By Jan A. Freund, Thorsten Pöschel

A set of brief articles which replicate and describe the fields within which stochastic tactics are so much fruitful. obtainable to graduate scholars and pros, this article is pedagogical and self-contained.

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**Example text**

63] with known a and b functions or stochastic processes. If so, the unique solution of the considered SDE takes the form: t t 0 0 ξ (t ) = ξ0 + ∫ a( s)ds + ∫ b( s)dB( s). 64] More generally, we can look for a transformation f in two variables x and t, monotone in t, satisfying the assumptions of Itô’s lemma and such that: df (ξ (t ), t ) = A(t )dt + B(t )dB(t ). 65] Finally, we find by inverse transformation in variable x the form of ξ (t ), t ∈ [ 0, T ]. 4. 1. 2. – The solution ξ = (ξ (t ), t ∈ [ 0, T ]) of this SDE is called an Itô process.

7. 134] and define, for every λ > 0, the following stochastic process X: t t ⎧⎪ ⎫⎪ 1 X (t ) = exp ⎨λξ (t ) − λ ∫ μ (ξ ( s ))ds − λ 2 ∫ σ 2 (ξ ( s ))ds ⎬ , t > 0. 135] 42 Applied Diffusion Processes from Engineering to Finance The main result of Stroock–Varadhan is that, under regular assumptions, the process X is a martingale with respect to the filtration generated by the Brownian motion B and conversely: if, for every λ, X is a martingale with respect to the filtration generated by the Brownian motion B, then the process ξ is a diffusion process.

As for the one-dimensional case, for the applications of such processes in finance, it is interesting to give the interpretations of these last properties: 1) The probability for the process ξ = (ξ (s), s ≥ 0) to have jumps of amplitude more than ε between t and t + h is o( h). Consequently, the process ξ = (ξ (s), s ≥ 0) is continuous in probability. 2) Properties i) and ii) can be rewritten as follows: i) E ⎡⎣ξ (t + h) − ξ (t ) ξ (t ) = x ⎤⎦ = a(x, t ) h + o( h), ii) E ⎡⎣ (ξ (t + h) − ξ (t ))(ξ (t + h) − ξ (t ))τ ξ (t ) = x ⎤⎦ = (bbτ )(x, t ) h + o( h).