Overview: Markov chains are a special class of stochastic processes where the future state depends only on the current state, not on the path taken to reach it. Transition Probabilities: These are central to Markov chains, indicating the likelihood of moving from one state to another.

Markov Properties: Definition: A stochastic process {Xn} has the Markov property if for every sequence of states and times, the conditional probability of future states depends only on the current state, not on past states. Mathematically, for all n and states i,j: (+1 = = , 1 = 1, . . . , = ) = (+1 = = ) 1. Transition Probabilities (Short term approximation): For a Markov chain, the transition probability from state i to state j in one step is denoted as . The probability of a assets pair's price moving from one level to another can be modeled with transition probabilities. For example, Pij would represent the probability of moving from price level i to j. 2. Chapman-Kolmogorov Equation (Long term approximation): For a Markov chain, it relates the n-step transition probabilities to the one-step transition probabilities.

(+) = () () ()is the probability of transitioning from state i to j in n steps. This equation helps in predicting where the longer-term behavior of an asset/currency pair by relating multi-step movements to single-step probabilities. Application of Theory Trading Strategy Parameters Initial Capital: $50,000 Win Probability (Pw): 55% Risk-Reward Ratio: 1:2 Maximum Drawdown: 5% Risk Per Trade: 1% to 10% Modelling Trading Outcome Objective: Determine the risk of reaching a 5% drawdown (ruin) for various risk levels. 5% drawdown is a common value for blowing an account for many prop firms; Apex, Topstep, FMTO etc. Simulation Approach Method: Running 100,000 simulations per risk level. Output: Risk of ruin for each risk level. Visualizing a simplified Markov Chain Results & Recommendations Results Low Risk per Trade = 1-3% Moderate Risk per Trade = 4-6% High Risk per Trade > 6% Ruin = 5% drawdown of equity value

Recommendations Based on the Markov chain analysis and the results obtained, the following detailed recommendations can be made for traders looking to determine their optimal risk per trade: Low Risk (1-3%) = 6-20% Risk of Ruin Suitability: Ideal for conservative traders, particularly those new to forex trading or with a low tolerance for risk. Advantages : This range significantly reduces the probability of ruin, allowing for more consistent trading and better capital preservation. Strategy: Focus on steady, small gains; avoid highly volatile currency pairs. Utilize tight stop-loss orders to minimize losses on individual trades. Long-Term View: Suitable for traders with a long-term perspective, where capital preservation and steady growth are prioritized over high short-term gains. This is the risk per trade any prop firm traders should commit; higher risk per trade will often result in a lost challenge Moderate Risk (4-6%) = 20-35% Risk of Ruin Suitability : higher