stochastic differential equations 5th ed - b oksendal

... open subsets of a topological space Ω (e.g. Ω = R n ), then B = H U is called the Borel σ-algebra on Ω and the elements B ∈ B are called Borel sets. B contains all open sets, all closed sets, ... not hard to show that H X = {X −1 (B) ; B ∈ B} , where B is the Borel σ-algebra on R n . Clearly, X will then be H X -measurable and H X is the smallest σ-algebra with this property. Th...

Ngày tải lên: 31/03/2014, 15:57

... Ornstein-Uhlenbeck (OU)-processes, covering both vector and matrix valued OU-processes, see Sect. 1.7. We will here construct the multi-dimensional OU-process as a time changed and scaled multi-dimensional ... standard d-dimensional OU-process can be obtained by the relation X t+Δ ∼ N d (X t e −Δ , Σ(1 −e −2Δ )). (2.3.2) The m × m Wishart process can be simulated from the non-central Wishart...

Ngày tải lên: 19/02/2014, 22:20

... :Ω→ R n be a random variable. Then U(X):={X −1 (B) |B B} is a σ-algebra, called the σ-algebra generated by X. This is the smallest sub-σ-algebra of U with respect to which X is measurable. Proof. ... of B as containing all the “nice, well-behaved” subsets of R n . DEFINTION. Let (Ω, U,P) be a probability space. A mapping X :Ω→ R n is called an n-dimensional random variable if for each B...

Ngày tải lên: 31/03/2014, 15:56

... :Ω→ R n be a random variable. Then U(X):={X −1 (B) |B B} is a σ-algebra, called the σ-algebra generated by X. This is the smallest sub-σ-algebra of U with respect to which X is measurable. Proof. ... of B as containing all the “nice, well-behaved” subsets of R n . DEFINTION. Let (Ω, U,P) be a probability space. A mapping X :Ω→ R n is called an n-dimensional random variable if for each B...

Ngày tải lên: 08/04/2014, 12:24

... R n ), B  (here B denotes the Borel - < /b> eld) such that the process Ψ (B) is  F B t  -adapted and Z =Ψ (B) Q-a.s. (ii) If  B is a m-dimensional (  F t )-Brownian motion on a filtered proba- bility ... (Z, B) of adapted processes on a filtered probability space  Ω, G, (G t ) t≥0 , Q  such that (a) B is a m-dimensional (G t )-Brownian motion, i.e., B is a m-dimensional Bro...

Ngày tải lên: 08/04/2014, 12:25

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Ngày tải lên: 08/04/2014, 12:25

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Ngày tải lên: 08/04/2014, 12:25

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Ngày tải lên: 31/03/2014, 15:16

... Garabedian (1964)). 2.1 Physical Classification Partial differential equations < /b> can be classi ed < /b> as equilibrium problems and marching prob- lems. The first class, equilibrium or steady state problems ... family of circles (see figure 6). -3 < /b> -2 < /b> -1 < /b> 0 1 2 3 y -3 < /b> -2 < /b> -1 < /b> 0 1 2 3 x -3 < /b> -2 < /b> -1 < /b> 0 1 2 3 y -3 < /b> -2 <...

Ngày tải lên: 31/03/2014, 15:57

... and Stochastic < /b> Integrals Moreover, since E(XB)t = 1 + X~~(XB) ,-~ < /b> B, we see that E(XB)t = e"t-gt is a continuous martingale. The process E(XB) is sometimes referred to as geometric Brownian ... of real-valued functions M defined on R, we let a{M) denote the space of functions defined on R which are measurable with respect to the a-algebra on R generated by {fP...

Ngày tải lên: 08/04/2014, 12:25