Markov Decision Processes with Applications to Finance (Universitext)

Markov Decision Processes with Applications to Finance
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Habermann and T. Sibbett, eds. Concepts, techniques and tools. MR [22] Thomas Mikosch , Non-life insurance mathematics , 2nd ed. An introduction with the Poisson process. MR [23] M.

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Qureshi and H. Sanders, Reward model solution methods with impulse and rate rewards: an algorithmic and numerical results , Performance Evaluation 20 , MR References [1] E. Andersen, An algebraic treatment of fluctuations of sums of random variables , Proc. California Press, Berkeley, MR [2] S. Asmussen, Risk theory in a Markovian environment , Scand. MR [3] S. Asmussen and H. Albrecher, Ruin Probabilities , Adv.

A risk minimization problem for finite horizon semi-Markov decision processes with loss rates

Citation: Qiuli Liu, Xiaolong Zou. Dassios A, Jang J-W. The book presents Markov decision processes in action and includes various state-of-the-art applications with a particular view towards finance. See all 1 customer reviews. For small c the risk loading is the same as in the case of no derivative held, since the probability that the derivative has a positive payoff is small.

Meyn, Risk-sensitive optimal control for Markov decision processes with monotone cost , Math. MR [5] R. Cavazos-Cadena and D. MR [6] W. Ching, X.

Huang, M. Ng, and T. MR [7] K. Chung and F. MR [8] H. MR [9] H. D'Amico, F.

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Gismondi, J. Manca, Discrete-time homogeneous Markov processes for the study of the basic risk processes , Methodol. MR [12] R. De Dominicis and R. Manca, Some new results on the transient behavior of semi-Markov reward processes , Methods Oper. MR [14] R. Howard and J.

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Matheson, Risk-sensitive Markov decision processes , Management Science 18 , MR [15] J. MR [16] J. Janssen and R.

nanicamo.cf MR [17] J. McNeil, R. Frey, and P. In the statement of [ 2 , Theorem 8. For proving Theorem 3. We need to check the conditions given in Definition 3. Now we need to show that the backward equation 11 has a solution and hence also 9 has a solution. Let F be the distribution function of Y 1,1. Expanding the above expression yields the first and the second claim of the lemma.

The right-hand side is finite by Assumption 2. The function h is a bounded linear operator. Thus h is bounded by Lemma 3. Since q is also continuous, the first statement follows. The claim that g is locally Lipschitz follows from Lemma 3. Denote the corresponding Lipschitz constant by L R. The first claim follows from Lemma 3. The latter follows from Lemmas 3. Now we prove that 11 has a unique maximal local solution. Then the backward equation 11 has a unique maximal local solution.

By Lemma 3. By Lemmas 3. Along the lines of the proof of [ 2 , Theorem 8. Given the value of the index c and the amount of wealth x at time t , the maximum expected utility of terminal wealth can be written as. Therefore 7 simplifies to. In particular, p does not depend on x and we omit that parameter from p henceforth.

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The function w 0 does not depend on c. So w 0 is the solution of an ordinary differential equation.

Since w 0 does not depend on c , the backward equation 11 becomes. From 15 we can derive a backward equation for p. Since w 0 does not depend on the second variable, we have. In this section, we present a convenient numerical method for computing the expected value in 17 for the case where the distribution of Y 1,1 has a smooth density.

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We have. The first claim now follows by linearity of the convolution. Our aim is to price a CAT spread option, i. We assume that there are M clients in the market who potentially contribute to the claims process. Let a be the fair annual premium for one client, i.

A risk minimization problem for finite horizon semi-Markov decision processes with loss rates

The annual premium for one contract therefore has to be greater or equal than a , since otherwise the insurance company will make an almost sure loss in the long run. Furthermore, the company faces an exogenously given demand curve d for insurance. For our numerical example we choose. We see that the price increases in c.

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Partially Observable Markov Decision Processes and Piecewise Determinsitic. Markov Decision .. to Finance, Universitext, DOI / 1. zeimurorenou.cf: Markov Decision Processes with Applications to Finance ( Universitext) (): Nicole Bäuerle, Ulrich Rieder: Books.

Further, we observe that in the SC-model the price is always lower than in the CC-model, since the latter more accurately accounts for a clustering of claims. Utility indifference prices p in the SC-model left and in the CC-model right units on the axes are 10 6 units of currency. For small c the risk loading is the same as in the case of no derivative held, since the probability that the derivative has a positive payoff is small.

As time increases the probability that the payoff of the derivative grows in c and hence compensates losses during the remaining time decreases, but also the expected number of claims before T decreases. An interesting observation is that for small c the first effect dominates and hence the risk loading increases, whereas for large c the latter effect dominates and hence the risk loading decreases.