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<title>Jump diffusion</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Jump diffusion</span></span>

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<p><b>Jump diffusion</b> is a <a href="Stochastic_process" title="Stochastic process">stochastic process</a> that involves <a href="Jump_process" title="Jump process">jumps</a> and <a href="Diffusion_process" title="Diffusion process">diffusion</a>. It has important applications in <a href="Magnetic_reconnection" title="Magnetic reconnection">magnetic reconnection</a>, <a href="Coronal_mass_ejection" title="Coronal mass ejection">coronal mass ejections</a>, <a href="Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>, and <a href="Pattern_theory" title="Pattern theory">pattern theory</a> and <a href="Computational_vision" class="mw-redirect" title="Computational vision">computational vision</a>.

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<div class="mw-heading mw-heading2"><h2 id="In_physics">In physics</h2></div>

<p>In crystals, <a href="Atomic_diffusion" title="Atomic diffusion">atomic diffusion</a> typically consists of jumps between vacant lattice sites. On time and length scales that average over many single jumps, the net motion of the jumping atoms can be described as regular <a href="Diffusion" title="Diffusion">diffusion</a>.

</p><p>Jump diffusion can be studied on a microscopic scale by <a href="Inelastic_neutron_scattering" class="mw-redirect" title="Inelastic neutron scattering">inelastic neutron scattering</a> and by <a href="M%C3%B6%C3%9Fbauer_spectroscopy" class="mw-redirect" title="Mößbauer spectroscopy">Mößbauer spectroscopy</a>. Closed expressions for the <a href="Autocorrelation_function" class="mw-redirect" title="Autocorrelation function">autocorrelation function</a> have been derived for several jump(-diffusion) models:

</p>

<ul><li>Singwi, Sjölander 1960:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> alternation between oscillatory motion and directed motion</li>

<li>Chudley, Elliott 1961:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> jumps on a lattice</li>

<li>Sears 1966,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> 1967:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> jump diffusion of rotational degrees of freedom</li>

<li>Hall, Ross 1981:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> jump diffusion within a restricted volume</li></ul>

<div class="mw-heading mw-heading2"><h2 id="In_economics_and_finance">In economics and finance</h2></div>

<p>A jump-diffusion model is a form of <a href="Mixture_model" title="Mixture model">mixture model</a>, mixing a <a href="Jump_process" title="Jump process">jump process</a> and a <a href="Diffusion_process" title="Diffusion process">diffusion process</a>. In finance, jump-diffusion models were first introduced by <a href="Robert_C._Merton" title="Robert C. Merton">Robert C. Merton</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Such models have a range of financial applications from <a href="Option_pricing" class="mw-redirect" title="Option pricing">option pricing</a>, to <a href="Credit_risk" title="Credit risk">credit risk</a>, to <a href="Time_series" title="Time series">time series</a> <a href="Forecasting" title="Forecasting">forecasting</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="In_pattern_theory,_computer_vision,_and_medical_imaging">In pattern theory, computer vision, and medical imaging</h2></div>

<p>In <a href="Pattern_theory" title="Pattern theory">pattern theory</a> and <a href="Computational_vision" class="mw-redirect" title="Computational vision">computational vision</a> in <a href="Medical_imaging" title="Medical imaging">medical imaging</a>, jump-diffusion processes were first introduced by Grenander and Miller<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

as a form of <a href="Random_sampling" class="mw-redirect" title="Random sampling">random sampling</a> algorithm that mixes "focus"-like motions, the <a href="Diffusion_process" title="Diffusion process">diffusion processes</a>, with <a href="Saccade" title="Saccade">saccade</a>-like motions, via <a href="Jump_process" title="Jump process">jump processes</a>.

The approach modelled sciences of electron-micrographs as containing multiple shapes, each having some fixed dimensional representation, with the collection of micrographs filling out the sample space corresponding to the unions of multiple finite-dimensional spaces.

