Risk, Uncertainty and Nonlinear Expectation
Shige Peng, Shandong University, China
Research School of Controllability
of Deterministic and Stochastic Systems and its Appl.
July 02, 2012, Iasi, Romania
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Uncertainty and Risk
How to understand uncertainty in order to quantitatively control risks
become a worldwide main concerned problem.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Uncertainty and Risk
How to understand uncertainty in order to quantitatively control risks
become a worldwide main concerned problem.
This problem is even urgent since 2008 after the last financial crisis
which caused a worldwide economic disaster.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Uncertainty and Risk
How to understand uncertainty in order to quantitatively control risks
become a worldwide main concerned problem.
This problem is even urgent since 2008 after the last financial crisis
which caused a worldwide economic disaster.
new mathematical concept and calculation tool called nonlinear
expectation theory which take the risk of model uncertainty
(Knightian uncertainty) into account.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Uncertainty and Risk
How to understand uncertainty in order to quantitatively control risks
become a worldwide main concerned problem.
This problem is even urgent since 2008 after the last financial crisis
which caused a worldwide economic disaster.
new mathematical concept and calculation tool called nonlinear
expectation theory which take the risk of model uncertainty
(Knightian uncertainty) into account.
Important: The existing results in probability theory, stochastic
controls, mathematical finance, risk measures and risk controls are
our rich sources.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Developments of research on uncertainty— an uncertain
process?
L. Bachelier (1900) Théorie de la Spéculation, Thesis.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Developments of research on uncertainty— an uncertain
process?
L. Bachelier (1900) Théorie de la Spéculation, Thesis.
H. Lebesgue (1901) Sur une generalisation de l’integrale definie,
CRAS, 132 1025-1028
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Developments of research on uncertainty— an uncertain
process?
L. Bachelier (1900) Théorie de la Spéculation, Thesis.
H. Lebesgue (1901) Sur une generalisation de l’integrale definie,
CRAS, 132 1025-1028
A. Einstein (1905) Investigations on the theory of the Brownian
movement (originally in German), Annalen der Physik, 17 549-560
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Developments of research on uncertainty— an uncertain
process?
L. Bachelier (1900) Théorie de la Spéculation, Thesis.
H. Lebesgue (1901) Sur une generalisation de l’integrale definie,
CRAS, 132 1025-1028
A. Einstein (1905) Investigations on the theory of the Brownian
movement (originally in German), Annalen der Physik, 17 549-560
P. Daniell (1918-1920): Daniell’s integral
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Developments of research on uncertainty— an uncertain
process?
L. Bachelier (1900) Théorie de la Spéculation, Thesis.
H. Lebesgue (1901) Sur une generalisation de l’integrale definie,
CRAS, 132 1025-1028
A. Einstein (1905) Investigations on the theory of the Brownian
movement (originally in German), Annalen der Physik, 17 549-560
P. Daniell (1918-1920): Daniell’s integral
R. Wiener (1921-1923, 1924) Brownian motion (Wiener process)
using Daniell’s integral
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Developments of research on uncertainty— an uncertain
process?
L. Bachelier (1900) Théorie de la Spéculation, Thesis.
H. Lebesgue (1901) Sur une generalisation de l’integrale definie,
CRAS, 132 1025-1028
A. Einstein (1905) Investigations on the theory of the Brownian
movement (originally in German), Annalen der Physik, 17 549-560
P. Daniell (1918-1920): Daniell’s integral
R. Wiener (1921-1923, 1924) Brownian motion (Wiener process)
using Daniell’s integral
A. N. Kolmogorov (1933) Grundbegrie der
Wahrscheinlichkeitsrechnung, Springer Berlin
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Developments of research on uncertainty— an uncertain
process?
L. Bachelier (1900) Théorie de la Spéculation, Thesis.
H. Lebesgue (1901) Sur une generalisation de l’integrale definie,
CRAS, 132 1025-1028
A. Einstein (1905) Investigations on the theory of the Brownian
movement (originally in German), Annalen der Physik, 17 549-560
P. Daniell (1918-1920): Daniell’s integral
R. Wiener (1921-1923, 1924) Brownian motion (Wiener process)
using Daniell’s integral
A. N. Kolmogorov (1933) Grundbegrie der
Wahrscheinlichkeitsrechnung, Springer Berlin
K. Itô (1942) Differential equations determining a Markov process, J.
Pan-Japan Math. Colloq. (in Japanese)
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Developments of research on uncertainty— an uncertain
process?
L. Bachelier (1900) Théorie de la Spéculation, Thesis.
H. Lebesgue (1901) Sur une generalisation de l’integrale definie,
CRAS, 132 1025-1028
A. Einstein (1905) Investigations on the theory of the Brownian
movement (originally in German), Annalen der Physik, 17 549-560
P. Daniell (1918-1920): Daniell’s integral
R. Wiener (1921-1923, 1924) Brownian motion (Wiener process)
using Daniell’s integral
A. N. Kolmogorov (1933) Grundbegrie der
Wahrscheinlichkeitsrechnung, Springer Berlin
K. Itô (1942) Differential equations determining a Markov process, J.
Pan-Japan Math. Colloq. (in Japanese)
W. Doeblin (1940, Pli cacheté)
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Kolmogorov’s Probability Space (Ω, F , P )
A fundamental and powerful theory and methodology to
treat uncertainties
A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung.
Springer, Berlin, 1933
(Foundations of the Theory of Probability, Chelsea, New York),
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Kolmogorov’s Probability Space (Ω, F , P )
A fundamental and powerful theory and methodology to
treat uncertainties
A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung.
Springer, Berlin, 1933
(Foundations of the Theory of Probability, Chelsea, New York),
Probability Space (Ω, F , P )
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Kolmogorov’s Probability Space (Ω, F , P )
A fundamental and powerful theory and methodology to
treat uncertainties
A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung.
Springer, Berlin, 1933
(Foundations of the Theory of Probability, Chelsea, New York),
Probability Space (Ω, F , P )
Hilbert’s 6th problem
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Impacts in Economics & Math. Finance
The von Neumann-Morgenstern utility axioms (1953) E [U (X )];
Theory of Games and Economic Behavior, Princeton;
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Impacts in Economics & Math. Finance
The von Neumann-Morgenstern utility axioms (1953) E [U (X )];
Theory of Games and Economic Behavior, Princeton;
P. A. Samuelson, Rational theory of warrant pricing, 1964
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Impacts in Economics & Math. Finance
The von Neumann-Morgenstern utility axioms (1953) E [U (X )];
Theory of Games and Economic Behavior, Princeton;
P. A. Samuelson, Rational theory of warrant pricing, 1964
Black-Scholes-Merton option pricing:
C (S, t ) = N (d1 ) − N (d2 )Ke −r (T −1) ,
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Impacts in Economics & Math. Finance
The von Neumann-Morgenstern utility axioms (1953) E [U (X )];
Theory of Games and Economic Behavior, Princeton;
P. A. Samuelson, Rational theory of warrant pricing, 1964
Black-Scholes-Merton option pricing:
C (S, t ) = N (d1 ) − N (d2 )Ke −r (T −1) ,
dS (t ) = µS (t )dt + σS (t )dW (t )
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Frank H. Knight (1921) “Risk, Uncertainty and Profit”
Knight, 1921
” Mathematical, or a priori, type of probability is practically never met
with in business ...”
