The Mark H.A. Davis festschrift: stochastics, control and finance

Mark Davis has been associated with numerous major advances, and has been influential to several generations of researchers working in the fields of stochastic analysis, stochastic control and financial mathematics. On a personal level, Mark has been an inspiring teacher and mentor as well as a great friend and colleague. It is therefore with great pleasure that we write the foreword to this special issue of Stochastics comprising papers presented at the Workshop on Stochastics, Control and Finance in honour of Mark Davis on the occasion of his 65th birthday at Imperial College London, 12–14 April 2010.1

After completing his BA degree in Electrical Engineering at the University of Cambridge, Mark pursued his PhD degree at UC Berkeley under the supervision of Pravin Varaiya. While at Berkeley, Mark had the opportunity to attend several influential courses, including his first-ever course in stochastic processes, given by Eugene Wong, and a course in probability given by Michel Loève. Indeed, this was a great period for stochastic analysis at Berkeley. Mark's cohort benefited by the interaction with several distinguished visitors such as Vic Beneš, and got access to ground-breaking works such as ‘On square integrable martingales’ by Kunita and Watanabe before they were published. At the same time, researchers such as Thomas Kailath and Tyrone Duncan were just across the Bay at Stanford.

Mark's PhD thesis initiated the martingale theory of stochastic control. This would soon become one of the main methodological approaches that we currently have for the study of stochastic control and optimization problems. In later years, Mark continued to work on several aspects of this theory, which has attracted the interest of numerous mathematicians, with Robert Elliott being among the first ones.

Returning from the USA in 1972, Mark joined the Control Group at Imperial College London. The group had been created by John Westcott and David Mayne; Martin Clark was already there, while Richard Vinter joined in 1974. Several research visits were influential to Mark's academic development over the next three decades. Notable ones include his visits at Harvard with Roger Brockett in 1974, at MIT with Sanjoy Mitter in 1978–1979 and in Vienna with Walter Schachermayer in 2000. His participation in a semester on control theory at the Banach Center in Warsaw in 1973 was another significant event during which he met Jerzy Zabcyk, Albert Shiryaev and several other mathematicians from Poland and Russia. In 1991, Mark spent six months at the University of Oslo visiting Bernt Øksendal, where he largely wrote his book Markov Models and Optimization.

After joining Imperial college, Mark worked in several areas beyond the martingale theory of stochastic control. His research contributions have been so multifold and significant that going in detail through all of them would go way beyond the scope of this brief introduction. Therefore, we will suffice ourselves to just listing a few of them. The general theory of jump processes was one of the first areas that Mark contributed to, while his fundamental research on the development of pathwise nonlinear filtering theory was motivated by a basic observation by Martin Clark. Starting from a collaboration with Michael Dempster and Domokos Vermes on capacity expansion problems in production planning, Mark came up with the theory of piecewise deterministic processes (PDPs). After the publication of his seminal paper in 1984, PDPs soon attracted most significant interest. In particular, their application to insurance was advanced by Paul Embrechts who was at Imperial college at that time. In the early 1990s, Mark originated the deterministic approach to stochastic control by means of appropriate Lagrange multipliers. Some important aspects of this theory were studied in collaboration with Ioannis Karatzas and Gabriel Burstein, and were later taken up by Chris Rogers.

In the second part of the 1980s, Mark developed a keen interest in mathematical finance. In his seminal paper with Andrew Norman, he derived a computable solution to the portfolio selection problem with transaction costs that has become one of the contributions that have shaped mathematical finance as the theory that we currently have. Mark made several further contributions to the theory of option pricing since then, with a notable one in collaboration with Thaleia Zariphopoulou. In the context of such applications in the area, Mark originated advances in the general theory of singular stochastic control in collaboration with Mihail Zervos.

In the early 1990s, Mark's contributions to mathematical finance had received wide recognition in the academic community. At that stage, Mark started feeling that he had done as much in the field as he possibly could without any first-hand practical experience. It was at that time in 1995 that he received a telephone call from a head-hunter acting on behalf of Mitsubishi Finance (MF), which would later become Tokyo-Mitsubishi International (TMI). He took the challenge and became Director and Head of Research and Product Development at MF in London to run a group of around six PhD quants working on pricing models and risk analysis for fixed income, equity and credit-related products. During his time in the City, Mark worked on several projects, including early models for credit derivatives, and came up with the ‘infectious defaults’ model with Violet Lo.

