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Construction and sample path properties of Brownian house-moving between two curves

Kensuke Ishitani, Daisuke Hatakenaka, Keisuke Suzuki

Abstract

This study aims to construct a stochastic process called "Brownian house-moving," which is a Brownian bridge conditioned to stay between two curves. To construct this process, statements are prepared on the weak convergence of conditioned Brownian motions, conditioned Brownian bridges, and conditioned three-dimensional Bessel bridges. Moreover, the sample path properties of Brownian house-moving are studied as well.

Construction and sample path properties of Brownian house-moving between two curves

Abstract

This study aims to construct a stochastic process called "Brownian house-moving," which is a Brownian bridge conditioned to stay between two curves. To construct this process, statements are prepared on the weak convergence of conditioned Brownian motions, conditioned Brownian bridges, and conditioned three-dimensional Bessel bridges. Moreover, the sample path properties of Brownian house-moving are studied as well.

Paper Structure

This paper contains 31 sections, 33 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Main results
  4. Construction and sample path properties of Brownian house-moving
  5. Construction of Brownian meander between two curves
  6. Construction of BES($3$)-bridge between two curves
  7. Proofs of Lemma \ref{['Lem_for_H_gm_to_gp']} and Lemma \ref{['Lem_WeakConv_Meander_btwn_2crvs']}
  8. Proof of Lemma \ref{['Lem_for_H_gm_to_gp']}
  9. Proof of Lemma \ref{['Lem_WeakConv_Meander_btwn_2crvs']}
  10. Preparation for proofs of the main results
  11. Proofs of the main results in Subsection \ref{['subsection_Moving_between_2curves']}
  12. Proof of Theorem \ref{['Thm_Def_and_Decomp_curved_Moving']}
  13. Proof of Corollary \ref{['Cor_Decomp_flat_Moving']}
  14. Proof of Corollary \ref{['Cor_30_curve_max_0t']}
  15. Proof of Corollary \ref{['Cor_30_curve_min_t1']}
  16. ...and 16 more sections

Key Result

Lemma 3.1

Let $0 \leq t_1 < t_2 \leq 1$. $X_{[t_1,t_2]}^{a, b,(g^-, g^+)}$ exists and its distribution is given as follows. For every $\mathbb{R}$-valued bounded continuous function $F$ on $C([t_1,t_2], \mathbb{R})$,

Theorems & Definitions (51)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.1
  • Corollary 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • ...and 41 more