key: cord-0540826-s320rnsy authors: Wang, Yunwen; Li, Jinfeng title: From Reflecting Brownian Motion to Reflected Stochastic Differential Equations: A Systematic Survey and Complementary Study date: 2020-09-08 journal: nan DOI: nan sha: fd508021b3c9b36f7417a7c526fa094e246c28f4 doc_id: 540826 cord_uid: s320rnsy This work contributes a systematic survey and complementary insights of reflecting Brownian motion and its properties. Extension of the Skorohod problem's solution to more general cases is investigated, based on which a discussion is further conducted on the existence of solutions for a few particular kinds of stochastic differential equations with a reflected boundary. It is proved that the multidimensional version of the Skorohod equation can be solved under the assumption of a convex domain (D). Brownian motion and reflecting Brownian motion have a set of properties that hold great potential for game-changing applications, ranging from the mathematical study of queuing models with heavy traffic (Reiman (1984) ), to statistical physics (Lang (1995) ), and the recent advances in statistical mechanics (Grebenkov (2019) ). Underpinned by the Itô's calculus (Malliaris (1983) ), the Brownian motion inspired models arguably lay a foundation for quantitative finance, as evidenced notably in the stock market forecasting (Guo and Li (2019) ), the valuations of options based on the BlackScholes Merton model (Calin (2012) ) in lattice-based computation or Monte Carlo based derivative investment instruments (Zhang (2020) ) with multiple sources of uncertainty. Fundamentally, the reflecting Brownian motion can be characterised by the Skorohod problem into solving a stochastic differential equation (SDE) with a reflecting boundary condition. Over decades, there has been a continuous research campaign to attempt solutions of the Skorohod stochastic differential equation with reflecting (Saisho (1987) ), semi-reflecting (Kanagawa (2009) ), or absorption (Berestycki et al. (2014) ) boundary conditions. By way of illustration, (DAuria and Kella (2012)) reports a reflected Markov-modulated Brownian motion with a two-sided reflection, which generalises the reflected Brownian motion to the Markov modulated case. More recently, Monte Carlo simulation was employed by (Malsagov and Mandjes (2019) ) for closed-form approximations of the mean and variance of fractional Brownian motion reflected at level 0, the problem of which explicit expressions and numerical methods struggle to address. Apart from the discretization error reported by (Asmussen et al. (1995) ), there remain massive technological gaps that challenge the conventional statistical thinking in tailoring the reflecting Brownian motion and its properties for real-life emerging applications, such as the contact tracing (Li and Guo (2020) ) for the coronavirus disease , the statistical clutter modelling and phased array signal processing for 5G communications (Li (2020) ) and beyond. This paper is organised into the following sections. First, the elements of Brownian Motion and the stochastic integral are elaborated in sections 2 and 3, respectively. Section 4 presents the survey and insights into the reflecting Brownian motion, followed by the in-depth discussion of reflected stochastic differential equations in section 5. In this section, we will introduce the definition of Brownian motion and reflecting brownian motion, which are the most important stochastic process that is widely used in applications. Definition 2.1.1(Brownian Motion): A stochastic process X=(X t ,t≥0) is called a d-dimensional Brownian motion(or Wiener