By Alexander N. Starkov

ISBN-10: 0821813897

ISBN-13: 9780821813898

A homogeneous stream is a dynamical process generated through the motion of a closed subgroup $H$ of a Lie team $G$ on a homogeneous house of $G$. The research of such platforms is of significant importance simply because they represent an algebraic version for extra common and extra advanced structures. additionally, there are plentiful functions to different fields of arithmetic, such a lot significantly to quantity concept. the current publication offers an intensive survey of the topic. within the first bankruptcy the writer discusses ergodicity and combining of homogeneous flows. the second one bankruptcy is concentrated on unipotent flows, for which huge development has been made over the last 10-15 years. The fruits of this development was once M. Ratner's celebrated facts of far-reaching conjectures of Raghunathan and Dani. The 3rd bankruptcy is dedicated to the dynamics of nonunipotent flows. the ultimate bankruptcy discusses functions of homogeneous flows to quantity conception, typically to the speculation of Diophantine approximations. particularly, the writer describes intimately the recognized facts of the Oppenheim-Davenport conjecture utilizing ergodic houses of homogeneous flows.

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**Additional info for Dynamical Systems on Homogeneous Spaces**

**Example text**

Suppose the price X(t) at time t of an asset (a property, a stock. . s. ν. 1) If we sell the asset at time s + τ we get the expected discounted net payoﬀ J τ (s, x) := E s,x e−ρ(s+τ ) (X(τ ) − a)X{τ <∞} , where ρ > 0 (the discounting exponent) and a > 0 (the transaction cost) are constants. 2) Φ(s, x) = sup J τ (s, x) = J τ (s, x). 2 to solve this problem as follows: Put S = R × (0, ∞) and s+t Y (t) = , t ≥ 0. X(t) Then ⎡ 1 ⎤ ⎡ 0 ⎦dt +⎣ dY (t) = ⎣ αX(t) βX(t) ⎤ ⎡ ⎢ ⎦dB(t) +⎢ ⎣ ⎤ ⎡ ⎤ s ⎥ ⎥, Y (0) = ⎣ ⎦ ⎦ z N (dt, dz) x 0 γX(t− ) R and the generator A of Y (t) is Aφ(s, x) = ∂φ ∂φ 1 2 2 ∂ 2 φ + αx + β x ∂s ∂x 2 ∂x2 + R φ(s, x + γxz) − φ(s, x) − γxz ∂φ ∂x ν(dz).

4) The corresponding wealth process V φ (t) is deﬁned by V φ (t) = φ0 (t)S0 (t) + φ1 (t)S1 (t), 0 ≤ t ≤ T. 6 Exercises 21 We say that (φ0 , φ1 ) is self-ﬁnancing if V φ (t) is also given by t V φ (t) = V φ (0) + t φ0 (s)dS0 (s) + 0 If, in addition, V φ (t) φ1 (s)dS1 (s). 7) we say that φ is admissible and write φ ∈ A0 . A portfolio φ ∈ A0 is called an arbitrage if V φ (0) = 0, V φ (T ) ≥ 0, and P [V φ (T ) > 0] > 0. 8) Does this market have an arbitrage? 6) to get φ0 (t) = e−rt (V φ (t) − φ1 (t)S1 (t)) and dV φ (t) = rV φ (t)dt + φ1 (t)S1 (t− ) (μ − r)dt + γ z N (dt, dz) .

5) and hence φ(y) = J (ˆu) (y) ≤ Φ(y) for all y ∈ S. 4). 2 (Optimal Consumption and Portfolio in a L´ evy Type Black–Scholes Market [Aa, FØS1]). Suppose we have a market with two possible investments: (i) A safe investment (bond, bank account) with price dynamics dP1 (t) = rP1 (t)dt, P1 (0) = p1 > 0. (ii) A risky investment (stock) with price dynamics dP2 (t) = P2 (t− ) μ dt + σ dB(t) + ∞ z N (dt, dz) , −1 P2 (0) = p2 > 0, where r > 0, μ > 0, and σ ∈ R are constants. We assume that ∞ −1 |z|dν(z) < ∞ and μ > r.

### Dynamical Systems on Homogeneous Spaces by Alexander N. Starkov

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