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Frank H. Knight (1921) “Risk, Uncertainty and Profit”
Knight, 1921
” Mathematical, or a priori, type of probability is practically never met
with in business ...”
”Uncertainty must be taken in a sense radically distinct from the
familiar notion of Risk, from which it has never been properly
separated.”
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Frank H. Knight (1921) “Risk, Uncertainty and Profit”
Knight, 1921
” Mathematical, or a priori, type of probability is practically never met
with in business ...”
”Uncertainty must be taken in a sense radically distinct from the
familiar notion of Risk, from which it has never been properly
separated.”
Knightian’s Risk
Probability (and prob. distribution) are known.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Frank H. Knight (1921) “Risk, Uncertainty and Profit”
Knight, 1921
” Mathematical, or a priori, type of probability is practically never met
with in business ...”
”Uncertainty must be taken in a sense radically distinct from the
familiar notion of Risk, from which it has never been properly
separated.”
Knightian’s Risk
Probability (and prob. distribution) are known.
Knightian uncertainty
The prob. and distr. are unknown— ”uncertainty of probability measures”.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
F. Knight (1921): Two types of uncertainty “risk”: given a
probability space (Ω, F , P ); “Knightian uncertainty” (ambiguity):
Probability measure P itself is uncertain;
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
F. Knight (1921): Two types of uncertainty “risk”: given a
probability space (Ω, F , P ); “Knightian uncertainty” (ambiguity):
Probability measure P itself is uncertain;
John Maynard Keynes (1921) A Treatise on Probability. Macmillan,
London, 1921.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
F. Knight (1921): Two types of uncertainty “risk”: given a
probability space (Ω, F , P ); “Knightian uncertainty” (ambiguity):
Probability measure P itself is uncertain;
John Maynard Keynes (1921) A Treatise on Probability. Macmillan,
London, 1921.
Allais paradox (1953) to vNM expected utility theory (1944);
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
F. Knight (1921): Two types of uncertainty “risk”: given a
probability space (Ω, F , P ); “Knightian uncertainty” (ambiguity):
Probability measure P itself is uncertain;
John Maynard Keynes (1921) A Treatise on Probability. Macmillan,
London, 1921.
Allais paradox (1953) to vNM expected utility theory (1944);
Ellsberg paradox (1961) to Savage’s expected utility (1954),
Ambiguity aversion (1961);
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
F. Knight (1921): Two types of uncertainty “risk”: given a
probability space (Ω, F , P ); “Knightian uncertainty” (ambiguity):
Probability measure P itself is uncertain;
John Maynard Keynes (1921) A Treatise on Probability. Macmillan,
London, 1921.
Allais paradox (1953) to vNM expected utility theory (1944);
Ellsberg paradox (1961) to Savage’s expected utility (1954),
Ambiguity aversion (1961);
Kahneman & Tversky (1979-1992): prospective theory by distorted
probability;
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
F. Knight (1921): Two types of uncertainty “risk”: given a
probability space (Ω, F , P ); “Knightian uncertainty” (ambiguity):
Probability measure P itself is uncertain;
John Maynard Keynes (1921) A Treatise on Probability. Macmillan,
London, 1921.
Allais paradox (1953) to vNM expected utility theory (1944);
Ellsberg paradox (1961) to Savage’s expected utility (1954),
Ambiguity aversion (1961);
Kahneman & Tversky (1979-1992): prospective theory by distorted
probability;
Gilboa & Schmeidler (1989) Maximin expected utility; Hansen &
Sargent (2000) Multiplier preference.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
F. Knight (1921): Two types of uncertainty “risk”: given a
probability space (Ω, F , P ); “Knightian uncertainty” (ambiguity):
Probability measure P itself is uncertain;
John Maynard Keynes (1921) A Treatise on Probability. Macmillan,
London, 1921.
Allais paradox (1953) to vNM expected utility theory (1944);
Ellsberg paradox (1961) to Savage’s expected utility (1954),
Ambiguity aversion (1961);
Kahneman & Tversky (1979-1992): prospective theory by distorted
probability;
Gilboa & Schmeidler (1989) Maximin expected utility; Hansen &
Sargent (2000) Multiplier preference.
Hansen & Sargent: Robust control method.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Motivated from g -Expectation [P.1994-1997]
Given r.v. X (ω ), solve the BSDE
dy (t ) = −g (y (t ), z (t ))dt + z (t )dB (t ), y (T ) = X (ω ).
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Motivated from g -Expectation [P.1994-1997]
Given r.v. X (ω ), solve the BSDE
dy (t ) = −g (y (t ), z (t ))dt + z (t )dB (t ), y (T ) = X (ω ).
Then define:
Eg [X ] := y (0), Eg [X |(B (s ))s ∈[0,t ] ] := y (t ).
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Artzner-Delbean-Eden-Heath1999, Coherent measures of risk, Math.
finance.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Artzner-Delbean-Eden-Heath1999, Coherent measures of risk, Math.
finance.
g -expectations provides dynamic coherent risk measure, Rosazza
(2005), S.P. (2004)
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Artzner-Delbean-Eden-Heath1999, Coherent measures of risk, Math.
finance.
g -expectations provides dynamic coherent risk measure, Rosazza
(2005), S.P. (2004)
Coquet-Hu-Memin-P.: dominated dynamic expectations are g expectations;
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Artzner-Delbean-Eden-Heath1999, Coherent measures of risk, Math.
finance.
g -expectations provides dynamic coherent risk measure, Rosazza
(2005), S.P. (2004)
Coquet-Hu-Memin-P.: dominated dynamic expectations are g expectations;
Delbaen-P.-Rosazza, 2008: If a dynamic expectation E is absolutely
continuous w.r.t. P then there exists a unique g such that E = Eg .
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Artzner-Delbean-Eden-Heath1999, Coherent measures of risk, Math.
finance.
g -expectations provides dynamic coherent risk measure, Rosazza
(2005), S.P. (2004)
Coquet-Hu-Memin-P.: dominated dynamic expectations are g expectations;
Delbaen-P.-Rosazza, 2008: If a dynamic expectation E is absolutely
continuous w.r.t. P then there exists a unique g such that E = Eg .