process) with the initial probability law µ, if (i) X t is continuous in t almost surely and X 0 has the distribution law µ on R d ; (ii) X t has independent increments, that is for any 0≤ t 0 < t 1 < ... < t n , the random variables, are independent with each other; (iii) for all 0≤ s < t, Then, for every 0< t 1 < ... < t m and A i ∈ B(R d ), i=1,2,...,m, we have the probability as being the probability distribution function of a d-dimensional Gaussian distribution. Then we are going to introduce a new concept called Brownian local time which will be used in the upcoming proof. Let X=(X t ) be a one-dimensional Brownian motion defined on the probability space (Ω,F,P). *Definition 2.1.2(Brownian local time) Ikeda and Watanabe (2014) :By the local time or the sojourn time density of X we mean family of nonnegative random variables φ(t, x, ω), t ∈ [0, ∞), x ∈ R 1 such that, with probability one, the following holds: It is clear that if such a family {φ(t,x)} exists, then it is unique and is given by Before moving on to reflecting Brownian motion, we will introduce a concept called Skorohod problem. It is useful when characterising the reflecting Brownian motion. In probability theory, the Skorokhod problem is the problem of solving a stochastic differential equation with a reflecting boundary condition. Let us firstly see the one dimensional version. We set t 0 I 0 (g(s))dh(s)=h(t), i.e., h(t) increases only on the set of t when g(t)=0. Proof: Set Then we will prove that g(t) and h(t) satisfy the above conditions (i), (ii) and (iii). According to the definition of g(t) and h(t), (i) apparently holds. So we only need to check (ii) and (iii). To prove (iii), we need to prove that the set I, which is the union of the intervals on which h(t) increases, is included by I 0 (g(t)). Assume that, min 0≤s≤t {x+f(s)} decreases on interval I', i.e. min 0≤s≤b {x+f(s)} ≤min 0≤s≤a {x+f(s)} for all b>a, where a,b ∈I'. When b−→a, since h(t) and f(t) are both continuous. Therefore, g(a)=0, g(b)=0. Then because a,b ∈I' are arbitrary, g(t)=0 on I', i.e. I ⊆ I 0 (g(t)). Hence (iii) holds. We shall prove the uniqueness. Suppose g 1 (t) and h 1 (t) ∈ C + also satisfy the condition (i), (ii) and (iii). Then If there exists t 1 >0 such that g(t 1 )-g 1 (t 1 ) >0, we set t 2 = max{ t< t 1 ; g(t)-g 1 (t)=0}. Then g(t)> g 1 (t)≥0 for all t ∈ (t 2 , t 1 ] and hence, by (iii), This is contradiction. Therefore g(t)≤ g 1 (t) for all t ≥0. By symmetry, g(t)≥ g 1 (t) for all t ≥0. Hence g(t)≡ g 1 (t) and so h(t)≡ h 1 (t). From the name, we can easily see that this Brownian motion has bounds and gets reflected when it goes beyond the bounds. Definition 2.3.1(Reflecting Brownian motion) Ikeda and Watanabe (2014) : Let X=(X t ) be a one-dimensional Brownian motion and let X + =(X + t ) be a continuous stochastic process on [0,∞) defined by Then we can easily infer that the event and µ + is the probability law of X + 0 = |X + 0 |. The process X + is called the one-dimensional reflecting Brownian motion. An illustrated one-dimensional Brownian motion and the associated reflected path are plotted in Figure 1 . There are different ways to characterise reflecting Brownian motion. We will present a characterisation due to Skorohod problem. (2014): Let {X(t), B(t),φ(t)} be a system of real continuous stochastic processes defined on a probability space such that B(t) is a one-dimensional Brownian motion with B(0)=0, X(0) and process B(t) are independent and with probability one the following holds: Then X=X(t) is a reflecting Brownian motion on [0,∞). Equation (2.1) is called the Skorohod equation which will be