Stochastics An International Journal of Probability and Stochastic Processes, 1998
Liptser and Stoyanov (Stochastics and Stochastics Reports, 32, (1990), pp. 145 163) study the lim... more Liptser and Stoyanov (Stochastics and Stochastics Reports, 32, (1990), pp. 145 163) study the limiting properties of the solution of the stochastic integral equation as when the drift and generator are regular enough to ensure existence of a unique strong solution for each is strictly stationary, ergodic, and independent of the Wiener process Among their results is one which shows
Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), 2000
We study a nonlinear filtering problem in which the signal to be estimated is conditioned by the ... more We study a nonlinear filtering problem in which the signal to be estimated is conditioned by the observations. The main result establishes pathwise uniqueness for the unnormalized (Zakai) filter equation
Stochastics An International Journal of Probability and Stochastic Processes, 2009
We show that a simple and elegant method of Bismut [J. Math. Analysis Appl., 44 (1973), pp. 384–4... more We show that a simple and elegant method of Bismut [J. Math. Analysis Appl., 44 (1973), pp. 384–404] for applying conjugate duality to convex problems of Bolza adapts directly to problems of utility maximization with portfolio constraints in mathematical finance. This gives a straightforward construction of an associated dual problem together with Euler–Lagrange and transversality relations, which are then used
Stochastics An International Journal of Probability and Stochastic Processes, 2005
Pardoux and Veretennikov [Ann. Probab., 29, (2001), 1061–1085], establish existence (in the Sobol... more Pardoux and Veretennikov [Ann. Probab., 29, (2001), 1061–1085], establish existence (in the Sobolev sense) and uniqueness of solutions of a Poisson equation for the differential operator of an ergodic diffusion in for which the covariance term is strictly non-degenerate and uniformly bounded. Our goal is to establish a similar solvability result, but for the complementary case of singular diffusions, in which the covariance is not necessarily of full-rank. In return for abandoning strict positive-definiteness of the covariance, we postulate second-order smoothness of the coefficients of the diffusion, and, to secure ergodicity, we postulate a “stability” condition on the eigenvalues of the symmetrized Jacobian matrix of the drift. We establish existence of solutions of the Poisson equation in the classical sense, and use this to characterize limits in a time-scales problem arising from perturbation of a stochastic differential equation by a rescaled singular diffusion; this is motivated by a stochastic averaging principle of Liptser and Stoyanov [Stochastics and Stochastics Reports, 32, (1990), 145–163].
ABSTRACT We study almost-sure limiting properties, taken as ε � 0, ofthe finite horizon sequence ... more ABSTRACT We study almost-sure limiting properties, taken as ε � 0, ofthe finite horizon sequence ofrandom estimates {θε 0 ,θ ε 1 ,θ ε 2 ,...,θ ε � T/ ε� } for the linear stochastic gradient algorithm θε n+1 = θ ε n + ε � an+1 − (θε n) � Xn+1 � Xn+1 ,θ ε 0 � = θ ∗ nonrandom, where T ∈ (0, ∞) is an arbitrary constant, ε ∈ (0, 1) is a (small) adaptation gain, and {an} and {Xn} are data sequences which drive the algorithm. These limiting properties are expressed in the form of a functional law of the iterated logarithm.
Journal of Mathematical Analysis and Applications, 1994
ABSTRACT In this note we consider the almost sure convergence (as epsilon --> 0) of soluti... more ABSTRACT In this note we consider the almost sure convergence (as epsilon --> 0) of solution X(epsilon)(.), defined over the interval 0 less-than-or-equal-to tau less-than-or-equal-to 1, of the random ordinary differential equation X(epsilon)(tau) = F(X(epsilon)(tau), tau/epsilon) subject to X(epsilon)(0) = x0. Here {F(x, t, omega), t greater-than-or-equal-to 0} is a strong mixing process for each x and (x, t) --> F(x, t, omega) is subject to regularity conditions which ensure the existence of a unique solution over 0 less-than-or-equal-to tau less-than-or-equal-to 1 for all epsilon > 0. Under rather weak conditions it is shown that the function X(epsilon)(., omega) converges a.s. to the solution x0(.) of a non-random averaged differential equation x0(tau) = FBAR(x0(tau)) subject to x0(0) = x0, the convergence being uniform over 0 less-than-or-equal-to tau less-than-or-equal-to 1. (C) 1994 Academic Press, Inc.
