Markov Processes

Ernst Eberlein; Jan Kallsen

doi:10.1007/978-3-030-26106-1_5

Mathematical Finance pp 299-336 | Cite as

Markov Processes

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Ernst Eberlein
Jan Kallsen

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First Online: 04 December 2019

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Part of the Springer Finance book series (FINANCE)

Abstract

Most processes in applications are Markov processes or can be viewed as components of multivariate Markov processes. As in discrete time the term Markov refers to a certain lack of memory.

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Appendix 1: Problems

The exercises correspond to the section with the same number.

5.1 (Reflection Principle)

For a Wiener process W set S(t) :=sup_s≤tW(s). Use the strong Markov property of W in order to show that P(W(t) ≤ a − x, S(t) ≥ a) = P(W(t) ≥ a + x) for any a > 0, x ≥ 0, t ≥ 0. Deduce that the joint law of (W(t), S(t)) has density

$$\displaystyle \begin{aligned}\varrho(x,y)=\sqrt{2\over\pi t^3}(2y-x)\exp\bigg(-{(2y-x)^2\over 2t}\bigg)1_{\mathbb R _+}(y-x)1_{\mathbb R _+}(y).\end{aligned}$$

Hint: Show that

$$\displaystyle \begin{aligned}\widetilde W(t):=\left\{\begin{array}{ll} W(t) & \mbox{ for }t\leq\tau,\\ 2a-W(t)& \mbox{ for }t>\tau \end{array}\right.\end{aligned}$$

is a Wiener process for a > 0 and the stopping time $\tau :=\inf \{t\geq 0:W(t)\geq a\}$.

5.2

Show that X is a time-inhomogeneous Markov process if and only if the space-time process $\overline X(t)=(t,X(t))$ is a Markov Process.

5.3

Show that the resolvent mapping R_λ of standard Brownian motion satisfies

$$\displaystyle \begin{aligned}R_\lambda f(x)=\int_{-\infty}^\infty {1\over\sqrt{2\lambda}}e^{-\sqrt{2\lambda}|x-y|}f(y)dy\end{aligned}$$

for $f\in C_0(\mathbb R)$.

5.4

Derive the characteristic function of X(t) for the square-root process solving $X(t)=x+\sqrt {X}\bullet W(t)$ with initial value x > 0 and some Wiener process W.

Hint: Try the ansatz $u(t,x)=\exp (\Psi _0(t,v)+\Psi _1(t,v)x)$ with $\Psi _0,\Psi _1:\mathbb R _+\to \mathbb R$ for the backward equation, where $v\in \mathbb R$ denotes the argument of the characteristic function.

5.5

Derive the probability density function of X(t) for an Ornstein–Uhlenbeck process as in ( 3.87) with L(t) = μt + σW(t) for $\mu \in \mathbb R,\sigma >0$ and some Wiener process W. Suppose that the initial distribution P^X(0) is Gaussian with mean μ₀ and variance $\sigma _0^2$.

Hint: Try the ansatz that X(t) is Gaussian and derive ODEs for its mean and variance.

5.6

Let X denote a solution process to the martingale problem related to (β, γ, 0) for some bounded continuous functions $\beta ,\gamma :\mathbb R\to \mathbb R$ such that γ > 0. Define the process W = γ(X)^−1∕2•X^c. Show that W is a Wiener process and that X solves the SDE $X=X(0)+\beta (X)\bullet I+\sqrt {\gamma (X)}\bullet W$.

5.7

Show that |W|^1∕4 is not a semimartingale for a Wiener process W.

Hint: If X = |W|^1∕4 is a semimartingale, apply Itōs formula to X⁴ = |W| in order to show that the local time of W in 0 equals L⁰ = 1_{W=0}•L⁰ = 0, which contradicts the fact that |W| is not a local martingale.

5.8

Verify the implication 2 ⇒ 1 in Proposition 5.34 and the growth condition in Example 5.44.

Appendix 2: Notes and Comments

For a thorough treatment of Markov process theory one may consult [100, 241, 251, 252]. For Example 5.10 see also [2, Theorem 3.1.9]. Concerning Sect. 5.2 we refer to [241, Exercise III.1.10]. Details on Remark 5.13(2) can be found in [241, Propositions III.2.4 and VII.1.4]. Our definition of the extended generator is close to the one in [241, Definition VII.1.8] and [152, Remarque 13.45]. For Proposition 5.20 see also [241, Definition VIII.3.2 and Proposition VIII.3.3] as well as [152, Remarque 13.46]. A result in the spirit of Remark 5.26 is stated in [226, Theorem 8.2.1]. For a version of Theorem 5.28 for continuous processes see [226, Exercise 8.3]. The approach to studying processes by their symbol is advocated in [36, 151]. Theorem 5.43 and the counterexample in Example 5.44 can be found in [201]. For Problem 5.1 see also [279, Section 3.7.3]. Problem 5.7 is based on [238, Theorem IV.71].

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Authors and Affiliations

Ernst Eberlein
- 1
Jan Kallsen
- 2

1.Department of Mathematical StochasticsUniversity of FreiburgFreiburg im BreisgauGermany
2.Department of MathematicsKiel UniversityKielGermany

Markov Processes

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References

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