introduced in the next section. Proof: By Lemma 2.2.1, X = X(t) and φ = φ(t) are uniquely determined by X(0) and B = B(t): To prove this theorem we only need to show that if x t is a one-dimensional Brownian motion, then X(t) = |x t | satisfies, with some processes B(t) and φ(t), the above properties. Let g n (x) be a nonnegative continuous function on R 1 with support Then we can see that u n ∈ C 2 (R), u n '(x) = d|x| dx |x| 0 g n (z)dz. And then |u n | ≤ | d|x| dx || ∞ 0 g n (z)dz| ≤1, u n (0)=0, u n (x) ↑ |x|. Then because lim n−→∞ 1 n =0 and non-negative function g n has support in (0,1/n), By Itô's formula (will be introduced in the following section), Therefore {X(t), B(t), φ(t)} satisfies all conditions in Theorem 2.1. Thus X-(X(t)) and φ=(φ(t)) are characterized as X = X(0)+B(t)-min 0≤s≤t {(X(0)+B(s))∧0} and φ = -min 0≤s≤t {(X(0)+B(s))∧0}. Proof : We will prove this theorem by using stochastic calculus. Let (F t ) = (F X t ) be the proper reference family of X. Then X is an (F t )-Brownian motion and X t − X 0 belong s to space M. Let g n (x) be a continuous function on R 1 such that its support is contained in (-1/n+a,1/n+a), g n (x) ≥ 0, g n (a+x)=g n (a-x) and However, the theorems stated above raise a new problem, i.e. the integration of stochastic process. Note that a continuous stochastic process can be nowhere differentiable, and the necessary condition of ordinary integration is not satisfied, hence the need to find a new integral. A way around the obstacle was found by Itô in the 1940s which will be introduced in the next section. We only give the relative definition and theorem of stochastic integral with respect to Brownian motion T 0 f(t)dB t which is also called Itô stochastic integral. The Itô integral is a random variable since B t and the integrand f(t)(to be precise, f(t,ω)) are random. In order to guarantee the regularity of the integral, we will give some restrictions on f(t). Definition 3.1.1: Denote L 2 to be the set of random variables X, in which E|X| 2 < ∞. Definition 3.1.2(M 2 Stochastic Process): Denote M 2 to be the class of stochastic processes f(t), t≥0, such that T be the class of stochastic processes f(t) such that f(t)∈ M 2 T for any T¿0. Definition 3.1.3(L 2 and M 2 T Norm): For a random variable X the L 2 norm is ||X|| L 2 = E(X 2 ). Also it is easy to verify that it satisfies (i) and (ii) of Theorem 3.1.6. Hence the Itô's Integral exists. Theorem 3.1.7: The following properties holds for any f, g∈ M 2 , any α, β ∈ R and any 0≤ s 0. Consider a partition of [0,T], 0=t n 0 < t n 1 < ... < t n n =T, where t n i = iT n . Denote B t n i by B n i ; the increments B n i+1 − B n i by ∆ n i B; and t n i+1 − t n i by ∆ n i t. n i in each interval [B n i , B n i+1 ] and a pointt n i in each interval [t n i , t n i+1 ] such that F (T, B T ) − F (0, B 0 ) = Σ n−1 i=0 [F (t n i+1 , B n i+1 ) − F (t n i , B n i )] = Σ n−1 i=0 [F (t n i+1 , B n i+1 ) − F (t n i , B n i+1 )] + Σ n−1 i=0 [F (t n i , B n i+1 ) − F (t n i , B n i )], = A 1,n + A 2,n + A 3,n + A 4,n + A 5,n , Since F t , F x , F xx and B t are continuous and bounded functions, we have Now, we have a look at the sum in (3.5): 1. From (3.6), (3.7) and the definition of the Riemann integral, we have in L 2 and thus in probability since the left quantity is the quadratic variation of Brownian motion. Together with the continuity result (3.7), we have the following convergence(in probability): ,...,5 involves different modes: A 1,n and A 2,n converge almost surely, A 3,n and A 4,n converge in L 2 , and A 5,n converges in probability. To combine the results, note that convergence in