## Bernoulli

• Bernoulli
• Volume 8, Number 6 (2002), 767-785.

### A technique for exponential change of measure for Markov processes

#### Abstract

We consider a Markov process $X(t)$ with extended generator ${\mathbf{A}}$ and domain $\cal{D}({\mathbf{A}})$. Let $\{{\cal F}_t\}$ be a right-continuous history filtration and ${\mathbb{P}}_t$ denote the restriction of ${\mathbb{P}}$ to ${\cal F}_t$. Let $\tilde{{\mathbb{P}}}$ be another probability measure on $(\Omega,{\cal F})$ such that $\rm d\tilde{{\mathbb{P}}}_t/\rm d{\mathbb{P}}_t=E^h(t)$, where $$E^h(t)=\frac{h(X(t))}{h(X(0))}\exp \left(-\int_0^t \frac{({\mathbf{A}}h)(X(s))}{h(X(s))}\rm ds \right)$$ is a true martingale for a positive function $h\in\cal{D}({\mathbf{A}})$. We demonstrate that the process $X(t)$ is a Markov process on the probability space $(\Omega, {\cal F}, \{{\cal F}_t\},\tilde{{\mathbb{P}}})$, we find its extended generator $\tilde{{\mathbf{A}}}$ and provide sufficient conditions under which $\cal{D}(\tilde{{\mathbf{A}}})=\cal{D}({\mathbf{A}})$. We apply this result to continuous-time Markov chains, to piecewise deterministic Markov processes and to diffusion processes (in this case a special choice of $h$ yields the classical Cameron--Martin--Girsanov theorem).

#### Article information

Source
Bernoulli, Volume 8, Number 6 (2002), 767-785.

Dates
First available in Project Euclid: 9 February 2004

https://projecteuclid.org/euclid.bj/1076364805

Mathematical Reviews number (MathSciNet)
MR1963661

Zentralblatt MATH identifier
1011.60054

#### Citation

Palmowski, Zbigniew; Rolski, Tomasz. A technique for exponential change of measure for Markov processes. Bernoulli 8 (2002), no. 6, 767--785. https://projecteuclid.org/euclid.bj/1076364805