Using techniques from <a href="Pattern_theory" title="Pattern theory">pattern theory</a>, a posterior probability model was constructed over the countable union of sample space; this is therefore a <a href="Hybrid_system" title="Hybrid system"> hybrid system model</a>, containing the discrete notions of object number along with the continuum notions of shape.

The jump-diffusion process was constructed to have <a href="Ergodic" class="mw-redirect" title="Ergodic">ergodic</a> properties so that after initially flowing away from its initial condition it would generate samples from the posterior probability model.

</p>

<ul><li><a href="Jump_process" title="Jump process">Jump process</a>, an example of jump diffusion</li>

<li><a href="Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic Markov process</a> (PDMP), an example of jump diffusion and a generalization of the jump process</li>

<li><a href="Hybrid_system" title="Hybrid system">Hybrid system</a> (in the context of <a href="Dynamical_system" title="Dynamical system">dynamical systems</a>), a generalization of jump diffusion</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFSingwiSjölander1960" class="citation journal cs1">Singwi, K.; Sjölander, A. (1960). "Resonance Absorption of Nuclear Gamma Rays and the Dynamics of Atomic Motions". <i>Physical Review</i>. <b>120</b> (4): 1093. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.120.1093">10.1103/PhysRev.120.1093</a>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFChudleyElliott1961" class="citation journal cs1">Chudley, C. T.; Elliott, R. J. (1961). "Neutron Scattering from a Liquid on a Jump Diffusion Model". <i>Proceedings of the Physical Society</i>. <b>77</b> (2): 353. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0370-1328%2F77%2F2%2F319">10.1088/0370-1328/77/2/319</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFSears1966" class="citation journal cs1">Sears, V. F. (1966). "Theory of Cold Neutron Scattering by Homonuclear Diatomic Liquids: I. Free Rotation". <i>Canadian Journal of Physics</i>. <b>44</b> (6): <span class="nowrap">1279–</span>1297. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1139%2Fp66-108">10.1139/p66-108</a>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFSears1967" class="citation journal cs1">Sears, V. F. (1967). "Cold Neutron Scattering by Molecular Liquids: Iii. Methane". <i>Canadian Journal of Physics</i>. <b>45</b> (2): <span class="nowrap">237–</span>254. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1139%2Fp67-025">10.1139/p67-025</a>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFHallRoss1981" class="citation journal cs1">Hall, P. L.; Ross, D. K. (1981). "Incoherent neutron scattering functions for random jump diffusion in bounded and infinite media". <i>Molecular Physics</i>. <b>42</b> (3): 673. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00268978100100521">10.1080/00268978100100521</a>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFMerton1976" class="citation journal cs1"><a href="Robert_C._Merton" title="Robert C. Merton">Merton, R. C.</a> (1976). "Option pricing when underlying stock returns are discontinuous". <i><a href="Journal_of_Financial_Economics" title="Journal of Financial Economics">Journal of Financial Economics</a></i>. <b>3</b> (<span class="nowrap">1–</span>2): <span class="nowrap">125–</span>144. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-405X%2876%2990022-2">10.1016/0304-405X(76)90022-2</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1721.1%2F1899">1721.1/1899</a></span>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFChristensen2012" class="citation journal cs1">Christensen, H. L. (2012). "Forecasting high-frequency futures returns using online Langevin dynamics". <i>IEEE Journal of Selected Topics in Signal Processing</i>. <b>6</b> (4): <span class="nowrap">366–</span>380. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FJSTSP.2012.2191532">10.1109/JSTSP.2012.2191532</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10.1109%2FJSTSP.2012.2191532">10.1109/JSTSP.2012.2191532</a></span>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFGrenanderMiller1994" class="citation journal cs1">Grenander, U.; Miller, M.I. (1994). "Representations of Knowledge in Complex Systems". <i>Journal of the Royal Statistical Society, Series B</i>. <b>56</b> (4): <span class="nowrap">549–</span>603. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2346184">2346184</a>.</cite></span>

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</ol></div></div>

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<ul><li><a href="Delta_neutral" title="Delta neutral">Delta neutral</a></li>

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<li><a href="Binary_option" title="Binary option">Binary</a></li>

