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 of stochastic calculation to investigate correlations of Brownian motion with partial differential equations, including the probabilistic solution of the classic Dirichlet problem. The construction of local time in these settings, the study of their regularity properties, and the proof of the formula for the density of the occupation are very convincing illustrations of the power of stochastic calculus techniques.
Gaussian Random Variables
Theorem 1.1 Let.Xn/n1 be a set of real random variables such that for each n1 n2/-distribution. The random variable X follows theN .m; 2/-distribution, where mDlimmn. ii) The convergence also holds in all Lp spaces,1p<1.
Gaussian Vectors
Furthermore, PX is absolutely continuous with respect to Lebesgue measure on E if and only if rDd, and in that case the density of X. Ifr Based on the definition of the independence of an infinite set of fields, it is sufficient to prove that, if; : : : ;ip2I am different, the fields. Hello Hip/are independent. A orR, the random variable PŒX2Aj.K/ is given by. a) Part (ii) of the statement can be interpreted to say that the conditional distribution ofXknowing.K/isN .pK.X/; 2/. B). On an appropriate probability space.˝;F;P/we can a set.Xi/i2I, indexed by the same index setI, of independentN .0; 1/random variables (see [64, Chapter III] for the existence of such a set – in the sequel we will only need the case when I is countable, and then an elementary construction that only the existence of of Lebesgue measure onŒ0 used; 1 is possible) , and we set. The more recent book [56] by Marcus and Rosen develops striking applications of the well-known results on Gaussian processes to Markov processes and their local times. We begin by introducing "pre-Brownian motion" (this is not canonical terminology), which is easily defined from Gaussian white noise on RC, whose intensity is a Lebesgue measure. Property (iv) of Proposition 2.3 shows that a process having the same finite-dimensional marginal distributions as pre-Brownian motion must also be pre-Brownian motion. It suffices to prove that for every fixed q/, X has a modification whose sample paths are Hölderian with exponent˛. According to the previous remarks, the sample paths of the XQ Hölder process are 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 (possibly random) initial value and to any dimension. Many of the 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 an arbitrary process .Xt/t0, and call the resulting filtration the completed canonical filtration ofX. The reader will easily be able to check whether all the results mentioned in Chapter 2, where we considered the canonical filtration of a Brownian motionB, remain valid if we treat the completed canonical filtration instead. Then one verifies that a process In this way, we get thatA\ fT tg 2FtC DGtand thusA2GT. u t Properties of stop time and related fields. 48 3 Filters and Martingales 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 use of the optional stop. Conversely, the difference of two nondecreasing continuous monotone functions that vanish at0 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. It will be useful for certain uniqueness statements that the initial value of a finite variational process is 74 4 Continuous Semimartingale. Proof From the observations preceding the statement of Theorem 4.2, we know that the sample paths of HA are finite variation functions. A continuous local martingale M such that there exists a random variable Z2 L1 withjMtj Z for each t 0 (in particular a bounded continuous local martingale) is a uniformly integrable martingale. iii). The desired result is an immediate consequence of (ii) since MTn is a continuous local martingale and jMTnj n. From Proposition4.7(ii) we get that this continuous local martingale is a uniformly integrable martingale, therefore. From Proposition 4.7 (ii) again, this continuous local martingale is a uniformly integrable martingale, therefore for every 0,. If A is a finite union of intervals, this follows from (4.7) and another application of the Cauchy-Schwarz inequality. Since every non-negative Borel function is a monotone 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 a continuous local martingale.). Show that the conclusion of the previous question still holds if one just assumes that this is a continuous local martingale with M0D0. Then, we note that the continuous martingales are orthogonal, and their corresponding quadratic variations are given by the orthogonality of the martingaleMias and the last representation formula are easily checked, for example using approximations ehM;Ni). 0 HsKsdhM;Nis: (5.5) The following "associativity" property of stochastic integrals, which is analogous to property (4.1) for integrals with respect to processes of finite variations, is very useful. To complete the proof of the casepD1 of the theorem, it is therefore sufficient to verify this. When XDM is a continuous local martingale, we know from the definition of the quadratic variation that M2 hM; Mi is a continuous 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, According to Theorem 3.7, the process.ˇr/r0 is adjusted with respect to the filtration.Gr/defined byGr DFr for everyr0, andG1D F1. To complete the proof, setqDp=22.0; 1/and integrate each side of the last limit with respect to meroq xq1dx. Proposition 4.7(ii) then shows that the federal local martingale M 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 understand that MDQ is a continuous local martingale underP, and so MQ is a continuous local martingale underQ. A). To see this, it suffices to consider the case in which X DM is a continuous local martingale (underP). The equation between the two ends of the last screen is the Cameron-Martin formula. Justify the definition of stochastic integrals that appear in the definition of eˇt, then show that the process .ˇt/t0is an.Ft/-Brownian motion started from 0. From the preceding formulas we see that the finite-dimensional marginals of the processX are completely determined by the semigroup.Qt/t0and the law forX0 (initial distribution). A central motivation for the introduction of the solver is the fact that it allows one to construct certain supermartingales associated with 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. Remark In general, it is very difficult to determine the exact domain of the generator. QXt.!/ are càdlàg asE-valued mappings, and not only asE-valued mappings (we already know that, for each fixed0,XQt.!/DXt.!/a.s. inEwith probability one, but this does not imply that the sample not paths, and their left borders, remain inE). In addition, we know that the example paths of.Yt/t0 are càdlàg (remember that./D0 by 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˚. 184 6 General Theory of Markov Processes The following theorem gives a complete description of the example paths fromX underPx. Many of the previous results can be extended to Feller Markov processes on a countable state space E. Note, however, that certain difficulties arise in the question of the existence of a process with given transition rates. We should also verify the measurability of the mapping.t;x/7!Qt.x;A/, but this will follow from stronger continuity properties that 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 the -solvent, for any > 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 connections between Brownian motion and partial differential equations. After a brief discussion of the heat equation, we focus on the Laplace equation and on the connections 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