... Hossain Pezeshki-Esfahani and Andrew J. Heunis ... According to Strassen [25, Theorem 2, p. 2... more ... Hossain Pezeshki-Esfahani and Andrew J. Heunis ... According to Strassen [25, Theorem 2, p. 216] there exists some process together with some standard Brownian motion , , defined on some common probability space , , such that i) the processes and are equal ...
ABSTRACT Consider the following random ordinary differential equation: $\dot{X}^\epsilon(\tau) = ... more ABSTRACT Consider the following random ordinary differential equation: $\dot{X}^\epsilon(\tau) = F(X^\epsilon(\tau), \tau/\epsilon, \omega) \text{subject to} X^\epsilon(0) = x_0$, where $\{F(x, t, \omega), t \geq 0\}$ are stochastic processes indexed by $x$ in $\mathfrak{R}^d$, and the dependence on $x$ is sufficiently regular to ensure that the equation has a unique solution $X^\epsilon(\tau, \omega)$ over the interval $0 \leq \tau \leq 1$ for each $\epsilon > 0$. Under rather general conditions one can associate with the preceding equation a nonrandom averaged equation: $\dot{x}^0(\tau) = \overline{F}(x^0(\tau)) \text{subject to} x^0(0) = x_0,$ such that $\lim_{\epsilon\rightarrow 0} \sup_{0\leq\tau\leq 1}E|X^\epsilon(\tau) - x^0(\tau)| = 0$. In this article we show that as $\epsilon \rightarrow 0$ the random function $(X^\epsilon(\cdot) - x^0(\cdot))/\sqrt{2\epsilon\log\log\epsilon^{-1}}$ almost surely converges to and clusters throughout a compact set $K$ of $C\lbrack 0, 1\rbrack$.
We apply conjugate duality to establish the existence of optimal portfolios in an asset-allocatio... more We apply conjugate duality to establish the existence of optimal portfolios in an asset-allocation problem, with the goal of minimizing the variance of the final wealth which results from trading over a fixed, finite horizon in a continuous-time, complete market, subject to the constraints that the expected final wealth equal a specified target value and the portfolio of the investor
Stochastics An International Journal of Probability and Stochastic Processes, 1998
Liptser and Stoyanov (Stochastics and Stochastics Reports, 32, (1990), pp. 145 163) study the lim... more Liptser and Stoyanov (Stochastics and Stochastics Reports, 32, (1990), pp. 145 163) study the limiting properties of the solution of the stochastic integral equation as when the drift and generator are regular enough to ensure existence of a unique strong solution for each is strictly stationary, ergodic, and independent of the Wiener process Among their results is one which shows
Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), 2000
We study a nonlinear filtering problem in which the signal to be estimated is conditioned by the ... more We study a nonlinear filtering problem in which the signal to be estimated is conditioned by the observations. The main result establishes pathwise uniqueness for the unnormalized (Zakai) filter equation
Stochastics An International Journal of Probability and Stochastic Processes, 2009
We show that a simple and elegant method of Bismut [J. Math. Analysis Appl., 44 (1973), pp. 384–4... more We show that a simple and elegant method of Bismut [J. Math. Analysis Appl., 44 (1973), pp. 384–404] for applying conjugate duality to convex problems of Bolza adapts directly to problems of utility maximization with portfolio constraints in mathematical finance. This gives a straightforward construction of an associated dual problem together with Euler–Lagrange and transversality relations, which are then used
Stochastics An International Journal of Probability and Stochastic Processes, 2005
Pardoux and Veretennikov [Ann. Probab., 29, (2001), 1061–1085], establish existence (in the Sobol... more Pardoux and Veretennikov [Ann. Probab., 29, (2001), 1061–1085], establish existence (in the Sobolev sense) and uniqueness of solutions of a Poisson equation for the differential operator of an ergodic diffusion in for which the covariance term is strictly non-degenerate and uniformly bounded. Our goal is to establish a similar solvability result, but for the complementary case of singular diffusions, in which the covariance is not necessarily of full-rank. In return for abandoning strict positive-definiteness of the covariance, we postulate second-order smoothness of the coefficients of the diffusion, and, to secure ergodicity, we postulate a “stability” condition on the eigenvalues of the symmetrized Jacobian matrix of the drift. We establish existence of solutions of the Poisson equation in the classical sense, and use this to characterize limits in a time-scales problem arising from perturbation of a stochastic differential equation by a rescaled singular diffusion; this is motivated by a stochastic averaging principle of Liptser and Stoyanov [Stochastics and Stochastics Reports, 32, (1990), 145–163].