<li><a href="Callable_bull/bear_contract" title="Callable bull/bear contract">Callable bull/bear contract</a></li>

<li><a href="Chooser_option" title="Chooser option">Chooser</a></li>

<li><a href="Cliquet_option" title="Cliquet option">Cliquet</a></li>

<li><a href="Compound_option" title="Compound option">Compound</a></li>

<li><a href="Forward_start_option" title="Forward start option">Forward start</a></li>

<li><a href="Interest_rate_option" title="Interest rate option">Interest rate</a></li>

<li><a href="Lookback_option" title="Lookback option">Lookback</a></li>

<li>Mountain range</li>

<li><a href="Rainbow_option" title="Rainbow option">Rainbow</a></li>

<li><a href="Spread_option" title="Spread option">Spread</a></li>

<li><a href="Swaption" title="Swaption">Swaption</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Options_strategy" title="Options strategy">Strategies</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Backspread" title="Backspread">Backspread</a></li>

<li><a href="Box_spread_(options)" class="mw-redirect" title="Box spread (options)">Box spread</a></li>

<li><a href="Butterfly_(options)" title="Butterfly (options)">Butterfly</a></li>

<li><a href="Calendar_spread" title="Calendar spread">Calendar spread</a></li>

<li><a href="Collar_(finance)" title="Collar (finance)">Collar</a></li>

<li><a href="Condor_(options)" title="Condor (options)">Condor</a></li>

<li><a href="Covered_option" title="Covered option">Covered option</a></li>

<li><a href="Credit_spread_(options)" title="Credit spread (options)">Credit spread</a></li>

<li><a href="Debit_spread" title="Debit spread">Debit spread</a></li>

<li><a href="Diagonal_spread" title="Diagonal spread">Diagonal spread</a></li>

<li><a href="Fence_(finance)" title="Fence (finance)">Fence</a></li>

<li><a href="Intermarket_spread" title="Intermarket spread">Intermarket spread</a></li>

<li><a href="Iron_butterfly_(options_strategy)" title="Iron butterfly (options strategy)">Iron butterfly</a></li>

<li><a href="Iron_condor" title="Iron condor">Iron condor</a></li>

<li><a href="Jelly_roll_(options)" title="Jelly roll (options)">Jelly roll</a></li>

<li><a href="Ladder_(option_combination)" title="Ladder (option combination)">Ladder</a></li>

<li><a href="Naked_option" title="Naked option">Naked option</a></li>

<li><a href="Straddle" title="Straddle">Straddle</a></li>

<li><a href="Strangle_(options)" title="Strangle (options)">Strangle</a></li>

<li><a href="Protective_option" title="Protective option">Protective option</a></li>

<li><a href="Ratio_spread" title="Ratio spread">Ratio spread</a></li>

<li><a href="Risk_reversal" title="Risk reversal">Risk reversal</a></li>

<li><a href="Vertical_spread" title="Vertical spread">Vertical spread</a> (<a href="Bear_spread" title="Bear spread">Bear</a>, <a href="Bull_spread" title="Bull spread">Bull</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Valuation_of_options" title="Valuation of options">Valuation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bachelier_model" title="Bachelier model">Bachelier</a></li>

<li><a href="Binomial_options_pricing_model" title="Binomial options pricing model">Binomial</a></li>

<li><a href="Black_model" title="Black model">Black</a></li>

<li><a href="Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a> (<a href="Black%E2%80%93Scholes_equation" title="Black–Scholes equation">equation</a>)</li>

<li><a href="Finite_difference_methods_for_option_pricing" title="Finite difference methods for option pricing">Finite difference</a></li>

<li><a href="Foreign_exchange_option#Valuation:_the_Garman–Kohlhagen_model" title="Foreign exchange option">Garman–Kohlhagen</a></li>

<li><a href="Heston_model" title="Heston model">Heston</a></li>

<li><a href="Lattice_model_(finance)" title="Lattice model (finance)">Lattices</a></li>