L 2 implies convergence in probability. Thus all A 3,n , A 4,n and A 5,n converge in probability. Note also that there is a subsequence {n k } k=1,2,... such that {A 3,n k } k=1,2,... converge a.s. Along this subsequence, we can nd a further subsequence n k l such that A 4,n k l converges a.s., and so forth. Finally, all A j,n , j = 1,...,5 converge a.s. with respect to some subsequence m 1 < m 2 ¡..., say. Then Example 3.2.1: F(t,x)=x 3 , the partial derivatives are F t (t, x) = 0, F x (t, x) = 3x 2 and F xx (t, x) = 6x. According to Itô's formula, we have dB 3 t = 3B t dt + 3B 2 t dB t provided 3B 2 t ∈ M ∈ (which has been proved). Looking carefully in to the proof above, we can find that Itô's Lemma also holds for F(t,X t ) where X t is a process with quadratic variation [X t ] satisfying d[X t ]=g(t)dt, for some g(t)∈ M 2 . The following theorem gives Itô's Lemma in general case. Theorem 3.2.2(Itô's formula in general case): Let X t be a stochastic process with quadratic variation x) are continuous for all t≥0 and x∈ R. Also the process g t F x (t, X t ) ∈ M 2 . Then F(t,X t ) can be expressed as Example 3.2.2: For a process X t which satisfying dX t = a t dt + b t dB t , where a t ∈ L 2 and b t ∈ M 2 , and d[ Omitting the term smaller than dt, we have As is introduced in section two, Skorohod equation describes a reflecting Brownian motion X(t,ω) on D=[0,∞) which satisfies that where B is a standard Brownian motion and φ is a continuous stochastic process increasing only when X(t)=0. However, there is a unique solution for (4,1) not only when B is a Brownain motion but also when it is a continuous function with B(0)∈ D. We firstly consider the case in which (4,1) is a multi-dimensional equation and D is a convex domain. An R d -valued function φ(t) = (φ 1 (t), ..., φ d (t)) defined on R + = [0, ∞) is said to be of bounded variation for simplicity if all φ i (t) are of bounded variation on each finite t-interval. For a right continuous function φ(t) with φ(0)=0, we define |φ|(t)= the total variation of φ on [0,t] = supΣ k |φ(t k ) − φ(t k−1 )|, where 0 = t 0 < t 1 < ... < t n = t is a partition. φ(t) can be expressed as where n(t) is a unit vector valued function and is uniquely determined almost everywhere with respect to the measure d|φ|. We Given a function X ∈ D(R + , D), a function φ is said to be associated with X if the following three conditions hold: (i) φ is a function in D(R + , R d ) with bounded variation and φ(0) = 0. (ii) The set {t ∈ R + : X(t) ∈ D} has d|φ|-measure zero. (iii) The function n(t) in (4,2) is a normal vector at X(t) for almost all t with respect to the measure d|φ|. We can also use the following one as a substitute: for any η ∈ C(R + , D), (η(t) − X(t), φ(dt)) ≥0. Then our problem can be expressed as follows: Given w∈ D(R + , R d ) with w(0)∈ D, find a solution X of and it is always assumed that X∈ D(R + , D) and φ is associated with X. In the general multi-dimensional case, the existence of a solution of (4.3) is not trivial. However when w is a step function, we can easily find the solution. For a given point x∈ R d − D, we denote the (unique) point on ∂D which gives the minimum distance between x and D by [x] ∂ . Lemma 4.1.1 Tanaka (2002) : If w is a step function with w(0)∈ D, there exists a solution of (4.3). Proof : To prove the existence, we can try to construct a X(t) satisfying the conditions. Luckily in the case, it is not hard to do that. Let T 1 = inf {t > 0 : w(t) / ∈ D and then define X to be X(t)=w(t) for 0 ≤ t < T 1 and X(T 1 ) = [w(T 1 )] ∂ , in this case φ(t) = 0 for t¡T 1 and φ(T 1 ) = [w(T 1 )] ∂ − w(T 1 ). It