The Shepp–Shiryaev Stochastic Game Driven by a Spectrally Negative Lévy Process

Abstract

This document is the author's final manuscript accepted version of the journal article, incorporating any revisions agreed during the peer review process. Some differences between this version and the published version may remain. You are advised to consult the publisher's version if you wish to cite from it.The Shepp-Shiryaev stochastic game driven by a spectrally negative Lévy process.

Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns

“…Then, we put y 0 (s) = s and define a decreasing sequence ( y n (s)) n∈N such that y 2l−1 (s) = sup{y < y 2l−2 (s) | b * (s, y) > s} and y 2l (s) = sup{y < y 2l−1 (s) | b * (s, y) ≤ s}, whenever they exist, and put y 2l−1 (s) = y 2l (s) = 0, l ∈ N, otherwise. Moreover, we can also define a decreasing sequence ( y n (s)) n∈N such that the boundary b * (s, y) exits the region E 3 from the side of d 3 2 at the points (s − y 2k−1 (s), s, y 2k−1 (s)) and enters E 3 downwards at the points (s − y 2k (s), s, y 2k (s)). Namely, we put y 0 (s) = s and define y 2k−1 (s) = sup{y < y 2k−2 (s) | b * (s, y) < s − y} and y 2k (s) = sup{y < y 2k−1 (s) | b * (s, y) ≥ s − y}, whenever such points exist, and put y 2k−1 (s) = y 2k (s) = 0 otherwise, for k ∈ N. Note that 0 < y 2k (s) < y 2k−1 (s) < s − K , k ∈ N, by construction.…”

“…In addition, the process (X, S, Y ) can exit the region R 3 2l in (3.30) passing to the stopping region D * from (2.9) only through the point (s(y), s(y), y), by hitting the plane d 3 1 , so that increasing its second component S until it reaches the value s(y) = inf{q > s | b * (q, y) ≤ q}. Observe that the boundary b * (q, y) provides the unique solution of the equation in (3.26) with the starting value b * (q, q−) = h * (q), for each q ≤ s(y), given that this solution stays strictly above the surface x = K ∨ (rK/δ(q, y)).…”

“…Then, the limits γ i (y+, y) can be identified with the functions β i (y), i = 1, 2, from Subsection 3.2 above, and the function in (3.36) should satisfy the property V (x, s, y; a * (s, y)) → V (x, y+, y; a * (y+, y)) as s ↓ y , for each s − y ≤ a * (s, y) < x ≤ s. Thus, we conclude that the equalities V (x, y+, y; a * (y+, y)) = U(x, y; a * (y+, y)) and a * (y+, y) = g * (y) (3.39) hold for 0 < a * (y+, y) < x ≤ y and U(x, s; g * (s)) given by (3.18) with g * (s) obtained in part (ii) of Subsection 3.2. To see this, we observe that the candidate value function evaluated at s ↓ y in (3.39) satisfies the normal reflection condition only at the diagonal d 3 3 = {(x, s, y) ∈ R 3 | 0 < x = s = y} of the plane d 3 1 , and thus, the function a * (y+, y) = g * (y) is the maximal solution of the equation in (3.20) with the boundary condition a * (∞, ∞) = g * (∞) of (3.21) as y = s → ∞ and such that this solution stays strictly below the curve x = L ∧ (rL/δ(y)).…”

“…Then, we put s 0 (y) = y and define an increasing sequence ( s n (y)) n∈N such that the boundary a * (s, y) exits the region E 3 from the side of the plane d 3 1 at the points ( s 2l−1 (y), s 2l−1 (y), y) and enters E 3 upwards at the points ( s 2l (y), s 2l (y), y). Namely, we define s 2l−1 (y) = inf{s > s 2l−2 (y) | a * (s, y) > s} and s 2l (y) = inf{s > s 2l−1 (y) | a * (s, y) ≤ s}, l ∈ N, whenever they exist, and put s 2l−1 (y) = s 2l (y) = ∞ otherwise, for l ∈ N. Note that y < s 2l−1 (y) < s 2l (y) ≤ L, l ∈ N, by construction.…”

Further Calculations for the McKean Stochastic Game for a Spectrally Negative Lévy Process: From a Point to an Interval

“…For other examples of stochastic games, see e.g. Kifer [12], Kyprianou [14], Baurdoux and Kyprianou [3], Gapeev and Kühn [10], Baurdoux et al [4]. Note that the McKean game can be seen as an extension of the classic McKean optimal stopping problem (cf.…”