<li><a href="Margrabe's_formula" title="Margrabe's formula">Margrabe</a></li>

<li><a href="Put%E2%80%93call_parity" title="Put–call parity">Put–call parity</a></li>

<li><a href="Monte_Carlo_methods_for_option_pricing" title="Monte Carlo methods for option pricing">MC Simulation</a></li>

<li><a href="Real_options_valuation" title="Real options valuation">Real options</a></li>

<li><a href="Trinomial_tree" title="Trinomial tree">Trinomial</a></li>

<li><a href="Vanna%E2%80%93Volga_pricing" title="Vanna–Volga pricing">Vanna–Volga</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Swap_(finance)" title="Swap (finance)">Swaps</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amortising_swap" title="Amortising swap">Amortising</a></li>

<li><a href="Asset_swap" title="Asset swap">Asset</a></li>

<li><a href="Basis_swap" title="Basis swap">Basis</a></li>

<li><a href="Commodity_swap" title="Commodity swap">Commodity</a></li>

<li><a href="Conditional_variance_swap" title="Conditional variance swap">Conditional variance</a></li>

<li><a href="Constant_maturity_swap" title="Constant maturity swap">Constant maturity</a></li>

<li><a href="Correlation_swap" title="Correlation swap">Correlation</a></li>

<li><a href="Credit_default_swap" title="Credit default swap">Credit default</a></li>

<li><a href="Currency_swap" title="Currency swap">Currency</a></li>

<li><a href="Dividend_swap" title="Dividend swap">Dividend</a></li>

<li><a href="Equity_swap" title="Equity swap">Equity</a></li>

<li><a href="Foreign_exchange_swap" title="Foreign exchange swap">Forex</a></li>

<li><a href="Forward_rate_agreement" title="Forward rate agreement">Forward Rate Agreement</a></li>

<li><a href="Inflation_swap" title="Inflation swap">Inflation</a></li>

<li><a href="Interest_rate_swap" title="Interest rate swap">Interest rate</a></li>

<li><a href="Overnight_indexed_swap" title="Overnight indexed swap">Overnight indexed</a></li>

<li><a href="Total_return_swap" title="Total return swap">Total return</a></li>

<li><a href="Variance_swap" title="Variance swap">Variance</a></li>

<li><a href="Volatility_swap" title="Volatility swap">Volatility</a></li>

<li><a href="Year-on-year_inflation-indexed_swap" title="Year-on-year inflation-indexed swap">Year-on-year inflation-indexed</a></li>

<li><a href="Zero_coupon_swap" title="Zero coupon swap">Zero Coupon</a></li>

<li><a href="Zero-coupon_inflation_swap" title="Zero-coupon inflation swap">Zero-coupon inflation-indexed</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="Forward_contract" title="Forward contract">Forwards</a></li><li><a href="Futures_contract" title="Futures contract">Futures</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Contango" title="Contango">Contango</a></li>

<li>Commodities future</li>

<li><a href="Currency_future" title="Currency future">Currency future</a></li>

<li><a href="Dividend_future" title="Dividend future">Dividend future</a></li>

<li><a href="Forward_market" title="Forward market">Forward market</a></li>

<li><a href="Forward_price" title="Forward price">Forward price</a></li>

<li><a href="Forward_contract#Spot–forward_parity" title="Forward contract">Forwards pricing</a></li>

<li><a href="Forward_rate" title="Forward rate">Forward rate</a></li>

<li><a href="Futures_contract#Pricing" title="Futures contract">Futures pricing</a></li>

<li><a href="Interest_rate_future" title="Interest rate future">Interest rate future</a></li>

<li><a href="Margin_(finance)" title="Margin (finance)">Margin</a></li>

<li><a href="Normal_backwardation" title="Normal backwardation">Normal backwardation</a></li>

<li><a href="Perpetual_futures" title="Perpetual futures">Perpetual futures</a></li>

<li><a href="Single-stock_futures" title="Single-stock futures">Single-stock futures</a></li>