Serious problem: under volatility uncertainty, it is impossible to find a
reference probability measure
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Artzner-Delbean-Eden-Heath1999, Coherent measures of risk, Math.
finance.
g -expectations provides dynamic coherent risk measure, Rosazza
(2005), S.P. (2004)
Coquet-Hu-Memin-P.: dominated dynamic expectations are g expectations;
Delbaen-P.-Rosazza, 2008: If a dynamic expectation E is absolutely
continuous w.r.t. P then there exists a unique g such that E = Eg .
Serious problem: under volatility uncertainty, it is impossible to find a
reference probability measure
State dependent Markovian case: Avellaneda, M., Levy, A. and Paras
A. (1995), T. Lyons (1995).
Longtime blockage...
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
G -Expectation and G -Brownian Motion
[Peng2004] Filtration consistent nonlinear expectations..., Applicatae
Sinica, 20(2), 1-24.
[Peng2005] Nonlinear expectations and nonlinear Markov chains,
Chin. Ann. Math. (paper for BSDE Weihai Conf. 2002)
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
G -Expectation and G -Brownian Motion
[Peng2004] Filtration consistent nonlinear expectations..., Applicatae
Sinica, 20(2), 1-24.
[Peng2005] Nonlinear expectations and nonlinear Markov chains,
Chin. Ann. Math. (paper for BSDE Weihai Conf. 2002)
[Peng2006] G–Expectation, G–Brownian Motion and Related
Stochastic Calculus of Ito’s type, Abel Symposium2005
(Springer2007).
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
G -Expectation and G -Brownian Motion
[Peng2004] Filtration consistent nonlinear expectations..., Applicatae
Sinica, 20(2), 1-24.
[Peng2005] Nonlinear expectations and nonlinear Markov chains,
Chin. Ann. Math. (paper for BSDE Weihai Conf. 2002)
[Peng2006] G–Expectation, G–Brownian Motion and Related
Stochastic Calculus of Ito’s type, Abel Symposium2005
(Springer2007).
Denis, L. and Martini, C. (2006) A theoretical framework for the
pricing of contingent claims in the presence of model uncertainty, The
Ann. of Appl. Probability
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
G -Expectation and G -Brownian Motion
[Peng2004] Filtration consistent nonlinear expectations..., Applicatae
Sinica, 20(2), 1-24.
[Peng2005] Nonlinear expectations and nonlinear Markov chains,
Chin. Ann. Math. (paper for BSDE Weihai Conf. 2002)
[Peng2006] G–Expectation, G–Brownian Motion and Related
Stochastic Calculus of Ito’s type, Abel Symposium2005
(Springer2007).
Denis, L. and Martini, C. (2006) A theoretical framework for the
pricing of contingent claims in the presence of model uncertainty, The
Ann. of Appl. Probability
[Peng2008-SPA] Multi-Dim G-Brownian Motion and Related
Stochastic Calculus.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
G -Expectation and G -Brownian Motion
[Peng2004] Filtration consistent nonlinear expectations..., Applicatae
Sinica, 20(2), 1-24.
[Peng2005] Nonlinear expectations and nonlinear Markov chains,
Chin. Ann. Math. (paper for BSDE Weihai Conf. 2002)
[Peng2006] G–Expectation, G–Brownian Motion and Related
Stochastic Calculus of Ito’s type, Abel Symposium2005
(Springer2007).
Denis, L. and Martini, C. (2006) A theoretical framework for the
pricing of contingent claims in the presence of model uncertainty, The
Ann. of Appl. Probability
[Peng2008-SPA] Multi-Dim G-Brownian Motion and Related
Stochastic Calculus.
[Denis-Hu-Peng2008] Capacity related to Sublinear Expectations:
appl. to G-Brownian Motion Paths.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
[Peng2009] Survey on G -normal distributions, central limit theorem,
Brownian motion and the related stochastic calculus under sublinear
expectations, Science in China Series A: Mathematics, Volume 52,
Number 7, 1391-1411.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
[Peng2009] Survey on G -normal distributions, central limit theorem,
Brownian motion and the related stochastic calculus under sublinear
expectations, Science in China Series A: Mathematics, Volume 52,
Number 7, 1391-1411.
[Peng2007-2010] G -Brownian motion· · ·
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
[Peng2009] Survey on G -normal distributions, central limit theorem,
Brownian motion and the related stochastic calculus under sublinear
expectations, Science in China Series A: Mathematics, Volume 52,
Number 7, 1391-1411.
[Peng2007-2010] G -Brownian motion· · ·
Peng, 2010, Tightness, weak compactness of nonlinear expectations
and application to CLT
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
[Peng2009] Survey on G -normal distributions, central limit theorem,
Brownian motion and the related stochastic calculus under sublinear
expectations, Science in China Series A: Mathematics, Volume 52,
Number 7, 1391-1411.
[Peng2007-2010] G -Brownian motion· · ·
Peng, 2010, Tightness, weak compactness of nonlinear expectations
and application to CLT
Cheridito, P., Soner, H.M. and Touzi, N., Victoir, N. (2007) Second
order BSDE’s and fully nonlinear PDE’s, Communications in Pure and
Applied Mathematics, 60, 1081- 1110.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
[Peng2009] Survey on G -normal distributions, central limit theorem,
Brownian motion and the related stochastic calculus under sublinear
expectations, Science in China Series A: Mathematics, Volume 52,
Number 7, 1391-1411.
[Peng2007-2010] G -Brownian motion· · ·
Peng, 2010, Tightness, weak compactness of nonlinear expectations
and application to CLT
Cheridito, P., Soner, H.M. and Touzi, N., Victoir, N. (2007) Second
order BSDE’s and fully nonlinear PDE’s, Communications in Pure and
Applied Mathematics, 60, 1081- 1110.
Soner, H. M. Touzi, N. and Zhang, J. (2011) Dual Formulation of
Second Order Target Problems, arxiv: 1003.6050.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
[Peng2009] Survey on G -normal distributions, central limit theorem,
Brownian motion and the related stochastic calculus under sublinear
expectations, Science in China Series A: Mathematics, Volume 52,
Number 7, 1391-1411.
[Peng2007-2010] G -Brownian motion· · ·
Peng, 2010, Tightness, weak compactness of nonlinear expectations
and application to CLT
Cheridito, P., Soner, H.M. and Touzi, N., Victoir, N. (2007) Second
order BSDE’s and fully nonlinear PDE’s, Communications in Pure and
Applied Mathematics, 60, 1081- 1110.
Soner, H. M. Touzi, N. and Zhang, J. (2011) Dual Formulation of
Second Order Target Problems, arxiv: 1003.6050.
2BSDE: by A. Matoussi, Possamai, Zhao, ...
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
[Peng2009] Survey on G -normal distributions, central limit theorem,
Brownian motion and the related stochastic calculus under sublinear
expectations, Science in China Series A: Mathematics, Volume 52,
Number 7, 1391-1411.
