# The Joint Law of the Extrema, Final Value and Signature of a Stopped Random Walk

@article{Duembgen2015TheJL, title={The Joint Law of the Extrema, Final Value and Signature of a Stopped Random Walk}, author={Moritz Duembgen and L. C. G. Rogers}, journal={arXiv: Probability}, year={2015}, pages={321-338} }

A complete characterization of the possible joint distributions of the maximum and terminal value of uniformly integrable martingale has been known for some time, and the aim of this paper is to establish a similar characterization for continuous martingales of the joint law of the minimum, final value, and maximum, along with the direction of the final excursion. We solve this problem completely for the discrete analogue, that of a simple symmetric random walk stopped at some almost-surely…

## 4 Citations

### On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale

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## References

SHOWING 1-10 OF 10 REFERENCES

### The joint law of the maximum and terminal value of a martingale

- Mathematics
- 1993

SummaryIn this paper, we characterise the possible joint laws of the maximum and terminal value of a uniformly-integrable martingale. We also characterise the joint laws of the maximum and terminal…

### The maximum maximum of a martingale constrained by an intermediate law

- Mathematics
- 2001

Abstract. Let (Mt) be any martingale with M0≡ 0, an intermediate law M1∼μ1, and terminal law M2∼μ2, and let M¯2≡ sup0≤t≤2Mt. In this paper we prove that there exists an upper bound, with respect to…

### The maximum maximum of a martingale

- Mathematics
- 1998

Let (M t )0≤t≤1 be any martingale with initial law M0 ~ μ0 and terminal law M1 ~ μ1 and let S≡sup0≤t≤1M t . Then there is an upper bound, with respect to stochastic ordering of probability measures,…

### A Stochastic Control Approach to No-Arbitrage Bounds Given Marginals, with an Application to Lookback Options

- Mathematics, Economics
- 2013

This work provides a dual formulation which converts the superhedging problem into a continuous martingale optimal transportation problem, and shows that this formulation allows to recover previously known results about Lookback options.

### Robust pricing and hedging of double no-touch options

- MathematicsFinance Stochastics
- 2011

This work establishes model-independent bounds on the price of double no-touch options based on the prices of more liquidly traded options (call and digital call options) and discusses two natural extensions of the notion of arbitrage, weak arbitrage and weak free lunch with vanishing risk which are needed to establish equivalence between the lack of Arbitrage and the existence of a market model.

### Robust Hedging of Barrier Options

- Economics, Mathematics
- 2001

This article considers the pricing and hedging of barrier options in a market in which call options are liquidly traded and can be used as hedging instruments. This use of call options means that…

### An Introduction to Probability Theory and Its Applications

- Mathematics
- 1950

Thank you for reading an introduction to probability theory and its applications vol 2. As you may know, people have look numerous times for their favorite novels like this an introduction to…

### ARBITRAGE BOUNDS FOR PRICES OF WEIGHTED VARIANCE SWAPS

- Economics, Mathematics
- 2010

We develop a theory of robust pricing and hedging of a weighted variance swap given market prices for a finite number of co‐maturing put options. We assume the put option prices do not admit…