<li><a href="Slippage_(finance)" title="Slippage (finance)">Slippage</a></li>

<li><a href="Stock_market_index_future" title="Stock market index future">Stock market index future</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Exotic_derivative" title="Exotic derivative">Exotic derivatives</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Commodity_derivative" class="mw-redirect" title="Commodity derivative">Commodity derivative</a></li>

<li><a href="Energy_derivative" title="Energy derivative">Energy derivative</a></li>

<li><a href="Freight_derivative" class="mw-redirect" title="Freight derivative">Freight derivative</a></li>

<li><a href="Inflation_derivative" title="Inflation derivative">Inflation derivative</a></li>

<li><a href="Property_derivative" title="Property derivative">Property derivative</a></li>

<li><a href="Weather_derivative" title="Weather derivative">Weather derivative</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other derivatives</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Collateralized_debt_obligation" title="Collateralized debt obligation">Collateralized debt obligation (CDO)</a></li>

<li><a href="Constant_proportion_portfolio_insurance" title="Constant proportion portfolio insurance">Constant proportion portfolio insurance</a></li>

<li><a href="Contract_for_difference" title="Contract for difference">Contract for difference</a></li>

<li><a href="Credit-linked_note" title="Credit-linked note">Credit-linked note (CLN)</a></li>

<li><a href="Credit_default_option" title="Credit default option">Credit default option</a></li>

<li><a href="Credit_derivative" title="Credit derivative">Credit derivative</a></li>

<li><a href="Equity-linked_note" title="Equity-linked note">Equity-linked note (ELN)</a></li>

<li><a href="Equity_derivative" title="Equity derivative">Equity derivative</a></li>

<li><a href="Foreign_exchange_derivative" title="Foreign exchange derivative">Foreign exchange derivative</a></li>

<li><a href="Fund_derivative" title="Fund derivative">Fund derivative</a></li>

<li><a href="Fund_of_funds" title="Fund of funds">Fund of funds</a></li>

<li><a href="Interest_rate_derivative" title="Interest rate derivative">Interest rate derivative</a></li>

<li><a href="Mortgage-backed_security" title="Mortgage-backed security">Mortgage-backed security</a></li>

<li><a href="Power_reverse_dual-currency_note" title="Power reverse dual-currency note">Power reverse dual-currency note (PRDC)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Market issues</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Consumer_debt" title="Consumer debt">Consumer debt</a></li>

<li><a href="Corporate_bond" title="Corporate bond">Corporate debt</a></li>

<li><a href="Government_debt" title="Government debt">Government debt</a></li>

<li><a href="Great_Recession" title="Great Recession">Great Recession</a></li>

<li><a href="Municipal_debt" class="mw-redirect" title="Municipal debt">Municipal debt</a></li>

<li><a href="Tax_policy" title="Tax policy">Tax policy</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="nowrap"><span class="noviewer" typeof="mw:File"></span> </span><a href="Portal%3ABusiness" title="Portal:Business">Business portal</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Volatility73" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Volatility73" style="font-size:114%;margin:0 4em"><a href="Volatility_(finance)" title="Volatility (finance)">Volatility</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modelling volatility</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Implied_volatility" title="Implied volatility">Implied volatility</a></li>

<li><a href="Volatility_smile" title="Volatility smile">Volatility smile</a></li>

<li><a href="Volatility_clustering" title="Volatility clustering">Volatility clustering</a></li>

<li><a href="Local_volatility" title="Local volatility">Local volatility</a></li>

<li><a href="Stochastic_volatility" title="Stochastic volatility">Stochastic volatility</a></li>

<li><a href="Jump-diffusion_models" class="mw-redirect" title="Jump-diffusion models">Jump-diffusion models</a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">ARCH and GARCH</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Trading volatility</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Volatility_arbitrage" title="Volatility arbitrage">Volatility arbitrage</a></li>

<li><a href="Straddle" title="Straddle">Straddle</a></li>

<li><a href="Volatility_swap" title="Volatility swap">Volatility swap</a></li>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Stochastic_processes496" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li>

<li><a href="Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li>

<li><a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Moran_process" title="Moran process">Moran process</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a>