Mark left TMI in November 1999 and returned to Imperial College London in August 2000 to build Imperial's Mathematical Finance group and programme. He launched the MSc in Mathematical Finance, originally designed by Terry Lyons, in that same year. He has been running the Mathematical Finance programme until recently. During this time, Mark has worked on numerous projects, including credit risk models with Giacomo Giampieri and Juan-Carlos Esparragoza, Malliavin Calculus with Martin Johansson, arbitrage conditions for option pricing with David Hobson and Jan Obłoj, interest rate models with Vicente Mataix-Pastor and risk-sensitive asset management with Sébastien Lleo. Hosting the fifth World Congress of the Bachelier Society in 2008 with about 600 participants was among the highlights of this period.

So far, Mark has authored five books on stochastic analysis, optimization and finance, and has written more than 100 publications listed in Mathematical Reviews. He was Editor-in-Chief of Stochastics and Stochastics Reports (1978–1995), a founding co-editor of Mathematical Finance (1990–1993) and an Associate Editor of Annals of Applied Probability (1995–1998). Currently, he is an Associate Editor of Quantitative Finance (since 2000) and SIAM Journal of Financial Mathematics (since 2009). Mark has been honoured by the award of the Naylor Prize in Applied Mathematics by the London Mathematical Society in 2002. He has been a Fellow of the Royal Statistical Society (elected in 1994), a Fellow of the Institute of Mathematical Statistics and an Honorary Fellow of the Institute of Actuaries (elected in 2001).

Outside academia, Mark has maintained a wide range of interests. Among them, his love of travelling and his involvement in music in the company of his wife Jessica have been harmonious signs and important reminders to everybody in his environment that academic life need not be dry, uneventful and lacking in sociability. We look forward to Mark's continued friendship as well as his future research contributions to our field!

19th September 2012

Mark H.A. Davis: books

1.	Linear Estimation and Stochastic Control, Chapman and Hall, London/Halsted Press, New York 1977.
2.	Lineinoe Otsenivanie i Stochasticheskoe Upravlenie, Nauka, Moscow 1984 (Russian translation of Linear Estimation and Stochastic Control with revisions).
3.	Lectures on Stochastic Control and Nonlinear Filtering, Narosa, New Delhi/Springer-Verlag, Berlin 1985.
4.	Stochastic Modelling and Control, Monographs on statistics and applied probability Vol. 24, Chapman and Hall, London, New York 1985 (with R.B. Vinter).
5.	Markov Models and Optimization, Monographs on statistics and applied probability Vol. 49, Chapman and Hall, London, New York 1993.
6.	Louis Bachelier's Theory of Speculation: The Origins of Modern Finance, Princeton University Press 2006 (with A. Etheridge).

