# STATISTICAL CAUSALITY AND QUASIMARTINGALES

### Abstract

**Concept of causality is very popular and applicable nowadays, especially when we consider the cases "what would happen if" and "what would have happened if". Here we consider the concept of causality based on the Granger’s definition of causality, introduced in Mykland (1986). Many of the systems to which it is natural to apply tests of causality take place in continuous time, so we will consider the continuous time processes. Here we consider the connection between the concept of causality and the property of being a quasimartingale. Quasimartingales were investigated by Fisk (1965), Orey and specially Rao (1969). Namely, in this paper we prove an equivalence between the given concept of causality and preservation of quasimartingale property if the filtration is getting larger. We prove the same equivalence for the stopped quasimartingale with respect to the truncated filtrations.**

### References

Bremaud, P., & Yor, M. 1978. Changes of filtrations and of probability measures. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 45(4), pp. 269-295. doi:10.1007/bf00537538

Comte, F., & Renault, E. 1996. Noncausality in Continuous Time Models. Econometric Theory, 12(02). doi:10.1017/s0266466600006575

Elliot, R. J. 1982. Stochastic Calculus and applications.New York: Springer-Verlag.

Fisk, D. L. 1965. Quasi-Martingales. Transactions of the American Mathematical Society, 120(3). doi:10.2307/1994531

Florens, J. P., & Mouchart, M. 1982. A Note on Noncausality. Econometrica, 50(3). doi:10.2307/1912602

Gill, J. B., & Petrovic, L. 1987. Causality and Stochastic Dynamic Systems. SIAM Journal on Applied Mathematics, 47(6), pp. 1361-1366. doi:10.1137/0147089

Granger, C. W. J. 1969. Investigating Causal Relations by Econometric Models and Cross-spectral Methods. Econometrica, 37(3). doi:10.2307/1912791

Mykland, P. A. 1986. Statistical Causality. Report, 2, pp. 1-21.

Petrović, L. 1996. Causality and Markovian representations. Statistics & Probability Letters, 29(3), pp. 223-227. doi:10.1016/0167-7152(95)00176-x

Petrović, L., Dimitrijević, S., & Valjarević, D. 2016. Granger causality and stopping times*. Lithuanian Mathematical Journal, 56(3), pp. 410-416. doi:10.1007/s10986-016-9325-0

Petrović, L., & Valjarević, D. 2013. Statistical causality and stable subspaces of the australian mathematical society. Bulletin of the Australian Mathematical Society, 88(01), pp. 17-25. doi:10.1017/s0004972712000482

Petrović, L., & Valjarević, D. 2014. Statistical causality and martingale representation property with application to stochastic differential equations. Bulletin of the Australian Mathematical Society, 90(02), pp. 327-338. doi:10.1017/s000497271400029x

Petrović, L., & Valjarević, D. 2015. Lecture Notes in Computer Science: Statistical Causality and Local Solutions of the Stochastic Differential Equations Driven with Semimartingales.Cham: Springer Nature America, Inc., pp. 261-269. doi:10.1007/978-3-319-15765-8_14

Protter, P. 2004. Stochastic Integration and Differential Equations.Berlin: Springer-Verlag.

Rao, K. M. 1969. Quasi-Martingales. Mathematica scandinavica, 24. doi:10.7146/math.scand.a-10921

Revuz, D., & Yor, M. 2005. Continuous martingales and Brownian motion. New York: Springer.

Rozanov, Y. A. 1974. Theory of Innovation Processes. Monographs in Probabolity Theory and Mathematical Statistics.Moscow: Izdat Nauka.

Sims, C. A. 1972. Money, income and causality. American Economic Review, 62, pp. 540-552.

Skorohod, I., & Gikhman, L. 2005. Stochastic processes. New York: Springer. 1.

Valjarević, D. 2012. Theory of statistical causality, stochastic differential equations and martingale representation property. University thought, 82, pp. 1326-1330.

Valjarević, D., & Petrović, L. 2012. Statistical causality and orthogonality of local martingales. Statistics & Probability Letters, 82(7), pp. 1326-1330. doi:10.1016/j.spl.2012.03.036