solves (4.3) for 0 ≤ t ≤ T 1 . Then we consider t> T 1 . Suppose that the solution of X(t) has been found for 0 ≤ t ≤ T n−1 . Let T n = inf t > T n−1 : w(t) + φ(T n−1 ) / ∈ D, By repeating this step we will get the solution of (4.3) since T n ↑ ∞ as n ↑ ∞. Lemma 4.1.2 Tanaka (2002):(i) Let w,w ∈ D(R + , R d ) with w(0),w(0) ∈ D and X,X be the solution of X=w + φ,X =w +φ respectively. Then we have Poof : Suppose X andX are both solutions of (4.3) and set w =w. By (i) of lemma(4.2) we have |X(t) −X(t)| ≤ |w(t) −w(t)| 2 + 2 t 0 (w(t) −w(t) − w(s) +w(s))(φ(ds) −φ(ds)) = 0. Lemma 4.1.4 Tanaka (2002) : If w is continuous, then the solution of (4.4) is also continuous. Use the inequation in (ii) of lemma(4.2), it is easy to prove the lemma. Lemma 4.1.5 Tanaka (2002) : Let {w n } n≥1 be a sequence in D(R + , R d ) such that for each n the equation X n = w n + φ n has a solution for 0 ≤ t ≤ T , T being a positive constant. If w n converges uniformly on [0,T] to some w ∈ C(R + , R d ) as n −→ ∞ and if {|φ n |(T )} n≥1 is bounded, then X n converges uniformly on [0,T] as n −→ ∞ to the solution X=w + φ for 0 ≤ t ≤ T . Proof : Assume |φ n |(t) ≤ K for each n. Then by (i) of lemma(4.2), we have Since the sequence {w n } n≥1 is uniformly convergent to w, X n = w n + φ n also converges uniformly to X=w + φ. Then we prove that X is the solution of (4.3). To prove this is just to prove φ is associated with X. The condition (i) is satisfied obviously since |φ n |(T) is bounded. Condition (ii) is also trivial. Then we try to verify condition (iii). Let η ∈ C(R + , D) and notice that for 0 ≤ t 1 < t 2 ≤ T | Proof : For any constant T>0, there is a constant N such that sup n max 0≤t≤T |X n (t)| < N For such N both X n and X are the solutions of (4.3) when 0 ≤ t ≤ T . Then we construct a domain D N = D ∩ {|x| < N } and it satisfies condition B. Hence by lemma 4.7 X is the solution of (4.3) for D N and so for D. We aim to verify the existence of solution of (4.3) without satisfying condition B. Let (Ω, F, P ) be a complete probability space with an increasing family {F} t≥0 of sub-σ-fields of F where F t = ∪ >0 F t+ . Let D be a convex domain. Then we will extend the theorems and lemmas before to stochastic case. Theorem 4.2.1 Tanaka (2002): Let M(t) be an R d -valued process with M(0)∈ D such that each component is a continuous local F t -martingale and A(t) be an R d valued, continuous and F t -adapted process of bounded variation with A(0)=0. then there exists a unique F t -adapted solution {X(t)} of Moreover, for f ∈ C 2 (R with f ≥ 0 on R + and 0 ≤ s ≤ t we have f (|X(t) − X(s)| 2 ) (17) where f' and f" are evaluated at |X(τ ) − X(s)| 2 and [M i , M j ] denotes the quadratic variation process. By a solution of (4.4), we mean a D-valued process {X(t)} which satisfies (4.4) almost surely, under the condition that almost all sample paths of {Φ(t)} are associated with those of {X(t)}. Let D be a convex domain in R d and {Ω, F, P ; F t } satisfy the same condition as in the last subsection. B(t)=(B 1 (t), ..., B r (t)) with B(0)=0 is an F t -adapted r-dimensional Brownian motion and for 0 ≤ s ≤ t, ε ∈ R d E[e i(ε,B(t)−B(s)) |F s ] = e −(t−s)|ε| 2 2 , a.s. Let σ(t, x) = {σ i k (t, x)} be an R d R r -valued function and b(t, x) = {b i (t, x)} be an R d -valued function, both being defined on R + × D. Then we consider the stochastic differential equation with reflection dX = σ(t, X)dB + b(t, X)st + dΦ, X(0) = x, where x=(x 1 , ..., x d ) ∈ D. The aim is to find an F t -adapted D-process {X(t)} under the condition that {Φ(t)} is an associated process of {X(t)}. σ(t, x) and b(t,x) are always assumed