Mark H.A. Davis: publications in refereed journals

1.	Dynamic programming conditions for partially-observed stochastic systems, SIAM J. Control Optim. 11 (1973) 226–261 (with P.P. Varaiya)
2.	On the existence of optimal policies in stochastic control, SIAM J. Control Optim. 11 (1973) 587–594
3.	Information states for linear stochastic systems, J. Math. Anal. Appl. (1972) 384–402 (with P.P. Varaiya)
4.	On the multiplicity of an increasing family of sigma-fields, Ann. Probab. 2 (1974) 958–963 (with P.P. Varaiya)
5.	Nonlinear filtering with counting observations, IEEE Trans. Inform. Theory IT-21 (1975) 143–149 (with A. Segall and T. Kailath)
6.	The application of nonlinear filtering to fault detection in linear systems, IEEE Trans. Automat. Contr. AC-20 (1975) 257–259
7.	On stochastic differentiation, Theory Probab. Appl. 20 (1975) 887–892
8.	The separation principle in stochastic control via Girsanov solutions, SIAM J. Control Optim. 14 (1976) 176–188
9.	The representation of martingales of jump processes, SIAM J. Control Optim. 14 (1976) 623–638
10.	Martingales of Wiener and Poisson processes, J. Lond. Math. Soc. (2) 13 (1976)
11.	Exact and approximate filtering in signal detection: An example, IEEE Trans. Inform. Theory IT-23 (1977) 768–772 (with E. Andreadakis)
12.	Optimal control of a jump process, Z Wahrscheinlichkeitstheorie ver Geb 40 (1977) 183–202 (with R.J. Elliott)
13.	The general point process disorder problem, IEEE Trans. Inform. Theory IT-23 (1977) 538–540 (with C.B. Wan)
14.	A direct proof of innovations/observations equivalence for Gaussian processes, IEEE Trans. Inform. Theory IT-24 (1978) 252–254
15.	Existence of optimal controls for stochastic jump processes, SIAM J. Control Optim. 17 (1979) 511–524 (with C.B. Wan)
16.	‘Predicted miss’ problems in stochastic optimal control, Stochastics 2 (1979) 197–209 (with J.M.C. Clark)
17.	Stochastic control by measure transformation: A general existence result, Inform. Sci. 21 (1980), 195–208 (with M. Kohlmann)
18.	Functionals of diffusion processes as stochastic integrals, Math. Proc. Cambridge Phil. Soc. 87 (1980) 157–166
19.	Capacity and cutoff rate for Poisson-type channels, IEEE Trans. Infor. Theory IT-26 (1980) 710–715
20.	On a multiplicative functional arising in nonlinear filtering theory, Z Wahrscheinlichkeitstheorie ver Geb 54 (1980) 125–139
21.	Optimal play in a stochastic differential game, SIAM J. Control Optim. 19 (1981) 543–554 (with R.J. Elliott)
22.	Factorization of a multiplicative functional of nonlinear filtering theory, Syst. Control Lett. 1 (1981) 49–53
23.	New approach to filtering for nonlinear systems, Proc. IEE D128 (1981) 166–172
24.	A note on a comparison theorem for equations with different diffusions, Stochastics 6 (1982) 147–149 (with L.I. Galchuk)
25.	A pathwise solution of the equations of nonlinear filtering, Theory Probab. Appl. (USSR) 27 (1982) 160–167
26.	On a problem of D.R. Cox, Ann. N. Y. Acad. Sci. 410 (1983) 129–132
27.	Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models (with discussion), J. R. Stat. Soc. B 46 (1984) 353–388
28.	Optimal timing of capacity expansion, J. Econ. Dynam. Control 10 (1986) 89–92 (with Z. Carvalhais)
29.	Pathwise nonlinear filtering for non-degenerate diffusions with noise correlation, SIAM J. Control Optim. 25 (1987) 260–278 (with M P Spathopoulos)
30.	Optimal capacity expansion under uncertainty, Adv. Appl. Probab. 19 (1987) 156–176 (with M.A.H. Dempster, S.P. Sethi and D. Vermes)
31.	The martingale maximum principle and the allocation of labour surplus, J. Econ. Dynam. Control 11 (1987) 210–217 (with G. Gomez)
32.	Approximations for optimal stopping of a piecewise-deterministic process, Math. Control Signals Systems 1 (1988) 123–146 (with O.L.V. Costa)
33.	Wiener space derivatives for functionals of diffusions on manifolds, Nonlinearity 1 (1988) 241–251
34.	Anticipative LQG control, IMA J. Math. Control Inform. 6 (1989) 259–265
35.	Impulse control of piecewise-deterministic processes, Math. Control Signals Systems 2 (1989) 187–206 (with O.L.V. Costa)
36.	On the minimum priciple principle for controlled diffusions on manifolds, SIAM J. Control Optim. 27 (1989) 1092–1107 (with M.P. Spathopoulos)