<ul><li><a href="Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li>

<li><a href="Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li>

<li><a href="Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li>

<li><a href="Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Additive_process" title="Additive process">Additive process</a></li>

<li><a href="Airy_process" title="Airy process">Airy process</a></li>

<li><a href="Bessel_process" title="Bessel process">Bessel process</a></li>

<li><a href="Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a>

<ul><li><a href="Birth_process" title="Birth process">pure birth</a></li></ul></li>

<li><a href="Wiener_process" title="Wiener process">Brownian motion</a>

<ul><li><a href="Brownian_bridge" title="Brownian bridge">Bridge</a></li>

<li><a href="Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson</a></li>

<li><a href="Brownian_excursion" title="Brownian excursion">Excursion</a></li>

<li><a href="Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li>

<li><a href="Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li>

<li><a href="Brownian_meander" title="Brownian meander">Meander</a></li></ul></li>

<li><a href="Cauchy_process" title="Cauchy process">Cauchy process</a></li>

<li><a href="Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li>

<li><a href="Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li>

<li><a href="Cox_process" title="Cox process">Cox process</a></li>

<li><a href="Diffusion_process" title="Diffusion process">Diffusion process</a></li>

<li><a href="Empirical_process" title="Empirical process">Empirical process</a></li>

<li><a href="Feller_process" title="Feller process">Feller process</a></li>

<li><a href="Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li>

<li><a href="Gamma_process" title="Gamma process">Gamma process</a></li>

<li><a href="Geometric_process" title="Geometric process">Geometric process</a></li>

<li><a href="Hawkes_process" title="Hawkes process">Hawkes process</a></li>

<li><a href="Hunt_process" title="Hunt process">Hunt process</a></li>

<li><a href="Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li>

<li><a href="It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li>

<li><a href="It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li>

<li><a href="Jump_process" title="Jump process">Jump process</a></li>

<li><a href="L%C3%A9vy_process" title="Lévy process">Lévy process</a></li>

<li><a href="Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li>

<li><a href="Markov_additive_process" title="Markov additive process">Markov additive process</a></li>

<li><a href="McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li>

<li><a href="Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson process</a>

<ul><li><a href="Compound_Poisson_process" title="Compound Poisson process">Compound</a></li>

<li><a href="Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li>

<li><a href="Quasimartingale" title="Quasimartingale">Quasimartingale</a></li>

<li><a href="Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li>

<li><a href="Semimartingale" title="Semimartingale">Semimartingale</a></li>

<li><a href="Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li>

<li><a href="Stable_process" title="Stable process">Stable process</a></li>

<li><a href="Superprocess" title="Superprocess">Superprocess</a></li>

<li><a href="Telegraph_process" title="Telegraph process">Telegraph process</a></li>

<li><a href="Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li>

<li><a href="Wiener_process" title="Wiener process">Wiener process</a></li>

<li><a href="Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Gaussian_process" title="Gaussian process">Gaussian process</a></li>

<li><a href="Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li>

<li><a href="Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li>

<li><a href="Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a>

<ul><li><a href="Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li>

<li><a href="Local_martingale" title="Local martingale">Local</a></li>

<li><a href="Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li>

<li><a href="Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li>

<li><a href="Regenerative_process" title="Regenerative process">Regenerative process</a></li>

<li><a href="Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li>

<li><a href="Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li>

<li><a href="White_noise" title="White noise">White noise</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li>

<li><a href="Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li>

<li><a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li>

<li><a href="Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li>

<li><a href="Ising_model" title="Ising model">Ising model</a>

<ul><li><a href="Potts_model" title="Potts model">Potts model</a></li>

<li><a href="Boolean_network" title="Boolean network">Boolean network</a></li></ul></li>

<li><a href="Markov_random_field" title="Markov random field">Markov random field</a></li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation</a></li>

<li><a href="Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li>

<li><a href="Point_process" title="Point process">Point process</a>

<li><a href="Determinantal_point_process" title="Determinantal point process">Determinantal</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li>