[Peng2007-2010] G -Brownian motion· · ·
Peng, 2010, Tightness, weak compactness of nonlinear expectations
and application to CLT
Cheridito, P., Soner, H.M. and Touzi, N., Victoir, N. (2007) Second
order BSDE’s and fully nonlinear PDE’s, Communications in Pure and
Applied Mathematics, 60, 1081- 1110.
Soner, H. M. Touzi, N. and Zhang, J. (2011) Dual Formulation of
Second Order Target Problems, arxiv: 1003.6050.
2BSDE: by A. Matoussi, Possamai, Zhao, ...
L. Epstein and S. Ji (2012) Ambiguous volatility, possibility and utility
in continuous time, (by random G -expectations).
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
M. Soner, N. Touzi, and J. Zhang (2011) Martingale representation
theorem for the G-expectation. in SPA.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
M. Soner, N. Touzi, and J. Zhang (2011) Martingale representation
theorem for the G-expectation. in SPA.
Soner, Touzi, Zhang, (2011) Quasi-sure stochastic analysis through
aggregation. Electron. J. Probab.,
Soner, Touzi, Zhang (2010) Well posedness of 2nd order BSDEs to
appear in PTRF
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
M. Soner, N. Touzi, and J. Zhang (2011) Martingale representation
theorem for the G-expectation. in SPA.
Soner, Touzi, Zhang, (2011) Quasi-sure stochastic analysis through
aggregation. Electron. J. Probab.,
Soner, Touzi, Zhang (2010) Well posedness of 2nd order BSDEs to
appear in PTRF
Song Y. 2007,2010, (2012Electronic JP) Uniqueness of the
representation for G-martingales, (2011, SPA) Properties of hitting
times for G-martingales
Y. Dolinsky, M. Nutz, M. Soner, Weak Approximation of
G-Expectations
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
M. Soner, N. Touzi, and J. Zhang (2011) Martingale representation
theorem for the G-expectation. in SPA.
Soner, Touzi, Zhang, (2011) Quasi-sure stochastic analysis through
aggregation. Electron. J. Probab.,
Soner, Touzi, Zhang (2010) Well posedness of 2nd order BSDEs to
appear in PTRF
Song Y. 2007,2010, (2012Electronic JP) Uniqueness of the
representation for G-martingales, (2011, SPA) Properties of hitting
times for G-martingales
Y. Dolinsky, M. Nutz, M. Soner, Weak Approximation of
G-Expectations
Natz (2010) Random G-expectations,
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
M. Soner, N. Touzi, and J. Zhang (2011) Martingale representation
theorem for the G-expectation. in SPA.
Soner, Touzi, Zhang, (2011) Quasi-sure stochastic analysis through
aggregation. Electron. J. Probab.,
Soner, Touzi, Zhang (2010) Well posedness of 2nd order BSDEs to
appear in PTRF
Song Y. 2007,2010, (2012Electronic JP) Uniqueness of the
representation for G-martingales, (2011, SPA) Properties of hitting
times for G-martingales
Y. Dolinsky, M. Nutz, M. Soner, Weak Approximation of
G-Expectations
Natz (2010) Random G-expectations,
S. Cohen (2011) Quasi-sure analysis, aggregation and dual
representations of sublinear expectations in general spaces.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
M. Soner, N. Touzi, and J. Zhang (2011) Martingale representation
theorem for the G-expectation. in SPA.
Soner, Touzi, Zhang, (2011) Quasi-sure stochastic analysis through
aggregation. Electron. J. Probab.,
Soner, Touzi, Zhang (2010) Well posedness of 2nd order BSDEs to
appear in PTRF
Song Y. 2007,2010, (2012Electronic JP) Uniqueness of the
representation for G-martingales, (2011, SPA) Properties of hitting
times for G-martingales
Y. Dolinsky, M. Nutz, M. Soner, Weak Approximation of
G-Expectations
Natz (2010) Random G-expectations,
S. Cohen (2011) Quasi-sure analysis, aggregation and dual
representations of sublinear expectations in general spaces.
P.-Song-Zhang (2012) A Complete Representation Theorem for
G-martingales;
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
M. Soner, N. Touzi, and J. Zhang (2011) Martingale representation
theorem for the G-expectation. in SPA.
Soner, Touzi, Zhang, (2011) Quasi-sure stochastic analysis through
aggregation. Electron. J. Probab.,
Soner, Touzi, Zhang (2010) Well posedness of 2nd order BSDEs to
appear in PTRF
Song Y. 2007,2010, (2012Electronic JP) Uniqueness of the
representation for G-martingales, (2011, SPA) Properties of hitting
times for G-martingales
Y. Dolinsky, M. Nutz, M. Soner, Weak Approximation of
G-Expectations
Natz (2010) Random G-expectations,
S. Cohen (2011) Quasi-sure analysis, aggregation and dual
representations of sublinear expectations in general spaces.
P.-Song-Zhang (2012) A Complete Representation Theorem for
G-martingales;
Nutz & van Handel (2012) Constructing Sublinear Expectations on
Research School of Controllability of Determ
Shige Peng,Path
Shandong
University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Space
Related works
Chen, Z. J. and Xiong, J., Large deviation principle for diffusion
processes under a sublinear expectation. Preprint 2010.
F. Gao, A variational representation and large deviations for
functionals of G -Brownian motion, 2012, preprint.
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
Chen, Z. J. and Xiong, J., Large deviation principle for diffusion
processes under a sublinear expectation. Preprint 2010.
F. Gao, A variational representation and large deviations for
functionals of G -Brownian motion, 2012, preprint.
F. Gao, Pathwise properties and homeomorphic for stochastic
differential equatios driven by G -Brownian motion. SPA, 119(2009)
Research School of Controllability of Determ
Shige Peng, Shandong University, China ()Risk, Uncertainty and Nonlinear Expectation
/ 39
Related works
Chen, Z. J. and Xiong, J., Large deviation principle for diffusion
processes under a sublinear expectation. Preprint 2010.
F. Gao, A variational representation and large deviations for
functionals of G -Brownian motion, 2012, preprint.
F. Gao, Pathwise properties and homeomorphic for stochastic
differential equatios driven by G -Brownian motion. SPA, 119(2009)
Large Deviations for Stochastic Differential Equations Driven by
G-Brownian Motion. Stoch. Proc. Appl., 120 (2010)
Research School of Controllability of Determ
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/ 39
Related works
Chen, Z. J. and Xiong, J., Large deviation principle for diffusion
processes under a sublinear expectation. Preprint 2010.
F. Gao, A variational representation and large deviations for
functionals of G -Brownian motion, 2012, preprint.
F. Gao, Pathwise properties and homeomorphic for stochastic
differential equatios driven by G -Brownian motion. SPA, 119(2009)
Large Deviations for Stochastic Differential Equations Driven by
G-Brownian Motion. Stoch. Proc. Appl., 120 (2010)
M. Hu, S. Ji, S. P. & S. Song, (2012) Backward Stochastic
Differential Equations driven by G -Brownian Motions.