to be Borel measurable in (t,x). Theorem 4.3.1 Tanaka (2002): If there exists a constant K>0 such that ||σ(t, x)|| ≤ K(1 + |x| 2 ) 1 2 , ||b(t, x)|| ≤ K(1 + |x| 2 ) 1 2 (20) then there exists a (pathwise) unique F t -adapted solution of (4.6) for any x∈ D. Before proving theorem 4.3.1, we introduce an inequation first. Lemma 4.3.2 Tanaka (2002): Replace w,w in (i) of lemma 4.1.2 by w + a,w +ã, respectively, where a andã are R d -valued right continuous functions of bounded variation with a(0)=ã(0)=0, then |X(t) −X(t)| 2 ≤ |w(t) −w(t)| 2 + 2 t 0 (X(s) −X(s))(a(ds) −ã(ds)) +2 t 0 (w(t) −w(t) − w(s) +w(s))(a(ds) −ã(ds)). By a similar replacement of w in (ii) of lemma 4.1.2 by w + a, we have the following inequation |X(t) − X(s)| 2 ≤ |w(t) − w(s)| 2 + 2 t 0 (X(τ ) − X(s))a(dτ ) +2 (s,t] (w(t) − w(τ ))(a(dτ ) + φ(dτ )). Theorem 4.3.3 Tanaka (2002): If σ(t,x) and b(t,x) are bounded continuous on R + × D, then on some probability space (Ω, F, P ) we can find an rdimensional Brownain motion {B(t)} in such a way that (4.6) has a solution. Discretization error in simulation of one-dimensional reflecting brownian motion Critical branching brownian motion with absorption: survival probability. Probability Theory and Related Fields An introduction to stochastic calculus with applications to finance Markov modulation of a two-sided reflected brownian motion with application to fluid queues Probability distribution of the boundary local time of reflected brownian motion in euclidean domains A novel twitter sentiment analysis model with baseline correlation for financial market prediction with improved efficiency Stochastic differential equations and diffusion processes Numerical analysis of reflecting brownian motion and a new model of semi-reflecting brownian motion with some domains Effective conductivity and skew brownian motion Low-loss tunable dielectrics for millimeter-wave phase shifter: from material modelling to device prototyping Global deployment mappings and challenges of contacttracing apps for covid-19. Available at SSRN 3609516 Itôs calculus in financial decision making Approximations for reflected fractional brownian motion Open queueing networks in heavy traffic Stochastic differential equations for multi-dimensional domain with reflecting boundary Stochastic differential equations with reflecting boundary condition in convex regions The value of monte carlo model-based variance reduction technology in the pricing of financial derivatives Now we move on to the problem of existence of solution for (4.3) under the assumption that w ∈ C(R + , R d ).Here, we only introduce two special conditions where the solution exists. Condition A: There is a unit vector e and a constant c>0 such that ( e, n) ≥c for any n ∈ ∪ y∈∂d N y (D. Lemma 4.1. 6 Tanaka (2002) : Assume that D satisfies the condition A. Then there exists a solution X of (4.3) for any w ∈ C(R + , R d ), and for 0 ≤ s 0 and δ >0 such that for any x ∈ ∂D we can find an open ball B (x 0 ) = y ∈ R d : |y − x 0 | < satisfying B (x 0 ) ⊂ D and |x − x 0 | ≤ δ. Condition B is always satisfied if D is bounded or if d=2. Lemma 4.1.7 Tanaka (2002) : Assume that D satisfies the condition B. Then there exists a unique solution of (4.3) if w ∈ C(R + , R d ), and the solution depends continuously on w with respect to the compact uniform topology. Theorem 4.1.8 Tanaka (2002) : Let D be a general convex domain and {w n } n≥1 be a sequence in C(R + , R d ) such that X n = w n + φ n has a solution for each n. Assume that w n and X n converge to w and X uniformly on compacts as n −→ ∞, respectively. Then X is a solution of (4.3).