<li><a href="Random_field" title="Random field">Random field</a></li>

<li><a href="Random_graph" title="Random graph">Random graph</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li>

<li><a href="Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li>

<li><a href="Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li>

<li><a href="Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li>

<li><a href="Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li>

<li><a href="Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li>

<li><a href="Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li>

<li><a href="Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li>

<li><a href="Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li>

<li><a href="Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li>

<li><a href="Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li>

<li><a href="Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li>

<li><a href="Heston_model" title="Heston model">Heston</a></li>

<li><a href="Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li>

<li><a href="Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li>

<li><a href="LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li>

<li><a href="Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li>

<li><a href="SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li>

<li><a href="Vasicek_model" title="Vasicek model">Vašíček</a></li>

<li><a href="Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li>

<li><a href="Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li>

<li><a href="Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li>

<li><a href="Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Fluid_queue" title="Fluid queue">Fluid</a></li>

<li><a href="G-network" title="G-network">Generalized queueing network</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li>

<li><a href="Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li>

<li><a href="Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li>

<li><a href="Ergodicity" title="Ergodicity">Ergodic</a></li>

<li><a href="Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li>

<li><a href="Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li>

<li><a href="Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li>

<li><a href="Markov_property" title="Markov property">Markov</a></li>

<li><a href="Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li>

<li><a href="Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li>

<li><a href="Predictable_process" title="Predictable process">Predictable</a></li>

<li><a href="Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li>

<li><a href="Self-similar_process" title="Self-similar process">Self-similar</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationary</a></li>

<li><a href="Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Donsker's_theorem" title="Donsker's theorem">Donsker's theorem</a></li>

<li><a href="Doob's_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li>

<li><a href="Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li>

<li><a href="Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li>

<li><a href="Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li>

<li><a href="Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li>

<li><a href="Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li>

<li><a href="Sanov's_theorem" title="Sanov's theorem">Sanov's theorem</a></li>

<li><a href="Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="Blumenthal's_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="Kolmogorov's_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="L%C3%A9vy's_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li>

<li><a href="Doob's_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li>

<li><a href="Doob's_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li>

<li><a href="Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li>

<li><a href="Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li>

<li><a href="Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li>

<li><a href="Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li>

<li><a href="Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li>

<li><a href="Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li>

<li><a href="Doob's_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li>

<li><a href="Dynkin's_formula" title="Dynkin's formula">Dynkin's formula</a></li>

<li><a href="Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li>

<li><a href="Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li>

<li><a href="Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li>

<li><a href="Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li>

<li><a href="It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li>

<li><a href="It%C3%B4's_lemma" title="Itô's lemma">Itô's lemma</a></li>

<li><a href="Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li>

<li><a href="Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li>

<li><a href="Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li>

<li><a href="L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li>

<li><a href="Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li>

<li><a href="Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li>

<li><a href="Prokhorov's_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li>

<li><a href="Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li>

<li><a href="Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li>

<li><a href="Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li>

<li><a href="Skorokhod's_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li>

<li><a href="Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li>

<li><a href="Snell_envelope" title="Snell envelope">Snell envelope</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a>

<ul><li><a href="Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li>

<li><a href="Stopping_time" title="Stopping time">Stopping time</a></li>

<li><a href="Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li>

<li><a href="Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li>

<li><a href="Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li>

<li><a href="Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a>

<ul><li><a href="Classical_Wiener_space" title="Classical Wiener space">Classical</a></li>

<li><a href="Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li>

<li><a href="Stochastic_control" title="Stochastic control">Control theory</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li>

<li><a href="Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li>

<li><a href="Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Queueing_theory" title="Queueing theory">Queueing theory</a></li>

<li><a href="Renewal_theory" title="Renewal theory">Renewal theory</a></li>

<li><a href="Ruin_theory" title="Ruin theory">Ruin theory</a></li>

<li><a href="Signal_processing" title="Signal processing">Signal processing</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li>

<li><a href="Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li>

<li><a href="Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li>

<li><a href="Machine_learning" title="Machine learning">Machine learning</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div>

<ul><li><a href="List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li>

<li>Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-03-19" href="https://en.wikipedia.org/wiki/?title=Jump_diffusion&oldid=1281276750">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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