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This talk is based on
Peng, S. Notes: Nonlinear Expectations and Stochastic Calculus
under Uncertainty, arxiv 2010.
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Expectation framework–G -framework
Ω: space of scenarios;
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Expectation framework–G -framework
Ω: space of scenarios;
H a linear space of risk positions or (risk losses) containing constants
(real functions defined on Ω) s.t.
X ∈ H =⇒ |X | ∈ H
Research School of Controllability of Determ
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Expectation framework–G -framework
Ω: space of scenarios;
H a linear space of risk positions or (risk losses) containing constants
(real functions defined on Ω) s.t.
X ∈ H =⇒ |X | ∈ H
We often ”equivalently” assume:
X1 , · · · , Xn ∈ H =⇒ ϕ(X1 , · · · , Xn ) ∈ H,
∀ ϕ ∈ CLip (Rn )
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Daniell’s Expectation: (Ω, H, E) v.s. (Ω, F , P)
X ∈ H =⇒ |X | ∈ H
(a) E [X ] ≥ E [Y ], if X ≥ Y
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Daniell’s Expectation: (Ω, H, E) v.s. (Ω, F , P)
X ∈ H =⇒ |X | ∈ H
(a) E [X ] ≥ E [Y ], if X ≥ Y
(b) E [c ] = c,
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Daniell’s Expectation: (Ω, H, E) v.s. (Ω, F , P)
X ∈ H =⇒ |X | ∈ H
(a) E [X ] ≥ E [Y ], if X ≥ Y
(b) E [c ] = c,
(c) E [X + Y ] = E [X ] + E [Y ],
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Daniell’s Expectation: (Ω, H, E) v.s. (Ω, F , P)
X ∈ H =⇒ |X | ∈ H
(a) E [X ] ≥ E [Y ], if X ≥ Y
(b) E [c ] = c,
(c) E [X + Y ] = E [X ] + E [Y ],
(d) E [λX ] = λE [X ], λ ≥ 0.
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Daniell’s Expectation: (Ω, H, E) v.s. (Ω, F , P)
X ∈ H =⇒ |X | ∈ H
(a) E [X ] ≥ E [Y ], if X ≥ Y
(b) E [c ] = c,
(c) E [X + Y ] = E [X ] + E [Y ],
(d) E [λX ] = λE [X ], λ ≥ 0.
E [Xi ] ↓ 0,
if Xi (ω ) ↓ 0, ∀ω
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Daniell’s Expectation: (Ω, H, E) v.s. (Ω, F , P)
X ∈ H =⇒ |X | ∈ H
(a) E [X ] ≥ E [Y ], if X ≥ Y
(b) E [c ] = c,
(c) E [X + Y ] = E [X ] + E [Y ],
(d) E [λX ] = λE [X ], λ ≥ 0.
E [Xi ] ↓ 0,
if Xi (ω ) ↓ 0, ∀ω
Theorem (Daniell-Stone Theorem)
There exists a unique prob. measure P on (Ω, σ(H)) s.t.
E [X ] =
Z
Ω
X ( ω )P ( ω ).
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Extension: Sublinear Expectation on (Ω, H, Ê)
X ∈ H =⇒ |X | ∈ H
(a) Ê[X ] ≥ Ê[Y ],
if X ≥ Y
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Extension: Sublinear Expectation on (Ω, H, Ê)
X ∈ H =⇒ |X | ∈ H
(a) Ê[X ] ≥ Ê[Y ], if X ≥ Y
(b) Ê[X + c ] = Ê[X ] + c,
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Extension: Sublinear Expectation on (Ω, H, Ê)
X ∈ H =⇒ |X | ∈ H
(a) Ê[X ] ≥ Ê[Y ], if X ≥ Y
(b) Ê[X + c ] = Ê[X ] + c,
(c) Ê[X + Y ]≤Ê[X ] + Ê[Y ] ” ≤ ” =⇒ sublinear
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Extension: Sublinear Expectation on (Ω, H, Ê)
X ∈ H =⇒ |X | ∈ H
(a)
(b)
(c)
(d)
Ê[X ] ≥ Ê[Y ], if X ≥ Y
Ê[X + c ] = Ê[X ] + c,
Ê[X + Y ]≤Ê[X ] + Ê[Y ] ” ≤ ” =⇒ sublinear
Ê[λX ] = λÊ[X ], λ ≥ 0.
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Extension: Sublinear Expectation on (Ω, H, Ê)
X ∈ H =⇒ |X | ∈ H
(a)
(b)
(c)
(d)
Ê[X ] ≥ Ê[Y ], if X ≥ Y
Ê[X + c ] = Ê[X ] + c,
Ê[X + Y ]≤Ê[X ] + Ê[Y ] ” ≤ ” =⇒ sublinear
Ê[λX ] = λÊ[X ], λ ≥ 0.
Ê[Xi ] ↓ 0, if Xi (ω ) ↓ 0, ∀ω
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Extension: Sublinear Expectation on (Ω, H, Ê)
X ∈ H =⇒ |X | ∈ H
(a)
(b)
(c)
(d)
Ê[X ] ≥ Ê[Y ], if X ≥ Y
Ê[X + c ] = Ê[X ] + c,
Ê[X + Y ]≤Ê[X ] + Ê[Y ] ” ≤ ” =⇒ sublinear
Ê[λX ] = λÊ[X ], λ ≥ 0.
Ê[Xi ] ↓ 0, if Xi (ω ) ↓ 0, ∀ω
Theorem (Robust Daniell-Stone Theorem)
There exists a family of {Pθ }θ ∈Θ of prob. measures on (Ω, σ(H)) s.t.
Ê[X ] = sup Eθ [X ] = sup
θ ∈Θ
Z
θ ∈Θ Ω
X (ω )Pθ (ω ), for each X ∈ H.
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Extension: Sublinear Expectation on (Ω, H, Ê)
X ∈ H =⇒ |X | ∈ H
(a)
(b)
(c)
(d)
Ê[X ] ≥ Ê[Y ], if X ≥ Y
Ê[X + c ] = Ê[X ] + c,
Ê[X + Y ]≤Ê[X ] + Ê[Y ] ” ≤ ” =⇒ sublinear
Ê[λX ] = λÊ[X ], λ ≥ 0.
Ê[Xi ] ↓ 0, if Xi (ω ) ↓ 0, ∀ω
Theorem (Robust Daniell-Stone Theorem)
There exists a family of {Pθ }θ ∈Θ of prob. measures on (Ω, σ(H)) s.t.
Ê[X ] = sup Eθ [X ] = sup
Z
θ ∈Θ Ω
θ ∈Θ
X (ω )Pθ (ω ), for each X ∈ H.
For each given X ∈ H,
Ê[ ϕ(X )] = sup
Z
θ ∈Θ R
ϕ(x )dFθ (x ), Fθ (x ) = Pθ (X ≤ x ).