37.	Strong consistency of the PLS criterion for order determination of autoregressive models, Ann. Stat. 17 (1989) 941–946 (with E.M. Hemerly)
38.	Recursive order estimation of stochastic control systems, Math. Syst. Theory 22 (1989) 323–346 (with E.M. Hemerly)
39.	Portfolio selection with transaction costs, Math. Oper. Res. 15 (1990) 676–713 (with A.R. Norman)
40.	Recursive order estimation of autoregressions without bounding the model set, J. R. Stat. Soc. B 53 (1991) 201–210 (with E.M. Hemerly)
41.	A deterministic approach to stochastic optimal control with application to anticipative control, Stochastics Stochastics Reports 40 (1992) 203–256 (with G. Burstein)
42.	Reducibility and unobservability of Markov processes, IEEE Trans. Autom. Control AC-37 (1992) 505–508 (with V. Lasdas)
43.	European option pricing with transaction costs, SIAM J. Control Optim. 31 (1993) 470–493 (with V.G. Panas and T. Zariphopoulou)
44.	A problem of singular control with discretionary stopping, Ann. Appl. Probab. 4 (1994) 226–240 (with M. Zervos)
45.	A note on super-replicating strategies, Phil. Trans. R. Soc. Lond. A 347 (1994) 485–494 (with J.M.C. Clark)
46.	The writing price of a European contingent claim under proportional transaction costs, Comput. Appl. Math. 13 (1994) 115–157 (with V.G. Panas)
47.	A new proof of the discrete-time LQG optimal control theorems, IEEE Trans. Autom. Control AC-49 (1995) 1450–1453 (with M. Zervos)
48.	Permanent health insurance: A case study in piecewise-deterministic Markov modelling, Mitteilungen der Schweiz Vereinigung der Versicherungsmathematiker, Heft 2 (1995) 177–212 (with M.H. Vellekoop)
49.	A target recognition problem: Sequential analysis and optimal control, SIAM J. Control Optim. 34 (1996) 2116–2132 (with M. Farid)
50.	A Markovian analysis of the M/D/1 Queue with finite buffer, Proc. R. Soc. A 453 (1997) 1947–1962 (with J.M. Howl)
51.	A new order estimation technique for time series modelling, IEEE Trans. Autom. Control 42 (1997) 402–403 (with W.X. Zheng)
52.	A note on the forward measure, Finance Stochastics 2 (1998) 19–28
53.	A pair of explicitly solvable singular stochastic control problems, Appl. Math. Optim. 38 (1998) 327–352 (with M. Zervos)
54.	Pricing weather derivatives by marginal value, Quantitative Finance 1 (2001) 305–308
55.	Infectious defaults, Quantitative Finance 1 (2001) 382–387 (with V. Lo)
56.	Pricing, no-arbitrage bounds and robust hedging of installment options, Quantitative Finance 1 (2001) 597–610 (with W. Schachermayer and R. Tompkins)
57.	Installment options and static hedging, Risk Finance 3 (2002) 46–52 (with W. Schachermayer and R. Tompkins)
58.	Complete-market models of stochastic volatility, Proc. R. Soc. Lond. A 460 (2004) 11–26
59.	A Hidden Markov Model of default interaction, Quantitative Finance 5 (2005) 27–34 (with G. Giampieri and M. Crowder)
60.	Malliavin Monte Carlo greeks for jump diffusions, Stoch. Proc. Appl. 116 (2006) 101–129 (with M. Johansson)
61.	The range of traded option prices, Math. Finance 17 (2007) 1–14 (with D. Hobson)
62.	Negative Libor rates in the swap market model, Finance Stochastics 11 (2007) 181–193 (with V. Mataix-Pastor)
63.	Risk-sensitive benchmarked asset management, Quantitative Finance 8 (2008) 415–426 (with S. Lleo)
64.	Large portfolio credit risk modelling, Int. J. Theor. Appl. Finance 10 (2007) 653–678 (with J.C. Esparragoza Rodriguez)
65.	Informed Traders, Proc. R. Soc. Lond. A 465 (2009) 1103–1122 (with D.C. Brody, R.L. Friedman and L.P. Hughston)
66.	Arbitrage-free interpolation of the Swap Curve, Int. J. Theor. Appl. Finance 12 (2009) 969–1005 (with V. Mataix-Pastor)
67.	Optimal investment under partial information, Math. Methods Oper. Res. 71 (2010) 371–399 (with T. Bjork and C. Landen)
68.	Impulse control of multidimensional jump-diffusions, SIAM J. Control Optim. 48 (2010) 5276–5293 (with X. Guo and G. Wu)
69.	Jump-diffusion risk-sensitive asset management I: Diffusion factor model, SIAM J. Financial Math. 2 (2011) 22–54 (with S. Lleo)
70.	Arbitrage bounds for weighted variance swap prices, Mathematical Finance to appear (with J. Obłoj and V. Raval)

Notes

¹ The workshop was supported by the European Science Foundation AMaMeF research network, the London Mathematical Society and Imperial College London.