#### References

• [1] Asmussen, S. (1994) Busy period analysis, rare events and transient behaviour in fluid models. J. Appl. Math. Stochastic Anal., 7(3), 269-299. Abstract can also be found in the ISI/STMA publication
• [2] Asmussen, S. (1995) Stationary distribution for fluid models with or without Brownian noise. Stochastic Models, 11, 21-49. Abstract can also be found in the ISI/STMA publication
• [3] Asmussen, S. (1998) Stochastic Simulation with a View towards Stochastic Processes, MaPhySto Lecture Notes 2. Aarhus: MaPhySto, University of Aarhus.
• [4] Asmussen, S. (2000) Ruin Probabilites. Singapore: World Scientific.
• [5] Asmussen, S. and Kella, O. (2000) Multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Probab., 32, 376-380.
• [6] Çinlar, E., Jacod, J., Protter, P. and Sharpe, M.J. (1980) Semimartingales and Markov process. Z. Wahrscheinlichkeitstheorie Verw. Geb., 54, 161-219.
• [7] Dassios, A. and Embrechts, P. (1989) Martingales and insurance risk. Stochastic Models, 5, 181-217. Abstract can also be found in the ISI/STMA publication
• [8] Davis, M.H.A. (1993) Markov Models and Optimization. London: Chapman & Hall.
• [9] Dellacherie, C. and Meyer, P. (1982) Probabilities and Potential B. New York: North-Holland.
• [10] Doob, J.L. (1984) Classical Potential Theory and Its Probabilistic Counterpart. New York: Springer- Verlag.
• [11] Dynkin, E.B. (1965) Markov Processes, Vol. I. Berlin: Springer-Verlag.
• [12] Ethier, H.J. and Kurtz, T.G. (1986) Markov Processes. Characterization and Convergence. New York: Wiley.
• [13] Ethier, H.J. and Kurtz, T.G. (1993) Fleming-Viot processes in population genetics. SIAM J. Control Optim., 31, 345-386.
• [14] Fukushima, M. and Stroock, D. (1986) Reversibility of solutions to martingale problems. In G.-C. Rota (ed.), Probability, Statistical Mechanics, and Number Theory, Adv. Math. Suppl. Stud. 9, pp. 107-123. Orlando, FL: Academic Press.
• [15] Gautam, N., Kulkarni, V., Palmowski, Z. and Rolski, T. (1999) Bounds for fluid models driven by semi-Markov inputs. Probab. Engrg. Inform. Sci., 13, 429-475. Abstract can also be found in the ISI/STMA publication
• [16] Itô, K. and Watanabe, S. (1965) Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier Grenoble, 15, 13-30.
• [17] Jacod, J. and Mémin, J. (1976) Characteristiques locales et conditions de continuité absolute pour les semi-martingales. Z. Wahrscheinlichkeitstheorie Verw. Geb., 35, 1-37.
• [18] Jacod, J. and Shiryaev, A.N. (1987) Limit Theorems for Stochastic Processes. Berlin: Springer-Verlag.
• [19] Karatzas, I. and Shreve, S.E. (1988) Brownian Motion and Stochastic Calculus. New York: Springer- Verlag.
• [20] Küchler, U. and Sørensen, M. (1997) Exponential Families of Stochastic Processes. New York: Springer-Verlag.
• [21] Kulkarni, V. and Rolski, T. (1994) Fluid model driven by an Ornstein-Uhlenbeck process. Probab. Engrg. Inform. Sci., 8, 403-417.
• [22] Kunita, H. (1969) Absolute continuity of Markov processes and generators. Nagoya Math. J., 36, 1-26.
• [23] Kunita, H. and Watanabe, T. (1963) Notes on transformations of Markov processes connected with multiplicative functionals. Mem. Fac. Sci. Kyushu Univ. Ser. A, 17, 181-191.
• [24] Liptser, R.S. and Shiryaev, A.N. (1986) Teorija Martingalov. Moskow: Nauka.
• [25] Palmowski, Z. (2002) Lundberg inequalities in a diffusion environment. Insurance Math. Econom. To appear.
• [26] Palmowski, Z. and Rolski, T. (1996) A note on martingale inequalities for fluid models. Statist. Probab. Lett., 31, 13-21. Abstract can also be found in the ISI/STMA publication
• [27] Palmowski, Z. and Rolski, T. (1998) Superposition of alternating on-off flows and a fluid model. Ann. Appl. Probab., 8, 524-541.
• [28] Parthasarathy, K.R. (1967) Probability Measures on Metric Spaces. New York: Academic Press.
• [29] Protter, P. (1990) Stochastic Integration and Differential Equations. A New Approach. New York: Springer-Verlag.
• [30] Revuz, D. and Yor, M. (1991) Continuous Martingales and Brownian Motion. Berlin: Springer-Verlag.
• [31] Ridder, A. (1996) Fast simulation of Markov fluid models. J. Appl. Probab., 33, 786-804. Abstract can also be found in the ISI/STMA publication
• [32] Ridder, A. and Walrand, J. (1992) Some large deviations results in Markov fluid models. Probab. Engrg. Inform. Sci., 6, 543-560. Abstract can also be found in the ISI/STMA publication
• [33] Rogers, L.C.G. and Williams, D. (1987) Diffusions, Markov Processes, and Martingales. Volume 2: Itô Calculus. New York: Wiley.
• [34] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J.L. (1999) Stochastic Processes for Insurance and Finance. New York: Wiley.
• [35] Schmidli, H. (1995) Cramér-Lundberg approximations for ruin functions of risk processes perturbed by diffusion. Insurance Math. Econom., 16, 135-149.
• [36] Schmidli, H. (1996) Lundberg inequalities for a Cox model with a piecewise constant intensity. J. Appl. Probab., 33, 196-210. Abstract can also be found in the ISI/STMA publication
• [37] Schmidli, H. (1997a) An extension to the renewal theorem and an application to risk theory. Ann. Appl. Probab. 7, 121-133. Abstract can also be found in the ISI/STMA publication
• [38] Schmidli, H. (1997b) Estimation of the Lundberg coefficient for a Markov modulated risk model. Scand. Actuar. J., 48-57. Abstract can also be found in the ISI/STMA publication
• [39] Schwartz, A. and Weiss, A. (1995) Large Deviations for Performance Analysis. London: Chapman & Hall.
• [40] Stroock, D.W. (1987) Lectures on Stochastic Analysis: Diffusion Theory. Cambrige: Cambridge University Press.