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Robust representation of a coherent risk measure
Huber Robust Statistics (1981), for finite state case.
Artzner-Delbean, Eber-Heath (1999), Delbean2002,
Föllmer & Schied (2002, 2004),
Fritelli & Rosazza-Gianin (2002)
Research School of Controllability of Determ
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Robust representation of a coherent risk measure
Huber Robust Statistics (1981), for finite state case.
Artzner-Delbean, Eber-Heath (1999), Delbean2002,
Föllmer & Schied (2002, 2004),
Fritelli & Rosazza-Gianin (2002)
Theorem (Robust Representation of coherent risk measure)
Ê[·] is a sublinear expectation iff there exists a family {Eθ }θ ∈Θ of linear
expectations s.t.
Ê[X ] = sup Eθ [X ], ∀X ∈ H.
θ ∈Θ
Research School of Controllability of Determ
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Robust representation of a coherent risk measure
Huber Robust Statistics (1981), for finite state case.
Artzner-Delbean, Eber-Heath (1999), Delbean2002,
Föllmer & Schied (2002, 2004),
Fritelli & Rosazza-Gianin (2002)
Theorem (Robust Representation of coherent risk measure)
Ê[·] is a sublinear expectation iff there exists a family {Eθ }θ ∈Θ of linear
expectations s.t.
Ê[X ] = sup Eθ [X ], ∀X ∈ H.
θ ∈Θ
Meaning:
Sublinear expectation corresponds the Knightian uncertainty of
probabilities: {Pθ }θ ∈Θ
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Uncertainty version of distributions in (Ω, H, Ê)
Definition
X ∼ Y if they have the same distribution uncertainty
X ∼Y
⇐⇒ Ê[ ϕ(X )] = Ê[ ϕ(Y )],
∀ ϕ ∈ Cb (Rn ).
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Uncertainty version of distributions in (Ω, H, Ê)
Definition
X ∼ Y if they have the same distribution uncertainty
X ∼Y
⇐⇒ Ê[ ϕ(X )] = Ê[ ϕ(Y )],
∀ ϕ ∈ Cb (Rn ).
Y Indenp. of X if each realization ”X = x” does not change the
distribution of Y :
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Uncertainty version of distributions in (Ω, H, Ê)
Definition
X ∼ Y if they have the same distribution uncertainty
X ∼Y
⇐⇒ Ê[ ϕ(X )] = Ê[ ϕ(Y )],
∀ ϕ ∈ Cb (Rn ).
Y Indenp. of X if each realization ”X = x” does not change the
distribution of Y :
Y indenp. of X ⇐⇒ Ê[ ϕ(X , Y )] = Ê[Ê[ ϕ(x, Y )]x =X ].
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Central Limit Theorem (CLT) under Knightian Uncertainty
Theorem
Let {Xi }i∞=1 in (Ω, H, Ê) be i.i.d.: Xi ∼ X1 and
Xi +1 Indep. (X1 , · · · , Xi ). Assume:
Ê[|X1 |2+α ] < ∞
, Ê[X1 ] = Ê[−X1 ] = 0.
Then:
lim Ê[ ϕ(
n→∞
X1 + · · · + Xn
√
)] = Ê[ ϕ(X )], ∀ ϕ ∈ Cb (R),
n
with X ∼ N (0, [σ2 , σ2 ]), where
σ2 = Ê[X12 ], σ2 = −Ê[−X12 ].
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Normal distributions under Knightian uncertainty
Definition
A loss position X in (Ω, H, Ê) is normally in uncertainty distribution if
p
aX + b X̄ ∼ a2 + b 2 X , ∀a, b ≥ 0.
where X̄ is an independent copy of X .
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Normal distributions under Knightian uncertainty
Definition
A loss position X in (Ω, H, Ê) is normally in uncertainty distribution if
p
aX + b X̄ ∼ a2 + b 2 X , ∀a, b ≥ 0.
where X̄ is an independent copy of X .
Ê[X ] = Ê[−X ] = 0.
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Normal distributions under Knightian uncertainty
Definition
A loss position X in (Ω, H, Ê) is normally in uncertainty distribution if
p
aX + b X̄ ∼ a2 + b 2 X , ∀a, b ≥ 0.
where X̄ is an independent copy of X .
Ê[X ] = Ê[−X ] = 0.
d
X = N (0, [σ2 , σ2 ]), where
σ2 := Ê[X 2 ], σ2 := −Ê[−X 2 ].
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G-normal distribution: under sublinear expectation E[·]
(1) For each convex ϕ, we have
Ê[ ϕ(X )] = √
1
2πσ2
Z ∞
−∞
ϕ(y ) exp(−
y2
)dy
2σ2
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G-normal distribution: under sublinear expectation E[·]
(1) For each convex ϕ, we have
Ê[ ϕ(X )] = √
1
2πσ2
Z ∞
−∞
ϕ(y ) exp(−
y2
)dy
2σ2
ϕ(y ) exp(−
y2
)dy
2σ2
(2) For each concave ϕ, we have,
Ê[ ϕ(X )] = p
1
2πσ2
Z ∞
−∞
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Remark.
If σ2 = σ2 , then N (0; [σ2 , σ2 ]) = N (0, σ2 ).
Remark.
The larger to [σ2 , σ2 ] the stronger the uncertainty.
Remark.
d
But X = N (0; [σ2 , σ2 ]) does not simply implies
1
Ê[ ϕ(X )] = sup √
2πσ
σ∈[σ2 ,σ2 ]
Z ∞
−∞
ϕ(x ) exp{
−x 2
}dx
2σ
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G -normal distribution characterized by nonlinear
infinitesimal generator
CLT converges in uncertainty distribution to N (0, [σ2 , σ2 ]):
Theorem
d
X = N (0, [σ2 , σ2 ]) in (Ω, H, Ê), then for each Cb function ϕ,
√
St ( ϕ)(x ) := Ê[ ϕ(x + tX )], x ∈ R, t ≥ 0
defines a nonlinear semigroup, since:S0 [ ϕ](x ) = Ê[ ϕ(x )] = ϕ(x ), and
√
St +s [ ϕ](x ) = Ê[ ϕ(x + t + sX )]
z }|
√ { √
= Ê[ ϕ(x + tX + s X̄ )]
"
#
z }|
√ { √
= Ê Ê[ ϕ(x + ty + s X̄ )]y =X
h
i
√
= Ê (Ss [ ϕ])(x + tX ) = St [Ss [ ϕ]](x ).
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A ϕ(x ) := lim
t →0
St ( ϕ)(x ) − ϕ(x )
= G (uxx ).
t
where
a
1
G (a) = Ê[ X 2 ] = (σ2 a+ − σ2 a− )
2
2
Thus we can solve the PDE
ut = G (∂2xx u ), t > 0, x ∈ R
u |t =0 = ϕ.