ABSTRACT We study almost-sure limiting properties, taken as ε � 0, ofthe finite horizon sequence ... more ABSTRACT We study almost-sure limiting properties, taken as ε � 0, ofthe finite horizon sequence ofrandom estimates {θε 0 ,θ ε 1 ,θ ε 2 ,...,θ ε � T/ ε� } for the linear stochastic gradient algorithm θε n+1 = θ ε n + ε � an+1 − (θε n) � Xn+1 � Xn+1 ,θ ε 0 � = θ ∗ nonrandom, where T ∈ (0, ∞) is an arbitrary constant, ε ∈ (0, 1) is a (small) adaptation gain, and {an} and {Xn} are data sequences which drive the algorithm. These limiting properties are expressed in the form of a functional law of the iterated logarithm.
Journal of Mathematical Analysis and Applications, 1994
ABSTRACT In this note we consider the almost sure convergence (as epsilon --> 0) of soluti... more ABSTRACT In this note we consider the almost sure convergence (as epsilon --> 0) of solution X(epsilon)(.), defined over the interval 0 less-than-or-equal-to tau less-than-or-equal-to 1, of the random ordinary differential equation X(epsilon)(tau) = F(X(epsilon)(tau), tau/epsilon) subject to X(epsilon)(0) = x0. Here {F(x, t, omega), t greater-than-or-equal-to 0} is a strong mixing process for each x and (x, t) --> F(x, t, omega) is subject to regularity conditions which ensure the existence of a unique solution over 0 less-than-or-equal-to tau less-than-or-equal-to 1 for all epsilon > 0. Under rather weak conditions it is shown that the function X(epsilon)(., omega) converges a.s. to the solution x0(.) of a non-random averaged differential equation x0(tau) = FBAR(x0(tau)) subject to x0(0) = x0, the convergence being uniform over 0 less-than-or-equal-to tau less-than-or-equal-to 1. (C) 1994 Academic Press, Inc.
... Hossain Pezeshki-Esfahani and Andrew J. Heunis ... According to Strassen [25, Theorem 2, p. 2... more ... Hossain Pezeshki-Esfahani and Andrew J. Heunis ... According to Strassen [25, Theorem 2, p. 216] there exists some process together with some standard Brownian motion , , defined on some common probability space , , such that i) the processes and are equal ...
ABSTRACT Consider the following random ordinary differential equation: $\dot{X}^\epsilon(\tau) = ... more ABSTRACT Consider the following random ordinary differential equation: $\dot{X}^\epsilon(\tau) = F(X^\epsilon(\tau), \tau/\epsilon, \omega) \text{subject to} X^\epsilon(0) = x_0$, where $\{F(x, t, \omega), t \geq 0\}$ are stochastic processes indexed by $x$ in $\mathfrak{R}^d$, and the dependence on $x$ is sufficiently regular to ensure that the equation has a unique solution $X^\epsilon(\tau, \omega)$ over the interval $0 \leq \tau \leq 1$ for each $\epsilon > 0$. Under rather general conditions one can associate with the preceding equation a nonrandom averaged equation: $\dot{x}^0(\tau) = \overline{F}(x^0(\tau)) \text{subject to} x^0(0) = x_0,$ such that $\lim_{\epsilon\rightarrow 0} \sup_{0\leq\tau\leq 1}E|X^\epsilon(\tau) - x^0(\tau)| = 0$. In this article we show that as $\epsilon \rightarrow 0$ the random function $(X^\epsilon(\cdot) - x^0(\cdot))/\sqrt{2\epsilon\log\log\epsilon^{-1}}$ almost surely converges to and clusters throughout a compact set $K$ of $C\lbrack 0, 1\rbrack$.
We apply conjugate duality to establish the existence of optimal portfolios in an asset-allocatio... more We apply conjugate duality to establish the existence of optimal portfolios in an asset-allocation problem, with the goal of minimizing the variance of the final wealth which results from trading over a fixed, finite horizon in a continuous-time, complete market, subject to the constraints that the expected final wealth equal a specified target value and the portfolio of the investor
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