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Stochastic Differential Equations Involving Fractional Brownian Motion

Part of the Lecture Notes in Mathematics book series (LNM, volume 1929)

Existence and Uniqueness of Solutions: the Results of Nualart and Răşcanu

Consider the function \(\sigma = \sigma \left( {t,x} \right):\left[ {0,T} \right] \times {\rm R} \to {\rm R}\) satisfying the assumptions: σ is differentiable in x, there exist and for any \(M > 0,0 < \gamma ,\kappa \le 1\) R > 0 there exists M R > 0 such that
  1. (i)
    σ is Lipschitz continuous in x
    $$\left| {\sigma \left( {t,x} \right) - \sigma \left( {t,y} \right)} \right| \le M\left| {x - y} \right|,\forall t \in \left[ {0,T} \right],x,y \in {\rm R};$$
     
  2. (ii)
    x-derivative of σ is local Hölder continuous in x:
    $$\left| {\sigma _x \left( {t,x} \right) - \sigma _x \left( {t,y} \right)} \right| \le M_R \left| {x - y} \right|^\kappa ,\forall \left| x \right|,\left| y \right| \le {\rm R,}t \in \left[ {0,T} \right];$$
     
  3. (iii)
    σ is Hölder continuous in time:
    $$\left| {\sigma \left( {t,x} \right) - \sigma \left( {s,x} \right)} \right| + \left| {\sigma _x \left( {t,x} \right) - \sigma _x \left( {s,x} \right)} \right| \le M\left| {t - s} \right|^\gamma ,\forall x \in {\rm R,}t,s \in \left[ {0,T} \right].$$
     

Let \(0 < \beta < 1/2,f \in W_0^\beta \left[ {0,T} \right],g \in W_1^{1 - \beta } \left[ {0,T} \right].\) We need some preliminary estimates, in addition to Lemmas 2.1.9 and 2.1.10.

Consider on
$$W_0^\beta \left[ {0,T} \right]$$
the norm, equivalent to \(\left\| \cdot \right\|_{0,\beta } :\)
$$\left\| f \right\|_{0,\beta ,\lambda } : = \frac{{\sup }}{{t \in \left[ {0,T} \right]}}e^{ - \lambda t} \varphi _f^\beta \left( t \right).$$

Keywords

Weak Solution Strong Solution Wiener Process Fractional Brownian Motion Orlicz Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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