property, which is a simple extension of the definition of a Markov process. Note The right side of the last display is the composition of Ysand from the mappingy7!EyŒ˚. 174 6 General Theory of Markov Processes The following proposition gives a complete description of the sample paths of X under Px. Many previous results can be extended to Feller Markov processes on the state counter spaceE. However, note that some difficulties arise in the question of the existence of a process with given transition rates. We also need to verify the measurability of mapping.t;x/7!Qt.x;A/, but this will follow from the stronger continuity properties we will establish to verify the Feller property. We write L for the generator of the semigroup.Qt/t0, D.L/ for the domain of L and R for de-resolvent, for each > 0. We refer to Bertoin's monograph [3] for a modern presentation of the theory of Lévy processes. In this chapter, we use the results of the previous two chapters to discuss relationships between Brownian motion and partial differential equations. After a brief discussion of the heat equation, we focus on the Laplace equation uD0 and on the relations between Brownian motion and harmonic functions on a domain of Rd. Brownian Motion and Harmonic Functions Harmonic Functions in a Ball and the Poisson Kernel Transience and Recurrence of Brownian Motion Planar Brownian Motion and Holomorphic Functions Asymptotic Laws of Planar Brownian Motion Motivation and General Definitions The Lipschitz Case Solutions of Stochastic Differential Equations as Markov The Ornstein–Uhlenbeck Process Geometric Brownian Motion Bessel Processes Tanaka’s Formula and the Definition of Local Times Continuity of Local Times and the Generalized Itô Formula Approximations of Local Times The Local Time of Linear Brownian Motion The Kallianpur–Robbins LawGaussian Processes and Gaussian Spaces
Gaussian White Noise
Pre-Brownian Motion
The Continuity of Sample Paths
Properties of Brownian Sample Paths
The Strong Markov Property of Brownian Motion
Filtrations and Processes
Stopping Times and Associated -Fields
Continuous Time Martingales and Supermartingales
Optional Stopping Theorems
Finite Variation Processes
Functions with Finite Variation
Finite Variation Processes
Continuous Local Martingales
The Quadratic Variation of a Continuous Local Martingale
The Bracket of Two Continuous Local Martingales
Continuous Semimartingales
The Construction of Stochastic Integrals
Itô’s Formula
A Few Consequences of Itô’s Formula
Lévy’s Characterization of Brownian Motion
Continuous Martingales as Time-Changed
The Burkholder–Davis–Gundy Inequalities
The Representation of Martingales as Stochastic Integrals
Girsanov’s Theorem
A Few Applications of Girsanov’s Theorem
General Definitions and the Problem of Existence
Feller Semigroups
The Regularity of Sample Paths
The Strong Markov Property
Three Important Classes of Feller Processes
Jump Processes on a Finite State Space
Lévy Processes
Continuous-State Branching Processes
Brownian Motion and the Heat Equation
A Few Examples of Stochastic Differential Equations