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Law of Large Numbers (LLN), Central Limit Theorem
(CLT)
Striking consequence of LLN & CLT
Accumulated independent and identically distributed random variables
tends to a normal distributed random variable, whatever the original
distribution.
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Maximal distribution M ([µ, µ]) under Knightian
uncertainty
Definition
A random variable Y in (Ω, H, Ê) is maximally distributed, denoted by
d
Y = M ([µ, µ]), if
d
aY + b Ȳ = (a + b )Y , a, b ≥ 0.
where Ȳ is an independent copy of Y ,
µ := Ê[Y ], µ := −Ê[−Y ].
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Maximal distribution M ([µ, µ]) under Knightian
uncertainty
Definition
A random variable Y in (Ω, H, Ê) is maximally distributed, denoted by
d
Y = M ([µ, µ]), if
d
aY + b Ȳ = (a + b )Y , a, b ≥ 0.
where Ȳ is an independent copy of Y ,
µ := Ê[Y ], µ := −Ê[−Y ].
We can prove that
Ê[ ϕ(Y )] = sup ϕ(y ).
y ∈[µ,µ]
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Case with mean-uncertainty E[·]
Definition
A pair of random variables (X , Y ) in (Ω, H, Ê) is
d
N ([µ, µ], [σ2 , σ2 ])-distributed ((X , Y ) = N ([µ, µ], [σ2 , σ2 ])) if
p
d
(aX + bX̄ , a2 Y + b2 Ȳ ) = ( a2 + b2 X , (a2 + b2 )Y ),
∀a, b ≥ 0.
where (X̄ , Ȳ ) is an independent copy of (X , Y ),
µ := Ê[Y ], µ := −Ê[−Y ]
σ2 := Ê[X 2 ], σ2 := −Ê[−X ], (Ê[X ] = Ê[ − X ] = 0).
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Theorem
d
(X , Y ) = N ([µ, µ], [σ2 , σ2 ]) in (Ω, H, Ê) iff for each ϕ ∈ Cb (R) the
function
√
u (t, x, y ) := Ê[ ϕ(x + tX , y + tY )], x ∈ R, t ≥ 0
is the solution of the PDE
ut = G (uy , uxx ), t > 0, x ∈ R
u |t =0 = ϕ,
where
a
G (p, a) := Ê[ X 2 + pY ].
2
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LLN + CLT under Knightian Uncertainty
Theorem
Let {Xi + Yi }i∞=1 be i.i.d. sequence. We assume furthermore that
Ê[|X1 |2+α ] + Ê[|Y1 |1+α ] < ∞, Ê[X1 ] = Ê[−X1 ] = 0.
Then, for each ϕ ∈ Cb (R),
lim Ê[ ϕ(
n→∞
X1 + · · · + Xn
Y1 + · · · + Yn
√
+
)] = Ê[ ϕ(X + Y )].
n
n
where (X , Y ) is N ([µ, µ], [σ2 , σ2 ])-distributed.
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Brownian Motion (Bt (ω ))t ≥0 in (Ω, F , Ê))
Definition
B is called aG -Brownian motion if:
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Brownian Motion (Bt (ω ))t ≥0 in (Ω, F , Ê))
Definition
B is called aG -Brownian motion if:
For each t1 ≤ · · · ≤ tn , Btn − Btn−1 is indep. of (Bt1 , · · · , Btn−1 ).
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Brownian Motion (Bt (ω ))t ≥0 in (Ω, F , Ê))
Definition
B is called aG -Brownian motion if:
For each t1 ≤ · · · ≤ tn , Btn − Btn−1 is indep. of (Bt1 , · · · , Btn−1 ).
d
Bt = Bs +t − Bs , for all s, t ≥ 0
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Brownian Motion (Bt (ω ))t ≥0 in (Ω, F , Ê))
Definition
B is called aG -Brownian motion if:
For each t1 ≤ · · · ≤ tn , Btn − Btn−1 is indep. of (Bt1 , · · · , Btn−1 ).
d
Bt = Bs +t − Bs , for all s, t ≥ 0
Ê[|Bt |3 ] = o (t ). .
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Theorem.
If (Bt (ω ))t ≥0 is a G –Brownian motion and Ê[Bt ] = Ê[−Bt ] ≡ 0 then:
d
Bt +s − Bs = N (0, [σ2 t, σ2 t ]), ∀ s, t ≥ 0
Sketch of Proof.
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Theorem.
If (Bt (ω ))t ≥0 is a G –Brownian motion and Ê[Bt ] = Ê[−Bt ] ≡ 0 then:
d
Bt +s − Bs = N (0, [σ2 t, σ2 t ]), ∀ s, t ≥ 0
Sketch of Proof.
St [ ϕ](x ) := Ê[ ϕ(x + Bt )] defines a nonlinear semigroup (St )t ≥0
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Theorem.
If (Bt (ω ))t ≥0 is a G –Brownian motion and Ê[Bt ] = Ê[−Bt ] ≡ 0 then:
d
Bt +s − Bs = N (0, [σ2 t, σ2 t ]), ∀ s, t ≥ 0
Sketch of Proof.
St [ ϕ](x ) := Ê[ ϕ(x + Bt )] defines a nonlinear semigroup (St )t ≥0
1
Ê[ ϕ(x + Bt )] − ϕ(x ) = Ê[ ϕx (x )Bt + ϕxx (x )Bt2 ] + o (t )
2
1
B2
= Ê[ ϕxx (x )Bt2 ] + o (t ), G (a) := Ê[ 1 a].
2
| 2 {z
}
=G ( ϕxx )t,
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Theorem.
If (Bt (ω ))t ≥0 is a G –Brownian motion and Ê[Bt ] = Ê[−Bt ] ≡ 0 then:
d
Bt +s − Bs = N (0, [σ2 t, σ2 t ]), ∀ s, t ≥ 0
Sketch of Proof.
St [ ϕ](x ) := Ê[ ϕ(x + Bt )] defines a nonlinear semigroup (St )t ≥0
1
Ê[ ϕ(x + Bt )] − ϕ(x ) = Ê[ ϕx (x )Bt + ϕxx (x )Bt2 ] + o (t )
2
1
B2
= Ê[ ϕxx (x )Bt2 ] + o (t ), G (a) := Ê[ 1 a].
2
| 2 {z
}
=G ( ϕxx )t,
Thus ∂t St [ ϕ](x )|t =0 = G ( ϕxx (x )): the infinitesimal generator of
(St )t ≥0 .
