Reflected Brownian motion with drift in a wedge

Abstract

We study reflecting Brownian motion with drift constrained to a wedge in the plane. Our first set of results provides necessary and sufficient conditions for existence and uniqueness of a solution to the corresponding submartingale problem with drift, and show that its solution possesses the Markov and Feller properties. Next, we study a version of the problem with absorption at the vertex of the wedge. In this case, we provide a condition for existence and uniqueness of a solution to the problem and some results on the probability of the vertex being reached.

Appendix

Lemma 8.1

There exists a function $f_{\varepsilon ,C}\in C_b^2(S)$ satisfying

$$\begin{aligned} f_{\varepsilon ,C}(x,y)= {\left\{ \begin{array}{ll} 0,\ \textrm{if}\ (x,y)\in S\setminus S^{\varepsilon /3},\\ y,\ \textrm{if}\ (x,y)\in S^{2\varepsilon /3}, y\le C, \end{array}\right. } \end{aligned}$$

such that in addition $f_{\varepsilon ,C}(x,0) =0$ for all $x\ge 0$, and $D_i f_{\varepsilon , C}\ge 0$ on $\partial S_i$.

Proof

Let $h_1\in C_b^2({{\mathcal {R}}})$ such that $h_1(x) \ge 0$ for all $x\in {{\mathbb {R}}}$ and

$$\begin{aligned} h_1(x)= {\left\{ \begin{array}{ll} 0,\ \textrm{if}\ x\le \varepsilon /3,\\ 1,\ \textrm{if}\ x\ge 2\varepsilon /3. \end{array}\right. } \end{aligned}$$

Let $h_2\in C_b^2({{\mathcal {R}}})$ such that $h_2(y)=y$ if $y\le C$. Then

$$\begin{aligned} f_{\varepsilon ,C} =h_1\left( x -\frac{y}{\tan \xi }\right) h_2(y)\quad (x,y)\in S \end{aligned}$$

satisfies the requirements of the lemma. Note that for any $\delta >0$ we have $(x,y)\in S^\delta$ if and only if $x-y/\tan \xi \ge \delta .$ Using this fact repeatedly one can verify the above statement by straightforward calculation. $\square$

Remark 8.2

There are slightly different definitions for the term “augmented filtration" in the literature; we use this term as defined in [32], Definition II.67.3. Let $\mathbb {P}$ be an arbitrary probability measure on ${{\mathcal {M}}}$, and let $(C_S, {{\mathcal {F}}}, ({{\mathcal {F}}}_t), \mathbb {P})$ be the augmentation of the probability space $(C_S,{{\mathcal {M}}},({{\mathcal {M}}}_t),\mathbb {P})$, in the above sense. It is known that right-continuous martingales (submartingales) on $(C_S,\mathcal{M},(\mathcal{M}_t),\mathbb {P})$ are also right-continuous martingales (submartingales) on $(C_S,\mathcal{F},(\mathcal{F}_t),\mathbb {P})$ (Lemma II.67.10 in [32]). The probability measure $\mathbb {P}$ in the second probability space is the extension of $\mathbb {P}$ from ${{\mathcal {M}}}$ to ${{\mathcal {F}}}$, without changing the notation. Also, it follows from the martingale characterization of Brownian motion (Theorem 3.3.16 in [22]) that a Brownian motion on $(C_S,\mathcal{M},(\mathcal{M}_t),\mathbb {P})$ is also a Brownian motion on $(C_S,\mathcal{F},(\mathcal{F}_t),\mathbb {P})$.

Remark 8.3

For the convenience of the reader in this remark we shall recall the notion of the Stochastic Differential Equation with Reflection (SDER) from Definition 2.4 of Kang and Ramanan [21]. In that paper the definition appears for arbitrary dimension and for a more general domain, but here we specify it to our case for the domain given by S in 2 dimensions. Let $d:S\rightarrow {{\mathbb {R}}}^2$ be a set-valued map from $\partial S$ to the class of subsets of ${{\mathbb {R}}}^2$ satisfying conditions (d1) and (d2) in Sect. 2.1, then extend it to the interior of S by defining $d(x)=\{0\}$ if x lies in the interior of S. The other inputs of the SDER are the drift and dispersion coefficients $b:S\mapsto {{\mathbb {R}}}^2$ and $\sigma :S\mapsto {{\mathbb {R}}}^{2\times 2}$. A weak solution to the SDER consists of a filtered probability space $(\Omega , {\mathcal F}, \{{{\mathcal {F}}_t}\})$, a family of probability measures $\{\mathbb {P}^z,z\in S\}$, and a pair of continuous adapted processes (Z, W) (each Z and W 2-dimensional), satisfying the following conditions for each $z\in S$:

• 1. W is a standard 2-dimensional Brownian motion on the above filtered probability space starting at zero under $\mathbb {P}^z$;

• 2. $\int _0^t|b(Z(s))|ds +\int _0^t|\sigma (Z(s))|^2ds<\infty$, for all $t\in [0,\infty )$, $\mathbb {P}^z$-a.s.;

• 3. Defining the 2-dimensional processes X and Y as

  $$\begin{aligned} X(t)= z+ \int _0^t b(Z(s))ds +\int _0^t\sigma (Z(s)) dW(s),\quad t\in [0,\infty ) \end{aligned}$$

  (48)

  and $Y=Z-X$, the couple (Z, Y) solves the ESP for X associated with $(S,d(\cdot ))$, almost surely under $\mathbb {P}^z$;

• 4. The set $\{t\in [0,\infty ): Z(t)\in \partial S\}$ has zero Lebesgue measure, $\mathbb {P}^z$-a.s.