Lie Symmetries Methods in Boundary Crossing Problems for Diffusion Processes
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Lie Symmetries Methods in Boundary Crossing Problems for Diffusion Processes Dmitry Muravey1
Received: 12 March 2019 / Accepted: 3 June 2020 © Springer Nature B.V. 2020
Abstract This paper uses Lie symmetry methods to analyze boundary crossing probabilities for a large class of diffusion processes. We show that if the Fokker–Planck–Kolmogorov equation has non-trivial Lie symmetry, then the boundary crossing identity exists and depends only on parameters of process and symmetry. For time-homogeneous diffusion processes we found the necessary and sufficient conditions of the symmetries’ existence. This paper shows that if a drift function satisfies one of a family of Riccati equations, then the problem has nontrivial Lie symmetries. For each case we present symmetries in explicit form. Based on obtained results, we derive two-parametric boundary crossing identities and prove its uniqueness. Further, we present boundary crossing identities between different process. We show, that if the problem has 6 or 4 group of symmetries then the first passage time density to any boundary can be explicitly represented in terms of the first passage time by a Brownian motion or a Bessel process. Many examples are presented to illustrate the method. Keywords Lie symmetry groups · Diffusion processes · Hitting time · Boundary-crossing probability · First hitting time density
1 Introduction Let b(t) be a function of time, W = {Wt : t ≥ 0} be a standard Brownian motion and X = {Xt : t ≥ 0} be a stochastic process that satisfies the following SDE dXt = μ(Xt , t)dt + σ (Xt , t)dWt ,
X0 = x0 > b(0).
(1)
We define the first passage time T b of the diffusion process X to the curved boundary b(t): T b = inf{t > 0; Xt ≤ b(t)},
B D. Muravey
[email protected]
1
Geolab, Ordzhonikidze street, 12, Moscow, Russia
inf ∅ = ∞.
(2)
D. Muravey
We consider the functions μ(x, t), σ (x, t) and the boundary b(t) belong to some spaces L and C 1 (R):
μ(x, t) ∈ L,
σ (x, t) ∈ L,
b(t) ∈ C 1 (R).
(3)
The spaces L and C (R) are set to guarantee that the equation (1) has a unique non-explosive solution for any initial value x0 and the probability density function of the stopping time T b exists. For example, we can define C 1 (R) and L analogous to [17]. More precisely, let C 1 (R) be the space of real-valued continuously differentiable functions and L be the space of Lipschitz continuous functions R × R+ → R of at most linear growth in x: 1
∀φ(x, t) ∈ L, ∃, K : |φ(x, t)| < (|x| + 1), |φ(x, t) − φ(y, t)| < K|x − y|, |φ(x, t) − φ(x, s)| < K|t − s|, ∀t, s ∈ R+ .
∀x, y ∈ R,
Under the assumption (3) the equation (1) has a unique non-explosive solution for any initial value x0 (see [21]). Using the same arguments as in [17] one can show that the probability density function of the stopping time T b exists. Let us note that the spaces L and C 1 (R) can also be defined as in the papers [27] (see formula (6)) or [40]. In the recent paper [22] authors showed that the first passage time density exists even in much relaxed conditions. We denote Px0
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