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Construct G -BM on a sublinear expectation space
(Ω, H, Ê)
Ω := C (0, ∞; R), Bt (ω ) = ωt
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Construct G -BM on a sublinear expectation space
(Ω, H, Ê)
Ω := C (0, ∞; R), Bt (ω ) = ωt
H := {X (ω ) = ϕ(Bt1 , Bt2 , · · · , Btn ), ti ∈ [0, ∞), ϕ ∈ CLip (Rn ), n ∈
Z}
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Construct G -BM on a sublinear expectation space
(Ω, H, Ê)
Ω := C (0, ∞; R), Bt (ω ) = ωt
H := {X (ω ) = ϕ(Bt1 , Bt2 , · · · , Btn ), ti ∈ [0, ∞), ϕ ∈ CLip (Rn ), n ∈
Z}
For each X (ω ) = ϕ(Bt1 , Bt2 − Bt1 , · · · , Btn − Btn−1 ), with ti < ti +1 ,
we set
√
√
√
Ê[X ] := Ẽ[ ϕ( t1 ξ 1 , t2 − t1 ξ 2 , · · · , tn − tn−1 ξ n )]
where
d
ξ i = N (0, [σ2 , σ2 ]), ξ i +1 is indep. of (ξ 1 , · · · , ξ i ) under Ẽ.
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Construct G -BM on a sublinear expectation space
(Ω, H, Ê)
Ω := C (0, ∞; R), Bt (ω ) = ωt
H := {X (ω ) = ϕ(Bt1 , Bt2 , · · · , Btn ), ti ∈ [0, ∞), ϕ ∈ CLip (Rn ), n ∈
Z}
For each X (ω ) = ϕ(Bt1 , Bt2 − Bt1 , · · · , Btn − Btn−1 ), with ti < ti +1 ,
we set
√
√
√
Ê[X ] := Ẽ[ ϕ( t1 ξ 1 , t2 − t1 ξ 2 , · · · , tn − tn−1 ξ n )]
where
d
ξ i = N (0, [σ2 , σ2 ]), ξ i +1 is indep. of (ξ 1 , · · · , ξ i ) under Ẽ.
Conditional expectation:
Êt1 [X ] = Ẽ[ ϕ(x,
√
t2 − t1 ξ 2 , · · · ,
√
tn − tn−1 ξ n )]x =Bt1
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Completion of H to LpG (Ω) under kX kLp := Ê[|X |p ]1/p , p ≥ 1
G
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Completion of H to LpG (Ω) under kX kLp := Ê[|X |p ]1/p , p ≥ 1
G
Ê[·] and Êt are extended to LpG (Ω) and keeping time consistency;
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Completion of H to LpG (Ω) under kX kLp := Ê[|X |p ]1/p , p ≥ 1
G
Ê[·] and Êt are extended to LpG (Ω) and keeping time consistency;
Itô’s integral, Itô’s calculus have been established
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Completion of H to LpG (Ω) under kX kLp := Ê[|X |p ]1/p , p ≥ 1
G
Ê[·] and Êt are extended to LpG (Ω) and keeping time consistency;
Itô’s integral, Itô’s calculus have been established
G-martingales, supermartingales, · · · have been established.
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Completion of H to LpG (Ω) under kX kLp := Ê[|X |p ]1/p , p ≥ 1
G
Ê[·] and Êt are extended to LpG (Ω) and keeping time consistency;
Itô’s integral, Itô’s calculus have been established
G-martingales, supermartingales, · · · have been established.
If G̃ is dominated by G : G̃ (a) − G̃ (b ) ≤ G (a − b ), then we can
establish a nonlinear expectation EG̃ on the same space LpG (Ω),
under which B is a G̃ -Brownian motion.
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Completion of H to LpG (Ω) under kX kLp := Ê[|X |p ]1/p , p ≥ 1
G
Ê[·] and Êt are extended to LpG (Ω) and keeping time consistency;
Itô’s integral, Itô’s calculus have been established
G-martingales, supermartingales, · · · have been established.
If G̃ is dominated by G : G̃ (a) − G̃ (b ) ≤ G (a − b ), then we can
establish a nonlinear expectation EG̃ on the same space LpG (Ω),
under which B is a G̃ -Brownian motion.
We don’t need to change stochastic calculus for these type of EG̃ .
Many Wiener measures and martingale measures dominated by Ê
work well in this fixed G -framework. (they maybe singular from each
others).
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Completion of H to LpG (Ω) under kX kLp := Ê[|X |p ]1/p , p ≥ 1
G
Ê[·] and Êt are extended to LpG (Ω) and keeping time consistency;
Itô’s integral, Itô’s calculus have been established
G-martingales, supermartingales, · · · have been established.
If G̃ is dominated by G : G̃ (a) − G̃ (b ) ≤ G (a − b ), then we can
establish a nonlinear expectation EG̃ on the same space LpG (Ω),
under which B is a G̃ -Brownian motion.
We don’t need to change stochastic calculus for these type of EG̃ .
Many Wiener measures and martingale measures dominated by Ê
work well in this fixed G -framework. (they maybe singular from each
others).
Note that if G1 ≤ G2 then LpG1 (Ω) ⊃ LpG2 (Ω).
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Probability v.s. Nonlinear Expectation
Probability Space
(Ω, F , P )
Nonlinear Expectation Space
(Ω, H, E): (sublinear is basic)
d
Distributions: X = Y
Independence: Y indep. of X
LLN and CLT
Normal distributions
Brownian motion Bt (ω ) = ωt
Qudratic variat. hB it = t
Lévy process
d
X = Y,
Y indep. of X , (non-symm.)
LLN + CTL
G-Normal distributions
G -B.M. Bt (ω ) = ωt ,
hB it : still a G -Brownian motion
G -Lévy process
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Probability v.s. Nonlinear Expectation
Probability Space
Itô’s calculus for BM
SDE dxt = b (xt )dt + σ(xt )dBt
Diffusion: ∂t u − Lu = 0
Markovian pro. and semi-grou
Martingales
RT
E [X |Ft ] = E [X ] + 0 zs dBs
Nonlinear Expectation Space
Itô’s calculus for G -BM
dxt = · · · + β(xt )d hB it
∂t u − G (Du, D 2 u ) = 0
Nonlinear Markovian
G -Martingales
Rt
E[X |Ft ] = E[X ] + 0 zs dBs + Kt
Rt
? Rt
Kt = 0 ηs d hB is − 0 2G (ηs )ds
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Probability Space
P-almost surely analysis
X (ω ): P-quasi continuous
⇐⇒ X is B(Ω)-meas.
Nonlinear Expectation Space
ĉ-quasi surely analysis
ĉ (A) = supθ EPθ [1A ]
X (ω ): ĉ-quasi surely
continuous =⇒ X is B(Ω)-meas.
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Program of IMS Singapore University
Nonlinear Expectations, Stochastic Calculus
under Knightian Uncertainty and Related Topics
Title:
Time period (around): 3rd Jun to 12th Jul 2013
(6 weeks, summer school and two workshops)
Proposed by: M. Dai, H. Föllmer, J. Hinz (NUS)
S. Peng, (SDU) J. Xia (AMSS, China) J. Zhang (USC)
Welcome to participate!!
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Thank you
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