Abstract
In the following section we discuss the most general stochastic differential equation considered here, whose solution is a diffusion. Then, linear differential equations (with variable coefficients) will be studied extensively. Here we obtain analytical solutions by Ito’s lemma. We discuss special cases that are widespread in the literature on finance. In the fourth section we turn to numerical solutions allowing to simulate processes.
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Notes
- 1.
Strictly speaking, this is a so-called “strong solution” in contrast to a “weak solution”. For a weak solution the behavior of X(t) is only characterized in distribution. We will not concern ourselves with weak solutions.
- 2.
Normally, one demands that they satisfy a Lipschitz condition. A function f is called Lipschitz continuous if it holds for all x and y that there exists a constant K with
$$\displaystyle{\vert \,f(x) - f(y)\vert \leq K\,\vert x - y\vert \,.}$$We can conceal this condition by requiring the stronger sufficient continuous differentiability.
- 3.
The renaming justifies the assumption regarding the starting value. Consider
$$\displaystyle{ dX(t) = c_{1}(t)\,X(t)\,dt +\sigma _{1}(t)X(t)\,dW(t)\,,\quad X(0)\neq 0\,, }$$with a starting value different from zero, then by division one can normalize \(Z(t) = X(t)/X(0)\).
- 4.
For this purpose we do not need an explicit expression for the process which, however, can be easily obtained from (12.13) with X(0) = 0:
$$\displaystyle{ X(t) = e^{-t}\int _{ 0}^{t} \frac{e^{s}} {\sqrt{1 + s}}dW(s). }$$ - 5.
There is the risk of confusing the symbols \(\sigma _{i},\,i = 1, 2\), from Eq. (12.3) with the ones from Eq. (11.1). Note that the volatility of X 1 (i.e. “\(\sigma _{1}\)”) is given by \(-\sigma _{1}\,X_{1}\) while the volatility term “\(\sigma _{2}\)” of X 2 just reads \(\sigma _{1}\,X_{2} +\sigma _{2}\)!
- 6.
In the literature, one speaks of an Euler approximation. An improvement is known under the keyword Milstein approximation. In order to explain what is meant by “improve” in this case, one would have to become more involved in numerics.
References
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Mikosch, Th. (1998). Elementary stochastic calculus with finance in view. Singapore: World Scientific Publishing.
Øksendal, B. (2003). Stochastic differential equations: An introduction with applications (6th ed.). Berlin/New York: Springer.
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Hassler, U. (2016). Stochastic Differential Equations (SDE). In: Stochastic Processes and Calculus. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-23428-1_12
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DOI: https://doi.org/10.1007/978-3-319-23428-1_12
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